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A Mixture of Experts Classifier with Learning Based on Both Labelled and Unlabelled Data David J. Miller and Hasan S. Uyar Department of Electrical Engineering The Pennsylvania State University University Park, Pa. 16802 miller@perseus.ee.psu.edu Abstract We address statistical classifier design given a mixed training set consisting of a small labelled feature set and a (generally larger) set of unlabelled features. This situation arises, e.g., for medical images, where although training features may be plentiful, expensive expertise is required to extract their class labels. We propose a classifier structure and learning algorithm that make effective use of unlabelled data to improve performance. The learning is based on maximization of the total data likelihood, i.e. over both the labelled and unlabelled data subsets. Two distinct EM learning algorithms are proposed, differing in the EM formalism applied for unlabelled data. The classifier, based on a joint probability model for features and labels, is a "mixture of experts" structure that is equivalent to the radial basis function (RBF) classifier, but unlike RBFs, is amenable to likelihood-based training. The scope of application for the new method is greatly extended by the observation that test data, or any new data to classify, is in fact additional, unlabelled data - thus, a combined learning/classification operation - much akin to what is done in image segmentation - can be invoked whenever there is new data to classify. Experiments with data sets from the UC Irvine database demonstrate that the new learning algorithms and structure achieve substantial performance gains over alternative approaches. 1 Introduction Statistical classifier design is fundamentally a supervised learning problem, wherein a decision function, mapping an input feature vector to an output class label, is learned based on representative (feature,class label) training pairs. While a variety of classifier structures and associated learning algorithms have been developed, a common element of nearly all approaches is the assumption that class labels are 572 D. J. Miller and H. S. Uyar known for each feature vector used for training. This is certainly true of neural networks such as multilayer perceptrons and radial basis functions (RBFs), for which classification is usually viewed as function approximation, with the networks trained to minimize the squared distance to target class values. Knowledge of class labels is also required for parametric classifiers such as mixture of Gaussian classifiers, for which learning typically involves dividing the training data into subsets by class and then using maximum likelihood estimation (MLE) to separately learn each class density. While labelled training data may be plentiful for some applications, for others, such as remote sensing and medical imaging, the training set is in principle vast but the size of the labelled subset may be inadequate. The difficulty in obtaining class labels may arise due to limited knowledge or limited resources, as expensive expertise is often required to derive class labels for features. In this work, we address classifier design under these conditions, i.e. the training set X is assumed to consist of two subsets, X = {Xl, Xu}, where Xl = {(Xl, cd, (X2' C2), ... ,(XNI,CNln is the labelled subset and Xu = {XNI+l, ... ,XN} is the unlabelled subset l. Here, Xi E R. k is the feature vector and Ci E I is the class label from the label set I = {I, 2, . ", N c }. The practical significance of this mixed training problem was recognized in (Lippmann 1989). However, despite this realization, there has been surprisingly little work done on this problem. One likely reason is that it does not appear possible to incorporate unlabelled data directly within conventional supervised learning methods such as back propagation. For these methods, unlabelled features must either be discarded or preprocessed in a suboptimal, heuristic fashion to obtain class label estimates. We also note the existence of work which is less than optimistic concerning the value of unlabelled data for classification (Castelli and Cover 1994). However, (Shashahani and Landgrebe 1994) found that unlabelled data could be used effectively in label-deficient situations. While we build on their work, as well as on our own previous work (Miller and Uyar 1996), our approach differs from (Shashahani and Landgrebe 1994) in several important respects. First, we suggest a more powerful mixture-based probability model with an associated classifier structure that has been shown to be equivalent to the RBF classifier (Miller 1996). The practical significance of this equivalence is that unlike RBFs, which are trained in a conventional supervised fashion, the RBF-equivalent mixture model is naturally suited for statistical training (MLE). The statistical framework is the key to incorporating unlabelled data in the learning. A second departure from prior work is the choice of learning criterion. We maximize the joint data likelihood and suggest two di"tinct EM algorithms for this purpose, whereas the conditional likelihood was considered in (Shashahani and Landgrebe 1994). We have found that our approach achieves superior results. A final novel contribution is a considerable expansion of the range of situations for which the mixed training paradigm can be applied. This is made possible by the realization that test data or new data to classify can al"o be viewed as an unlabelled set, available for "training". This notion will be clarified in the sequel. 2 Unlabelled Data and Classification Here we briefly provide some intuitive motivation for the use of unlabelled data. Suppose, not very restrictively, that the data is well-modelled by a mixture density, lThis problem can be viewed as a type of "missing data" problem, wherein the missing items are class labels. As such, it is related to, albeit distinct from supervised learning involving missing and/or noisy jeaturecomponents, addressed in (Ghahramani and Jordan 1995),(Tresp et al. 1995). A Mixture of Experts Classifier for Label-deficient Data 573 in the following way. The feature vectors are generated according to the density L f(z/9) = 2: ad(z/9t), where f(z/Oc) is one of L component densities, with non1=1 L negative mixing parameters 0.1, such that 2: 0.1 = 1. Here, 01 is the set of parameters 1=1 specifying the component density, with 9 = {Ol}. The class labels are also viewed as random quantities and are assumed chosen conditioned on the selected mixture component 7'7I.i E {I, 2, ... , L} and possibly on the feature value, i.e. according to the probabilities P[CdZi,7'7I.iJ 2. Thus, the data pairs are assumed generated by selecting, in order, the mixture component, the feature value, and the class label, with each selection depending in general on preceding ones. The optimal classification rule for this model is the maximum a posteriori rule: S(z) = arg max L P[c .. = k/7'7I.i = i, Zi]P[7'7I.i = i/Zi], k . (1) j where P[7'7I.i = i/Zi] = LajJ(~./6,) , and where S(z) is a selector function with 2: atf(~i/61) 1=1 range in T. Since this rule is based on the a posteriori class probabilities, one can argue that learning should focus solely on estimating these probabilities. However, if the classifier truly implements (1), then implicitly it has been assumed that the estimated mixture density accurately models the feature vectors. If this is not true, then presumably estimates of the a posteriori probabilities will also be affected. This suggests that even in the ab8ence of cla88 label8, the feature vectors can be used to better learn a posteriori probabilities via improved estimation of the mixture-based feature density. A commonly used measure of mixture density accuracy is the data likelihood. 3 Joint Likelihood Maximization for a Mixtures of Experts Classifier The previous section basically argues for a learning approach that uses labelled data to directly estimate a posteriori probabilities and unlabelled data to estimate the feature density. A criterion which essentially fulfills these objectives is the joint data likelihood, computed over both the labelled and unlabelled data subsets. Given our model, the joint data log-likelihood is written in the form L L log L = L log L ad(z,i/O,) + L log L aIP[cdzi, 7'7I.i = l]f(Zi/91). (2) 1=1 1=1 This objective function consists of a "supervised" term based on XI and an "unsupervised" term based on Xu. The joint data likelihood was previously considered in a learning context in (Xu et al. 1995). However, there the primary justification was simplification of the learning algorithm in order to allow parameter estimation based on fixed point iterations rather than gradient descent. Here, the joint likelihood allows the inclusion of unlabelled samples in the learning. We next consider two special cases of the probability model described until now. 2The usual assumption made is that components are "hard-partitioned", in a deterministic fashion, to classes. Our random model includes the "partitioned" one as a special case. We have generally found this model to be more powerful than the "partitioned" one (Miller Uyar 1996). 574 D. J. Miller and H. S. Uyar The "partitioned" mixture (PM) model: This is the previously mentioned case where mixture components are "hard-partitioned" to classes (Shashahani and Landgrebe 1994). This is written Mj E C/e, where Mj denotes mixture component j and C/e is the subset of components owned by class k. The posterior probabilities have the form 2: ajf(3!/Oj) P[Ci = k/3!] = )_·;_M_'L,-EC_,, ___ _ (3) 2: azf(3!/Or) 1=1 The generalized mixture (G M) model: The form of the posterior for each mixture component is now P[c,:/1'7l.i, 3!il = P[c,:/1'7l.il == {3c,/m,, i.e., it is independent of the feature value. The overall posterior probability takes the form [ 1 '" ( ad(3!i/Oj) ) P C,:/3!i = ~ '2t azf(3!dOI ) {3c,lj. (4) This model was introduced in (Miller and Uyar 1996) and was shown there to lead to performance improvement over the PM model. Note that the probabilities have a "mixture of experts" structure, where the "gating units" are the probabilities P[1'7l.i = jl3!il (in parentheses), and with the "expert" for component j just the probability {3c,Ii' Elsewhere (Miller 1996), it has been shown that the associated classifier decision function is in fact equivalent to that of an RBF classifier (Moody and Darken 1989). Thus, we suggest a probability model equivalent to a widely used neural network classifier, but with the advantage that, unlike the standard RBF, the RBF-equivalent probability model is amenable to statistical training, and hence to the incorporation of unlabelled data in the learning. Note that more powerful models P[cilTn.i, 3!i] that do condition on 3!i are also possible. However, such models will require many more parameters which will likely hurt generalization performance, especially in a label-deficient learning context. Interestingly, for the mixed training problem, there are two Expectation-Maximization (EM) (Dempster et al. 1977) formulations that can be applied to maximize the likelihood associated with a given probability model. These two formulations lead to di8tinct methods that take different learning "trajectories", although both ascend in the data likelihood. The difference between the formulations lies in how the "incomplete" and "complete" data elements are defined within the EM framework. We will develop these two approaches for the suggested G M model. EM-I (No class labels assumed): Distinct data interpretations are given for XI and Xu' In this case, for Xu, the incomplete data consists of the features {3!o.} and the complete data consists of {(3!i' 1'7l.iH. For XI, the incomplete data consists of {(3!;, Ci)}, with the complete data now the triple {(3!o., Co., Tn.i)}. To clarify, in this case mizture labels are viewed as the sole missing data elements, for Xu as well as for XI' Thus, in effect class labels are not even postulated to exist for Xu' EM-II (Class labels assumed): The definitions for XI are the same as before. However, for Xu, the complete data now consists of the triple {( 3!o., Ci, 1'7l.i H, i.e. class labels are also assumed missing for Xu' For Gaussian components, we have 01 = {I-'I , EI}, with 1-'1 the mean vector and EI the covariance matrix. For EM-I, the resulting fixed point iterations for updating the parameters are: A Mixture of Experts Classifier for Label-deficient Data 575 + L S};)P[ffli =j/Xi,O(t)]) z.EX" + L P[ffli = j/Xi, ott)]) z.EX,. Vj I: P[ffli = j / Xi, Ci, ott)] .B(Hl) = ziEX,nCi=k kIJ I: P[ffli = j/Xi,Ci,O(t)] Vk,j (5) ziEX, Here, S~;) == (Xi ~;t»)(Xi ~;t»)T. New parameters are computed at iteration t+ 1 based on their values at iteration t. In these equations, P[ffli = j/Xi, Ci, ott)] = ",(')p(') ·f(z )e('» ",(.) f(~ )e(") I:~("C\~) 'zJ e(.) andP[ffli=j/Xi,O(t)]= M J • J • For EM-II, it can be .... Pcil .... f ( .1 .... ) I: ",~., f(zile~") ,",=1 shown that the resulting re-estimation equations are identical to those in (5) except regarding the parameters {.Bk/}}' The updates for these parameters now take the form ,q(t+l) _ 1 ( " P[. _ 'j . . il(t)l "P[· -' . _ k/ . il(t)]) fJklj --(t-) ~ ffli -) X" C,,!7 J + ~ ffli ), c, X,,!7 N a j z.EX,nCi=k ZiEX,. (t)~(.) ( le(") H 'd t'f P[ . k/ il(t)] "'J "Io/,f Zi J I h' J: l' ere, we 1 en 1 y ffli = ), Ci = Xi, !7 =" i.) (.). n t 1S !ormu atlOn, L. "'~ f(z.le .... ) ... joint probabilities for class and mixture labels are computed for data in Xu and used in the estimation of {.Bkfj}, whereas in the previous formulation {.Bklj} are updated solely on the basis of X,. While this does appear to be a significant qualitative difference between the two methods, both do ascend in log L, and in practice we have found that they achieve comparable performance. 4 Combined Learning and Classification The range of application for mixed training is greatly extended by the following observation: te~t data (with label~ withheld), or for that matter, any new batch of data to be cla~~ified, can be viewed ~ a new, unlabelled data ~et, Hence, this new data can be taken to be Xu and used for learning (based on EM-I or EM-II) prior to its classification, What we are suggesting is a combined learning/classification operation that can be applied whenever there is a new batch of data to classify. In the usual supervised learning setting, there is a clear division between the learning and classification (use) phases, In this setting, modification of the classifier for new data is not possible (because the data is unlabelled), while for test data such modification is a form of "cheating". However, in our suggested scheme, this learning for unlabelled data is viewed simply as part of the classification operation. This is analogous to image segmentation, wherein we have a common energy function that is minimized for each new image to be segmented. Each such minimization determines a model local to the image and a segmentation for the image, Our "segmentation" is just classification, with log L playing the role of the energy function. It may consist of one term which is always fixed (based on a given labelled training set) and one term which is modified based on each new batch of unlabelled data to classify. We can envision several distinct learning contexts where this scheme can 576 D. 1. Miller and H. S. Uyar be used, as well as different ways of realizing the combined learning/classification operation3 One use is in classification of an image/speech archive, where each image/speaker segment is a separate data "batch". Each batch to classify can be used as an unlabelled "training" set, either in concert with a representative labelled data set, or to modify a design based on such a set4 . Effectively, this scheme would adapt the classifier to each new data batch. A second application is supervised learning wherein the total amount of data is fixed. Here, we need to divide the data into training and test sets with the conflicting goals of i) achieving a good design and ii) accurately measuring generalization performance. Combined learning and classification can be used here to mitigate the loss in performance associated with the choice of a large test set. More generally, our scheme can be used effectively in any setting where the new data to classify is either a) sizable or b) innovative relative to the existing training set. 5 Experimental Results Figure 1a shows results for the 40-dimensional, 3-class wa.veform- +noise data set from the UC Irvine database. The 5000 data pairs were split into equal-size training and test sets. Performance curves were obtained by varying the amount of labelled training data. For each choice of N/, various learning approaches produced 6 solutions based on random parameter initialization, for each of 7 different labelled subset realizations. The test set performance was then averaged over these 42 "trials". All schemes used L = 12 components. DA-RBF (Miller et at. 1996) is a deterministic annealing method for RBF classifiers that has been found to achieve very good results, when given adequate training datas. However, this supervised learning method is forced to discard unlabelled data, which severely handicaps its performance relative to EM-I, especially for small NI , where the difference is substantial. TEM-I and TEM-II are results for the EM methods (both I and II) in combined learning and classification mode, i.e., where the 2500 test vectors were also used as part of Xu. As seen in the figure, this leads to additional, significant performance gains for small N/. ~ ote also that performance of the two EM methods is comparable. Figure 1b shows results of similar experiments performed on 6-class satellite imagery data ("at), also from the UC Irvine database. For this set, the feature dimension is 36, and we chose L = 18 components. Here we compared EM-I with the method suggested in (Shashahani and Landgrebe 1994) (SL), based on the PM model. EM-I is seen to achieve substantial performance gains over this alternative learning approach. Note also that the EM-I performance is nearly constant, over the entire range of N/. Future work will investigate practical applications of combined learning and classification, as well as variations on this scheme which we have only briefly outlined. Moreover, we will investigate possible extensions of the methods described here for the regression problem. 3The image segmentation analogy in fact suggests an alternative scheme where we perform joint likelihood maximization over both the model parameters and the "hard", missing class labels. This approach, which is analogous to segmentation methods such as ICM, would encapsulate the classification operation directly within the learning. Such a scheme will be investigated in future work. ~Note that if the classifier is simply modified based on Xu, EM-I will not need to update {,8kl;}, while EM-II must update the entire model. 5 We assumed the same number of basis functions as mixture components. Also, for the DA design, there was only one initialization, since DA is roughly insensitive to this choice. A Mixture of Experts Classifier for Label-deficient Data 1 02< " I ~022 Ii 1 02 " 1°,8 io.,6 . J 0,,, . " ' ! ' •• . . • •. ! .••..••• ,. M-H . !EI'~I .... .... . , '0' Acknowledgement s 1021 ., .. ,.. . ..... , .. I lO.2< Ii 1022 " , .... ... ; ....... . ; . . . . 0" .... , .. ...... , .... . 577 'hi This work was supported in part by National Science Foundation Career Award IRI-9624870. References V. Castelli and T. M. Cover. On the exponential value of labeled samples. Pattern Recognition Letters, 16:105-111, 1995. A.P. Dempster, N.M. Laird, and D.B. Rubin. Maximum-likelihood from incomplete data via the EM algorithm. Journal of the Roy. Stat. Soc. I Ser. B, 39:1-38, 1977. Z. Ghahramani and M. I. Jordan. Supervised learning from incomplete data via an EM approach. In Neural Information Processing Systems 6, 120-127, 1994. M. 1. Jordan and R. A. Jacobs. Hierarchical mixtures of experts and the EM algorithm. Neural Computation, 6:181-214, 1994. R. P. Lippmann. Pattern classification using neural networks. IEEE Communications Magazine, 27,47-64, 1989. D. J. Miller, A. Rao, K. Rose, and A. Gersho. A global optimization method for statistical classifier design. IEEE Transactions on Signal Processing, Dec. 1996. D. J. Miller and H. S. Uyar. A generalized Gaussian mixture classifier with learning based on both labelled and unlabelled data. Conf. on Info. Sci. and Sys., 1996. D. J. Miller. A mixture model equivalent to the radial basis function classifier. Submitted to Neural Computation, 1996. J. Moody and C. J. Darken. Fast learning in locally-tuned processing units. Neural Computation, 1:281-294, 1989. B. Shashahani and D. Landgrebe. The effect of unlabeled samples in reducing the small sample size problem and mitigating the Hughes phenomenon. IEEE Transactions on Geoscience and Remote Sensing, 32:1087-1095, 1994. V. Tresp, R. N euneier, and S. Ahmad. Efficient methods for dealing with missing data in supervised learning. In Neural Information Processing Systems 7, 689696,1995. L. Xu, M. I. Jordan, and G. E. Hinton. An alternative model for mixtures of experts. In Neural Information Processing Systems 7, 633-640, 1995.
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Prior Knowledge in Support Vector Kernels Bernhard Scholkopf*t, Patrice Simardt, Alex Smola t, & Vladimir Vapnikt * Max-Planck-Institut fur biologische Kybernetik, Tiibingen, Gennany t GMD FIRST, Rudower Chaussee 5, 12489 Berlin, Gennany t AT&T Research, 100 Schulz Drive, Red Bank, NJ, USA bS@first.gmd.de Abstract We explore methods for incorporating prior knowledge about a problem at hand in Support Vector learning machines. We show that both invariances under group transfonnations and prior knowledge about locality in images can be incorporated by constructing appropriate kernel functions. 1 INTRODUCTION When we are trying to extract regularities from data, we often have additional knowledge about functions that we estimate. For instance, in image classification tasks, there exist transfonnations which leave class membership invariant (e.g. local translations); moreover, it is usually the case that images have a local structure in that not all correlations between image regions carry equal amounts of infonnation. The present study investigates the question how to make use of these two sources of knowledge by designing appropriate Support Vector (SV) kernel functions. We start by giving a brief introduction to SV machines (Vapnik & Chervonenkis, 1979; Vapnik, 1995) (Sec. 2). Regarding prior knowledge about invariances, we present a method to design kernel functions for invariant classification hyperplanes (Sec. 3). The method is applicable to invariances under the action of differentiable local 1-parameter groups of local transfonnations, e.g. translational invariance in pattern recognition. In Sec. 4, we describe kernels which take into account image locality by using localized receptive fields. Sec. 5 presents experimental results on both types of kernels, followed by a discussion (Sec. 6). 2 OPTIMAL MARGIN HYPERPLANES For linear hyperplane decision functions f(x) = sgn((w· x) + b), the VC-dimension can be controlled by controlling the nonn of the weight vector w. Given training data (xl,yd, ... ,(Xl,Yl), Xi E RN,Yi E {±1}, a separating hyperplane which generalizes Prior Knowledge in Support Vector Kernels 641 well can be found by minimizing ~llwl12 subject to Yi' ((Xi' w) + b) ~ 1 for i = 1, ... , f, (I) the latter being the conditions for separating the training data with a margin. Nonseparable cases are dealt with by introducing slack variables (Cortes & Vapnik 1995), but we shall omit this modification to simplify the exposition. All of the following also applies for the nonseparable case. To solve the above convex optimization problem, one introduces a Lagrangian with multipliers Qi and derives the dual form of the optimization problem: maximize iIi i L Qi - '2 L QiYiQkYk(Xi . Xk) subject to Qi ~ 0, L QiYi = 0. (2) i=l i,k=l i=l It turns out that the solution vector has an expansion in terms of training examples, W = L:=l QiYiXi, where only those Qi corresponding to constraints (1) which are met can become nonzero; the respective examples Xi are called Support Vectors. Substituting this expansion for W yields the decision function f(x) = sgn (t o,y,(x. Xi) + b) . (3) It can be shown that minimizing (2) corresponds to minimizing an upper bound on the VC dimension of separating hyperplanes, or, equivalently, to maximizing the separation margin between the two classes. In the next section, we shall depart from this and modify the dot product used such that the minimization of (2) corresponds to enforcing transformation invariance, while at the same time the constraints (1) still hold. 3 INVARIANT HYPERPLANES Invariance by a self-consistency argument. We face the following problem: to express the condition of invariance of the decision function, we already need to know its coefficients which are found only during the optimization, which in turn should already take into account the desired invariances. As a way out of this circle, we use the following ansatz: consider decision functions f = (sgn 0 g), where g is defined as i g(Xj) := L QiYi(Bxj . BXi) + b, (4) i=l with a matrix B to be determined below. This follows Vapnik (1995), who suggested to incorporate invariances by modifying the dot product used. Any nonsingular B defines a dot product, which can equivalently be written as (Xj . AXi), with a positive definite matrix A = BTB. Clearly, invariance of g under local transformations of all Xj is a sufficient condition for the local invariance of f, which is what we are aiming for. Strictly speaking, however, invariance of g is not necessary at points which are not Support Vectors, since these lie in a region where (sgn 0 g) is constant however, before training, it is hard to predict which examples will turn out to become SVs. In the Virtual SV method (Scholkopf, Burges, & Vapnik, 1996), a first run of the standard SV algorithm is carried out to obtain an initial SV set; similar heuristics could be applied in the present case. Local invariance of g for each pattern Xj under transformations of a differentiable local I-parameter group of local transformations Lt, ~ I g(LtXj) = 0, (5) ut t=o 642 B. Schllikopf, P. Simard, A. 1 Smola and V. Vapnik can be approximately enforced by minimizing the regularizer 1 i (8 )2 eL 8tlt=og(.ct Xj) j=1 (6) Note that the sum may run over labelled as well as unlabelled data, so in principle one could also require the decision function to be invariant with respect to transformations of elements of a test set. Moreover, we could use different transformations for different patterns. For (4), the local invariance term (5) becomes using the chain rule. Here, 81 (B.coxj . BXi) denotes the gradient of (x· y) with respect to x, evaluated at the point (x . y) = (B.coxj . BXi). Substituting (7) into (6), using the facts that.co = I and 81 (x, y) = Y T, yields the regularizer 1 i (i 8)2 i - '" '" OWi(Bxi)T B8 1 .ctXj = '" (};iYi(};kYk(Bxi' BCBT BXk) (8) e ~ ~ t t=O ~ j=1 ~=1 i,k=1 where 1i (8 )(8 )T C:=-'" -I .ctx· -I .ctx· e ~ 8t t=O J 8t t=O J j=1 (9) We now choose B such that (8) reduces to the standard SV target function IlwW in the form obtained by substituting the expansion w = 2::=1 (};iYiXi into it (cf. the quadratic term of (2», utilizing the dot product chosen in (4), i.e. such that (BXi . BCBT BXk) = (f3Xi . BXk). Assuming that the Xi span the whole space, this condition becomes BT BC B B = B T B, or, by requiring B to be nonsingular, i.e. that no information get lost during the preprocessing, BCBT = I. This can be satisfied by a preprocessing (whitening) matrix B =C-t (10) (modulo a unitary matrix, which we disregard), the nonnegative square root of the inverse of the nonnegative matrix C defined in (9). In practice, we use a matrix C>. := (1 - A)C + AI, (11 ) o < A ~ 1, instead of C. As C is nonnegative, C>. is invertible. For A = 1, we recover the standard SV optimal hyperplane algorithm, other values of A determine the trade-off between invariance and model complexity control. It can be shown that using C>. corresponds to using an objective function 4>(w) = (1 - A) 2:i(W' ttlt=0.ct Xi)2 + Allw112. By choosing the preprocessing matrix B according to (10), we have obtained a formulation of the problem where the standard SV quadratic optimization technique does in effect minimize the tangent regularizer (6): the maximum of (2), using the modified dot product as in (4), coincides with the minimum of (6) subject to the separation conditions Yi . g(Xi) 2: I, where 9 is defined as in (4). Note that preprocessing with B does not affect classification speed: since (Bxj . BXi) = (Xj . BT BXi), we can precompute BT BXi for all SVs Xi and thus obtain a machine (with modified SVs) which is as fast as a standard SV machine (cf. (4». Relationship to Principal Component Analysis (PCA). Let us now provide some interpretation of (10) and (9). The tangent vectors ± tt It=o.ctxj have zero mean, thus C is a Prior Knowledge in Support Vector Kernels 643 sample estimate of the covariance matrix of the random vector s . %t It=OLtX, s E {±1} being a random sign. Based on this observation, we call C (9) the Tangent Covariance Matrix of the data set {Xi: i = 1, . .. ,f} with respect to the transformations Lt. Being positive definite,1 C can be diagonalized, C = SDST, with an orthogonal matrix S consisting of C's Eigenvectors and a diagonal matrix D containing the corresponding positive Eigenvalues. Then we can compute B = C-! = SD-! ST, where D- ~ is the diagonal matrix obtained from D by taking the inverse square roots of the diagonal elements. Since the dot product is invariant under orthogonal transformations, we may drop the leading S and (4) becomes l g(Xj) = 2:>~iYi(D-t ST Xj . D-~ sT Xi) + b. ( 12) i=l A given pattern X is thus first transformed by projecting it onto the Eigenvectors of the tangent covariance matrix C, which are the rows of ST. The resulting feature vector is then rescaled by dividing by the square roots of C's Eigenvalues.2 In other words, the directions of main variance of the random vector %t It=OLtX are scaled back, thus more emphasis is put on features which are less variant under Lt. For example, in image analysis, if the Lt represent translations, more emphasis is put on the relative proportions of ink in the image rather than the positions of lines. The peA interpretation of our preprocessing matrix suggests the possibility to regularize and reduce dimensionality by discarding part of the features, as it is common usage when doing peA. In the present work, the ideas described in this section have only been tested in the linear case. More generally, SV machines use a nonlinear kernel function which can be shown to compute a dot product in a high-dimensional space F nonlinearly related to input space via some map <P, i.e. k(x, y) = (<fl(x) . <fl(y)). In that case, the above analysis leads to a tangent covariance matrix C in P, and it can be shown that (12) can be evaluated in terms of the kernel function (Scholkopf, 1997). To this end, one diagonalizes C using techniques of kernel peA (Scholkopf, Smola, & Muller, 1996). 4 KERNELS USING LOCAL CORRELATIONS By using a kernel k(x,y) = (x· y)d, one implicitly constructs a decision boundary in the space of all possible products of d pixels. This may not be desirable, since in natural images, correlations over short distances are much more reliable as features than long-range correlations are. To take this into account, we define a kernel k~l ,d2 as follows (cf. Fig. 1): 1. compute a third image z, defined as the pixel-wise product of x and y 2. sample Z with pyramidal receptive fields of diameter p, centered at ~ 11 locations (i,j), to obtain the values Zij 3. raise each Zij to the power d1 , to take into account local correlations within the range of the pyramid 4. sum ztJ over the whole image, and raise the result to the power d2 to allow for longe-range correlations of order d2 lIt is understood that we use C>. if C is not definite (cf. (11)). Alternatively, we can below use the pseudoinverse. 2 As an aside, note that our goal to build invariant SV machines has thus serendipitously provided us with an approach for an open problem in SV learning, namely the one of scaling: in SV machines, there has so far been no way of automatically assigning different weight to different directions in input space in a trained SV machine, the weights of the first layer (the SV s) form a subset of the training set. Choosing these Support Vectors from the training set only gives rather limited possibilities for appropriately dealing with different scales in different directions of input space. 644 B. SchOlkopf, P. Simard, A. 1. Smola and V. Vapnik ~d j C.'.) I (. )dl Figure I: Kernel utilizing local correlations in images, corresponding to a dot product in a polynomial space which is spanned mainly by local correlations between pixels (see text). The resulting kernel will be of order d1 • d2 , however, it will not contain all possible correlations of d1 . d2 pixels. 5 EXPERIMENTAL RESULTS In the experiments, we used a subset of the MNIST data base of handwritten characters (Bottou et aI., 1994), consisting of 5000 training examples and 10000 test examples at a resolution of 20x20 pixels, with entries in [-1, 1]. Using a linear SV machine (i.e. a separating hyperplane), we obtain a test error rate of 9.8% (training 10 binary classifiers, and using the maximum value of 9 (cf. (4» for lO-class classification); by using a polynomial kernel of degree 4, this drops to 4.0%. In all of the following experiments, we used degree 4 kernels of various types. The number 4 was chosen as it can be written as a product of two integers, thus we could compare results to a kernel k~l ,d2 with d1 = d2 = 2. For the considered classification task, results for higher polynomial degrees are very similar. In a series of experiments with a homogeneous polynomial kernel k(x, y) = (x· y)4, using preprocessing with Gaussian smoothing kernels of standard deviation 0.1, 0.2, ... ,1.0, we obtained error rates which gradually increased from 4.0% to 4.3%; thus no improvement of this performance was possible by a simple smoothing operation. Applying the Virtual SV method (retraining the SV machine on translated SVs; Scholkopf, Burges, & Vapnik,1996) to this problem results in an improved error rate of 2.8%. For training on the full 60000 pattern set, the Virtual SV performance is 0.8% (Scholkopf, 1997). Invariant hyperplanes. Table 1 reports results obtained by preprocessing all patterns with B (cf. (10», choosing different values of ..\ (cf. (11». In the experiments, the patterns were first rescaled to have entries in [0,1], then B was computed, using horizontal and vertical translations, and preprocessing was carried out; finally, the resulting patterns were scaled back again. This was done to ensure that patterns and derivatives lie in comparable regions of RN (note that if the pattern background level is a constant -1, then its derivative is 0). The results show that even though (9) was derived for the linear case, it can lead to improvements in the nonlinear case (here, for a degree 4 polynomial), too. Dimensionality reduction. The above [0, 1] scaling operation is affine rather than linear, hence the argument leading to (12) does not hold for this case. We thus only report results on dimensionality reduction for the case where the data is kept in [0, 1] scaling from the very Prior Knowledge in Support Vector Kernels 645 Table I: Classification error rates for modifying the kernel k(x, y) = (X·y)4 with the invariI ant hyperplane preprocessing matrix B).. = C~ 'i ; cf. (10) and (11). Enforcing invariance with 0.1 < A < 1 leads to improvements over the original performance (A = 1). A 0.1 0.2 0.4 0.6 error rate in % 4.2 3.8 3.6 3.8 Table 2: Dropping directions corresponding to smaIl Eigenvalues of C (cf. (12)) leads to substantial improvements. AIl results given are for the case A = 0.4 (cf. Table 1); degree 4 homogeneous polynomial kernel. principal components discarded error rate in % beginning on. Dropping principal components which are less important leads to substantial improvements (Table 2); cf. the explanation foIlowing (12). The results in Table 2 are somewhat distorted by the fact that the polynomial kernel is not translation invariant, and performs poorly on the [0, 1] data, which becomes evident in the case where none of the principal components are discarded. Better results have been obtained using translation invariant kernels, e.g. Gaussian REFs (Scholkopf, 1997). Kernels using local correlations. To exploit locality in images, we used a pyramidal receptive field kernel k;l,d 2 with diameter p = 9 (cf. Sec. 4). For d1 = d2 = 2, we obtained an improved error rate of 3.1%, another degree 4 kernel with only local correlations (dl = 4, d2 = 1) led to 3.4%. Albeit significantly better than the 4.0% for the degree 4 homogeneous polynomial (the error rates on the 10000 element test set have an accuracy of about 0.1%, cf. Bottouet aI., 1994), this is still worse than the Virtual SV resultof2.8%. As the two methods, however, exploit different types of prior knowledge, it could be expected that combining them leads to still better performance; and indeed, this yielded the best performance of all (2.0%). For the purpose of benchmarking, we also ran our system on the US postal service database of 7291 +2007 handwritten digits at a resolution of 16 x 16. In that case, we obtained the foIlowing test error rates: SV with degree 4 polynomial kernel 4.2%, Virtual SV (same kernel) 3.5%, SV with k~,2 3.6%, Virtual SV with k~,2 3.0%. The latter compares favourably to almost all known results on that data base, and is second only to a memory-based tangentdistance nearest neighbour classifier at 2.6% (Simard, LeCun, & Denker, 1993). 6 DISCUSSION With its rather general class of admissible kernel functions, the SV algorithm provides ample possibilities for constructing task-specific kernels. We have considered an image classification task and used two forms of domain knowledge: first, pattern classes were required to be locally translationaIly invariant, and second, local correlations in the images were assumed to be more reliable than long-range correlations. The second requirement can be seen as a more general form of prior knowledge it can be thought of as arising partiaIly from the fact that patterns possess a whole variety of transformations; in object recognition, for instance, we have object rotations and deformations. Typically, these transformations are continuous, which implies that local relationships in an image are fairly stable, whereas global relationships are less reliable. We have incorporated both types of domain knowledge into the SV algorithm by constructing appropriate kernel functions, leading to substantial improvements on the considered pattern recognition tasks. Our method for constructing kernels for transformation invariant SV machines, put forward to deal with the first type of domain knowledge, so far has 646 B. SchOlkopf, P. Simard, A. 1. Smola and V. Vapnik only been applied in the linear case, which partially explains why it only led to moderate improvements (also, we so far only used translational invariance). It is applicable for differentiable transformations other types, e.g. for mirror symmetry, have to be dealt with using other techniques, e.g. Virtual SVs (Scholkopf, Burges, & Vapnik, 1996). Its main advantages compared to the latter technique is that it does not slow down testing speed, and that using more invariances leaves training time almost unchanged. The proposed kernels respecting locality in images led to large improvements; they are applicable not only in image classification but in all cases where the relative importance of subsets of products features can be specified appropriately. They do, however, slow down both training and testing by a constant factor which depends on the specific kernel used. Both described techniques should be directly applicable to other kernel-based methods as SV regression (Vapnik, 1995) and kernel PCA (Scholkopf, Smola, & Muller, 1996). Future work will include the nonlinear case (cf. our remarks in Sec. 3), the incorporation of invariances other than translation, and the construction of kernels incorporating local feature extractors (e.g. edge detectors) different from the pyramids described in Sec. 4. Acknowledgements. We thank Chris Burges and Uon Bottou for parts of the code and for helpful discussions, and Tony Bell for his remarks. References B. E. Boser, I .M. Guyon, and V. N. Vapnik. A training algorithm for optimal margin classifiers. In D. Haussler, editor, Proceedings of the 5th Annual ACM Workshop on Computational Learning Theory, pages 144-152, PittSburgh, PA, 1992. ACM Press. L. Bottou, C. Cortes, J. S. Denker, H. Drucker, I. Guyon, L. D. Jackel, Y. LeCun, U. A. Muller, E. Sackinger, P. Simard, and V. Vapnik. Comparison of classifier methods: a case study in handwritten digit recognition. In Proceedings of the J 2th International Conference on Pattern Recognition and Neural Networks, Jerusalem, pages 77 - 87. IEEE Computer Society Press, 1994. C. Cortes and V. Vapnik. Support vector networks. Machine Learning, 20:273 - 297, 1995. B. Scholkopf. Support Vector Learning. R. Oldenbourg Verlag, Munich, 1997. ISBN 3-486-24632-1. B. Scholkopf, C. Burges, and V. Vapnik. Incorporating invariances in support vector learning machines. In C. von der Malsburg, W. von Seelen, J. C. Vorbriiggen, and B. Sendhoff, editors, Artificial Neural Networks-ICANN'96, pages 47 - 52, Berlin, 1996a. Springer Lecture Notes in Computer Science, Vol. 1112. B. Scholkopf, A. Smola, and K.-R. MulIer. Nonlinear component analysis as a kernel eigenvalue problem. Technical Report 44, Max-Planck-Institut fUr biologische Kybernetik, 1996b. in press (Neural Computation). P. Simard, Y. LeCun, and J. Denker. Efficient pattern recognition using a new transformation distance. In S. 1. Hanson, J. D. Cowan, and C. L. Giles, editors, Advances in Neural Information Processing Systems 5, pages 50-58, San Mateo, CA, 1993. Morgan Kaufmann. P. Simard, B. Victorri, Y. LeCun, and 1. Denker. Tangent prop a formalism for specifying selected invariances in an adaptive network. In J. E. Moody, S. J. Hanson, and R. P. Lippmann, editors, Advances in Neural Information Processing Systems 4, pages 895903, San Mateo, CA, 1992. Morgan Kaufmann. V. Vapnik. The Nature of Statistical Learning Theory. Springer Verlag, New York, 1995. V. Vapnik and A. Chervonenkis. Theory of Pattern Recognition [in Russian}. Nauka, Moscow, 1974. (German Translation: W. Wapnik & A. Tscherwonenkis, Theorie der Zeichenerkennung, Akademie-Verlag, Berlin, 1979).
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Incorporating Test Inputs into Learning Zebra Cataltepe Learning Systems Group Department of Computer Science California Institute of Technology Pasadena, CA 91125 zehra@cs.caltech.edu Malik Magdon-Ismail Learning Systems Group Department of Electrical Engineering California Institute of Technology Pasadena, CA 91125 magdon@cco.caltech.edu Abstract In many applications, such as credit default prediction and medical image recognition, test inputs are available in addition to the labeled training examples. We propose a method to incorporate the test inputs into learning. Our method results in solutions having smaller test errors than that of simple training solution, especially for noisy problems or small training sets. 1 Introduction We introduce an estimator of test error that takes into consideration the test inputs. The new estimator, augmented error, is composed of the training error and an additional term computed using the test inputs. In some applications, such as credit default prediction and medical image recognition, we do have access to the test inputs. In our experiments, we found that the augmented error (which is computed without looking at the test outputs but only test inputs and training examples) can result in a smaller test error. In particular, it tends to increase when the test error increases (overtraining) even if the simple training error does not. (see figure (1)). In this paper, we provide an analytic solution for incorporating test inputs into learning in the case oflinear, noisy targets and linear hypothesis functions. We also show experimental results for the nonlinear case. Previous results on the use of unlabeled inputs include Castelli and Cover [2] who show that the labeled examples are exponentially more valuable than unlabeled examples in reducing the classification error. For mixture models, Shahshahani and Landgrebe [7] and Miller and Uyar [6] investigate incorporating unlabeled examples into learning for classification problems and using EM algorithm, and show that unlabeled examples are useful especially when input dimensionality is high and the number of examples is small. In our work we only concentrate on estimating the test error better using the test inputs and our method 438 Z Cataltepe and M. MagMn-Ismail 2·~----~======~==~~------1 1:l1----__ I!! ~ w 1 5 Training error - - Test error -- Augmented error '. , \ \ , \ __ • J I I ,. --,. , .~~------~5~------~6--------~7--------~8 log(pass) Figure I: The augmented error, computed not looking at the test outputs at all, follows the test error as overtraining occurs. extends to the case of unlabeled inputs or input distribution information. Our method is also applicable for regression or classification problems. In figure 1, we show the training, test and augmented errors, while learning a nonlinear noisy target function with a nonlinear hypothesis. As overtraining occurs, the augmented error follows the test error. In section 2, we explain our method of incorporating test inputs into learning and give the analytical solutions for linear target and hypothesis functions. Section 3 includes theory about the existence and general form of the new solution. Section 4 discusses experimental results. Section 5 extends our solution to the case of knowing the input distribution, or knowing extra inputs that are not necessarily test inputs. 2 Incorporating Test Inputs into Learning In learning-from-examples, we assume we have a training set: {(Xl, II), ., . ,(XN, IN)} with inputs Xn and possibly noisy targets In. Our goal is to choose a hypothesis gv, among a class of hypotheses G, minimizing the test error on an unknown test set {(YI, hd,···, (YM, hM)}' Using the sample mean square error as our error criterion, the training error of hypothesis gv is: Similarly the test error of gv is: E(gv) Expanding the test error: Incorporating Test InpuJs into Learning 439 The main observation is that. when we know the test inputs. we know the first term exactly. Therefore we need only approximate the remaining terms using the training set: 1 M 2 N 1 N M L9~(Ym)- NL9v (Xn)!n+ NL!~ m=1 n=1 n=1 (1) We scale the addition to the training error by an augmentation parameter a to obtain a more general error function that we call the augmented error: Eo (g.) + <> (~ ~ g; (Ym) - ~ t. g; (Xn)) where a = a corresponds to the training error Eo and a = 1 corresponds to equation (1). The best value of the augmentation parameter depends on a number offactors including the target function. the noise distribution and the hypothesis class. In the following sections we investigate properties of the best augmentation parameter and give a method of finding the best augmentation parameter when the hypothesis is linear. 3 Augmented Solution for the Linear Hypothesis In this section we assume hypothesis functions of the form 9v(X) = v T x. From here onwards we will denote the functions by the vector that multiplies the inputs. When the hypothesis is linear we can find the minimum of the augmented error analytical1y. Let X dxN be the matrix of training inputs. YdxM be the matrix of test inputs and fNXI contain the training targets. The solution Wo minimizing the training error Eo is the least squares solution [5]: Wo = (X~T) -1 x:. The augmented error Ea (v) = Eo (v) + avT (Y[/ - x ~T ) v is minimized at the augmented error Wa : (2) ( T) -1 T where R = 1 x ~ Y1,; . When a = O. the augmented solution Wa is equal to the least mean squares solution Woo 4 Properties of the Augmentation Parameter Assume a linear target and possibly noisy training outputs: f = w·T X +e where (eeT ) = u;INxN . Since the specific realization of noise e is unknown. instead of minimizing the test error directly. we focus on minimizing (E (wa))e. the expected value of the test error of the augmented solution with respect to the noise distribution: (E (wa))e w·T ((I - aRT) -1 - I) Y~T ((I - aR)-1 - I) w· + ~tr ((I -aRT' Y~T (I - <>R)-' (X;Tr') (3) 440 Z Cataltepe and M. Magdon-Ismail where we have used (eT Ae) e = cr;tr (A) and tr(A) denotes the trace of matrix A. When o = 0, we have: -tr -- --cr; (YYT (XXT) -1) N M N (4) Now, we prove the existence of a nonzero augmentation parameter 0 when the outputs are noisy. Theorem 1: If cr; > 0 and tr (R (I - R)) =1= 0, then there is an 0 =1= 0 that minimizes the expected test error (E (wa))e' Proof: Since &B;~(a) = _B-l(o)&~~a)B-l(o) for any matrix B whose elements are scalar functions of 0 [3], the derivative of (E (wa))e with respect to 0 at 0 = 0 is: dIE t·)). I.~. = 2~tr (R( X;TfY~T) = 2~tr (R(I -R» If the derivative is < 0 (> 0 respectively), then (E (wa))e is minimized at some 0 > 0 (0 < 0 respectively). 0 The following proposition gives an approximate formula for the best o. Theorem 2: If Nand M are large, and the traini~ and test inputs are drawn i.i.d from an input distribution with covariance matrix (xx ) = cr;l, then the 0* minimizing (E (wa))e,x,y' the expected test error of the augmented solution with respect to noise and inputs, is approximately: (5) Proof: is given in the appendix. 0 This formula determines the behavior of the best o. The best 0: • decreases as the signal-to-noise ratio increases. • increases as ~ increases, i.e. as we have less examples per input dimension. 4.1 Wa as an Estimator ofw* The mean squared error (m.s.e.) of any estimator W ofw*, can be written as [1]: IIW* - (w)eI1 2 + (11w - (w)eI1 2 ) e m.s.e(w) bias2(w) + variance(w) When 0 is independent of the specific realization e of the noise: W*T (1 - (1 - oRT)-l) (1 - (I - oR)-I) w* + ~ tr ( (X;T) -1 (I _ aRTr1(I _ aR)-I) 1.5 1.4 1.3 w !! ~ 12 • ! CD 1.1 0> ! ~ < 0.9 0.8 Incorporating Test Inputs into Learning 441 Hence the m.s.e. of the least square estimator Wo is: m.s.e.(wo) Wo is the minimum variance unbiased linear estimator of W·. Although w 0< is a biased estimator if exR =/:. 0, the following proposition shows that, when there is noise, there is an ex =/:. o minimizing them.s.e. ofwo<: Theorem 3: If 0'; > 0 and tT ( ( X~T) -1 (R + RT)) =/:. 0, then there is an ex =/:. 0 that minimizes the m.s.e. ofwo<' Proof: is similar to the proof of proposition 1 and will be skipped D. As Nand M get large, R = 1 (X~T) -1 Y,{/ -+ 0 and Wo< = (1 - aR)-lwo -+ woo Hence, for large Nand M, the bias and variance of w 0< approach 0, making w 0< an un biased and consistent estimator of w· . 5 A Method to Find the Best Augmentation Parameter Uver data. <1=0. M=CO Bond da1a. d=11. 1.1=50 i .. least squares >+-< augmented error with estimated alpha 6.5 k 1 \ w 6 , "e \ • • 5.S ! • I .. '. '. ' . 0> ! ~ < ..... ~:f.::~ 4.5 4 0 20 40 60 80 100 120 140 160 40 60 80 100 120 140 N:number of tranng exal!'Clies N romber of tranng exal!'Clies Figure 2: Using the augmented error results in smaller test error especially when the number of training examples is small. Given only the training and test inputs X and Y, and the training outputs f, in this section we propose a method to find the best ex minimizing the test error of w 0<' Equation (3) gives a formula for the expected test error which we want to minimize. However, we do not know the target w· and the noise variance 0';. In equation (3), we replace w'" by Wo< and 0'; by (XTwa-:L:~~Twa-f), where Wo< is given by equation (2). Then we find the ex minimizing the resulting approximation to the expected test error. We experimented with this method of finding the best a on artificial and real data. The results of experiments for liver datal and bond data2 are shown in figure 2. In the liver Iftp:llftp.ics.uci.edulpub/machine-Iearning-databaseslliver-disorders/bupa.data 2We thank Dr. John Moody for providing the bond data. 442 Z Cataltepe and M. Magdon-Ismail database the inputs are different blood test results and the output is the number of drinks per day. The bond data consists of financial ratios as inputs and rating of the bond from AAA to B- or lower as the output. We also compared our method to the least squares (wo) and early stopping using different validation set sizes for linear and noisy problems. The table below shows the results. SNR mean E(wol E(wo) mean 1>(Wearlll .top) N -!i E~wo.l 'v 3 mean 1>(Wearlll .top) N =!i E Wn 'v 6 O.oI 0.650 ± 0.006 0.126 ± 0.003 0.192 ± 0.004 I 0.830 ± 0.007 1.113 ± 0.021 1.075 ± 0.020 100 1.001 ± 0.002 2.373 ± 0.040 2.073 ± 0.042 Table 1: Augmented solution is consistently better than the least squares whereas early stopping gives worse results as the signal-to-noise ratio (SNR) increases. Even averaging early stopping solutions did not help when SNR = 100 (E(wE(~~;top) = 1.245 ± 0.018 when Nv = ~ and 1.307 ± 0.021 for Nv = ~). For the results shown, d = 11, N = 30 training examples were used, N v is the number of validation examples. 6 Extensions When the input probability distribution or the covariance matrix of inputs, instead of test inputs are known, YJ/ can be replaced by (xxT) = E and our methods are still applicable. If the inputs available are not test inputs but just some extra inputs, they can still be incorporated into learning. Let us denote the extra K inputs {ZI, ... , ZK} by the matrix ZdxK. Then the augmented error becomes: Ea(v) Eo (v) + a v T -- - -v K (ZZT XXT) K+N K N The augmented new solution and its expected test error are same as in equations (2) and (3), except we have Rz = 1 ( X~T) -1 Z;T instead of R. Note that for the linear hypothesis case, the augmented error is not necessarily a regularized version of the training error, because the matrix Yl;T - ~ is not necessarily a positive definite matrix. 7 Conclusions and Future Work We have demonstrated a method of incorporating inputs into learning when the target and hypothesis functions are linear, and the target is noisy. We are currently working on extending our method to nonlinear target and hypothesis functions. Appendix Proof of Theorem 2: When the spectral radius of o.R is less than I (a is small and/or Nand M are large), we can approximate (1 - aR)-l :::::: 1 + aR [4], and similarly, (1 - aRT) -1 :::::: 1 + aRT. Discarding any terms with powers of a greater than 1, and Incorporating Test Inputs into Learning 443 solving for 0 in d(E(W;l)·'X'y = (d(E~"'»). ) = 0: x ,Y 0* The last step follows since we can write y~T = u; (I + h ), X ~T = u; (I - .IN ) and (X ~T) -1 = th (I + .IN + Yi-) + 0 (Nb) for matrices Vx and Vy such that (Vx}x = (Vy}y = 0 and (Vx2)x and (V;)y are constant with respect to Nand M. For large M we can approximate (Y~T R2) = u; (R2) . x ,Y x ,Y Ignoring terms of 0 (Nt 5 ) , (R2 - R) x,y (Yi- + ~) X ,y' It can be shown that (Yi) x (2Yi + ~) and (R2) x ,y x,y = ~ I for a constant .A depending on the input distribution. Similarly (~ ) y = ttl. Therefore: 0* o Acknowledgments We would like to thank the Caltech Learning Systems Group: Prof. Yaser Abu-Mostafa, Dr. Amir Atiya, Alexander Nicholson, Joseph Sill and Xubo Song for many useful discussions. References [1] Bishop, c. (1995) Neural Networks for Pattern Recognition, Clarendon Press, Oxford, 1995. [2] Castelli, V. & Cover T. (1995) On the Exponential Value of Labeled Samples. Pattern Recognition Letters, Vol. 16, Jan. 1995, pp. 105-111. [3] Devijver, P. A. & Kittler, J. (1982) Pattern Recognition: A Statistical Approach, pp. 434. Prentice-Hall International, London. [4] Golub, G. H. & Van Loan C. F. (1993) Matrix Computations, The Johns-Hopkins University Press, Baltimore, MD. [5] Hocking, R. R. (1996) Methods and Applications of Linear Models. John Wiley & Sons, NY. [6] Miller, D. J. & Uyar, S. (1996), A Mixture of Experts Classifier with Learning Based on Both Labeled and Unlabeled Data. In G. Tesauro, D. S. Touretzky and T.K. Leen (eds.), Advances in Neural Information Processing Systems 9. Cambridge, MA: MIT Press. [7] Shahshahani, B. M. & Landgrebe, D. A. (1994) The Effect of Unlabeled Samples in Reducing Small Sample Size Problem and Mitigating the Hughes Phonemenon. IEEE Transactions on Geoscience and Remote Sensing, Vol. 32 No. 5, Sept 1994, pp. 10871095.
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Analytical study of the interplay between architecture and predictability Avner Priel, Ido Kanter, David A. Kessler Minerva Center and Department of Physics, Bar Ilan University, Ramat-Gan 52900, Israel. e-mail: priel@mail.cc.biu.ac.il (web-page: http://faculty.biu.ac.il/ ""'priel) Abstract We study model feed forward networks as time series predictors in the stationary limit. The focus is on complex, yet non-chaotic, behavior. The main question we address is whether the asymptotic behavior is governed by the architecture, regardless the details of the weights. We find hierarchies among classes of architectures with respect to the attract or dimension of the long term sequence they are capable of generating; larger number of hidden units can generate higher dimensional attractors. In the case of a perceptron, we develop the stationary solution for general weights, and show that the flow is typically one dimensional. The relaxation time from an arbitrary initial condition to the stationary solution is found to scale linearly with the size of the network. In multilayer networks, the number of hidden units gives bounds on the number and dimension of the possible attractors. We conclude that long term prediction (in the non-chaotic regime) with such models is governed by attractor dynamics related to the architecture. Neural networks provide an important tool as model free estimators for the solution of problems when the real model is unknown, or weakly known. In the last decade there has been a growing interest in the application of such tools in the area of time series prediction (see Weigand and Gershenfeld, 1994). In this paper we analyse a typical class of architectures used in this field, i.e. a feed forward network governed by the following dynamic rule: S t+l S 1 = out ; S t+l = st 1 J' 2 N ] ]-, .. . , (1) where Sout is the network's output at time step t and Sj are the inputs at that time; N is the size of the delayed input vector. The rational behind using time delayed vectors as inputs is the theory of state space reconstruction of a dynamic system 316 A. Priel, 1. Kanter and D. A. Kessler using delay coordinates (Takens 1981, Sauer Yorke and Casdagli 1991). This theory address the problem of reproducing a set of states associated with the dynamic system using vectors obtained from the measured time series, and is widely used for time series analysis. A similar architecture incorporating time delays is the TDNN - time-delay neural network with a recurrent loop (Waibel et. al. 1989). This type of networks is known to be appropriate for learning temporal sequences, e.g. speech signal. In the context of time series, it is mostly used for short term predictions. Our analysis focuses on the various long-time properties of the sequence generated by a given architecture and the interplay between them. The aim of such an investigation is the understanding and characterization of the long term sequences generated by such architectures, and the time scale to reach this asymptotic behavior. Such knowledge is necessary to define adequate measures for the transition between a locally dependent prediction and the long term behavior. Though some work has been done on characterization of a dynamic system from its time series using neural networks, not much analytical results that connect architecture and long-time prediction are available (see M. Mozer in Weigand and Gershenfeld, 1994). Nevertheless, practical considerations for choosing the architecture were investigated extensively (Weigand and Gershenfeld, 1994 and references therein). It has been shown that such networks are capable of generating chaotic like sequences. While it is possible to reconstruct approximately the phase space of chaotic attractors (at least in low dimension), it is clear that prediction of chaotic sequences is limited by the very nature of such systems, namely the divergence of the distance between nearby trajectories. Therefore one can only speak about short time predictions with respect to such systems. Our focus is the ability to generate complex sequences, and the relation between architecture and the dimension of such sequences. 1 Perceptron We begin with a study of the simplest feed forward network, the perceptron. We analyse a perceptron whose output Sout at time step t is given by: Sou. = tanh [13 (t,(W; + WO)Sj) ] (2) where {3 is a gain parameter, N is the input size. The bias term ,Wo, plays the same role as the common 'external field' used in the literature, while preserving the same qualitative asymptotic solution. In a previous work (Eisenstein et. al. , 1995) it was found that the stationary state (of a similar architecture but with a "sign" activation function instead of the "tanh", equivalently (3 --t 00) is influenced primarily by one of the larger Fourier components in the power spectrum of the weights vector W of the perceptron. This observation motivates the following representation of the vector W. Let us start with the case of a vector that consists of a singl€ biased Fourier component of the form: Wj = acos(27fKjjN) j = 1, ... ,N ; Wo =b (3) where a, b are constants and K is a positive integer. This case is generalized later on, however for clarity we treat first the simple case. Note that the vector W can always be represented as a Fourier decomposition of its values. The stationary solution for the sequence (SI) produced by the output of the percept ron , when inserting this choice of the weights into equation (2), can be shown to be of the form: SI = tanh [A({3) cos(27fKljN) + B({3)] There are two non-zero solutions possible for the variables (A, B): (4) The Interplay between Architecture and Predictability A B t{3N a I:~l D(p)(A/2)2P-l (p!)-2 {3Nb I:~l D(p)S2p-l ((2p)!)-1 317 B =0 (5) where D(p) = 22p (22p - 1)B2p and B2p are the Bernoulli numbers. Analysis of equations (5) reveals the following behavior as a function of the parameter {3. Each of the variables is the amplitude of an attractor. The attractor represented by (A i- 0, B = 0) is a limit cycle while the attractor represented by (B i- 0, A = 0) is !l fixed point of the dynamics. The onset of each of the attractors A(B) is at {3cl = 2(aN)-1 ({3c2 = (bN)-l) respectively. One can identify three regimes: (1) {3 < {3cl,c2 - the stable solution is Sl = O. (2) min({3cl, (3c2) < {3 < max({3cl, (3c2) the system flows for all initial conditions into the attractor whose {3c is smaller. (3) {3 > {3cl,c2 - depending on the initial condition of the input vector, the system flows into one of the attractors, namely, the stationary state is either a fixed point or a periodic flow. {3cl is known as a Hopf bifurcation point. Naturally, the attractor whose {3c is smaller has a larger basin of attraction, hence it is more probable to attract the flow (in the third regime). 1.0 0.5 0.0 -0.5 -1.0 -1.0 000000000 -0.5 0.0 Sl o o o o o o o o o o 00 00 000 00 0.5 1.0 Figure 1: Embedding of a sequence generated by a perceptron whose weights follow eq. 3 (6) . Periodic sequence (outer curve) N = 128, k = 17, b = 0.3, {3 = 1/40 and quasi periodic (inner) k = 17, ¢ = 0.123, (3 1/45 respectively. Next we discuss the more general case where the weights of eq. (3) includes an arbitrary phase shift of the form: Wj = acos(27fKj/N - 7f¢) ¢ E (-1,1) (6) The leading term of the stationary solution in the limit N » 1 is of the form: Sl = tanh [A({3) cos(27f(K - ¢)l/N) + B({3)] (7) where the higher harmonic corrections are of O( 1 / K). A note should be made here that the phase shift in the weights is manifested as a frequency shift in the solution. In addition, the attractor associated with A i- 0 is now a quasi-periodic flow in the generic case when ¢ is irrational. The onset value ofthe fixed point ({3c2) is the same as before, however the onset of the quasi-periodic orbit is (3cl = sin'(!4» 2(aN)-1. The variables A, B follow similar equations to (5): A (3Na SinJ;4» I:~l D(p)(A/2)2P-l(p!)-2 B =0 (8) A=O The three regimes discussed above appear in this case as well. Figure 1 shows the attractor associated with (A i- 0, B = 0) for the two cases where the series generated by the output is embedded as a sequence of two dimensional vectors (Sl+l, Sl). 318 A. Priel, I Kanter and D. A. Kessler The general weights can be written as a combination of their Fourier components with different K's and ¢'s: m Wj = Laicos(27fKd/N-7f¢i) ¢i E (-1,1) (9) i=l When the different K's are not integer divisors of each other, the general solution is similar to that described above: Sl = tanh [t, A,({3) cos(27r(K, - <pi)l / N) + B({3) 1 (10) where m is the number of relevant Fourier components. As above, the variables Ai ,B are coupled via self consistent equations. Nevertheless, the generic stationary flow is one of the possible attractors, depending on /3 and the initial condition; i.e. (Aq i- 0, Ai = 0 Vi i- q ,B = 0) or (B i- 0, Ai = 0). By now we can conclude that the generic flow for the perceptron is one of three: a fixed point, periodic cycle or quasi-periodic flow. The first two have a zero dimension while the last describes a one dimensional flow. we stress that more complex flows are possible even in our solution (eq. 10), however they require special relation between the frequencies and a very high value of /3, typically more than an order of magnitude greater than bifurcation value. 2 Relaxation time At this stage the reader might wonder about the relation between the asymptotic results presented above and the ability of such a model to predict. In fact, the practical use of feed forward networks in time series prediction is divided into two phases. In the first phase, the network is trained in an open loop using a given time series. In the second phase, the network operates in a closed loop and the sequence it generates is also used for the future predictions. Hence, it is clear from our analysis that eventually the network will be driven to one of the attractors. The relevant question is how long does it takes to arrive at such asymptotic behavior? We shall see that the characteristic time is governed by the gap between the largest and the second largest eigenvalues of the linearized map. Let us start by reformulating eqs. (1,2) in a matrix form, i.e. we linearize the map. Denote st = (Sf, s~, ... , Sj.,,) -t -t -t+1 and (S )' is the transposed vector. The map is then T(S)' = (S )' where CN-l CN o 0 T= o 0 (11) o 0 1 o The first row of T gives the next output value = si+ 1 while the rest of the matrix is just the shift defined by eq. (1) . This matrix is known as the "companion matrix" (e.g. Ralston and Rabinowitz, 1978). The characteristic function of T can be written as follows: N /3 '" Cn = 1 ~ ).n n=l (12) from which it is possible to extract the eigenvalues. At (3 = /3c the largest eigenvalue of T is 1).11 = 1. Denote the second largest eigenvalue ).2 such that 1).21 = 1 - 6. . The Interplay between Architecture and Predictability 0.002 <l 0.001 o o 0.01 1/N 0.02 Figure 2: Scaling of ~ for a perceptron with two Fourier components, (eq. 9), with ai = 1, Kl = 3, 1>1 = 0.121, K2 = 7, 1>2 = 0 , Wo = 0.3 . The dashed line is a linear fit of 0.1/ N, N = 50, ... ,400. 319 Applying T T - times to an initial state vector results in a vector whose second largest component is of order: (13) therefore we can define the characteristic relaxation time in the vicinity of an attractor to be T = ~ -1 . 1 We have analysed eq. (12) numerically for various cases of Ci, e.g. Wi composed of one or two Fourier components. In all the cases (3 was chosen to be the minimal f3c to ensure that the linearized form is valid. We found that ~,....., I/N. Figure 2 depicts one example of two Fourier components. Next, we have simulated the network and measured the average time ( T S ) it takes to flow into an attractor starting from an arbitrary initial condition. The following simulations support the analytical result ( T ,....., N ) for general (random) weights and high gain (13) value as well. The threshold we apply for the decision whether the flow is already close enough to the attractor is the ratio between the component with the largest power in the -t spectrum and the total power spectrum of the current state (S ), which should exceed 0.95. The results presented in Figure 3 are an average over 100 samples started from random initial condition. The weights are taken at random, however we add a dominant Fourier component with no phase to control the bifurcation point more easily. This component has an amplitude which is about twice the other components to make sure that its bifurcation point is the smallest. We observe a clear linear relation between this time and N (T S ,....., N ). The slope depends on the actual values of the weights, however the power law scaling does not change. On general principles, we expect the analytically derived scaling law for ~ to be valid even beyond the linear regime. Indeed the numerical simulations (Figure 3) support this conjecture. 3 Multilayer networks For simplicity, we restrict the present analysis to a multilayer network (MLN) with N inputs, H hidden units and a single linear output, however this restriction can be removed, e.g. nonlinear output and more hidden layers. The units in the hidden layer are the perceptrons discussed above and the output is given by: (14) INote that if one demand the L.R.S. of eq. (13) to be of O(~), then T '" ~ -11og(~ -1). 320 800 600 I"'p 400 200 0 0 100 200 N 300 A. Priel, 1. Kanter and D. A. Kessler Figure 3: Scaling of r S for random weights with a dominant component at K = 7, ¢ = 0, a = 1; All other amplitudes are randomly taken between (0,0.5) and the phases are random as well. {3 = 3.2/N. The dashed line is a linear fit of eN, e = 2.73 ± 0.03. N = 16, ... , 256. The dynamic rule is defined by eq. (1). First consider the case where the weights of each hidden unit are of the form described by eq. (6), Le. each hidden unit has only one (possibly biased) Fourier component: m=l, ... ,H. (15) Following a similar treatment as for the perceptron, the stationary solution is a combination of the perceptron-like solution: H Sl = L tanh [Am({3) cos(27r(Km - ¢m)l/N) + Bm({3)] (16) m=l The variables Am, Bm are the solution of the self consistent coupled equations, however by contrast with the single perceptron, each hidden unit operates independently and can potentially develop an attractor of the type described in section 1. The number of attractors depends on {3 with a maximum of H attractors. The number of non-zero Am's defines the attractor's dimension in the generic case of irrational ¢'s associated with them. If different units do not share Fourier components with a common divisor or harmonics of one another, it is easy to define the quantitative result, otherwise, one has to analyse the coupled equations more carefully to find the exact value of the variables. Nevertheless, each hidden unit exhibits only a single highly dominant component (A 1= 0 or B 1= 0). Generalization of this result to more than a single biased Fourier component is straightforward. Each vector is of the form described in eq. (9) plus an index for the hidden unit. The solution is a combination of the general perceptron solution, eq. (10). This solution is much more involved and the coupled equations are complicated but careful study of them reveals the same conclusion, namely each hidden unit possess a single dominant Fourier component (possibly with several other much smaller due to the other components in the vector). As the gain parameter {3 becomes larger, more components becomes available and the number of possible attractors increases. For a very large value it is possible that higher harmonics from different hidden units might interfere and complicate considerably the solution. Still, one can trace the origin of this behavior by close inspection of the fields in each hidden unit. We have also measured the relaxation time associated with MLN's in simulations. The preliminary results are similar to the perceptron, Le. r S '" N but the constant prefactor is larger when the weights consist of more Fourier components. The Interplay between Architecture and Predictability 321 4 Discussion Neural networks were proved to be universal approximators (e.g. Hornik, 1991), hence they are capable of approximating the prediction function of the delay coordinate vector. The conclusion should be that prediction is indeed possible. This observation holds only for short times in general. As we have shown, long time predictions are governed by the attractor dynamics described above. The results point out the conclusion that the asymptotic behavior for this networks is dictated by the architecture and not by the details of the weights. Moreover, the attractor dimension of the asymptotic sequence is typically bounded by the number of hidden units in the first layer (assuming the network does not contain internal delays) . To prevent any misunderstanding we note again that this result refers to the asymptotic behavior although the short term sequence can approximate a very complicated attractor. The main result can be interpreted as follows. Since the network is able to approximate the prediction function, the initial condition is followed by reasonable predictions which are the mappings from the vicinity of the original manifold created by the network. As the trajectory evolves, it flows to one of the attractors described above and the predictions are no longer valid. In other words, the initial combination of solutions described in eq. (10) or its extension to MLN (with an arbitrary number of non-zero variables, A's or B's) serves as the approximate mapping. Evolution of this approximation is manifested in the variables of the solution, which eventually are attracted to a stable attractor (in the non-chaotic regime). The time scale for the transition is given by the relaxation time developed above. The formal study can be applied for practical purposes in two ways. First, taking into account this behavior by probing the generated sequence and looking for its indications. One such indication is stationarity of the power spectrum. Second, one can incorporate ideas from local linear models in the reconstructed space to restrict the inputs in such a way that they always remain in the vicinity of the original manifold (Sauer, in Weigand and Gershenfeld, 1994). Acknowledgments This research has been supported by the Israel Science Foundation. References Weigand A. S. and Gershenfeld N. A. ; Time Series Prediction, Addison-Wesley, Reading, MA, 1994. E. Eisenstein, I. Kanter, D. A. Kessler and W. Kinzel; Generation and prediction of time series by a neural network, Phys. Rev. Lett. 74,6 (1995). Waibel A., Hanazawa T., Hinton G., Shikano K. and Lang K.; Phoneme recognition using TDNN, IEEE Trans. Acoust., Speech & Signal Proc. 37(3), (1989). Takens F., Detecting strange attractors in turbulence, in Lecture notes in mathematics vol. 898, Springer-Verlag, 1981. T. Sauer, J. A. Yorke and M. Casdagli; Embedology, J. Stat. Phys. 65(3), (1991) . Ralston A. and Rabinowitz P. ; A first course in numerical analysis, McGraw-Hill, 1978. K. Hornik; Approximation capabilities of multilayer feed forward networks, Neural Networks 4, (1991).
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A Hippocampal Model of Recognition Memory Randall C. O'Reilly Department of Psychology University of Colorado at Boulder Campus Box 345 Boulder, CO 80309-0345 oreilly@psych.colorado.edu Kenneth A. Norman Department of Psychology Harvard University 33 Kirkland Street Cambridge, MA 02138 nonnan@wjh.harvard.edu James L. McClelland Department of Psychology and Center for the Neural Basis of Cognition Carnegie Mellon University Pittsburgh, PA 15213 jlm@cnbc.cmu.edu Abstract A rich body of data exists showing that recollection of specific information makes an important contribution to recognition memory, which is distinct from the contribution of familiarity, and is not adequately captured by existing unitary memory models. Furthennore, neuropsychological evidence indicates that recollection is sub served by the hippocampus. We present a model, based largely on known features of hippocampal anatomy and physiology, that accounts for the following key characteristics of recollection: 1) false recollection is rare (i.e., participants rarely claim to recollect having studied nonstudied items), and 2) increasing interference leads to less recollection but apparently does not compromise the quality of recollection (i.e., the extent to which recollected infonnation veridically reflects events that occurred at study). 1 Introduction For nearly 50 years, memory researchers have known that our ability to remember specific past episodes depends critically on the hippocampus. In this paper, we describe our initial attempt to use a mechanistically explicit model of hippocampal function to explain a wide range of human memory data. Our understanding of hippocampal function from a computational and biological perspec74 R. C. 0 'Reilly, K. A. Norman and 1. L McClelland tive is based on our prior work (McClelland, McNaughton, & O'Reilly, 1995; O'Reilly & McClelland, 1994). At the broadest level, we think that the hippocampus exists in part to provide a memory system which can learn arbitrary information rapidly without suffering undue amounts of interference. This memory system sits on top of, and works in conjunction with, the neocortex, which learns slowly over many experiences, producing integrative representations of the relevant statistical features of the environment. The hippocampus accomplishes rapid, relatively interference-free learning by using relatively non-overlapping (pattern separated) representations. Pattern separation occurs as a result of 1) the sparseness of hippocampal representations (relative to cortical representations), and 2) the fact that hippocampal units are sensitive to conjunctions of cortical features given two cortical patterns with 50% feature overlap, the probability that a particular conjunction of features will be present in both patterns is much less than 50%. We propose that the hippocampus produces a relatively high-threshold, high-quality recollective response to test items. The response is "high-threshold" in the sense that studied items sometimes trigger rich recollection (defined as "retrieval of most or all of the test probe's features from memory") but lures never trigger rich recollection. The response is "high-quality" in the sense that, most of the time, the recollection signal consists of part or all of a single studied pattern, as opposed to a blend of studied patterns. The highthreshold, high-quality nature of recollection can be explained in terms of the conjunctivity of hippocampal representations: Insofar as recollection is a function of whether the features of the test probe were encountered together at study, lures (which contain many novel feature conjunctions, even if their constituent features are familiar) are unlikely to trigger rich recollection; also, insofar as the hippocampus stores feature conjunctions (as opposed to individual features), features which appeared together at study are likely to appear together at test. Importantly, in accordance with dual-process accounts of recognition memory (Yonelinas, 1994; Jacoby, Yonelinas, & Jennings, 1996), we believe that hippocampally-driven recollection is not the sole contributor to recognition memory performance. Rather. extensive evidence exists that recollection is complemented by a "fallback" familiarity signal which participants consult when rich recollection does not occur. The familiarity signal is mediated by as-yet unspecified areas (likely including the parahippocampal temporal cortex: Aggleton & Shaw, 1996; Miller & Desimone, 1994). Our account differs substantially from most other computational and mathematical models of recognition memory. Most of these models compute the "global match" between the test probe and stored memories (e.g .• Hintzman, 1988; Gillund & Shiffrin, 1984); recollection in these models involves computing a similarity-weighted average of stored memory patterns. In other memory models, recollection of an item depends critically on the extent to which the components of the item's representation were linked with that of the study context (e.g., Chappell & Humphreys, 1994). Critically, recollection in all of these models lacks the high-threshold, high-quality character of recollection in our model. This is most evident when we consider the effects of manipulations which increase interference (e.g., increasing the length of the study list. or increasing inter-item similarity). As interference increases, global matching models predict increasingly "blurry" recollection (reflecting the contribution of more items to the composite output vector), while the other models predict that false recollection of lures will increase. In contrast, our model predicts that increasing interference should lead to decreased correct recollection of studied test probes, but there should be no concomitant increase in "erroneous" types of recollection (i.e., recollection of details which mismatch studied test probes, or rich recollection of lures). This prediction is consistent with the recent finding that correct recollection of studied items decreases with increasing list length (Yonelinas, 1994). Lastly, although extant data certainly do not contradict the claim that the veridicality of recollection is robust to interference, we acknowledge that additional, focused experimentation is needed to definitively resolve this issue. A Hippocampal Model of Recognition Memory /" ,\ / I --- -'---;----; C~-:;~ _ 1 L __ oL::..J' '75 Figure I: The model. a) Shows the areas and connectivity, and the corresponding columns within the Input, EC. and CAl (see text). b) Shows an example activity pattern. Note the sparse activity in the DG and CA3, and intermediate sparseness of the CAL 2 Architecture and Overall Behavior Figure I shows a diagram of our model, which contains the basic anatomical regions of the hippocampal formation, as well as the entorhinal cortex (EC), which serves as the primary cortical input/output pathway for the hippocampus. The model as described below instantiates a series of hypotheses about the structure and function of the hippocampus and associated cortical areas, which are based on anatomical and physiological data and other models as described in O'Reilly and McClelland (1994) and McClelland et al. (1995), but not elaborated upon significantly here. The Input layer activity pattern represents the state of the EC resulting from the presentation of a given item. We assume that the hippocampus stores and retrieves memories by way of reduced representations in the EC, which have a correspondence with more elaborated representations in other areas of cortex that is developed via long-term cortical learning. We further assume that there is a rough topology to the organization of EC, with different cortical areas and/or sub-areas represented by different slots, which can be thought of as representing different feature dimensions of the input (e.g_, color, font, semantic features, etc.). Our EC has 36 slots with four units per slot; one unit per slot was active (with each unit representing a particular "feature value"). Input patterns were constructed from prototypes by randomly selecting different feature values for a random subset of slots. There are two functionally distinct layers of the EC, one which receives input from cortical areas and projects into the hippocampus (superficial or ECin ), and another which receives projections from the CAl and projects back out to the cortex (deep or ECout ). While the representations in these layers are probably different in their details, we assume that they are functionally equivalent, and use the same representations across both for convenience. ECin projects to three areas of the hippocampus: the dentate gyrus (DO), area CA3, and area CAL The storage of the input pattern occurs through weight changes in the feedforward and recurrent projections into the CA3, and the CA3 to CAl connections. The CA3 and CAl contain the two primary representations of the input pattern, while the DO plays an important but secondary role as a pattern-separation enhancer for the CA3. The CA3 provides the primary sparse, pattern-separated, conjunctive representation described above. This is achieved by random, partial connectivity between the EC and CA3, and a high threshold for activation (i.e., sparseness), such that the few units which are activated in the CA3 (5% in our model) are those which have the most inputs from active EC units. The odds of a unit having such a high proportion of inputs from even two relatively similar EC patterns is low, resulting in pattern separation (see O'Reilly & McClelland, 76 R. C. O'Reilly, K. A. Norman and 1. L. McClelland 1994 for a much more detailed and precise treatment of this issue, and the role of the DO in facilitating pattern separation). While these CA3 representations are useful for allowing rapid learning without undue interference, the pattern-separation process eliminates any systematic relationship between the CA3 pattern and the original EC pattern that gave rise to it. Thus, there must be some means of translating the CA3 pattern back into the language of the EC. The simple solution of directly associating the CA3 pattern with the corresponding EC pattern is problematic due to the interference caused by the relatively high activity levels in the EC (around 15%, and 25% in our model). For this reason, we think that the translation is formed via the CAl, which (as a result of long-term learning) is capable of expanding EC representations into sparser patterns that are more easily linked to CA3, and then mapping these sparser patterns back onto the EC. Our CAl has separate representations of small combinations of slots (labeled columns); columns can be arbitrarily combined to reproduce any valid EC representation. Thus, representations in CAl are intermediate between the fully conjunctive CA3, and the fully combinatorial EC. This is achieved in our model by training a single CAl column of 32 units with slightly less than 10% activity levels to be able to reproduce any combination of patterns over 3 ECin slots (64 different combinations) in a corresponding set of3 ECout slots. The resulting weights are replicated across columns covering the entire EC (see Figure la). The cost of this scheme is that more CAl units are required (32 vs 12 per column in the EC), which is nonetheless consistent with the relatively greater expansion of this area relative to other hippocampal areas as a function of cortical size. After learning, our model recollects studied items by simply reactivating the original CA3, CAl and ECout patterns via facilitated weights. With partial or noisy input patterns (and with interference), these weights and two forms of recurrence (the "short loop" within CA3, and the "big loop" out to the EC and back through the entire hippocampus) allow the hippocampus to bootstrap its way into recalling the complete original pattern (pattern completion). If the EC input pattern corresponds to a nonstudied pattern, then the weights will not have been facilitated for this particular activity pattern, and the CAl will not be strongly driven by the CA3. Even if the ECin activity pattern corresponds to two components that were previously studied, but not together (see below), the conjunctive nature of the CA3 representations will minimize the extent to which recall occurs. Recollection is operationalized as successful recall of the test probe. This raises the basic problem that the system needs to be able to distinguish between the EC out activation due to the item input on ECin (either directly or via the CAl), and that which is due to activation coming from recall in the CA3-CAl pathway. One solution to this problem, which is suggested by autocorrelation histograms during reversible CA3 lesions (Mizumori et aI., 1989), is that the CA3 and CAl are 1800 out of phase with respect to the theta rhythm. Thus, when the CA3 drives the CAl, it does so at a point when the CAl units would otherwise be silent, providing a means for distinguishing between EC and CA3 driven CA 1 activation. We approximate something like this mechanism by simply turning off the ECin inputs to CAl during testing. We assess the quality of hippocampal recall by comparing the resulting ECout pattern with the ECin cue. The number of active units that match between ECin and ECout (labeled C) indicates how much of the test probe was recollected. The number of units that are active in EC out but not in ECin (labeled E) indicates the extent to which the model recollected an item other than the test probe. 3 Activation and Learning Dynamics Our model is implemented using the Leabra framework, which provides a robust mechanism for producing controlled levels of sparse activation in the presence of recurrent activaA Hippocampal Model of Recognition Memory 77 tion dynamics, and a simple, effective Hebbian learning rule (O'Reilly, 1996)1. The activation function is a simple thresholded single-compartment neuron model with continuousvalued spike rate output. Membrane potential is updated by dVd't(t) = T L:c gc (t)gc (Ec Vm(t)), with 3 channels (c) corresponding to: e excitatory input; lleak current; and i inhibitory input. Activation communicated to other cells is a simple thresholded function of the membrane potential: Yj(t) = 1/ (1 + 'Y[v>n(:)-9J+)' As in the hippocampus (and cortex), all principal weights (synaptic efficacies) are excitatory, while the local-circuit inhibition controls positive feedback loops (i.e., preventing epileptiform activity) and produces sparse representations. Leabra assumes that the inhibitory feedback has an approximate set-point (i.e., strong activity creates compensatorially stronger inhibition, and vice-versa), resulting in roughly constant overall activity levels, with a firm upper bound. Inhibitory current is given by gi = g~+l + q(gr g~+l)' where 0 < q < 1 is typically .25, and 8 L:. 9c9c(Ec-8) .. . ' . 9 = ct· 8-Ei for the UnIts With the k th and k + 1 th highest excitatory mputs. A simple, appropriately normalized Hebbian rule is used in Leabra: f).wij = XiYj - YjWij, which can be seen as computing the expected value of the sending unit's activity conditional on the receiver's activity (if treated like a binary variable active with probability Yj): Wij ~ (xiIYj}p' This is essentially the same rule used in standard competitive learning or mixtures-of-Gaussians. 4 Interference and List-Length, Item Similarity Here, we demonstrate that the hippocampal recollection system degrades with increasing interference in a way that preserves its essential high-threshold, high-quality nature. Figure 2 shows the effects of list length and item similarity on our C and E measures. Only studied items appear in the high C, low E comer representing rich recollection. As length and similarity increase, interference results in decreased C for studied items (without increased E), but critically there is no change in responding to new items. Interference in our model arises from the reduced but nevertheless extant overlap between representations in the hippocampal system as a function of item similarity and number of items stored. To the extent that increasing numbers of individual CA3 units are linked to mUltiple contradictory CAl representations, their contribution is reduced, and eventually recollection fails. As for the frequently obtained finding that decreased recollection of studied items is accompanied by an increase in overall false alarms, we think this results from subjects being forced to rely more on the (less reliable) fallback familiarity mechanism. 5 Conjunctivity and Associative Recognition Now, we consider what happens when lures are constructed by recombining elements of studied patterns (e.g., study ''window-reason'' and "car-oyster", and test with "windowoyster"). One recent study found that participants are much more likely to claim to recollect studied pairs than re-paired lures (Yonelinas, 1997). Furthermore, data from this study is consistent with the idea that re-paired lures sometimes trigger recollection of the studied word pairs that were re-combined to generate the lure; when this happens (assuming that each word occurred in only one pair), the participant can confidently reject the lure. Our simulation data is consistent with these findings: For studied word pairs, the model (richly) recollected both pair components 86% of the time. As for re-paired lures, both pair components were never recalled together, but 16% of the time the model recollected one of the pair components, along with the component that it was paired with at study. The I Note that the version of Leabra described here is an update to the cited version, which is currently being prepared for publication. 78 R. C. O'Reilly, K. A. Nonnan and 1. L McClelland Figure 2: Effects of list length and similarity on recollection perfonnance. Responses can be categorized according to the thresholds shown, producing three regions: rich recollection (RR), weak recollection (WR), and misrecollection (MR). Increasing list length and similarity lead to less rich recollection of studied items (without increasing misrecollection for these items), and do not significantly affect the model's responding to lures. model responded in a similar fashion to pairs consisting of one studied word and a new word (never recollecting both pair components together, but recollecting the old item and the item it was paired with at study 13% of the time). Word pairs consisting of two new items failed to trigger recollection of even a single pair component. Similar findings were obtained in our simulation of the (Hintzman, Curran, & Oppy, 1992) experiment involving recombinations of word and plurality cues. 6 Discussion While the results presented above have dealt with the presentation of complete probe stimuli for recognition memory tests, our model is obviously capable of explaining cued recall and related phenomena such as source or context memory by virtue of its pattern completion abilities. There are a number of interesting issues that this raises. For example, we predict that successful item recollection will be highly correlated with the ability to recall additional information from the study episode, since both rely on the same underlying memory. Further, to the extent that elderly adults form less distinct encodings of stimuli (Rabinowitz & Ackerman, 1982), this explains both their impaired recollection on recognition tests (Parkin & Walter, 1992) and their impaired memory for contextual ("source") details (Schacter et aI., 1991). In summary, existing mathematical models of recognition memory are most likely incorrect in assuming that recognition is performed with one memory system. Global matching models may provide a good account of familiarity-based recognition, but they fail to account for the contributions of recollection to recognition, as discussed above. For example, global matchil).g models predict that lures which are similar to studied items will always trigger a stronger signal than dissimilar lures; as such, these models can not account for the fact that sometimes subjects can reject similar lures with high levels of confidence (due, in our model, to recollection ofa similar studied item; Brainerd, Reyna, & Kneer, 1995; Hintzman et aI., 1992). Further, global matching models confound the signal for the extent to which individual components of the test probe were present at all during study, and signal for the A Hippocampal Model of Recognition Memory 79 extent to which they occurred together. We believe that these signals may be separable, with recollection (implemented by the hippocampus) showing sensitivity to conjunctions of features, but not the occurrence of individual features, and familiarity (implemented by cortical regions) showing sensitivity to component occurrence but not co-occurence. This division of labor is consistent with recent data showing that familiarity does not discriminate well between studied item pairs and lures constructed by conjoining items from two different studied pairs (so long as the pairings are truly novel) (Yonelinas, 1997), and with the point, set forth by (McClelland et aI., 1995), that catastrophic interference would occur if rapid learning (required to learn feature co-occurrences) took place in the neocortical structures which generate the familiarity signal. 7 References Aggleton, J. P., & Shaw, C. (1996). Amnesia and recognition memory: are-analysis of psychometric data. Neuropsychologia, 34, 51. Brainerd, C. J., Reyna, V. F., & Kneer, R. (1995). False-recognition reversal: When similarity is distinctive. Journal of Memory and Language, 34, 157-185. Chappell, M., & Humphreys, M. S. (1994). An auto-associative neural network for sparse representations: Analysis and application to models of recognition and cued recall. Psychological Review, 101, 103-128. Gillund, G., & Shiffrin, R. M. (1984). A retrieval model for both recognition and recall. Psychological Review, 91, 1-67. Hintzman, D. L. (1988). Judgments of frequency and recognition memory in a multiple-trace memory model. Psychological Review, 95, 528-551. Hintzman, D. L., Curran, T., & Oppy, B. (1992). Effects of similiarity and repetition on memory: Registration without learning. Journal of Experimental Psychology: Learning. Memory. and Cognition, 18, 667-680. Jacoby, L. L., Yonelinas, A. P., & Jennings, J. M. (1996). The relation between conscious and unconscious (automatic) influences: A declaration of independence. In J. D. Cohen, & J. W. Schooler (Eds.), Scientific approaches to the question of consciousness (pp. 13-47). Hi1lsdale, NJ: Lawrence Erbaum Associates. McClelland, J. L., MCNaughton, B. L., & O'Reilly, R. C. (1995). Why there are complementary learning systems in the hippocampus and neocortex: Insights from the successes and failures of connectionst models of learning and memory. Psychological Review, 102,419-457. Miller, E. K., & Desimone, R. (1994). Parallel neuronal mechanisms for short-term memory. Science, 263, 520--522. Mizumori, S. J. Y., McNaughton, B. L., Barnes, C. A., & Fox, K. B. (1989). Preserved spatial coding in hippocampal CAl pyramidal cells during reversible suppression ofCA3c output: Evidence for pattern completion in hippocampus. Journal of NeuroSCience, 9( II), 3915-3928. O'Reilly, R. C. (1996). The leabra model of neural interactions and learning in the neocortex. PhD thesis, Carnegie Mellon University, Pittsburgh, PA, USA. O'Reilly, R. C., & McClelland, J. L. (1994). Hippocampal conjunctive encoding, storage, and recall: Avoiding a tradeoff. Hippocampus, 4(6), 661-682. Parkin, A. J., & Walter, B. M. (1992). Recollective experience, normal aging, and frontal dysfunction. Psychology and Aging, 7,290--298. Rabinowitz, J. C., & Ackerman, B. P. (1982). General encoding of episodic events by elderly adults. In F. I. M. C. S. Trehub (Ed.), Aging and cognitive processes. Plenum Publishing Corporation. Schacter, D. L., Kaszniak, A. W., Kihlstrom, J. F., & Valdiserri, M. (1991). The relation between source memory and aging. Psychology and Aging, 6, 559-568. Yonelinas, A. P. (1994). Receiver-operating characteristics in recognition memory: Evidence for a dual-process model. Journal of Experimental Psychology: Learning. Memory. and Cognition, 20, 1341-1354. Yonelinas, A. P. (1997). Recognition memory ROCs for item and associative information: The contribution of recollection and familiarity. Memory and Cognition, 25,747-763.
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Wavelet Models for Video Time-Series Sheng Ma and Chuanyi Ji Department of Electrical, Computer, and Systems Engineering Rensselaer Polytechnic Institute, Troy, NY 12180 e-mail: shengm@ecse.rpi.edu, chuanyi@ecse.rpi.edu Abstract In this work, we tackle the problem of time-series modeling of video traffic. Different from the existing methods which model the timeseries in the time domain, we model the wavelet coefficients in the wavelet domain. The strength of the wavelet model includes (1) a unified approach to model both the long-range and the short-range dependence in the video traffic simultaneously, (2) a computationally efficient method on developing the model and generating high quality video traffic, and (3) feasibility of performance analysis using the model. 1 Introduction As multi-media (compressed Variable Bit Rate (VBR) video, data and voice) traffic is expected to be the main loading component in future communication networks, accurate modeling of the multi-media traffic is crucial to many important applications such as video-conferencing and video-on-demand. From modeling standpoint, multi-media traffic can be regarded as a time-series, which can in principle be modeled by techniques in time-seres modeling. Modeling such a time-series, however, turns out to be difficult, since it has been found recently that real-time video and Ethernet traffic possesses the complicated temporal behavior which fails to be modeled by conventional methods[3] [4]. One of the significant statistical properties found recently on VBR video traffic is the co-existence of the long-range (LRD) and the short-range (SRD) dependence (see for example [4][6] and references therein). Intuitively, this property results from scene changes, and suggests a complex behavior of video traffic in the time domain[7]. This complex temporal behavior makes accurate modeling of video traffic a challenging task. The goal of this work is to develop a unified and computationally efficient method to model both the long-range and the short-range dependence in real video sources. Ideally, a good traffic model needs to be (a) accurate enough to characterize pertinent statistical properties in the traffic, (b) computationally efficient, and (c) fea916 s. Ma and C. Jj sible for the analysis needed for network design. The existing models developed to capture both the long-range and the short-range dependence include Fractional Auto-regressive Integrated Moving Average (FARIMA) models[4]' a model based on Hosking's procedure[6], Transform-Expand-Sample (TES) model[9] and scenebased models[7]. All these methods model both LRD and SRD in the time domain. The scene-based modeling[7] provides a physically interpretable model feasible for analysis but difficult to be made very accurate. TES method is reasonably fast but too complex for the analysis. The rest of the methods suffer from computational complexity too high to be used to generate a large volume of synthesized video traffic. To circumvent these problems, we will model the video traffic in the wavelet domain rather than in the time domain. Motivated by the previous work on wavelet representations of (the LRD alone) Fractional Gaussian Noise (FGN) process (see [2] and references therein), we will show in this paper simple wavelet models can simultaneously capture the short-range and the long-rage dependence through modeling two video traces. Intuitively, this is due to the fact that the (deterministic) similar structure of wavelets provides a natural match to the (statistical) self-similarity of the long-range dependence. Then wavelet coefficients at each time scale is modeled based on simple statistics. Since wavelet transforms and inverse transforms is in the order of O(N), our approach will be able to attain the lowest computational complexity to generate wavelet models. Furthermore, through our theoretical analysis on the buffer loss rate, we will also demonstrate the feasibility of using wavelet models for theoretical analysis. 1.1 Wavelet Transforms In L2(R) space, discrete wavelets ¢j(t)'s are ortho-normal basis which can be represented as ¢j(t) = 2-j / 2¢(2-i t - m), for t E [0,2 K 1] with K ~ 1 being an integer. ¢(t) is the so-called mother wavelet. 1 ~ j ~ K and 0 ~ m ~ 2K -j - 1 represent the time-scale and the time-shift, respectively. Since wavelets are the dilation and shift of a mother wavelet, they possess a deterministic similar structure at different time scales. For simplicity, the mother wavelet in this work is chosen to be the Haar wavelet, where ¢(t) is 1 for 0 ~ t < 1/2, -1 for 1/2 ~ t < 1 and 0 otherwise. Let dj's be wavelet coefficients of a discrete-time process x(t) (t E [0,2K 1]). Then dj can be obtained through the wavelet transform dj = K L:;=O-l x(t)¢j(t). x(t) can be represented through the inverse wavelet transform ( K 2K - , 1 X t) = L:j=l L:m=O - dj¢j(t) + ¢o, where ¢o is equal to the average of x(t). 2 Wavelet Modeling of Video Traffic 2.1 The Video Sources Two video sources are used to test our wavelet models: (1) "Star Wars"[4]' where each frame is encoded by JPEG-like encoder, and (2) MPEG coded videos at Group of Pictures (GOP) level[7][ll] called "MPEG GOP" in the rest of the paper. The modeling is done at either the frame level or the GOP level. Wavelet Models for Video Time-Series 917 31 & 20 31 • .. .+ • AMIA('.0.4.o) 34 • • .1 dR{') . :. + 32 .. -.' ,. . ~ AR1IIA{O,Q.4.o) " 9 .. ;30 , ~.O ~ i21 • ~ .0 • i . . • • . .. > +. .. i 21 • js • • .. a -u .. • ':QOP ... 22 .:Si90Soutlt . , .• :.: .. 0 20 . ~ " .,\ 0 • 10 12 14 .. 0 8 .0 .2 -I TWoScoIoI Figure 1: Log 2 of Variance of dJ versus the time scale j Figure 2: Log 2 of Variance of dJ versus the time scale j 0 .• 0.8 j J 0 •• 0.2 0 -0.20 2 8 - : StarW.,. .. :GOP 8 10 12 1. 18 18 20 Lag Figure 3: The sample auto correlations of ds. 2.2 The Variances and Auto-correlation of Wavelet Coefficients As the first step to understand how wavelets capture the LRD and SRD, we plot in Figure (1) the variance of the wavelet coefficients dj's at different time scales for both sources. To understand what the curves mean, we also plot in Figure (2) the variances of wavelet coefficients for three well-known processes: FARIMA(O, 0.4, 0), FARIMA(l, 0.4, 0), and AR(l). FARIMA(O, 0.4,0) is a long-range dependent ptocess with Hurst parameter H = 0.9. AR(l) is a short-range dependent process, and FARIMA(l, 0.4,0) is a mixture of the long-range and the short-range dependent process. As observed, for FARIMA(O, 0.4, 0) process (LRD alone), the variance increases with j exponentially for all j. For AR(l) (SRD alone), the variance increases at an even faster rate than that of FARIMA(O, 0.4, 0) when j is small but saturates when j is large. For FARIMA(l, 0.4,0), the variance shows the mixed properties from both AR(l) and FARIMA(O, 0.4, 0). The variance of the video sources behaves similarly to that of FARIMA(l, 0.4,0), and thus demonstrate the co-existence of the SRD and LRD in the video sources in the wavelet domain. Figure 3 gives the sample auto-correlation of ds in terms of m's. The autocorrelation function of the wavelet coefficients approaches zero very rapidly, and 918 I . . , ~ ~ m s. Ma and C. Ji . ; -... -2 0 so ... Quantll._ of Stand.rd Norm •• Figure 4: Quantile-Quantile of d';' for j = 3. Left: Star Wars. Right: GOP. thus indicates the short-range dependence in the wavelet domain. This suggests that although the autocorrelation of the video traffic is complex in the time-domain, modeling wavelet coefficients may be done using simple statistics within each time scale. Similar auto-correlations have been observed for the other j's. 2.3 Marginal Probability Density Functions Is variance sufficient for modeling wavelet coefficients? Figure (4) plots the Q - Q plots for the wavelet coefficients of the two sources at j = 31. The figure shows that the sample marginal density functions of wavelet coefficients for both the "Star Wars" and the MPEG GOP source at the given time scale have a much heavier tail than that of the normal distribution. Therefore, the variance alone is only sufficient when the marginal density function is normal, and in general a marginal density function should be considered as another pertinent statistical property. It should be noted that correlation among wavelet coefficients at different time scales is neglected in this work for simplicity. We will show both empirically and theoretically that good performance in terms of sample auto-correlation and sample buffer loss probability can be obtained by a corresponding simple algorithm. More careful treatment can be found in [8]. 2.4 An Algorithm for Generating Wavelet Models The algorithm we derive include three main steps: (a) obtain sample variances of wavelet coefficients at each time scale, (b) generate wavelet coefficients independently from the normal marginal density function using the sample mean and variance 2, and (c) perform a transformation on the wavelet coefficients so that the ISimilar behaviors have been observed at the other time scales. A Q - Q plot is a standard statistical tool to measure the deviation of a marginal density function from a normal density. The Q - Q plots of a process with a normal marginal is a straight line. The deviation from the line indicates the deviation from the normal density. See [4] and references therein for more details. 2The mean of the wavelet coefficients can be shown to be zero for stationary processes. Wavelet Models for Video Time-Series 919 resulting wavelet coefficients have a marginal density function required by the traffic. The obtained wavelet coefficients form a wavelet model from which synthesized video traffic can be generated. The algorithm can be summarized as follows. Let x(t) be the video trace oflength N. Algorithm 1. Obtain wavelet coefficients from x(t) through the wavelet transform. 2. Compute the sample variance Uj of wavelet coefficients at each time scale j. 3. Generate new wavelet coefficients dj's for all j and m independently through Gaussian distributions with variances Uj 's obtained at the previous step. 4. Perform a transformation on the wavelet coefficients so that the marginal density function of wavelet coefficients is consistent with that determined by the video traffic ( see [6] for details on the transformation). 5. Do inverse wavelet transform using the wavelet coefficients obtained at the previous step to get the synthesized video traffic in the time domain. The computational complexity of both the wavelet transform (Step 1) and the inverse transform (Step 5) is O(N). So is for Steps 2, 3 and 4. Then O(N) is the computational cost of the algorithm, which is the lowest attainable for traffic models. 2.5 Experimental Results Video traces of length 171, 000 for "Star Wars" and 66369 for "MPEG GOP" are used to obtain wavelet models. FARIMA models with 45 parameters are also obtained using the same data for comparison. The synthesized video traffic from both models are generated and used to obtain sample auto-correlation functions in the time-domain, and to estimate the buffer loss rate. The results3 are given in Figure (6). Wavelet models have shown to outperform the FARIMA model. For the computation time, it takes more than 5-hour CPU time4 on a SunSPARC 5 workstation to develop the FARIMA model and to generate synthesized video traffic of length 171, 0005 . It only takes 3 minutes on the same machine for our algorithm to complete the same tasks. 3 Theory It has been demonstrated empirically in the previous section that the wavelet model, which ignores the correlation among wavelet coefficients of a video trace, can match well the sample auto-correlation function and the buffer loss probability. To further evaluate the feasibility of the wavelet model, the buffer overflow probability has been analyzed theoretically in [8]. Our result can be summarized in the following theorem. 3Due to page limit, we only provide plots for JPEG. GOP has similar results and was reported in [8]. 4Computation time includes both parameter estimation and synthesized traffic generation. 5The computational complexity to generate synthesized video traffic of length N is O(N2) for an FARIMA model[5][4]. 920 .. OJ 01 0.2 0.1 Figure 5: "-": Autocorrelation of "Star Wars"; "- -": ARIMA(25,d,20); " ". Our Algorithm -2 -2.5 -4.5 -5 ~.5 I I I S. Ma and C. Ii 4~1 O.li 0.4 0.45 o.s 0.&6 0.8 085 07 0.71 OJ Figure 6: Loss rate attained via simulation. Vertical axis: loglO (Loss Rate); horizontal axis: work load. "-": the single video source; "". FARIMA(25,d,20); "-" Our algorithm. The normalized buffer size: 0.1, 1, 10,30 and 100 from the top down. Theorem Let BN and EN be the buffer sizes at the Nth time slot due to the synthesized traffic by the our wavelet model, and by the FGN process, respectively. Let C and B represent the capacity, and the maximum allowable buffer size respectively. Then InPr(BN > B) InPr(EN > B) (C JL)2(i!:;?(1-H)e~7f-)2H 20-2(1- H)2 (1) where ~ < H < 1 is the Hurst parameter. JL and 0-2 is the mean and the variance of the traffic, respectively. B is assume to be (C - It )2ko, where ko is a positive integer. This result demonstrates that using our simple wavelet model which neglects the correlations among wavelet coefficients, buffer overflow probability obtained is similar to that of the original FGN process as given in[10]. In other words, it shows that the wavelet model for a FG N process can have good modeling performance in terms of the buffer overflow criterion. We would like to point out that the above theorem is held for a FGN process. Further work are needed to account for more general processes. 4 Concl usions In this work, we have described an important application on time-series modeling: modeling video traffic. We have developed a wavelet model for the timeseries. Through analyzing statistical properties of the time-series and comparing the wavelet model with FARIMA models, we show that one of the key factors to successfully model a time-series is to choose an appropriate model which naturally fits the pertinant statistical properties of the time-series. We have shown wavelets are particularly feasible for modeling the self-similar time-series due to the video traffic. Wavelet Models for Video Time-Series 921 We have developed a simple algorithm for the wavelet models, and shown that the models are accurate, computationally efficient and simple enough for analysis. References [1] I. Daubechies, Ten Lectures on Wavelets. Philadelphia: SIAM, 1992. [2] Patrick Flandrin, "Wavelet Analysis and Synthesis of Fractional Brownian Motion", IEEE transactions on Information Theory, vol. 38, No.2, pp.910-917, 1992. [3] W.E Leland, M.S. Taqqu, W. Willinger and D.V. Wilson, "On the Self-Similar Nature of Ethernet Traffic (Extended Version)", IEEE/ACM Transactions on Networking, vo1.2, 1-14, 1994. [4] Mark W. Garrett and Walter Willinger. "Analysis, Modeling and Generation of Self-Similar VBR Video Traffic.", in Proceedings of ACM SIGCOMM'94, London, U.K, Aug., 1994 [5] J .R.M. Hosking, "Modeling Persistence in Hydrological Time Series Using Fractional Differencing", Water Resources Research, 20, pp. 1898-1908, 1984. [6] C. Huang, M. Devetsikiotis, I. Lambadaris and A.R. Kaye, "Modeling and Simulation of Self-Similar Variable Bit Rate Compressed Video: A Unified Approach", in Proceedings of ACM SIGCOMM'95, pp. 114-125. [7] Predrag R. Jelenlnovic, Aurel A. Lazar, and Nemo Semret. The effect of multiple time scales and subexponentiality in mpeg video streams on queuing behavior. IEEE Journal on Selected Area of Communications, 15, to appear in May 1997. [8] S. Ma and C. Ji, "Modeling Video Traffic in Wavelet Domain" , to appear IEEE INFO COM, 1998. [9] B. Melamed, D. Raychaudhuri, B. Sengupta, and J. Zdepski. Tes-based video source modeling for performance evaluation of integrated networks. IEEE Transactions on Communications, 10, 1994. [10] Ilkka Norros, "A storage model with self-similar input," Queuing Systems, vol.16, 387-396, 1994. [11] O. Rose. "Statistical properties of mpeg video traffic and their impact on traffic modeling in atm traffic engineering", Technical Report 101, University of Wurzburg, 1995. , )
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Multi-time Models for Temporally Abstract Planning Doina Precup, Richard S. Sutton University of Massachusetts Amherst, MA 01003 {dprecuplrich}@cs.umass.edu Abstract Planning and learning at multiple levels of temporal abstraction is a key problem for artificial intelligence. In this paper we summarize an approach to this problem based on the mathematical framework of Markov decision processes and reinforcement learning. Current model-based reinforcement learning is based on one-step models that cannot represent common-sense higher-level actions, such as going to lunch, grasping an object, or flying to Denver. This paper generalizes prior work on temporally abstract models [Sutton, 1995] and extends it from the prediction setting to include actions, control, and planning. We introduce a more general form of temporally abstract model, the multi-time model, and establish its suitability for planning and learning by virtue of its relationship to the Bellman equations. This paper summarizes the theoretical framework of multi-time models and illustrates their potential advantages in a grid world planning task. The need for hierarchical and abstract planning is a fundamental problem in AI (see, e.g., Sacerdoti, 1977; Laird et aI., 1986; Korf, 1985; Kaelbling, 1993; Dayan & Hinton, 1993). Model-based reinforcement learning offers a possible solution to the problem of integrating planning with real-time learning and decision-making (Peng & Williams, 1993, Moore & Atkeson, 1993; Sutton and Barto, 1998). However, current model-based reinforcement learning is based on one-step models that cannot represent common-sense, higher-level actions. Modeling such actions requires the ability to handle different, interrelated levels of temporal abstraction. A new approach to modeling at multiple time scales was introduced by Sutton (1995) based on prior work by Singh, Dayan, and Sutton and Pinette. This approach enables models of the environment at different temporal scales to be intermixed, producing temporally abstract models. However, that work was concerned only with predicting the environment. This paper summarizes an extension of the approach including actions and control of the environment [Precup & Sutton, 1997]. In particular, we generalize the usual notion of a Multi-time Models for Temporally Abstract Planning 1051 primitive, one-step action to an abstract action, an arbitrary, closed-loop policy. Whereas prior work modeled the behavior of the agent-environment system under a single, given policy, here we learn different models for a set of different policies. For each possible way of behaving, the agent learns a separate model of what will happen. Then, in planning, it can choose between these overall policies as well as between primitive actions. To illustrate the kind of advance we are trying to make, consider the example shown in Figure 1. This is a standard grid world in which the primitive actions are to move from one grid cell to a neighboring cell. Imagine the learning agent is repeatedly given new tasks in the form of new goal locations to travel to as rapidly as possible. If the agent plans at the level of primitive actions, then its plans will be many actions long and take a relatively long time to compute. Planning could be much faster if abstract actions could be used to plan for moving from room to room rather than from cell to cell. For each room, the agent learns two models for two abstract actions, one for traveling efficiently to each adjacent room. We do not address in this paper the question of how such abstract actions could be discovered without help; instead we focus on the mathematical theory of abstract actions. In particular, we define a very general semantics for them-a property that seems to be required in order for them to be used in the general kind of planning typically used with Markov decision processes. At the end of this paper we illustrate the theory in this example problem, showing how room-to-room abstract actions can substantially speed planning. 4 unreliable primitive actions up 18«+ fight FaU33% 01 th&tlm8 down 8 abstract actions (to each room's 2 hallways) Figure 1: Example Task. The Natural abstract actions are to move from room to room. 1 Reinforcement Learning (MDP) Framework In reinforcement learning, a learning agent interacts with an environment at some discrete, lowest-level time scale t = 0,1,2, ... On each time step, the agent perceives the state of the environment, St , and on that basis chooses a primitive action, at. In response to each primitive action, at, the environment produces one step later a numerical reward, Tt+l, and a next state, St+l. The agent's objective is to learn a policy, a mapping from states to probabilities of taking each action, that maximizes the expected discounted future reward from each state s: 00 v"{s) = E7r{L: ';lTt+l I So = s}, t=O where'Y E [0, 1) is a discount-rate parameter, and E7r {} denotes an expectation implicitly conditional on the policy 7f being followed. The quantity v7r( s) is called the value of state S under policy 7f, and v7r is called the value function for policy 7f. The value under the optimal policy is denoted: v*(S) = maxv7r(s}. 7r Planning in reinforcement learning refers to the use of models of the effects of actions to compute value functions, particularly v*. ]052 D. Precup and R. S. Sutton We assume that the states are discrete and fonn a finite set, St E {1,2, ... ,m}. This is viewed as a temporary theoretical convenience; it is not a limitation of the ideas we present. This assumption allows us to alternatively denote the value functions, v7r and v*, as column vectors, v7r and v*, each having m components that contain the values of the m states. In general, for any m-vector, x, we will use the notation x( s) to refer to its sth component. The model of an action, a, whether primitive or abstract, has two components. One is an m x m matrix, Pa , predicting the state that will result from executing the action in each state. The other is a vector, ga, predicting the cumulative reward that will be received along the way. In the case of a primitive action, Pa is the matrix of I-step transition probabilities of the environment, times ,: P;(s) = ,E {St+! 1st = s, at = a}, Vs where P;(s) denotes the sth column of P; (these are the predictions corresponding to state s) and St denotes the unit basis m-vector corresponding to St. The reward prediction, ga, for a primitive action contains the expected immediate rewards: ga(s) = E {rt+l 1st = s, at = a}, Vs For any stochastic policy, 1f, we can similarly define its I-step model, g7r, P7r as: and Vs (1) 2 Suitability for Planning In conventional planning, one-step models are used to compute value functions via the Bellman equations for prediction and control. In vector notation, the prediction and control Bellman equations are and v* = max{ga + PaV*}, a (2) respectively, where the max function is applied component-wise in the control equation. In planning, these equalities are turned into updates, e.g., vk+! ~ g7r + P7r vk' which converge to the value functions. Thus, the Bellman equations are usually used to define and compute value functions given models of actions. Following Sutton (1995), here we reverse the roles: we take the value functions as given and use the Bellman equations to define and compute models of new, abstract actions. In particular, a model can be used in planning only if it is stable and consistent with the Bellman equations. It is useful to define special tenns for consistency with each Bellman equation. Let g, P denote an arbitrary model (an m-vector and an m x m matrix). Then this model is said to be vaLid for policy 1f [Sutton, 1995] if and only if limk-+oo pk = 0 and v7r = g + P v 7r. (3) Any valid model can be used to compute v7r via the iteration algorithm v k+1 t- g + Pvk. This is a direct sense in which the validity of a model implies that it is suitable for planning. We introduce here a parallel definition that expresses consistency with the control Bellman equation. The model g, P is said to be non-overpromising (NaP) if and only if P has only positive elements, limk-+oo pk = 0, and V* ~ g + Pv*, (4) where the ~ relation holds component-wise. If a Nap model is added inside the max operator in the control Bellman equation (2), this condition ensures that the true value, v*, will not be exceeded for any state. Thus, any model that does not promise more than it Multi-time Models for Temporally Abstract Planning 1053 is achievable (is not (;>verpromising) can serve as an option for planning purposes. The one-step models of primitive actions are obviously NOP, due to (2). It is similarly straightforward to show that the one-step model of any policy is also NOP. For some purposes, it is more convenient to write a model g, P as a single (m+ 1) x (m+ 1) matrix: o P We say that the model M has been put in homogeneous coordinates. The vectors corresponding to the value functions can also be put into homogeneous coordinates, by adding an initial element that is always 1. Using this notation, new models can be combined using two basic operations: composition and averaging. Two models Ml and M2 can be composed by matrix multiplication, yielding a new model M = M1M2 . A set of models Mi can be averaged, weighted by a set of diagonal matrices Di , such that I::i Di = I, to yield a new model M = I::i DiMi. Sutton (1995) showed that the set of models that are valid for a policy 7r is closed under composition and averaging. This enables models acting at different time scales to be mixed together, and the resulting model can still be used to compute v 1T• We have proven that the set of NOP models is also closed under composition and averaging [Precup & Sutton, 1997]. These operations permit a richer variety of combinations for NOP models than they do for valid models because the NOP models that are combined need not correspond to a particular policy. 3 Multi-time models The validity and NOP-ness of a model do not imply each other [Precup & Sutton, 1997]. Nevertheless, we believe a good model should be both valid and NOP. We would like to describe a class of models that, in some sense, includes all the "interesting" models that are valid and non-overpromising, and which is expressive enough to include common-sense notions of abstract action. These goals have led us to the notion of a multi-time model. The simplest example of multi-step model, called the n-step model for policy 7r, predicts the n-step truncated return and the state n steps into the future (times Tn). If different nstep models of the same policy are averaged, the result is called a mixture model. Mixtures are valid and non-overpromising due to the closure properties established in the previous section. One kind of mixture suggested in [Sutton, 1995] allows an exponential decay of the weights over time, controlled by a parameter {3. Figure 2: Two hypothetical Markov environments Are mixture models expressive enough for capturing the properties of the environment? In order to get some intuition about the expressive power that a model should have, let us consider the example in figure 2. If we are only interested if state G is attained, then the two environments presented shOUld be characterized by significantly different models. However, n-step models, or 2ny linear mixture of n-step models cannot achieve this goal. In order to remediate this problem, models should average differently over all the different trajectories that are possible through the state space. A full {3-model [Sutton, 1995] can 1054 D. Precup and R. S. Sutton distinguish between these two situations. A ,B-model is a more general form of mixture model, in which a different ,B parameter is associated with each state. For a state i, ,Bi can be viewed as the probability that the trajectory through the state space ends in state i. Although ,B-models seem to have more expressive power, they cannot describe n-step models. We would like to have a more general form of model, that unifies both classes. This goal is achieved by accurate multi-time models. Multi-time models are defined with respect to a policy. Just as the one-step model for a policy is defined by (1), we define g, P to be an accurate multi-time model if and only if 00 pT (s) = Ell'{ 2: Wt 'l St I So = s}, t=l 00 g(s) = Ell'{2: wdrl + ,r2 + ... + ,t-Irt) I So = s} t=l for some Jr, for all s, and for some sequence of random weights, WI, W2, •.. such that Wt > 0 and 2::1 Wt = 1. The weights are random variables chosen according to a distribution that depends only on states visited at or before time t. The weight Wt is a measure of the importance given to the t-th state of the trajectory. In particular, if Wt = 0, then state t has no weight associated with it. If Wt = 1- 2:~:~ Wi, all the remaining weight along the trajectory is given to state t. The effect is that state St is the "outcome" state for the trajectory. The random weights along each trajectory make this a very general form of model. The only necessary constraint is that the weights depend only on previously visited states. In particular, we can choose weighting sequences that generate the types of multi-step models described in [Sutton, 1995]. If the weighting variables are such that wn=l, and Wt = O;v't i= n , we obtain n-step models. A weighting sequence of the form Wt = rr~:6,Bi 'tit, where ,Bi is the parameter associated to the state visited on time step i, describes a full ,B-model. The main result for multi-time models is that they satisfy the two criteria defined in the previous section. Any accurate multi-time model is also NOP and valid for Jr. The proofs of these results are too long to include here. 4 Illustrative Example In order to illustrate the way in which multi-time models can be used in practice, let us return to the grid world example (Figure I). The cells of the grid correspond to the states of the environment. From any state the agent can perform one of four primitive actions, up, down, left or right. With probability 2/3, the actions cause the agent to move one cell in the corresponding direction (unless this would take the agent into a wall, in which case it stays in the same state). With probability 1/3, the agent instead moves in one of the other three directions (unless this takes it into a wall of course). There is no penalty for bumping into walls. In each room, we also defined two abstract actions, for going to each of the adjacent hallways. Each abstract action has a set of input states (the states in the room) and two outcome states: the target hallway, which corresponds to a successful outcome, and the state adjacent to the other hallway, which corresponds to failure (the agent has wandered out of the room). Each abstract action is given by its complete model g:;-', P:;, where Jr is the optimal policy for getting into the target hallway, and the weighting variables W along any trajectory have the value I for the outcome states and 0 everywhere else. Multi-time Models for Temporally Abstract Planning J055 I I I .. .... . ... • •• • • •• •• • • • • • • • • Iteration #1 Iteration #2 Iteration #3 Iteration #4 Iteration #5 Iteration #6 Figure 3: Value iteration using primitive and abstract actions The goal state can have an arbitrary position in any of the rooms, but for this illustration let us suppose that the goal is two steps down from the right hallway. The value of the goal state is 1, there are no rewards along the way, and the discounting factor is , = 0.9. We perfonned planning according to the standard value iteration method: where vo(s) = 0 for all the states except the goal state (which starts at 1). In one experiment, a ranged only over the primitive actions, in the other it ranged over the set including both the primitive and the abstract actions. When using only primitive actions, the values are propagated one step away on each iteration. After six iterations, for instance, only the states that are at most six steps away from the goal will be attributed non-zero values. The models of abstract actions produce a significant speed-up in the propagation of values at each step. Figure 3 shows the value function after each iteration, using both primitive and abstract actions for planning. The area of the circle drawn in each state is proportional to the value attributed to the state. The first three iterations are identical with the case when only primitive actions are used. However, once the values are propagated to the first hallway, all the states in the rooms adjacent to that hallway will receive values as well. For the states in the room containing the goal, these values correspond to perfonning the abstract action of getting into the right hallway, and then following the optimal primitive actions to get to the goal. At this point, a path to the goal is known from each state in the right half of the environment, even if the path is not optimal for all states. After six iterations, an optimal policy is known for all the states in the environment. The models of the abstract actions do not need to be given a priori, they can be learned from experience. In fact, the abstract models that were used in this experiment have been learned during a I,OOO,DOO-step random walk in the environment. The starting point for 1056 D. Precup and R. S. Sutton learning was represented by the outcome states of each abstract action, along with the hypothetical utilities U associated with these states. We used Q-Iearning [Watkins, 1989] to learn the optimal state-action value function Q'U B associated with each abstract action. The greedy policy with respect to Q'U,B is the pol'icy associated with the abstract action. At the same time, we used the ,B-model learning algorithm presented in [Sutton, 1995] to compute the model corresponding to the policy. The learning algorithm is completely online and incremental, and its complexity is comparable to that of regular I-step TDlearning. Models of abstract actions can be built while an agent is acting in the environment without any additional effort. Such models can then be used in the planning process as if they would represent primitive actions, ensuring more efficient learning and planning, especially if the goal is changing over time. Acknowledgments The authors thank Amy McGovern and Andy Fagg for helpful discussions and comments contributing to this paper. This research was supported in part by NSF grant ECS-951 1805 to Andrew G. Barto and Richard S. Sutton, and by AFOSR grant AFOSR-F49620-96-1-0254 to Andrew G. Barto and Richard S. Sutton. Doina Precup also acknowledges the support of the Fulbright foundation. References Dayan, P. (1993). Improving generalization for temporal difference learning: The successor representation. Neural Computation, 5, 613-624. Dayan, P. & Hinton, G. E. (1993). Feudal reinforcement learning. In Advances in Neural Information Processing Systems, volume 5, (pp. 271-278)., San Mateo, CA. Morgan Kaufmann. Kaelbling, L. P. (1993). Hierarchical learning in stochastic domains: Preliminary results. In Proceedings of the Tenth International Conference on Machine Learning ICML'93, (pp. 167-173)., San Mateo, CA. Morgan Kaufmann. Korf, R. E. (1985). Learning to Solve Problems by Searching for Macro-Operators. London: Pitman Publishing Ltd. Laird, J. E., Rosenbloom, P. S., & Newell, A. (1986). Chunking in SOAR: The anatomy of a general learning mechanism. Machine Learning, I, 11-46. Moore, A. W. & Atkeson, C. G. (1993). Prioritized sweeping: Reinforcement learning with less data and less real time. Machine Learning, 13, 103-130. Peng, J. & Williams, J. (1993). Efficient learning and planning within the Dyna framework. Adaptive Behavior, 4, 323-334. Precup, D. & Sutton, R. S. (1997). Multi-Time models for reinforcement learning. In ICML'97 Workshop: The Role of Models in Reinforcement Learning. Sacerdoti, E. D. (1977). A Structure for Plans and Behavior. North-Holland, NY: Elsevier. Singh, S. P. (1992). Scaling reinforcement learning by learning variable temporal resolution models. In Proceedings of the Ninth International Conference on Machine Learning ICML'92, (pp. 202207)., San Mateo, CA. Morgan Kaufmann. Sutton, R. S. (1995). TD models: Modeling the world as a mixture of time scales. In Proceedings of the Twelfth International Conference on Machine Learning ICML'95, (pp. 531-539)., San Mateo, CA. Morgan Kaufmann. Sutton, R. S. & Barto, A. G. (1998). Reinforcement Learning. An Introduction. Cambridge, MA: MIT Press. Sutton, R. S. & Pinette, B. (1985). The learning of world models by connectionist networks. In Proceedings of the Seventh Annual Conference of the Cognitive Science Society, (pp. 54-64). Watkins, C. 1. C. H. (1989). Learning with Delayed Rewards. PhD thesis, Cambridge University.
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Linear concepts and hidden variables: An empirical study Adam J. Grove NEC Research Institute 4 Independence Way Princeton NJ 08540 grove@research.nj.nec.com Dan Rothe Department of Computer Science University of Illinois at Urbana-Champaign 1304 W. Springfield Ave. Urbana 61801 danr@cs.uiuc.edu Abstract Some learning techniques for classification tasks work indirectly, by first trying to fit a full probabilistic model to the observed data. Whether this is a good idea or not depends on the robustness with respect to deviations from the postulated model. We study this question experimentally in a restricted, yet non-trivial and interesting case: we consider a conditionally independent attribute (CIA) model which postulates a single binary-valued hidden variable z on which all other attributes (i.e., the target and the observables) depend. In this model, finding the most likely value of anyone variable (given known values for the others) reduces to testing a linear function of the observed values. We learn CIA with two techniques: the standard EM algorithm, and a new algorithm we develop based on covariances. We compare these, in a controlled fashion, against an algorithm (a version of Winnow) that attempts to find a good linear classifier directly. Our conclusions help delimit the fragility of using the CIA model for classification: once the data departs from this model, performance quickly degrades and drops below that of the directly-learned linear classifier. 1 Introduction We consider the classic task of predicting a binary (0/1) target variable zo, based on the values of some n other binary variables ZI ••• Zft,. We can distinguish between two styles of learning approach for such tasks. Parametric algorithms postulate some form of probabilistic model underlying the data, and try to fit the model's parameters. To classify an example we can compute the conditional probability distribution for Zo given the values of the known variables, and then predict the most probable value. Non-parametric algorithms do not assume that the training data has a particular form. They instead search directly in the space of possible classification functions, attempting to find one with small error on the training set of examples. An important advantage of parametric approaches is that the induced model can be used to support a wide range of inferences, aside from the specified classification task. On the other hand, to postulate a particular form of probabilistic model can be a very strong assumption. "Partly supported by ONR grant NOOOI4-96-1-0550 while visiting Harvard University. Linear Concepts and Hidden Variables: An Empirical Study 501 So it is important to understand how robust such methods are when the real world deviates from the assumed model. In this paper, we report on some experiments that test this issue. We consider the specific case of n + 1 conditionally independent attributes Zi together with a single unobserved variable z, also assumed to be binary valued, on which the Zi depend (henceforth, the binary CIA model); see Section 2. In fact, such models are plausible in many domains (for instance, in some language interpretation tasks; see [GR96]). We fit the parameters of the CIA model using the well-known expectation-maximization (EM) technique [DLR77], and also with a new algorithm we have developed based on estimating covariances; see Section 4. In the nonparametric case, we simply search for a good linear separator. This is because the optimal predictors for the binary CIA model (i.e., for predicting one variable given known values for the rest) are also linear. This means that our comparison is "fair" in the sense that neither strategy can choose from classifiers with more expressive power than the other. As a representative of the non-parametric class of algorithms, we use the Winnow algorithm of [Lit881, with some modifications (see Section 6). Winnow works directly to find a "good" linear separator. It is guaranteed to find a perfect separator if one exists, and empirically seems to be fairly successful even when there is no perfect separator [GR96, Blu9?]. It is also very fast. Our experimental methodology is to first generate synthetic data from a true CIA model and test performance; we then study various deviations from the model. There are various interesting issues involved in constructing good experiments, including the desirability of controlling the inherent "difficulty" of learning a model. Since we cannot characterize the entire space, we consider here only deviations in which the data is drawn from a CIA model in which the hidden variable can take more than two values. (Note that the optimal classifier given Zo is generally not linear in this case.) Our observations are not qualitatively surprising. CIA does well when the assumed model is correct, but performance degrades when the world departs from the model. But as we discuss, we found it surprising how fragile this model can sometimes be, when compared against algorithms such as Winnow. This is even though the data is not linearly separable either, and so one might expect the direct learning techniques to degrade in performance as well. But it seems that Wmnow and related approaches are far less fragile. Thus the main contribution of this work is that our results shed light on the specific tradeoff between fitting parameters to a probabilistic model, versus direct search for a good classifier. Specifically, they illustrate the dangers of predicting using a model that is even "slightly" simpler than the distribution actually generating the data, vs. the relative robustness of directly searching for a good predictor. This would seem to be an important practical issue, and highlights the need for some better theoretical understanding of the notion of "robustness". 2 Conditionally Independent Attributes Throughout we assume that each example is a binary vector z E {O, 1 }n+l, and that each example is generated independently at random according to some unknown distribution on {O, 1 }n+l. We use Xi to denote the i'th attribute, considered as a random variable, and Zi to denote a value for Xi. In the conditionally independent attribute (CIA) model, examples are generated as follows. We postulate a "hidden" variable Z with Ie values, which takes values z for 0 $ z < Ie with probability a. ~ O. Since we must have E::~ a. = 1 there are Ie - 1 independent parameters. Having randomly chosen a value z for the hidden variable, we choose the value Zi for each observable Xi: the value is 1 with probability p~.}, and 0 otherwise. Here p~.} E [0,1). The attributes' values are chosen independently of each other, although z remains fixed. Note that there are thus (n + 1)1e probability parameters p~.). In the following, let l' denote the set of all (n + 1)1e + Ie - 1 parameters in the model. From this point, and until Section 7, we always assume that Ie = 2 and in this case, to simplify notation, we write al as a, ao (= 1 - a) as ai, p! as Pi and p~ as qi. 502 A. 1. Grove and D. Roth 3 The Expectation-Maximization algorithm (EM) One traditional unsupervised approach to learning the parameters of this model is to find the maximum-likelihood parameters of the distribution given the data. That is, we attempt to find the set of parameters that maximizes the probability of the data observed. Finding the maximum likelihood parameterization analytically appears to be a difficult problem, even in this rather simple setting. However, a practical approach is to use the wellknown Expectation-Maximization algorithm (EM) [DLR77], which is an iterative approach that always converges to a local maximum of the likelihood function. In our setting, the procedure is as follows. We simply begin with a randomly chosen parameterization p, and then we iterate until (apparent) convergence: 1 Expectation: For all zi, compute Ui = p-p(zi 1\ Z = 1) and Vi = p-p(zi 1\ Z = 0). Maximization: Reestimate P as follows (writing U = Ei Ui and V = Ei Vi): a f- E:=I Ui/(U + V) P; fE{i::i~=I} u;./U qj fE{i::i~=I} Vi/V. After convergence has been detected all we kno'w is that we are near a [ocdi minima of the likelihood function. Thus it is prudent to repeat the process with many different restarts. (All our experiments were extremely conservative concerning the stopping criteria at each iteration, and in the number of iterations we tried.) But in practice, we are never sure that the true optimum has been located. 4 Covariances-Based approach Partly in response to concern just expressed, we also developed another heuristic technique for learning P. The algorithm, which we call COY, is based on measuring the covariance between pairs of attributes. Since we do not see Z, attributes will appear to be correlated. In fact, if the CIA model is correct, it is easy to show that covariance between Xi and X j (defined as Yi,; = ~,; ~I-'; where~, 1-';, ~,; are the expectations of Xi, Xj, (Xi and Xj), respectively), will be Yi,j = aa'did; where di denotes Pi - qi. We also know that the expected value of Xi is ~ = aPi + a'qi. Furthermore, we will be able to get very accurate estimates of ~ just by observing the proportion of samples in which Zi is 1. Thus, if we could estimate both a and di it would be trivial to solve for estimates of Pi and qi. To estimate di, suppose we have computed all the pairwise covariances using the data; we use fli,; to denote our estimate of Yi,j' For any distinct j, Ie i= i we clearly have aa l5; = IV'rd.r"/o1 so we could estimate d; using this equation. A better estimate would be "/o to consider all pairs j, Ie and average the individual estimates. However, not all individual estimates are equally good. It can be shown that the smaller Y;,II is, the less reliable we should expect the estimate to be (and in the limit, where X; and XII are perfectly uncorrelated, we get no valid estimate at all). This suggests that we use a weighted average, with the weights proportional to Yj,II. Using these weights leads us to the next equation for determining 5i , which, after simplification, is: E j,II:j;t1l;ti IYi,jYi,II I E;,II:;;tll;ti IY;,II I (E;:#i IYi,; 1)2 - E;:;;ti if,; E;,II:;;tll IY;,II I - 2 Ej:j;ti IYj,i I By substituting the estimates 'Oi,; we get an estimate for aa' dl. This estimate can be computed in linear time except for the determination of Ej,II:j;tll IYj,II I which, although quadratic, does not depend on i and so can be computed once and for all. Thus it takes O(n2) time in total to estimate aa'd; for all i. It remains only to estimate a and the signs of the di'S. Briefly, to determine the signs we first stipulate that do is positive. (Because we never see z, one sign can be chosen at random.) IThe maximization phase works as though we were estimating parameters by taking averages based on weighted labeled data (Le., in which we see z). If ii is a sample point, these fictional data points are (ii,Z = 1) with weight Ui/U and (ii, z = 0) with weight Vi/V. Linear Concepts and Hidden Variables: An Empirical Study 503 In principle, then, the sign of 0; will then be equal to the sign of Yo,;, which we have an estimate for. In practice, this can statistically unreliable for small sample sizes and so we use a more involved ''voting'' procedure (details omitted here). Finally we estimate Q. We have found no better method of doing this than to simply search for the optimal value, using likelihood as the search criterion. However, this is only a I-dimensional search and it turns out to be quite efficient in practice. 5 Linear Separators and CIA Given a fully parameterized CIA model, we may be interested in predicting the value of one variable, say Xo, given known values for the remaining variables. One can show that in fact the optimal prediction region is given by a linear separator in the other variables, although we omit details of this derivationhere.2 This suggest an obvious learning strategy: simply try to find the line which minimizes this loss on the training set. Unfortunately, in general the task of finding a linear separator that minimizes disagreements on a collection of examples is known to be NP-hard [HS92]. So instead we use an algorithm called Winnow that is known to produce good results when a linear separator exists, as well as under certain more relaxed assumptions [Lit9I], and appears to be quite effective in practice. 6 Learning using a Winnow-based algorithm The basic version of the Winnow algorithm [Lit88] keeps an n-dimensional vector w = (1011" .1On ) of positive weights (Le., w, is the weight associated with the ith feature), which it updates whenever a mistake is made. Initially, the weight vector is typically set to assign equal positive weight to all features. The algorithm has 3 parameters, a promotion parameter Q > I, a demotion parameter 0 < f3 < 1 and a threshold 8. For a given instance (:1:1, • "1 :l:n) the algorithm predicts that :1:0 = 1 iff E~l W,:I:, > 8. If the algorithm predicts 0 and the label (Le., :1:0) is 1 (positive example) then the weights which correspond to active attributes (:1:, = 1) are promoted-the weight 10, is replaced by a larger weight Q • Wi. Conversely, if algorithm predicts 1 and the received label is 0, then the weights which correspond to active features are demoted by factor {3. We allow for negative weights as follows. Given an example (:1:1" "1 :l:n), we rewrite it as an example over 2n variables (Y1, 'Y21 •.. I 'Y2n) where y, = :1:, and Yn+, = 1 - :1:,. We then apply Winnow just as above to learn 2n (positive) weights. If wi is the weight associated with :1:, and wi is the weight associated with :l:n+i (Le., 1 - :1:,), then the prediction rule is simply to compare E~=l(wi:l:, + wi(1 - :1:,)) with the threshold. In the experiments described here we have made two significant modifications to the basic algorithm. To reduce variance, our final classifier is a weighted average of several classifiers; each is trained using a subs ample from the training set, and its weight is based based on how well it was doing on that sample. Second, we biased the algorithm so as to look for "thick" classifiers. To understand this, consider the case in which the data is perfectly linearly separable. Then there will generally be many linear concepts that separate the training data we actually see. Among these, it seems plausible that we have a better chance of doing well on the unseen test data if we choose a linear concept that separates the positive and negative training examples as "widely" as possible. The idea of having a wide separation is less clear when there is no perfect separator, but we can still appeal to the basic intuition. To bias the search towards "thick" separators, we change Wmnow's training rule somewhat. We now have a new margin parameter T. As before, we always update when our current hypothesis makes a mistake, but now we also update if I E~=l Wi:l:, - 8 I is less than T, even if the prediction is correct. In our experiments, we found that performance when using this version of Winnow is better than that of the basic algorithm, so in this paper we present results for the former. 2 A derivation for the slightly different case, for predicting z, can be found in [MP69J. 504 A. J Grove and D. Roth 7 Experimental Methodology Aside from the choice of algorithm used, the number of attributes n, and the sample size 8, our experiments also differed in two other dimensions. These are the type of process generating the data (we will be interested in various deviations from CIA), and the "difficulty" of the problem. These features are determined by the data model we use (i.e., the distribution over {O, I} ft used to generate data sets). Our first experiments consider the case where the data really is drawn from a binary CIA distribution. We associated with any such distribution a "difficulty" parameter B, which is the accuracy with which one could predict the value of Z if one actually knew the correct model. (Of course, even with knowledge of the correct model we should not expect 100% accuracy.) The ability to control B allows us to select and study models with different qualitative characteristics. In particular, this has allowed us concentrated most of our experiments on fairly "hard" instances3, and to more meaningfully compare trials with differing numbers of attributes. We denote by CIA( n, 2, b) the class of all data models which are binary CIA distributions over n variables with difficulty b.4 The next family of data models we used are also CIA models, but now using more than two values for the hidden variable. We denote the family using Ie values as CIA(n, Ie, b) where n and b are as before. When Ie > 2 there are more complex correlation patterns between the Xi than when Ie = 2. Furthermore, the optimal predictor is not necessarily linear. The specific results we discuss in the next section have concentrated on this case. Given any set of parameters, including a particular class of data models, our experiments are designed with the goal of good statistical accuracy. We repeatedly (typically 100 to 300 times) choose a data model at random from the chosen class, choose a sample of the appropriate size from this model, and then run all our algorithms. Each algorithm produces a (linear) hypothesis. We measure the success rate Salg (i.e., the proportion of times a hypothesis makes the correct prediction of :1:0) by drawing yet more random samples from the data model being used. In the test phase we always draw enough new samples so that the confidence interval for Salg, for the results on a single model, has width at most ± 1 %. We use the Salg values to construct a normalized measure of performance (denoted T) as follows. Let Sbest be the best possible accuracy attainable for predicting:l:o (i.e., the accuracy achieved by the actual model generating the data). Let Sconst denote the performance of the best possible constant prediction rule (i.e., the rule that predicts the most likely a priori value for :1:0). Note that Sconst and Sbest can vary from model to model. For each model we compute :alg--;onst ,and our normalized statistic T is the average of these values. It can be best- const thought of as measuring the percentage of the possible predictive power, over a plausible baseline, that an algorithm achieves. 8 Results We only report on a small, but representative, selection of our experiments in any detail. For instance, although we have considered many values of n ranging from 10 to 500, here we show six graphs giving the learning curves for CIA(n, Ie, 0.90) for n = 10,75, and for Ie = 2,3,5; as noted, we display the T statistic. The error bars show the standard error,s providing a rough indication of accuracy. Not surprisingly, when the data model is binary 3Note that if one simply chooses parameters of a CIA model independently at random, without examining the difficulty of the model or adjusting for n, one will get many trivial problems, in which it is easy to predict Z with nearly 100% accuracy, and thus predict optimally for Xo. 41t is nontrivial to efficiently select random models from this class. Briefly, our scheme is to choose each parameter in a CIA model independently from a symmetric beta distribution. Thus, the model parameters will have expected value 0.5. We choose the parameter of the beta distribution (which determines concentration about 0.5) so that the average B value, of the models thus generated, equals b. Finally, we use rejection sampling to find CIA models with B values that are exactly b ± 1 %. 5Computed as the observed standard deviation, divided by the square root of the number of trials. Linear Concepts and Hidden Variables: An Empirical Study 505 CIA, the EM algorithm does extremely well, learning significantly (if not overwhelmingly) faster than Winnow. But as we depart from the binary CIA assumption, the performance of EM quickly degrades. CI"(10.2.0.1IO) ,"" '00 --_ .. loo ... .... Joo I ·· z .. t 40 J: m -EM - oa>I ...... ......, ,0 .. '00 ... '000 .oIT~~ Figure 1: CIA(10,2,0.9) OA,(10 •• ,O.IO) '00 .. 00 l }oo j40 I"" m -E .. - oa>I ...... ,0 .. '00 ... '000 .dT"INng~ Figure 3: CIA(1O,3,0.9) CIAC10.a,O.1O) ........... loo '-----' 140 too I 0 -20 ......,~,o~----~ .. ~~,~~~-----=*~~,_ .oIT' .... ExemPM '00 00 lOO , 40 j: )......, .... '0 00 00 l40 I"" t 0 J ...... ......, ,0 10 40 l , 20 j a )......, '0 Cl"(78.2.0.1IO) , , ., ,1,1 of ,.·f ~i'.~/ .... ~ _ ~~! ~ .. ! -,-4· m ·· ""-EM - oa>I .. '00 ... '000 .oIT,.......~ Figure 2: CIA(75,2,0.9) CIA(78,',O.1O) m ···""-EM - oa>I .. '00 ... '000 .01 Tralring ~ Figure 4: CIA(75,3,0.9) CIA(7 •••• o..a) " -1--_---t--rI -r' m -E .. - oa>I eo '00 ... 'ODD .oIT~~ Figure 5: CIA(10,5,0.9) Figure 6: CIA(75,5,0.9) When Ie = 3 performances is, on the whole, very similar for Winnow and EM. But when Ie = 5 Winnow is already superior to EM; significantly and uniformly so for n = 10. For fixed Ie the difference seems to become somewhat less dramatic as n increases; in Figure 6 (for n = 75) Winnow is less obviously dominant, and in fact is not better than EM until the sample size has reached 100. (But when 8 ~ n, meaning that we have fewer samples than attributes, the performance is unifonnly dismal anyway.) Should we attribute this degradation to the binary CIA assumption, or to the EM itself? This question is our reason for also considering the covariance algorithm. We see that the results for COY are generally similar to EM's, supporting our belief that the phenomena we see are properties inherent to the model rather than to the specific algorithm being used. Similarly (the results are omitted) we have tried several other algorithms that try to find good linear separators directly, including the classic Perceptron algorithm [MP69); our version of Winnow was the best on the experiments we tried and thus we conjecture that its performance is (somewhat) indicative of what is possible for any such approach. As the comparison between n = 10 and n = 75 illustrates, there is little qualitative differ506 A. J. Grove and D. Roth ence between the phenomena observed as the number of attributes increases. Nevertheless, as n grows it does seem that Winnow needs more examples before its performance surpasses that of the other algorithms (for any fixed k). As already noted, this may be due simply to the very "noisy" nature of the region 8 $ n. We also have reasons to believe that this is partially an artifact of way we select models. As previously noted, we also experimented with varying "difficulty" (B) levels. Although we omit the corresponding figures we mentioned that the main difference is that Winnow is a little faster in surpassing EM when the data deviates from the assumed model, but when the data model really is binary CIA, and EM converge even faster to an optimal performance. These patterns were confinned when we tried to compare the approaches on real data. We have used data that originates from a problem in which assuming a hidden "context" variable seems somewhat plausible. The data is taken from the context-sensitive spelling correction domain. We used one data set from those that were used in [GR96]. For example, given sentences in which the word passed or past appear, the task is to determine, for each such occurrence, which of the two it should be. This task may be modeled by thinking of the "context" as a hidden variable in our sense. Yet when we tried to learn in this case under the CIA model, with a binary valued hidden variable, the results were no better than just predicting the most likely classification (around 70%). Winnow, in contrast, performed extremely well and exceeds 95% on this task. We hesitate to read much into our limited real-data experiments, other than to note that so far they are consistent witli the more careful experiments on synthetic data. 9 Conclusion By restricting to a binary hidden variable, we have been able to consider a "fair" comparison between probabilistic model construction, and more traditional algorithms that directly learn a classification-at least in the sense that both have the same expressive power. Our conclusions concerning the fragility of the former should not be surprising but we believe that given the importance of the problem it is valuable to have some idea of the true significance of the effect. As we have indicated, in many real-world cases, where a model of the sort we have considered here seems plausible, it is impossible to nail down more specific characterizations of the probabilistic model. Our results exhibit how important it is to use the correct model and how sensitive are the results to deviations from it, when attempting to learn using model construction. The purpose of this paper is not to advocate that in practice one should use either Winnow or binary CIA in exactly the form considered here. A richer probabilistic model should be used along with a model selection phase. However, studying the problem in a restricted and controlled environment in crucial so as to understand the nature and significance of this fundamental problem. References [Blu97] A. Blum. Empirical support for winnow and weighted majority based algorithms: results on a calendar scheduling domain. Machine Learning, 26: 1-19, 1997. [DLR 77] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Royal Statistical SOCiety B, 39: 1-38, 1977. [GR96] A. R. Golding and D. Roth. Applying winnow to context-sensitive spelling correcton. In Proc. 13th International Conference on Machine Learning (ML' 96), pages 182-190, 1996. [HS92] K. HOffgen and H. Simon. Robust trainability of single neurons. In Proc. 5th Annu. Workshop on Comput. Learning Theory, pages 428-439, New York, New York, 1992. ACM Press. [Lit88] N. Littlestone. Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning, 2:285-318,1988. [Lit91] N. Littlestone. Redundant noisy attributes, attribute errors, and linear threshold learning using Winnow. In Proc. 4th Annu. Workshop on Comput. Learning Theory, pages 147-156, San Mateo, CA, 1991. Morgan Kaufmann. [MP69] M. L. Minsky and S. A. Papert. Perceptrons. MIT Press, Cambridge, MA, 1969.
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Comparison of Human and Machine Word Recognition M. Schenkel Dept of Electrical Eng. University of Sydney Sydney, NSW 2006, Australia schenkel@sedal.usyd.edu.au M. Jabri C. Latimer Dept of Psychology University of Sydney Sydney, NSW 2006, AustTalia Dept of Electrical Eng. University of Sydney Sydney, NSW 2006, Australia marwan@sedal. usyd.edu.au Abstract We present a study which is concerned with word recognition rates for heavily degraded documents. We compare human with machine reading capabilities in a series of experiments, which explores the interaction of word/non-word recognition, word frequency and legality of non-words with degradation level. We also study the influence of character segmentation, and compare human performance with that of our artificial neural network model for reading. We found that the proposed computer model uses word context as efficiently as humans, but performs slightly worse on the pure character recognition task. 1 Introduction Optical Character Recognition (OCR) of machine-print document images ·has matured considerably during the last decade. Recognition rates as high as 99.5% have been reported on good quality documents. However, for lower image resolutions (200 Dpl and below), noisy images, images with blur or skew, the recognition rate declines considerably. In bad quality documents, character segmentation is as big a problem as the actual character recognition. fu many cases, characters tend either to merge with neighbouring characters (dark documents) or to break into several pieces (light documents) or both. We have developed a reading system based on a combination of neural networks and hidden Markov models (HMM), specifically for low resolution and degraded documents. To assess the limits of the system and to see where possible improvements are still to be Comparison of Human and Machine Word Recognition 95 expected, an obvious comparison is between its performance and that of the best reading system known, the human reader. It has been argued, that humans use an extremely wide range of context information, such as current topics, syntax and semantic analysis in addition to simple lexical knowledge during reading. Such higher level context is very hard to model and we decided to run a. first comparison on a word recognition task, excluding any context beyond word knowledge. The main questions asked for this study are: how does human performance compare with our system when it comes to pure character recognition (no context at all) of bad quality documents? How do they compare when word context can be used? Does character segmentation information help in reading? 2 Data Preparation We created as stimuli 36 data sets, each containing 144 character strings, 72 words and 72 non-words, all lower case. The data sets were generated from 6 original sets, each COI!taining 144 unique wordsjnon-words. For each original set we used three ways to divide the words into the different degradation levels such that each word appears once in each degradation level. We also had two ways to pick segmented/non-segmented so that each word is presented once segmented and once non-segmented. This counterbalancing creates the 36 sets out of the six original ones. The order of presentation within a test set was randomized with respect to degradation, segmentation and lexical status. All character strings were printed in 'times roman 10 pt' font. Degradation was achieved by photocopying and faxing the printed docJiment before scanning it at 200Dpl. Care was taken to randomize the print position of the words such that as few systematic degradation differences as possible were introduced. Words were picked from a dictionary of the 44,000 most frequent words in the 'Sydney Morning Herald'. The length of the words was restricted to be between 5 and 9 characters. They were divided in a 3x3x2 mixed factorial model containing 3 word-frequency groups, 3 stimulus degradation levels and visually segmented/non-segmented words. The three word-frequency groups were: 1 to 10 occurences/million (o/m) as low frequency, 11 to 40 ojm as medium frequency and 41 or more ojm as high frequency. Each participant was presented with four examples per stimulus class (e.g. four high frequency words in medium degradation level, not segmented). The non-words conformed to a 2x3x2 model containing legal/illegal non-words, 3 stimulus degradation levels and visually segmented/non-segmented strings. The illegal non-words (e.g. 'ptvca') were generated by randomly selecting a word length between 5 and 9 characters (using the same word length frequencies as the dictionary has) and then randomly picking characters (using the same character frequencies as the dictionary has) and keeping the unpronouncable sequences. The legal non-words (e.g. 'slunk') were generated by using trigrams (using the dictionary to compute the trigram probabilities) and keeping pronouncable sequences. Six examples per non-word stimulus class were used in each test set. (e.g. six illegal non-words in high degradaton level, segmented). 3 Human Reading There were 36 participants in the study. Participants were students and staff of the University of Sydney, recruited by advertisement and paid for their service. They were all native English speakers, aged between 19 and 52 with no reported uncorrected visual deficits. The participants viewed the images, one at a time, on a computer monitor and were asked to type in the character string they thought would best fit the image. They had been 96 M. Schenkel, C. Latimer and M. Jabri instructed that half of the character strings were English words and half non-words, and they were informed about the degradation levels and the segmentation hints. Participants were asked to be as fast and as accurate as possible. After an initial training session of 30 randomly picked character strings not from an independent training set, the participants had a short break and were then presented with the test set, one string at a time. After a Carriage Return was typed, time was recorded and the next word was displayed. Training and testing took about one hour. The words were about 1-1.5cm large on the screen and viewed at a distance of 60cm, which corresponds to a viewing angle of 1°. 4 Machine Reading For the machine reading tests, we used our integrated segmentation/recognition system, using a sliding window technique with a combination of a neural network and an HMM [6). In the following we describe the basic workings without going into too much detail on the specific algorithms. For more detailed description see (6]. A sliding window approach to word recognition performs no segmentation on the input data of the recognizer. It consists basically of sweeping a window over the input word in small steps. At each step the window is taken to be a tentative character and corresponding character class scores are produced. Segmentation and recognition decisions are then made on the basis of the sequence of character scores produced, possibly taking contextual information into account. In the preprocessing stage we normalize the word to a fixed height. The result is a grey-normalized pixel map of the word. This pixel map is the input to a neural network which estimates a posteriori probabilities of occurrence for each character given the input in the sliding window whose length corresponds approximately to two characters. We use a space displacement neural network (SDNN) which is a multi-layer feed-forward network with local connections and shared weights, the layers of which perform successively higherlevel feature extraction. SDNN's are derived from Time Delay Neural Networks which have been successfully used in speech recognition (2] and handwriting recognition (4, 1]. Thanks to its convolutional structure the computational complexity of the sliding window approach is kept tractable. Only about one eighth of the network connections are reevaluated for each new input window. The outputs of the SDNN are processed by an HMM. In our case the HMM implements character duration models. It tries to align the best scores of the SDNN with the corresponding expected character durations. The Viterbi algorithm is used for this alignment, determining simultaneously the segmentation and the recognition of the word. Finding this state sequence is equivalent to finding the most probable path through the graph which represents the HMM. Normally additive costs are used instead of multiplicative probabilities. The HMM then selects the word causing the smallest costs. Our best architecture contains 4 convolutional layers with a total of 50,000 parameters (6]. The training set consisted of a subset of 180,000 characters from the SEDAL database, a low resuloution degraded document database which was collected earlier and is independent of any data used in this experiment. 4.1 The Dictionary. Model A natural way of including a dictionary in this process, is to restrict the solution space of the HMM to words given by the dictionary. Unfortunately this means calculating the cost for each word in the dictionary, which becomes prohibitively slow with increasing dictionary size (we use a combination of available dictionaries, with a total size of 98,000 words). We thus chose a two step process for the dictionary search: in a first step a list of the most probable words is generated, using a fast-matcher technique. In the second step the HMM costs are calculated for the words in the proposed list. Comparison of Human and Machine Word Recognition 97 To generate the word list, we take the character string as found by the HMM without the dictionary and calculate the edit-distance between that string and all the words in the dictionary. The edit-distance measues how many edit operations (insertion, deletion and substitution) are necessary to convert a given input string into a target word [3, 5]. We now select all dictionary words that have the smallest edit-distance to the string recognized without using the dictionary. The composed word list contains on average 10 words, and its length varies considerably depending on the quality of the initial string. For all words in the word list the HMM cost is now calculated and the word with the smallest cost is the proposed dictionary word. As the calculation of the edit-distance is much faster than the calculation of the HMM costs, the recognition speed is increased substantially. In a last step the difference in cost between the proposed dictionary word and the initial string is calculated. If this difference is smaller than a threshold, the system will return the dictionary word, otherwise the original string is returned. This allows for the recognition of non-dictionary words. The value for the threshold determines the amount of reliance on the dictionary. A high value will correct most words but will also force non-words to be recognized as words. A low value, on the other hand, leaves the non-words unchanged but doesn't help for words either. Thus the value of the threshold influences the difference between word and non-word recognition. We chose the value such that the over-all error rate is optimized. 4.2 The Case of Segmented data When character segmentation is given, we know how many characters we have and where to look for them. There is no need for an HMM and we just sum up the character probabilities over the x-coordinate in the region corresponding to a segment. This leaves a vector of 26 scores {the whole alphabet) for each character in the input string. With no dictionary constraints, we simply pick the label corresponding to the highest probability for each character. The dictionary is used in the same way, replacing the HMM scores by calculating the word scores directly from the corresponding character probabilities. 5 Results Recognition Performance 0.6 :::::=---Non-Words 0.1 Words Human Reading -- ... o~------------~2------------~ Degradation Figure 1: Human Reading Performance. Machine Reading 0.6 X Non-Segmental 0 S.pncnled -~0.5 ~ ::;0.4 1:: -------lC' ,..IX'! r·3 ao.2 0.1 Words 0·~------------~2------------~3~ Degradation Figure 2: Machine Reading Performance. 98 M. Schenkel, C. Latimer and M. Jabri Figure 1 depicts the recognition results for human readers. All results are per character error rates counted by the edit-distance. All results reported as significant pass an F -test with p < .01. As expected there was a significant interaction between error rate and degradation and clearly non-words have higher error rates than words. Also character segmentation has also an influence on the error rate. Segmentation seems to help slightly more for higher degradations. Figure 2 shows performance of the machine algorithm. Again greater degradation leads to higher error rates and non-words have higher error rates than words. Segmentation hints lead to significantly better recognition for all degradation levels; in fact there is no interaction between degradation and segmentation for the machine algorithm. In general the machine benefited more from segmentation than humans. One would expect a smaller gain from lexical knowledge for higher en;or rates (i.e. higher degradation) as in the limit of complete degradation all error rates will be 100%. Both humans and machine show this 'closing of the gap . Segmented Recognition 0.6 -Human - - - • Mac;binc 0.1 Words o~------------~2------------~3~ Degradation Figure 3: Segmented Data. Non-Segmented Recognition 0.6 -Human o~------------~2------------~3~ Degradation Figure 4: Non-Segmented Data. More interesting is the direct comparison between the error rates for humans and machine as shown in figure 3 and figure 4. The difference for non-words reflects the difference in ability to recognize the geometrical shape of characters without context. For degradation levels 1 and 2, the machine has the same reading abilities as humans for segmented data and looses only about 7% in the non-'segmented case. For degradation level 3; the machine clearly performs worse than human readers. The difference between word and non-word error rates reflects the ability of the participant to use lexical knowledge. Note that the task contains word/non-word discrimination as well as recognition. It is striking how similar the behaviour for humans and machine is for degradation levels 1 and 2. Timing Results Figure 5 shows the word entry times for humans. As the main goal was to compare recognition rates, we did not emphasize entry speed when instructing the participants. However, we recorded the word entry time for each word (which includes inspection time and typing). When analysing the timimg data the only interest was in relative difference between word groups. Times were therefore· converted for each participant into a z-score (zero mean with a standard deviation of one) and statistics were made over the z-scores of all participants. Non-words generally took longer to recognize than words and segmented data took longer Comparison of Human and Machine Woni Recognition Humau Reading Times o.s,..-------,---------.--, ·----------0 ___ .. ----------====---------------x -{).S'-:-1 ------2:-------~3--l Desradation Figure 5: Human Reading Times. Non-Segmented 0.5,..-------,.--------~ -Human -----· ;..., 0.3 ~ -------- ------=~~~= -.!:!. 0 I ~ . b Jj -{).I "E ~ -{)3 -{).S'-;-------2:-------~3:-' Degradation Figure 6: Non-Segmented Reading Times. 99 to recognize than non-segmented for humans which we believe stems from participants not being used to reading segmented data. When asked, participants reported difficulties in using the segmentation lines. Interestingly this segmentation effect is significant only for words but not for non-words. As predicted there is also an interaction between time and degradation. Greater degradations take longer to recognize. Again, the degradation effect for time is only significant for words but barely for non-words. Our machine reading algorithm behaves differently in segmented and non-segmented mode with respect to time consumption. In segmented mode, the time for evaluating the word list in our system is very short compared to the base recognition time, as there is no HMM involved. Accordingly we found no or very little effects on timing for our system for segmented data. All the timing information for the machine refer to the non-segmented case (see Figure 6). Frequency and Legality Table 5 shows word frequencies, legality of non-words and entry-time. Our experiment confirmed the well known frequency and legality effect for humans in recognition rate as well as time and respectively for frequency. The only exception is that there is no difference in error rate for middle and low frequency words. The machine shows (understandably) no frequency effect in error rate or time, as all lexical words had the same prior probability. Interestingly even when using the correct prior probabilities we could not produce a strong word frequency· effect for the machine. Also no legality effect was observed for the error rate. One way to incorporate legality effects would be the use of Markov chains such as n-grams. Note however, how the recognition time for non-words is higher than for words and the legality effect for the recognition time. Recognition times for our system in non-segmented mode depend mainly on the time it takes to evaluate the word list. Non-words generally produce longer word lists than words, because there are no good quality matches for a non-word in the dictionary (on average a word list length of 8.6 words was found for words and of 14.5 for non-words). Also illegal non-words produce longer word lists than legal ones, again because the match quality for illegal non-words is worse than for legal ones (average length for illegal non-words 15.9 and for legal non-words 13.2). The z-scores for the word list length parallel nicely the recognition time scores. In segmented mode, the time for evaluating the word list is very short compared to the 100 M. Schenkel, C. Latimer and M. Jabri base recognition time, as there is no HMM involved. Accordingly we found no or very little effects on timing for our system in the segmented case. Error l%J Humans Machine Error z-Time Error z-Time Words 41+ 0.22 -0.37 0.36 -0.14 Words 11-40 0.27 -0.13 0.34 -0.19 Words 1-10 0.26 -0.06 0.33 -0.22 Legal Non-W. 0.36 0.07 0.47 0.09 lllegal Non-W. 0.46 0.31 0.49 0.28 Table 1: Human and Machine Error rates for the different word and non-word classes. The z-times for the machine are for the non-segmented data only. 6 Discussion . The ability to recognize the geometrical shape of characters without the possibility to use any sort of context information is reflected in the error rate of illegal non-words. The difference between the error rate for illegal non-words and the one for words reflects the ability to use lexical knowledge. To our surprise the behavior of humans and machine is very similar for both tasks, indicating a near to optimal machine recognition system. Clearly this does not mean our system is a good model for human reading. Many effects such as semantic and repetition priming are not reproduced and call for a system which is able to build semantic classes and memorize the stimuli presented. Nevertheless, we believe that our experiment validates empirically the verification model we implemented, using real world data. Acknowledgments This research is supported by a grant from the Australian Research Council (grant No A49530190). References [1] I. Guyon, P. Albrecht, Y. Le Cun, J. Denker, and W. Hubbard. Design of a neural network character recognizer for a touch terminal. Pattern Recognition, 24(2):105-119, 1991. [2] K. J. Lang and G. E. Hinton. A Time Delay Neural Network architecture for speech recognition. Technical Report CMU-cs-88-152, Carnegie-Mellon University, Pittsburgh PA, 1988. [3] V.I. Levenshtein. Binary codes capable of correcting deletions, insertions and reversals. Soviet Physics-Doklady, 10(8):707-710, 1966. [4] 0. Matan, C. J. C. Burges, Y. Le Cun, and J. Denker. Multi-digit recognition using a Space Dispacement Neural Network. In J. E. Moody, editor, Advances in Neural Information Processing Systems 4, pages 488-495, Denver, 1992. Morgan Kaufmann. f5] T. Okuda, E. Tanaka, and K. Tamotsu. A method for the correction of garbled words based on the Levenshtein metric. IEEE Transactions on Computers, c-25(2):172-177, 1976. [6] M. Schenkel and M. Jabri. Degraded printed document recognition using convolutional neural networks and hidden markov models. In Proceedings of the A CNN, Melbourne, 1997.
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Characterizing Neurons in the Primary Auditory Cortex of the Awake Primate U sing Reverse Correlation R. Christopher deC harms decharms@phy.ucsf.edu Michael M . Merzenich merz@phy.ucsf.edu w. M. Keck Center for Integrative Neuroscience University of California, San Francisco CA 94143 Abstract While the understanding of the functional role of different classes of neurons in the awake primary visual cortex has been extensively studied since the time of Hubel and Wiesel (Hubel and Wiesel, 1962), our understanding of the feature selectivity and functional role of neurons in the primary auditory cortex is much farther from complete. Moving bars have long been recognized as an optimal stimulus for many visual cortical neurons, and this finding has recently been confirmed and extended in detail using reverse correlation methods (Jones and Palmer, 1987; Reid and Alonso, 1995; Reid et al., 1991; llingach et al., 1997). In this study, we recorded from neurons in the primary auditory cortex of the awake primate, and used a novel reverse correlation technique to compute receptive fields (or preferred stimuli), encompassing both multiple frequency components and ongoing time. These spectrotemporal receptive fields make clear that neurons in the primary auditory cortex, as in the primary visual cortex, typically show considerable structure in their feature processing properties, often including multiple excitatory and inhibitory regions in their receptive fields. These neurons can be sensitive to stimulus edges in frequency composition or in time, and sensitive to stimulus transitions such as changes in frequency. These neurons also show strong responses and selectivity to continuous frequency modulated stimuli analogous to visual drifting gratings. 1 Introduction It is known that auditory neurons are tuned for a number of independent feature parameters of simple stimuli including frequency (Merzenich et al., 1973), intensity (Sutter and Schreiner, 1995), amplitude modulation (Schreiner and Urbas, 1988), and Characterizing Auditory Cortical Neurons Using Reverse Correlation 125 others. In addition, auditory cortical responses to multiple stimuli can enhance or suppress one another in a time dependent fashion (Brosch and Schreiner, 1997; Phillips and Cynader, 1985; Shamma and Symmes, 1985), and auditory cortical neurons can be highly selective for species-specific vocalizations (Wang et al., 1995; Wollberg and Newman, 1972), suggesting complex acoustic processing by these cells. It is not yet known if these many independent selectivities of auditory cortical neurons reflect a discernible underlying pattern of feature decomposition, as has often been suggested (Merzenich et al., 1985; Schreiner and Mendelson, 1990; Wang et al., 1995). Further, since sustained firing rate responses in the auditory cortex to tonal stimuli are typically much lower than visual responses to drifting bars (deCharms and Merzenich, 1996b), it has been suggested that the preferred type of auditory stimulus may still not be known (Nelken et al., 1994). We sought to develop an unbiased method for determining the full feature selectivity of auditory cortical neurons, whatever it might be, in frequency and time based upon reverse correlation. 2 Methods Recordings were made from a chronic array of up to 49 individually placed ultrafine extracellular Iridium microelectrodes, placed in the primary auditory cortex of the adult owl monkey. The electrodes had tip lengths of 10-25microns, which yield impedance values of .5-SMOhm and good isolation of signals from individual neurons or clusters of nearby neurons. We electrochemically activated these tips to add an ultramicroscopic coating of Iridium Oxide, which leaves the tip geometry unchanged, but decreases the tip impedance by more than an order of magnitude, resulting in substantially improved recording signals. These signals are filtered from .3-8kHz, sampled at 20kHz, digitized, and sorted. The stimuli used were a variant of random VlsuII Cortn: Reveree Correlltlon U.lng 2·D VI.nl Pltternl In Time t - 40m.ec x II ! SplkeT .. ln. Spltlotemporal Receptive Field Auditory Cortex: Rever.e Correlltlon U.lng 1·D Auditory Pltternl (Chordl) In Tim. t -Om.ec t- 20m.ec t-40msec x II Spectrotempoul Receptive Field Figure 1: Schematic of stimuli used for reverse correlation. white noise which was designed to allow us to characterize the responses of neurons in time and in frequency. As shown in figure 1, these stimuli are directly analogous to stimuli that have been used previously to characterize the response properties of neurons in the primary visual cortex (Jones and Palmer, 1987; Reid and Alonso, 1995; Reid et al., 1991). In the visual case, stimuli consist of spatial checkerboards that span some portion of the two-dimensional visual field and change pattern with a short sampling interval. In the auditory case, which we have studied here, the stimuli chosen were randomly selected chords, which approximately evenly span a 126 R C. deChanns and M M. Merzenich portion of the one-dimensional receptor surface of the cochlea. These stimuli consist of combinations of pure tones, all with identical phase and all with 5 msec cosineshaped ramps in amplitude when they individually turn on or off. Each chord was created by randomly selecting frequency values from 84 possible values which span 7 octaves from 110Hz to 14080Hz in even semitone steps. The density of tones in each stimulus was 1 tone per octave on average, or 7 tones per chord, but the stimuli were selected stochastically so a given chord could be composed of a variable number of tones of randomly selected frequencies. We have used sampling rates of 10-100 chords/second, and the data here are from stimuli with 50 chords/second. Stimuli with random, asynchronous onset times of each tone produce similar results. These stimuli were presented in the open sound field within an acoustical isolation chamber at 44. 1kHz sampling rate directly from audio compact disk, while the animal sat passively in the sound field or actively performed an auditory discrimination task, receiving occasional juice rewards. The complete characterization set lasted for ten minutes, thereby including 30,000 individual chords. Spike trains were collected from mUltiple sites in the cortex simultaneously during the presentation of our characterization stimulus set, and individually reverse correlated with the times of onset of each of the tonal stimuli. The reverse correlation method computes the number of spikes from a neuron that were detected, on average, during a given time preceding, during, or following a particular tonal stimulus component from our set of chords. These values are presented in spikes/s for all of the tones in the stimulus set, and for some range of time shifts. This method is somewhat analogous in intention to a method developed earlier for deriving spectrotemporal receptive fields for auditory midbrain neurons (Eggermont et al., 1983), but previous methods have not been effective in the auditory cortex. 3 Results Figure 2 shows the spectrotemporal responses of neurons from four locations in the primary auditory cortex. In each panel, the time in milliseconds between the onset of a particular stimulus component and a neuronal spike is shown along the horizontal axis. Progressively greater negative time shifts indicate progressively longer latencies from the onset of a stimulus component until the neuronal spikes. The frequency of the stimulus component is shown along the vertical axis, in octave spacing from a 110Hz standard, with twelve steps per octave. The brightness corresponds to the average rate of the neuron, in spk/s, driven by a particular stimulus component. The reverse-correlogram is thus presented as a stimulus triggered spike rate average, analogous to a standard peristimulus time histogram but reversed in time, and is identical to the spectrogram of the estimated optimal stimulus for the cell (a spike triggered stimulus average which would be in units of mean stimulus denSity). A minority of neurons in the primary auditory cortex have spectrotemporal receptive fields that show only a single region of increased rate, which corresponds to the traditional characteristic frequency of the neuron, and no inhibitory region. We have found that cells of this type (less than 10%, not shown) are less common than cells with multimodal receptive field structure. More commonly, neurons have regions of both increased and decreased firing rate relative to their mean rate within their receptive fields. For terminological convemence, these will be referred to as excitatory and inhibitory regions, though these changes in rate are not diagnostic of an underlying mechanism. Neurons with receptive fields of this type can serve as detectors of stimulus edges in both frequency space, and in time. The neuron shown in figure 2a has a receptive field structure indicative of lateral inhibition in frequency space. This cell prefers a very narrow range of frequencies, and decreases its firing rate for nearby frequencies, giving the characteristic of a sharply-tuned bandpass filter. This Characterizing Auditory Cortical Neurons Using Reverse Correlation 127 a) b) 3 3.5 40 2.5 3 .. 2 !: 2.5 30 > .. !!! g 1.5 3 8 2 20 1.5 0.5 10 -100 -50 0 -SO 0 msec d) msec c) 3.5 15 3 N 3 10 J: 0 2.5 )! 10;::: 2.5 5 ., '" > 2 > 2 0 .. .8 g 1.5 < -5 5 ., 1.5 ~ -10 CD U -100 -50 0 0 ~O -40 -20 msec msec Figure 2: Spectrotemporal receptive fields of neurons in the primary auditory cortex of the awake primate. These receptive fields are computed as described in methods. Receptive field structures read from left to right correspond to a preferred stimulus for the neuron, with light shading indicating more probable stimulus components to evoke a spike, and dark shading indicating less probable components. Receptive fields read from right to left indicate the response of the neuron in time to a particular stimulus component. The colorbars correspond to the average firing rates of the neurons in Hz at a given time preceding, during, or following a particular stimulus component. type of response is the auditory analog of a visual or tactile edge detector with lateral inhibition. Simple cells in the primary visual cortex typically show similar patterns of center excitation along a short linear segment, surrounded by inhibition (Jones and Palmer, 1987;·Reid and Alonso, 1995; Reid et al., 1991). The neuron shown in figure 2b shows a decrease in firing rate caused by a stimulus frequency which at a later time causes an increase in rate. This receptive field structure is ideally suited to detect stimulus transients; and can be thought of as a detector of temporal edges. Neurons in the auditory cortex typically prefer this type of stimulus, which is initially soft or silent and later loud. This corresponds to a neuronal response which shows an increase followed by a decrease in firing rate. This is again analogous to neuronal responses in the primary visual cortex, which also typically show a firing rate pattern to an optimal stimulus of excitation followed by inhibition, and preference for stimulus transients such as when a stimulus is first off and then comes on. The neuron shown in figures 2c shows an example which has complex receptive field structure, with multiple regions. Cells of this type would be indicative of selectivity for feature conjunctions or quite complex stimuli, perhaps related to sounds in the animal's learned environment. Cells with complex receptive field structures are common in the awake auditory cortex, and we are in the process of quantifying the percentages of cells that fit within these different categories. Neurons were observed which respond with increased rate to one frequency range at one time, and a different frequency range at a later time, indicative of selectivity for frequency modulations(Suga, 1965). Regions of decreased firing rate can show similar patterns. The neuron shown in figure 2d is an example of this type. This pattern is strongly analogous to motion energy detectors in the visual system (Adelson and Bergen, 1985), which detect stimuli moving in space, and these cells are selective for changes in frequency. 128 R. C. deCharms and M M. Merzenich 2 oct/sec 6 oct/sec 10 oct/sec 14 oct/sec 30 oct/sec 100 oct/sec ·2 oct/sec ·6 oct/sec ·10 oct/sec ·14 oct/sec ·30 oct/sec ·100 oct/sec Figure 3: Parametric stimulus set used to explore neuronal responses to continuously changing stimulus frequency. Images axe spectrograms of stimuli from left to right in time, and spanning seven octaves of frequency from bottom to top. Each stimulus is one second. Numbers indicate the sweep rate of the stimuli in octaves per second. Based on the responses shown, we wondered whether we could find a more optimal class of stimuli for these neuron, analogous to the use of drifting bars or gratings in the primary visual cortex. We have created auditory stimuli which correspond exactly to the preferred stimulus computed for a paxticulax cell from the cell's spectrotemporal receptive field (manuscript in prepaxation), and we have also designed a paxametric class of stimuli which are designed to be particularly effective for neurons selective for stimuli of changing amplitude or frequency, which are presented here. The stimuli shown in figure 3 are auditory analogous of visual drifting grating stimuli. The stimuli axe shown as spectrograms, where time is along the horizontal axis, frequency content on an octave scale is along the vertical axis, and brightness corresponds to the intensity of the signal. These stimuli contain frequencies that change in time along an octave frequency scale so that they repeatedly pass approximately linearly through a neurons receptive field, just as a drifting grating would pass repeatedly through the receptive field of a visual neuron. These stimuli axe somewhat analogous to drifting ripple stimuli which have recently been used by Kowalski, et.al. to characterize the linearity of responses of neurons in the anesthetized ferret auditory cortex (Kowalski et al., 1996a; Kowalski et al., 1996b). Neurons in the auditory cortex typically respond to tonal stimuli with a brisk onset response at the stimulus transient, but show sustained rates that axe far smaller than found in the visual or somatosensory systems (deCharms and Merzenich, 1996a). We have found neurons in the awake animal that respond with high firing rates and significant selectivity to the class of moving stimuli shown in figure 3. An outstanding example of this is shown in figure 4. The neuron in this example showed a very high sustained firing rate to the optimal drifting stimulus, as high as 60 Hz· for one second. The neuron shown in this example also showed considerable selectivity for stimulus velocity, as well as some selectivity for stimulus direction. 4 Conclusions These stimuli enable us to efficiently quantify the response characteristics of neurons in the awake primaxy auditory cortex, as well as producing optimal stimuli for particular neurons. The data that we have gathered thus far extend our knowledge about the complex receptive field structure of cells in the primary auditory cortex, Characterizing Auditory Cortical Neurons Using Reverse Correlation 129 2 oct/sec 6 oct/sec 10 octIsec 14 oct/sec 30 oct/sec 100 oct/sec -2 oct/sec -6 oct/sec -10 oct/sec -14 oct/sec -30 oct/sec -100 oct/sec Figure 4: Responses of a neuron in the primary auditory cortex of the awake primate to example stimuli take form our characterization set, as shown in figure 3. In each panel, the average response rate histogram in spikes per second is shown below rastergrams showing the individual action potentials elicited on,each of twenty trials. and show some considerable analogy with neurons in the primary visual cortex. In addition, they indicate that it is possible to drive auditory cortical cells to high rates of sustained firing, as in the visual cortex. This method will allow a number of future questions to be addressed. Since we have recorded many neurons simultaneously, we are interested in the interactions among large populations of neurons and how these relate to stimuli. We are also recording responses to these stimuli while monkeys are performing cognitive tasks involving attention and learning, and we hope that this will give us insight into the effects on cell selectivity of the context provided by other stimuli, the animal's behavioral state or awareness of the stimuli, and the animal's prior learning of stimulus sets. 5 References Adelson EH, Bergen JR (1985) Spatiotemporal energy models for the perception of motion. J. Opt. Soc. Am. A, 2, 284-299. Brosch M, Schreiner CE (1997) Time course of forward masking tuning curves in cat primary auditory cortex. J Neurophysiol, 77, 923-43. deCharms Re, Merzenich MM (1996a) Primary cortical representation of sounds by the coordination of action-potential timing. Nature, 381, 610-3. deCharms RC, Merzenich MM (1996b) Primary cortical representation of sounds by the coordination of action-potential timing. Nature, 381, 610-613. EggeI1I).ont JJ, Aertsen AM, Johannesma PI (1983) Quantitative characterisation procedure for auditory neurons based on the spectro-temporal receptive field. Hear Res, 10, 167-90. Hubel DH, Wiesel TN (1962) Receptive fields, binocular interaction and functional archtecture in the cat's visual cortex. J. Physiol., 160, 106-154. Jones JP, Palmer LA (1987) The two-dimensional spatial structure of simple receptive 130 R. C. deCharms and M M. Merzenich fields in cat striate cortex. J Neurophysiol, 58, 1187-211. Kowalski N, Depireux DA, Shamma SA (1996a) Analysis of dynamic spectra in ferret primary auditory cortex. I. Characteristics of single-unit responses to moving ripple spectra. J Neurophysiol, 76, 3503-23. Kowalski N, Depireux DA, Shamma SA (1996b) Analysis of dynamic spectra in ferret primary auditory cortex. II. Prediction of unit responses to arbitrary dynamic spectra. J Neurophysiol, 76, 3524-34. Merzenich MM, Jenkins WM, Middlebrooks JC (1985) Observations and hypotheses on special organizational features of the central auditory nervous system. In: Dynamic Aspects of Neocortical Function. Edited by E. G. a. W. M. C. G. Edelman. New York: Wiley, pp. 397-423. Merzenich MM, Knight PL, Roth GL (1973) Cochleotopic organization of primary auditory cortex in the cat. Brain Res, 63, 343-6. Nelken I, Prut Y, Vaadia E, Abeles M (1994) In search of the best stimulus: an optimization procedure for finding efficient stimuli in the· cat auditory cortex. Hear Res, 72, 237-53. Phillips DP, Cynader MS (1985) Some neural mechanisms in the cat's auditory cortex underlying sensitivity to combined tone and wide-spectrum noise stimuli. Hear Res, 18, 87-102. Reid RC, Alonso JM (1995) Specificity of monosynaptic connections from thalamus to visual cortex. Nature, 378,281-4. Reid RC, Soodak RE, Shapley RM (1991) Directional selectivity and spatiotemporal structure of receptive fields of simple cells in cat striate cortex. J Neurophysiol, 66, 505-29. Ringach DL, Hawken MJ, Shapley R (1997) Dynamics of orientation tuning in macaque primary visual cortex. Nature, 387, 281-4. Schreiner CE, Mendelson JR (1990) Functional topography of cat primary auditory cortex: distribution of integrated excitation. J Neurophysiol, 64, 1442-59. Schreiner CE, Urbas JV (1988) Representation of amplitude in the auditory cortex of the cat. II. Comparison between cortical fields. Hear. Res., 32, 49-64. Shamma SA, Symmes D (1985) Patterns of inhibition in auditory cortical cells in awake squirrel monkeys. Hear Res, 19, 1-13. Suga N (1965) Responses of cortical auditory neurones to frequency modulated sounds in echo-locating bats. Nature, 206, 890-l. Sutter ML, Schreiner CE (1995) Topography of intensity tuning in cat primary auditory cortex: single-neuron versus multiple-neuron recordings. J Neurophysiol, 73, 190-204. Wang X, Merzenich MM, Beitel R, Schreiner CE (1995) Representation of a speciesspecific vocalization in the primary auditory cortex of the common marmoset: temporal and spectral characteristics. J Neurophysiol, 74, 2685-706. Wollberg Z, Newman JD (1972) Auditory cortex of squirrel monkey: response patterns of single cells to species-specific vocalizations. Science, 175, 212-214.
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Phase transitions and the perceptual organization of video sequences Yair Weiss Dept. of Brain and Cognitive Sciences Massachusetts Institute of Technology ElO-120, Cambridge, MA 02139 http://www-bcs.mit.edu;-yweiss Abstract Estimating motion in scenes containing multiple moving objects remains a difficult problem in computer vision. A promising approach to this problem involves using mixture models, where the motion of each object is a component in the mixture. However, existing methods typically require specifying in advance the number of components in the mixture, i.e. the number of objects in the scene. • Here we show that the number of objects can be estimated automatically in a maximum likelihood framework, given an assumption about the level of noise in the video sequence. We derive analytical results showing the number of models which maximize the likelihood for a given noise level in a given sequence. We illustrate these results on a real video sequence, showing how the phase transitions correspond to different perceptual organizations of the scene. Figure la depicts a scene where motion estimation is difficult for many computer vision systems. A semi-transparent surface partially occludes a second surface, and the camera is translating horizontally. Figure 1 b shows a slice through the horizontal component of the motion generated by the camera - points that are closer to the camera move faster than those further away. In practice, the local motion information would be noisy as shown in figure lc and this imposes conflicting demands on a motion analysis system - reliable estimates require pooling together many measurements while avoiding mixing together measurements derived from the two different surfaces. Phase Transitions and the Perceptual Organization of Video Sequences 851 rd rd 'Cd . ...... .. . .. .. . ., .. . ,. ".... ... .. ' . r r ' I: ____________ , I:l .... ".'-, .. ~ . .'~ ... -:".. '..". . .... : .. . .. .. _.. .... ... .. -" .... " . ...... .:. ... .... .. a b c d Figure 1: a: A simple scene that can cause problems for motion estimation. One surface partially occludes another surface. b: A cross section through the horizontal motion field generated when the camera translates horizontally. Points closer to the camera move faster. c: Noisy motion field. In practice each local measurement will be somewhat noisy and pooling of information is required. d: A cross section through the output of a multiple motion analysis system. Points are assigned to surfaces (denoted by different plot symbols) and the motion of each surface is estimated. . .' .' • • w' • " 0. II. • W. • .' I. .1,. I'· •• • • I• I, •• 11 •• 11 •• .. ..' . ·1 -0 .. -01 -04 -a.2 0. 0..1 0.. O. •• 1 -I -OJ! ..Q. -0. -OJ! 0 0., GAl O' 0 1 I Figure 2: The "correct" number of surfaces in a given scene is often ambiguous. Was the motion here generated by one or two surfaces? Significant progress in the analysis of such scenes has been achieved by multiple motion analyzers - systems that simultaneously segment the scene into surfaces and estimating the motion of each surface [9]. Mixture models are a commonly used framework for performing mUltiple motion estimation [5, 1, 10]. Figure 1d shows a slice through the output of a multiple motion analyzer on this scene - pixels are assigned to one of two surfaces and motion information is only combined for pixels belonging to the same surface. The output shown in figure 1d was obtained by assuming the scene contains two surfaces. In general, of course, one does not know the number of surfaces in the scene in advance. Figure 2 shows the difficulty in estimating this number. It is not clear whether this is very noisy data generated by a single surface, or less noisy data generated by two surfaces. There seems no reason to prefer one description over another. Indeed, the description where there are as many surfaces as pixels is also a valid interpretation of this data. Here we take the approach that there is no single "correct" number of surfaces for a given scene in the absence of any additional assumptions. However, given an assumption about the noise in the sequence, there are more likely and less likely interpretations. Intuitively, if we know that the data in figure 2a was taken with a very noisy camera, we would tend to prefer the one surface solution - adding additional surfaces would cause us to fit the noise rather than the data. However, if we know that there is little noise in the sequence, we would prefer solutions that use many surfaces, there is a lot less danger of "overfitting". In this paper} we show, 1 A longer version of this paper is available on the author's web page. 852 Y. Weiss following [6, 8] that this intuition regarding the dependence of number of surfaces to assumed noise level is captured in the maximum likelihood framework. We derive analytical results for the critical values of noise levels where the likelihood function undergoes a "phase transition" - from being maximized by a single model to being maximized by mUltiple models. We illustrate these transitions on synthetic and real video data. 1 Theory 1.1 Mixture Models for optical flow In mixture models for optical flow (cf. [5, 1]) the scene is modeled as composed of K surfaces with the velocity of each vsurface at location (x, y) given by (uk(x,y),vk(x,y). The velocity field is parameterized by a vector f)k. A typical choice [9] is the affine representation: Uk (x, y) = f)~ + f)~ x + f)~ Y vk(x, y) = f)! + f)~x + f):y (1) (2) The affine family of motions includes rotations, translations, scalings and shears. It corresponds to the 2D projection of a plane undergoing rigid motion in depth. Corresponding pixels in subsequent frames are assumed to have identical intensity values, up to imaging noise which is modeled as a Gaussian with variance a2• The task of multiple motion estimation is to find the most likely motion parameter values given the image data. A standard derivation (see e.g. [1]) gives the following log likelihood function for the parameters e: K lee) = L log(L e-R~(x.Y)/2u2) (3) x,y k=l With Rk(X, y) the residual intensity at pixel (x, y) for velocity k: Rk(x, y) = Ix (x, y)uk(x, y) + Iy(x, y)vk(x, y) + It(x, y) (4) where Ix, Iy,It denote the spatial and temporal derivatives of the image sequence. Although our notation does not make it explicit, Rk(X, y) is a function of f)k through equations 1-2. As in most mixture estimation applications, equation 3 is not maximized directly, but rather an Expectation-Maximization (EM) algorithm is used to iteratively increase the likelihood [3]. 1.2 Maximum Likelihood not necessarily with maximum number of models It may seem that since K is fixed in the likelihood function (equation 3) there is no way that the number of surfaces can be found by maximizing the likelihood. However, maximizing over the likelihood may lead to a a solution in which some of the f) parameters are identical [6, 5, 8]. In this case, although the number of surfaces is still K, the number of distinct surfaces may be any number less than K. Consider a very simple case where K = 2 and the motion of each surface is restricted to horizontal translation u(x, y) = u, vex, y) = O. The advantage of this simplified Phase Transitions and the Perceptual Organization of Video Sequences 853 Figure 3: The log likelihood for the data in figure 2 undergoes a phase transition when a is varied. For small values of a the likelihood has two maxima, and at both these maxima the two motions are distinct. For large a 2 the likelihood function has single maximum at the origin, corresponding to the solution where both velocities are equal to zero, or only one unique surface. case is that the likelihood function is a function of two variables and can be easily visualized. Figure 3 shows the likelihood function for the data in figure 2 as (7 is varied. Observe that for small values of (72 the likelihood has two maxima, and at both these maxima the two motions are distinct. For large (72 the likelihood function has single maximum at the origin, corresponding to the solution where both velocities are equal to zero, or only one unique surface. This is a simple example where the ML solution corresponds to a small number of unique surfaces. Can we predict the range of values for (7 for which the likelihood function has a maximum at the origin? This happens when the gradient of the likelihood at the origin is zero and the Hessian has two negative eigenvalues. It is easy to show that the if the data has zero mean, the gradient is zero regardless of (7. As for the Hessian, H, direct calculation gives: -~ ) A.-I 20-~ (5) where E is the mean squared residual of a single motion and c is a positive constant. The two eigenvalues are proportional to -1 and E / (72 -1. So the likelihood function has a local maximum at the origin if and only if E < (72. (see [6, 4, 8] for a similar analysis in other contexts). This result makes intuitive sense. Recall that (72 is the expected noise variance. Thus if the mean squared residual is less than (72 with a single surface, there is no need to add additional surfaces. The result on the Hessian shows that this intuition is captured in the likelihood function. There is no need to introduce additional "complexity costs" to avoid overfitting in this case. More generally, if we assume the velocity fields are of general parametric form, the Hessian evaluated at the point where both surfaces are identical has the form: -~ ) b-F 20(6) where E and F are matrices: (7) z ,y 854 o "0: . .. -.. 0":" .. " a Y. Weiss b Figure 4: a: data generated by two lines. b: the predicted phase diagram for the likelihood of this dataset in a four component mixture. The phase transitions are at (J" = 0.084, 0.112, 0.8088 F = L d(x, y)d(x, y)t (8) :t,Y with d(x, y) = aR~~,y), and R(x, y) the residual as before. A necessary and sufficient condition for the Hessian to have only negative eigenvalues is: (9) Thus when the maximal eigenvalue of F- 1 E is less than (12 the fit with a single model is a local maximum of the likelihood. Note that F- 1 E is very similar to a weighted mean squared error, with every residual weighted by a positive definite matrix (E sums all the residuals times their weight, and F sums all the weights, so F- 1 E is similar to a weighted average). The above analysis predicts the phase transition of a two component mixture likelihood, i.e. the critical value of (12 such that above this critical value, the maximum likelihood solution will have identical motion parameters for both surfaces. This analysis can be straightforwardly generalized to finding the first phase transition of a K component mixture, although the subsequent transitions are harder to analyze. 2 Results The fact that the likelihood function undergoes a phase transition as (1 is varied predicts that a ML technique will converge to different number of distinct models as (1 is varied. We first illustrate these phase transitions on a ID line fitting problem which shares some of the structure of multiple motion analysis and is easily visualized. Figure 4a shows data generated by two lines with additive noise, and figure 4b shows a phase diagram calculated using repeated application of equation 9; i.e. by solving equation 9 for all the data, taking the two line solution obtained after the transition, and repeating the calculation separately for points assigned to each of the two lines. Figure 5 shows the output of an EM algorithm on this data set. Initial conditions are identical in all runs, and the algorithm converges to one, two, three or four distinct lines depending on (1. We now illustrate the phase transitions on a real video sequence. Figures 6- 8 show the output of an EM motion segmentation algorithm with four components on the MPEG flower garden sequence (cf. [9, 10]). The camera is translating in Phase Transitions and the Perceptual Organization of Video Sequences 855 x ······· . . ..... .078 .089 0.1183 1.0 Figure 5: The data in figure 1 are fit with one, two, three or four models depending on a. The results of EM with identical initial conditions are shown, only a is varied. The transitions are consistent with the theoretical predictions . .. ;v~~ \; "" ; • • .. ,~ a b Figure 6: The first phase transition. The algorithm finds two segments corresponding to the tree and the rest of the scene. The critical value of a 2 for which this transition happens is consistent with the theoretical prediction. the scene, and objects move with different velocities due to parallax. The phase transitions correspond to different perceptual organizations of the scene - first the tree is segmented from the background, then branches are split from the tree, and finally the background splits into the flower bed and the house. 3 Discussion Estimating the number of components in a Gaussian mixture is a well researched topic in statistics and data mining [7]. Most approaches involve some tradeoff parameter to balance the benefit of an additional component versus the added complexity [2]. Here we have shown how this tradeoff parameter can be implicitly specified by the assumed level of noise in the image sequence. While making an assumption regarding a may seem rather arbitrary in the abstract Gaussian mixture problem, we find it quite reasonable in the context of motion estimation, where the noise is often a property of the imaging system, not of the underlying surfaces. Furthermore, as the phase diagram in figure 4 shows, a wide range of assumed a values will give similar answer, suggesting that an exact specification of a is not needed. In current work we are exploring the use of weak priors on a as well as comparing our method to those based on cross validation [7] . • ,' ,h ~3 t > ' , • > Figure 7: The second phase transition. The algorithm finds three segments - branches which are closer to the camera than the rest of the tree are segmented from it. Since the segmentation is based solely on motion, portions of the flower bed that move consistently with the branches are erroneously grouped with them. 856 Y. Weiss Figure 8: The third phase transition. The algorithm finds four segments - the Bower bed and the house are segregated. Our analytical and simulation results show that an assumption of the noise level in the sequence enables automatic determination of the number of moving objects using well understood maximum likelihood techniques. Furthermore, for a given scene, varying the assumed noise level gives rise to different perceptually meaningful segmentations. Thus mixture models may be a first step towards a well founded probabilistic framework for perceptual organization. Acknowledgments I thank D. Fleet, E. Adelson, J. Tenenbaum and G. Hinton for stimulating discussions. Supported by a training grant from NIG MS. References [1] Serge Ayer and Harpreet S. Sawhney. Layered representation of motion video using robust maximum likelihood estimation of mixture models and MDL encoding. In Proc. Int'l Con/. Comput. Vision, pages 777-784, 1995. [2] J. Buhmann. Data clustering and learning. In M. Arbib, editor, Handbook of Brain Theory and Neural Networks. MIT Press, 1995. [3] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. J. R. Statist. Soc. B, 39:1-38, 1977. [4] R. Durbin, R. Szeliski, and A. Yuille. An analysis of the elastic net approach to the travelling salesman problem. Neural Computation, 1(3):348-358, 1989. [5] A. Jepson and M. J. Black. Mixture models for optical Bow computation. In Proc. IEEE Con/. Comput. Vision Pattern Recog., pages 760-761, New York, June 1993. [6] K. Rose, F. Gurewitz, and G. Fox. Statistical mechanics and phase transitions in clustering. Physical Review Letters, 65:945-948, 1990. [7] P. Smyth. Clustering using monte-carlo cross-validation. In KDD-96, pages 126-133, 1996. [8] J. B. Tenenbaum and E. V. Todorov. Factorial learning by clustering features. In G. Tesauro, D.S. Touretzky, and K. Leen, editors, Advances in Neural Information Processing Systems 7, 1995. [9] J. Y. A. Wang and E. H. Adelson. Representing moving images with layers. IEEE Transactions on Image Processing Special Issue: Image Sequence Compression, 3(5):625-638, September 1994. [10] Y. Weiss and E. H. Adelson. A unified mixture framework for motion segmentation: incorporating spatial coherence and estimating the number of models. In Proc. IEEE Con/. Comput. Vision Pattern Recog., pages 321-326, 1996.
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Regression with Input-dependent Noise: A Gaussian Process Treatment Paul W. Goldberg Department of Computer Science University of Warwick Coventry, CV 4 7 AL, UK pvgGdcs.varvick.ac.uk Christopher K.I. Williams Neural Computing Research Group Aston University Birmingham B4 7ET, UK c.k.i.villiamsGaston.ac.uk Christopher M. Bishop Microsoft Research St. George House 1 Guildhall Street Cambridge, CB2 3NH, UK cmbishopOmicrosoft.com Abstract Gaussian processes provide natural non-parametric prior distributions over regression functions. In this paper we consider regression problems where there is noise on the output, and the variance of the noise depends on the inputs. If we assume that the noise is a smooth function of the inputs, then it is natural to model the noise variance using a second Gaussian process, in addition to the Gaussian process governing the noise-free output value. We show that prior uncertainty about the parameters controlling both processes can be handled and that the posterior distribution of the noise rate can be sampled from using Markov chain Monte Carlo methods. Our results on a synthetic data set give a posterior noise variance that well-approximates the true variance. 1 Background and Motivation A very natural approach to regression problems is to place a prior on the kinds of function that we expect, and then after observing the data to obtain a posterior. The prior can be obtained by placing prior distributions on the weights in a neural 494 P. W Goldberg, C. K. L Williams and C. M. Bishop network, although we would argue that it is perhaps more natural to place priors directly over functions. One tractable way of doing this is to create a Gaussian process prior. This has the advantage that predictions can be made from the posterior using only matrix multiplication for fixed hyperparameters and a global noise level. In contrast, for neural networks (with fixed hyperparameters and a global noise level) it is necessary to use approximations or Markov chain Monte Carlo (MCMC) methods. Rasmussen (1996) has demonstrated that predictions obtained with Gaussian processes are as good as or better than other state-of-the art predictors. In much of the work on regression problems in the statistical and neural networks literatures, it is assumed that there is a global noise level, independent of the input vector x. The book by Bishop (1995) and the papers by Bishop (1994), MacKay (1995) and Bishop and Qazaz (1997) have examined the case of input-dependent noise for parametric models such as neural networks. (Such models are said to heteroscedastic in the statistics literature.) In this paper we develop the treatment of an input-dependent noise model for Gaussian process regression, where the noise is assumed to be Gaussian but its variance depends on x. As the noise level is nonnegative we place a Gaussian process prior on the log noise level. Thus there are two Gaussian processes involved in making predictions: the usual Gaussian process for predicting the function values (the y-process), and another one (the z-process) for predicting the log noise level. Below we present a Markov chain Monte Carlo method for carrying out inference with this model and demonstrate its performance on a test problem. 1.1 Gaussian processes A stochastic process is a collection of random variables {Y(x)lx E X} indexed by a set X. Often X will be a space such as 'R,d for some dimension d, although it could be more general. The stochastic process is specified by giving the probability distribution for every finite subset of variables Y(Xl), ... , Y(Xk) in a consistent manner. A Gaussian process is a stochastic process which can be fully specified by its mean function J.L(x) = E[Y(x)] and its covariance function Cp(x,x') = E[(Y(x)-J.L(x»)(Y(x')-J.L(x'»]; any finite set of points will have a joint multivariate Gaussian distribution. Below we consider Gaussian processes which have J.L(x) == O. This assumes that any known offset or trend in the data has been. removed. A non-zero I' (x ) is easily incorporated into the framework at the expense of extra notational complexity. A covariance junction is used to define a Gaussian process; it is a parametrised function from pairs of x-values to their covariance. The form of the covariance function that we shall use for the prior over functions is given by Cy(x(i),xU» =vyexp (-~ tWYl(x~i) _x~j»2) + Jy8(i,j) (1) 1=1 where vy specifies the overall y-scale and W;:/2 is the length-scale associated with the lth coordinate. Jy is a "jitter" term (as discussed by Neal, 1997), which is added to prevent ill-conditioning of the covariance matrix of the outputs. Jy is a typically given a small value, e.g. 10-6 . For the prediction problem we are given n data points 1) = ((Xl,t1),(X2,t2), Input-dependent Noise: A Gaussian Process Treatment 495 ... , (xn, tn»), where ti is the observed output value at Xi. The t's are assumed to have been generated from the true y-values by adding independent Gaussian noise whose variance is x-dependent. Let the noise variance at the n data points be r = (r(xl),r(x2), ... ,r(xn)). Given the assumption of a Gaussian process prior over functions, it is a standard result (e.g. Whittle, 1963) that the predictive distribution P(t*lx*) corresponding to a new input x* is t* "'" N(t(X*),0'2(X*)), where i(x*) k~(x*)(Ky + KN)-lt (2) 0'2(X*) Cy(x*, x*) + r(x*) k~(x*)(Ky + KN )-lky(x*) (3) where the noise-free covariance matrix K y satisfies [K Y] ij = Cy (x i, X j ), and ky(x*) = (Cy(x*,xd, ... ,Cy(x*,xn»T, KN = diag(r) and t = (tb ... ,tn)T, and V0'2(X*) gives the "error bars" or confidence interval of the prediction. In this paper we do not specify a functional form for the noise level r(x) but we do place a prior over it. An independent Gaussian process (the z-process) is defined to be the log of the noise level. Its values at the training data points are denoted by z = (zl, . .. ,zn),sothatr = (exp(zl), ... ,exp(zn». The priorforz has a covariance function CZ(X(i), xU» similar to that given in equation 1, although the parameters vz and the WZI'S can be chosen to be different to those for the y-process. We also add the jitter term Jz t5(i,j) to the covariance function for Z, where Jz is given the value 10-2 • This value is larger than usual, for technical reasons discussed later. We use a zero-mean process for z which carries a prior assumption that the average noise rate is approximately 1 (being e to the power of components of z). This is suitable for the experiment described in section 3. In general it is easy to add an offset to the z-process to shift the prior noise rate. 2 An input-dependent noise process We discuss, in turn, sampling the noise rates and making predictions with fixed values of the parameters that control both processes, and sampling from the posterior on these parameters. 2.1 Sampling the Noise Rates The predictive distribution for t*, the output at a point x*, is P(t*lt) = f P(t*lt,r(z»P(zlt)dz. Given a z vector, the prediction P(t*lt,r(z» is Gaussian with mean and variance given by equations 2 and 3, but P(zlt) is difficult to handle analytically, so we use a Monte Carlo approximation to the integral. Given a representative sample {Zb ... ' Zk} of log noise rate vectors we can approximate the integral by the sum i E j P(t*lt,r(zj». We wish to sample from the distribution P(zlt). As this is quite difficult, we sample instead from P(y, zit); a sample for P(zlt) can then be obtained by ignoring the y values. This is a similar approach to that taken by Neal (1997) in the case of Gaussian processes used for classification or robust regression with t-distributed noise. We find that P(y, zit) oc P(tly, r(z»P(y)P(z). (4) We use Gibbs sampling to sample from P(y, zit) by alternately sampling from P(zly, t) and P(ylz, t). Intuitively were are alternating the "fitting" of the curve (or 496 P. W. Goldberg, C. K. 1. Williams and C. M Bishop y-process) with "fitting" the noise level (z-process). These two steps are discussed in turn . • Sampling from P(ylt, z) For y we have that P(ylt, z) ex P(tly, r(z»P(y) (5) where n 1 ( (ti - Yi)2 ) P(tly, r(z» = TI (21l'Ti)l/2 exp 2Ti . (6) Equation (6) can also be written as P(tly,r(z» '" N(t,KN)' Thus P(ylt,z) is a multivariate Gaussian with mean (Kyl + Ki/)-l K;/t and covariance matrix (Kyl + KN1)-1 which can be sampled by standard methods . • Sampling from P(zlt,y) For fixed y and t we obtain P(zly, t) ex P(tly, z)P(z). (7) The form of equation 6 means that it is not easy to sample z as a vector. Instead we can sample its components separately, which is a standard Gibbs sampling algorithm. Let Zi denote the ith component of z and let Z-i denote the remaining components. Then (8) P(Zilz-i) is the distribution of Zi conditioned on the values of Z-i' The computation of P(zilz-i) is very similar to that described by equations (2) and (3), except that Cy ( " .) is replaced by C z ( " .) and there is no noise so that T (.) will be identically zero. We sample from P(zilz-i' y, t) using rejection sampling. We first sample from P(zilz-i), and then reject according to the term exp{ -Zi/2 - Hti - Yi)2 exp( -Zi)} (the likelihood of local noise rate Zi), which can be rescaled to have a maximum value of lover Zi. Note that it is not necessary to perform a separate matrix inversion for each i when computing the P(zilz-i) terms; the required matrices can be computed efficiently from the inverse of K z. We find that the average rejection rate is approximately two-thirds, which makes the method we currently use reasonably efficient. Note that it is also possible to incorporate the term exp( -Zi/2) from the likelihood into the mean of the Gaussian P(zilz-i) to reduce the rejection rate. As an alternative approach, it is possible to carry out Gibbs sampling for P(zilz-i' t) without explicitly representing y, using the fact that 10gP(tlz) = -~logIKI !tT K-1t + canst, where K = K y + K N . We have implemented this and found similar results to those obtained using sampling of the y's. However, explicitly representing the y-process is useful when adapting the parameters, as described in section 2.3. Input-dependent Noise: A Gaussian Process Treatment 497 2.2 Making predictions So far we have explained how to obtain a sample from P(zlt). To make predictions we use P(t*lt) ~ ~ l: P(t*lt, r(zj)). (9) j However, P(t*lt,r(zj)) is not immediately available, as z*, the noise level at x* is unknown. In fact (10) P(z*IZj, t) is simply a Gaussian distribution for z* conditioned on Zj, and is obtained in a similar way to P(zilz-i). As P(t*lz*, t, r(zj)) is a Gaussian distribution as given by equations (2) and (3), P(t*\t, r(z j)) is an infinite mixture of Gaussians with weights P(z*IZj). Note, however, that each ofthese components has the same mean i(x*) as given by equation (2), but a different variance. We approximate P(t*lt, r(zj)) by taking s = 10 samples of P(z*lzj) and thus obtain a mixture of s Gaussians as the approximating distribution. The approximation for P(t*lt) is then obtained by averaging these s-component mixtures over the k samples Z1> ••• , Zk to obtain an sk-component mixture of Gaussians. 2.3 Adapting the parameters Above we have described how to obtain a sample from the posterior distribution P(z\t) and to use this to make predictions, based on the assumption that the parameters Oy (Le. Vy,Jy,WYl, . .. ,WYd) and Oz (Le. vz,JZ,WZl, ... ,WZd) have been set to the correct values. In practice we are unlikely to know what these settings should be, and so introduce a hierarchical model, as shown in Figure l. This graphical model shows that the joint probability distribution decomposes as P(Oy,OZ, y, z, t) = P(Oy)P(Oz)P(yIOy )P(z\Oz)P(t\y, z). Our goal now becomes to obtain a sample from the posterior P(Oy,Oz,y,zlt), which can be used for making predictions as before. (Again, the y samples are not needed for making predictions, but they will turn out to be useful for sampling Oy .) Sampling from the joint posterior can be achieved by interleaving updates of Oy and Oz with y and Z updates. Gibbs sampling for Oy and Oz is not feasible as these parameters are buried deeply in the K y and K N matrices, so we use the Metropolis algorithm for their updates. As usual, we consider moving from our current state 0 = (Oy,Oz) to a new state 0 using a proposal distribution J(O,O). In practice we take J to be an isotropic Gaussian centered on 0°. Denote the ratio of P(Oy)P(Oz)P(yIOy)P(z\Oz) in states 9 and 0 by r. Then the proposed state 0 is accepted with probability min{r, 1}. It would also be possible to use more sophisticated MCMC algorithms such as the Hybrid Monte Carlo algorithm which uses derivative information, as discussed in Neal (1997). 3 Results We have tested the method on a one-dimensional synthetic problem. 60 data points 498 P. W. Goldberg, C. K. l Williams and C. M Bishop Figure 1: The hierarchical model including parameters. were generated from the function y = 2 sin(271"x) on [0, 1] by adding independent Gaussian noise. This noise has a standard deviation that increases linearly from 0.5 at x = 0 to 1.5 at x = 1. The function and the training data set are illustrated in Figure 2(a). As the parameters are non-negative quantities, we actually compute with their log values. logvy, logvz, logwy and log Wz were given N(O, 1) prior distributions. The jitter values were fixed at Jy = 10-6 and J z = 10-2 • The relatively large value for J z assists the convergence of the Gibbs sampling, since it is responsible for most of the variance of the conditional distribution P(Zi/Z-i}. The broadening of this distribution leads to samples whose likelihoods are more variable, allowing the likelihood term (used for rejection) to be more influential. In our simulations, on each iteration we made three Metropolis updates for the parameters, along with sampling from all of the y and Z variables. The Metropolis proposal distribution was an isotropic Gaussian with variance 0.01. We ran for 3000 iterations, and discarded the first one-third of iterations as "burn-in", after which plots of each of the parameters seemed to have settled down. The parameters and Z values were stored every 100 iterations. In Figure 2(b) the average standard deviation of the inferred noise has been plotted, along with with two standard deviation error-bars. Notice how the standard deviation increases from left to right, in close agreement with the data generator. Studying the posterior distributions of the parameters, we find that the ylength scale A y d;j (wy) -1/2 is well localized around 0.22 ± 0.1, in good agreement with the wavelength of the sinusoidal generator. (For the covariance function in equation 1, the expected number of zero crossings per unit length is 1/7I"Ay.) (WZ)-1/2 is less tightly constrained, which makes sense as it corresponds to a longer wavelength process, and with only a short segment of data available there is still considerable posterior uncertainty. 4 Conclusions We have introduced a natural non-parametric prior on variable noise rates, and given an effective method of sampling the posterior distribution, using a MCMC Input-dependent Noise: A Gaussian Process Treatment 499 .' " .. .' .. - 2 o. °O~~ O I~~ 0 2--0~'~O~'~ O ~'~"--O~7~O~'~"~ (a) (b) Figure 2: (a) shows the training set (crosses); the solid line depicts the x-dependent mean of the output. (b) The solid curve shows the average standard deviation of the noise process, with two standard deviation error bars plotted as dashed lines. The dotted line indicates the true standard deviation of the data generator. method. When applied to a data set with varying noise, the posterior noise rates obtained are well-matched to the known structure. We are currently experimenting with the method on some more challenging real-world problems. Acknowledgements This work was carried out at Aston University under EPSRC Grant Ref. GR/K 51792 Validation and Verification of Neural Network Systems. References [1] C.M. Bishop (1994). Mixture Density Networks. Technical report NCRG/94/001, Neural Computing Research Group, Aston University, Birmingham, UK. [2] C.M. Bishop (1995). Neural Networks for Pattern Recognition. Oxford University Press. [3] C.M. Bishop and C. Qazaz (1997). Regression with Input-dependent Noise: A Bayesian Treatment. In M. C. Mozer, M. I. Jordan and T. Petsche (Eds) Advances in Neural Information Processing Systems 9 Cambridge MA MIT Press. [4] D. J. C. MacKay (1995). Probabilistic networks: new models and new methods. In F. Fogelman-Soulie and P. Gallinari (Eds), Proceedings ICANN'95 International Conference on Neural Networks, pp. 331-337. Paris, EC2 & Cie. [5] R. Neal (1997). Monte Carlo Implementation of Gaussian Process Models for Bayesian Regression and Classification. Technical Report 9702, Department of Statistics, University of Toronto. Available from http://vvv.cs.toronto.edurradford/. [6] C.E. Rasmussen (1996). Evaluation of Gaussian Processes and Other Methods for Nonlinear Regression. PhD thesis, Department of Computer Science, University of Toronto. Available from http://vwv . cs .utoronto. carcarl/. [7] C.K.I. Williams and C.E. Rasmussen (1996). Gaussian Processes for Regression. In D. S. Touretzky, M. C. Mozer and M. E. Hasselmo Advances in Neural Information Processing Systems 8 pp. 514-520, Cambridge MA MIT Press. [8] P. Whittle (1963). Prediction and regulation by linear least-square methods. English Universities Press.
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A Generic Approach for Identification of Event Related Brain Potentials via a Competitive Neural Network Structure Daniel H. Lange Department of Electrical Engineering Technion - liT Haifa 32000 Israel e-mail: lange@turbo.technion.ac.il Hillel Pratt Evoked Potential Laboratory Technion - liT Haifa 32000 Israel e-mail: hillel@tx.technion.ac.il Hava T. Siegelmann Department of Industrial Engineering Technion - liT Haifa 32000 Israel e-mail: iehava@ie.technion.ac.il Gideon F. Inbar Department of Electrical Engineering Technion - liT Haifa 32000 Israel e-mail: inbar@ee.technion.ac.il Abstract We present a novel generic approach to the problem of Event Related Potential identification and classification, based on a competitive N eural Net architecture. The network weights converge to the embedded signal patterns, resulting in the formation of a matched filter bank. The network performance is analyzed via a simulation study, exploring identification robustness under low SNR conditions and compared to the expected performance from an information theoretic perspective. The classifier is applied to real event-related potential data recorded during a classic odd-ball type paradigm; for the first time, withinsession variable signal patterns are automatically identified, dismissing the strong and limiting requirement of a-priori stimulus-related selective grouping of the recorded data. 902 D. H. Lange, H. T. Siegelmann, H. Pratt and G. F. Inbar 1 INTRODUCTION 1.1 EVENT RELATED POTENTIALS Ever since Hans Berger's discovery that the electrical activity of the brain can be measured and recorded via surface electrodes mounted on the scalp, there has been major interest in the relationship between such recordings and brain function. The first recordings were concerned with the spontaneous electrical activity of the brain, appearing in the form of rhythmic voltage oscillations, which later received the term electroencephalogram or EEG. Subsequently, more recent research has concentrated on time-locked brain activity, related to specific events, external or internal to the subject. This time-locked activity, referred to also as Event Related Potentials (ERP's), is regarded as a manifestation of brain processes related to preparation for or in response to discrete events meaningful to the subject. The ongoing electrical activity of the brain, the EEG, is comprised of relatively slow fluctuations, in the range of 0.1 - 100 Hz, with magnitudes of 10 - 100 uV. ERP's are characterized by overlapping spectra with the EEG, but with significantly lower magnitudes of 0.1 - 10 uV. The unfavorable Signal to Noise Ratio (SNR) requires filtering of the raw signals to enable analysis of the time-locked signals. The common method used for this purpose is signal averaging, synchronized to repeated occurrences of a specific event. Averaging-based techniques assume a deterministic signal within the averaged session, and thus signal variability can not be modeled unless a-priori stimulus- or response-based categorization is available; it is the purpose of this paper to provide an alternative working method to enhance conventional averaging techniques, and thus facilitating identification and analysis of variable brain responses. 1.2 COMPETITIVE LEARNING Competitive learning is a well-known branch of the general unsupervised learning theme. The elementary principles of competitive learning are (Rumelhart & Zipser, 1985): (a) start with a set of units that are all the same except for some randomly distributed parameter which makes each of them respond slightly differently to a set of input patterns, (b) limit the strength of each unit, and (c) allow the units to compete in some way for the right to respond to a given subset of inputs. Applying these three principles yields a learning paradigm where individual units learn to specialize on sets of similar patterns and thus become feature detectors. Competitive learning is a mechanism well-suited for regularity detection (H aykin , 1994), where there is a popUlation of input patterns each of which is presented with some probability. The detector is supposed to discover statistically salient features of the input population, without a-priori categorization into which the patterns are to be classified. Thus the detector needs to develop its own featural representation of the population of input patterns capturing its most salient features. 1.3 PROBLEM STATEMENT The complicated, generally unknown relationships between the stimulus and its associated brain response, and the extremely low SNR of the brain responses which are practically masked by the background brain activity, make the choice of a self organizing structure for post-stimulus epoch analysis most appropriate. The competitive network, having the property that its weights converge to the actual embedded signal patterns while inherently averaging out the additive background EEG, is thus an evident choice. A Generic Approach for Identification of Event Related Brain Potentials 903 2 THE COMPETITIVE NEURAL NETWORK 2.1 THEORY The common architecture of a competitive learning system appears in Fig. 1. The system consists of a set of hierarchically layered neurons in which each layer is connected via excitatory connections with the following layer. Within a layer, the neurons are divided into sets of inhibitory clusters in which all neurons within a cluster inhibit all other neurons in the cluster, which results in a competition among the neurons to respond to the pattern appearing on the previous layer. Let Wji denote the synaptic weight connecting input node i to neuron j. A neuron learns by shifting synaptic weights from its inactive to active input nodes. If a neuron does not respond to some input pattern, no learning occurs in that neuron. When a single neuron wins the competition, each of its input nodes gives up some proportion of its synaptic weight, which is distributed equally among the active input nodes, fulfilling: 2:i Wji = 1. According to the standard competitive learning rule, for a winning neuron to an input vector Xi, the change llWji is defined by: llWji = 7J(Xi Wji), where 7J is a learning rate coefficient. The effect of this rule is that the synaptic weights of a winning neuron are shifted towards the input pattern; thus assuming zero-mean additive background EEG, once converged, the network operates as a matched filter bank classifier. 2.2 MATCHED FILTERING From an information theoretic perspective, once the network has converged, our classification problem coincides with the general detection problem of known signals in additive noise. For simplicity, we shall limit the discussion to the binary decision problem of a known signal in additive white Gaussian noise, expandable to the M-ary detection in colored noise (Van Trees, 1968). Adopting the common assumption of EEG and ERP additivity (Gevins, 1984), and distinct signal categories, the competitive NN weights inherently converge to the general signal patterns embedded within the background brain activity; therefore the converged network operates as a matched filter bank. Assuming the simplest binary decision problem, the received signal under one hypothesis consists of a completely known signal, VEs(t), representing the EP, corrupted by an additive zero-mean Gaussian noise w(t) with variance (72; the received signal under the other hypothesis consists of the noise w(t) alone. Thus: Ho: ret) = wet), 0 $ t $ T HI: ret) = ../Es(t) + wet), 0 $ t $ T For convenience we assume that JoT s2(t)dt = 1, so that E represents the signal energy. The problem is to observe r(t) over the interval [0, T] and decide whether Ho or Hl is true. It can be shown that the matched filter is the optimal detector, its impulse response being simply the signal reversed in time and shifted: her) = s(T - r) (1) Assuming that there is no a-priori knowledge of the probability of signal presence, the total probability of error depends only on the SNR and is given by (Van Trees, 1968): 1 fOO 2 Pe = r.c exp( - ~ )dz V 21r IJi: 2 V-;;2 (2) Fig. 2 presents the probability of true detection: (a) as a function of SNR, for minimized error probability, and (b) as a function of the probability of false detection. These 904 D. H. Lange, H. T. Siegelmann, H. Pratt and G. F. Inbar results are applicable to our detection problem assuming approximate Gaussian EEG characteristics (Gersch, 1970), or optimally by using a pre-whitening approach (Lange et. al., 1997). L llyer 1 Illhlb,"n ry C lul le rs Layer :! Inhibi to ry CJ u~ lcn. 1 •• lyel I I" pull lnils. • 0 00 0 0 00 0 0. 0 o · oooo ~ 0 . 0 0 . 0 • t t INPUT PATffiRN 1.xIllIlOry ("""nUnOIlS ).Kllillory ('Ollonec.uons Figure 1: The architecture of a competitive learning structure: learning takes place in hierarchically layered units, presented as filled (active) and empty (inactive) dots. Probability of True Detection with Minimum Error J~~r : ? : : I ~·~0----~3~ 0 ----~ 20~---~ '0~--~ 0 --~1~ 0--~2~ 0 --~3~ 0 --~40 SNR In dB Detection performance: SNA "" +20 • • ,0. 0, - 10 and -20 dB ~ F,~~~;;~::~====~~~~~~~~;:~ 9 1 }! 20dB ~O.B ~0 . 6 "0 ~04 ~ 0.2 £ 0.1 0.2 0 .3 0.4 05 O.B 0 .7 0 ,8 0 .9 Probability of False Detection Figure 2: Detection performance. Top: probability of detection as a function of the SNR. Bottom: detection characteristics. 2.3 NETWORK TRAINING AND CONVERGENCE Our net includes a 300-node input layer and a competitive layer consisting of singlelayered competing neurons. The network weights are initialized with random values and trained with the standard competitive learning rule, applied to the normalized input vectors: z · AWji = 77( ~ - Wji) (3) L.Ji Xi The training is applied to the winning neuron of each epoch, while increasing the bias of the frequently winning neuron to gradually reduce its chance of winning consecutively (eliminating the dead neuron effect (Freeman & Skapura, 1992)). Symmetrically, its bias is reduced with the winnings of other neurons. In order to evaluate the network performance, we explore its convergence by analyzing the learning process via the continuously adapting weights: pj(n) ~ J~!lWl. ; ; ~ 1,2, .•• ,C (4) where C represents the pre-defined number of categories. We define a set of classification confidence coefficients of the converged network: (5) Assuming existence of a null category, in which the measurements include only background noise (EEG), maxj{pj(N)} corresponds to the noise variance. Thus the values of r j, the confidence coefficients, ranging from 0 to 1 (random classification to completely separated categories), indicate the reliability of classification, which breaks down with the fall of SNR. Finally, it should be noted that an explicit statistical evaluation of the network convergence properties can be found in (Lange, 1997). A Generic Approachfor Identification of Event Related Brain Potentials 905 2.4 SIMULATION STUDY A simulation study was carried out to assess the performance of the competitive network classification system. A moving average (MA) process of order 8 (selected according to Akaike's condition applied to ongoing EEG (Gersch, 1970)), driven by a deterministic realization of a Gaussian white noise series, simulated the ongoing background activity x(n). An average of 40 single-trials from a cognitive odd-ball type experiment (to be explained in the Experimental Study), was used as the signal s(n). Then, five 100-trial ensembles were synthesized, to study the classification performance under variable SNR conditions. A sample realization and its constituents, at an SNR of ° dB, is shown in Fig. 3. The simulation included embedding the signal s(n) in the synthesized background activity x(n) at five SNR levels (-20,-10,0,+10, and +20 dB), and training the network with 750 sweeps (per SNR level). Fig. 4 shows the convergence patterns and classification confidences of the two neurons, where it can be seen that for SNR's lower than -10dB the classification confidence declines sharply. .. Templat., ~d NOIU de5hed RNllzallon dolted Figure 3: A sample single realization (dotted) and its constituents (signal solid, noise - dashed). SNR = 0 dB. SNR..od8 C 100 com.cMn-o g ito'z ... _--. __ ._----_ .. _---100 200 300 '0' .---"'''''"''"'''''''''''''''"''P''''-="----, SNR __ 20 dB oon""'_006 100 200 300 "2100 cotalu:lanoe..o ~7 i,o'z ..... ~- -~Il __ ..r..l...:.. ___ _ 10-0 100 200 300 i·j):~··-- I -20 - ,0 0 20 SNR Figure 4: Convergence patterns and classification confidence values for varying SNR levels. The classification results, tested on 100 input vectors, 50 of each category, for each SNR, are presented in the table below; due to the competitive scheme, Positives and False Negatives as well as Negatives and False Positives are complementary. These empirical results are in agreement with the analytical results presented in the above Matched Filtering sectioll. Table 1: Classification Results II Pos I Neg I FP I FN 3 EXPERIMENTAL STUDY 3.1 MOTIVATION An important task in ERP research is to identify effects related to cognitive processes triggered by meaningful versus non-relevant stimuli. A common procedure to study these effects is the classic odd-ball paradigm, where the subject is exposed to a random 906 D. H. Lange, H. T. Siegelmann, H. Pratt and G. F Inbar sequence of stimuli and is instructed to respond only to the task-relevant (Target) ones. Typically, the brain responses are extracted via selective averaging of the recorded data, ensembled according to the types of related stimuli. This method of analysis assumes that the brain responds equally to the members of each type of stimulus; however the validity of this assumption is unknown in this case where cognition itself is being studied. Using our proposed approach, a-priori grouping of the recorded data is not required, thus overcoming the above severe assumption on cognitive brain function. The results of applying our method are described below. 3.2 EXPERIMENTAL PARADIGM Cognitive event-related potential data was acquired during an odd-ball type paradigm from pz referenced to the mid-lower jaw, with a sample frequency of 250 Hz (Lange et. al., 1995). The subject was exposed to repeated visual stimuli, consisting of the digits '3' and '5', appearing on a PC screen. The subject was instructed to press a pushbutton upon the appearance of '5' - the Target stimulus, and ignore the appearances of the digit '3'. With odd-ball type paradigms, the Target stimulus is known to elicit a prominent positive component in the ongoing brain activity, related to the identification of a meaningful stimulus. This component has been labeled P300, indicating its polarity (positive) and timing of appearance (300 ms after stimulus presentation). The parameters of the P 300 component (latency and amplitude) are used by neurophysiologists to assess effects related to the relevance of stimulus and level of attention (Lange et. al., 1995). 3.3 IDENTIFICATION RESULTS The competitive network was trained with 80 input vectors, half of which were Target ERP's and the other half were Non Target. The network converged after approximately 300 iterations (per neuron), yielding a reasonable confidence coefficient of 0.7. A sample of two single-trial post-stimulus sweeps, of the Target and Non-Target averaged ERP templates and of the NN identified signal categories, are presented in Fig. 5. The convergence pattern is shown in Fig. 6. The automatic identification procedure has provided two signal categories, with almost perfect matches to the stimulus-related selective averaged signals. The obtained categorization confirms the usage of averaging methods for this classic experiment, and thus presents an important result in itself. 4 DISCUSSION AND CONCLUSION A generic system for identification and classification of single-trial ERP's was presented. The simulation study demonstrated the powerful capabilities of the competitive neural net in classifying the low amplitude signals embedded within the large background noise. The detection performance declined rapidly for SNR's lower than -10dB, which is in general agreement with the theoretical statistical results, where loss of significance in detection probability is evident for SNR's lower than -20dB. Empirically, high classification performance was maintained with SNR's of down to -10dB, yielding confidences in the order of 0.7 or higher. The experimental study presented an unsupervised identification and classification of the raw data into Target and Non-Target responses, dismissing the requirement of stimulus-related selective data grouping. The presented results indicate that the noisy brain responses may be identified and classified objectively in cases where relevance of A Generic Approachfor Identification of Event Related Brain Potentials 907 the stimuli is unknown or needs to be determined, e.g. in lie-detection scenarios (Lange & Inbar, 1996), and thus open new possibilities in ERP research. J;:s::1 ~F?'S2 ,:F?tl ,:F=l -20~ -20L~;:--_-;:-;;o.'-----:, .~g,~g o 05 I 0 06 , Soc Soc Figure 5: Top row: sample raw Target and Non- Target sweeps. Middle row: Target and Non- Target ERP templates. Bottom row: the NN categorized patterns. References SNR _ _ 6 dB '0' 10-40!;-----;:;;,--------; 200:;;----==""';;---=----;;:;;--~ ~ of~. Figure 6: Convergence pattern of the ERP categorization process; convergence is achieved after 300 iterations per neuron. [1] Freeman J.A. and Skapura D.M. Neural Network6: Algorithm6, Application6, and Programming Technique6: Addison-Wesley Publishing Company, USA, 1992. [2] Gersch W., "Spectral Analysis of EEG's by Autoregressive Decomposition of Time Series," Math. Bi06c., vol. 7, pp. 205-222, 1970. [3] Gevins A.S., "Analysis of the Electromagnetic Signals of the Human Brain: Milestones, Obstacles, and Goals," IEEE Tran6. Biomed. Eng., vol. BME-31, pp. 833-850, 1984. [4] Haykin S. Neural Network6: A Comprehen6ive Foundation. Macmillan College Publishing Company, Inc., USA, 1994. [5] Lange D. H. Modeling and E6timation of Tran6ient, Evoked Brain Potential6. D.Sc. dissertation, Techion - Israel Institute of Technology, 1997. [6] Lange D.H. and Inbar G.F., "Brain Wave Based Polygraphy," Proceeding6 of the IEEE EMBS96 - the 18th Annual International Conference of the IEEE Engineering on Medicine and Biology Society, Amsterdam, October 1996. [7] Lange D.H., Pratt H. and Inbar G.F., "Modeling and Estimation of Single Evoked Brain Potential Components", IEEE. Tran6. Biomed. Eng., vol. BME-44, pp. 791-799, 1997. [8] Lange D.H., Pratt H., and Inbar G.F., "Segmented Matched Filtering of Single Event Related Evoked Potentials," IEEE. Tran6. Biomed. Eng., vol. BME-42, pp. 317-321, 1995. [9] Rumelhart D.E. and Zipser D., "Feature Discovery by Competitive Learning," Cognitive Science, vol. 9, pp. 75-112, 1985. [10] Van Trees H.L. Detection, E6timation, and Modulation Theory: Part 1: John Wiley and Sons, Inc., USA, 1968.
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An application of Reversible-J ump MCMC to multivariate spherical Gaussian mixtures Alan D. Marrs Signal & Information Processing Dept. Defence Evaluation & Research Agency Gt. Malvern, UK WR14 3PS marrs@signal.dra.hmg.gb Abstract Applications of Gaussian mixture models occur frequently in the fields of statistics and artificial neural networks. One of the key issues arising from any mixture model application is how to estimate the optimum number of mixture components. This paper extends the Reversible-Jump Markov Chain Monte Carlo (MCMC) algorithm to the case of multivariate spherical Gaussian mixtures using a hierarchical prior model. Using this method the number of mixture components is no longer fixed but becomes a parameter of the model which we shall estimate. The Reversible-Jump MCMC algorithm is capable of moving between parameter subspaces which correspond to models with different numbers of mixture components. As a result a sample from the full joint distribution of all unknown model parameters is generated. The technique is then demonstrated on a simulated example and a well known vowel dataset. 1 Introduction Applications of Gaussian mixture models regularly appear in the neural networks literature. One of their most common roles in the field of neural networks, is in the placement of centres in a radial basis function network. In this case the basis functions are used to model the distribution of input data (Xi == [Xl, X2, .•• , Xd]T, (i = l,n)), and the problem is one of mixture density estimation. 578 A D. Marrs k p(Xi ) = L 7Ijp(Xi I9j), (1) j=1 where k is the number of mixture components, 7rj the weight or mixing proportion for component j and 8 j the component parameters (mean & variance in this case). The mixture components represent the basis functions of the neural network and their parameters (centres & widths) may be estimated using the expectationmaximisation (EM) algorithm. One of the key issues arising in the use of mixture models is how to estimate the number of components. This is a model selection problem: the problem of choosing the 'correct' number of components for a mixture model. This may be thought of as one of comparing two (or more) mixture models with different components, and choosing the model that is 'best' based upon some criterion. For example, we might compare a two component model to one with a single component. (2) This may appear to be a case of testing of nested hypotheses. However, it has been noted [5] that the standard frequentist hypothesis testing theory (generalised likelihood ratio test) does not apply to this problem because the desired regularity conditions do not hold. In addition, if the models being tested have 2 and 3 components respectively, they are not strictly nested. For example, we could equate any pair of components in the three component model to the components in the two component model, yet how do we choose which component to 'leave out'? 2 Bayesian approach to Gaussian mixture models A full Bayesian analysis treats the number of mixture components as one of the parameters of the model for which we wish to find the conditional distribution. In this case we would represent the joint distribution as a hierarchical model where we may introduce prior distributions for the model parameters, ie. p(k, 7r, Z, 9, X) = p(k)p(7rlk)p(zl7r, k)p(9Iz, 7r, k)p(XI9, z, 7r, k), (3) where7r = (7rj)J=1, 9 = (9j )J=1 and z = (Zi)f::l are allocation variables introduced by treating mixture estimation as a hidden data problem with Zi allocating the ith observation to a particular component. A simplified version of this model can be derived by imposing further conditional independencies, leading to the following expression for the joint distribution p(k, 7r, Z, 9, X) = p(k)p(7rlk)p(zl7r, k)p(9Ik)p(XI9, z). (4) In addition, we add an extra layer to the hierarchy representing priors on the model parameters giving the final form for the joint distribution peA, 6, T}, k, 7r, Z, 9, X) = p(A)p(6)p(T})p(kIA)p(7rlk, 6)p(zl7r, k) x p(9Ik, T})p(XI9, z). (5) Until recently a full Bayesian analysis has been mathematically intractable. Model comparison was carried out by conducting an extensive search over all possible Reversible-Jump MCMC for Multivariate Spherical Gaussian Mixtures 579 model orders comparing Bayes factors for all possible pairs of models. What we really desire is a method which will estimate the model order along with the other model parameters. Two such methods based upon Markov Chain Monte Carlo (MCMC) techniques are reversible-jump MCMC [2] and jump-diffusion [3]. In the following sections, we extend the reversible-jump MCMC technique to multivariate spherical Gaussian mixture models. Results are then shown for a simulated example and an example using the Peterson-Barney vowel data. 3 Reversible-jump MCMC algorithm Following [4) we define the priors for our hierarchical model and derive a set of 5 move types for the reversible jump MCMC sampling scheme. To simplify some of the MCMC steps we choose a prior model where the prior on the weights is Dirichlet and the prior model for IJ.j = [JLji' ... ,JLjclV and U;2 is that they are drawn independently with normal and gamma priors, (6) where for the purposes of this study we follow[4] and define the hyper-parameters thus: 6 = 1.0; 'TJ is set to be the mean of the data; A is the diagonal precision matrix for the prior on IJ.j with components aj which are taken to be liT] where Tj is the data range in dimension j; a = 2.0 and (3 is some small multiple of liT;' The moves then consist of: I: updating the weights; II: updating the parameters (IJ., u); III: updating the allocation; IV: updating the hyper-parameters; V: splitting one component into two, or combining two into one. The first 4 moves are relatively simple to define, since the conjugate nature of the priors leads to relatively simple forms for the full conditional distribution of the desired parameter. Thus the first 4 moves are Gibbs sampling moves and the full conditional distributions for the weights 1rj, means Jij, variances Uj and allocation variables Zi are given by: (7) where nk is the number of observations allocated to component k; d -2 II njXimUj + am'f/m -2 -1 p(ltjl .. ·) = P(JLjml .. ·) : p(JLj .. '!-.. ) '" N( -2 ,(njuj + am) ), m=1 (njuj + am) (8) where we recognise that IJ.j is an d dimensional vector with components JL;m (m = 1, d), 'f/m are the components of the Itj prior mean and am represent the diagonal components of A. n -2 1 ~ p(uj \ ... ) == r(lI + nj - 1, '2 Li=I :Zi;=l (9) and (10) 580 A. D. Marrs The final move involves splitting/combining model components. The main criteria which need to be met when designing these moves are that they are irreducible, aperiodic, form a reversible pair and satisfy detailed balance [1]. The MCMC step for this move takes the form of a Metropolis-Hastings step where a move from state y to state y' is proposed, with 1r(Y) the target probability distribution and qm(Y, Y') the proposal distribution for the move m. The resulting move is then accepted with probability am _ . {I 1r(Y')qm(y/,y)} am - mtn , () ( ') . 1r Y qm y, Y (11) In the case of a move from state Y to a state y' which lies in a higher dimensional space, the move may be implemented by drawing a vector of continuous random variables u, independent of y. The new state y' is then set using an invertible deterministic function of x and u. It can be shown [2] that the acceptance probability is then given by . { 1r(y')Tm{y') 8y' } am=mm 1'1r(y)Tm(y)q{u)1 8(y,u)1 , (12) where Tm(Y) is the probability of choosing move type m when in state y, and q(u) is the density function of u. The initial application of the reversible jump MCMC technique to normal mixtures [4J was limited to the univariate case. This yielded relatively simple expressions for the split/combine moves, and, most importantly, the determinant ofthe Jacobian of the tra~formation from a model with k components to one with k + 1 components was simple to derive. In the more general case of multivariate normal models care must be taken in prescribing move transformations. A complicated transformation will lead to problems when the !Jacobian I for a d-dimensional model is required. For multivariate spherical Gaussian models, we randomly choose a model component from the current k component model. The decision is then made to split or combine with one of its neighbours with probability P'k and PCIr respectively (where PCk = 1-Pile)' If the choice is to combine the component, we label the chosen component Zl, and choose Z2 to be a neighbouring component i with probability Q( l/T; where Tj is the distance from the component Zl. The new component resulting from the combination of Zl and Z2 is labelled Zc and its parameters are calculated from: (13) If the decision is to split, the chosen component is labelled Zc and it is used to define two new model components Zl and Z2 with weights and parameters conforming to (13). In making this transformation there are 2 + d degrees of freedom, so we need to generate 2 + d random numbers to enable the specification of the new component parameters. The random numbers are denoted u}, U2 = [U211 ... , u2dlT and U3. All are drawn from Beta{2,2) distributions while the components of U2 each have probability 0.5 of being negative. The split transformation is then defined by: Reversible-Jwnp MCMC for Multivariate Spherical Gaussian Mixtures 581 2 (1 ) 2 7r ZI , U Z2 = - U3 U Zc -. 7r Z2 (14) Once the new components have been defined it is necessary to evaluate the probability of choosing to combine component ZI with component Z2 in this new model. Having proposed the split/combine move all that remains is to calculate the Metropolis-Hastings acceptance probability (t, where (t = min(I, R) for the split move and (t = min(I, 1/ R) for the combine move. Where in the case of a split move from a model with k components to one with k+ 1 components, or a combine move from k + 1 to k, R is given by: n ~ p(X, le,e) n~ p(X,le,e) R = ._l:·ij-·hn p~~I,:~e,:;2 X ;-I:·,,-.c o-l+nl 0-I+n2 11' '" 1r "2 'II"!c 1+nl +n2 B(6,k6) x n~;::1 J (~;) exp ( -~am ((/LZlm -11m)2 + (/LZ2m - 11m? - (JLzcm - 11m)) ) X &(c7(';~2) (a-I) exp ( -f3(u;..2 +u~2 -u;:2)) X p:c::;;oc (g2,2(Ut)gl,1 (U3) n;=1 g2,2(U2,)) X 'II" c7d+1 ·c ·c (15) (2«I-uI)uI)(d+ 1)/2 J(I-u3)u3) , where g2,20 denotes a Beta(2,2) density function. The first line on the R.H.S is due to the ratio of likelihoods for those observations assigned to the components in question, the subsequent three lines are due to the prior ratios, the fifth line is due to the the proposal ratio and the last line due to the I Jacobian I of the transformation. The term Palloe represents a combination of the probability of obtaining the current allocation of data to the components in question and the probability of choosing to combine components Zl and Z2. 4 Results To assess this approach to the estimation of multivariate spherical Gaussian mixture models, we firstly consider a toy problem where 1000 bivariate samples were generated from a known 20 component mixture model. This is followed by an analysis of the Peterson-Barney vowel data set comprising 780 samples of the measured amplitUde of four formant frequencies for 10 utterances. For this mixture estimation example, we ignore the class labels and consider the straight forward density estimation problem. 4.1 Simulated data The resulting reversible-jump MCMC chain of model order can be seen in figure 1, along with the resulting histogram (after rejecting the first 2000 MCMC sampies). The histogram shows that the maximum a posteriori value for model order is 17. The MAP estimate of model parameters was obtained by averaging all the 17 component model samples, the estimated model is shown in figure 2 alongside the original generating model. The results are rather encouraging given the large number of model components and the relatively small number of samples. 582 A. D. Marrs 200 100 19 20 Iteration {k} Figure 1: Reversible-jump MCMC chain and histogram of model order for simulated data. ]0 .' ~~ •• ::0:''. ....... .' ',a,.1 \. .' 20 ~..... • .,:, .~. '. \0" ., '»" .' . . ~. -. \ •• ..... tI.-' 10.·.~···· •••• ; ..... • ' ~'&."" .,' "~'~:"""''''~' .. w: .. . ~" " :. . ,~, ' 'Ie" ,:": - 10 ' .,. ".!'J" • , .. ~ • ' rl .... -20 S,.' .;~:! : . :,~' • f#. itt . . . ,.'~, -30 .~.:' Cenerating Modal : I. -]0 -20 -10 0 10 20 ]0 ]0 ,,;,~., :.'0 . .~. .1....... \... _ .. 20 • • .~. /I /I. • ,:~:. .'. tI •• :_ ~ /1.\ .,. .. ~. . .. .... .. 10 "'~" ~"" • ~:':;.,., : ... ~ • ~ .. ~. .. .yt.' , '. . .~~ . ie' ';':', , -10 •• l' ,T...,. ~ . -. ~~:! :' :,~' -20 ',~ • -]0 " ,.' •. ":~.:~ MAP E.tilllllta Modal : I. -]0 -20 -10 0 10 20 ]0 Figure 2: Example of model estimation for simulated data. 4.2 Vowel data The reversible-jump MCMC chain of model order for the Peterson-Barney vowel data example is shown in figure 3, alongside the resulting MAP model estimate. For ease of visualisation, the estimated model and data samples have been projected onto the first two principal components of the data. Again, the results are encouraging. 5 Conclusion One of the key problems when using Gaussian mixture models is estimation of the optimum number of components to include in the model. In this paper we extend the reversible-jump MCMC technique for estimating the parameters of Gaussian mixtures with an unknown number of components to the multivariate spherical Gaussian case. The technique is then demonstrated on a simulated data example and an example using a well known dataset. The attraction of this approach is that the number of mixture components is not fixed at the outset but becomes a parameter of the model. The reversible-jump MCMC approach is then capable of moving between parameter subspaces which .. Reversible-Jwnp MCMC for Multivariate Spherical Gaussian Mixtures 583 Figure 3: Reversible-jump MCMC chain of model order and MAP estimate of model (projected onto first two principal components) for vowel data. correspond to models with different numbers of mixture components. As a result a sample of the full joint distribution is generated from which the posterior distribution for the number of model components can be derived. This information may then either be used to construct a Bayesian classifier or to define the centres in a radial basis function networ k. References [1] W.R. Gilks, S. Richardson, and D.J. Spiegelhalter Eds. Markov Chain Monte Carlo in Practice. Chapman and Hall, 1995. [2] P.J. Green. Reversible jump MCMC computation and Bayesian model determination. Boimetrika, 82:711-732, 1995. [3] D.B. Phillips and A.F.M. Smith. Bayesian model comparison via jump diffusions. In W.R. Gilks, S. Richardson, and D.J. Spiegelhalter, editors, Markov Chain Monte Carlo in Practice. Chapman and Hall, 1995. [4] S. Richardson and P. J. Green. On Bayesian analysis of mixtures with an unknown number of components. J. Royal Stat. Soc. Series B, 59(4), 1997. [5] D.M. Titterington, A.F.M. Smith, and U.E. Makov. Statistical Analysis of Finite Mixture Distributions. Wiley, 1985. ©British Crown Copyright 1998 /DERA. Published with the permission of the controller of Her Britannic Majesty's Stationary Office.
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On the infeasibility of training neural networks with small squared errors Van H. Vu Department of Mathematics, Yale University vuha@math.yale.edu Abstract We demonstrate that the problem of training neural networks with small (average) squared error is computationally intractable. Consider a data set of M points (Xi, Yi), i = 1,2, ... , M, where Xi are input vectors from Rd, Yi are real outputs (Yi E R). For a network 10 in some class F of neural networks, (11M) L~l (fO(Xi)Yi)2)1/2 - inlfEF(l/ M) "2:f!1 (f(Xi) - YJ2)1/2 is the (avarage) relative error occurs when one tries to fit the data set by 10. We will prove for several classes F of neural networks that achieving a relative error smaller than some fixed positive threshold (independent from the size of the data set) is NP-hard. 1 Introduction Given a data set (Xi, Yi), i = 1,2, ... , M. Xi are input vectors from Rd , Yi are real outputs (Yi E R). We call the points (Xi, Yi) data points. The training problem for neural networks is to find a network from some class (usually with fixed number of nodes and layers), which fits the data set with small error. In the following we describe the problem with more details. Let F be a class (set) of neural networks, and a be a metric norm in RM. To each 1 E F, associate an error vector Ef = (1/(Xd - Yil)f;l (EF depends on the data set, of course, though we prefer this notation to avoid difficulty of having too many subindices). The norm of Ej in a shows how well the network 1 fits the data regarding to this particular norm. Furthermore, let eo:,F denote the smallest error achieved by a network in F, namely: eo: F = min liEf 110: , fEF In this context, the training problem we consider here is to find 1 E F such that 372 v.n Vu IIEfila - ea,F ~ fF, where fF is a positive number given in advance, and does not depend on the size M of the data set. We will call fF relative error. The norm a is chosen by the nature of the training process, the most common norms are: 100 norm: Ilvll oo = maxlvi/ (interpolation problem) 12 norm: IIvl12 = (l/M2::;l v[)1/2, where v = (Vi)t;l (least square error problem). The quantity liEf 1112 is usually referred to as the emperical error of the training process. The first goal of this paper is to show that achieving small emperical error is NP-hard. From now on, we work with 12 norm, if not otherwise specified. A question of great importance is: given the data set, F and fF in advance, could one find an efficient algorithm to solve the training problem formulated above. By efficiency we mean an algorithm terminating in polynomial time (polynomial in the size of the input). This question is closely related to the problem of learning neural networks in polynomial time (see [3]). The input in the algorithm is the data set, by its size we means the number of bits required to write down all (Xi, Yi). Question 1. Given F and fF and a data set. Could one find an efficient algorithm which produces a function f E F such that liEf II < eF + fF Question 1 is very difficult to answer in general. In this paper we will investigate the following important sub-question: Question 2. Can one achieve arbitrary small relative error using polynomial algorithms ? Our purpose is to give a negative answer for Question 2. This question was posed by 1. Jones in his seminar at Yale (1996). The crucial point here is that we are dealing with 12 norm, which is very important from statistical point of view. Our investigation is also inspired by former works done in [2], [6], [7], etc, which show negative results in the 100 norm case. Definition. A positive number f is a threshold of a class F of neural networks if the training problem by networks from F with relative error less than f is NP-hard (i.e., computationally infeasible). In order to provide a negative answer to Question 2, we are going to show the existence of thresholds (which is independent from the size of the data set) for the following classes of networks. • Fn = {flf(x) = (l/n)(2:~=l step (ai x - bi)} • F~ = {flf(x) = (2:7=1 Cistep (ai x - bd} • On = {glg(x) = 2:~1 ci<!>i(aix - bi)} where n is a positive integer, step(x) = 1 if x is positive and zero otherwise, ai and x are vectors from Rd , bi are real numbers, and Ci are positive numbel's. It is clear that the class F~ contains Fn; the reason why we distinguish these two cases is that the proof for Fn is relatively easy to present, while contains the most important ideas. In the third class, the functions 1>i are sigmoid functions which satisfy certain Lipchitzian conditions (for more details see [9]) Main Theorem (i) The classes F1, F2, F~ and 02 have absolute constant (positive) thresholds On the Infeasibility of Training Neural Networks with Small Squared Errors (ii) For ellery class Fn+2, n > 0, there is a threshold of form (n- 3/'2d- 1/'2. (iii) For every F~+'2' 11 > 0, there is a threshold of form (n-3/2d-3/'2 . (iv) For every class 9n+2, n > 0, there is a threshold of form (n- 5/ 2d- 1/ 2 . In the last three statements. ( is an absolute positive constant. 373 Here is the key argument of the proof. Assume that there is an algorithm A which solves the training problem in some class (say Fn ) with relative error f. From some (properly chosen) NP-hard problem. we will construct a data set so that if f is sufficiently small, then the solution found by A (given the constructed data set as input) in Fn implies a solution for the original NP-hard problem. This will give a lower bound on f, if we assume that the algorithm A is polynomial. In all proofs the leading parameter is d (the dimension of data inputs). So by polynomial we mean a polynomial with d as variable. All the input (data) sets constructed ""ill have polynomial size in d. The paper is organized as follow. In the next Section, we discuss earlier results concerning the 100 norm. In Section 3, we display the NP-hard results we will use in the reduction. In Section 4, we prove the main Theorem for class F2 and mention the method to handle more general cases. We conclude with some remarks and open questions in Section 5. To end this Section, let us mention one important corollary. The Main Theorem implies that learning Fn, F~ and 9n (with respect to 12 norm) is hard. For more about the connection between the complexity of training and learning problems, we refer to [3], [5]. Notation: Through the paper Ud denotes the unit hypercube in Rd. For any number x, Xd denotes the vector (x, X,." x) of length d. In particular, Od denotes the origin of Rd. For any half space H, fI is the complement of H. For any set A, IAI is the number of elements in A. A function y( d) is said to have order of magnitude 0(F(d)), if there are c < C positive constants such that c < y(d)jF(d) < C for all d. 2 Previous works in the loo case The case Q = 100 (interpolation problem) was considered by several authors for many different classes of (usually) 2-layer networks (see [6],[2], [7], [8]). Most of the authors investigate the case when there is a perfect fit, i.e., eleo,F = O. In [2], the authors proved that training 2-layer networks containing 3 step function nodes with zero relative error is NP-hard. Their proof can be extended for networks with more inner nodes and various logistic output nodes. This generalized a former result of Maggido [8] on data set with rational inputs. Combining the techniques used in [2] with analysis arguments, Lee Jones [6] showed that the training problem with relative error 1/10 by networks with two monotone Lipschitzian Sigmoid inner nodes and linear output node, is also NP-hard (NP-complete under certain circumstances). This implies a threshold (in the sense of our definition) (1/10)M- 1/ 2 for the class examined. However, this threshold is rather weak, since it is decreasing in M. This result was also extended for the n inner nodes case [6]. It is also interesting to compare our results with Judd's. In [7] he considered the following problem "Given a network and a set of training examples (a data set), does there exist a set of weights so that the network gives correct output for all training examples ?" He proved that this problem is NP-hard even if the network is 374 V. H. Vu required to produce the correct output for two-third of the traing examples. In fact, it was shown that there is a class of networks and a data sets so that any algorithm will produce poorly on some networks and data sets in the class. However, from this result one could not tell if there is a network which is "hard to train" for all algorithms. Moreover, the number of nodes in the networks grows with the size of the data set. Therefore, in some sense, the result is not independent from the size of the data set. In our proofs we will exploit many techniques provided in these former works. The crucial one is the reduction used by A. Blum and R. Rivest, which involves the NP-hardness of the Hypergraph 2-Coloring problem. 3 Sonle NP hard problems Definition Let B be a CNF formula, where each clause has at most k literals. Let max(B) be the maximum number of clauses which can be satisfied by a truth assignment. The APP MAX k-SAT problem is to find a truth assignment which satisfies (1 - f)max(B) clauses. The following Theorem says that this approximation problem is NP -hard, for some small f. Theorem 3.1.1 Fix k 2: 2. There is fl > 0, such that finding a truth assignment. which satisfies at least (1- fdmax(B) clauses is NP-h a rd. The problem is still hard, when every literal in B appears in only few clauses, and every clause contains only few literals. Let B3(5) denote the class of CNFs with at most 3 literals in a clause and every literal appears in at most 5 clauses (see [1]). Theorem 3.1.2 There is t2 > 0 such that finding a truth assignment, which satisfies at least (1 - f)max(B) clauses in a formula B E B3(5) is NP-hard. The optimal thresholds in these theorems can be computed, due to recent results in Thereotical Computer Science. Because of space limitation, we do not go into this matter. Let H = (V, E) be a hypergraph on the set V, and E is the set of edges (collection of subsets of V). Elements of V are called vertices. The degree of a vertex is the number of edges containing the vertex. We could assume that each edge contains at least two vertices. Color the vertices with color Blue or Red. An edge is colorful if it contains vertices of both colors, otherwise we call it monochromatic. Let c( H) be the maximum number of colorful edges one can achieve by a coloring. By a probabilistic argument, it is easy to show that c(H) is at least IEII2 (in a random coloring, an edge will be colorful with probability at least 1/2). Using 3.1.2, we could prove the following theorem (for the proof see [9]) Theorem 3.1.3 There is a constant f3 > 0 such that finding a coloring with at least (1 - t3)c(H) colorful edges is NP-hard. This statement holds even in the case when every but one degree in H is at most 10 4 Proof for :F2 We follow the reduction used in [2]. Consider a hypergraph H(V, E) described Theorem 3.2.1. Let V = {I, 2, . . " d + I}, where with the possible exception of the vertex d + 1, all other vertices have degree at most 10. Every edge will have at least 2 and at most 4 vertices. So the number of edges is at least (d + 1) /4. On the Infeasibility of Training Neural Networks with Small Squared Errors 375 Let Pi be the ith unit vector in R d+l , Pi = (0,0, . .. ,0,1,0, .. . ,0). Furthermore, Xc = LiE C Pi for every edge C E E. Let S be a coloring with maximum number of colorful edges. In this coloring denote by Al the set of colorful edges and by A2 the set of monochromatic edges. Clearly IAII = e(H). Our data set will be the following (inputs are from Rd+l instead of from Rd , but it makes no difference) where (Pd+1,1/2)t and (Od+l , l)t means (Pd+1, 1/2) and (Od+l, 1) are repeated t times in the data set, resp. Similarly to [2], consider two vectors a and b in R d+l where a = (al,"" ad+l), ai = -1 if i is Red and ai = d + 1 otherwise b = (b l , . .. , bd+l) , bi = -1 if i is Blue and bi = d + 1 otherwise It is not difficult to verify that the function fa = (1/2)(step (ax + 1/2) + step (bx + 1/2)) fits the data perfectly, thus e:F2 = IIEjal1 = O. Suppose f = (1/2) (step (ex - I) + step (dx - 6» satisfies M MllEjW = 2)f(Xd - Yi)2 < Mc2 i=l Since if f(Xi ) # Yi then U(Xi ) - Yi)2 2: 1/4, the previous inequality implies: Po = l{i.J(Xd # Ydl < 4Mc2 = p The ratio po/Mis called misclassification ratio, and we will show that this ratio cannot be arbitrary small. In order to avoid unnecessary ceiling and floor symbols, we assume the upper-bound p is an integer. We choose t = P so that we can also assume that (Od+l, 1) and (Pd+l, 1/2) are well classified. Let Hl (H2) be the half space consisting of x: ex 'Y > 0 (dx - 6 > 0). Note that Od E HI n H2 and Pd+l E fI I U fI 2. Now let Pl denote the set of i where Pi t/:. HI, and P2 the set of i such that Pi E Hl n H 2• Clearly, if j E P2 , then f(pj) # Yj, hence: IP21::; p. Let Q = {C E EIC n P2 # 0}. Note that for each j E P2, the degree of j is at most 10, thus: IQI ::; 10!?:?1 ::; lOp Let A~ = {Clf(xc) = I}. Since less than p points are misclassified, IA~ .0. A I I < p. Color V by the following rule: (1) if Pi E PI, then i is Red; (2) if Pi E P2 , color i arbitrarily, either Red or Blue; (3) if Pi t/:. Pl U P2 , then i is Blue. N ow we can finish the proof by the following two claims: Claim 1: Every edge in A~ \Q is colorful. It is left to readers to verify this simple statement. Claim 2: IA~ \QI is close to IAII · Notice that: IAI \(A~ \Q)I ::; IAI.0.A~ 1+ IQI ::; p + lOp = IIp Observe that the size of the data set is M = d + 2t + lEI, so lEI + d 2: M - 2t = M - 2p. Moreover, lEI 2: (d + 1)/4, so lEI 2: (1/5)(M - 2p). On the other hand, IAII2: (1/2)IEI, all together we obtain; IAII2: (1/10)(M - p), which yields: 376 V. H. Vu Choose f = f4 such that k(f4) ~ f3 (see Theortm 3.1.3). Then f4 will be a threshold for the class ;:2. This completes the proof. Q.E.D. Due to space limitation, we omit the proofs for other classes and refer to [9]. However, let us at least describe (roughly) the general method to handle these cases. The method consists of following steps: • Extend the data set in the previous proof by a set of (special) points. • Set the multiplicities of the special points sufficiently high so that those points should be well-classified. • If we choose the special points properly, the fact that these points are well-classified will determine (roughly) the behavior of all but 2 nodes. In general we will show that all but 2 nodes have little influence on the outputs of non-special data points. • The problem basically reduces to the case of two nodes. By modifying the previous proof, we could achieve the desired thresholds. 5 Remarks and open problems • Readers may argue about the existence of (somewhat less natural) data points of high multiplicities. We can avoid using these data points by a combinatorial trick described in [9]. • The proof in Section 4 could be carried out using Theorem 3.1.2. However, we prefer using the hypergraph coloring terminology (Theorem 3.1.3), which is more convenient and standard. Moreover, Theorem 3.1.3 itself is interesting, and has not been listed among well known "approximation is hard" theorems. • It remains an open question to determine the right order of magnitude of thresholds for all the classes we considered. (see Section 1). By technical reasons, in the Main theorem, the thresholds for more than two nodes involve the dimension (d). We conjecture that there are dimension-free thresholds. Acknowledgement We wish to thank A. Blum, A. Barron and 1. Lovasz for many useful ideas and discussions. References [1] S. Arora and C. Lund Hardness of approximation, book chapter, preprint [2] A. Blum, R. Rivest Training a 3-node neural network is NP-hard Neutral Networks, Vol 5., p 117-127, 1992 [3] A. Blumer, A. Ehrenfeucht, D. Haussler, M. Warmuth, Learnability and the Vepnik-Chervonenkis Dimension, Journal ofthe Association for computing Machinery, Vol 36, No.4, 929-965, 1989. [4] M. Garey and D. Johnson, Computers and intractability: A guide to the theory of NP-completeness, San Francisco, W.H.Freeman, 1979 On the Infeasibility o/Training Neural Networks with Small Squared Errors 377 [5] D. Haussler, Generalizing the PAC model for neural net and other learning applications (Tech. Rep. UCSC-CRL-89-30). Santa Cruz. CA: University of California 1989. [6] L. J ones, The computational intractability of training sigmoidal neural networks (preprint) [7] J. Judd Neutral Networks and Complexity of learning, MIT Press 1990. [8] N. Meggido, On the complexity of polyhedral separability (Tech. Rep. RJ 5252) IBM Almaden Research Center, San Jose, CA [9] V. H. Vu, On the infeasibility of training neural networks with small squared error. manuscript.
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Structure Driven Image Database Retrieval Jeremy S. De Bonet &, Paul Viola Artificial Intelligence Laboratory Learning & Vision Group 545 Technology Square Massachusetts Institute of Technology . Cambridge, MA 02139 EMAIL: jsdCOaLmit. edu & violaCOaLmit. edu HOMEPAGE: http://www.ai . mit. edu/pro j ects/l v Abstract A new algorithm is presented which approximates the perceived visual similarity between images. The images are initially transformed into a feature space which captures visual structure, texture and color using a tree of filters. Similarity is the inverse of the distance in this perceptual feature space. Using this algorithm we have constructed an image database system which can perform example based retrieval on large image databases. Using carefully constructed target sets, which limit variation to only a single visual characteristic, retrieval rates are quantitatively compared to those of standard methods. 1 Introduction Without supplementary information, there exists no way to directly measure the similarity between the content of images. In general, one cannot answer a question of the form: "is image A more like image B or image C?" without defining the criteria by which this comparison is to be made. People perform such tasks by inferring some criterion, based on their visual experience or by complex reasoning about the situations depicted in the images. Humans are very capable database searchers. They can perform simple searches like, "find me images of cars", or more complex or loosely defined searches like, "find me images that depict pride in America". In either case one must examine all, or a large portion, of the database. As the prevalence and size of multimedia databases increases, automated techniques will become critical in the successful retrieval of relevant information. Such techniques must be able to measure the similarity between the visual content of natural images. Structure Driven Image Database Retrieval 867 Many algorithms have been proposed for image database retrieval. For the most part these techniques compute a feature vector from an image which is made up of a handful of image measurements. Visual or semantic distance is then equated with feature distance. Examples include color histograms, texture histograms, shape boundary descriptors, eigenimages, and hybrid schemes (QBIC, ; Niblack et aI., 1993; Virage, ; Kelly, Cannon and Hush, 1995; Pentland, Picard and ScIaroff, 1995; Picard and Kabir, 1993; Santini and Jain, 1996). A query to such a system typically consists of specifying two types of parameters: the target values of each of the measurements; and a set of weights, which determine the relative importance of deviations from the target in each measurement dimension. The features used by these systems each capture some very general property of images. As a result of their lack of specificity however, many images which are actually very different in content generate the same feature responses. In contrast our approach extracts thousands of very specific features. These features measure both local texture and global structure. The feature extraction algorithm computes color, edge orientation, and other local properties at many resolutions. This sort of multi-scale feature analysis is of critical importance in visual recognition and has been used successfully in the context of object recognition (von der Malsburg, 1988; Rao and Ballard, 1995; Viola, 1996) Our system differs from others because it detects not only first order relationships, such as edges or color, but also measures how these first order relationships are related to one another. Thus by finding patterns between image regions with particular local properties, more complex - and therefore more discriminating - features can be extracted. This type of repeated, non-linear, feature detection bears a strong resemblance to the response properties of visual cortex cells (Desimone et al., 1984). While the mechanism for the responses of these cells is not yet clear, this work supports the conclusion that this type of representation is very useful in practical visual processing. 2 Computing the Characteristic Signature The "texture-of-texture" measurements are based on the outputs of a tree of nonlinear filtering operations. Each path through the tree creates a particular filter network, which responds to certain structural organization in the image. Measuring the appropriately weighted difference between the signatures of images in the database and the set of query-images, produces a similarity measure which can be used to rank and sort the images in the database. The computation of the characteristic signature is straightforward. At the highest level of resolution the image is convolved with a set of 25 local linear features including oriented edges and bars. The results of these convolutions are 25 feature response images. These images are then rectified by squaring, which extracts the texture energy in the image, and then downsampled by a factor of two. At this point there are 25 half scale output images which each measure a local textural property of the input image. For example one image is sensitive to vertical edges, and responds strongly to both skyscrapers and picket fences. Convolution, rectification and downsampling is then repeated on each of these 25 half resolution images producing 625 quarter scale "texture-of-texture" images. The second layer will respond strongly to regions where the texture specified in the first layer has a particular spatial arrangement. For example if horizontal alignments of vertical texture are detected, there will be a strong response to a picket fence and little response to a skyscraper. With additional layers additional specificity is 868 J. S. D. Bonet and P. A. Viola achieved; repeating this procedure a third time yields 15,625 meta-texture images at eighth scale. Each of the resulting meta-textures is then summed to compute a single value and provides one element in the characteristic signature. When three channels of color are included there are a total of 46,875 elements in the characteristic signature. Once computed, the signature elements are normalized to reduce the effects of contrast changes. More formally the characteristic signature of an image is given by: Si,j,k,e(I) = L Ei,j,k(Ie) (1) pixels where I is the image, i, j and k index over the different types of linear filters, and Ie are the different color channels of the image. The definition of E is: Ei(I) 2 .j.. [(Fi 0 1)2] (2) Ei,j (I) 2.j.. [(Fj 0 Ei(I))2] (3) Ei,j,k(I) 2.j.. [(Fk 0 Ei,j(I))2] . (4) where F j is the ith filter and 2.j.. is the downsampling operation. 3 U sing Characteristic Signatures To Form Image Queries In the "query by image" paradigm, we describe similarity in terms of the difference between an image and a group of example query images. This is done by comparing the characteristic signature of the image to the mean signature of the query images. The relative importance of each element of the characteristic signature in determining similarity is proportional to the inverse variance of that element across the example-image group: L = - L L L L [S;,j,k,O( Iq) - S;,j,k,o (I.".) r i j k e Var[Si,j,k,e(lq )] (5) where Si,j,k,e(Iq ) and Var [Si,j,k,c(Iq )] are the mean and variance of the characteristic signatures computed over the query set. This is a diagonal approximation of the Mahalanobis distance (Duda and Hart, 1973). It has the effect of normalizing the vector-space defined by the characteristic signatures, so that characteristic elements which are salient within the group of query images contribute more to the overall similarity of an image. In Figure 1 three 2D projections of these 46,875 dimensional characteristic signature space are shown. The data points marked with circles are generated by the 10 images shown at the top of Figure 3. The remaining points are generated by 2900 distract or images. Comparing (a) and (b) we see that in some projections the images cluster tightly, while in others they are distributed. Given a sample of images from the target set we can observe the variation in each possible projection axis. Most of the time the axes shown in (a) will be strongly discounted by the algorithm because these features are not consistent across the query set. Similarly the axes from (b) will receive a large weight because the target images have very consistent values. The axes along which target groups cluster, however, differ from target group to target group. As a result it is not possible to conclude that the axes in (b) are simply better than the axes in (a). In Figure 1 (c) the same projection is shown again this time with a different target set highlighted (with asterisks). Structure Driven Image Database Retrieval 9 ~i~~:;:,~·~~~~'·:.~ .' .' ~~~" 9.:'.): t'K .. o o (a) 869 (b) (c) Figure 1: In some projections target groups do not cluster (a), and they do in others (b). However, different target groups will not necessarily cluster in the same projections (c). 4 Experiments In the first set of experiments we use a database of 2900 images from 29 Corel Photo CD (collections 1000-2900.) Figure 2 shows the results of typical user query on this system. The top windows in each Figure contain the query-images submitted by the user. The bottom windows show the thirty images found to be most similar; similarity decreases from upper left (most similar) to lower right. Though these examples provide an anecdotal indication that the system is generating similarity measures which roughly conform to human perception, it is difficult to fully characterize the performance of this image retrieval technique. This is a fundamental problem of the domain. Images vary from each other in an astronomical number of ways, and similarity is perceived by human observers based upon complex interactions between recognition, cognition, and assumption. It seems unlikely that an absolute criterion for image similarity can ever be determined, if one truly exists. However using sets of images which we believe are visually similar, we can establish a basis for comparing algorithms. To better measure the performance of the system we added a set of 10 images to the 2900 image database and attempted to retrieve these new images. We compare the performance of the present system to ten other techniques. Though these techniques are not as sophisticated as those used in systems developed by other researchers, they are indicative of the types of methods which are prevalent in the literature. In each experiment we measure the retrieval rates for a set of ten target images which we believe to be visually similar because they consist of images of a single scene. Images in the target set differ due to variation of a single visual characteristic. In some of the target sets the photographic conditions have been changed, either by moving the camera, the objects or the light. In other target sets post photograph image manipulation has been performed. Two example target sets are shown in Figure 3. In each experiment we perform 45 database queries using every possible pair of images from the target set. The retrieval methods compared are: ToT The current textures-of-textures system; RGB-216(or 512)C R,G,B color histograms using 216 (or 512) bins by dividing each color dimension into 6 (or 8) regions. The target histogram is generated by combining the histograms from the two model images; HSV-216(or 512)C same using H,S,V color space; RGB(or HSV)-216(or 870 J. s. D. Bonet and P. A. Viola Figure 2: Sample queries and top 30 responses. Structure Driven Image Database Retrieval 'i'~,.' ~ ~ .. . ";"~. . "r;IiB ,~~-o: ~ ., ~,~ p .. ~. . "t- '. '\\ -' ' .. ~ ,L1 J, rf. ~< -" .,'" . . " :". : ... . ' 871 , .~t . • . , 1:: :tic '" ~ ~~ ~~ ~' .. Figure 3: Two target sets used in the retrieval experiments. Top: Variation of object location. BOTTOM: Variation of hard shadows. c:: o +=' U Q) 0.5 CD ~ ~ O~--~L-~LL~ -c:: o U Q) 0.5 Q) :3~ -c:: o U o Q) 0.5 Q) :3~ Bnghtness Camera Position Contrast Hard Shadows NOise 0.25 NOise 0 .5 NOise 0.75 Object Location Occlusion Rotation (360deg) Rotation (60deg) Translation Soft shadows I ROe-l16C RGe-O,, 2C "lGa-2I SNN ~GB - ~l2N .'" . ~S\l-2 16C H'>V-2"i~ I"+S"'~ l 2N" COr-lU"M c.~ Object Pose ~ ,,' Zoom Figure 4: The percentage of target images ranked above all the distract or images, shown for 15 target sets such as those in Figure 3. The textures-of-textures model presented here achieves perfect performance in 13 of the 15 experiments. 512)NN histograms in which similarity is measured using a nearest neighbor metric; COR-full(and low)res full resolution (and 4x downsampled) image correlation. 872 J. S. D. Bonet and P. A. Viola Rankings for all 10 target images in each of the 45 queries are obtained for each variation. To get a comparative sense of the overall performance of each technique, we show the number of target images retrieved with a Neyman-Pearson criterion of zero, i.e. no false positives Figure 4. The textures-of-textures model substantially outperforms all of the other techniques, achieving perfect performance in 13 experiments. 5 Discussion We have presented a technique for approximating perceived visual similarity, by measuring the structural content similarity between images. Using the high dimensional "characteristic signature" space representation, we directly compare database-images to a set of query-images. A world wide web interface to system has been created and is available via the URL: http://www.ai.mit.edu/~jsd/Research/lmageDatabase/Demo Experiments indicate that the present system can retrieve images which share visual characteristics with the query-images, from a large non-homogeneous database. Further, it greatly outperforms many of the standard methods which form the basis of other systems. Though the results presented here are encouraging, on real world queries, the retrieved images often contain many false alarms, such as those in Figure 2; however, we believe that with additional analysis performance can be improved. References Desimone, R., Albright, T. D., Gross, C. G., and Bruce, C. (1984). Stimulus selective properties of inferior temporal neurons in the macaque. Journal of Neuroscience, 4:2051-2062. Duda, R. and Hart, P. (1973). Pattern Classification and Scene Analysis. John Wiley and Sons. Kelly, M., Cannon, T. M., and Hush, D. R. (1995). Query by image example: the candid approach. SPIE Vol. 2420 Storage and Retrieval for Image and Video Databases III, pages 238-248. Niblack, V., Barber, R ., Equitz, W., Flickner, M., Glasman, E., Petkovic, D., Yanker, P., Faloutsos, C., and Taubin, G. (1993). The qbic project: querying images by content using color, texture, and shape. ISf3TjSPIE 1993 International Symposium on Electronic Imaging: Science f3 Technology, 1908:173-187. Pentland, A., Picard, R. W., and Sclaroff, S. (1995). Photobook: Content-based manipulation of image databases. Technical Report 255, MIT Media Lab. Picard, R. W. and Kabir, T. (1993). Finding similar patterns in large image databases. ICASSP, V:161-164. QBIC. The ibm qbic project. Web: http://wwwqbic.almaden.ibm.comf. Rao, R. P. N. and Ballard, D. (1995). Object indexing using an iconic sparse distributed memory. Technical Report TR-559, University of Rochester. Santini, S. and Jain, R. (1996). Gabor space and the development of preattentive similarity. In Proceedings of ICPR 96. International Conference on Pattern Recognition, Vienna. Viola, P. (1996). Complex feature recognition: A bayesian approach for learning to recognize objects. Technical Report 1591, MIT AI Lab. Virage. The virage project. Web: http://www.virage.com/. von der Malsburg, C. (1988). Pattern recognition by labeled graph matching. Neural Networks, 1:141-148.
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Bayesian Robustification for Audio Visual Fusion Javier Movellan * movellanOcogsci.ucsd.edu Department of Cognitive Science University of California, San Diego La Jolla, CA 92092-0515 Paul Mineiro pmineiroOcogsci.ucsd.edu Department of Cognitive Science University of California, San Diego La Jolla, CA 92092-0515 Abstract We discuss the problem of catastrophic fusion in multimodal recognition systems. This problem arises in systems that need to fuse different channels in non-stationary environments. Practice shows that when recognition modules within each modality are tested in contexts inconsistent with their assumptions, their influence on the fused product tends to increase, with catastrophic results. We explore a principled solution to this problem based upon Bayesian ideas of competitive models and inference robustification: each sensory channel is provided with simple white-noise context models, and the perceptual hypothesis and context are jointly estimated. Consequently, context deviations are interpreted as changes in white noise contamination strength, automatically adjusting the influence of the module. The approach is tested on a fixed lexicon automatic audiovisual speech recognition problem with very good results. 1 Introduction In this paper we address the problem of catastrophic fusion in automatic multimodal recognition systems. We explore a principled solution based on the Bayesian ideas of competitive models and inference robustification (Clark & Yuille, 1990; Box, 1980; O'Hagan, 1994). For concreteness, consider an audiovisual car telephony task which we will simulate in later sections. The task is to recognize spoken phone numbers based on input from a camera and a microphone. We want the recognition system to work on environments with non-stationary statistical properties: at times the video signal (V) may be relatively clean and the audio signal (A) may be contaminated by sources like the radio, the engine, and friction with the road. At other times the A signal may be more reliable than the V signal, e.g., the radio is off, but the talker's mouth is partially occluded. Ideally we want the audio-visual system to combine the A and V sources optimally given the conditions at hand, e.g., give more weight to whichever channel is more reliable at that time. At a minimum we expect that for a wide variety of contexts, the performance after fusion should not be worse than the independent unimodal systems (Bernstein & Benoit, 1996). When component modules can Significantly outperform the overall system after fusion, catastrophic fusion is said to have occurred. • To whom correspondence should be addressed. Bayesian Robustification for Audio Visual Fusion 743 Fixed vocabulary audiovisual speech recognition (AVSR) systems typically consist of two independent modules, one dedicated to A signals and one to V signals (Bregler, Hild, Manke & Waibel, 1993; Wolff, Prasad, Stork & Hennecke, 1994; Adjondani & Benoit, 1996; Movellan & Chadderdon, 1996). From a Bayesian perspective this modularity reflects an assumption of conditional independence of A and V signals (i.e., the likelihood function factorizes) P(XaXvIWi.Aa.Av) <X P(XaIWi.Aa)P(xvlwi.Av), (1) where Xa and Xv are the audio and video data, Wi is a perceptual interpretation of the data (e.g., the word "one") and {.Aa,.Av} are the audio and video models according to which these probabilities are calculated, e.g., a hidden Markov model, a neural network, or an exemplar model. Training is also typically modularized: the A module is trained to maximize the likelihood of a sample of A signals while the V module is trained on the corresponding sample of V signals. At test time new data are presented to the system and each module typically outputs the log probability of its input given each perceptual alternative. Assuming conditional independence, Bayes' rule calls for an affine combination of modules 'Ii; argmax {logp(wilxaxv.Aa.Av)} Wi Wi where 'Ii; is the interpretation chosen by the system, and p( Wi) is the prior probability of each alternative. This fusion rule is optimal in the sense that it minimizes the expected error: no other fusion rule produces smaller error rates, provided the models {.Aa,.Av} and the assumption of conditional independence are correct. Unfortunately a naive application of Bayes' rule to AVSR produces catastrophic fusion. The A and V modules make assumptions about the signals they receive, either explicitly, e.g., a well defined statistical model, or implicitly, e.g., a blackbox trained with a particular data sample. In our notation these assumptions are reflected by the fact that the log-likelihoods are conditional on models: {.A a , .Av}. The fact that modules make assumptions implies that they will operate correctly only within a restricted context, i.e, the collection of situations that meet the assumptions. In practice one typically finds that Bayes' rule assigns more weight to modules operating outside their valid context, the opposite of what is desired. 2 Competitive Models and Bayesian Robustification Clark and Yuille (1990) and Yuille and Bulthoff (1996) analyzed information integration in sensory systems from a Bayesian perspective. Modularity is justified in their view by the need to make assumptions that disambiguate the data available to the perceptual system (Clark & Yuille, 1990, p. 5). However, this produces modules which are valid only within certain contexts. The solution proposed by Clark and Yuille (1990) is the creation of an ensemble of models each of which specializes on a restricted context and automatically checks whether the context is correct. The hope is that by working with such an ensemble of models, robustness under a variety of contexts can be achieved (Clark & Yuille, 1990, p. 13). Box (1980) investigated the problem of robust statistical inference from a Bayesian perspective. He proposed extending inference models with additional "nuisance" parameters a, a process he called Bayesian robustification. The idea is to replace an implicit assumption about the specific value of a with a prior distribution over a, representing uncertainty about that parameter. The approach here combines the ideas of competitive models and robustification. Each of the channels in the multimodal recognition system is provided with extra 744 1. Movellan and P. Mineiro parameters that represent non-stationary properties of the environment, what we call a context model. By doing so we effectively work with an infinite ensemble of models each of which compete on-line to explain the data. As we will see later even unsophisticated context models provide superior performance when the environment is non-stationary. We redefine the estimation problem as simultaneously choosing the most probable A and V context parameters and the most probable perceptual interpretation w = argmax {max P(WwaO'vIXaXvAaAv)} Wi Uo,U v (3) where 0' a and 0' v are the context parameters for the audio and visual channels and Wi are the different perceptual interpretations. One way to think of this joint decision approach is that we let all context models compete and we let only the most probable context models have an influence on the fused percept. Hereafter we refer to this approach as competitive fusion. Assuming conditional independence of the audio and video data and uninformative priors for (0' a, 0' v), we have w = ar~ax {logp(Wi) + (4) [~~IOgP(XalwwaAa)] + [~::XIOgP(xvIWiO'vAv)]}. Thus conditional independence allows a modular implementation of competitive fusion, Le., the A and V channels do not need to talk to each other until the time to make a joint decision, as follows. 1. For each Wi obtain conditional estimates of the context parameters for the audio and video signals: o-~IWi ~ argmax { 10gp(xalwWaAa) } , (5) (To. and o-~IWi ~ argmax{ 10gp(xvlwWvAv) }. (6) (Tv 2. Find the best Wi using the conditional context estimates. w = argmax {IOgp(Wi) + logp(xalwio-alwi Aa) + logp(xv IWio-vlwi Av)} (7) Wi 3 Application to A VSR Competitive fusion can be easily applied to Hidden Markov Models (HMM), an architecture closely related to stochastic neural networks and arguably the most successful for AVSR. Typical hidden Markov models used in AVSR are defined by • Markovian state dynamics: p(qt+ll2.t ) = p(qt+llqt), where qt is the state at time t and 2.t = (ql," . qt), • Conditionally independent sensor models linking observations to states ! (Xt Iqt), typically a mixture of multivariate Gaussian densities !(xtlqt) = L p(mi Iqt)(27l') -N/2 1~I-l/2 exp(d(xt, qt, Pi, ~)), (8) Bayesian Robustification for Audio VIsual Fusion 745 where N is the dimensionality of the data, mi is the mixture label, p(milqt) is the mixture distribution for state qt, Pi is the centroid for mixture mi, E is a covariance matrix, and d is the Mahalanobis norm (9) The approach explored here consists on modeling contextual changes as variations on the variance parameters. This corresponds to modeling non-stationary properties of the environments as variations in white noise power within each channel. Competitive fusion calls for on-line maximization of the variance parameters at the same time we optimize with respect to the response alternative. 11; = arg:ax{ logp(wi) + [ ~~logp(xalwiEaAa)] + [ ~~logp(xvlwiEv'\v)] }. (10) The maximization with respect to the variances can be easily integrated into standard HMM packages by simply applying the EM learning algorithm (Dampster, Laird & Rubin, 1977) on the variance parameters at test time. Thus the only difference between the standard approach and competitive fusion is that we retrain the variance parameters of each HMM at test time. In practice this training takes only one or two iterations of the EM algorithm and can be done on-line. We tested this approach on the following AVSR problem. Training database We used Tulips1 (Movellan, 1995) a database consisting of 934 images of 9 male and 3 female undergraduate students from the Cognitive Science Department at the University of California, San Diego. For each of these, two samples were taken for each of the digits "one" through "four". Thus, the total database consists of 96 digit utterances. The specifics of this database are explained in (Movellan, 1995). The database is available at http://cogsci.ucsd.edu. Visual processing We have tried a wide variety of visual processing approaches on this database, including decomposition with local Gaussian templates (Movellan, 1995), PCA-based templates (Gray, Movellan & Sejnowski, 1997), and Gabor energy templates (Movellan & Prayaga, 1996). To date, best performance was achieved with the local Gaussian approach. Each frame of the video track is soft-thresholded and symmetrized along the vertical axis, and a temporal difference frame is obtained by subtracting the previous symmetrized frame from the current symmetrized frame. We calculate the inner-products between the symmetrized images and a set of basis images. Our basis images were lOx15 shifted Gaussian kernels with a standard deviation of 3 pixels. The loadings of the symmetrized image and the differential image are combined to form the final observation frame. Each of these composite frames has 300 dimensions (2xlOx15). The process is explained in more detail in Movellan (1995). Auditory processing LPC /cepstral analysis is used for the auditory front-end. First, the auditory signal is passed through a first-order emphasizer to spectrally flatten it. Then the signal is separated into non-overlapping frames at 30 frames per second. This is done so that there are an equal number of visual and auditory feature vectors for each utterance, which are then synchronized with each other. On each frame we perform the standard LPC / cepstral analysis. Each 30 msec auditory frame is characterized by 26 features: 12 cepstral coefficients, 12 delta-cepstrals, 1 log-power, and 1 delta-log-power. Each of the 26 features is encoded with 8-bit accuracy. 746 J Movellan and P. Mineiro Figure 1: Examples of the different occlusion levels, from left to right: 0%, 10%, 20%, 40%, 60%, 80%. Percentages are in terms of area. Recognition Engine In previous work (Chadderdon & Movellan, 1995) a wide variety of HMM architectures were tested on this database including architectures that did not assume conditional independence. Optimal performance was found with independent A and V modules using variance matrices of the form uI, where u is a scalar and I the identity matrix. The best A models had 5 states and 7 mixtures per state and the best V models had 3 states and 3 mixtures per state. We also determined the optimal weight of A and V modules. Optimal performance is obtained by weighting the output of V times 0.18. Factorial Contamination Experiment In this experiment we used the previously optimized architecture and compared its performance under 64 different conditions using the standard and the competitive fusion approaches. We used a 2 x 8 x 8 factorial design, the first factor being the fusion rule, and the second and third factors the context in the audio and video channels. To our knowledge this is the first time an AVSR system is tested with a factorial experimental design with both A and V contaminated at various levels. The independent variables were: 1. Fusion rule: Classical, and competitive fusion. 2. Audio Context: Inexistent, clean, or contaminated at one of the following signal to noise ratios: 12 Db, 6 Db, 0 Db, -6 Db, -12 Db and -100 Db. The contamination was done with audio digitally sampled from the interior of a car while running on a busy highway with the doors open and the radio on a talk-show station. 3. Video Context: Inexistent, clean or occluded by a grey level patch. The percentages of visual area occupied by the patch were 10%,20%,40%,60%, 80% and 100% (see Figure 1). The dependent variable was performance on the digit recognition task evaluated in terms of generalization to new speakers. In all cases training was done with clean signals and testing was done with one of the 64 contexts under study. Since the training sample is small, generalization performance was estimated using a jackknife procedure (Efron, 1982). Models were trained with 11 subjects, leaving a different subject out for generalization testing. The entire procedure was repeated 12 times, each time leaving a different subject out for testing. Statistics of generalization performance are thus based on 96 generalization trials (4 digits x 12 subjects x 2 observations per subject). Standard statistical tests were used to compare the classical and competitive context rules. The results of this experiment are displayed in Table 1. Note how the experiment replicates the phenomenon of catastrophic fusion. With the classic approach, when one of the channels is contaminated, performance after fusion can be significantly Bayesian Robustificationfor Audio Visual Fusion 747 Video one -6 -1 None 95.83 95.83 90.62 80.20 67.70 42.70 19.80 Clean 84.37 97.92 97.92 94.80 90.62 89.58 81.25 1 82.20 I 10% 73.95 93.75 93.75 94.79 87.50 80.20 71.87 64.58 20% 62.50 96.87 96.87 94.79 89.58 80.20 162.501 41.66 40% 37.50 93.75 89.58 87.50 83.30 70.83 43.75 30.20 60% 34.37 93.75 91.66 82.29 65.62 42.70 26.04 80% 27.00 95.83 90.62 79.16 64.58 146.871 25.00 100% 25.00 93.75 92.71 84.37 78.12 63.54 44.79 26.04 Performance wIth C/a .. 1e Fu" on Audio Video one ean None 95.83 94.79 89.58 79.16 65.62 40.62 20.83 Clean 86.45 98.95 96.87 95.83 93.75 87.50 79.16 1 70.83 / 10% 73.95 93.75 93.75 93.75 89.58 79.16 70.83 52.58 20% 54.16 89.58 84.37 84.37 75.00 ~ 43.00 40% 29.16 81.25 78.12 67.20 52 .08 38.54 34.37 60% 32.29 77.08 72.91 62.50 47.91 37.50 29. 16 80% 29.16 70.83 68.75 54.16 44.79 133 .831 28.12 100% 25.00 61.46 61.45 58.33 51.04 42.70 38.54 29.16 Table 1: Average generalization performance with standard and competitive fusion. Boxed cells indicate a statistically significant difference a = 0.05 between the two fusion approaches. worse than performance with the clean channel alone. For example, when the audio is clean, the performance of the audio-only system is 95.83%. When combined with bad video (100% occlusion), this performance drops down to 61.46%, a statistically significant difference, F(l,ll) = 132.0, p < 10-6 . Using competitive fusion, the performance of the joint system is 93.75%, which is not significantly different from the performance of the A system only, F(l,ll) = 2.4, p= 0.15. The table shows in boxes the regions for which the classic and competitive fusion approaches were significantly different (a = 0.05). Contrary to the classic approach, the competitive approach behaves robustly in all tested conditions. 4 Discussion Catastrophic fusion may occur when the environment is non-stationary forcing modules to operate outside their assumed context. The reason for this problem is that in the absence of a context model, deviations from the expected context are interpreted as information about the different perceptual interpretations instead of information about contextual changes. We explored a principled solution to this problem inspired by the Bayesian ideas of robustification (Box, 1980) and competitive models (Clark & Yuille, 1990). Each module was provided with simple white-noise context models and the most probable context and perceptual hypothesis were jointly estimated. Consequently, context deviations are interpreted as changes in the white noise contamination strength, automatically adjusting the influence of the module. The approach worked very well on a fixed lexicon AVSR problem. References Adjondani, A. & Benoit, C. (1996). On the Integration of Auditory and Visual Parameters in an HMM-based ASR. In D. G. Stork & M. E. Hennecke (Eds.), Speechreading by Humans and Machines: Models, Systems, and Applications, pages 461-471. New York: NATO/Springer-Verlag. Bernstein, L. & Benoit, C. (1996). For Speech Perception Three Senses are Bettern 748 1. Movellan and P. Mineiro than One. In Proc. of the 4th Int. Conf. on Spoken Language Processing, Philadelphia, PA., USA. Box, G. E. P. (1980). Sampling and Bayes inference in scientific modeling. J. Roy. Stat. Soc., A., 143, 383-430. Bregler, C., Hild, H., Manke, S., & Waibel, A. (1993). Improving Connected Letter Recognition by Lipreading. In Proc. Int. Conf. on Acoust., Speech, and Signal Processing, volume 1, pages 557-560, Minneapolis. IEEE. B iilt hoff, H. H. & Yuille, A. L. (1996). A Bayesian framework for the integration of visual modules. In T. Inui & J. L. McClelland (Eds.), Attention and performance XVI: Information integmtion in perception and communication, pages 49-70. Cambridge, MA: MIT Press. Chadderdon, G. & Movellan, J. (1995). Testing for Channel Independence in Bimodal Speech Recognition. In Proceedings of 2nd Joint Symposium on Neuml Computation, pages 84-90. Clark, J. J. & Yuille, A. L. (1990). Data Fusion for Sensory Information Processing Systems. Boston: Kluwer Academic Publishers. Dampster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Stat. Soc., 39, 1-38. Efron, A. (1982). The jacknife, the bootstmp and other resampling plans. Philadelphia, Pennsylvania: SIAM. Gray, M. S., Movellan, J. R., & Sejnowski, T. (1997). Dynamic features for visual speechreading: A systematic comparison. In Mozer, Jordan, & Petsche (Eds.), Advances in Neuml Information Processing Syste"ms, volume 9. MIT Press. Movellan, J. R. (1995). Visual speech recognition with stochastic neural networks. In G. Tesauro, D. Touretzky, & T. Leen (Eds.), Advances in neuml information processing systems. Cambridge,Massacusetts: MIT Press. Movellan, J. R. & Chadderdon, G. (1996). Channel Separability in the Audio Visual Integration of Speech: A Bayesian Approach. In D. G. Stork & M. E. Hennecke (Eds.), Speechreading by Humans and Machines: Models, Systems, and Applications, pages 473-487. New York: NATO/Springer-Verlag. Movellan, J. R. & Prayaga, R. S. (1996). Gabor Mosaics: A description of Local Orientation Statistics with Applications to Machine Perception. In G. W. Cottrell (Ed.), proceedings of the Eight Annual Conference of the Cognitive Science Society, page 817. Mahwah, New Jersey: LEA. O'Hagan, A. (1994). Kendall's Advanced Theory of Statistics: Volume 2B, Bayesian Inference. volume 2B. Cambridge University Press. Wolff, G. J., Prasad, K. V., Stork, D. G., & Hennecke, M. E. (1994). Lipreading by Neural Networks: Visual Preprocessing, Learning and Sensory Integration. In J. D. Cowan, G. Tesauro, & J. Alspector (Eds.), Advances in Neuml Information Processing Systems, volume 6, pages 1027-1034. Morgan Kaufmann.
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A Neural Network Model of Naive Preference and Filial Imprinting in the Domestic Chick Lucy E. Hadden Department of Cognitive Science University of California, San Diego La Jolla, CA 92093 hadden@cogsci.ucsd.edu Abstract Filial imprinting in domestic chicks is of interest in psychology, biology, and computational modeling because it exemplifies simple, rapid, innately programmed learning which is biased toward learning about some objects. Hom et al. have recently discovered a naive visual preference for heads and necks which develops over the course of the first three days of life. The neurological basis of this predisposition is almost entirely unknown; that of imprinting-related learning is fairly clear. This project is the first model of the predisposition consistent with what is known about learning in imprinting. The model develops the predisposition appropriately, learns to "approach" a training object, and replicates one interaction between the two processes. Future work will replicate more interactions between imprinting and the predisposition in chicks, and analyze why the system works. 1 Background Filial imprinting iIi domestic chicks is of interest in psychology, biology, and computational modeling (O'Reilly and Johnson, 1994; Bateson and Hom, 1994) because it exemplifies simple, rapid, innately programmed learning which is biased toward learning about some particular objects, and because it has a sensitive period in which learning is most efficient. Domestic chicks will imprint on almost anything (including boxes, chickens, and humans) which they see for enough time (Hom, 1985). Hom and his colleagues (Hom, 1985) have recently found a naive visual preference (predisposition) for heads and necks which develops over the course of the first three days of life. In particular, the birds prefer to approach objects shaped like heads and necks, even if they are the heads and necks of other species, including ducks and polecats (Hom, 1985). This preference interacts interestingly with filial imprinting, or learning to recognize a parent. Chicks can sti1llearn about (and imprint 32 L E.Hadden on) other objects even in the presence of this predisposition, and the predisposition can override previously learned preferences (Johnson et aI., 1985), which is usually hard with imprinted chicks. These interactions are like other systems which rely on naive preferences and learning. While the neurological basis of imprinting is understood to some extent, that of the predisposition for heads and necks is only beginning to be investigated. Imprinting learning is known to take place in IMHY (intermediate and medial portions of the hyperstriatum ventrale) (Hom, 1985), and to rely on noradrenaline (Davies et ai., 1992). The predisposition's location is currently unknown, but its strength correlates with plasma testosterone levels (Hom, 1985). 1.1 Previous Models No previous models of imprinting have incorporated the predisposition in any meaningful way. O'Reilly & Johnson's (1994) model focussed on accounting for the sensitive period via an interaction between hysteresis (slow decay of activation) and a Hebbian learning rule, and ignored the predisposition. The only model which did try to include a predisposition (Bateson and Hom, 1994) was a 3-layer Hebbian network with real-valued input vectors, and outputs which represented the strength of an "approach" behavior. Bateson and Hom (1994) found a "predisposition" in their model by comparing networks trained on input vectors of Os and Is (High) to vectors where non-zero entries were 0.6 (Low). Untrained networks preferred (produced a higher output value for) the high-valued input ("hen"), and trained networks preferred the stimulus they were trained on ("box"). Of course, in a network with identical weights, an input with higher input values will naturally excite an output unit more than one with lower input values. Thus, this model's predisposition is implicit in the input values, and is therefore hard to apply to chicks. In this project, I develop a model which incorporates both the predisposition and imprinting, and which is as consistent as possible with the known neurobiology. The overall goals of the project are to clarify how this predisposition might be implemented, and to examine more generally the kinds of representations that underlie naive preferences that interact with and facilitate, rather than replace, learning. These particular simulations show that the model exhibits the same qualitative behavior as chicks under three important sets of conditions. The rest of the paper first describes the architecture of the current model (in general terms and then in more detail). It goes on to describe the particular simulations, and then compares the results of those simulations with the data gathered from chicks. 2 Architecture The neural network model's architecture is shown in Figure 1. The input layer is a 6x6 pixel "retina" to which binary pictures are presented. The next layer is a feature detector. The predisposition serves as the home of the network's naive preference, while the IMLL (intermediate learning layer) is intended to correspond to a chick's IMHY, and is where the network stores its learned representations. The output layer consists of two units which are taken to represent different action patterns (following Bateson and Hom (1994): an "approach" unit and a "withdraw" unit. These are the two chick behaviors which researchers use to assess a chick's degree of preference for a particular stimulus. The feature detector provides input to the predisposition and IMLL layers; they in tum provide input to the output layer. Where there are connections, layers (and subparts) are fully interconnected. The feature detector uses a linear activation function; the rest of the network has a hyperbolic tangent activation function. All activations and all connections can be either positive Model of Predispositions in Chicks Feature Predisp. Input IMLL · .... ········· .. ·····0 = Fixed Weights ........... = Modifiable Weights Figure 1: The network architecture sketched. All connections are feedforward (from input toward output) only. 33 Ou ut box head/neck cylinder Figure 2: The three input patterns used by the network. They have between 16 and 18 pixels each, and the central moment of each image is the same. or negative; the connections are limited to ±0.9. Most of the learning takes place via a covariant Hebb rule, because it is considered to be plausible neurally. The lowest level ofthe network is a feature-detecting preprocessor. The current implementation of this network takes crude 6x6 binary pictures (examples of which can be seen in Fig. 2), and produces a IS-place floating-point vector. The output units of the feature detector are clustered into five groups of three units each; each group of three units operates under a winner-take-all rule, in order to increase the difference between preprocessed patterns for the relevant pictures. The feature detector was trained on random inputs for 400 cycles with a learning rate of .01, and its weights were then frozen. Training on random input was motivated by the finding that the lower levels of the visual system require some kind of input in order to organize; Miller et al. (1989) suggest that, at least in cats, the random firing of retinal neurons is sufficient. The predisposition layer was trained via backprop using the outputs of the feature detector as its inputs. The pattern produced by the "head-neck" picture in the feature detector was trained to excite the "approach" output unit and inhibit the "withdraw" unit; other patterns were trained to a neutral value on both output units. These weights were stored, and treated as fixed in the larger network. (In fact, these weights were scaled down by a constant factor (.8) before being used in the larger network.) Since this is a naive preference, or predisposition, these weights are assumed to be fixed evolutionarily. Thus, the method of setting them is irrelevant; they could also have been found by a genetic algorithm. The IMLL layer is a winner-take-all network of three units. Its connections with the feature detector's outputs are learned by a Hebb rule with learning rate .01 and a weight decay (to 0) term of .0005. For these simulations, its initial weights were fixed by hand, in a pattern which insured that each IMLL unit received a substantially different value for the same input pattern. This pattern of initial weights also increased the likelihood that the three patterns of interest in the simulations maximally affected different IMLL units. As previously mentioned, the output layer consists of an "approach" and a "withdraw" unit. It also learns via a Hebb rule, with the same learning rate and decay term as IMLL. Its connections with IMLL are learned; those with the predisposition layer are fixed. Initial weights between IMLL and the output layer are random, and vary from -0.3 to 0.3. The bias to the approach unit is 0; that to the withdraw unit is 0.05. 34 L E. Hadden 2.1 Training In the animal experiments on which this model is based, chicks are kept in the dark (and in isolation) except for training and testing periods. Training periods involve visual exposure to an object (usually a red box); testing involves allowing the chick to choose between approaching the training object and some other object (usually either a stuffed hen or a blue cylinder) (Hom, 1985). The percentage of time the chick approaches the training object (or other object of interest) is its preference score for that object (Hom, 1985). A preference score of 50% indicates indifference; scores above 50% indicate a preference for the target object, and those below indicate a preference for the other object. For the purposes of modeling, the most relevant information is the change (particularly the direction of change) in the preference score between two conditions. Following this approach, the simulations use three preset pictures. One, a box, is the only one for which weights are changed; it is the training pattern. The other two pictures are test patterns; when they are shown, the network's weights are not altered. One of these test patterns is the head/neck picture on which the predisposition network was trained; the other is a cylinder. As with chicks, the behavioral measure is the preference score. For the network, this is calculated as pref. score = 100 x at / (at + ac ), where at is the activation of the approach unit when the network is presented with the training (or target) picture, and ac is the activation of the approach unit given the comparison picture. It is assumed that both values are positive; otherwise. the approach unit is taken to be off. In these simulations, the network gets the training pattern (a "box") during training periods, and random input patterns (simulating the random firing of retinal neurons) otherwise. The onset of the predisposition is modeled by allowing the predisposition layer to help activate the outputs only after the network receives an "experience" signal. This signal models the sharp rise in plasma testosterone levels in dark-reared chicks following any sort of handling (Hom, 1985). Once the network has received the "experience" signal, the weights are modified for random input as well as for the box picture. Until then, weights are modified only for the box picture. Real chicks can be tested only once because of the danger of one-trial learning, so all chick data compares the behavior of groups of chicks under different conditions. The network's weights can be kept constant during testing, and the same network's responses can be measured before and after it is exposed to the relevant condition. All simulations were 100 iterations long. 3 Simulations The simulations using this model currently address three phenomena which have been studied in chicks. First, in simple imprinting chicks learn to recognize a training object, and usually withdraw from other objects once they have imprinted on the training object. This simulation requires simply exposing the network to the training object and measuring its responses. The model "imprints" on the box if its preference for the box relative to both the head/neck and cylinder pictures increases during training. Ideally, the value of the approach unit for the cylinder and box will also decrease, to indicate the network's tendency to withdraw from "unfamiliar" stimuli. Second, chicks with only the most minimal experience (such as being placed in a dark running wheel) develop a preference for a stuffed fowl over other stimuli. That is, they will approach the fowl significantly more than another object (Hom, 1985). This is modeled by turning on the "predisposition" and allowing the network to develop with no training whatsoever. The network mimics chick behavior if the preference score for the head/neck picture increases relative to the box and the cylinder pictures. Third, after the predisposition has been allowed to develop, training on a red box decreases Model of Predispositions in Chicks 35 +50 +50 -50 a d -50 Figure 3: A summary of the results of the model. All bars are differences in preference scores between conditions for chicks (open bars) and the model (striped bars). a: Imprinting (change in preference for training object): trained - untrained. b: Predisposition (change in preference for fowl): experience - no experience (predisposition - no predisposition). c: Change in preference for fowl vs. box: trained - predisposition only. d: Change in preference for box vs. cylinder: trained - predisposition only. (Chick data adapted from (Hom, 1985; Bolhuis et aI., 1989).) a chick's preference for the fowl relative to the box. It also increases the chick's preference for the box relative to a blue cylinder or other unfamiliar object (Bolhuis et aI., 1989). In the model, the predisposition layer is allowed to activate the output layer for 20 iterations before training starts. Then the model is exposed to the network for 25 iterations. If its preference score for the fowl decreases after training, the network has shown the same pattern as chicks. 4 Results and Discussion A summary of the results is shown in Figure 3. Since these simulations try to capture the qualitative behavior of chicks, all results are shown as the change in preference scores between two conditions. For the chick data, the changes are approximate, and calculated from the means only. The network data is the average of the results for 10 networks, each with a different random seed (and therefore initial weight patterns). For the three conditions tested, the model's preference scores moved in the same direction as the chicks. The interaction between imprinting and the predisposition cannot be investigated computationally unless the model displays both behaviors. These baseline behaviors are shown in Fig. 3-a and b. Trained chicks prefer the training object more after training than before (Hom, 1985); so does the model (Fig. 3-a). In the case of the predisposition (Fig. 3-b), the bird data is a difference between preferences for a stuffed fowl in chicks which had developed the predisposition (and therefore preferred the fowl) and those which had not (and therefore did not). Similarly, the network preferred the head/neck picture more after the predisposition had been allowed to develop than at the beginning of the simulation. The interactions between imprinting and the predisposition are the real measures of the model's success. In Fig. 3-c, the predisposition has been allowed to develop before training begins. Trained birds with the predisposition were compared with untrained birds also with the predisposition (trained - untrained). Trained birds preferred the stuffed fowl less than their untrained counterparts (Bolhuis et aI., 1989). The network's preference score just before training is subtracted from its score after training. As with the real chicks, the network prefers the head/neck picture less after training than it did before. Fig. 3-d shows that, as with chicks, the network's preference for the box increased relative to that for the cylinder during the course of training. For these three conditions, then, the model is 36 L E.Hadden qualitatively a success. 4.1 Discussion of Basic Results The point of network models is that their behavior can be analyzed and understood morc easily than animals'. The predisposition's behavior is quite simple: to the extent that a random input pattern is similar to the head/neck picture, it activates the predisposition layer, and through it the approach unit. Thus the winning unit in IMLL is correlated with the approach unit, and the connections are strengthened by the Hebb rule. Imprinting is similar, but only goes through the IMLL layer, so the approach unit may not be on. In both cases, the weights from the other units decay slowly during training, so that usually the other input patterns fail to excite the approach unit, and even excite the withdraw unit slightly because of its small positive bias. Only one process is required to obtain both the predisposition and imprinting, since both build representations in IMLL. The interaction between imprinting and the predisposition first increases the preference for the predisposition, and then alters the weights affecting the reaction to the box picture. The training phase acts just like the ordinary imprinting phase, so that preference for both the head/neck and the cylinder decrease during training. Some exploration of the relevant parameters suggests that the predisposition's behavior does not depend simply on its strength. Because IMLL is a winner-take-alliayer, changing the predisposition's strength can, by moving the winning node around during training, cause previous learning to be lost. Such motion obviously has a large effect on the outcome of the simulation. 4.2 Temporal aspects of imprinting The primary weakness of the model is its failure to account for some critical temporal aspects of imprinting. It is premature to draw many conclusions about chicks from this model, because it fails to account for either long-term sensitive periods or the short-term time course of the predisposition. Neither the predisposition nor imprinting in the model have yet been shown to have sensitive periods, though both do in real birds (Hom, 1985; Johnson et aI., 1985). Preliminary results, however, suggest that imprinting in the networks does have a sensitive period, presumably because of weight saturation during learning. It is not yet clear whether the predisposition's sensitive period will require an exogenous process. Second, the model does not yet show the appropriate time course for the development of the predisposition. In chicks, the predisposition develops fairly slowly over the course of five or so hours (Johnson et al., 1985). In chicks for which the first experience is training, the predisposition's effect is to increase the bird's preference for the fowl regardless of training object, over the course of the hours following training (Johnson et aI., 1985). In the model, the predisposition appears quickly and, because of weight decay and other factors, the strength of the predisposition slowly decreases over the iterations following training, rather than increasing. Increasing the learning rate of IMLL over time could solve this problem. Once it exhibits time course behaviors, especially if no further processes need to be postulated, the model will facilitate interesting analyses of how a simple set of processes and assumptions can interact to produce highly complicated behavior. 5 Conclusion This model displays some important interactions between learning and a predisposition in filial imprinting. It is the first which accounts for the predisposition at all. Other models Model of Predispositions in Chicks 37 of imprinting have either ignored the issue or built in the predisposition by hand. In this model, the interaction between two simple systems, a fixed predisposition and a learned approach system, gives rise to one important more complex behavior. In addition, the two representations of the head/neck predisposition can account for lesion studies in which lesioning IMLL removes a chick's memory of its training object or prevents it from learning anything new about specific objects, but leaves the preference for heads and necks intact (Hom, 1985). Clearly, if the IMLL layer is missing, the network loses any infonnation it might have learned about training objects, and is unable to learn anything new from future training. The predisposition, however, is still intact and able to influence the network's behavior. The nature of predispositions like chicks' naive preference for heads and necks, and how they interact with learning, are interesting in a number of fields. Morton and Johnson (1991) have already explored the similarities between chicks' preferences for heads and necks and human infants' preferences for human faces. Such naive preferences are also important in any discussion of innate infonnation, and the number of processes needed to handle innate and learned infonnation. Although this model and its successors cannot directly address these issues, I hope that their explication of how fairly general predispositions can influence learning will improve understanding of some of the mechanisms underlying them. Acknowledgements This work was supported by a fellowship from the National Physical Sciences Consortium. References P. Bateson and G. Hom. Imprinting and recognition memory: A neural net model. Animal Behaviour, 48(3):695-715,1994. J. J. Bolhuis, M. H. Johnson, and G. Hom. Interacting mechanisms during the fonnation of filial preferences: The development of a predisposition does not prevent learning. Journal of Experimental Psychology: Animal Behavior Processes, 15(4):376-382, 1989. D. C. Davies, M. H. Johnson, and G. Hom. The effect of the neurotoxin dsp4 on the development of a predisposition in the domestic chick. Developmental Psychobiology, 25(2):251-259, 1992. G. Hom. Memory, Imprinting, and the Brain: An inquiry into mechanisms. Clarendon Press, Oxford, 1985. M. H. Johnson, J. 1. Bolhuis, and G. Horn. Interaction between acquired preferences and developing predispositions during imprinting. Animal Behaviour, 33(3): 1 000-1 006, 1985. K. Miller, J. Keller, and M. Stryker. Ocular dominance column development: analysis and simulation. Science, 245:605-615, 1989. J. Morton and M. H. Johnson. Conspec and conlern: a two-process theory of infant face recognition. Psychological Review, 98(2): 164-181, 1991. R. C. O'Reilly and M. H. Johnson. Object recognition and sensitive periods: A computational analysis of visual imprinting. Neural Computation, 6(3):357-389,1994.
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Shared Context Probabilistic Transducers Yoshua Bengio* Dept. IRO, Samy Bengio t Microcell Labs, Universite de Montreal, Montreal (QC) , Canada, H3C 3J7 bengioyOiro.umontreal.ca 1250, Rene Levesque Ouest, Montreal (QC) , Canada, H3B 4W8 samy.bengioOmicrocell.ca Jean-Fran~ois Isabellet Microcell Labs, 1250, Rene Levesque Ouest, Montreal (QC), Canada, H3B 4W8 jean-francois.isabelleCmicrocell.ca Abstract Yoram Singer AT&T Laboratories, Murray Hill, NJ 07733, USA, singerOresearch.att.com Recently, a model for supervised learning of probabilistic transducers represented by suffix trees was introduced. However, this algorithm tends to build very large trees, requiring very large amounts of computer memory. In this paper, we propose anew, more compact, transducer model in which one shares the parameters of distributions associated to contexts yielding similar conditional output distributions. We illustrate the advantages of the proposed algorithm with comparative experiments on inducing a noun phrase recogmzer. 1 Introduction Learning algorithms for sequential data modeling are important in many applications such as natural language processing and time-series analysis, in which one has to learn a model from one or more sequences of training data. Many of these algorithms can be cast as weighted transducers (Pereira, Riley and Sproat, 1994), which associate input sequences to output sequences, with weights for each input/output * Yoshua Bengio is also with AT&T Laboratories, Holmdel, NJ 07733, USA. t This work was performed while Samy Bengio was at INRS-Telecommunication, Iledes-Soeurs, Quebec, Canada, H3E IH6 t This work was performed while Jean-Franc;ois Isabelle was at INRS-Telecommunication, Ile-des-Soeurs, Quebec, Canada, H3E IH6 410 y. Bengio, S. Bengio, J-F. Isabelle and Y. Singer sequence pair. When these weights are interpreted as probabilities, such models are called probabilistic transducers. In particular, a probabilistic transducer can represent the conditional probability distribution of output sequences given an input sequence. For example, algorithms for combining several transducers were found useful in natural language and speech processing (Riley and Pereira, 1994). Very often, weighted transducers use an intermediate variable that represents "context", such as the state variable of Hidden Markov Models (Baker, 1975; Jelinek, 1976). A particular type of weighted transducer, called Input/Output Hidden Markov Model, is one in which the input-to-context distribution and context-to-output distribution are represented by flexible parameterized models (such as neural networks) (Bengio and Frasconi, 1996). In this paper, we will study probabilistic transducers with a deterministic input-to-state mapping (i.e., a function from the past input subsequence to the current value of the context variable). One such transducer is the one which assigns a value of the context variable to every value of the past input subsequence already seen in the data. This input-to-state mapping can be efficiently represented by a tree. Such transducers are called suffix tree transducers (Singer, 1996). A problem with suffix tree transducers is that they tend to yield very large trees (whose size may grow as O(n2) for a sequence of data of length n). For example, in the application studied in this paper, one obtains trees requiring over a gigabyte of memory. Heuristics may be used to limit the growth of the tree (e.g., by limiting the maximum depth of the context, i.e., of the tree, and by limiting the maximum number of contexts, i.e., nodes of the tree). In this paper, instead, we propose a new model for a probabilistic transducer with deterministic input-to-state function in which this function is compactly represented, by sharing parameters of contexts which are associated to similar output distributions. Another way to look at the proposed algorithm is that it searches for a clustering of the nodes of a suffix tree transducer. The data structure that represents the contexts is not anymore a tree but a single-root acyclic directed graph. 2 Background: Suffix Tree Probabilistic Transducers The learning algorithm for suffix tree probabilistic transducers (Singer, 1996) constructs the model P(Yilxi) from discrete input sequences xi = {Xl,X2, ... ,Xn} to output sequences yi = {Y1, Y2, ... , Yn}, where Xt are elements of a finite alphabet Ein. This distribution is represented by a tree in which each internal node may have a child for every element of Ein, therefore associating a label E Ein to each arc. A node at depth d is labeled with the sequence ut of labels on arcs from root to node, corresponding to a particular input context, e.g., at some position n in the sequence a context of length d is the value ut of the preceding subsequence x~_d l' Each node at depth d is therefore associated with a model of the output distribution in this context, P(Yn IX~-d+1 = ut) (independent of n). To obtain a local output probability for Yn (i.e., given xi), one follows the longest possible path from the root to a node a depth d according to the labels xn, xn -1, ... Xn-d+1. The local output probability at this node is used to model Yn' Since p(yflxf) can always be written n~=1 P(Yn Ixi)' the overall input/output conditional distribution can be decomposed, according to this model, as follows: T p(yflxf) = II P(YnIX~_d(x~)+l)' (1) n=1 where d(xi) is the depth of the node of the tree associated with the longest suffix uf = x~_d+1 of xi. Figure 1 gives a simple example of a suffix tree transducer. Shared Context Probabilistic Transducers Pfaj"O.S Pfb)~OJ P(c)", 0 2 11 p(a)",O.6 p(b) _O.) p(c:)= O I 411 Figure 1: ExampJe of suffix tree transducer (Singer, 1996). The input alphabet, E in = {O, I} and the output aJphabet, EotJt = {a, b, c}. For instance, P(aIOOllO) = P(alllO) = 0.5. 3 Proposed Model and Learning Algorithm In the model proposed here, the input/output conditional distribution p(yT I xI) is represented by a single-root acyclic directed graph. Each node of this graph is associated with a set of contexts Cnode = {(jt·}, corresponding to all the paths i (of various lengths di ) from the root of the tree to this node. All these contexts are associated with the same local output distribution P(Yn Ix? has a suffix in Cnode). Like in suffix tree transducers, each internal node may have a child for every element of E in . The arc is labeled with the corresponding element of ~in . Also like in suffix tree transducers, to obtain P(Ynlx~), one follows the path from the root to the deepest node called deepest(x?) according to the labels Xn, Xn-l, etc .. . The local output distribution at this node is used to predict Yn or its probability. The overall conditional distribution is therefore given by T P(yilxf) = II P(Ynldeepest(x~)) (2) n=l where the set of contexts Cdeepe3t(x~) associated to the deepest node deepest(xl) contains a suffix of x? The model can be used both to compute the conditional probability of a given input/output sequence pair, or to guess an output sequence given an input sequence. Note that the input variable can contain delayed values of the output variable (as in Variable Length Markov Models). 3.1 Proposed Learning Algorithm We present here a constructive learning algorithm for building the graph of the model and specify which data points are used to update each local output model (associated to nodes of the graph). The algorithm is on-line and operates according to two regimes: (1) adding new nodes and simply updating the local output distributions at existing nodes, and (2) merging parts of the graph which represent similar distributions. If there are multiple sequences in the training data they are concatenated in order to obtain a single input/output sequence pair. (1) After every observation (xn, Yn), the algorithm updates the output distributions 412 y. Bengio, S. Bengio, J-F. Isabelle and Y. Singer of the nodes for which Cnode(x~) contains a suffix of Xl, possibly adding new nodes (with labels x~_d.) until xl E Cnode for some node. (2) Every Tmerge observations, the algorithm attempts to merge sub-graphs which are found similar enough, by comparing the N (N - 1) /2 pairs of sub-graphs rooted at the N nodes that have seen at least minn observations. Merging two subgraphs is equivalent to forcing them to share parameters (as well as reducing the size of the representation of the distribution). A merge is performed between the graphs rooted at nodes a and b if Ll(a, b) < mina and the merge succeeds. The details of the similarity measure and merging algorithm are given in the next subsections. 3.2 Similarity Measure Between Rooted Subgraphs In order to compare (asymmetrically) output distributions P(yla) two nodes a and b, one can use the Kullback-Liebler divergence: _ ~ P(ylb) Ii. L(a, b) = L...J P(ylb) log P(yla) yEEout and P(ylb) at (3) However, we want to compare the whole acyclic graphs rooted at these 2 nodes. In order to do so, let us define the following. Let s be a string of input labels, and b a node. Define desc(b, s) as the most remote descendant of b obtained by following from b the arcs whose labels correspond to the sequence s. Let descendents(a) be the set of strings obtained by following the arcs starting from node a until reaching the leaves which have a as an ancestor. Let P(sla) be the probability offollowing the arcs according to string s, starting from node a. This distribution can be estimated by counting the relative number of descendents through each of the children of each node. To compare the graphs rooted at two nodes a and b, we extend the KL divergence by weighing each of the descendents of a, as follows: W K L(a, b) = L P(sla)K L(desc(a, s), desc(b, s)) (4) 3 E de3cendent8 (a) Finally, to obtain a symmetric measure, we define Ll(a,b) = WKL(a,b) + WKL(b,a) (5) that is used in the merge phase of the constructive learning algorithm to decide whether the subgraphs rooted at a and b should be merged. 3.3 Merging Two Rooted Subgraphs If Ll (a, b) < mina (a predefined threshold) we want to merge the two subgraphs rooted at a and b and create a new subgraph rooted at c. The local output distribution at c is obtained from the local output distributions at a and b as follows: P(Yn Ic) = P(Ynla)P(ala or b) + P(Ynlb)P(bla or b) (6) where we define ad(a) P(ala or b) = ad(a) + ad(b) , (7) where d(a) is the length ofthe longest path from the root to node a, and a represents a prior parameter (between 0 and 1) on the depth of the acyclic graphs. This prior parameter can be used to induce a prior distribution over possible rooted acyclic graphs structures which favors smaller graphs and shorter contexts (see the mixture of probabilistic transducers of (Singer, 1996)). The merging algorithm can then be summarized as follows: Shared Context Probabilistic Transducers 413 • The parents of a and b become parents for c. • Some verifications are made to prevent merges which would yield to cycles in the graph. The nodes a and b are not merged if they are parents of one another. • We make each child of a a child of c. For each child u of b (following an arc labeled l), look for the corresponding child v of c (also following the arc labeled l) . If there is no such child, and u is not a parent of c, make u a new child of c. Else, if u and v are not parents of each other, recursively merge them. • Delete nodes a and b, as well as all the links from and to these nodes. This algorithm is symmetric with respect to a and b except when a merge cannot be done because a and b are parents of one another. In this case, an asymmetric decision must be taken: we chose to keep only a and reject b. Figure 2 gives a simple example of merge. Figure 2: This figure shows how two nodes are merged. The result is no longer a tree, but a directed graph. Some verifications are done to avoid cycles in the graph. Each node can have multiple labels, corresponding to the multiple possible paths from the root to the node. 4 Comparative Experiments We compared experimentally our model to the one proposed in (Singer, 1996) on mixtures of suffix tree transducers, using the same task. Given a text where each word is assigned an appropriate part-of-speech value (verb, noun, adjective, etc), the task is to identify the noun phrases in the text. The UPENN tree-bank corpus database was used in these experiments. The input vocabulary size, IEinl = 41, is the number of possible part-of-speech tags, and the output vocabulary size is IEouti = 2. The model was trained over 250000 marked tags, constraining the tree to be of maximal depth 15. The model was then tested (freezing the model structure and its parameters) over 37000 other tags. Using the mixture of suffix tree transducers (Singer, 1996) and thresholding the output probability at 0.5 to take output decisions, yielded an accuracy rate of 97.6% on the test set, but required over 1 gigabyte of computer memory. To make interesting comparisons with the shared context transducers, we chose the following experimental scheme. Not only did we fix the maximal depth of the directed graph to 15, but we also fixed the maximal number of allocated nodes, i.e., simulating fixed memory resources. When this number was reached, we froze the structure but continued to update the parameters of the model until the end of the training database was reached. For the shared context version, whenever a merge freed some nodes, we let the graph grow again to its maximal node size. At the end of this process, we evaluated the model on the test set. 414 Y. Bengio, S. Bengio, J-F. Isabelle and Y. Singer We tested this method for various values of the maximum number of nodes in the graph. For each experiment, we tried different values of the other parameters (the similarity threshold min~ for merging, the minimum number of observations miIln at a node before it can be considered for a merge, and the delay Tmerge between two merging phases), and we picked the one which performed the best on the training set. Results are reported in figure 3. maximal with without 005 number merge merge 09 of nodes (%) (%) 085 20 0.762 0.584 01 50 0.827 0.624 075 100 0.861 0.727 500 0.924 0.867 07 1000 0.949 0.917 085 2000 0.949 0.935 01 5000 0.952 0.948 055 Figure 3: This figure shows the generalization accuracy rate of a transducer with merges (shared contexts graph) against one without merges (suffix tree), with different maximum number of nodes. The maximum number of nodes are in a logarithmic scale, and the accuracy rates are expressed in relative frequency of correct classification. As can be seen from the results, the accuracy rate over the test set is better for transducers with shared contexts than without. More precisely, the gain is greater when the maximum number of nodes is smaller. When we fix the maximum number of nodes to a very small value (20), a shared context transducer performs 1.3 times better (in classification error) than a non-shared one. This gain becomes smaller and smaller as the maximum size increases. Beyond a certain maximum size, there is almost no gain, and one could probably observe a loss for some large sizes. We also need to keep in mind that the larger the transducer is, the slower the program to create the shared context transducer is, compared to the non-shared one. Finally, it is interesting to note that using only 5000 nodes, we were able to obtain 95.2% accuracy, which is only 2.4% less than those obtained with no constraint on the number of nodes. 5 Conclusion In this paper, we have presented the following: • A new probabilistic model for probabilistic transducers with deterministic input-to-state function, represented by a rooted acyclic directed graph with nodes associated to a set of contexts and children associated to the different input symbols. This is a generalization of the suffix tree transducer. • A constructive learning algorithm for this model, based on construction and merging phases. The merging is obtained by clustering parts of the graph which represent a similar conditional distribution. • Experimental results on a natural-language task showing that when the size of the graph is constrained, this algorithm performs better than the purely constructive (no merge) suffix tree algorithm. Shared Context Probabilistic Transducers 4}5 References Baker, J. (1975). Stochastic modeling for automatic speech understanding. In Reddy, D., editor, Speech Recognition, pages 521-542. Academic Press, New York. Bengio, S. and Bengio, Y. (1996). An EM algorithm for asynchronous input/output hidden markov models. In Proceedings of the International Conference on Neural Information Processing, Honk Kong. Bengio, Y. and Frasconi, P. (1996). Input/Output HMMs for sequence processing. IEEE Transactions on Neural Networks, 7(5):1231-1249. Jelinek, F. (1976). Continuous speech recognition by statistical methods. Proceedings of the IEEE, 64:532-556. Pereira, F., Riley, M., and Sproat, R. (1994). Weighted rational transductions and their application to human language processing. In ARPA Natural Language Processing Workshop. Riley, M. and Pereira, F. (1994). Weighted-finite-automata tools with applications to speech and language processing. Technical Report Technical Memorandum 11222-931130-28TM, AT&T Bell Laboratories. Singer, Y. (1996). Adaptive mixtures of probabilistic transducers. In Mozer, M., Touretzky, D., and Perrone, M., editors, Advances in Neural Information Processing Systems 8. MIT Press, Cambridge, MA.
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Learning to Schedule Straight-Line Code Eliot Moss, Paul Utgoff, John Cavazos Doina Precup, Darko Stefanovic . Dept. of Compo Sci., Univ. of Mass. Amherst, MA 01003 Abstract Carla Brodley, David Scheeff Sch. of Elec. and Compo Eng. Purdue University W. Lafayette, IN 47907 Program execution speed on modem computers is sensitive, by a factor of two or more, to the order in which instructions are presented to the processor. To realize potential execution efficiency, an optimizing compiler must employ a heuristic algorithm for instruction scheduling. Such algorithms are painstakingly hand-crafted, which is expensive and time-consuming. We show how to cast the instruction scheduling problem as a learning task, obtaining the heuristic scheduling algorithm automatically. Our focus is the narrower problem of scheduling straight-line code (also called basic blocks of instructions). Our empirical results show that just a few features are adequate for quite good performance at this task for a real modem processor, and that any of several supervised learning methods perform nearly optimally with respect to the features used. 1 Introduction Modem computer architectures provide semantics of execution equivalent to sequential execution of instructions one at a time. However, to achieve higher execution efficiency, they employ a high degree of internal parallelism. Because individual instruction execution times vary, depending on when an instruction's inputs are available, when its computing resources are available, and when it is presented, overall execution time can vary widely. Based on just the semantics of instructions, a sequence of instructions usually has many permutations that are easily shown to have equivalent meaning-but they may have considerably different execution time. Compiler writers therefore include algorithms to schedule instructions to achieve low execution time. Currently, such algorithms are hand-crafted for each compiler and target processor. We apply learning so that the scheduling algorithm is constructed automatically. Our focus is local instruction scheduling, i.e., ordering instructions within a basic block. A basic block is a straight-line sequence of code, with a conditional or unconditional branch instruction at the end. The scheduler should find optimal, or good, orderings of the instructions prior to the branch. It is safe to assume that the compiler has produced a semantically correct sequence of instructions for each basic block. We consider only reorderings of each sequence 930 E. Moss, P. UtgofJ, J Cavazos, D. Precup, D. Stefanovic, C. Bradley and D. Scheeff (not more general rewritings), and only those reorderings that cannot affect the semantics. The semantics of interest are captured by dependences of pairs of instructions. Specifically, instruction Ij depends on (must follow) instruction Ii if: it follows Ii in the input block and has one or more of the following dependences on Ii: (a) Ij uses a register used by Ii and at least one of them writes the register (condition codes, if any, are treated as a register); (b) Ij accesses a memory location that may be the same as one accessed by Ii, and at least one of them writes the location. From the input total order of instructions, one can thus build a dependence DAG, usually a partial (not a total) order, that represents aU the semantics essential for scheduling the instructions of a basic block. Figure 1 gives a sample basic block and its DAG. The task of scheduli.ng is to find a least-cost total order of each block's DAG. X~V ; Y=*P; P =P+ I; I: STQ RI.X 2: LDQ R2,O(RIO) 3: STQ R2, Y 4: ADDQ RIO,RIO,S v\ ~ A~~\ Not Available Available (a) C Code (b) Instruction Sequence to be Scheduled (c) Dependence Dag of Instructions (d) Partial Schedule Figure 1: Example basic block code, DAG, and partial schedule 2 Learning to Schedule The learning task is to produce a scheduling procedure to use in the performance task of scheduling instructions of basic blocks. One needs to transform the partial order of instructions into a total order that will execute as efficiently as possible, assuming that all memory references "hit" in the caches. We consider the class of schedulers that repeatedly select the apparent best of those instructions that could be scheduled next, proceeding from the beginning of the block to the end; this greedy approach should be practical for everyday use. Because the scheduler selects the apparent best from those instructions that could be selected next, the learning task consists of learning to make this selection well. Hence, the notion of 'apparent best instruction' needs to be acquired. The process of selecting the best of the alternatives is like finding the maximum of a list of numbers, One keeps in hand the current best, and proceeds with pairwise comparisons, always keeping the better of the two. One can view this as learning a relation over triples (P,Ii,Ij), where P is the partial schedule (the total order of what has been scheduled, and the partial order remaining), and I is the set of instructions from which the selection is to be made. Those triples that belong to the relation define pairwise preferences in which the first instruction is considered preferable to the second. Each triple that does not belong to the relation represents a pair in which the first instruction is not better than the second. One must choose a representation in which to state the relation, create a process by which correct examples and counter-examples of the relation can be inferred, and modify the expression of the relation as needed. Let us consider these steps in greater detail. 2.1 Representation of Scbeduling Preference The representation used here takes the form of a logical relation, in which known examples and counter-examples of the relation are provided as triples. It is then a matter of constructing or revising an expression that evaluates to TRUE if (P,Ii,!j) is a member of the relation, and FALSE if it is not. If (P, Ii, Ij) is considered to be a member of the relation, then it is safe to infer that (P,Ij,Ii) is not a member, For any representation of preference, one needs to represent features of a candidate instruction and of the partial schedule. There is some art in picking useful features for a state. The method Learning to Schedule Straight-line Code 931 used here was to consider the features used in a scheduler (called DEC below) supplied by the processor vendor, and to think carefully about those and other features that should indicate predictive instruction characteristics or important aspects of the partial schedule. 2.2 Inferring Examples and Counter-Examples One would like to produce a preference relation consistent with the examples and counterexamples that have been inferred, and that generalizes well to triples that have not been seen. A variety of methods exist for learning and generalizing from examples, several of which are tested in the experiments below. Of interest here is how to infer the examples and counterexamples needed to drive the generalization process. The focus here is on supervised learning (reinforcement learning is mentioned later), in which one provides a process that produces correctly labeled examples and counter-examples of the preference relation. For the instruction-scheduling task, it is possible to search for an optimal schedule for blocks of ten or fewer instructions. From an optimal schedule, one can infer the correct preferences that would have been needed to produce that optimal schedule when selecting the best instruction from a set of candidates, as described above. It may well be that there is more than one optimal schedule, so it is important only to infer a preference for a pair of instructions when the first can produce some schedule better than any the second can. One should be concerned whether training on preference pairs from optimally scheduled small blocks is effective, a question the experiments address. It is worth noting that for programs studied below, 92% of the basic blocks are of this small size, and the average block size is 4.9 instructions. On the other hand, larger blocks are executed more often, and thus have disproportionate impact on program execution time. One could learn from larger blocks by using a high quality scheduler that is not necessarily optimal. However, the objective is to be able to learn to schedule basic blocks well for new architectures, so a useful learning method should not depend on any pre-existing solution. Of course there may be some utility in trying to improve on an existing scheduler, but that is not the longer-term goal here. Instead, we would like to be able to construct a scheduler with high confidence that it produces good schedules. 2.3 Updating the Preference Relation A variety of learning algorithms can be brought to bear on the task of Updating the expression of the preference relation. We consider four methods here. The first is the decision tree induction program m (Utgoff, Berkman & Clouse, in press). Each triple that is an example of the relation is translated into a vector offeature values, as described in more detail below. Some of the features pertain to the current partial schedule, and others pertain to the pair of candidate instructions. The vector is then labeled as an example of the relation. For the same pair of instructions, a second triple is inferred, with the two instructions reversed. The feature vector for the triple is constructed as before, and labeled as a counterexample of the relation. The decision tree induction program then constructs a tree that can be used to predict whether a candidate triple is a member of the relation. The second method is table lookup OLU), using a table indexed by the feature values of a triple. The table has one cell for every possible combination of feature values, with integer valued features suitably discretized. Each cell records the number of positive and negative instances from a training set that map to that cell. The table lookup function returns the most frequently seen value associated with the corresponding cell. It is useful to know that the data set used is large and generally covers all possible table cells with mUltiple instances. Thus, table lookup is "unbiased" and one would expect it to give the best predictions possible for the chosen features, assuming the statistics of the training and test sets are consistent. The third method is the ELF function approximator (Utgoff & Precup, 1997), which constructs 932 E. Moss, P Utgoff, J Cavazos, D. Precup, D. Stefanovic, C. Brodley and D. Scheeff additional features (much like a hidden unit) as necessary while it updates its representation of the function that it is learning. The function is represented by two layers of mapping. The first layer maps the features of the triple, which must be boolean for ELF, to a set of boolean feature values. The second layer maps those features to a single scalar value by combining them linearly with a vector of real-valued coefficients called weights. Though the second layer is linear in the instruction features, the boolean features are nonlinear in the instruction features. Finally, the fourth method considered is a feed-forward artificial neural network (NN) (Rumelhart, Hinton & Williams, 1986). Our particular network uses scaled conjugate gradient descent in its back-propagation, which gives results comparable to back-propagation with momentum, but converges much faster. Our configuration uses 10 hidden units. 3 Empirical Results We aimed to answer the following questions: Can we schedule as well as hand-crafted algorithms in production compilers? Can we schedule as well as the best hand-crafted algorithms? How close can we come to optimal schedules? The first two questions we answer with comparisons of program execution times, as predicted from simulations of individual basic blocks (multiplied by the number of executions of the blocks as measured in sample program runs). This measure seems fair for local instruction scheduling, since it omits other execution time factors being ignored. Ultimately one would deal with these factors, but they would cloud the issues for the present enterprise. Answering the third question is harder, since it is infeasible to generate optimal schedules for long blocks. We offer a partial answer by measuring the number of optimal choices made within small blocks. To proceed, we selected a computer architecture implementation and a standard suite of benchmark programs (SPEC95) compiled for that architecture. We extracted basic blocks from the compiled programs and used them for training, testing, and evaluation as described below. 3.1 Architecture and Benchmarks We chose the Digital Alpha (Sites, 1992) as our architecture for the instruction scheduling problem. When introduced it was the fastest scalar processor available, and from a dependence analysis and scheduling standpoint its instruction set is simple. The 21064 implementation of the instruction set (DEC, 1992) is interestingly complex, having two dissimilar pipelines and the ability to issue two instructions per cycle (also called dual issue) if a complicated collection of conditions hold. Instructions take from one to many tens of cycles to execute. SPEC95 is a standard benchmark commonly used to evaluate CPU execution time and the impact of compiler optimizations. It consists of 18 programs, 10 written in FORTRAN and tending to use floating point calculations heavily, and 8 written in C and focusing more on integers, character strings, and pointer manipulations. These were compiled with the vendor's compiler, set at the highest level of optimization offered, which includes compile- or linktime instruction scheduling. We call these the Orig schedules for the blocks. The resulting collection has 447,127 basic blocks, composed of 2,205,466 instructions. 3.2 Simulator, Schedulers, and Features Researchers at Digital made publicly available a simulator for basic blocks for the 21064, which will indicate how many cycles a given block requires for execution, assuming all memory references hit in the caches and translation look-aside buffers, and no resources are busy when the basic block starts execution. When presenting a basic block one can also request that the simulator apply a heuristic greedy scheduling algorithm. We call this scheduler DEC. By examining the DEC scheduler, applying intuition, and considering the results of various Learning to Schedule Straight-Une Code 933 preliminary experiments, we settled on using the features of Table 1 for learning. The mapping from triples to feature vectors is: odd: a single boolean 0 or 1; wep, e, and d: the sign ( -, 0, or +) of the value for Ij minus the value for Ii; ie: both instruction's values, expressed as 1 of 20 categories. For ELF and NN the categorical values for ie, as well as the signs, are mapped to a l-of-n vector of bits, n being the number of distinct values. Table I: Features for Instructions and Partial Schedule Heuristic Name Heuristic Description Intuition for Use Odd Partial (odd) Is the current number of instructions schedIf TRUE, we're interested in scheduling inuled odd or even? structions that can dual-issue with the previous instruction. Instruction Class (ic) The Alpha's instructions can be divided into The instructions in each class can be exeequivalence classes with respect to timing cuted only in certain execution pipelines, etc. properties. Weighted Critical Path (wcp) The height ofthe instruction in the DAG (the Instructions on longer critical paths should length of the longest chain of instructions debe scheduled first, since they affect the lower pendent on this one), with edges weighted by bound of the schedule cost. expected latency of the result produced by the instruction Actual Dual (d) Can the instruction dual-issue with the previIf Odd Partial is TRUE, it is important that ous scheduled instruction? we find an instruction, if there is one, that can issue in the same cycle with the previous scheduled instruction. Max Delay (e) The earliest cycle when the instruction can We want to schedule instructions that will begin to execute, relative to the current cycle; have their data and functional unit available this takes into account any wait for inputs for earliest. functional units to become available This mapping of triples to feature values loses information. This does not affect learning much (as shown by preliminary experiments omitted here), but it reduces the size of the input space, and tends to improve both speed and quality of learning for some learning algorithms. 3.3 Experimental Procedures From the 18 SPEC95 programs we extracted aU basic blocks, and also determined, for sample runs of each program, the number of times each basic block was executed. For blocks having no more than ten instructions, we used exhaustive search of all possible schedules to (a) find instruction decision points with pairs of choices where one choice is optimal and the other is not, and (b) determine the best schedule cost attainable for either decision. Schedule costs are always as judged by the DEC simulator. This procedure produced over 13,000,000 distinct choice pairs, resulting in over 26,000,000 triples (given that swapping Ii and Ij creates a counter-example from an example and vice versa). We selected I % of the choice pairs at random (always insuring we had matched example/counter-example triples). For each learning scheme we performed an 18-fold cross-validation, holding out one program's blocks for independent testing. We evaluated both how often the trained scheduler made optimal decisions, and the simulated execution time of the resulting schedules, The execution time was computed as the sum of simulated basic block costs, weighted by execution frequency as observed in sample program runs, as described above. To summarize the data, we use geometric means across the 18 runs of each scheduler. The geometric mean g(XI, ... ,Xn) of XI, ... ,Xn is (XI· ... ·xn)l/n. It has the nice property that g(xI/YI, ... ,Xn/Yn) = g(XI, ... ,Xn)/g(YI"",Yn), which makes it particularly meaningful for comparing performance measures via ratios, It can also be written as the anti-logarithm of the mean of the logarithms of the Xi; we use that to calculate confidence intervals using traditional measures over the logarithms of the values. In any case, geometric means are preferred for aggregating benchmark results across differing programs with varying execution times. 934 E. Moss, P. Utgoff, 1. Cavazos, D. Precup, D. Ste!anovic, C. Brodley and D. Scheeff 3.4 Results and Discussion Our results appear in Table 2. For evaluations based on predicted program execution time, we compare with Drig. For evaluations based directly on the learning task, i.e., optimal choices, we compare with an optimal scheduler, but only over basic blocks no more than 10 instructions long. Other experiments indicate that the DEC scheduler almost always produces optimal schedules for such short blocks; we suspect it does well on longer blocks too. Table 2: Experimental Results: Predicted Execution Time Relevant Blocks Only All Blocks Small Blocks Schecycles ratio to Orig cycles ratio to Orig % Optimal duler (x 109) (95% conf. int.) (x 109) (95% conf. int.) Choices DEC 24.018 0.979 (0.969,0.989) 28.385 0.983 (0.975,0.992) TLU 24.338 0.992 (0.983,1.002) 28.710 0.995 (0.987,1.003) 98.1 m 24.395 0.995 (0.984,1.006) 28.758 0.996 (0.987,1.006) 98.2 NN 24.410 0.995 (0.983,1.007) 28.770 0.997 (0.986,1.008) 98.1 ELF 24.465 0.998 (0.985,1.010) 28.775 0.997 (0.988,1.006) 98.1 Orig 24.525 1.000 (1.000,1.000) 28.862 1.000 (1.000,1.000) Rand 31.292 1.276 (1.186,1.373) 36.207 1.254 (1.160,1.356) The results show that all supervised learning techniques produce schedules predicted to be better than the production compilers, but not as good as the DEC heuristic scheduler. This is a striking success, given the small number of features. As expected, table lookup performs the best of the learning techniques. Curiously, relative performance in terms of making optimal decisions does not correlate with relative performance in terms of producing good schedules. This appears to be because in each program a few blocks are executed very often, and thus contribute much to execution time, and large blocks are executed disproportionately often. Still, both measures of performance are quite good. What about reinforcement learning? We ran experiments with temporal difference (ID) learning, some of which are described in (Scheeff, et at., 1997) and the results are not as good. This problem appears to be tricky to cast in a form suitable for ID, because ID looks at candidate instructions in isolation, rather than in a preference setting. It is also hard to provide an adequate reward function and features predictive for the task at hand. 4 Related Work, Conclusions, and Outlook Instruction scheduling is well-known and others have proposed many techniques. Also, optimal instruction scheduling for today's complex processors is NP-complete. We found two pieces of more closely related work. One is a patent (Tarsy & Woodard, 1994). From the patent's claims it appears that the inventors trained a simple perceptron by adjusting weights of some heuristics. They evaluate each weight setting by scheduling an entire benchmark suite, running the resulting programs, and using the resulting times to drive weight adjustments. This approach appears to us to be potentially very time-consuming. It has two advantages over our technique: in the learning process it uses measured execution times rather than predicted or simulated times, and it does not require a simulator. Being a patent, this work does not offer experimental results. The other related item is the application of genetic algorithms to tuning weights of heuristics used in a greedy scheduler (Beaty, S., Colcord, & Sweany, 1996). The authors showed that different hardware targets resulted in different learned weights, but they did not offer experimental evaluation of the qUality of the resulting schedulers. While the results here do not demonstrate it, it was not easy to cast this problem in a form suitable for machine learning. However, once that form was accomplished, supervised learnLearning to Schedule Straight-line Code 935 ing produced quite good results on this practical problem-better than two vendor production compilers, as shown on a standard benchmark suite used for evaluating such optimizations. Thus the outlook for using machine learning in this application appears promising. On the other hand, significant work remains. The current experiments are for a particular processor; can they be generalized to other processors? After all, one of the goals is to improve and speed processor design by enabling more rapid construction of optimizing compilers for proposed architectures. While we obtained good performance predictions, we did not report performance on a real processor. (More recently we obtained those results (Moss, et al., 1997); ELF tied Orig for the best scheme.) This raises issues not only of faithfulness of the simulator to reality, but also of global instruction scheduling, i.e., across basic blocks, and of somewhat more general rewritings that allow more reorderings of instructions. From the perspective of learning, the broader context may make supervised learning impossible, because the search space will explode and preclude making judgments of optimal vs. suboptimal. Thus we will have to find ways to make reinforcement learning work better for this problem. A related issue is the difference between learning to make optimal decisions (on small blocks) and learning to schedule (all) blocks well. Another relevant issue is the cost not of the schedules, but of the schedulers: are these schedulers fast enough to use in production compilers? Again, this demands further experimental work. We do conclude, though, that the approach is promising enough to warrant these additional investigations. Acknowledgments: We thank various people of Digital Equipment Corporation, for the DEC scheduler and the ATOM program instrumentation tool (Srivastava & Eustace, 1994), essential to this work. We also thank Sun Microsystems and Hewlett-Packard for their support. References Beaty, S., Colcord, S., & Sweany, P. (1996). Using genetic algOrithms to fine-tune instructionscheduling heuristics. In Proc. of the Int'l Con! on Massively Parallel Computer Systems. Digital Equipment Corporation, (1992). DECchip 2I064-AA Microprocessor Hardware Reference Manual, Maynard, MA, first edition, October 1992. Haykin, S. (1994). Neural networks: A comprehensivefoundation. New York, NY: Macmillan. Moss, E., Cavazos, J., Stefanovic, D., Utgoff, P., Precup, D., Scheeff, D., & Brodley, C. (1997). Learning Policies for Local Instruction Scheduling. Submitted for publication. Rumelhart, D. E., Hinton, G. E., & Williams, RJ. (1986). Learning internal representations by error propagation. In Rumelhart & McClelland (Eds.), Parallel distributed processing: Explorations in the microstructure of cognition. Cambridge, MA: MIT Press. Scheeff, D., Brodley, C., Moss, E., Cavazos, J., Stefanovic. D. (1997). Applying Reinforcement Learning to Instruction Scheduling within Basic Blocks. Technical report. Sites, R. (1992). Alpha Architecture Reference Manual. Digital Equip. Corp., Maynard, MA. Srivastava, A. & Eustace, A. (1994). ATOM: A system for building customized program analysis tools. In Proc. ACM SIGPLAN '94 Con! on Prog. Lang. Design and Impl., 196-205. Sutton, R. S. (1988). Learning to predict by the method of temporal differences. Machine Learning, 3,9-44. Tarsy, G. & Woodard, M. (1994). Method and apparatus for optimizing cost-based heuristic instruction schedulers. US Patent #5,367,687. Filed 7/7/93, granted 11122/94. Utgoff. P. E., Berkman, N. C., & Clouse, J. A. (in press). Decision tree induction based on efficient tree restructuring. Machine Learning. Utgoff, P. E., & Precup. D. (1997). Constructive function approximation, (Technical Report 97 -04), Amherst. MA: University of Massachusetts, Department of Computer Science.
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Approximating Posterior Distributions in Belief Networks using Mixtures Christopher M. Bishop Neil Lawrence Neural Computing Research Group Dept. Computer Science & Applied Mathematics Aston University Binningham, B4 7ET, U.K. Tommi Jaakkola Michael I. Jordan Center for Biological and Computational Learning Massachusetts Institute of Technology 79 Amherst Street, ElO-243 Cambridge, MA 02139, U.S.A. Abstract Exact inference in densely connected Bayesian networks is computationally intractable, and so there is considerable interest in developing effective approximation schemes. One approach which has been adopted is to bound the log likelihood using a mean-field approximating distribution. While this leads to a tractable algorithm, the mean field distribution is assumed to be factorial and hence unimodal. In this paper we demonstrate the feasibility of using a richer class of approximating distributions based on mixtures of mean field distributions. We derive an efficient algorithm for updating the mixture parameters and apply it to the problem of learning in sigmoid belief networks. Our results demonstrate a systematic improvement over simple mean field theory as the number of mixture components is increased. 1 Introduction Bayesian belief networks can be regarded as a fully probabilistic interpretation of feedforward neural networks. Maximum likelihood learning for Bayesian networks requires the evaluation of the likelihood function P(VIO) where V denotes the set of instantiated (visible) variables, and 0 represents the set of parameters (weights and biases) in the network. Evaluation of P(VIO) requires summing over exponentially many configurations of Approximating Posterior Distributions in Belief Networks Using Mixtures 417 the hidden variables H, and is computationally intractable except for networks with very sparse connectivity, such as trees. One approach is to consider a rigorous lower bound on the log likelihood, which is chosen to be computationally tractable, and to optimize the model parameters so as to maximize this bound instead. If we introduce a distribution Q (H), which we regard as an approximation to the true posterior distribution, then it is easily seen that the log likelihood is bounded below by F[Q] = L Q(H) In P(V ~H). (1) {H} Q( ) The difference between the true log likelihood and the bound given by (1) is equal to the Kullback-Leibler divergence between the true posterior distribution P(HIV) and the approximation Q(H). Thus the correct log likelihood is reached when Q(H) exactly equals the true posterior. The aim of this approach is therefore to choose an approximating distribution which leads to computationally tractable algorithms and yet which is also flexible so as to permit a good representation of the true posterior. In practice it is convenient to consider parametrized distributions, and then to adapt the parameters to maximize the bound. This gives the best approximating distribution within the particular parametric family. 1.1 Mean Field Theory Considerable simplification results if the model distribution is chosen to be factorial over the individual variables, so that Q(H) = ni Q(hd, which gives meanfieid theory. Saul et al. (1996) have applied mean field theory to the problem of learning in sigmoid belief networks (Neal. 1992). These are Bayesian belief networks with binary variables in which the probability of a particular variable Si being on is given by P(S, = Ilpa(S,» =" ( ~ ],;S; + b) (2) where u(z) == (1 + e-Z)-l is the logistic sigmoid function, pa(Si) denote the parents of Si in the network. and Jij and bi represent the adaptive parameters (weights and biases) in the model. Here we briefly review the framework of Saul et ai. (1996) since this forms the basis for the illustration of mixture modelling discussed in Section 3. The mean field distribution is chosen to be a product of Bernoulli distributions of the form Q(H) = II p,~i (1 _ p,;)l-hi (3) in which we have introduced mean-field parameters J.Li. Although this leads to considerable simplification of the lower bound, the expectation over the log of the sigmoid function. arising from the use of the conditional distribution (2) in the lower bound (I), remains intractable. This can be resolved by using variational methods (Jaakkola, 1997) to find a lower bound on F(Q), which is therefore itself a lower bound on the true log likelihood. In particular, Saul et al. (1996) make use of the following inequality (In[l + e Zi ]) ::; ei(Zi) + In(e-~iZi + e(1-~;)Zi) (4) where Zi is the argument of the sigmoid function in (2), and ( ) denotes the expectation with respect to the mean field distribution. Again, the quality of the bound can be improved by adjusting the variational parameter ei. Finally, the derivatives of the lower bound with respect to the Jij and bi can be evaluated for use in learning. In summary. the algorithm involves presenting training patterns to the network. and for each pattern adapting the P,i and ei to give the best approximation to the true posterior within the class of separable distributions of the form (3). The gradients of the log likelihood bound with respect to the model parameters Jij and bi can then be evaluated for this pattern and used to adapt the parameters by taking a step in the gradient direction. 418 C. M. Bishop, N. LAwrence, T. Jaakkola and M I. Jordan 2 Mixtures Although mean field theory leads to a tractable algorithm, the assumption of a completely factorized distribution is a very strong one. In particular, such representations can only effectively model posterior distributions which are uni-modal. Since we expect multi-modal distributions to be common, we therefore seek a richer class of approximating distributions which nevertheless remain computationally tractable. One approach (Saul and Jordan, 1996) is to identify a tractable substructure within the model (for example a chain) and then to use mean field techniques to approximate the remaining interactions. This can be effective where the additional interactions are weak or are few in number, but will again prove to be restrictive for more general, densely connected networks. We therefore consider an alternative approach I based on mixture representations of the form M Qmix(H) = L amQ(Hlm) (5) m=l in which each of the components Q(Hlm) is itself given by a mean-field distribution, for example of the form (3) in the case of sigmoid belief networks. Substituting (5) into the lower bound (1) we obtain F[Qmix] = L amF[Q(Hlm)] + f(m, H) (6) m where f(m, H) is the mutual information between the component label m and the set of hidden variables H, and is given by Q(Hlm) f(m,H) = L L amQ(Hlm) In Q . (H)' m {H} mix (7) The first tenn in (6) is simply a convex combination of standard mean-field bounds and hence is no greater than the largest of these and so gives no useful improvement over a single mean-field distribution. It is the second term, i.e. the mutual infonnation, which characterises the gain in using mixtures. Since f(m, H) ~ 0, the mutual information increases the value of the bound and hence improves the approximation to the true posterior. 2.1 Smoothing Distributions As it stands, the mutual infonnation itself involves a summation over the configurations of hidden variables, and so is computationally intractable. In order to be able to treat it efficiently we first introduce a set of 'smoothing' distributions R(Hlm), and rewrite the mutual infonnation (7) in the form f(m, H) LLamQ(Hlm)lnR(Hlm) - LamInam m {H} m - L L amQ(Hlm) In {R(H1m) Qmix(H) } . (8) m {H} am Q(Hlm) It is easily verified that (8) is equivalent to (7) for arbitrary R(Hlm). We next make use of the following inequality - In x ~ - ~x + In ~ + 1 (9) lHere we outline the key steps. A more detailed discussion can be found in Jaakkola and Jordan (1997). Approximating Posterior Distributions in Belief Networks Using Mixtures 419 to replace the logarithm in the third term in (8) with a linear function (conditionally on the component label m). This yields a lower bound on the mutual information given by J(m,H) ~ J),(m,H) where h(m,H) I:I:amQ(Hlm)lnR(Hlm)- I: am In am m {H} m - I: Am I: R(Hlm)Qmix(H) + I: am InAm + 1. (10) m {H} m With J),(m, H) substituted for J(m, H) in (6) we again obtain a rigorous lower bound on the true log likelihood given by F),[Qmix(H)] = I: amF[Q(Hlm)] + h(m, H). (11) m The summations over hidden configurations {H} in (10) can be performed analytically if we assume that the smoothing distributions R(Hlm) factorize. In particular, we have to consider the following two summations over hidden variable configurations I: R(Hlm)Q(Hlk) II I: R(hilm)Q(hilk) ~ 7rR,Q(m, k) (12) {H} i h. I: Q(Hlm) InR(Hlm) I: I: Q(hilm) InR(hilm) ~f H(QIIRlm). (13) {H} h. We note that the left hand sides of (12) and (13) represent sums over exponentially many hidden configurations, while on the right hand sides these have been re-expressed in terms of expressions requiring only polynomial time to evaluate by making use of the factorization of R(Hlm). It should be stressed that the introduction of a factorized form for the smoothing distributions still yields an improvement over standard mean field theory. To see this, we note that if R(Hlm) = const. for all {H, m} then J(m, H) = 0, and so optimization over R(Hlm) can only improve the bound. 2.2 Optimizing the Mixture Distribution In order to obtain the tightest bound within the class of approximating distributions, we can maximize the bound with respect to the component mean-field distributions Q(Hlm), the mixing coefficients am, the smoothing distributions R(Hlm) and the variational parameters Am' and we consider each of these in turn. We will assume that the choice of a single mean field distribution leads to a tractable lower bound, so that the equations 8F[Q] 8Q(hj ) = const (14) can be solved efficiently2. Since h(m, H) in (10) is linear in the marginals Q(hjlm), it follows that its derivative with respect to Q(hj 1m) is independent of Q(hjlm), although it will be a function of the other marginals, and so the optimization of (11) with respect to individual marginals again takes the form (14) and by assumption is therefore soluble. Next we consider the optimization with respect to the mixing coefficients am. Since all of the terms in (11) are linear in am, except for the entropy term, we can write F),[Qmix(H)] = I:am(-Em) - I:amlnam + 1 (15) m m 2In standard mean field theory the constant would be zero, but for many models of interest the slightly more general equations given by (14) will again be soluble. 420 C. M. Bishop, N. Lawrence, T. Jaakkola and M. L Jordan where we have used (10) and defined F[Q(Hlm)] + L Q(Hlm) InR(Hlm) {H} + LAk LR(Hlk)Q(Hlm) +lnAm . k {H} (16) Maximizing (15) with respect to am, subject to the constraints 0 ~ am ~ 1 and Lm am = 1, we see that the mixing coefficients which maximize the lower bound are given by the Boltzmann distribution exp(-Em) am = Lk exp(-Ek)' (17) We next maximize the bound (11) with respect to the smoothing marginals R(hj 1m). Some manipulation leads to the solution R(hilm) = amQA~ilm) [~>'~'Q(m'k)Q(hi'k)l-1 (18) in which 7r~,Q(m, k) denotes the expression defined in (12) but with the j term omitted from the product. The optimization of the JLmj takes the form of a re-estimation formula given by an extension of the result obtained for mean-field theory by Saul et al. (1996). For simplicity we omit the details here. Finally, we optimize the bound with respect to the Am, to give 1 1 ~ = - L 7rR,Q(m, k). m am k (19) Since the various parameters are coupled, and we have optimized them individually keeping the remainder constant, it will be necessary to maximize the lower bound iteratively until some convergence criterion is satisfied. Having done this for a particular instantiation of the visible nodes, we can then determine the gradients of the bound with respect to the parameters governing the original belief network, and use these gradients for learning. 3 Application to Sigmoid Belief Networks We illustrate the mixtures formalism by considering its application to sigmoid belief networks of the form (2). The components of the mixture distribution are given by factorized Bernoulli distributions of the form (3) with parameters JLmi. Again we have to introduce variational parameters ~mi for each component using (4). The parameters {JLmi, ~mi} are optimized along with {am, R(hjlm), Am} for each pattern in the training set. We first investigate the extent to which the use of a mixture distribution yields an improvement in the lower bound on the log likelihood compared with standard mean field theory. To do this, we follow Saul et al. (1996) and consider layered networks having 2 units in the first layer, 4 units in the second layer and 6 units in the third layer, with full connectivity between layers. In all cases the six final-layer units are considered to be visible and have their states clamped at zero. We generate 5000 networks with parameters {Jij, bi } chosen randomly with uniform distribution over (-1, 1). The number of hidden variable configurations is 26 = 64 and is sufficiently small that the true log likelihood can be computed directly by summation over the hidden states. We can therefore compare the value of Approximating Posterior Distributions in Belief Networks Using Mixtures 421 the lower bound F with the true log likelihood L, using the nonnalized error (L - F)/ L. Figure 1 shows histograms of the relative log likelihood error for various numbers of mixture components, together with the mean values taken from the histograms. These show a systematic improvement in the quality of the approximation as the number of mixture components is increased. 5 components, mean: 0.011394 3~r-----~----~----~---. 0.02 0.04 0.06 0.08 3 components, mean: 0.01288 0.02 0.04 0.06 0.08 1 component, mean: 0.015731 ~r-----~----~----~--~ 0.04 0.06 0.08 4 components, mean: 0.012024 3000r-----~----~----~--_. 0.02 0.04 0.06 0.08 2 components, mean: 0.013979 3000r-----~----~----~--_. 2000 0.04 0.06 0.08 0.01 Gn..:-p----~----~----~----..., g 0.014 o III c as ~ 0.012 o o 0.01 '------~----~----~-----' 1 2 3 4 5 no. of components Figure 1: Plots of histograms of the normalized error between the true log likelihood and the lower bound. for various numbers of mixture components. Also shown is the mean values taken from the histograms. plotted against the number of components. Next we consider the impact of using mixture distributions on learning. To explore this we use a small-scale problem introduced by Hinton et al. (1995) involving binary images of size 4 x 4 in which each image contains either horizontal or vertical bars with equal probability, with each of the four possible locations for a bar occupied with probability 0.5. We trained networks having architecture 1-8-16 using distributions having between 1 and 5 components. Randomly generated patterns were presented to the network for a total of 500 presentations, and the J-tmi and ~mi were initialised from a unifonn distribution over (0,1). Again the networks are sufficiently small that the exact log likelihood for the trained models can be evaluated directly. A Hinton diagram of the hidden-to-output weights for the eight units in a network trained with 5 mixture components is shown in Figure 2. Figure 3 shows a plot of the true log likelihood versus the number M of components in the mixture for a set of experiments in which, for each value of M, the model was trained 10 times starting from different random parameter initializations. These results indicate that, as the number of mixture components is increased, the learning algorithm is able to find a set of network parameters having a larger likelihood, and hence that the improved flexibility of the approximating distribution is indeed translated into an improved training algorithm. We are currently applying the mixture fonnalism to the large-scale problem of hand-written digit classification. 422 .• 0. t •••• •••• • •• j ••• jO 0.0 100.0 10 ••. C. M. Bishop, N. Lawrence, T. Jaakkola and M. I. Jordan o • • ••• • o - 0 0.0 0 •• 0 o -0 Figure 2: Hinton diagrams of the hidden-to-output weights for each of the 8 hidden units in a network trained on the 'bars' problem using a mixture distribution having 5 components. -5 -6 -7 '8 0 :£ -8 Q) ~ CI .S! Q) -9 E -10 -11 -12 0 8 0 ~ 0 0 0 o o 8 o 8 o o 6 B 8 o o o o o o o 234 no. of components o o 5 6 Figure 3: True log likelihood (divided by the number of patterns) versus the number M of mixture components for the 'bars' problem indicating a systematic improvement in performance as M is increased. References Hinton, G. E., P. Dayan, B. 1. Frey, and R. M. Neal (1995). The wake-sleep algorithm for unsupervised neural networks. Science 268, 1158-1161. Jaakkola, T. (1997). Variational Methods for Inference and Estimation in Graphical Models. Ph.D. thesis, MIT. Jaakkola, T. and M. I. Jordan (1997). Approximating posteriors via mixture models. To appear in Proceedings NATO ASI Learning in Graphical Models, Ed. M. I. Jordan. Kluwer. Neal, R. (1992). Connectionist learning of belief networks. Artificial Intelligence 56, 71-113. Saul, L. K., T. Jaakkola, and M. I. Jordan (1996). Mean field theory for sigmoid belief networks. Journal of Artificial Intelligence Research 4,61-76. Saul, L. K. and M. I. Jordan (1996). Exploiting tractable substructures in intractable networks. In D. S. Touretzky, M . C. Mozer, and M. E. Hasselmo (Eds.), Advances in Neural Information Processing Systems, Volume 8, pp. 486-492. MIT Press.
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The Canonical Distortion Measure in Feature Space and I-NN Classification Jonathan Baxter*and Peter Bartlett Department of Systems Engineering Australian National University Canberra 0200, Australia {jon,bartlett}@syseng.anu.edu.au Abstract We prove that the Canonical Distortion Measure (CDM) [2, 3] is the optimal distance measure to use for I nearest-neighbour (l-NN) classification, and show that it reduces to squared Euclidean distance in feature space for function classes that can be expressed as linear combinations of a fixed set of features. PAC-like bounds are given on the samplecomplexity required to learn the CDM. An experiment is presented in which a neural network CDM was learnt for a Japanese OCR environment and then used to do I-NN classification. 1 INTRODUCTION Let X be an input space, P a distribution on X, F a class of functions mapping X into Y (called the "environment"), Q a distribution on F and (J' a function (J': Y X Y -t [0, ."1]. The Canonical Distortion Measure (CDM) between two inputs x, Xl is defined to be: p(x, Xl) = L (J'(f(x) , f(x l)) dQ(f). (1) Throughout this paper we will be considering real-valued functions and squared loss, so Y = ~ and (J'(y, yl) := (y - yl)2. The CDM was introduced in [2, 3], where it was analysed primarily from a vector quantization perspective. In particular, the CDM was proved to be the optimal distortion measure to use in vector quantization, in the sense of producing the best approximations to the functions in the environment F. In [3] some experimental results were also presented (in a toy domain) showing how the CDM may be learnt. The purpose of this paper is to investigate the utility of the CDM as a classification tool. In Section 2 we show how the CDM for a class of functions possessing a common feature *The first author was supported in part by EPSRC grants #K70366 and #K70373 246 1. Baxter and P. Bartlett set reduces, via a change of variables, to squared Euclidean distance in feature space. A lemma is then given showing that the CDM is the optimal distance measure to use for 1nearest-neighbour (l-NN) classification. Thus, for functions possessing a common feature set, optimall-NN classification is achieved by using squared Euclidean distance in feature space. In general the CDM will be unknown, so in Section 4 we present a technique for learning the CDM by minimizing squared loss, and give PAC-like bounds on the sample-size required for good generalisation. In Section 5 we present some experimental results in which a set of features was learnt for a machine-printed Japanese OCR environment, and then squared Euclidean distance was used to do I-NN classification in feature space. The experiments provide strong empirical support for the theoretical results in a difficult real-world application. 2 THE CDM IN FEATURE SPACE Suppose each f E F can be expressed as a linear combination of a fixed set of features ~ := (¢l, ... , ¢k). That is, for all f E F, there exists w := (WI,···, Wk) such that f = w . ~ = 2:7=1 Wi¢i. In this case the distribution Q over the environment F is a distribution over the weight vectors w. Measuring the distance between function values by ()(y, y') := (y - yl)2, the CDM (1) becomes: p(x, x') = r [w· ~(x) - w· ~(X,)]2 dQ(w) = (~(x) - ~(X'))W(~(x) ~(X'))' iE.k (2) where W = fw w'w dQ(w). is a k x k matrix. Making the change of variable ~ -t ~JW, we have p(x, x') = 11~(x) ~(x')112 . Thus, the assumption that the functions in the environment can be expressed as linear combinations of a fixed set of features means that the CDM is simply squared Euclidean distance in a feature space related to the original by a linear transformation. 3 I-NN CLASSIFICATION AND THE CDM Suppose the environment F consists of classifiers, i.e. {O, 1 }-valued functions. Let f be some function in F and z := (Xl, f(Xl)), ... , (Xn, f(x n)) a training set of examples of f. In I-NN classification the classification of a novel x is computed by f(x*) where X* = argminx • d(x, Xi)), i.e. the classification of X is the classification of the nearest training point to x under some distance measure d. If both f and x are chosen at random, the expected misclassification error of the 1-NN scheme using d and the training points x := (xl, ... ,xn)is er(x, d) := EF Ex [J(x) - f(x* )]2 , (3) where x* is the nearest neighbour to x from {Xl, . . . , xn }. The following lemma is now immediate from the definitions. Lemma 1. For all sequences x = (Xl, . .. , X n ). er{x, d) is minimized ifd is the CDM p. Remarks. Lemma 1 combined with the results of the last section shows that for function classes possessing a common feature set, optimall-NN classification is achieved by using squared Euclidean distance in feature space. In Section 5 some experimental results on Japanese OCR are presented supporting this conclusion. The property of optimality of the CDM for I-NN classification may not be stable to small perturbations. That is, if we learn an approximationg to p, then even ifExxx (g(x, x') The Canonical Distortion Measure in Feature Space and I-NN Classification 247 p(x, x'))2 is small it may not be the case that l-NN classification using 9 is also small. However, one can show that stability is maintained for classifier environments in which positive examples of different functions do not overlap significantly (as is the case for the Japanese OCR environment of Section 5, face recognition environments, speech recognition environments and so on). We are currently investigating the general conditions under which stability is maintained. 4 LEARNING THE CDM For most environments encountered in practice (e.g speech recognition or image recognition), P will be unknown. In this section it is shown how p may be estimated or learnt using function approximation techniques (e.g. feedforward neural networks). 4.1 SAMPLING THE ENVIRONMENT To learn the CDM p, the learner is provided with a class of functions (e.g. neural networks) 9 where each 9 E 9 maps X x X ~ [0, M]. The goal of the learner is to find a 9 such that the error between 9 and the CDM p is small. For the sake of argument this error will be measured by the expected squared loss: erp(g) := Exxx [g(x, x') - p(x, x')f , (4) where the expectation is with respect to p2. Ordinarily the learner would be provided with training data in the form (x, x', p( x, x'}) and would use this data to minimize an empirical version of (4). However, p is unknown so to generate data of this form p must be estimated for each training pair x, x'. Hence to generate training sets for learning the CDM, both the distribution Q over the environment :F and the distribution P over the input space X must be sampled. So let f := (it, ... , f m) be m i.i.d. samples from :F according to Q and let x := (Xl, ... , x n ) be n i.i.d. samples from X according to P. For any pair Xi, X j an estimate of p( Xi, X j) is given by 1 m P(Xi' Xj} := m ~ (J'(fdxd,fk(Xj )). (5) k=l This gives n (n - 1) /2 training triples, {(xi,Xj,p(xi,xj)),l::; i < j::; n} , which can be used as data to generate an empirical estimate of er p (g): (6) Only n(n - 1)/2 of the possible n 2 training triples are used because the functions 9 E 9 are assumed to already be symmetric and to satisfy 9 (x, x) = 0 for all x (if this is not the case then set g'(X, x') := (g(x, x') + g(x', x))/2 if x =j:. x' and g'(X, x) = 0 and use g' := {g': 9 E g} instead). In [3] an experiment was presented in which 9 was a neural network class and (6) was minimized directly by gradient descent. In Section 5 we present an alternative technique in which a set of features is first learnt for the environment and then an estimate of p in feature space is constructed explicitly. 248 J. Baxter and P. Bartlett 4.2 UNIFORM CONVERGENCE We wish to ensure good generalisation from a 9 minimizing e~r r, in the sense that (for x , small 6, 5), Pr { x, r : :~~ lerx,f(g) - erp(g) I > 6} < 5, The following theorem shows that this occurs if both the number of functions m and the number of input samples n are sufficiently large. Some exotic (but nonetheless benign) measurability restrictions have been ignored in the statement of the theorem. In the statement of the theorem, N (E , 9) denotes the smallest 6 -cover of 9 under the L 1 ( P 2) norm, where {gl , . . . , gN} is an 6-cover of9 iffor all 9 E 9 there exists gi such that Ilgi - gil ~ 6. Theorem 2. Assume the range of the functions in the environment :F is no more than [-J B /2, J B /2) and in the class 9 (used to approximate the CDM) is no more than [0, VB). For all 6 > 0 and 0 < 5 ~ 1. if 32B4 4 m> --log(7) 6 2 5 and 512B2 ( 512B 2 8) n 2: 6 2 logN(6,9) + log 62 + log;5 then Proof For each 9 E 9 , define erx(g) := (2 ) nn-1 l~i<j~n If for any x = (Xl, . . . , Xn ), Pr {r: sup ler r(g) - erx(g) I > ~} ~ ~ , gE9 x , 2 2 and (8) (9) (10) (II) Pr {x: sup lerx(g) - erp(g)1 > ~} ~ ~ , (12) gE9 2 2 then by the triangle inequality (9) will hold. We treat (11) and (12) separately. Equation (11). To simplify the notation let gij , Pij and Pij denote 9 (Xi, X j), p( Xi, X j) and p(Xi' Xj) respectively. Now, 2 n(n - 1) 4B < -,----:- n(n - 1) L (gij - pij )2 2: (gij - Pij)2 l~i<j~n 19<j~n 2: (Pij - Pij) (2gij - Pij - Pij) l~i<j~n L (Pij - Pij) 19<j~n 1 m E.rx(J) m 2: X(Jk) k=l The Canonical Distortion Measure in Feature Space and J-NN Classification 249 where x: :F -+ [0, 4B2] is defined by Thus, 4B x (f) : = ----,----n(n 1) 1 :Si<J:S n Pr {f: :~g I"r.,f(g) - e-r.(g) I > n S Pr {f EJ'x(f) - ~ t, xU,) > ~ } which is ~ 2 exp (_m€2/ (32B4)) by Hoeffding's inequality. Setting this less than 6/2 gives the bound on m in theorem 2. Equation (12). Without loss of generality, suppose that n is even. The trick here is to split the sum over all pairs (Xi, X j) (with i < j) appearing in the definition of er x (g) into a double sum: ~ 2 erx(g) = ( )" [g(Xj, Xj) - p(Xi, Xj)]2 nn-1 6 1 ::;i<j:S n 1 n-1 2 n /2 2 = n _ 1 L ;; L [g(xo ,U), xO:(j)) - p(xo.(j), xO:(j))] , i=l j=l where for each i = 1, ... , n - 1, (J"i and (J"~ are permutations on {I, ... , n} such that {(J"d 1), ... , (J"i (n/2)) n {(J"H 1), ... , (J"~( n/2)} is empty. That there exist permutations with this property such that the sum can be broken up in this way can be proven easily by induction. Now, conditional on each (J"i, the n/2 pairs Xi := {(Xo.(j), xO:(j)), j = 1, ... , n/2} are an i.i.d. sample from X x X according to p2. So by standard results from real-valued function learning with squared loss [4]: { 2 n/2 } Pr Xi: suP ;;?= [g(XO.(j), Xo:u)) - p(xo.U). XO:(j))]2 - erp(g) > ~ gEQ J=l ~ 4N (48~2 ' g) exp ( - 2;:~2 ) . Hence, by the union bound, Pr { x: ~~g /erx(g) - erp(g) I > ~} ~ 4(n - l)N (48~2 ' g) exp ( - 2;:~2 ) . Setting n as in the statement of the theorem ensures this is less than 6/2. D Remark. The bound on m (the number of functions that need to be sampled from the environment) is independent of the complexity of the class g. This should be contrasted with related bias learning (or equivalently, learning to learn) results [1] in which the number of functions does depend on the complexity. The heuristic explanation for this is that here we are only learning a distance function on the input space (the CDM), whereas in bias learning we are learning an entire hypothesis space that is appropriate for the environment. However, we shall see in the next section how for certain classes of problems the CDM can also be used to learn the functions in the environment. Hence in these cases learning the CDM is a more effective method of learning to learn. 5 EXPERIMENT: JAPANESE OCR To verify the optimality of the CDM for I-NN classification, and also to show how it can be learnt in a non-trivial domain (only a toy example was given in [3]), the 250 1. Baxter and P Bartlett COM was learnt for a Japanese OCR environment. Specifically, there were 3018 functions I in the environment F, each one a classifier for a different Kanji character. A database containing 90,918 segmented, machine-printed Kanji characters scanned from various sources was purchased from the CEDAR group at the State University of New York, Buffalo The quality of the images ranged from clean to very degraded (see http://www . cedar .buffalo. edu/Databases/JOcR/). The main reason for choosing Japanese OCR rather than English OCR as a test-bed was the large number of distinct characters in Japanese. Recall from Theorem 2 that to get good generalisation from a learnt COM, sufficiently many functions must be sampled from the environment. If the environment just consisted of English characters then it is likely that "sufficiently many" characters would mean all characters, and so it would be impossible to test the learnt COM on novel characters not seen in training. Instead of learning the COM directly by minimizing (6), it was learnt implicitly by first learning a set of neural network features for the functions in the environment. The features were learnt using the method outlined in [1], which essentially involves learning a set of classifiers with a common final hidden layer. The features were learnt on 400 out of the 3000 classifiers in the environment, using 90% of the data in training and 10% in testing. Each resulting classifier was a linear combination of the neural network features. The average error of the classifiers was 2.85% on the test set (which is an accurate estimate as there were 9092 test examples). Recall from Section 2 that if all f E F can be expressed as I = W . 4> for a fixed feature set 4>, then the COM reduces to p{x, x') = (4)(x) - 4>(x' ))W(4>{x) 4>(XI ))' where W = fw w/w dQ(w). The result of the learning procedure above is a set of features ci> and 400 weight vectors w l, . . . , W 400, such that for each of the character classifiers fi used in training, Ii :: Wi . ci>. Thus, g(x, x') := (ci>(x) - ci>(X'))W(ci>(x) - 4>(X'))' is an empirical estimate of the true CDM, where W := L;~~ W:Wi. With a linear change of variable ci> -+ ci>VW, 9 becomes g(x, x') = 114>(x) - ci>(x')112. This 9 was used to do I-NN classification on the test examples in two different experiments. In the first experiment, all testing and training examples that were not an example of one of the 400 training characters were lumped into an extra category for the purpose of classification. All test examples were then given the label of their nearest neighbour in the training set under 9 (i.e. , initially all training examples were mapped into feature space to give {ci>( Xl)' ... , ci>( Xn )}. Then each test example was mapped into feature space and assigned the same label as argminx.llci>( x) - ci>( Xi) 11 2).The total misclassification error was 2.2%, which can be directly compared with the misclassification error of the original classifiers of 2.85%. The COM does better because it uses the training data explicitly and the information stored in the network to make a comparison, whereas the classifiers only use the information in the network. The learnt COM was also used to do k-NN classification with k > 1. However this afforded no improvement. For example, the error of the 3-NN classifier was 2.54% and the error of the 20-NN classifier was 3.99%. This provides an indication that the COM may not be the optimal distortion measure to use if k-NN classification (k > 1) is the aim. In the second experiment 9 was again used to do I-NN classification on the test set, but this time all 3018 characters were distinguished. So in this case the learnt COM was being asked to distinguish between 2618 characters that were treated as a single character when it was being trained. The misclassification error was a surprisingly low 7.5%. The 7.5% error compares favourably with the 4.8% error achieved on the same data by the CEDAR group, using a carefully selected feature set and a hand-tailored nearest-neighbour routine [5]. In our case the distance measure was learnt from raw-data input, and has not been the subject of any optimization or tweaking. The Canonical Distortion Measure in Feature Space and I-NN Classification 251 Figure 1: Six Kanji characters (first character in each row) and examples of their four nearest neighbours (remaining four characters in each row). As a final, more qualitative assessment, the learnt CDM was used to compute the distance between every pair of testing examples, and then the distance between each pair of characters (an individual character being represented by a number of testing examples) was computed by averaging the distances between their constituent examples. The nearest neighbours of each character were then calculated. With this measure, every character turned out to be its own nearest neighbour, and in many cases the next-nearest neighbours bore a strong subjective similarity to the original. Some representative examples are shown in Figure 1. 6 CONCLUSION We have shown how the Canonical Distortion Measure (CDM) is the optimal distortion measure for I-NN classification, and that for environments in which all the functions can be expressed as a linear combination of a fixed set of features, the Canonical Distortion Measure is squared Euclidean distance in feature space. A technique for learning the CDM was presented and PAC-like bounds on the sample complexity required for good generalisation were proved. Experimental results were presented in which the CDM for a Japanese OCR environment was learnt by first learning a common set of features for a subset of the character classifiers in the environment. The learnt CDM was then used as a distance measure in I-NN neighbour classification, and performed remarkably well, both on the characters used to train it and on entirely novel characters. References [1] Jonathan Baxter. Learning Internal Representations. In Proceedings of the Eighth International Conference on Computational Learning Theory, pages 311-320. ACM Press, 1995. [2] Jonathan Baxter. The Canonical Metric for Vector Quantisation. Technical Report NeuroColt Technical Report 047, Royal Holloway College, University of London, July 1995. [3] Jonathan Baxter. The Canonical Distortion Measure for Vector Quantization and Function Approximation. In Proceedings of the Fourteenth International Conference on Machine Learning, July 1997. To Appear. [4] W S Lee, P L Bartlett, and R C Williamson. Efficient agnostic learning of neural networks with bounded fan-in. IEEE Transactions on Information Theory, 1997. [5] S.N. Srihari, T. Hong, and Z. Shi. Cherry Blossom: A System for Reading Unconstrained Handwritten Page Images. In Symposium on Document Image Understanding Technology (SDIUT), 1997.
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A Non-parametric Multi-Scale Statistical Model for Natural Images Jeremy S. De Bonet & Paul Viola Artificial Intelligence Laboratory Learning & Vision Group 545 Technology Square Massachusetts Institute of Technology Cambridge, MA 02139 EMAIL: jsd@ai.mit.edu & viola@ai.mit.edu HOMEPAGE: http://www.ai .mit. edu/projects/lv Abstract The observed distribution of natural images is far from uniform. On the contrary, real images have complex and important structure that can be exploited for image processing, recognition and analysis. There have been many proposed approaches to the principled statistical modeling of images, but each has been limited in either the complexity of the models or the complexity of the images. We present a non-parametric multi-scale statistical model for images that can be used for recognition, image de-noising, and in a "generative mode" to synthesize high quality textures. 1 Introduction In this paper we describe a multi-scale statistical model which can capture the structure of natural images across many scales. Once trained on example images, it can be used to recognize novel images, or to generate new images. Each of these tasks is reasonably efficient, requiring no more than a few seconds or minutes on a workstation. The statistical modeling of images is an endeavor which reaches back to the 60's and 70's (Duda and Hart, 1973). Statistical approaches are alluring because they provide a unified view of learning, classification and generation. To date however, a generic, efficient and unified statistical model for natural images has yet to appear. Nevertheless, many approaches have shown significant competence in specific areas. Perhaps the most influential statistical model for generic images is the Markov random field (MRF) (Geman and Geman, 1984). MRF's define a distribution over 774 J. S. D. Bonet and P. A. Viola images that is based on simple and local interactions between pixels. Though MRF's have been very successfully used for restoration of images, their generative properties are weak. This is due to the inability of the MRF's to capture long range (low frequency) interactions between pixels. Recently there has been a great deal of interest in hierarchical models such as the Helmholtz machine (Hinton et al., 1995; Dayan et al., 1995). Though the Helmholtz machine can be trained to discover long range structure, it is not easily applied to natural images. Multi-scale wavelet models have emerged as an effective technique for modeling realistic natural images. These techniques hypothesize that the wavelet transform measures the underlying causes of natural images which are assumed to be statistically independent. The primary evidence for this conjecture is that the coefficients of wavelet transformed images are uncorrelated and low in entropy (hence the success of wavelet compression). These insights have been used for noise reduction (Donoho and Johnstone, 1993; Simoncelli and Adelson, 1996), and example driven texture synthesis (Heeger and Bergen, 1995). The main drawback of wavelet algorithms is the assumption of complete independence between coefficients. We conjecture that in fact there is strong cross-scale dependence between the wavelet coefficients of an image, which is consistent with observations in (De Bonet, 1997) and (Buccigrossi and Simoncelli, 1997). 2 Multi-scale Statistical Models Multi-scale wavelet techniques assume that images are a linear transform of a collection of statistically independent random variables: 1= W-1C, where I is an image, W- 1 is the inverse wavelet transform, and C = {Ck} is a vector of random variable "causes" which are assumed to be independent. The distribution of each cause Ck is Pk ( .), and since the Ck'S are independent it follows that: p( C) = nk Pk (Ck). Various wavelet transforms have been developed, but all share the same type of multi-scale structure each row of the wavelet matrix W is a spatially localized filter that is a shifted and scaled version of a single basis function. The wavelet transform is most efficiently computed as an iterative convolution using a bank of filters. First a "pyramid" of low frequency downsampled images is created: Go = I , G1 = 2 ..!-(9 ® Go), and Gi+l = 2 ..!-(9 ® Gi ), where 2..!- downsamples an image by a factor of 2 in each dimension, ® is the convolution operation, and 9 is a low pass filter. At each level a series offilter functions are applied: Fj = h (j!) Gj, where the Ii's are various types of filters. Computation of the Fj's is a linear transformation that can thought of as a single matrix W. With careful selection of 9 and h this matrix can be constructed so that W- 1 = W T (Simoncelli et al., 1992)1. Where convenient we will combine the pixels of the feature images Fj(x, y) into a single cause vector C. The expected distribution of causes, Ck, is a function of the image classes that are being modeled. For example it is possible to attempt to model the space of all natural images. In that case it appears as though the most accurate Pk (.) 's are highly kurtotic which indicates that the Ck ' S are most often zero but in rare cases take on very large values (Donoho and Johnstone, 1993; Simoncelli and Adelson, 1996). This is in direct contrast to the distribution of Ck'S for white-noise images which is gaussian. The difference in these distributions can be used as the basis of noise reduction algorithms, by reducing the wavelet coefficients which are more lComputation of the inverse wavelet transform is algorithmically similar to the computation of the forward wavelet transform. Non-Parametric Multi-Scale Statistical Image Model 775 likely to be noise than signal. Specific image classes can be modeled using similar methods (Heeger and Bergen, 1995)2. For a given set of input images the empirical distribution of the Ck'S is observed. To generate a novel example of a texture a new set of causes, (}, is sampled from the assumed independent empirical distributions Pk (.). The generated images are computed using the inverse wavelet transform: l' = W- 1(},. Bergen and Heeger have used this approach to build a probabilistic model of a texture from a ::;ingle example image. To do this they assume that textures are spatially ergodic - that the expected distribution is not a function of position in the image. As a result the pixels in anyone feature image, Fj(x, y), are samples from the same distribution and can be combined3 . Heeger and Bergen's work is at or near the current state of the art in texture generation. Figure 1 contains some example textures. Notice however, that this technique is much better at generating smooth or noise-like textures than those with well defined structure. Image structures, such as the sharp edges at the border of the tiles in the rightmost texture can not be modeled with their approach. These image features directly contradict the assumption that the wavelet coefficients, or causes, of the image are independent. For many types of natural images the coefficients of the wavelet transform are not independent, for example images which contain long edges. While wavelets are local both in frequency and space, a long edge is not local in frequency nor in space. As a result the wavelet representation of such a feature requires many coefficients. The high frequencies of the edge are captured by many small high frequency wavelets. The long scale is captured by a number of larger low frequency wavelets. A model which assumes these coefficients are independent can never accurately model images which contain these non-local features. Conversely a model which captures the conditional dependencies between coefficients will be much more effective. We chose to approximate the joint distribution of coefficients as a chain, in which coefficients that occur higher in the wavelet pyramid condition the distribution of coefficients at lower levels (Le. low frequencies condition the generation of higher frequencies). For every pixel in an image define the parent vector of that pixel: .... [ 0 1 N V (x, y) = Fo (x, y), Fo (x, y), . .. ,Fo (x, y), -nO X y 1 X y) N X YJ) l'{(l2"J, l2"J),F1 (L2"J, L2"J , ... ,Fl (l2"J, L2" , ... o x Y 1 X J Y) N L x J L Y J)] FM(L2MJ,l2MJ),FM(l2M ,L2MJ , ... ,FM ( 2M ' 2M (1) where M is the top level of the pyramid and N is the number of features. Rather than generating each of these coefficients independently, we define a chain across scale. In this chain the generation of the lower levels depend on the higher levels: p(V(x, y)) = p(VM(x, y)) x p(VM- 1 (x, y)IVM(x, y)) x p(VM-2(X, y)!VM-l (x, y), VM(x, y)) x ... x p(Vo(x, y)IVl (x, y), ... , VM-l (x, y), VM(x, y)) (2) 2See (Zhu, Wu and Mumford, 1996) for a related but more formal model. 3Their generation process is slightly more complex than this, involving a iteration designed to match the pixel histogram. The implementation used for generating the images in Figure 1 incorporates this, but we do not discuss it here. 776 1. S. D. Bonet and P. A. Viola Figure 1: Synthesis results for the Heeger and Bergen (1995) model. Top: Input textures. BOTTOM: Synthesis results. This technique is much better at generating fine or noisy textures then it is at generating textures which require co-occurrence of wavelets at multiple scales. Figure 2: Synthesis results using our technique for the input textures shown in Figure 1 (Top). where Yt(x, y) is the a subset of the elements of Vex, y) computed from C/. Usually we will assume ergodicity, i.e. that p(V(x, y)) is independent of x and y. The generative process starts from the top of the pyramid, choosing values for the V M (x, y) at all points. Once these are generated the values at the next level, V M -1 (x , y), are generated. The process continues until all of the wavelet coefficients are generated. Finally the image is computed using the inverse wavelet transform. It is important to note that this probabilistic model is not made up of a collection of independent chains, one for each Vex, y). Parent vectors for neighboring pixels have substantial overlap as coefficients in the higher pyramid levels (which are Non-Parametric Multi-Scale Statistical Image Model 777 lower resolution) are shared by neighboring pixels at lower pyramid levels. Thus, the generation of nearby pixels will be strongly dependent. In a related approach a similar arrangement of generative chains has been termed a Markov tree (Basseville et al., 1992). 2.1 Estimating the Conditional Distributions The additional descriptive power of our generative model does not come without cost. The conditional distributions that appear in (2) must be estimated from observations. We choose to do this directly from the data in a non-parametric fashion. Given a sample of parent vectors {8(x, y)} from an example image we estimate the conditional distribution as a ratio of Parzen window density estimators: (~( )1v,M ( )) _ p(ViM(x,y)) '" Lx',y' R(V;M(X,y), 8r(x', y')) p I x,y /+1 x,y .... M '" "'M .... M p(V/+1(x,y)) Lx',y' R(V/+l (x, y), S/+1 (x',y')) (3) where Vik(x,y) is a subset of the parent vector V(x,y) that contains information from level I to level k, and R(·) is a function of two vectors that returns maximal values when the vectors are similar and smaller values when the vectors are dissimilar. We have explored various R(·) functions. In the results presented the R( .) function returns a fixed constant 1/ z if all of the coefficients of the vectors are within some threshold () and zero otherwise. Given this simple definition for R(·) sampling from p(Vz(x,Y)IV;~1(X,y)) is very straightforward: find all x', y' such that R(8#1 (x', y'), 8#1 (x, y)) = 1/ z and pick from among them to set Vz(x,y) = SI(X',y'). 3 Experiments We have applied this approach to the problems of texture generation, texture recognition, target recognition, and signal de-noising. In each case our results are competitive with the best published approaches. In Figure 2 we show the results of our technique on the textures from Figure 1. For these textures we are better able to model features which are caused by a conjunction of wavelets. This is especially striking in the rightmost texture where the geometrical tiling is almost, but not quite, preserved. In our model, knowledge of the joint distribution provides constraints which are critical in the overall perceived appearance of the synthesized texture. Using this same model, we can measure the textural similarity between a known and novel image. We do this by measuring the likelihood of generating the parent vectors in the novel image under the chain model of the known image. On "easy" data sets, such as the the MeasTex Brodatz texture test suite, performance is slightly higher than other techniques, our approach achieved 100% correct classification compared to 97% achieved by a gaussian MRF approach (Chellappa and Chatterjee, 1985). The MeasTex lattice test suite is slightly more difficult because each texture is actually a composition of textures containing different spatial frequencies. Our approach achieved 97% while the best alternate method, in this case Gabor Convolution Energy method (Fogel and Sagi, 1989) achieved 89%. Gaussian MRF's explicitly assume that the texture is a unimodal distribution and as a result achieve only 79% correct recognition. We also measured performance on a set of 20 types of natural texture and compared the classification power of this model to that of human observers (humans discriminate textures extremely accurately.) On this 778 1. S. D. Bonet and P. A. Viola Original Denoise Shrinkage Shrinkage Residual Noised Denoise Ours Our Residual Figure 3: (Original) the original image; (Noised) the image corrupted with white gaussian noise (SNR 8.9 dB); (Denoise Shrinkage) the results of de-noising using wavelet shrinkage or coring (Donoho and Johnstone, 1993; Simoncelli and Adelson, 1996) (SNR 9.8 dB); (Shrinkage Residual) the residual error between the shrinkage de-noised result and the original notice that the error contains a great deal of interpretable structure; (Denoise Ours) our de-noising approach (SNR 13.2 dB); and (Our Residual) the residual error these errors are much less structured. test, humans achieved 86% accuracy, our approach achieved an accuracy of 81%, and GMRF's achieved 68%. A strong probabilistic model for images can be used to perform a variety of image processing tasks including de-noising and sharpening. De-noising of an observed image i can be performed by Monte Carlo averaging: draw a number of sample images according to the prior density P(I), compute the likelihood of the noise for each image P(v = (1) - 1), and then find the weighted average over these images. The weighted average is the estimated mean over all possible ways that the image might have been generated given the observation. Image de-noising frequently relies on generic image models which simply enforce image smoothness. These priors either leave a lot of residual noise or remove much of the original image. In contrast, we construct a probability density model from the noisy image itself. In effect we assume that the image is redundant, containing many examples of the same visual structures, as if it were a texture. The value of this approach is directly related to the redundancy in the image. If the redundancy in the image is very low, then the parent structures will be everywhere different, and the only resampled images with significant likelihood will be the original image. But if there is some redundancy in the image that might arise from a regular texture or smoothly varying patch the resampling will freely average across these similar regions. This will have the effect of reducing noise in these images. In Figure 3 we show results of this de-noising approach. Non-Parametric Multi-Scale Statistical Image Model 779 4 Conclusions We have presented a statistical model of texture which can be trained using example images. The form of the model is a conditional chain across scale on a pyramid of wavelet coefficients. The cross scale condtional distributions are estimated non-parametrically. This is important because many of the observed conditional distributions are complex and contain multiple modes. We believe that there are two main weaknesses of the current approach: i) the tree on which the distributions are defined are fixed and non-overlapping; and ii) the conditional distributions are estimated from a small number of samples. We hope to address these limitations in future work. Acknowledgments In this research, Jeremy De Bonet is supported by the DOD Multidisciplinary Research Program of the University Research Initiative, and Paul Viola by Office of Naval Research Grant No. N00014-96-1-0311. References Basseville, M., Benveniste, A., Chou, K. C., Golden, S. A., Nikoukhah, R., and Will sky, A. S. (1992). Modeling and estimation of multiresolution stochastic processes. IEEE Transactions on Information Theory, 38(2):766-784. Buccigrossi, R. W. and Simoncelli, E. P. (1997). Progressive wavelet image coding based on a conditional probability model. In Proceedings ICASSP-97, Munich, Germany. Chellappa, R. and Chatterjee, S. (1985). Classification of textures using gaussian markov random fields. In Proceedings of the International Joint Conference on Acoustics, Speech and Signal Processing, volume 33, pages 959-963. Dayan, P., Hinton, G., Neal, R., and Zemel, R. (1995). The helmholtz machine. Neural Computation, 7:1022-1037. De Bonet, J. S. (1997). Multiresolution sampling procedure for analysis and synthesis of texture images. In Computer Graphics. ACM SIGGRAPH. Donoho, D. L. and Johnstone, 1. M. (1993). Adaptation to unknown smoothness via wavelet shrinkage. Technical report, Stanford University, Department of Statistics. Also Tech. Report 425. Duda, R. and Hart, P. (1973). Pattern Classification and Scene Analysis. John Wiley and Sons. Fogel, I. and Sagi, D. (1989). Gabor filters as texture discriminator. Biological Cybernetics, 61: 103- 113. Geman, S. and Geman, D. (1984). Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:721-741. Heeger, D. J. and Bergen, J. R. (1995). Pyramid-based texture analysis/synthesis. In Computer Graphics Proceedings, pages 229-238. Hinton, G., Dayan, P., Frey, B., and Neal, R. (1995). The "wake-sleep" algorithm for unsupervised neural networks. Science, 268:1158-116l. Simoncelli, E. P. and Adelson, E. H. (1996). Noise removal via bayesian wavelet coring. In IEEE Third Int'l Conf on Image Processing, Laussanne Switzerland. IEEE. Simoncelli, E. P., Freeman, W. T., Adelson, E. H., and Heeger, D. J. (1992). Shiftable multiscale transforms. IEEE Transactions on Information Theory, 38(2):587-607. Zhu, S. C., Wu, Y., and Mumford, D. (1996). Filters random fields and maximum entropy(frame): To a unified theory for texture modeling. To appear in Int'l Journal of Computer Vision.
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A Solution for Missing Data in Recurrent Neural Networks With an Application to Blood Glucose Prediction Volker Tresp and Thomas Briegel * Siemens AG Corporate Technology Otto-Hahn-Ring 6 81730 Miinchen, Germany Abstract We consider neural network models for stochastic nonlinear dynamical systems where measurements of the variable of interest are only available at irregular intervals i.e. most realizations are missing. Difficulties arise since the solutions for prediction and maximum likelihood learning with missing data lead to complex integrals, which even for simple cases cannot be solved analytically. In this paper we propose a specific combination of a nonlinear recurrent neural predictive model and a linear error model which leads to tractable prediction and maximum likelihood adaptation rules. In particular, the recurrent neural network can be trained using the real-time recurrent learning rule and the linear error model can be trained by an EM adaptation rule, implemented using forward-backward Kalman filter equations. The model is applied to predict the glucose/insulin metabolism of a diabetic patient where blood glucose measurements are only available a few times a day at irregular intervals. The new model shows considerable improvement with respect to both recurrent neural networks trained with teacher forcing or in a free running mode and various linear models. 1 INTRODUCTION In many physiological dynamical systems measurements are acquired at irregular intervals. Consider the case of blood glucose measurements of a diabetic who only measures blood glucose levels a few times a day. At the same time physiological systems are typically highly nonlinear and stochastic such that recurrent neural networks are suitable models. Typically, such networks are either used purely free running in which the networks predictions are iterated, or in a teacher forcing mode in which actual measurements are substituted • {volker.tresp, thomas.briegel}@mchp.siemens.de 972 V. Tresp and T. Briegel if available. In Section 2 we show that both approaches are problematic for highly stochastic systems and if many realizations of the variable of interest are unknown. The traditional solution is to use a stochastic model such as a nonlinear state space model. The problem here is that prediction and training missing data lead to integrals which are usually considered intractable (Lewis, 1986). Alternatively, state dependent linearizations are used for prediction and training, the most popular example being the extended Kalman filter. In this paper we introduce a combination of a nonlinear recurrent neural predictive model and a linear error model which leads to tractable prediction and maximum likelihood adaptation rules. The recurrent neural network can be used in all generality to model the nonlinear dynamics of the system. The only limitation is that the error model is linear which is not a major constraint in many applications. The first advantage of the proposed model is that for single or multiple step prediction we obtain simple iteration rules which are a combination of the output of the iterated neural network and a linear Kalman filter which is used for updating the linear error model. The second advantage is that for maximum likelihood learning the recurrent neural network can be trained using the real-time recurrent learning rule RTRL and the linear error model can be trained by an EM adaptation rule, implemented using forward-backward Kalman filter equations. We apply our model to develop a model of the glucose/insulin metabolism of a diabetic patient in which blood glucose measurements are only available a few times a day at irregular intervals and compare results from our proposed model to recurrent neural networks trained and used in the free running mode or in the teacher forcing mode as well as to various linear models. 2 RECURRENT SYSTEMS WITH MISSING DATA Y, / reasonable estimate for 1 =:t. 6 /ncosuremcnt at (,me t=7 y. .. • 0 ~ .. '_'m~__ I o teacher forC ing free running ~CQsurcment at lIm~ 1= 7 ' .. -. UIIIIdrrrrI 1 2 " 4 S 6 7 8 9 10 J I 12 13 Figure 1: A neural network predicts the next value of a time-series based on the latest two previous measurements (left). As long as no measurements are available (t = 1 to t = 6), the neural network is iterated (unfilled circles). In a free-running mode, the neural network would ignore the measurement at time t = 7 to predict the time-series at time t = 8. In a teacher forcing mode, it would substitute the measured value for one of the inputs and use the iterated value for the other (unknown) input. This appears to be suboptimal since our knowledge about the time-series at time t = 7 also provides us with information about the time-series at time t = 6. For example the dotted circle might be a reasonable estimate. By using the iterated value for the unknown input, the prediction of the teacher forced system is not well defined and will in general lead to unsatisfactory results. A sensible response is shown on the right where the first few predictions after the measurement are close to the measurement. This can be achieved by including a proper error model (see text). Consider a deterministic nonlinear dynamical model of the form Yt = !w(Yt-l," ·,Yt-N,Ut) of order N, with input Ut and where ! w (.) is a neural network model with parametervector w. Such a recurrent model is either used in a free running mode in which network predictions are used in the input of the neural network or in a teacher forcing mode where measurements are substituted in the input of the neural network whenever these are available. Missing Data in RNNs with an Application to Blood Glucose Prediction 973 - - .. -.:::...:..- -~-~ ----Figure 2: Left: The proposed architecture. Right: Linear impulse response. Both can lead to undesirable results when many realizations are missing and when the system is highly stochastic. Figure 1 (left) shows that a free running model basically ignores the measurement for prediction and that the teacher forced model substitutes the measured value but leaves the unknown states at their predicted values which also might lead to undesirable responses. The traditional solution is to include a model of the error which leads to nonlinear stochastical models, the simplest being Yt = fw (Yt-l,.'" Yt-N, utJ + lOt where lOt is assumed to be additive uncorrelated zero-mean noise with probability density P€ (f) and represents unmodeled system dynamics. For prediction and learning with missing values we have to integrate over the unknowns which leads to complex integrals which, for nonlinear models, have to be approximated. for example, using Monte Carlo integration.l In general, those integrals are computationally too expensive to solve and, in practice, one relies on locally linearized approximations of the nonlinearities typically in form of the extended Kalman filter. The extended Kalman filter is suboptimal and summarizes past data by an estimate of the means and the covariances of the variables involved (Lewis, 1986). In this paper we pursue an alternative approach. Consider the model with state updates * Yt Xt Yt fW(Y;-l"' " Y;-N' ut} K 2: (JiXt-i + lOt i=l K Y; + Xt = fw (Y;-l' ... , Y;-N, ue) + L (JiXt-i + lOt i=l and with measurement equation (1) (2) (3) Zt=Yt+Ot. (4) where lOt and Ot denote additive noise. The variable of interest Yt is now the sum of the deterministic response of the recurrent neural network Y; and a linear system error model Xt (Figure 2). Zt is a noisy measurement of Yt. In particular we are interested in the special cases that Yt can be measured with certainty (variance of Ot is zero) or that a measurement is missing (variance of Ot is infinity). The nice feature is now that Y; can be considered a deterministic input to the state space model consisting of the equations (2)- (3). This means that for optimal one-step or multiple-step prediction, we can use the linear Kalman filter for equations (2)- (3) and measurement equation (4) by treating Y; as deterministic input. Similarly, to train the parameters in the linear part of the system (i.e. {Oi }f:l) we can use an EM adaptation rule, implemented using forward-backward Kalman filter equations (see the Appendix). The deterministic recurrent neural network is adapted with the residual error which cannot be explained by the linear model, i.e. target~nn = y": - :Wnear 1 For maximum likelihood learning of linear models we obtain EM equations which can be solved using forward-backward Kalman equations (see Appendix). 974 V. Tresp and T. Briegel where Y~ is a measurement ofYt at time t and where f)/near is the estimate of the linear model. After the recurrent neural network is adapted the linear model can be retrained using the residual error which cannot be explained by the neural network. then again the neural network is retrained and so on until no further improvement can be achieved. The advantage of this approach is that all of the nonlinear interactions are modeled by a recurrent neural network which can be trained deterministically. The linear model is responsible for the noise model which can be trained using powerful learning algorithms for linear systems. The constraint is that the error model cannot be nonlinear which often might not be a major limitation. 3 BLOOD GLUCOSE PREDICTION OF A DIABETIC The goal of this work is to develop a predictive model of the blood glucose ofa person with type 1 Diabetes mellitus. Such a model can have several useful applications in therapy: it can be used to warn a person of dangerous metabolic states, it can be used to make recommendations to optimize the person's therapy and, finally, it can be used in the design of a stabilizing control system for blood glucose regulation, a so-called "artificial beta cell" (Tresp, Moody and Delong, 1994). We want the model to be able to adapt using patient data collected under normal every day conditions rather than the controlled conditions typical of a clinic. In a non-clinical setting, only a few blood glucose measurements per day are available. Our data set consists of the protocol ofa diabetic over a period of almost six months. During that time period, times and dosages of insulin injections (basal insulin ut and normal insulin u;), the times and amounts of food intake (fast u~, intermediate ut and slow u~ carbohydrates), the times and durations of exercise (regular u~ or intense ui) and the blood glucose level Yt (measured a few times a day) were recorded. The u{, j = 1, ... ,7 are equal to zero except if there is an event, such as food intake, insulin injection or exercise. For our data set, inputs u{ were recorded with 15 minute time resolution. We used the first 43 days for training the model (containing 312 measurements of the blood glucose) and the following 21 days for testing (containing 151 measurements of the blood glucose). This means that we have to deal with approximately 93% of missing data during training. The effects on insulin, food and exercise on the blood glucose are delayed and are approximated by linear response functions. v{ describes the effect of input u{ on glucose. As an example, the response vt of normal insulin u; after injection is determined by the diffusion of the subcutaneously injected insulin into the blood stream and can be modeled by three first order compartments in series or, as we have done, by a response function of the form vt = l:T g2(t - r)u; withg2(t) = a2t2e-b2t (see figure 2 for a typical impulse response). The functional mappings gj (.) for the digestive tract and for exercise are less well known. In our experiments we followed other authors and used response functions of the above form. The response functions 9 j ( .) describe the delayed effect of the inputs on the blood glucose. We assume that the functional form of gj (.) is sufficient to capture the various delays of the inputs and can be tuned to the physiology of the patient by varying the parameters aj ,bj . To be able to capture the highly nonlinear physiological interactions between the response functions vi and the blood glucose level Yt, which is measured only a few times a day, we employ a neural network in combination with a linear error model as described in Section 2. In our experiments fw (.) is a feedforward multi-layer perceptron with three hidden units. The five inputs to the network were insulin (in; = vi + v;>, food (in; = vf + vt + vt), exercise (inr = vf + vi) and the current and previous estimate of the blood glucose. To be specific, the second order nonlinear neural network model is * * f (. * . 1 . 2 . 3) Yt = Yt-l + W Yt_llYt_2,lnt ,znt ,znt (5) Missing Data in RNNs with an Application to Blood Glucose Prediction 975 For the linear error model we also use a model of order 2 (6) Table 1 shows the explained variance of the test set for different predictive models. 2 In the first experiment (RNN-FR) we estimate the blood glucose at time t as the output of the neural network Yt = y;. The neural network is used in the free running mode for training and prediction. We use RTRL to both adapt the weights in the neural network as well as all parameters in the response functions 9j (.). The RNN-FR model explains 14.1 percent of the variance. The RNN-TF model is identical to the previous experiment except that measurements are substituted whenever available. RNN-TF could explain more of the variance (18.8%). The reason for the better performance is, of course, that information about measurements of the blood glucose can be exploited. The model RNN-LEM2 (error model with order 2) corresponds to the combination of the recurrent neural network and the linear error model as introduced in Section 2. Here, Yt = Xt + Y; models the blood glucose and Zt = Yt + 8t is the measurement equation where we set the variance of 8t = 0 for a measurement of the blood glucose at time t and the variance of 8t = 00 for missing values. For ft we assume Gaussian independent noise. For prediction, equation (5) is iterated in the free running mode. The blood glucose at time t is estimated using a linear Kalman filter, treating Y; as deterministic input in the state space model Yt = x t + Y; ,Zt = Yt + 8t . We adapt the parameters in the linear error model (i.e. (h, O2 , the variance of ft) using an EM adaptation rule, implemented using forwardbackward Kalman filter equations (see Appendix). The parameters in the neural network are adapted using RTRL exactly the same way as in the RNN-FR model, except that the target is now target~nn = yr - iftinear where yr is a measurement of Yt at time t and where iftinear is the estimate of the linear error model (based on the linear Kalman filter). The adaptation of the linear error model and the neural network are performed aIternatingly until no significant further improvement in performance can be achieved. As indicated in Table 1, the RNN-LEM2 model achieves the best prediction performance with an explained variance of 44.9% (first order error model RNN-LEMI: 43.7%). As a comparison, we show the performance of just the linear error model LEM (this model ignores all inputs), a linear model (LM-FR) without an error model trained with RTRL and a linear model with an error model (LM-LEM). Interestingly, the linear error model which does not see any of the inputs can explain more variance (12.9%) than the LM-FR model (8.9%). The LM-LEM model, which can be considered a combination of both can explain more than the sum of the individual explained variances (31.5%) which indicates that the combined training gives better perfonnance than training both submodels individually. Note also, that the nonlinear models (RNN-FR, RNN-TF, RNN-LEM) give considerably better results than their linear counterparts, confirming that the system is highly nonlinear. Figure 3 (left) shows an example of the responses of some of the models. We see that the free running neural network (dotted line) has relatively small amplitudes and cannot predict the three measurements very well. The RNN-TF model (dashed line) shows a better response to the measurements than the free running network. The best prediction of all measurements is indeed achieved by the RNN-LEM model (continuous line). Based on the linear iterated Kalman filter we can calculate the variance of the prediction. As shown in Figure 3 (right) the standard deviation is small right after a measurement is available and then converges to a constant value. Based on the prediction and the estimated variance, it will be possible to do a risk analysis for the diabetic (i.e a warning of dangerous metabolic states). 2MSPE(model) is the mean squared prediction error on the test set of the model and MSPE( mean) is the mean squared prediction error of predicting the mean. 976 V. Tresp and T. Briegel 240~--~----~--~--~ 230 p. 220 ~\ ..,-210 1\\ ~ I , =-200 I \ I J \ \ \ ~ 190 '." ~ 1 . '. ~ -g lBO • " , "_ ~ - ': '.~'::"" 170· ........ 160 ••••••••••••••• ::;-;~ .. -~..,,"150 2 . 5 5 7 .5 time [hOurS] 2.5 5 7.5 time [hOurS] Figure 3: Left: Responses of some models to three measurements. Note, that the prediction of the first measurement is bad for all models but that the RNN-LEM model (continuous line) predicts the following measurements much better than both the RNN -FR (dotted) and the RNN-TF (dashed) model. Right: Standard deviation of prediction error ofRNN-LEM. Table 1: Explained variance on test set [in percent]: 100 . (1 - ~;~~ ::~~ ) MODEL % MODEL % mean 0 RNN-TF 18.8 LM 8.9 LM-LEM 3l.4 LEM 12.9 RNN-LEMl 43.7 RNN-FR 14.1 RNN-LEM2 44.9 4 CONCLUSIONS We introduced a combination of a nonlinear recurrent neural network and a linear error model. Applied to blood glucose prediction it gave significantly better results than both recurrent neural networks alone and various linear models. Further work might lead to a predictive model which can be used by a diabetic on a daily bases. We believe that our results are very encouraging. We also expect that our specific model can find applications in other stochastical nonlinear systems in which measurements are only available at irregular intervals such that in wastewater treatment, chemical process control and various physiological systems. Further work will include error models for the input measurements (for example, the number of food calories are typically estimated with great uncertainty). Appendix: EM Adaptation Rules for Training the Linear Error Model Model and observation equations of a general model are3 Xt = eXt-l + ft Zt = MtXt + 8t . (7) where e is the K x K transition matrix ofthe K -order linear error model. The K x 1 noise terms (t are zero-mean uncorrelated normal vectors with common covariance matrix Q. 8t is m-dimensional 4 zero-mean uncorrelated normal noise vector with covariance matrix Rt • Recall that we consider certain measurements and missing values as special cases of 3Note, that any linear system of order K can be transformed into a first order linear system of dimension K. 4 m indicates the dimension of the output of the time-series. Missing Data in RNNs with an Application to Blood Glucose Prediction 977 noisy measurements. The initial state of the system is assumed to be a normal vector with mean Jl and covariance E. We describe the EM equations for maximizing the likelihood of the model. Define the estimated parameters at the (r+ l)st iterate of EM as the values Jl, E, e, Q which maximize G(Jl, E, e, Q) = Er (log Llzl, ... , zn) (8) where log L is log-likelihood of the complete data Xo, Xl, ••• , X n , ZI, • .• , Zn and Er denotes the conditional expectation relative to a density containing the rth iterate values Jl(r), E(r), e(r) and Q(r). Recall that missing targets are modeled implicitly by the definition of M t and Rt . For calculating the conditional expectation defined in (8) the following set of recursions are used (using standard Kalman filtering results, see (Jazwinski, 1970)). First, we use the forward recursion t-l e t-l X t - x t _ l pt-l ept-leT + Q t t-l K t P/-lMtT(MtP/-IMtT + Rt)-1 (9) Xt t-l + r.' (* M t-l) t xt I'q Yt tXt Pi ptt-l - K t M t P;-1 where we take xg = Jl and p3 = E. Next, we use the backward recursion J Pt-leT(pt-l)-l t-l t-l t X~_l x~=~ + Jt-l(X~ ex~=D Ptn_l p:~t + Jt-dPr - p;-l)J?'_l (10) pr-l,t-2 P/~11JL2 + Jt-dPtt-l ePtt~l)Jt~2 with initialization p;: n-l = (1 - KnMn)ep;:::l. One forward and one backward recursion completes the E-'step of the EM algorithm. To derive the M-step first realize that the conditional expectations in (8) yield to the following equation: G = -~ log IEI- !tr{E-l(Pon + (xo - Jl)(xo - Jl)T)} -% log IQI-1tr{Q-l(C - BeT - eBT - eAeT)} -% log IRtl- !tr{R;-l E~l[(Y; - Mtxt)(y; - Mtxd T + MtprMtT]} where tr{.} denotes the trace, A = E~=l (pr-l + xr_lx~:d, B = E~=l(Pt~t-l + X~X~:l) and C = E~=l (pr + x~x~ T). (11 ) e(r + 1) = BA- 1 and Q(r + 1) = n-1(C - BA- l BT) maximize the log-likelihood equation (11). Jl (r + 1) is set to Xo and E may be fixed at some reasonable baseline level. The derivation of these equations can be found in (Shumway & Stoffer, 1981). The E- (forward and backward Kalman filter equations) and M-steps are alternated repeatedly until convergence to obtain the EM solution. References Jazwinski, A. H. (1970) Stochastic Processes and Filtering Theory, Academic Press, N.Y. Lewis, F. L. (1986) Optimal Estimation, John Wiley, N.Y. Shumway, R. H. and Stoffer, D. S. (1981) TIme Series Smoothing and Forecasting Using the EM Algorithm, Technical Report No. 27, Division of Statistics, UC Davis. Tresp, v., Moody, 1. and Delong, W.-R. (1994) Neural Modeling of Physiological Processes, in Comput. Leaming Theory and Natural Leaming Sys. 2, S. Hanson et al., eds., MIT Press.
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A 1,OOO-Neuron System with One Million 7-bit Physical Interconnections Yuzo Hirai Institute of Information Sciences and Electronics University of Tsukuba 1-1-1 Ten-nodai, Tsukuba, Ibaraki 305, Japan e-mail: hirai@is.tsukuba.ac.jp Abstract An asynchronous PDM (Pulse-Density-Modulating) digital neural network system has been developed in our laboratory. It consists of one thousand neurons that are physically interconnected via one million 7-bit synapses. It can solve one thousand simultaneous nonlinear first-order differential equations in a fully parallel and continuous fashion. The performance of this system was measured by a winner-take-all network with one thousand neurons. Although the magnitude of the input and network parameters were identical for each competing neuron, one of them won in 6 milliseconds. This processing speed amounts to 360 billion connections per second. A broad range of neural networks including spatiotemporal filtering, feedforward, and feedback networks can be run by loading appropriate network parameters from a host system. 1 INTRODUCTION The hardware implementation of neural networks is crucial in order to realize the real-time operation of neural functions such as spatiotemporal filtering, learning and constraint processings. Since the mid eighties, many VLSI chips and systems have been reported in the literature, e.g. [1] [2]. Most of the chips and the systems including analog and digital implementations, however, have focused on feedforward neural networks. Little attention has been paid to the dynamical aspect of feedback neural networks, which is especially important in order to realize constraint processings, e.g. [3]. Although there were a small number of exceptions that used analog circuits [4] [5], their network sizes were limited as compared to those of their feedforward counterparts because of wiring problems that are inevitable in regard to full and physical interconnections. To relax this problem, a pulse-stream system has been used in analog [6] and digital implementations [7]. 706 Y. Hirai The author developed a fully interconnected 54-neuron system that uses an asynchronous PDM (Pulse-Density-Modulating) digital circuit system [8]. The present paper describes a thousand-neuron system in which all of the neurons are physically interconnected via one million 7-bit synapses in order to create a fully parallel feedback system. The outline of this project was described in [10]. In addition to the enlargement of system size, synapse circuits were improved and time constant of each neuron was made variable. The PDM system was used because it can accomplish faithful analog data transmission between neurons and can relax wiring problems. An asynchronous digital circuit was used because it can solve scaling problems, and we could also use it to connect more than one thousand VLSI chips, as described below. 2 NEURON MODEL AND THE CIRCUITS 2.1 SINGLE NEURON MODEL The behavior of each neuron in the system can be described by the following nonlinear first-order differential equation: dyi(t) N = -viet) + L WijYj{t) + li{t), (1) Iti--;];t j=l Yi{ t) = <p[yi{t)], and (2) <pta] { ~ if a > 0 (3) = otherwise, where Iti is a time constant of the i-th neuron, y;(t) is an internal potential of the i-th neuron at time t, Wij is a synaptic weight from the j-th to the i-th neurons, and li(t) is an external input to the i-th neuron. <pta] is an analog threshold output function which becomes saturated at a given maximum val~e. The system solves Eq.{l) in the following integral form: yi{t) = (t {-VieT) + t WijYj(T) + h(T)} d~ + yi(O), 10 j=l Itl (4) where y;(O) is an initial value. An analog output of a neuron is expressed by a pulse stream whose frequency is proportional to the positive, instantaneous internal potential. 2.2 SINGLE NEURON CmCUIT 2.2.1 Synapse circuits The circuit diagrams for a single neuron are shown in Fig. 1. As shown in Fig.l(a), it consists of synapse circuits, excitatory and inhibitory dendrite OR circuits, and a cel~ body circuit. Each synapse circuit transforms the instantaneous frequency of the input pulses to a frequency that is proportional to the synaptic weight. This transformation is carried out by a 6-bit rate multiplier, as shown in Fig.l{b). The behavior of a rate multiplier is illustrated in Fig.l(c) using a 3-bit case for brevity. A rate multiplier is a counter and its state transits to the next state when an input pulse occurs. Each binary bit of a given weight specifies at which states the output pulses are generated. When the LSB is on, an output pulse is generated at the fourth state. When the second bit is on, output pulses are generated at the second A I,OOO-Neuron System with One Million 7-bit Physical Interconnections Neuron circuit Dendrite extension terminals Synapse circuit 3-blt rate multiplier states 707 '.:;g::. multiplier wt nIl!.. Rate tp@ weight 01234567 InJ)Ut pulses JUL .® ~ ~ ~ ~~ j ~ Synapse ~ j -'L U I circuit I - U r,:,. a: a: ~ 0_0 J J J JUL : i .. -~ (&-bit) 1 + @ Synaptic weight (7-blt) sign (Osw<1) (b) 000 001 1 010 1 1 011 1 1 1 100 1 1 1 1 101 1 1 1 1 1 110 1 1 1 11 1 1 111 1111111 (c) sign i _ . I Sy~pse I e ~ L Cell body circuit a (Circuit r ~ t.::\@, UDr-----., r-=-~+'_:_--, ~ C """ ,.=. Sampling ~ Up-down f-Rate 2f Output circuit Down counter ~ ~ multiplier ;. gate and CI@l®rJUL 'w f~~~_(1~2-b-+t=) __ I--(_12~-br .... It)_ =~ I. f<:IrI Main clock = 4fma (2OMHz) -:-='112 I ~ II Cell body circuit \51 r=----,~-..~ I Down • r Rate I If for2f @. JlJlJL L-----:~---II X (-1) multiplier f.J H slgn=O Output pulses Up' (&-bit) (a) (d) Figure 1: Circuit diagram of a single neuron. (a) Circuit diagram of a single neuron and (b) that of a synapse circuit. (c) To illustrate the function of a rate multiplier, the multiplication table for a 3-bit case is shown. (d) Circuit diagram of a cell body circuit. See details in text. and at the sixth states. When the MSB is on, they are generated at all of the odd states. Therefore, the magnitude of synaptic weight that can be represented by a rate multiplier is less than one. In our circuit, this limitation was overcome by increasing the frequency of a neuron output by a factor of two, as described below. 2.2.2 Dendrite circuits Output pulses from a synapse circuit are fed either to an excitatory dendrite OR circuit or to an inhibitory one, according to the synaptic weight sign. In each dendrite OR circuit, the synaptic output pulses are summed by OR gates, as is shown along the right side of Fig.1(a). Therefore, if these output pulses are synchronized, they are counted as one pulse and linear summation cannot take place. In our circuit, each neuron is driven by an individual clock oscillator. Therefore, they will tend to become desynchronized. The summation characteristic was analysed in [9], and it was shown to have a saturation characteristic that is similar to the positive part of a hyperbolic tangent function. 2.2.3 Cell body circuit A cell body circuit performs the integration given by Eq.( 4) as follows. As shown in Fig.1(d), integration was performed by a 12-bit up-down counter. Input pulses from an excitatory dendrite OR circuit are fed into the up-input of the counter and those from an inhibitory one are fed into the down-input after conflicts between 708 Y. Hirai excitatory and inhibitory pulses have been resolved by a sampling circuit. A 12-bit rate multiplier produces internal pulses whose frequency is 21, where 1 is proportional to the absolute value of the counter. The rate multiplier is driven by a main clock whose frequency is 4/max, Imax being the maximum output frequency. When the counter value is positive, an output pulse train whose frequency is either 1 or 2/, according to the scale factor is transmitted from a cell body circuit. The negative feedback term that appeared in the integrand of Eq.( 4) can be realized by feeding the internal pulses into the down-input of the counter when the counter value is positive and feeding them into the up-input when it is negative. The 6bit rate multiplier inserted in this feedback path changes the time constant of a neuron. Let f3i be the rate value of the rate multiplier, where 0 ~ f3i < 26 . The Eq.(4) becomes: yi(t) (5) 26 211 Therefore, the time constant changes to l!i£, where Pi was given by -,- seconds. /3 max It should be noted that, since the magnitUde of the total input was increased by a factor of ~, the strength of the input should be decreased by the inverse of that factor in order to maintain an appropriate output level. If it is not adjusted, we can increase the input strength. Therefore, the system has both input and output scaling functions. The time constant varies from about 416psec for f3 = 63 to 26.2msec for f3 = 1. When f3 = 0, the negative feedback path is interrupted and the circuit operates as a simple integrator, and every feedforward network can be run in this mode of operation. 3 THE 1,OOO-NEURON SYSTEM 3.1 VLSI CHIP A single type of VLSI chip was fabricated using a 0.7 pm CMOS gate array with 250,000 gates. A single chip contains 18 neurons and 51 synapses for each neuron. Therefore, each chip has a total of 918 synapses. About 85% of the gates in a gate array could be used, which was an extremely efficient value. A chip was mounted on a flat package with 256 pins. Among them, 216 pins were used for signals and the others were used for twenty pairs of V cc( =3.3V) and GND. 3.2 THE SYSTEM As illustrated in Fig.2(a), this system consists of 56 x 20 = 1,120 chips. 56 chips are used for both cell bodies and synapses, and the others are used to extend dendrite circuits and increase the number of synapses. In order to extend the dendrites, the dendrite signals in a chip can be directly transmitted to the dendrite extention terminals of another chip by bypassing the cell body circuits. There are 51 x 20 = 1,020 synapses per neuron. Among them, 1,008 synapses are used for fully hard-wired interconnections and the other 12 synapses are used to receive external signals. There are a total of 1,028,160 synapses in this system. It is controlled by a personal computer. The synaptic weights, the contents of the up-down counters A 1,OOO-Neuron System with One Million 7-bit Physical Interconnections Host computer High-speed bus 12 external inputs m Neural chips for neurons o Neural chips for synapses (a) (b) 709 Figure 2: Structure of the system. (a) System configuration. The down arrows emitted from the open squares designate signal lines that are extending dendrites. The others designate neuron outputs. (b) Exterior of the system. It is controlled by a personal computer. and the control registers can be read and written by the host system. It takes about 6 seconds to set all the network parameters from the host system. The exterior of this system is shown in Fig.2(b). Inside the cabinet, there are four shelves. In each shelf, fourteen circuit boards were mounted and on each board 20 chips were mounted. One chip was used for 18 neurons and the other chips were used to extend the dendrites. Each neuron is driven by an individual 20MHz clock oscillator. 4 SYSTEM PERFORMANCE In order to measure the performance of this system, one neuron was used as a signal generator. By setting all the synaptic weights and the internal feedback gain of a signal neuron to zero, and by setting the content of the up-down counter to a given value, it can produce an output with a constant frequency that is proportional to the counter value. The input strength of the other neurons can be adjusted by changing the counter value of a signal neuron or the synaptic weights from it. The step reponses of a neuron to different inputs are shown in Fig.3(a). As seen in the figure, the responses exactly followed Eq.(l) and the time constant Was about 400psec. Figure 3(b) shows responses with different time constants. The inputs were identical for all cases. Figure 3( c) shows the response of a temporal filter that was obtained by the difference between a fast and a slow neuron. By combining two low-pass filters that had different cutoff frequencies, a band-pass filter was created. A variety of spatiotem710 _------1792 r: rc-------:: '&. 768 ::J 500 512 256 OL--~--~--~--~--~--~ o 10000 20000 30000 40000 50000 time (x 100 nanoeeconds) (a) 2000 Bela_53 ---::=--------:j i 1500 S ! 1000 500 OIL....--~---'---~......::::.==--..-J o 10000 20000 30000 40000 50000 lime (x 100 nanoeeconds) (c) Y. Hirai s;.-~ /~.~------... ,I ...•. ~ 1500 " 32 .. / 16 8 S i.· ----1 1000 /.'/:/. /~"--_.---. -1 / _____ _ __ ---I' /' - --~ ~ ~~-:::-::~~-----'--o 10000 20000 30000 40000 50000 line (x 100 Il8IlO58conds) (b) 1~~-~~-~--~---~--. 1000 800 1&600 1 400 i ~P--"""IiiiiiiiOi;~ % ·200 -400 :L_'---_'---_~~~ o 20000 40000 60000 80000 time (x 100 nanoseconds) (d) Figure 3: Responses obtained by the system. (a) Step responses to different input levels. Parameters are the values that are set in the up-down counter of a signal neuron. (b) Step responses for different time constants. Parameters are the values of f3i in Eq.5. Inputs were identical in all cases. (c) Response of a temporal filter that was ol>taind by the difference between a fast and a slow neuron. (d) Response of a winner-take-all network among 1,007 neurons. The responses of a winner neuron and 24 of the 1,006 defeated neurons are shown. poral filters can be implemented in this way. Figure 3( d) shows the responses of a winner-take-all network among 1,007 neurons. The time courses of the responses of a winner neuron and 24 of the 1,006 defeated neurons are shown in the figure. The strength of all of the inhibitory synaptic weights between neurons was set to 2 x (- ::), where 2 is an output scale factor. The synaptic weights from a signal neuron to the 1,007 competing ones were identical and were ~;. Although the network parameters and the inputs to all competing neurons were identical, one of them won in 6 msec. Since the system operates asynchronously and the spatial summation of the synaptic output pulses is probabilistic, one of the competing neurons can win in a stochastic manner. In order to derive the processing speed in terms of connections per second, the same winner-take-all network was solved by the Euler method on a latest workstation. Since it took about 76.2 seconds and 2,736 iterations to converge, the processing speed of the workstation was about 36 million connections per second ( l007xI0<17x2736) S' h' . 10000' f h h k . ::::::! 76 .2 ! • mce t 1S system IS , times aster t an t e wor statlOn, A I,OOO-Neuron System with One MillioK7-bit Physical Interconnections 711 the processing speed amounts to 360 billion connections per second. Various kinds of neural networks including spatiotemporal filtering, feedforward and feedback neural networks can be run in this single system by loading appropriate network parameters from the host system. The second version of this system, which can be used via the Internet, will be completed by the end of March, 1998. Acknowledgements The author is grateful to Mr. Y. Kuwabara and Mr. T. Ochiai of Hitachi Microcomputer System Ltd. for their collaboration in developing this system and to Dr. M. Yasunaga and Mr. M. Takahashi for their help in testing it. The author is also grateful to Mr. H. Toda for his collaboration in measuring response data. This work was supported by "Proposal-Based Advanced Industrial Technology R&D Program" from NEDO in Japan. References [1] C. Mead: Analog VLSI and Neural Systems. Addison-Wesley Publishing Company, Massachusetts, 1989 [2] K.W.Przytula and V.K.Prasanna, Eds.: Parallel Digital Implementations of Neural Networks. Prentice Hall, New Jersey, 1993 [3] J.J. Hopfield: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. U.S.A., 81, pp.3088-3092, 1984 [4] P. Mueller, J. van der Spiegel, V. Agami, D. Blackman, P. Chance, C. Donham, R. Etienne, J. Kim. M. Massa and S. Samarasekera: Design and performance of a prototype analog neural computer. Proc. the 2nd International Conf. on Microelectronics for Neural Networks, pp.347-357, 1991 [5] G. Cauwenberghs: A learning analog neural network chip with continuous-time recurrent dynamics. In J. D. Cowan, G. Tesauro and J. Alspector, Eds., Advances in Neural Information Processing Systems 6, Morgan Kaufmann Publishers, San Mateo, CA, pp.858-865, 1994 [6] S. Churcher, D. J. Baxter, A. Hamilton, A. F. Murry, and H. M. Reekie: Generic analog neural computation - The EPSILON chip. In S. J. Hanson, J. D. Cowan and C. L. Giles, Eds., Advances in Neural Information Processing Systems 6, Morgan Kaufmann Publishers, San Mateo, CA, pp.773-780, 1993 [7] H. Eguchi, T. Furuta, H. Horiguchi, S. Oteki and T. Kitaguchi: Neural network LSI chip with on-chip learning. Proceedings of IJCNN'91 Seattle, Vol.I/453-456, 1991 [8] Y. Hirai, et al.: A digital neuro-chip with unlimited connectability for large scale neural networks. Proc. International Joint Conf. on Neural Networks'89 Washington D.C., Vo1.11/163-169, 1989 [9] Y.Hirai, VLSI Neural Network Systems (Gordon and Breach Science Publishers, Birkshire, 1992) [10] Y. Hirai and M. Yasunaga: A PDM digital neural network system with 1,000 neurons fully interconnected via 1,000,000 6-bit synapses. Proc. International Conference on Neural Information Processings'96, Vo1.ll/1251, 1996
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Regularisation in Sequential Learning Algorithms J oao FG de Freitas Cambridge University Engineering Department Cambridge CB2 IPZ England jfgf@eng.cam.ac.uk [Corresponding author] Mahesan Niranjan Cambridge University Engineering Department Cambridge CB2 IPZ England niranjan@eng.cam.ac.uk Andrew H Gee Cambridge University Engineering Department Cambridge CB2 IPZ England ahg@eng.cam.ac.uk Abstract In this paper, we discuss regularisation in online/sequential learning algorithms. In environments where data arrives sequentially, techniques such as cross-validation to achieve regularisation or model selection are not possible. Further, bootstrapping to determine a confidence level is not practical. To surmount these problems, a minimum variance estimation approach that makes use of the extended Kalman algorithm for training multi-layer perceptrons is employed. The novel contribution of this paper is to show the theoretical links between extended Kalman filtering, Sutton's variable learning rate algorithms and Mackay's Bayesian estimation framework. In doing so, we propose algorithms to overcome the need for heuristic choices of the initial conditions and noise covariance matrices in the Kalman approach. 1 INTRODUCTION Model estimation involves building mathematical representations of physical processes using measured data. This problem is often referred to as system identification, time-series modelling or machine learning. In many occasions, the system being modelled varies with time. Under this circumstance, the estimator needs to be Regularisation in Sequential Learning Algorithms 459 updated sequentially. Online or sequential learning has many applications in tracking and surveillance, control systems, fault detection, communications, econometric systems, operations research, navigation and other areas where data sequences are often non-stationary and difficult to obtain before the actual estimation process. To achieve acceptable generalisation, the complexity of the estimator needs to be judiciously controlled. Although there are various reliable schemes for controlling model complexity when training en bloc (batch processing), the same cannot be said about sequential learning. Conventional regularisation techniques cannot be applied simply because there is no data to cross-validate. Consequently, there is ample scope for the design of sequential methods of controlling model complexity. 2 NONLINEAR ESTIMATION A dynamical system may be described by the following discrete, stochastic state space representation: Wk +dk g(Wk, tk) + Vk (1) (2) where it has been assumed that the model parameters (Wk E R<J) constitute the states of the system, which in our case represent the weights of a multi-layer perceptron (MLP). g is a nonlinear vector function that may change at each estimation step k, tk denotes the time at the k-th estimation step and dk and Vk represent zero mean white noise with covariances given by Qk and Rk respectively. The noise terms are often called the process noise (dk) and the measurement noise (Vk). The system measurements are encoded in the output vector Yk E Rm. The estimation problem may be reformulated as having to compute an estimate Wk of the states Wk using the set of measurements Yk = {Yl, Y2, "', Yk}. The estimate Wk can be used to predict future values of the output y. We want Wk to be an unbiased, minimum variance and consistent estimate (Gelb 1984). A minimum variance (unbiased) estimate is one that has its variance less than or equal to that of any other unbiased estimator. Since the variance of the output Y depends directly on the variance of the parameter estimates (Astrom 1970), the minimum variance framework constitutes a regularisation scheme for sequential learning. The conditional probability density function of Wk given Yk (p(wkIYk)) constitutes the complete solution of the estimation problem (Bar-Shalom and Li 1993, Ho and Lee 1964, Jazwinski 1970). This is simply because p(wkIYk) embodies all the statistical information about Wk given the measurements Yk and the initial condition Woo This is essentially the Bayesian approach to estimation, where instead of describing a model by a single set of parameters, it is expressed in terms of the conditional probability p(wkIYk) (Jaynes 1986, Jazwinski 1970). The estimate Wk can be computed from p(wklY k) according to several criteria, namely MAP estimation (peak of the posterior), minimum variance estimation (centroid of the posterior) and minimax estimation (median of the posterior). The Bayesian solution to the optimal estimation problem is (Ho and Lee 1964): P(Wk+1,Yk+I IYk) p(Yk+1IYk) J p(Yk+1IYk, Wk+l )p(wk+1lwk)P(Wk IY k)dwk J J p(Yk+lIYk, Wk+1 )p(Wk+llwk)p(Wk IY k)dwk+l dWk (3) where the integrals run over the parameter space. This functional integral difference equation governing the evolution of the posterior density function is not suitable 460 1. R G. d. Freitas, M. Niranjan andA. H. Gee for practical implementation (Bar-Shalom and Li 1993, Jazwinski 1970). It involves propagating a quantity (the posterior density function) that cannot be described by a finite number of parameters. The situation in the linear case is vastly simpler. There the mean and covariance are sufficient statistics for describing the Gaussian posterior density function. In view of the above statements, it would be desirable to have a framework for nonlinear estimation similar to the one for linear-Gaussian estimation. The extended Kalman filter (EKF) constitutes an attempt in this direction (Bar-Shalom and Li 1993, Gelb 1984). The EKF is a minimum variance estimator based on a Taylor series expansion of the nonlinear function g(w) around the previous estimate. The EKF equations for a linear expansion are given by: (Pk + Qk)Gk+l [Rk + Gk+1 (Pk + Qk)Gk+1]-1 Wk + Kk+l(Yk+l - Gk+l Wk) Pk + Qk - Kk+lGk+l (Pk + Qk) (4) (5) (6) where Pk denotes the covariance of the weights. In the general multiple input, multiple output (MIMO) case, g E ~m is a vector function and G represents the Jacobian of the network outputs with respect to the weights. The EKF provides a minimum variance Gaussian approximation to the posterior probability density function. In many cases, p(wkIYk) is a multi-modal (several peaks) function. In this scenario, it is possible to use a committee of Kalman filters, where each individual filter approximates a particular mode, to produce a more accurate approximation (Bar-Shalom and Li 1993, Kadirkamanathan and Kadirkamanathan 1995). The parameter covariances of the individual estimators may be used to determine the contribution of each estimator to the committee. In addition, the parameter covariances serve the purpose of placing confidence intervals on the output prediction. 3 TRAINING MLPs WITH THE EKF One of the earliest implementations of EKF trained MLPs is due to Singhal and Wu (Singhal and Wu 1988). In their method, the network weights are grouped into a single vector w that is updated in accordance with the EKF equations. The entries of the Jacobian matrix are calculated by back-propagating the m output values through the network. The algorithm proposed by Singhal and Wu requires a considerable computational effort. The complexity is of the order mq2 multiplications per estimation step. Shah, Palmieri and Datum (1992) and Puskorius and Feldkamp (1991) have proposed strategies for decoupling the global EKF estimation algorithm into local EKF estimation sub-problems, thereby reducing the computational time. The EKF is an improvement over conventional MLP estimation techniques, such as back-propagation, in that it makes use of second order statistics (covariances). These statistics are essential for placing error bars on the predictions and for combining separate networks into committees of networks. Further, it has been proven elsewhere that the back-propagation algorithm is simply a degenerate of the EKF algorithm (Ruck, Rogers, Kabrisky, Maybeck and Oxley 1992). However, the EKF algorithm for training MLPs suffers from serious difficulties, namely choosing the initial conditions (wo, Po) and the noise covariance matrices Rand Q. In this work, we propose the use of maximum likelihood techniques, such as back-propagation computed over a small set of initial data, to initialise the Regularisation in Sequential Learning Algorithm~ 461 EKF-MLP estimator. The following two subsections· describe ways of overcoming the difficulty of choosing R and Q. 3.1 ELIMINATING Q BY UPDATING P WITH BACK-PROPAGATION . To circumvent the problem of choosing the process noise covariance Q, while at the same time increasing computational efficiency, it is possible to extend an algorithm proposed by Sutton (Sutton 1992) to the nonlinear case. In doing so, the weights covariance is approximated by a diagonal matrix with entries given by pqq = exp(,8q), where,8 is updated by error back-propagation (de Freitas, Niranjan and Gee 1997). The Kalman gain K k and the weights estimate Wk are updated using a variation of the Kalman equations, where the Kalman gain and weights update equations are independent of Q (Gelb 1984), while the weights covariance P is updated by backpropagation. This algorithm lessens the burden of choosing the matrix Q by only having to choose the learning rate scalar 1]. The performance of the EKF algorithm with P updated by back-propagation will be analysed in Section 4. 3.2 KALMAN FILTERING AND BAYESIAN TECHNIQUES A further improvement on the EKF algorithm for training MLPs would be to update Rand Q automatically each estimation step. This can be done by borrowing some ideas from the Bayesian estimation field. In particular, we shall attempt to link Mackay's work (Mackay 1992, Mackay 1994) on Bayesian estimation for neural networks with the EKF estimation framework. This theoretical link should serve to enhance both methods. Mackay expresses the prior, likelihood and posterior density functions in terms of the following Gaussian approximations: 1 (a 2 p(w) = (27r)q/2a -q/2 exp - "2llwll ) (7) 1 ,8~ A 2 p(Yklw) = (27r)n/2,8-n/2 exp ( - "2 L.,..(Yk - fn,q(w, <Pk)) ) k=l (8) 1 1 T p(wIYk ) = (27r)q/2IAI-1/ 2 exp ( - 2(w - WMP) A(w - WMP)) (9) where in,q(w, <Pk) represents the estimator and the hyper-parameters a and ,8 control the variance of the prior distribution of weights and the variance of the measurement noise. a also plays the role of the regularisation coefficient. The posterior is obtained by approximating it with a Gaussian function, whose mean wMP is given by a minimum of the following regularised error function: a ,8~ A 2 S(w) = "2llwl12 + "2 L.,..(Yk - fn,q(w, <Pk)) k==l (10) The posterior covariance A is the Hessian of the above error function. In Mackay's estimation framework, also known as the evidence framework, the parameters ware obtained by minimising equation (10), while the hyper-parameters a and ,8 are obtained by maximising the evidence p(Yk la,,8) after approximating the posterior density function by a Gaussian function. In doing so, the following recursive formulas for a and ,8 are obtained: '1 n-'1 ak+1 = L:q 2 and ,8k+1 = n A 2 i=l Wi L::k=l (Yk - in,q(Wk, <Pk)) 462 J. F. G. d. Freitas, M. Niranjan andA. H. Gee The quantity 'Y represents the effective number of parameters 'Y = 2J~=1 >.:~a' where the Ai correspond to the eigenvalues of the Hessian of the error function without the regularisation term. Instead of adopting Mackay's evidence framework, it is possible to maximise the posterior density function by performing integrations over the hyper-parameters analytically (Buntine and Weigend 1991, Mackay 1994). The latter approach is known as the MAP framework for 0 and {3. The hyper-parameters computed by the MAP framework differ from the ones computed by the evidence framework in that the former makes use of the total number of parameters and not only the effective number of parameters. That is, 0 and {3 are updated according to: q n Ok+l = ",q 2 and {3k+1 = n A 2 L.,..i=l Wi l:k=l (Yk - /n,q(Wk , <Pk)) By comparing the equations for the prior, likelihood and posterior density functions in the Kalman filtering framework (Ho and Lee 1964) with equations (7), (8) and (9) we can establish the following relations: P=A- 1 , Q=o-IIq_A- 1 and R={3-1Im where Iq and 1m represent identity matrices of sizes q and m respectively. Therefore, it is possible to update Q and R sequentially by expressing them in terms of the sequential updates of 0 and {3. -60~--:':: 10---::2'=""0 ::'::30--=4'=""0 -::'::5o----:6O':--7=o----:8o~-90 :'::---,-' 100 I -~-8~10~-82~0-~ ~-M~0--~ ~-860~-8~70--880~-8~90-~900 I Figure 1: Prediction using the conventional EKF algorithm for a network with 20 hidden neurons. Actual output [ . . . J and estimated output [-J. 4 RESULTS To compare the performance of the conventional EKF algorithm, the EKF algorithm with P updated by back-propagation, and the EKF algorithm with Rand Q updated sequentially according to the Bayesian MAP framework, noisy data was generated from the following nonlinear, non-stationary, multivariate process: (t) - { Xl (t) + X2 (t) + v(t) 1 ::; t ::; 200 Y 4sin(xdt)) + X2(t) sin(0.03(t - 200)) + v(t} 200 < t ::; 1000 Regularisation in Sequential Learning Algorithms 463 where the inputs Xi are uniformly distributed random sequences with variance equal . to 1 and v(t) corresponds to uniformly distributed noise with variance equal to 0.1. Figure 1 shows the prediction obtained using the conventional EKF algorithm. To 3.S ~ 3 .§ r s i 2 ~ 1.5 a: 05 \ " " , , , , I , I , I , I , I , I 10 12 14 16 18 20 Tnal Figure 2: Output error for the conventional EKF algorithm [ . .. ], the EKF algorithm with P updated by back-propagation [- . -], the EKF algorithm with Rand Q updated sequentially according to the Bayesian MAP framework [-], and the EKF algorithm with the Bayesian evidence framework [- - -]. compare the four estimation frameworks, an MLP with 20 neurons in the hidden layer was selected. The initial conditions were obtained by using back-propagation on the first 100 samples and assigning to P a diagonal matrix with diagonal elements equal to 10. The matrices R and Q in the conventional EKF algorithm were chosen, by trial and error, to be identity matrices. In the EKF algorithm with P updated by back-propagation, R was chosen to be equal to the identity matrix, while the learning rate was set to 0.01. Finally, in the EKF algorithm with Rand Q updated sequentially, the initial Rand Q matrices were chosen to be identity matrices. The prediction errors obtained for each method with random input data are shown in Figure 2. It is difficult to make a fair comparison between the four nonlinear estimation methods because their parameters were optimised independently. However, the results suggest that the prediction obtained with the conventional EKF training outperforms the predictions of the other methods. This may be attributed to the facts that, firstly, in this simple problem it is possible to guess the optimal values for Rand Q and, secondly, the algorithms to update the noise covariances may affect the regularisation performance of the EKF algorithm. This issue, and possible solutions, is explored in depth by the authors in (de Freitas et al. 1997). 5 Conclusions In this paper, we point out the links between Kalman filtering, gradient descent algorithms with variable learning rates and Bayesian estimation. This results in two algorithms for eliminating the problem of choosing the initial conditions and the noise covariance matrices in the training of MLPs with the EKF. These algorithms are illustrated on a toy problem here, but more extensive experiments have been reported in (de Freitas et al. 1997). Improved estimates may be readily obtained by combining the estimators into com464 J. R G. d. Freitas, M. Niranjan and A. H. Gee mit tees or extending the training methods to recurrent networks. Finally, the computational time may be reduced by decoupling the network weights. Acknowledgements Joao FG de Freitas is financially supported by two University of the Witwatersrand Merit Scholarships, a Foundation for Research Development Scholarship (South Africa) and a Trinity College External Studentship (Cambridge). References Astrom, K. J. (1970). Introduction to Stochastic Control Theory, Academic Press. Bar-Shalom, Y. and Li, X. R. (1993). Estimation and Tracking: Principles, Techniques and Software, Artech House, Boston. Buntine, W. L. and Weigend, A. S. (1991). Bayesian back-propagation, Complex Systems 5: 603-643. de Freitas, J., Niranjan, M. and Gee, A. (1997). Hierarchichal BayesianKalman models for regularisation and ARD in sequential learning, Technical Report CUED/F-INFENG/TR 307, Cambridge University, http:j jsvrwww.eng.cam.ac.ukj-jfgf. Gelb, A. (ed.) (1984). Applied Optimal Estimation, MIT Press. Ho, Y. C. and Lee, R. C. K. (1964). A Bayesian approach to problems in stochastic estimation and control, IEEE Transactions on Automatic Control AC-9: 333339. Jaynes, E. T. (1986). Bayesian methods: General background, in J. H. Justice (ed.), Maximum Entropy and Bayesian Methods in Applied Statistics, Cambridge University Press, pp. 1-25. Jazwinski, A. H. (1970). Stochastic Processes and Filtering Theory, Academic Press. Kadirkamanathan, V. and Kadirkamanathan, M. (1995). Recursive estimation of dynamic modular RBF networks, in D. S. Touretzky, M. C. Mozer and M. E. Hasselmo (eds), Advances in Neural Information Processing Systems 8, pp. 239-245. Mackay, D. J. C. (1992). Bayesian interpolation, Neural Computation 4(3): 415-447. Mackay, D. J. C. (1994). Hyperparameters: Optimise or integrate out?, in G. Heidbreder (ed.), Maximum Entropy and Bayesian Methods. Puskorius, G. V. and Feldkamp, 1. A. (1991). Decoupled extended Kalman filter training of feedforward layered networks, International Joint Conference on Neural Networks, Seattle, pp. 307-312. Ruck, D. W., Rogers, S. K., Kabrisky, M., Maybeck, P. S. and Oxley, M. E. (1992). Comparative analysis of backpropagation and the extended Kalman filter for training multilayer perceptrons, IEEE Transactions on Pattern Analysis and Machine Intelligence 14(6): 686-690. Shah, S., Palmieri, F. and Datum, M. (1992). Optimal filtering algorithms for fast learning in feedforward neural networks, Neural Networks 5: 779-787. Singhal, S. and Wu, 1. (1988). Training multilayer perceptrons with the extended Kalman algorithm, in D. S. Touretzky (ed.), Advances in Neural Information Processing Systems, Vol. 1, San Mateo, CA, pp. 133-140. Sutton, R. S. (1992). Gain adaptation beats least squares?, Proceedings of the Seventh Yale Workshop on Adaptive Learning Systems, pp. 161-166.
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An Improved Policy Iteratioll Algorithm for Partially Observable MDPs Eric A. Hansen Computer Science Department University of Massachusetts Amherst, MA 01003 hansen@cs.umass.edu Abstract A new policy iteration algorithm for partially observable Markov decision processes is presented that is simpler and more efficient than an earlier policy iteration algorithm of Sondik (1971,1978). The key simplification is representation of a policy as a finite-state controller. This representation makes policy evaluation straightforward. The paper's contribution is to show that the dynamic-programming update used in the policy improvement step can be interpreted as the transformation of a finite-state controller into an improved finite-state controller. The new algorithm consistently outperforms value iteration as an approach to solving infinite-horizon problems. 1 Introduction A partially observable Markov decision process (POMDP) is a generalization of the standard completely observable Markov decision process that allows imperfect information about the state of the system. First studied as a model of decision-making in operations research, it has recently been used as a framework for decision-theoretic planning and reinforcement learning with hidden state (Monahan, 1982; Cassandra, Kaelbling, & Littman, 1994; Jaakkola, Singh, & Jordan, 1995). Value iteration and policy iteration algorithms for POMDPs were first developed by Sondik and rely on a piecewise linear and convex representation of the value function (Sondik, 1971; Smallwood & Sondik,1973; Sondik, 1978). Sondik's policy iteration algorithm has proved to be impractical, however, because its policy evaluation step is extremely complicated and difficult to implement. As a result, almost all subsequent work on dynamic programming for POMDPs has used value iteration. In this paper, we describe an improved policy iteration algorithm for POMDPs that avoids the difficulties of Sondik's algorithm. We show that these difficulties hinge on the choice of a policy representation and can be avoided by representing a policy as a finite-state 1016 E. A. Hansen controller. This representation makes the policy evaluation step easy to implement and efficient. We show that the policy improvement step can be interpreted in a natural way as the transformation of a finite-state controller into an improved finite-state controller. Although it is not always possible to represent an optimal policy for an infinite-horizon POMDP as a finite-state controller, it is always possible to do so when the optimal value function is piecewise linear and convex. Therefore representation of a poiicy as a finite-state controller is no more limiting than representation of the value function as piecewise linear and convex. In fact, it is the close relationship between representation of a policy as a finite-state controller and representation of a value function as piecewise linear and convex that the new algorithm successfully exploits. The paper is organized as follows. Section 2 briefly reviews the POMDP model and Sondik's policy iteration algorithm. Section 3 describes an improved policy iteration algorithm. Section 4 illustrates the algorithm with a simple example and reports a comparison of its performance to value iteration. The paper concludes with a discussion of the significance of this work. 2 Background Consider a discrete-time POMDP with a finite set of states 5, a finite set of actions A, and a finite set of observations e. Each time period, the system is in some state i E 5, an agent chooses an action a E A for which it receives a reward with expected value ri, the system makes a transition to state j E 5 with probability pij' and the agent observes () E e with probability tje. We assume the performance objective is to maximize expected total discounted reward over an infinite horizon. Although the state of the system cannot be directly observed, the probability that it is in a given state can be calculated. Let 7r denote a vector of state probabilities, called an information state, where 7ri denotes the probability that the system is in state i. If action a is taken in information state 7r and () is observed, the successor information state is determined by revising each state probability using Bayes' theorem: trj = LiEs 7riPijQje/ Li,jES 7riPijQje' Geometrically, each information state 7r is a point in the (151 - I)-dimensional unit simplex, denoted II. It is well-known that an information state 7r is a sufficient statistic that summarizes all information about the history of a POMDP necessary for optimal action selection. Therefore a POMDP can be recast as a completely observable MDP with a continuous state space II and it can be theoretically solved using dynamic programming. The key to practical implementation of a dynamic-programming algorithm is a piecewise-linear and convex representation of the value function. Smallwood and Sondik (1973) show that the dynamic-programming update for POMDPs preserves the piecewise linearity and convexity of the value function. They also show that an optimal value function fot a finite-horizon POMDP is always piecewise linear and convex. For infinite-horizon POMDPs, Sondik (1978) shows that an optimal value function is sometimes piecewise linear and convex and can be aproximated arbitrarily closely by a piecewise linear and convex function otherwise. A piecewise linear and convex value function V can be represented by a finite set of lSI-dimensional vectors, r = {aO,ai , •.. }, such that V(7r) = maxkLi s7riaf. A dynamic-programming update transforms a value function V representedEfiy a set r of a-vectors into an improved value function V' represented by a set r' of a-vectors. Each possible a-vector in r' corresponds to choice of an action, and for each possible observation, choice of a successor vector in r. Given the combinatorial number of choices that can be made, the maximum n4mber of vectors in r' is IAllfll91. However most of these potential vectors are not needed to define the updated value function and can be pruned. Thus the dynamic-programming update problem is to find a An Improved Policy Iteration Algorithmfor Partially Observable MDPs J017 minimal set of vectors r' that represents V', given a set of vectors r that represents V . Several algorithms for performing this dynamic-programming update have been developed but describing them is beyond the scope of this paper. Any algorithm for performing the dynamic-programming update can be used in the policy improvement step of policy iteration. The algorithm that is presently the fastest is described by (Cassandra, Littman, & Zhang, 1997). For value iteration, it is sufficient to have a representation of the value function because a policy is defined implicitly by the value function, as follows, 8(11") = a(arg mF L 1I"iof), (1) iES where a(k) denotes the action associated with vector ok. But for policy iteration, a policy must be represented independently of the value function because the policy evaluation step computes the value function of a given policy. Sondik's choice of a policy representation is influenced by Blackwell's proof that for a continuous-space infinite-horizon MDP, there is a stationary, deterministic Markov policy that is optimal (Blackwell, 1965). Based on this result, Sondik restricts policy space to stationary and deterministic Markov policies that map the continuum of information space II into action space A. Because it is important for a policy to have a finite representation, Sondik defines an admissible policy as a mapping from a finite number of polyhedral regions of II to A. Each region is represented by a set of linear inequalities, where each linear inequality corresponds to a boundary of the region. This is Sondik's canonical representation of a policy, but his policy iteration algorithm makes use of two other representations. In the policy evaluation step, he converts a policy from this representation to an equivalent, or approximately equivalent, finitestate controller. Although no method is known for computing the value function of a policy represented as a mapping from II to A, the value function of a finite-state controller can be computed in a straightforward way. In the policy improvement step, Sondik converts a policy represented implicitly by the updated value function and equation (1) back to his canonical representation. The complexity of translating between these different policy representations - especially in the policy evaluation step - makes Sondik's policy iteration algorithm difficult to implement and explains why it is not used in practice. 3 Algorithm We now show that policy iteration for POMDPs can be simplified - both conceptually and computationally - by using a single representation of a policy as a finite-state controller. 3.1 Policy evaluation As Sondik recognized, policy evaluation is straightforward when a policy is represented as a finite-state controller. An o-vector representation of the value function of a finitestate controller is computed by solving the system of linear equations, k _ a(k) + (3'"' a(k) a(k) s(k ,8) 0i ri L.JPij qj8 OJ , (2) j ,8 where k is an index of a state of the finite-state controller, a(k) is the action associated with machine state k, and s(k,O) is the index of the successor machine state if 0 is observed. This value function is convex as well as piecewise linear because the expected value of an information state is determined by assuming the controller is started in the machine state that optimizes it. 1018 E. A. Hansen 1. Specify an initial finite-state controller, <5, and select f. for detecting convergence to an f.-optimal policy. 2. Policy evaluation: Calculate a set r of a-vectors that represents the value function for <5 by solving the system of equations given by equation 2. 3. Policy improvement: Perform a dynamic-programming update and use the new set of vectors r' to transform <5 into a new finite-state controller, <5', as follows: (a) For each vector a in r': l. If the action and successor links associated with a duplicate those of a machine state of <5, then keep that machine state unchanged in 8'. ii. Else if a pointwise dominates a vector associated with a machine state of <5, change the action and successor links of that machine state to those used to create a. (If it pointwise dominates the vectors of more than one machine state, they can be combined into a single machine state.) iii. Otherwise add a machine state to <5' that has the same action and successor links used to create a. (b) Prune any machine state for which there is no corresponding vector in r', as long as it is not reachable from a machine state to which a vector in r' does correspond. 4. Termination test. If the Bellman residual is less than or equal to f.(1 - /3)//3, exit with f.-optimal policy. Otherwise set <5 to <5' and go to step 2. Figure 1: Policy iteration algorithm. 3.2 Policy improvement The policy improvement step uses the dynamic-programming update to transform a value function V represented by a set r of a-vectors into an improved value function V' represented by a set r' of a-vectors. We now show that the dynamic-programming update can also be interpreted as the transformation of a finite-state controller 8 into an improved finite-state controller <5'. The transformation is made based on a simple comparison of r' and r. First note that some of the a-vectors in r' are duplicates of a-vectors in r, that is, their action and successor links match (and their vector values are pointwise equal). Any machine state of <5 for which there is a duplicate vector in r' is left unchanged. The vectors in r' that are not duplicates of vectors in r indicate how to change the finite-state controller. If a non-duplicate vector in r' pointwise dominates a vector in r, the machine state that corresponds to the pointwise dominated vector in r is changed so that its action and successor links match those of the dominating vector in r'. If a non-duplicate vector in r' does not pointwise dominate a vector in r, a machine state is added to the finite-state controller with the same action and successor links used to generate the vector. There may be some machine states for which there is no corresponding vector in r' and they can be pruned, but only if they are not reachable from a machine state that corresponds to a vector in r'. This last point is important because it preserves the integrity of the finite-state controller. A policy iteration algorithm that uses these simple transformations to change a finitestate controller in the policy improvement step is summarized in Figure 1. An algorithm that performs this transformation is easy to implement and runs very efficiently because it simply compares the a-vectors in r' to the a-vectors in r and modifies the finite-state controller accordingly. The policy evaluation step is invoked to compute the value function of the transformed finite-state controller. (This is only necessary An Improved Policy Iteration Algorithmfor Partially Observable MDPs 1019 if a machine state has been changed, not if machine states have simply been added.) It is easy to show that the value function of the transformed finite-state controller /j' dominates the value function of the original finite-state controller, /j, and we omit the proof which appears in (Hansen, 1998). Theorem 1 If a finite-state controller is not optimal, policy improvement transforms it into a finite-state controller with a value function that is as good or better for every information state and better for some information state. 3.3 Convergence If a finite-state controller cannot be improved in the policy improvement step (Le., all the vectors in r' are duplicates of vectors in r), it must be optimal because the value function satisfies the optimality equation. However policy iteration does not necessarily converge to an optimal finite-state controller after a finite number of iterations because there is not necessarily an optimal finite-state controller. Therefore we use the same stopping condition used by Sondik to detect t-optimality: a finite-state controller is t-optimal when the Bellman residual is less than or equal to t(l- {3) / {3, where {3 denotes the discount factor. Representation of a policy as a finite-state controller makes the following proof straightforward (Hansen, 1998). Theorem 2 Policy iteration converges to an t-optimal finite-state controller after a finite number of iterations. 4 Example and performance We illustrate the algorithm using the same example used by Sondik: a simple twostate, two-action, two-observation POMDP that models the problem of finding an optimal marketing strategy given imperfect information about consumer preferences (Sondik,1971,1978). The two states of the problem represent consumer preference or lack of preference for the manufacturers brand; let B denote brand preference and ....,B denote lack of brand preference. Although consumer preferences cannot be observed, they can be infered based on observed purchasing behavior; let P denote purchase of the product and let ....,p denote no purchase. There are two marketing alternatives or actions; the company can market a luxury version of the product (L) or a standard version (S). The luxury version is more expensive to market but can bring greater profit. Marketing the luxury version also increases brand preference. However consumers are more likely to purchase the less expensive, standard product. The transition probabilities, observation probabilities, and reward function for this example are shown in Figure 2. The discount factor is 0.9. Both Sondik's policy iteration algorithm and the new policy iteration algorithm converge in three iterations from a starting policy that is equivalent to the finite-state AClions Transilion Observalion Expecled probabililies probabililies reward B -B P -p Markel B/O.8/0.2\ B 10.81 0.2\ B§j luxury -B 0.5 0.5 -B 0.60.4 -B ·4 producl (L) B -B P -p Markel B~ B~ Bbj slandard producl (S) -B 0.4 o. -B O. 0. -B ·3 Figure 2: Parameters for marketing example of Sondik (1971,1978) . 1020 (.) (b) (e) ~; .. " -~ '''' .. " a = L \ ~ 9,96 : '- 18.86 <,8=-P '~~9~~_:~~~;~<~._ :' .1 = S " \\ : 14.82! \ \ 18.20 / \ ', __ __ - '~~ P \., ''', ,.;,\ /""---... ,9=-p,y: " a.= S \ .. ,'" : 14.86 t ____ .... \_: 8.1~/ 8=P (d) E A. Hansen (e) Figure 3: (a) shows the initial finite-state controller, (b) uses dashed circles to show the vectors in r' generated in the first policy improvement step and (c) shows the transformed finite-state controller, (d) uses dashed circles to show the vectors in r' generated in the second policy improvement step and (e) shows the transformed finite-state controller after policy evaluation. The optimality of this finite-state controller is detected on the third iteration, which is not shown. Arcs are labeled with one of two possible observations and machine states are labeled with one of two possible actions and a 2-dimensional vector that contains a value for each of the two possible system states. controller shown in Figure 3a. Figure 3 shows how the initial finite-state controller is transformed into an optimal finite-state controller by the new algorithm. In the first iteration, the updated set of vectors r' (indicated by dashed circles in Figure 3b) includes two duplicate vectors and one non-duplicate that results in an added machine state. Figure 3c shows the improved finite-state controller after the first iteration. In the second iteration, each of the three vectors in the updated set of vectors r' (indicated by dashed circles in Figure 3d) pointwise dominates a vector that corresponds to a current machine state. Thus each of these machine states is changed. Figure 4e shows the improved finite-state controller after the second iteration. The optimality of this finite-state controller is detected in the third iteration. This is the only example for which Sondik reports using policy iteration to find an optimal policy. For POMDPs with more than two states, Sondik's algorithm is especially difficult to implement. Sondik reports that his algorithm finds a suboptimal policy for an example described in (Smallwood & Sondik, 1973). No further computational experience with his algorithm has been reported. The new policy iteration algorithm described in this paper easily finds an optimal finite-state controller for the example described in (Smallwood & Sondik, 1973) and has been used to solve many other POMDPs. In fact, it consistently outperforms value iteration. We compared its performance to the performance of value iteration on a suite of ten POMDPs that represent a range of problem sizes for which exact dynamicprogramming updates are currently feasible. (Presently, exact dynamic-prorgramming updates are not feasible for POMDPs with more than about ten or fifteen states, actions, or observations.) Starting from the same point, we measured how soon each algorithm converged to f-optimality for f values of 10.0, 1.0, 0.1, and 0.01. Policy iteration was consistently faster than value iteration by a factor that ranged from a low of about 10 times faster to a high of over 120 times faster. On average, its rate of convergence was between 40 and 50 times faster than value iteration for this set of examples. The finite-state controllers it found had as many as several hundred machine states, although optimal finite-state controllers were sometimes found with just a few machine states. An Improved Policy Iteration Algorithm for Partially Observable MDPs 1021 5 Discussion We have demonstrated that the dynamic-programming update for POMDPs can be interpreted as the improvement of a finite-state controller. This interpretation can be applied to both value iteration and policy iteration. It provides no computational speedup for value iteration, but for policy iteration it results in substantial speedup by making policy evaluation straightforward and easy to implement. This representation also has the advantage that it makes a policy easier to understand and execute than representation as a mapping from regions of information space to actions. In particular, a policy can be executed without maintaining an information state at run-time. It is well-known that policy iteration converges to f-optimality (or optimality) in fewer iterations than value iteration. For completely observable MDPs, this is not a clear advantage because the policy evaluation step is more computationally expensive than the dynamic-programming update. But for POMDPs, policy evaluation has loworder polynomial complexity compared to the worst-case exponential complexity of the dynamic-programming update (Littman et al., 1995). Therefore, policy iteration appears to have a clearer advantage over value iteration for POMDPs. Preliminary testing bears this out and suggests that policy iteration significantly outperforms value iteration as an approach to solving infinite-horizon POMDPs. Acknowledgements Thanks to Shlomo Zilberstein and especially Michael Littman for helpful discussions. Support for this work was provided in part by the National Science Foundation under grants IRI-9409827 and IRI-9624992. References Blackwell, D. {1965} Discounted dynamic programming. Ann. Math. Stat. 36:226235. Cassandra, A.; Kaelbling, L.P.; Littman, M.L. {1994} Acting optimally in partially observable stochastic domains. In Proc. 13th National Conf. on AI, 1023-1028. Cassandra, A.; Littman, M.L.; & Zhang, N.L. (1997) Incremental pruning: A simple, fast, exact algorithm for partially observable Markov decision processes. In Proc. 13th A nnual Con/. on Uncertainty in AI. Hansen, E.A. (1998). Finite-Memory Control of Partially Observable Systems. PhD thesis, Department of Computer Science, University of Massachusetts at Amherst. Jaakkola, T.; Singh, S.P.; & Jordan, M.I. (1995) Reinforcement learning algorithm for partially observable Markov decision problems. In NIPS-7. Littman, M.L.; Cassandra, A.R.; & Kaebling, L.P. (1995) Efficient dynamicprogramming updates in partially observable Markov decision processes. Computer Science Technical Report CS-95-19, Brown University. Monahan, G.E. (1982) A survey of partially observable Markov decision processes: Theory, models, and algorithms. Management Science 28:1-16. Smallwood, R.D. & Sondik, E.J. (1973) The optimal control of partially observable Markov processes over a finite horizon. Operations Research 21:1071-1088. Sondik, E.J. (1971) The Optimal Control of Partially Observable Markov Processes. PhD thesis, Department of Electrical Engineering, Stanford University. Sondik, E.J. (1978) The optimal control of partially observable Markov processes over the infinite horizon: Discounted costs. Operations Research 26:282-304.
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The Asymptotic Convergence-Rate of Q-Iearning es. Szepesvari* Research Group on Artificial Intelligence, "Jozsef Attila" University, Szeged, Aradi vrt. tere 1, Hungary, H-6720 szepes@math.u-szeged.hu Abstract In this paper we show that for discounted MDPs with discount factor, > 1/2 the asymptotic rate of convergence of Q-Iearning is O(1/tR(1-1') if R(1 - ,) < 1/2 and O( Jlog log tit) otherwise provided that the state-action pairs are sampled from a fixed probability distribution. Here R = Pmin/Pmax is the ratio of the minimum and maximum state-action occupation frequencies. The results extend to convergent on-line learning provided that Pmin > 0, where Pmin and Pmax now become the minimum and maximum state-action occupation frequencies corresponding to the stationary distribution. 1 INTRODUCTION Q-Iearning is a popular reinforcement learning (RL) algorithm whose convergence is well demonstrated in the literature (Jaakkola et al., 1994; Tsitsiklis, 1994; Littman and Szepesvari, 1996; Szepesvari and Littman, 1996). Our aim in this paper is to provide an upper bound for the convergence rate of (lookup-table based) Q-Iearning algorithms. Although, this upper bound is not strict, computer experiments (to be presented elsewhere) and the form of the lemma underlying the proof indicate that the obtained upper bound can be made strict by a slightly more complicated definition for R. Our results extend to learning on aggregated states (see (Singh et al., 1995» and other related algorithms which admit a certain form of asynchronous stochastic approximation (see (Szepesv<iri and Littman, 1996». Present address: Associative Computing, Inc., Budapest, Konkoly Thege M. u. 29-33, HUNGARY-1121 The Asymptotic Convergence-Rate of Q-leaming 1065 2 Q-LEARNING Watkins introduced the following algorithm to estimate the value of state-action pairs in discounted Markovian Decision Processes (MDPs) (Watkins, 1990): Here x E X and a E A are states and actions, respectively, X and A are finite. It is assumed that some random sampling mechanism (e.g. simulation or interaction with a real Markovian environment) generates random samples of form (Xt, at, Yt, rt), where the probability of Yt given (xt,at) is fixed and is denoted by P(xt,at,Yt), E[rt I Xt, at] = R(x, a) is the immediate average reward which is received when executing action a from state x, Yt and rt are assumed to be independent given the history of the learning-process, and also it is assumed that Var[rt I Xt, at] < C for some C > O. The values 0 ~ at(x,a) ~ 1 are called the learning rate associated with the state-action pair (x, a) at time t. This value is assumed to be zero if (x,a) =J (xt,at), i.e. only the value of the actual state and action is reestimated in each step. If 00 L at(x, a) = 00 (2) t=l and 00 L a;(x, a) < 00 (3) t=l then Q-Iearning is guaranteed to converge to the only fixed point Q* of the operator T : lRX x A ~ lRXxA defined by (TQ)(x,a) = R(x, a) +, L P(x,a,y)mFQ(y,b) yEX (convergence proofs can be found in (Jaakkola et al., 1994; TSitsiklis, 1994; Littman and Szepesv.hi, 1996; Szepesvari and Littman, 1996)). Once Q* is identified the learning agent can act optimally in the underlying MDP simply by choosing the action which maximizes Q* (x, a) when the agent is in state x (Ross, 1970; Puterman, 1994). 3 THE MAIN RESULT Condition (2) on the learning rate at(x, a) requires only that every state-action pair is visited infinitely often, which is a rather mild condition. In this article we take the stronger assumption that {(Xt, at) h is a sequence of independent random variables with common underlying probability distribution. Although this assumption is not essential it simplifies the presentation of the proofs greatly. A relaxation will be discussed later. We further assume that the learning rates take the special form ( ) { ~ l , if (x,a) = (xt,a); at x, a = Ol,x,a, 0, otherwise, where St (x, a) is the number of times the state-action pair was visited by the process (xs, as) before time step t plus one, i.e. St(x, a) = 1 + #{ (xs, as) = (x, a), 1 ~ s ~ 1066 C. Szepesvari t }. This assumption could be relaxed too as it will be discussed later. For technical reasons we further assume that the absolute value of the random reinforcement signals Tt admit a common upper bound. Our main result is the following: THEOREM 3.1 Under the above conditions the following relations hold asymptotically and with probability one: IQt(x, a) - Q*(x, a)1 ~ tR(~-'Y) (4) and * JIOg log t IQt(x,a) - Q (x,a)1 ~ B t' (5) for some suitable constant B > O. Here R = Pmin/Pmax, where Pmin = min(z,a) p(x, a) and Pmax = max(z,a) p(x, a), where p(x, a) is the sampling probability of (x, a). Note that if'Y 2: 1 - Pmax/2pmin then (4) is the slower, while if'Y < 1 - Pmax/2Pmin then (5) is the slower. The proof will be presented in several steps. Step 1. Just like in (Littman and Szepesvari, 1996) (see also the extended version (Szepesvciri and Littman, 1996)) the main idea is to compare Qt with the simpler process Note that the only (but rather essential) difference between the definition of Qt and that of Qt is the appearance of Q* in the defining equation of Qt. Firstly, notice that as a consequence of this change the process Qt clearly converges to Q* and this convergence may be investigated along each component (x, a) separately using standard stochastic-approximation techniques (see e.g. (Was an , 1969; Poljak and Tsypkin, 1973)). Using simple devices one can show that the difference process At(x, a) = IQt(x, a)at(x, a)1 satisfies the following inequality: A t+1 (x, a) ~ (1 - Ot(x, a))At(x, a) + 'Y0t(x, a)(IIAtll + lIat - Q*II). (7) Here 11·11 stands for the maximum norm. That is the task of showing the convergence rate of Qt to Q* is reduced to that of showing the convergence rate of At to zero. Step 2. We simplify the notation by introducing the abstract process whose update equation is (8) where i E 1,2, ... , n can be identified with the state-action pairs, Xt with At, f.t with Qt - Q*, etc. We analyze this process in two steps. First we consider processes when the "perturbation-term" f.t is missing. For such processes we have the following lemma: LEMMA 3.2 Assume that 771,1]2, ... ,'TIt, . .. are independent random variables with a common underlying distribution P{TJt = i) = Pi > O. Then the process Xt defined The Asymptotic Convergence-Rate of Q-leaming 1067 by (9) satisfies IIxtil = OCR(~--Y») wi~h probability one (w.p.1), where R = mini Pi/ maxi Pi. Proof. (Outline) Let To = 0 and Tk+l = min{ t ~ Tk I Vi = 1 . . . n, 3s = s(i) : 1]8 = i}, i.e. Tk+1 is the smallest time after time Tk such that during the time interval [Tk + 1, Tk+d all the components of XtO are "updated" in Equation (9) at least once. Then (10) where Sk = maxi Sk(i) . This inequality holds because if tk(i) is the last time in [Tk + 1, Tk+1] when the ith component is updated then XT"+l+1(i) = Xtk(i)+l(i) = (1-1/St/o(i»Xt,,(i)(i) + ,/St,,(i) IIXt,,(i) 011 < (l-l/St,,(i»lIxt/o(i)OIl +,/St,,(i)lIxt,,(i)OIl = (1 -1 -,) IIXt,,(i) 011 St,,(i) < (1- 1 ;k') IIXT,,+1011, where it was exploited that Ilxtll is decreasing. Now, iterating (10) backwards in time yields X7Hl(-)::: IIxolin (1- 1 ~ 'Y). Now, consider the following approximations: Tk ~ Ck, where C ~ 1/Pmin (C can be computed explicitly from {Pi}), Sk ~ PmaxTk+1 ~ Pmax/Pmin(k + 1) ~ (k + 1)/ Ro, where Ro = 1/CPmax' Then, using Large Deviation's Theory, k-l ( 1 _ ,) k-l ( Ro(1 _ ,») (1) Ro(l--Y) IT 1-~II 1. ~j=O Sj j=O J + 1 k (11) holds w.p.1. Now, by defining s = Tk + 1 so that siC ~ k we get which holds due to the monotonicity of Xt and l/kRo(l--y) and because R Pmin/Pmax ~ Ro. 0 Step 3. Assume that, > 1/2. Fortunately, we know by an extension of the Law of the Iterated Logarithm to stochastic approximation processes that the convergence 1068 C. Szepesvari rate of IIOt -Q*II is 0 (y'loglogt/t) (the uniform boundedness ofthe random reinforcement signals must be exploited in this step) (Major, 1973). Thus it is sufficient to provide a convergence rate estimate for the perturbed process, Xt, defined by (8), when f.t = Cy'loglogt/t for some C > O. We state that the convergence rate of f.t is faster than that of Xt. Define the process ZHI (i) = { (1 ~~l)) Zt(i), if 7Jt = i; Zt (i), if 7Jt f. i. (12) This process clearly lower bounds the perturbed process, Xt. Obviously, the convergence rate of Zt is O(l/tl-'Y) which is slower than the convergence rate of f.t provided that, > 1/2, proving that f.t must be faster than Xt. Thus, asymptotically f.t ~ (1/, - l)xt, and so Ilxtll is decreasing for large enough t. Then, by an argument similar to that of used in the derivation of (10), we get XTIo+1+1(i) ~ (1- 1 ~k') II XTk +1 II ~ ~ f.Tk, (13) where Sk = mini Sk(i). By some approximation arguments similar to that of Step 2, together with the bound (l/n71) 2:: s71-3/ 2Jloglogs ~ s-1/2Jloglogs, 1 > 7J > 0, which follows from the mean-value theorem for integrals and the law of integration by parts, we get that Xt ~ O(l/tR (l-'Y»). The case when , ~ 1/2 can be treated similarly. Step 5. Putting the pieces together and applying them for At = Ot - Qt yields Theorem 3.1. 4 DISCUSSION AND CONCLUSIONS The most restrictive of our conditions is the assumption concerning the sampling of (Xt, at). However, note that under a fixed learning policy the process (Xt, at) is a (non-stationary) Markovian process and if the learning policy converges in the sense that limt-+oo peat 1Ft) = peat I Xt) (here Ft stands for the history of the learning process) then the process (Xt, at) becomes eventually stationary Markovian and the sampling distribution could be replaced by the stationary distribution of the underlying stationary Markovian process. If actions become asymptotically optimal during the course of learning then the support of this stationary process will exclude the state-action pairs whose action is sub-optimal, i.e. the conditions of Theorem 3.1 will no longer be satisfied. Notice that the proof of convergence of such processes still follows very similar lines to that of the proof presented here (see the forthcoming paper (Singh et al., 1997)), so we expect that the same convergence rates hold and can be proved using nearly identical techniques in this case as well. A further step would be to find explicit expressions for the constant B of Theorem 3.1. Clearly, B depends heavily on the sampling of (Xt, at), as well as the transition probabilities and rewards of the underlying MDP. Also the choice of harmonic learning rates is arbitrary. If a general sequence at were employed then the artificial "time" Tt (x, a) = 1 /IT}=o (1 - at (x, a)) should be used (note that for the harmonic sequence Tt(x, a) ~ t). Note that although the developed bounds are asymptotic in their present forms, the proper usage of Large Deviation's Theory would enable us to develope non-asymptotic bounds. The Asymptotic Convergence-Rate ofQ-learning 1069 Other possible ways to extend the results of this paper may include Q-Iearning when learning on aggregated states (Singh et al., 1995), Q-Iearning for alternating/simultaneous Markov games (Littman, 1994; Szepesvari and Littman, 1996) and any other algorithms whose corresponding difference process At satisfies an inequality similar to (7). Yet another application of the convergence-rate estimate might be the convergence proof of some average reward reinforcement learning algorithms. The idea of those algorithms follows from a kind of Tauberian theorem, Le. that discounted sums converge to the average value if the discount rate converges to one (see e.g. Lemma 1 of (Mahadevan, 1994; Mahadevan, 1996) or for a value-iteration scheme relying on this idea (Hordjik and Tijms, 1975)). Using the methods developed here the proof of convergence of the corresponding Q-learning algorithms seems quite possible. We would like to note here that related results were obtained by Bertsekas et al. et. al (see e.g. (Bertsekas and Tsitsiklis, 1996)). Finally, note that as an application of this result we immediately get that the convergence rate of the model-based RL algorithm, where the transition probabilities and rewards are estimated by their respective averages, is clearly better than that of for Q-Iearning. Indeed, simple calculations show that the law of iterated logarithm holds for the learning process underlying model-based RL. Moreover, the exact expression for the convergence rate depends explicitly on how much computational effort we spend on obtaining the next estimate of the optimal value function, the more effort we spend the faster is the convergence. This .bound thus provides a direct way to control the tradeoff between the computational effort and the convergence rate. Acknowledgements This research was supported by aTKA Grant No. F20132 and by a grant provided by the Hungarian Educational Ministry under contract no. FKFP 1354/1997. I would like to thank Andras Kramli and Michael L. Littman for numerous helpful and thought-provoking discussions. References Bertsekas, D. and Tsitsiklis, J. (1996). Neuro-Dynamic Programming. Athena Scientific, Belmont, MA. Hordjik, A. ~nd Tijms, H. (1975). A modified form of the iterative method of dynamic programming. Annals of Statistics, 3:203-208. Jaakkola, T., Jordan, M., and Singh, S. (1994). On the convergence of stochastic iterative dynamic programming algorithms. Neural Computation, 6(6):11851201. Littman, M. (1994). Markov games as a framework for multi-agent reinforcement learning. In Proc. of the Eleventh International Conference on Machine Learning, pages 157-163, San Francisco, CA. Morgan Kauffman. Littman, M. and Szepesvciri, C. (1996). A Generalized Reinforcement Learning Model: Convergence and applications. In Int. Con/. on Machine Learning. http://iserv.ikLkfki.hu/ asl-publs.html. 1070 C. Szepesvari Mahadevan, S. (1994). To discount or not to discount in reinforcement learning: A case study comparing R learning and Q learning. In Proceedings of the Eleventh International Conference on Machine Learning, pages 164-172, San Francisco, CA. Morgan Kaufmann. Mahadevan, S. (1996). Average reward reinforcement learning: Foundations, algorithms, and empirical results. Machine Learning, 22(1,2,3):124-158. Major, P. (1973). A law of the iterated logarithm for the Robbins-Monro method. Studia Scientiarum Mathematicarum Hungarica, 8:95-102. Poljak, B. and Tsypkin, Y. (1973). Pseudogradient adaption and training algorithms. Automation and Remote Control, 12:83-94. Puterman, M. L. (1994). Markov Decision Processes Discrete Stochastic Dynamic Programming. John Wiley & Sons, Inc., New York, NY. Ross, S. (1970). Applied Probability Models with Optimization Applications. Holden Day, San Francisco, California. Singh, S., Jaakkola, T., and Jordan, M. (1995). Reinforcement learning with soft state aggregation. In Proceedings of Neural Information Processing Systems. Singh, S., Jaakkola, T., Littman, M., and Csaba Szepesva ri (1997). On the convergence of single-step on-policy reinforcement-learning al gorithms. Machine Learning. in preparation. Szepesvari, C. and Littman, M. (1996). Generalized Markov Decision Processes: Dynamic programming and reinforcement learning algorithms. Machine Learning. in preparation, available as TR CS96-10, Brown Univ. Tsitsiklis, J. (1994). Asynchronous stochastic approximation and q-learning. Machine Learning, 8(3-4):257-277. Wasan, T. (1969). Stochastic Approximation. Cambridge University Press, London. Watkins, C. (1990). Learning /rom Delayed Rewards. PhD thesis, King's College, Cambridge. QLEARNING.
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Learning Human-like Knowledge by Singular Value Decomposition: A Progress Report Thomas K. Landauer Darrell Laham Department of Psychology & Institute of Cognitive Science University of Colorado at Boulder Boulder, CO 80309-0345 {landauer, dlaham}@psych.colorado.edu Peter Foltz Department of Psychology New Mexico State University Las Cruces, NM 88003-8001 pfoltz@crl.nmsu.edu Abstract Singular value decomposition (SVD) can be viewed as a method for unsupervised training of a network that associates two classes of events reciprocally by linear connections through a single hidden layer. SVD was used to learn and represent relations among very large numbers of words (20k-60k) and very large numbers of natural text passages (lk70k) in which they occurred. The result was 100-350 dimensional "semantic spaces" in which any trained or newly aibl word or passage could be represented as a vector, and similarities were measured by the cosine of the contained angle between vectors. Good accmacy in simulating human judgments and behaviors has been demonstrated by performance on multiple-choice vocabulary and domain knowledge tests, emulation of expert essay evaluations, and in several other ways. Examples are also given of how the kind of knowledge extracted by this method can be applied. 1 INTRODUCTION Traditionally, imbuing machines with human-like knowledge has relied primarily on explicit coding of symbolic facts into computer data structures and algorithms. A serious limitation of this approach is people's inability to access and express the vast reaches of unconscious knowledge on which they rely, knowledge based on masses of implicit inference and irreversibly melded data. A more important deficiency of this state of affairs is that by coding the knowledge ourselves, (as we also do when we assign subjectively hypothesized rather than objectively identified features to input or output nodes in a neural net) we beg important questions of how humans acquire and represent the cOOed knowledge in the fIrSt place. 46 T. K. Landauer; D. Laham and P. Foltz Thus. from both engineering and scientific perspectives. there are reasons to try to design learning machines that can ocquire human-like quantities of human-like knowledge from the same sources as humans. The success of such techniques would not prove that the same mechanisms are used by humans. but because we presently do not know how the problem can be solved in principle, successful simulation may offer theoretical insights as well as practical applications. In the work reported here we have found a way to induce significant amounts of knowledge about the meanings of passages and of their constituent vocabularies of words by training on large bodies of natural text. In general terms, the method simultaneously extracts the similarity between words (the likelihood of being used in passages that convey similar ideas) and the similarity of passages (the likelihood of containing words of similar meaning). The conjoint estimation of similarity is accomplished by a fundamentally simple representational technique that exploits mutual constraints implicit in the occurrences of very many words in very many contexts. We view the resultant system both as a means for automatically learning much of the semantic content of words and passages. and as a potential computational model for the process that underlies the corresponding human ability. While the method starts with data about the natural contextual co-occurrences of words. it uses them in a different manner than has previously been applied. A long -standing objection to co-occurrence statistics as a source of linguistic knowledge (Chomsky's 1957) is that many grammatically acceptable expressions. for example sentences with potentially unlimited embedding structures. cannot be produced by a finite Markov process whose elements are transition probabilities from word to word. If word-word probabilities are insufficient to generate language. then. it is argued, acquiring estimates of such probabilities cannot be a way that language can be learned. However, our approach to statistical knowledge learning differs from those considered in the past in two ways. First. the basic associational data from which knowledge is induced are not transition frequencies between successive individual words or phrases. but rather the frequencies with which particular words appear as components of relatively large natural passages, utterances of the kind that humans use to convey complete ideas. The result of this move is that the statistical regularities reflected are relations among unitary expressions of meaning. rather than syntactic constraints on word order that may serve additional purposes such as output and input processing efficiencies. error protection. or esthetic style. Second, the mutual constraints inherent in a multitude of such local ~ occurrence relations are jointly satisfied by being forced into a global representation of lower dimensionality. This constraint satisfaction. a form of induction. was accomplished by singular value decomposition. a linear factorization technique that produces a representational structure equivalent to a three layer neural network model with linear activation functions. 2 THE TEXT ANALYSIS MODEL AND METHOD The text analysis process that we have explored is called Latent Semantic Analysis (LSA) (Deerwester et al .• 1990; Landauer and Dumais. 1997). It comprises four steps: (1) A large body of text is represented as a matrix [ij], in which rows stand for individual word types. columns for meaning-bearing passages such as sentences or paragraphs. mel cells contain the frequency with which a word occurs in a passage. (2) Cell entries (freqi) are transformed to: log(freqi, + I) L(( fref/,] {freq. ]'l -1-, ~fr:~v *10 ~fre~v ) a measure of the first order association of a word and its context. Learning Human Knowledge by Singular Value Decomposition: A Progress Report 47 (3) The matrix is then subjected to singular value decomposition (Berry, 1992): [ij] = [ik] [kk] Uk]' in which [ik] and Uk] have orthonormal columns, [kk] is a diagonal matrix of singular values, and k <= max (ij). (4) Finally, all but the d largest singular values are set to zero. Pre-multiplication of the right-hand matrices produces a least-squares best approximation to the original matrix given the number of dimensions, d, (hidden units in a corresponding neural net model representation) that are retained. The SVD with dimension reduction constitutes a constraint-satisfaction induction process in that it predicts the original observations on the basis of linear relations among the abstracted representations of the data that it has retained. By hypothesis, the analysis induces human-like relationships among passages and words because humans also make inferences about semantic relationships from abstracted representations based on limited data, and do so by an analogous process. In the result, each word and passage is represented as a vector of length d. Performance depends strongly on the choice of number of dimensions. The optimal number is typically around 300. The similarity of any two words, any two text passages, or any word and any text passage, are computed by measures on their vectors. We have most often used the cosine (of the contained angle between the vectors in semantic d-space) which we interpret as the degree of qualitative similarity of meaning. The length of vectors is also useful and interpretable. 3 TESTS OF LSA'S PERFORMANCE LSA's ability to simulate human knowledge and meaning relations has been tested in a variety of ways. Here we describe two relatively direct sources of evidence and briefly list several others. 3.1 VOCABULARY & DOMAIN KNOWLEDGE TESTS In all cases, LSA was ftrst trained on a large text corpus intended to be representative of the text from which humans gain most of the semantic knowledge to be simulated. In a previously reported test (Landauer and Dumais, 1997), LSA was trained on approximately five million words of text sampled from a high-school level encyclopedia, then tested on multiple choice items from the Educational Testing Service Test of English as a Foreign Language (TOEFL). These test questions present a target word or short phrase and ask the student to choose the one of four alternative words or phrases that is most similar in meaning. LSA's answer was determined by computing the cosine between the derived vector for the target word or phrase and each of the alternatives and choosing the largest. LSA was correct on 64% of the 80 items, identical to the average of a large sample of students from non-English speaking countries who had applied for admission to U. S. colleges. When in error, LSA made choices positively correlated (product-moment r = .44) with those preferred by students. We have recently replicated this result with training on a similar sized sample from the Associated Press newswire In a new set of tests, LSA was trained on a popular introductory psychology textbook (Myers, 1995) and tested with the same four-alternative multiple choice tests used for students in two large classes. In these experiments, LSA's score was about 6O%-lower than the class averages but above passing level, and far above guessing probability. Its errors again resembled those of students; it got right about half as many of questions rated difftcult by the test constructors as ones rated easy (Landauer, Foltz and Laham, 1997). 3.2 ESSA Y TESTS 48 T. K. Landauer, D. Laham and P. Foltz Word-wad meaning similarities are a good test of knowledge-indeed, vocabulary tests are the best single measure of human intelligence. However, they are not sufficient to assess the correspondence of LSA and human knowledge because people usually express knowledge via larger verbal strings, such as sentences, paragraphs and articles. Thus, just as multiple choice tests of student knowledge are often supplemented by essay tests whose content is then judged by humans, we wished to evaluate the adequacy of LSA' s representation of knowledge in complete passages of text. We could not have LSA write essays because it has no means for producing sentences. However, we were able to assess the accum::y with which LSA could extract and represent the knowledge expressed in essays written by students by simulating judgments about their content that were made by human readers (Landauer, Laham, Rehder, & Schreiner, in press). In these tests, students were asked to write short essays to cover an assigned topic or to answer a posed question. In various experiments, the topics included anatomy axI function of the heart, phenomena from introductory psychology, the history of the Panama Canal, and tolerance of diversity in America. In each case, LSA was first trained either on a large sample of instructional text from the same domain or, in the latter case, on combined text from the very large number of essays themselves, to produce a highdimensional (100-350 dimensions in the various tests) semantic space. We then represented each essay simply as the vector average of the vectors for the words it contained. Two properties of these average vectors were then used to measure the quality and quantity of knowledge conveyed by an essay: (1) the similarity (measured as the cosine of the angle between vectors) of the student essay and one or more standard essays, and (2) the total amount of domain specific content, measured as the vector length. In each case, two human experts independently rated the overall quality of each essay on a five or ten point scale. The judges were either university course instructors or professional exam readers from Educational Testing Service. The LSA measures were calibrated with respect to the judges' rating scale in several different ways, but because they gave nearly the same results only one will be described here. In this method, each student essay was compared to a large (90-200) set of essays previously scored by experts, and the ten most similar (by cosine) identified. The target essay was then assigned a "quality" score component consisting of the cosine-weighted average of the ten. A second, "relevant quantity", score component was the vector length of the student essay. Finally, regression on expert scores was used to weight the quality and quantity scores (However, the weights in all cases were so close to equal that merely adding them would have given comparable results). Calibration was performed on data independent of that used to evaluate the relation between LSA and expert ratings. The correlation between the LSA score for an essay and that assigned by the average of the human readers was .80, .64, .XX and .84 for the four sets of exams. The comparable correlation between one reader and the other was .83, .65, .XX and .82, respectively. In the heart topic case, each student had also taken a carefully constructed "objective" test over the same material (a short answer test with near perfect scoring agreement). The correlation between the LSA essay score and the objective test was .81, the average correlation for the two expert readers .74. A striking aspect of these results is that the LSA representations were based on analyses of the essays that took no account of word order, each essay was treated as a "bag of words". In extracting meaning from a text, human readers presumably rely on syntax as well as the mere combination of words it contains, yet they were no better at agreeing on an essay's quality or in assigning a score that predicted a performance on a separate test of knowledge. Apparently, either the relevant information conveyed by word order in sentences is redundant with the information that can be inferred from the combination of words in the essay, or the processes used by LSA and humans extract different but compensatingly useful information. Learning HumanKnowledgc by Singular Value Decomposition: A Progress Report 49 3.3 OTHER EVIDENCE LSA has been compared with human knowledge in several additional ways, some confrrming the correspondence, others indicating limitations. Here are some examples, all based on encyclopedia corpus training. (1) Overall LSA similarity between antonyms (mean cos = .18) was equivalent to that between synonyms (mean cos = .17) in triplets sampled from an antonym/synonym dictionary (W & R Chambers, 1989), both of which significantly exceeded that for unrelated pairs (mean cos = .01; ps < .0001). However for antonym (but not for synonym) pairs a dominant dimension of difference could easily be extracted by computing a one dimensional unfolding using the LSA cosines from a set of words listed in Rogel's (1992) thesaurus as related respectively to the two members of the pair. (2) Anglin (1970) asked children and adults to sort words varying in concept relations ad parts of speech. LSA wont-word similarity correlates .50 with children and .32 with adults for the number of times they sorted two words together. Conceptual structure is reflected, but grammatical classification, strong in the adult data, is not. (3) When people are asked to decide that a letter string is a wont, they do so faster if they have just read a sentence that does not contain the word but implies a related concept (e.g. Till, Mross & Kintsch, 1988). LSA mirrors this result with high similarities between the same sentences and words. (Landauer & Dumais, 1997). (4) People frequently make a logical error, called the conjunction error by Tversky ad Kahneman (1974), in which they estimate that the probability that an object is a member of a class is greater than that it is a member of a superset class when the description of the object is "similar" to the description of the subset. For example, when told that "Linda is a young woman who is single, outspoken ... deeply concerned with issues of discrimination and social justice," over 80% of even statistically sophisticated subjects rate it more likely that Linda is a feminist bank teller than that Linda is a bank teller (Tversky & Kahneman, 1980). LSA similarities between descriptions of people ad occupations of this kind taken from Shafir, Smith and Osherson (1990) were computed as the cosine between the vector averages of words in the paired person-occupation descriptions. Conjunction error statements were more similar to the subset than superset statement in 12 out of 14 cases (p<.01), showing that LSA's representation of sentential meaning reflected similarity relations of the sort that have been hypothesized to underlie the conjunction fallacy in human judgment. (5) A semantic subspace was constructed for words from natural kind and artifact categories whose differential preservation is characteristic of agnosias due to local damage from herpes simplex encephalitis (Warrington & Shallice, 1984). Principal components analysis of the similarities among these words as represented by LSA revealed that categories that tend to be lost contain words that are more highly inter-related than those in preserved categories (Laham, in press). Of course, LSA does not capture all of the human knowledge conveyed by text. Some of the shortfall is probably due merely to the use of training corpora that are still imperfectly representative of the language experience of a typical person, and to lack of knowledge from non-textual sources. For example, in all these studies, less total text was used for LSA training than even a single educated adult would have read However, a more fundamental restriction is that the analysis does not reflect order relations between words, and therefore cannot extract infonnation that depends on syntax. Because the analysis discovers and represents only unsigned continuous similarities, it can be used to induce only certain classes of structural relations, not including ones that express Boolean, causal or other non-commutative logical relations. As we have seen, this lack does not prevent accurate simulation of human cognition in many cases, possibly because humans also 50 T. K Landauer, D. Laham and P. Foltz frequently rely on similarity rather than syntax-based. discrete logic (fversky axl Kahneman, 1983); however, it does limit the utility of the results for populating the symbolic data structures commonly used to represent knowledge in traditional AI. On the other hand, as examples in the next section show, continuous-valued similarity relations can be fruitfully applied if appropriate computational use is made of them. 4 SAMPLE APPLICATIONS LSA has been used successfully in a variety of experimental applications, including the essay scoring techniques described earlier. Here are some additional examples: (1) The technique has been used to improve automatic information retrieval by 20-30% over otherwise identical methods by allowing users' queries to match documents with the desired conceptual meaning but expressed in different words (Dumais, 1991, 1994). (2) By training on corpora of translated text in which the words of corresponding paragraphs in the two languages are combined in the "bags of words", LSA has been able to provide at least as good retrieval when queries and documents are in a different language as when in the same language (Landauer and Littman, 1990). (3) LSA-based measures of the similarity of student essays on a topic to instructional texts can predict how much an individual student will learn from a particular text (Wolfe et al., in press; Rehder et al., in press). To do this, the full set of student essays and the texts in question are aligned along a single dimension that best accommodates the LSA similarities among them. Estimates from one such experiment showed that using LSA to choose the optimal one of four texts for each student (a text that is slightly more sophisticated than the student) rather than assigning all students the overall best text (which LSA also picked correctly) increased the average amount learned by over 40%. (4) LSA-based measures of conceptual similarity between successive sentences accurately predict differences in judged coherence and measured comprehensibility of text (Foltz, Kintsch and Landauer, in press). 5 SUMMARY SVD-based learning of the structure underlying the use of words in meaningful contexts has been found capable of deriving and representing the similarity of meaning of words and text passages in a manner that accurately simulates corresponding similarity relations as reflected in several sorts of human judgments and behavior. The validity of the resulting representation of meaning similarities has been established in a variety of ways, and the utility of its knowledge representation illustrated by several educational axl cognitive psychological research applications. It is obviously too early to assess whether the particular computational model is a true analog of the process used by the human brain to accomplish the same things. However, the basic process, the representation of myriad local associative relations between components and larger contexts of experience in a joint space of lower dimensionality, offers, for the first time, a candidate for such a mechanism that has been shown sufficient to approximate human knowledge acquisition from natural sources at natural scale. Acknowledgments We thank members of the LSA research group at the University of Colorado for valuable collaboration and advice: Walter Kintsch, Bob Rehder, Mike Wolfe, & M. E. Shreiner. We especially acknowledge two participants from the Spring 1997 LSA seminar at CU whose unpublished work is described: Alan Sanfey (3.3.4) and Michael Emerson (3.3.1). Thanks also to Susan T. Dumais of Bellcore. Learning Human Knowledge by Singular Value Decomposition: A Progress Repol1 References Anglin, J. M. (1970). The growth of word meaning. Cambridge, MA: MIT. Berry, M. W. (1992). Large scale singular value computations. International Journal of Supercomputer Applications, 6, 13-49. 51 Deerwester, S., Dumais, S. T., Furnas, G. W., Landauer, T. K., & Harshman, R. (1990). Indexing By Latent Semantic Analysis. Journal of the American Society For Information Science. 41, 391-407. Dumais, S. T. (1991). Improving the retrieval of infonnation from external sources. Behavior Research Methods, Instruments and Computers. 23,229-236. Dumais, S. T. (1994). Latent semantic indexing (LSI) and TREC-2. In D. Harman (Ed.), National Institute of Standards and Technology Text Retrieval Conference. NIST special publication. Foltz, P. W., Kintsch, W., & Landauer, T. K. (in press). Analysis of text coherence using Latent Semantic Analysis. Discourse Processes. Laham, D. (in press). Latent Semantic Analysis approaches to categorization. Proceedings of the Cognitive Science Society. 1997. Landauer, T. K., & Dumais, S. T. (1997). A solution to Plato's problem: The Latent Semantic Analysis theory of the acquisition, induction, and representation of knowledge. Psychological Review, 104,211-240. Landauer, T. K., Foltz, P. W., & Laham, D. (1997). Latent Semantic Analysis passes the test: knowledge representation and multiple-choice testing. Manuscript in preparation. Landauer, T. K., Laham, D., Rehder, B. & Schreiner, M .E. (in press). How well can passage meaning be derived without using word order: A comparison of Latent Semantic Analysis and humans. Proceedings of the Cognitive Science Society, 1997. Landauer, T. K., & Littman, M. L. (1990). Fully automatic cross-language document retrieval using latent semantic indexing. In Proceedings of the Sixth Annual Conference of the UW Centre for the New Oxford English Dictionary and Text Research (pp. 31-38). Waterloo, Ontario: UW Centre for the New OED. Myers, D. G. (1995). Psychology. Fourth Edition. NY, NY: Worth. Rehder, B., Schreiner, M. E., Wolfe, B. W., Laham, D., Landauer, T. K., & Kintsch, W. (in press). Using Latent Semantic Analysis to assess knowledge: Some technical considerations. Discourse Processes. ShafIf, E., Smith, E. E., & Osherson, D. N. (1990). Typicality and reasoning judgments. Memory & Cognition, 3, 229-239. Till, R. E., Mross, E. F., & Kintsch. W. (1988). Time course of priming for associate and inference words in discourse context. Memory and Cognition, 16, 283-299. Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 18S, 1124-1131. Tversky, A., & Kahneman, D. (1980). Judgments of and by representativeness. In D. Kahneman, P. Slovic, & A. Tversky (Eds.), Judgment under uncertainty: Heuristics and biases. New York: Cambridge University Press. Tversky, A., & Kahneman, D. (1983). Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment. Psychological Review, 90,293-315. Warrington, E. K., & Shallice, T. (1984). Category-specific semantic impairments. Brain, 107, 829-853. Wolfe, M. B., Schreiner, M. E., Rehder, B., Laham, D., Foltz, P. W., Kintsch, W., & Landauer, T. K. (in press). Learning from text: Matching readers and text by Latent Semantic Analysis. Discourse Processes.
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2D Observers for Human 3D Object Recognition? Zili Liu NEC Research Institute . Abstract Daniel Kersten University of Minnesota Converging evidence has shown that human object recognition depends on familiarity with the images of an object. Further, the greater the similarity between objects, the stronger is the dependence on object appearance, and the more important twodimensional (2D) image information becomes. These findings, however, do not rule out the use of 3D structural information in recognition, and the degree to which 3D information is used in visual memory is an important issue. Liu, Knill, & Kersten (1995) showed that any model that is restricted to rotations in the image plane of independent 2D templates could not account for human performance in discriminating novel object views. We now present results from models of generalized radial basis functions (GRBF), 2D nearest neighbor matching that allows 2D affine transformations, and a Bayesian statistical estimator that integrates over all possible 2D affine transformations. The performance of the human observers relative to each of the models is better for the novel views than for the familiar template views, suggesting that humans generalize better to novel views from template views. The Bayesian estimator yields the optimal performance with 2D affine transformations and independent 2D templates. Therefore, models of 2D affine matching operations with independent 2D templates are unlikely to account for human recognition performance. 1 Introduction Object recognition is one of the most important functions in human vision. To understand human object recognition, it is essential to understand how objects are represented in human visual memory. A central component in object recognition is the matching of the stored object representation with that derived from the image input. But the nature of the object representation has to be inferred from recognition performance, by taking into account the contribution from the image information. When evaluating human performance, how can one separate the con830 Z Liu and D. Kersten tributions to performance of the image information from the representation? Ideal observer analysis provides a precise computational tool to answer this question. An ideal observer's recognition performance is restricted only by the available image information and is otherwise optimal, in the sense of statistical decision theory, irrespective of how the model is implemented. A comparison of human to ideal performance (often in terms of efficiency) serves to normalize performance with respect to the image information for the task. We consider the problem of viewpoint dependence in human recognition. A recent debate in human object recognition has focused on the dependence of recognition performance on viewpoint [1, 6]. Depending on the experimental conditions, an observer's ability to recognize a familiar object from novel viewpoints is impaired to varying degrees. A central assumption in the debate is the equivalence in viewpoint dependence and recognition performance. In other words, the assumption is that viewpoint dependent performance implies a viewpoint dependent representation, and that viewpoint independent performance implies a viewpoint independent representation. However, given that any recognition performance depends on the input image information, which is necessarily viewpoint dependent, the viewpoint dependence of the performance is neither necessary nor sufficient for the viewpoint dependence of the representation. Image information has to be factored out first, and the ideal observer provides the means to do this. The second aspect of an ideal observer is that it is implementation free. Consider the GRBF model [5], as compared with human object recognition (see below). The model stores a number of 2D templates {Ti} of a 3D object 0, and reco~nizes or rejects a stimulus image S by the following similarity measure ~iCi exp UITi - SI1 2 j2(2), where Ci and a are constants. The model's performance as a function of viewpoint parallels that of human observers. This observation has led to the conclusion that the human visual system may indeed, as does the model, use 2D stored views with GRBF interpolation to recognize 3D objects [2]. Such a conclusion, however, overlooks implementational constraints in the model, because the model's performance also depends on its implementations. Conceivably, a model with some 3D information of the objects can also mimic human performance, so long as it is appropriately implemented. There are typically too many possible models that can produce the same pattern of results. In contrast, an ideal observer computes the optimal performance that is only limited by the stimulus information and the task. We can define constrained ideals that are also limited by explicitly specified assumptions (e.g., a class of matching operations). Such a model observer therefore yields the best possible performance among the class of models with the same stimulus input and assumptions. In this paper, we are particularly interested in constrained ideal observers that are restricted in functionally Significant aspects (e.g., a 2D ideal observer that stores independent 2D templates and has access only to 2D affine transformations). The key idea is that a constrained ideal observer is the best in its class. So if humans outperform this ideal observer, they must have used more than what is available to the ideal. The conclusion that follows is strong: not only does the constrained ideal fail to account for human performance, but the whole class of its implementations are also falsified. A crucial question in object recognition is the extent to which human observers model the geometric variation in images due to the projection of a 3D object onto a 2D image. At one extreme, we have shown that any model that compares the image to independent views (even if we allow for 2D rigid transformations of the input image) is insufficient to account for human performance. At the other extreme, it is unlikely that variation is modeled in terms of rigid transformation of a 3D object 2D Observers/or Hwnan 3D Object Recognition? 831 template in memory. A possible intermediate solution is to match the input image to stored views, subject to 2D affine deformations. This is reasonable because 2D affine transformations approximate 3D variation over a limited range of viewpoint change. In this study, we test whether any model limited to the independent comparison of 2D views, but with 2D affine flexibility, is sufficient to account for viewpoint dependence in human recognition. In the following section, we first define our experimental task, in which the computational models yield the provably best possible performance under their specified conditions. We then review the 2D ideal observer and GRBF model derived in [4], and the 2D affine nearest neighbor model in [8]. Our principal theoretical result is a closed-form solution of a Bayesian 2D affine ideal observer. We then compare human performance with the 2D affine ideal model, as well as the other three models. In particular, if humans can classify novel views of an object better than the 2D affine ideal, then our human observers must have used more information than that embodied by that ideal. 2 The observers Let us first define the task. An observer looks at the 2D images of a 3D wire frame object from a number of viewpoints. These images will be called templates {Td. Then two distorted copies of the original 3D object are displayed. They are obtained by adding 3D Gaussian positional noise (i.i.d.) to the vertices of the original object. One distorted object is called the target, whose Gaussian noise has a constant variance. The other is the distract or , whose noise has a larger variance that can be adjusted to achieve a criterion level of performance. The two objects are displayed from the same viewpoint in parallel projection, which is either from one of the template views, or a novel view due to 3D rotation. The task is to choose the one that is more similar to the original object. The observer's performance is measured by the variance (threshold) that gives rise to 75% correct performance. The optimal strategy is to choose the stimulus S with a larger probability p (OIS). From Bayes' rule, this is to choose the larger of p (SIO). Assume that the models are restricted to 2D transformations of the image, and cannot reconstruct the 3D structure of the object from its independent templates {Ti}. Assume also that the prior probability p(Td is constant. Let us represent S and Ti by their (x, y) vertex coordinates: (X Y )T, where X = (Xl, x2, ... , xn), y = (yl, y2 , ... , yn). We assume that the correspondence between S and T i is solved up to a reflection ambiguity, which is equivalent to an additional template: Ti = (xr yr )T, where Xr = (xn, ... ,x2,xl ), yr = (yn, ... ,y2,yl). We still denote the template set as {Td. Therefore, (1) In what follows, we will compute p(SITi)p(Ti ), with the assumption that S = F (Ti) + N (0, crI2n), where N is the Gaussian distribution, 12n the 2n x 2n identity matrix, and :F a 2D transformation. For the 2D ideal observer, :F is a rigid 2D rotation. For the GRBF model, F assigns a linear coefficient to each template T i , in addition to a 2D rotation. For the 2D affine nearest neighbor model, :F represents the 2D affine transformation that minimizes liS - Ti11 2 , after Sand Ti are normalized in size. For the 2D affine ideal observer, :F represents all possible 2D affine transformations applicable to T i. 832 Z Liu and D. Kersten 2.1 The 2D ideal observer The templates are the original 2D images, their mirror reflections, and 2D rotations (in angle ¢) in the image plane. Assume that the stimulus S is generated by adding Gaussian noise to a template, the probability p(SIO) is an integration over all templates and their reflections and rotations. The detailed derivation for the 2D ideal and the GRBF model can be found in [4]. Ep(SITi)p(Ti) ex: E J d¢exp (-liS - Ti(¢)112 /2(2 ) • (2) 2.2 The GRBF model The model has the same template set as the 2D ideal observer does. Its training requires that EiJ;7r d¢Ci(¢)N(IITj - Ti(¢)II,a) = 1, j = 1,2, ... , with which {cd can be obtained optimally using singular value decomposition. When a pair of new stimuli is} are presented, the optimal decision is to choose the one that is closer to the learned prototype, in other words, the one with a smaller value of 111- E 127r d¢ci(¢)exp (_liS -2:~(¢)1I2) II. (3) 2.3 The 2D affine nearest neighbor model It has been proved in [8] that the smallest Euclidean distance D(S, T) between S and T is, when T is allowed a 2D affine transformation, S ~ S/IISII, T ~ T/IITII, D2(S, T) = 1 - tr(S+S . TTT)/IITII2, (4) where tr strands for trace, and S+ = ST(SST)-l. The optimal strategy, therefore, is to choose the S that gives rise to the larger of E exp (_D2(S, Ti)/2a2) , or the smaller of ED2(S, Ti). (Since no probability is defined in this model, both measures will be used and the results from the better one will be reported.) 2.4 The 2D affine ideal observer We now calculate the Bayesian probability by assuming that the prior probability distribution of the 2D affine transformation, which is applied to the template T i, AT + Tr = (~ ~) Ti + (~: ::: ~:), obeys a Gaussian distribution N(Xo,,,,/16), where Xo is the identity transformation xl' = (a,b,c,d,tx,ty) = (1,0,0,1,0,0). We have Ep(SITi ) = E i: dX exp (-IIATi + Tr - SII2/2(2) (5) = EC(n, a, ",/)deC 1 (QD exp (tr (KfQi(QD-1QiKi) /2(12), (6) where C(n, a, ",/) is a function of n, a, "'/; Q' = Q + ",/-212, and Q _ ( XT . XT XT · Y T ) QK _ ( XT· Xs Y T . Xs) -21 YT ·XT YT ·YT ' XT ·Ys YT .Ys +"'/ 2· (7) The free parameters are "'/ and the number of 2D rotated copies for each T i (since a 2D affine transformation implicitly includes 2D rotations, and since a specific prior probability distribution N(Xo, ",/1) is assumed, both free parameters should be explored together to search for the optimal results). 2D Observers for Hwnan 3D Object Recognition? 833 • • • • • • Figure 1: Stimulus classes with increasing structural regularity: Balls, Irregular, Symmetric, and V-Shaped. There were three objects in each class in the experiment. 2.5 The human observers Three naive subjects were tested with four classes of objects: Balls, Irregular, Symmetric, and V-Shaped (Fig. 1). There were three objects in each class. For each object, 11 template views were learned by rotating the object 60° /step, around the X- and Y-axis, respectively. The 2D images were generated by orthographic projection, and viewed monocularly. The viewing distance was 1.5 m. During the test, the standard deviation of the Gaussian noise added to the target object was (J"t = 0.254 cm. No feedback was provided. Because the image information available to the humans was more than what was available to the models (shading and occlusion in addition to the (x, y) positions of the vertices), both learned and novel views were tested in a randomly interleaved fashion. Therefore, the strategy that humans used in the task for the learned and novel views should be the same. The number of self-occlusions, which in principle provided relative depth information, was counted and was about equal in both learned and novel view conditions. The shading information was also likely to be equal for the learned and novel views. Therefore, this additional information was about equal for the learned and novel views, and should not affect the comparison of the performance (humans relative to a model) between learned and novel views. We predict that if the humans used a 2D affine strategy, then their performance relative to the 2D affine ideal observer should not be higher for the novel views than for the learned views. One reason to use the four classes of objects with increasing structural regularity is that structural regularity is a 3D property (e.g., 3D Symmetric vs. Irregular), which the 2D models cannot capture. The exception is the planar V-Shaped objects, for which the 2D affine models completely capture 3D rotations, and are therefore the "correct" models. The V-Shaped objects were used in the 2D affine case as a benchmark. If human performance increases with increasing structural regularity of the objects, this would lend support to the hypothesis that humans have used 3D information in the task. 2.6 Measuring performance A stair-case procedure [7] was used to track the observers' performance at 75% correct level for the learned and novel views, respectively. There were 120 trials for the humans, and 2000 trials for each of the models. For the GRBF model, the standard deviation of the Gaussian function was also sampled to search for the best result for the novel views for each of the 12 objects, and the result for the learned views was obtained accordingly. This resulted in a conservative test of the hypothesis of a GRBF model for human vision for the following reasons: (1) Since no feedback was provided in the human experiment and the learned and novel views were randomly intermixed, it is not straightforward for the model to find the best standard deviation for the novel views, particularly because the best standard deviation for the novel views was not the same as that for the learned 834 Z Liu and D. Kersten ones. The performance for the novel views is therefore the upper limit of the model's performance. (2) The subjects' performance relative to the model will be defined as statistical efficiency (see below). The above method will yield the lowest possible efficiency for the novel views, and a higher efficiency for the learned views, since the best standard deviation for the novel views is different from that for the learned views. Because our hypothesis depends on a higher statistical efficiency for the novel views than for the learned views, this method will make such a putative difference even smaller. Likewise, for the 2D affine ideal, the number of 2D rotated copies of each template Ti and the value I were both extensively sampled, and the best performance for the novel views was selected accordingly. The result for the learned views corresponding to the same parameters was selected. This choice also makes it a conservative hypothesis test. 3 Results Learned Views 25 • Human IJ 20 Ideal eO GRBF O 20 Affine Nearest NtMghbor .£. rn 20 Affine kIoai :g 0 ~ 1.5 81 l! l0.5 Object Type e.£. :!2 0 ~ 1.5 ~ ~ INovel Views • Human EJ 20 Ideal o GRBF o 20 Affine Nearesl N.tghbor ~ 2DAfllna~ Object Type Figure 2: The threshold standard deviation of the Gaussian noise, added to the distractor in the test pair, that keeps an observer's performance at the 75% correct level, for the learned and novel views, respectively. The dotted line is the standard deviation of the Gaussian noise added to the target in the test pair. Fig. 2 shows the threshold performance. We use statistical efficiency E to compare human to model performance. E is defined as the information used by humans relative to the ideal observer [3] : E = (d~uman/d~deal)2, where d' is the discrimination index. We have shown in [4] that, in our task, E = ((a~1!f;actor)2 - (CTtarget)2) / ((CT~~~~~tor)2 - (CTtarget)2) , where CT is the threshold. Fig. 3 shows the statistical efficiency of the human observers relative to each of the four models. We note in Fig. 3 that the efficiency for the novel views is higher than those for the learned views (several of them even exceeded 100%), except for the planar V-Shaped objects. We are particularly interested in the Irregular and Symmetric objects in the 2D affine ideal case, in which the pairwise comparison between the learned and novel views across the six objects and three observers yielded a significant difference (binomial, p < 0.05). This suggests that the 2D affine ideal observer cannot account for the human performance, because if the humans used a 2D affine template matching strategy, their relative performance for the novel views cannot be better than for the learned views. We suggest therefore that 3D information was used by the human observers (e.g., 3D symmetry). This is supported in addition by the increasing efficiencies as the structural regularity increased from the Balls, Irregular, to Symmetric objects (except for the V-Shaped objects with 2D affine models). 2D Observers for Hwnan 3D Object Recognition? 835 300 "" l 300 >300 20 Ideal GRBF Modol j 20 Aftlne Nearest Ighbor l 20 Affine Ideal l 250 250 250 o Learned I 0 l&arnedl ~ o Learned ,.. 250 l " o Learned .. • Novel .Noval • Novel j • Novel " 200 '" ~ 200 200 .. f " ..! '50 t ~ " "" 150 '50 $: ~ j w i "- .'" ---------------II! " '" I Q ! N ~ 0 ObJect Type Q Object Type ObjOGtType ObjoctTypo N Figure 3: Statistical efficiencies of human observers relative to the 2D ideal observer, the GRBF model, the 2D affine nearest neighbor model, and the 2D affine ideal observer_ 4 Conclusions Computational models of visual cognition are subject to information theoretic as well as implementational constraints. When a model's performance mimics that of human observers, it is difficult to interpret which aspects of the model characterize the human visual system. For example, human object recognition could be simulated by both a GRBF model and a model with partial 3D information of the object. The approach we advocate here is that, instead of trying to mimic human performance by a computational model, one designs an implementation-free model for a specific recognition task that yields the best possible performance under explicitly specified computational constraints. This model provides a well-defined benchmark for performance, and if human observers outperform it, we can conclude firmly that the humans must have used better computational strategies than the model. We showed that models of independent 2D templates with 2D linear operations cannot account for human performance. This suggests that our human observers may have used the templates to reconstruct a representation of the object with some (possibly crude) 3D structural information. References [1] Biederman I and Gerhardstein P C. Viewpoint dependent mechanisms in visual object recognition: a critical analysis. J. Exp. Psych.: HPP, 21: 1506-1514, 1995. [2] Biilthoff H H and Edelman S. Psychophysical support for a 2D view interpolation theory of object recognition. Proc. Natl. Acad. Sci., 89:60-64, 1992. [3] Fisher R A. Statistical Methods for Research Workers. Oliver and Boyd, Edinburgh, 1925. [4] Liu Z, Knill D C, and Kersten D. Object classification for human and ideal observers. Vision Research, 35:549-568, 1995. [5] Poggio T and Edelman S. A network that learns to recognize three-dimensional objects. Nature, 343:263-266, 1990. [6] Tarr M J and Biilthoff H H. Is human object recognition better described by geon-structural-descriptions or by multiple-views? J. Exp. Psych.: HPP, 21:1494-1505,1995. [7] Watson A B and Pelli D G. QUEST: A Bayesian adaptive psychometric method. Perception and Psychophysics, 33:113-120, 1983. [8] Werman M and Weinshall D. Similarity and affine invariant distances between 2D point sets. IEEE PAMI, 17:810-814,1995. Toward a Single-Cell Account for Binocular Disparity Tuning: An Energy Model May be Hiding in Your Dendrites Bartlett W. Mel Department of Biomedical Engineering University of Southern California, MC 1451 Los Angeles, CA 90089 mel@quake.usc.edu Daniel L. Ruderman The Salk Institute 10010 N. Torrey Pines Road La Jolla, CA 92037 ruderman@salk.edu Kevin A. Archie Neuroscience Program University of Southern California Los Angeles, CA 90089 karchie@quake.usc.edu Abstract Hubel and Wiesel (1962) proposed that complex cells in visual cortex are driven by a pool of simple cells with the same preferred orientation but different spatial phases. However, a wide variety of experimental results over the past two decades have challenged the pure hierarchical model, primarily by demonstrating that many complex cells receive monosynaptic input from unoriented LGN cells, or do not depend on simple cell input. We recently showed using a detailed biophysical model that nonlinear interactions among synaptic inputs to an excitable dendritic tree could provide the nonlinear subunit computations that underlie complex cell responses (Mel, Ruderman, & Archie, 1997). This work extends the result to the case of complex cell binocular disparity tuning, by demonstrating in an isolated model pyramidal cell (1) disparity tuning at a resolution much finer than the the overall dimensions of the cell's receptive field, and (2) systematically shifted optimal disparity values for rivalrous pairs of light and dark bars-both in good agreement with published reports (Ohzawa, DeAngelis, & Freeman, 1997). Our results reemphasize the potential importance of intradendritic computation for binocular visual processing in particular, and for cortical neurophysiology in general. A Single-Cell Accountfor Binocular Disparity Tuning 209 1 Introduction Binocular disparity is a powerful cue for depth in vision. The neurophysiological basis for binocular disparity processing has been of interest for decades, spawned by the early studies of Rubel and Wiesel (1962) showing neurons in primary visual cortex which could be driven by both eyes. Early qualitative models for disparity tuning held that a binocularly driven neuron could represent a particular disparity (zero, near, or far) via a relative shift of receptive field (RF) centers in the right and left eyes. According to this model, a binocular cell fires maximally when an optimal stimulus, e.g. an edge of a particular orientation, is simultaneously centered in the left and right eye receptive fields, corresponding to a stimulus at a specific depth relative to the fixation point. An account of this kind is most relevant to the case of a cortical "simple" cell, whose phase-sensitivity enforces a preference for a particular absolute location and contrast polarity of a stimulus within its monocular receptive fields. This global receptive field shift account leads to a conceptual puzzle, however, when binocular complex cell receptive fields are considered instead, since a complex cell can respond to an oriented feature nearly independent of position within its monocular receptive field. Since complex cell receptive field diameters in the cat lie in the range of 1-3 degrees, the excessive "play" in their monocular receptive fields would seem to render complex cells incapable of signaling disparity on the much finer scale needed for depth perception (measured in minutes). Intriguingly, various authors have reported that a substantial fraction of complex cells in cat visual cortex are in fact tuned to left-right disparities much finer than that suggested by the size of the monocular RF's. For such cells, a stimulus delivered at the proper disparity, regardless of absolute position in either eye, produces a neural response in excess of that predicted by the sum of the monocular responses (Pettigrew, Nikara, & Bishop, 1968; Ohzawa, DeAngelis, & Freeman, 1990; Ohzawa et al., 1997). Binocular responses of this type suggest that for these cells, the left and right RF's are combined via a correlation operation rather than a simple sum (Nishihara & Poggio, 1984; Koch & Poggio, 1987). This computation has also been formalized in terms of an "energy" model (Ohzawa et al., 1990, 1997), building on the earlier use of energy models to account for complex cell orientation tuning (Pollen & Ronner, 1983) and direction selectivity (Adelson & Bergen, 1985). In an energy model for binocular disparity tuning, sums of linear Gabor filter outputs representing left and right receptive fields are squared to produce the crucial multiplicative cross terms (Ohzawa et al., 1990, 1997). Our previous biophysical modeling work has shown that the dendritic tree of a cortical pyramidal cells is well suited to support an approximative high-dimensional quadratic input-output relation, where the second-order multiplicative cross terms arise from local interactions among synaptic inputs carried out in quasi-isolated dendritic "subunits" (Mel, 1992b, 1992a, 1993). We recently applied these ideas to show that the position-invariant orientation tuning of a monocular complex cell could be computed within the dendrites of a single cortical cell, based exclusively upon excitatory inputs from a uniform, overlapping population of unoriented ON and OFF cells (Mel et al., 1997). Given the similarity of the "energy" formulations previously proposed to account for orientation tuning and binocu~ar disparity tuning, we hypothesized that a similar type of dendritic subunit computation could underlie disparity tuning in a binocularly driven complex cell. 210 B. W. Mel, D. L Ruderman and K A. Archie Parameter Value Rm IOkOcm:l Ra 2000cm em 1.0ILF/cm~ Vrest -70 mV Compartments 615 Somatic !iNa, YnR 0.20,0.12 S/cm:l Dendritic !iNa, YnR 0.05,0.03 S/cm:t. Input frequency 0- 100 Hz gAMPA 0.027 nS - 0.295 nS TAMPA (on, of f) 0.5 ms, 3 ms gNMDA 0.27 nS - 2.95 nS 7'NMDA (on, off) 0.5 ms, 50 ms Esyn OmV Table 1: Biophysical simulation parameters. Details of HH channel implementation are given elsewhere (Mel, 1993); original HH channel implementation courtesy Ojvind Bernander and Rodney Douglas. In order that local EPSP size be held approximately constant across the dendritic arbor, peak synaptic conductance at dendritic location x was approximately scaled to the local input resistance (inversely), given by 9syn(X) = C/Rin(X), where c was a constant, and Rin(X) = max(Rin(X),200MO). Input resistance Rin(X) was measured for a passive cell. Thus 9syn was identical for all dendritic sites with input resistance below 200MO, and was given by the larger conductance value shown; roughly 50% of the tree fell within a factor of 2 of this value. Peak conductances at the finest distal tips were smaller by roughly a factor of 10 (smaller number shown). Somatic input resistance was near 24MO. The peak synaptic conductance values used were such that the ratio of steady state current injection through NMDA vs. AMPA channels was 1.2±0.4. Both AMPA and NMDA-type synaptic conductances were modeled using the kinetic scheme of Destexhe et al. (1994); synaptic activation and inactivation time constants are shown for each. 2 Methods Compartmental simulations of a pyramidal cell from cat visual cortex (morphology courtesy of Rodney Douglas and Kevan Martin) were carried out in NEURON (Hines, 1989); simulation parameters are summarized in Table 1. The soma and dendritic membrane contained Hodgkin-Huxley-type (HH) voltage-dependent sodium and potassium channels. Following evidence for higher spike thresholds and decremental propagation in dendrites (Stuart & Sakmann, 1994), HH channel density was set to a uniform, 4-fold lower value in the dendritic membrane relative to that of the cell body. Excitatory synapses from LGN cells included both NMDA and AMPAtype synaptic conductances. Since the cell was considered to be isolated from the cortical network, inhibitory input was not modeled. Cortical cell responses were reported as average spike rate recorded at the cell body over the 500 ms stimulus period, excluding the 50 ms initial transient. The binocular LGN consisted of two copies of the monocular LGN model used previously (Mel et al., 1997), each consisting of a superimposed pair of 64x64 ON and OFF subfields. LGN cells were modeled as linear, half-rectified center-surround filters with centers 7 pixels in width. We randomly subsampled the left and right LGN arrays by a factor of 16 to yield 1,024 total LGN inputs to the pyramidal cell. A Single-Cell Account for Binocular Disparity Tuning 211 A developmental principle was used to determine the spatial arrangement of these 1,024 synaptic contacts onto the dendritic branches of the cortical cell, as follows. A virtual stimulus ensemble was defined for the cell, consisting of the complete set of single vertical light or dark bars presented binocularly at zero-disparity within the cell's receptive field. Within this ensemble, strong pairwise correlations existed among cells falling into vertically aligned groups of the same (ON or OFF) type, and cells in the vertical column at zero horizontal disparity in the other eye. These binocular cohorts of highly correlated LGN cells were labeled mutual "friends". Progressing through the dendritic tree in depth first order, a randomly chosen LG N cell was assigned to the first dendritic site. A randomly chosen "friend" of hers was assigned to the second site, the third site was assigned to a friend of the site 2 input, etc., until all friends in the available subsample were assigned (4 from each eye, on average). If the friends of the connection at site i were exhausted, a new LGN cell was chosen at random for site i + 1. In earlier work, this type of synaptic arrangement was shown to be the outcome of a Hebb-type correlational learning rule, in which random, activity independent formation of synaptic contacts acted to slowly randomize the axo-dendritic interface, shaped by Hebbian stabilization of synaptic contacts based on their short-range correlations with other synapses. 3 Results Model pyramidal cells configured in this way exhibited prominent phase-invariant orientation tuning, the hallmark response property of the visual complex cell. Multiple orientation tuning curves are shown, for example, for a monocular complex cell, giving rise to strong tuning for light and dark bars across the receptive field (fig. 1). The bold curve shows the average of all tuning curves for this cell; the half-width at half max is 25°, in the normal range for complex cells in cat visual cortex (Orban, 1984). When the spatial arrangement of LGN synaptic contacts onto the pyramidal cell dendrites was randomly scrambled, leaving all other model parameters unchanged, orientation tuning was abolished in this cell (right frame), confirming the crucial role of spatially-mediated nonlinear synaptic interactions (average curve from left frame is reproduced for comparison). Disparity-tuning in an orientation-tuned binocular model cell is shown in fig. 2, compared to data from a complex cell in cat visual cortex (adapted from Ohzawa et al. (1997)). Responses to contrast matched (light-light) and contrast non-matched (light-dark) bar pairs were subtracted to produce these plots. The strong diagonal structure indicates that both the model and real cells responded most vigorously when contrast-matched bars were presented at the same horizontal position in the left and right-eye RF's (Le. at zero-disparity), whereas peak responses to contrastnon-matched bars occured at symmetric near and far, non-zero disparities. 4 Discussion The response pattern illustrated in fig. 2A is highly similar to the response generated by an analytical binocular energy model for a complex cell (Ohzawa et al., 1997): {exp (-kXi) cos (271' f XL) + exp (-kX'kJ cos (271' f XR)}2 + {exp (-kxiJ sin (271' f XL) + exp (-kXh) sin (271' f XR)}2, (1) where XL and X R are the horizontal bar positions to the two eyes, k is the factor 212 70 60 '0 50 Ql ~ Ql 40 "'" '5. ~ .e Ql 30 (/) c: 8. 20 (/) Ql ex: 10 0 -90 Orientation Tuning average +lightO -+dark 4 -€Ilight 8 ,,*light16 .... dark 16 -ll-60 -30 0 30 60 90 Orientabon (degrees) U Ql (/) Us Ql "'" '5. .e Ql (/) c: 8. (/) Ql ex: B. W. Mel, D_ L Ruderman and K. A. Archie Ordered vs. Scrambled 55 50 ordered scrambled -+45 40 35 30 25 20 "' + , 15 /'--/ , ~ '+---- / 10 + I ' +_+- ~ 5 -90 -60 -30 0 30 60 90 Orientation (degrees) Figure 1: Orientation tuning curves are shown in the left frame for light and dark bars at 3 arbitrary positions_ Essentially similar responses were seen at other receptive field positions, and for other complex cells_ Bold trace indicates average of tuning curves at positions 0, 1, 2, 4, 8, and 16 for light and dark bars. Similar form of 6 curves shown reflects the translation-invariance of the cell's response to oriented stimuli, and symmetry with respect to ON and OFF input. Orientation tuning is eliminated when the spatial arrangement of LGN synapses onto the model cell dendrites is randomly scrambled (right frame). Complex Cell Model Right eye position Complex Cell in Cat VI Ohzawa, Deangelis, & Freeman, 1997 Right eye position Figure 2: Comparison of disparity tuning in model complex cell to that of a binocular complex cell from cat visual cortex. Light or dark bars were presented simultaneously to the left and right eyes. Bars could be of same polarity in both eyes (light, light) or different polarity (light, dark); cell responses for these two cases were subtracted to produce plot shown in left frame. Right frame shows data similarly displayed for a binocular complex cell in cat visual cortex (adapted from Ohzawa et al. (1997)). A Single-Cell Account for Binocular Disparity Tuning 213 that determines the width of the subunit RF's, and f is the spatial frequency. In lieu of literal simple cell "subunits" , the present results indicate that the subunit computations associated with the terms of an energy model could derive largely from synaptic interactions within the dendrites of the individual cortical cell, driven exclusively by excitatory inputs from unoriented, monocular ON and OFF cells drawn from a uniform overlapping spatial distribution. While lateral inhibition and excitation play numerous important roles in cortical computation, the present results suggest they are not essential for the basic features of the nonlinear disparity tuned responses of cortical complex cells. Further, these results address the paradox as to how inputs from both unoriented LGN cells and oriented simple cells can coexist without conflict within the dendrites of a single complex cell. A number of controls from previous work suggest that this type of subunit processing is very robustly computed in the dendrites of an individual neuron, with little sensitivity to biophysical parameters and modeling assumptions, including details of the algorithm used to spatially organize the genicula-cortical projection, specifics of cell morphology, synaptic activation density across the dendritic tree, passive membrane and cytoplasmic parameters, and details of the kinetics, voltage-dependence, or spatial distribution of the voltage-dependent dendritic channels. One important difference between a standard energy model and the intradendritic responses generated in the present simulation experiments is that the energy model has oriented RF structure at the linear (simple-cell-like) stage, giving rise to oriented, antagonistic ON-OFF subregions (Movshon, Thompson, & Tolhurst, 1978), whereas the linear stage in our model gives rise to center-surround antagonism only within individual LGN receptive fields. Put another way, the LGN-derived subunits in the present model cannot provide all the negative cross-terms that appear in the energy model equations, specifically for pairs of pixels that fall outside the range of a single LG N receptive field. While the present simulations involve numerous simplifications relative to the full complexity of the cortical microcircuit, the results nonetheless emphasize the potential importance of intradendritic computation in visual cortex. Acknowledgements Thanks to Ken Miller, Allan Dobbins, and Christof Koch for many helpful comments on this work. This work was funded by the National Science Foundation and the Office of Naval Research, and by a Slo~n Foundation Fellowship (D.R.). References Adelson, E., & Bergen, J. (1985). Spatiotemporal energy models for the perception of motion. J. Opt. Soc. Amer., A 2, 284-299. Rines, M. (1989). A program for simulation of nerve equations with branching geometries. Int. J. Biomed. Comput., 24, 55-68. Rubel, D., & Wiesel, T. (1962) . Receptive fields, binocular interaction and functional architecture in the cat's visual cortex. J. Physiol., 160, 106- 154. Koch, C., & Poggio, T . (1987) . Biophysics of computation: Neurons, synapses, and membranes. In Edelman, G., Gall, W., & Cowan, W. (Eds.), Synaptic junction, pp. 637-697. Wiley, New York. Mel, B. (1992a). The clusteron: Toward a simple abstraction for a complex neuron. In Moody, J., Hanson, S., & Lippmann, R. (Eds.), Advances in Neural 214 B. W. Mel, D. L Ruderman and K. A Archie Information Processing Systems, vol. 4, pp. 35-42. Morgan Kaufmann, San Mateo, CA. Mel, B. (1992b). NMDA-based pattern discrimination in a modeled cortical neuron. Neural Computation, 4, 502-516. Mel, B. (1993). Synaptic integration in an excitable dendritic tree. J. Neurophysiol., 70(3), 1086-110l. Mel, B., Ruderman, D., & Archie, K. (1997). Complex-cell responses derived from center-surround inputs: the surprising power of intradendritic computation. In Mozer, M., Jordan, M., & Petsche, T. (Eds.), Advances in Neural Information Processing Systems, Vol. 9, pp. 83-89. MIT Press, Cambridge, MA. Movshon, J., Thompson, I., & Tolhurst, D. (1978). Receptive field organization of complex cells in the cat's striate cortex. J. Physiol., 283, 79-99. Nishihara, H., & Poggio, T. (1984). Stereo vision for robotics. In Brady, & Paul (Eds.), Proceedings of the First International Symposium of Robotics Research, pp. 489-505. MIT Press, Cambridge, MA. Ohzawa, I., DeAngelis, G., & Freeman, R. (1990). Stereoscopic depth discrimination in the visual cortex: Neurons ideally suited as disparity detectors. Science, 249, 1037- 104l. Ohzawa, I., DeAngelis, G., & Freeman, R. (1997). Encoding of binocular disparity by complex cells in the cat's visual cortex. J. Neurophysiol., June. Orban, G. (1984). Neuronal operations in the visual cortex. Springer Verlag, New York. Pettigrew, J., Nikara, T., & Bishop, P. (1968). Responses to moving slits by single units in cat striate cortex. Exp. Brain Res., 6, 373-390. Pollen, D., & Ronner, S. (1983). Visual cortical neurons as localized spatial frequency filters. IEEE Trans. Sys. Man Cybero., 13, 907-916. Stuart, G., & Sakmann, B. (1994). Active propagation of somatic action potentials into neocortical pyramidal cell dendrites. Nature, 367, 69-72.
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Recurrent Neural Networks Can Learn to Implement Symbol-Sensitive Counting Paul Rodriguez Department of Cognitive Science University of California, San Diego La Jolla, CA. 92093 prodrigu@cogsci.ucsd.edu Janet Wiles School of Information Technology and Department of Psychology University of Queensland Brisbane, Queensland 4072 Australia janetw@it.uq.edu.au Abstract Recently researchers have derived formal complexity analysis of analog computation in the setting of discrete-time dynamical systems. As an empirical constrast, training recurrent neural networks (RNNs) produces self -organized systems that are realizations of analog mechanisms. Previous work showed that a RNN can learn to process a simple context-free language (CFL) by counting. Herein, we extend that work to show that a RNN can learn a harder CFL, a simple palindrome, by organizing its resources into a symbol-sensitive counting solution, and we provide a dynamical systems analysis which demonstrates how the network: can not only count, but also copy and store counting infonnation. 1 INTRODUCTION Several researchers have recently derived results in analog computation theory in the setting of discrete-time dynamical systems(Siegelmann, 1994; Maass & Opren, 1997; Moore, 1996; Casey, 1996). For example, a dynamical recognizer (DR) is a discrete-time continuous dynamical system with a given initial starting point and a finite set of Boolean output decision functions(pollack. 1991; Moore, 1996; see also Siegelmann, 1993). The dynamical system is composed of a space,~n , an alphabet A, a set of functions (1 per element of A) that each maps ~n -+ ~n and an accepting region H lie, in ~n. With enough precision and appropriate differential equations, DRs can use real-valued variables to encode contents of a stack or counter (for details see Siegelmann, 1994; Moore, 1996). As an empirical contrast, training recurrent neural networks (RNNs) produces selforganized implementations of analog mechanisms. In previous work we showed that an RNN can learn to process a simple context-free language, anbn , by organizing its resources into a counter which is similar to hand-coded dynamical recognizers but also exhibits some 88 P. Rodriguez and J. Wiles novelties (Wlles & Elman, 1995). In particular, similar to band-coded counters, the network developed proportional contracting and expanding rates and precision matters - but unexpectedly the network distributed the contraction/expansion axis among hidden units, developed a saddle point to transition between the first half and second half of a string, and used oscillating dynamics as a way to visit regions of the phase space around the fixed points. In this work we show that an RNN can implement a solution for a harder CFL, a simple palindrome language(desaibed below), which requires a symbol-sensitive counting solution. We provide a dynamical systems analysis which demonstrates how the network can not only count, but also copy and store counting information implicitly in space around a fixed point. 2 TRAINING an RNN TO PROCESS CFLs We use a discrete-time RNN that has 1 hidden layer with recurreot connections, and 1 output layer withoutrecurreot connections so that the accepting regions are determined by the output units. The RNN processes output in Tune(n), where n is the length of the input, and it can recognize languages that are a proper subset of context-sensitive languages and a proper superset of regular languages(Moore, 1996). Consequently, the RNN we investigate can in principle embody the computational power needed to process self-recursion. Furthermore, many connectionist models of language processing have used a prediction task(e.g. Elman, 1990). Hence, we trained an RNN to be a real-time transducer version of a dynamical recognizer that predicts the next input in a sequence. Although the network does not explicitly accept or reject strings, if our network makes all the right predictions possible then perlorming the prediction task subsumes the accept task, and in principle one could simply reject unmatched predictions. We used a threshbold criterion of .5 such that if an ouput node has a value greater than .5 then the network is considered to be making that prediction. If the network makes all the right predictions possible for some input string, then it is correctly processing that string. Although a finite dimensional RNN cannot process CFLs robustly with a margin for error (e.g.Casey, 1996;Maass and Orponen,I997), we will show that it can acquire the right kind of trajectory to process the language in a way that generalizes to longer strings. 2.1 A SIMPLE PALINDROME LANGUAGE A palindrome language (mirror language) consists of a set of strings, S, such that each string, 8 eS, 8 = wwr , is a concatenation of a substring, w, and its reverse, wr • The relevant aspect of this language is that a mechanism cannot use a simple counter to process the string but must use the functional equivalent of a stack that enables it to match the symbols in second half of the string with the first half. We investigated a palindrome language that uses only two symbols for w, two other symboIs for w r , such that the second half of the string is fully predictable once the change in symbols occurs. The language we used is a simple version restricted such that one symbol is always present and precedes the other, for example: w = anbm , wr = Bm An, e.g. aaaabbbBBBAAAA, (where n > 0, m >= 0). Note that the embedded subsequence bm B m is just the simple-CFL used in Wlles & Elman (1995) as mentioned above, hence, one can reasonably expect that a solution to this task has an embedded counter for the subsequence b ... B. 202 LINEAR SYSTEM COUNTERS A basic counter in analog computation theory uses real-valued precision (e.g. Siegelman 1994; Moore 1996). For example, a l-dimensional up/down counter for two symbols { a I b} RNNs Can Learn Symbol-Sensitive Counting 89 is the system J(z) = .5z + .5a, J(z) = 2z - .5b where z is the state variable, a is the input variable to count up(push), and b is the variable to count down(pop). A sequence of input aaabbb has state values(starting at 0): .5,.75,.875, .75,.5,0. Similarly, for our transducer version one can develop piecewise linear system equations in which counting takes place along different dimensions so that different predictions can be made at appropriate time stepSI. The linear system serves as a hypothesis before running any simulations to understand the implementation issues for an RNN. For example, using the function J(z) = z for z E [0,1], ° for z < 0, 1 for z > 1, then for the simple palindrome task one can explicitly encode a mechanism to copy and store the count for a across the b ... B subsequences. If we assign dimension-l to a, dimension-2 to b, dimension-3 to A, dimension-4 to B, and dimension-5 to store the a value, we can build a system so that for a sequence aaabbBBAAA we get state variables values: initial, (0,0,0,0,0), (.5,0,0,0,0), (.75,0,0,0,0), (.875,0,0,0,0), (0,.5,0,0,.875), (0,.75,0,0,.875), (0,0,0,.5,.875), (0,0,0,0,.875), (0,0,.75,0,0), (0,0,.5,0,0), (0,0,0,0,0). The matrix equations for such a system could be: Xt = J( [~5 .~ ~ ~ ~ 1 * Xt- 1 + [~5 1 ~l ::: 1 * It} o 2 ° 2 ° ° -5 ° -1 1 ° ° ° 1 -5 ° -5 ° where t is time, X t is the 5-dimensional state vector, It is the 4-dimensional input vector using l-hotencodingofa = [1,0,0,0];6 = [O,I,O,O];A = [O,O,I,O],B = [0,0,0,1]. The simple trick is to use the input weights to turn on or off the counting. For example, the dimension-5 state variable is turned off when input is a or A, but then turned on when b is input, at which time it copies the last a value and holds on to it. It is then easy to add Boolean output decision functions that keep predictions linearly separable. However, other solutions are possible. Rather than store the a count one could keep counting up in dimension-l for b input and then cancel it by counting down for B input. The questions that arise are: Can an RNN implement a solution that generalizes? What kind of store and copy mechanism does an RNN discover? 1.3 TRAINING DATA & RESULTS The training set consists of 68 possible strings of total length $ 25, which means a maximum of n + m = 12, or 12 symbols in the first half, 12 symbols in the second half, and 1 end symbol 2. The complete training set has more short strings so that the network does not disregard the transitions at the end of the string or at the end of the b ... B subsequence. The network consists of 5 input, 5 hidden, 5 output units, with a bias node. The hidden and recurrent units are updated in the same time step as the input is presented. The recurrent layer activations are input on the next time step. The weight updates are performed using back-propagation thru time training with error injected at each time step backward for 24 time stepS for each input. We found that about half our simulations learn to make predictions for transitions, and most will have few generalizations on longer strings not seen in the training set. However, no network learned the complete training set perfectly. The best network was trained for 250K sweeps (1 per character) with a learning parameter of .001, and 136K more sweeps with .0001, for a total of about 51K strings. The network made 28 total prediction errors on 28 l1bese can be expanded relatively easily to include more symbols, different symbol representations, harder palindrome sequences, or different kind of decision planes. 2We removed training strings w = a"b,for n > 1; it turns out that the network interpolates on the B-to-A transition for these. Also, we added an end symbol to help reset the system to a consistent starting value. 90 P. Rodriguez and 1. Wiles different strings in the test set of 68 possible strings seen in training. All of these errors were isolated to 3 situations: when the number of a input = 2or4 the error occurred at the B-toA transition, when the number of a input = 1, for m > 2, the error occurred as an early A-to-end transition. Importantly, the networlcmade correct predictions on many strings longer than seen in training, e.g. strings that have total length > 25 (or n + m > 12). It counted longer strings of a .. As with or without embedded b .. Bs; such as: w = a13 ; w = a13b2 ; w = anb7 , n = 6, 7or8 (recall that w is the first half of the string). It also generalized to count longer subsequences ofb .. Bs withorwithoutmorea .. As; suchasw = a5h'" where n = 8,9,10,11,12. The longest string it processed correctly was w = a9 b9 , which is 12 more characters than seen during training. The network learned to store the count for a 9 for up to 9bs, even though the longest example it had seen in training had only 3bs - clearly it's doing something right. 2.4 NETWORK EVALUATION Our evaluation will focus on how the best network counts, copies, and stores information. We use a mix of graphical analysis and linear system analysis, to piece together a global picture of how phase space trajectories hold informational states. The linear system analysis consists of investigating the local behaviour of the Jacobian at fixed points under each input condition separately. We refer to Fa as the autonomous system under a input condition and similarly for Fb, FA, and FB. The most salient aspect to the solution is that the network divides up the processing along different dimensions in space. By inspection we note that hidden unitl (HUI) takes on low values for the first half of the string and high values for the second half, which helps keep the processing linearly separable. Therefore in the graphical analysis of the RNN we can set HUI to a constant. FIrst, we can evaluate how the network counts the b .. B subsequences. Again, by inspection the network uses dimensions HU3,HU4. The graphical analysis in FIgure Ia and Figure Ib plots the activity ofHU3xHU4. It shows how the network counts the right number of Bs and then makes a transition to predict the first A. The dominant eigenvalues at the Fb attracting point and F B saddle point are inversely proportional, which indicates that the contraction rate to and expansion rate away from the fixed points are inversely matched. The FB system expands out to a periodic-2 fixed point in HU3xHU4 subspace, and the unstable eigenvector corresponding to the one unstable eigenvalue has components only in HU3,HU4. In Fi~ure 2 we plot the vector field that describes the flow in phase space for the composite F B' which shows the direction where the system contracts along the stable manifold, and expands on the unstable manifold. One can see that the nature of the transition after the last b to the first B is to place the state vector close to saddle point for FB so that the number of expansion steps matches the number of the Fb contraction steps. In this way the b count is copied over to a different region of phase space. Now we evaluate how the network counts a ... A, first without any b ... B embedding. Since the output unit for the end symbol bas very high weight values for HU2, and the Fa system bas little activity in HU4, we note that a is processed in HU2xHU3xHU5. The trajectories in Figure 3 show a plot of a 13 A 13 that properly predicts all As as well as the transition at the end. Furthermore, the dominant eigenvalues for the Fa attracting point and the FA saddle point are nearly inversely proportional and the FA system expands to a periodic-2 fixed point in 4-dimensions (HUI is constant, whereas the other HU values are periodic). The Fa eigenvectors have strong-moderate components in dimensions HU2, HU3, HU5; and likewise in HU2, HU3, HU4, HU5 for FA. The much harder question is: How does the network maintain the information about the count of as that were input while it is processing the b .. B subsequence? Inspection shows RNNs Can Learn Symbol-Sensitive Counting 91 that after processing an the activation values are not directly copied over any HU values, nor do they latch any HU values that indicate how many as were processed. Instead, the last state value after the last a affects the dynamics for b ... B in such a way that clusters the last state value after the last B, but only in HU3xHU4 space (since the other HU dimensions were unchanging throughout b ... B processing). We show in Figure 4 the clusters for state variables in HU3xHU4 space after processing an bm B m , where n = 2,3,4, 50r6; m = 1 .. 10. The graph shows that the information about how many a's occurred is "stored" in the HU3xHU4 region where points are clustered. Figure 4 includes the dividing line from Figure Ib for the predict A region. The network does not predict the B-to-A transition after a4 or a2 because it ends up on the wrong side of the dividing line of Figure Ib, but the network in these cases still predicts the A-to-end transition. We see that if the network did not oscillate around the F B saddle point while exanding then the trajectory would end up correctly on one side of the decision plane. It is important to see that the clusters themselves in Figure 4 are on a contracting trajectory toward a fixed point, which stores information about increasing number of as when matched by an expansion of the FA system. For example, the state values after a5 AA and a5bm Bm AA, m = 2 .. 10 have a total hamming distance for all 5 dimensions that ranged from .070 to .079. Also, the fixed point for the Fa. system, the estimated fixed point for the composite F'B 0 Fb 0 F;: , and the saddle point of the FA system are colinear 3. in all the relevant counting dimensions: 2,3,4, and 5. In other words, the FA system contracts the different coordinate points, one for an and one for anbm B m, towards the saddle point to nearly the same location in phase space, treating those points as having the same information. Unfortunately, this is a contraction occuring through a 4 dimensional subspace which we cannot easily show grapbically. 3 CONCLUSION In conclusion, we have shown that an RNN can develop a symbol-sensitive counting s0lution for a simple palindrome. In fact, this solution is not a stack but consists of nonindependent counters that use dynamics to visit different regions at appropriate times. Furthermore, an RNN can implement counting solutions for a prediction task that are functionally similar to that prescribed by analog computation theory, but the store and copy functions rely on distance in phase space to implicitly affect other trajectories. Acknowledgements This research was funded by the UCSD, Center for Research in Language Training Grant to Paul Rodriguez, and a grant from the Australian Research Council to Janet Wlles. References Casey, M. (1996) The Dynamics of Discrete-TIme Computation, With Application to Recurrent Neural Networks and Fmite State Machine Extraction. Neural Computation, 8. Elman, JL. (1990) Finding Structure in TIme. Cognitive Science, 14, 179-211. Maass, W. ,Orponen, P. (1997) On the Effect of Analog Noise in Discrete-TIme Analog Computations. Proceedings Neural Information Processing Systems, 1996. Moore, C. (1996) Dynamical Recognizers: Real-TIme Language Recognition by Analog Computation. Santa Fe InstituteWorking Paper 96-05-023. )Relative to the saddle point, the vector for one fixed point, multiplied by a constant had the same value(to within .OS) in each of 4 dimensions as the vector for the other fixed point 92 HU4 1 0.8 0.6 0 . 4 0.2 o o p~ct b regiOD 0.2 0.4 0.6 0.8 HU4 1 0.8 0 . 6 0 . 4 p~ctB ~giOD P. Rodriguez and 1. Wiles 0.2 ,--", o o~~':"'o .~2--0:-.~4 -"":0-. 6=----=-0"": . 8:---~1 HU3 Figure 1: la)Trajectory of 610 (after 0 5) in HU3xHU4. Each arrow represents a trajectory step:the base is a state vector at time t, the head is a state at time t + 1. The first b trajectory step has a base near (.9,.05), which is the previous state from the last o. The output node b is> .5 above the dividing line. Ib) Trajectory of BI0 (after 05b10) in HU3xHU4. The output node B is > .5 above the dashed dividing line, and the output node A is > .5 below the solid dividing line. The system crosses the line on the last B step, hence it predicts the B-to-A transitioo. Pollack, J.B. (1991) The Induction of Dynamical Recognizers. Machine Learning, 7, 227252. Siegelmann, H.(1993) Foundations of Recurrent Neural Networks. PhD. dissertation, unpublished. New Brunswick Rutgers, The State University of New Jersey. Wtles, 1., Elman, J. (1995) Counting Without a Counter: A Case Study in Activation Dynamics. Proceedings of the Seventeenth Annual Conference of the Cognitive Science Society. Hillsdale, N J .: Lawrence Erlbaum Associates. HU4 0 . 8 \ \ ", ....... _-..,.-/ 0.6 t , \ , .... ,. " / t \ \ \ ~ .. , ~ " / , l ~ , , I , " I I ~ , ... • , , I ~ Figure 2: Vector field that desaibes the flow of Fj projected onto HU3xHU4. The graph shows a saddle point near (.5,.5)and a periodic-2 attracting point. RNNs Can Learn Symbol-Sensitive Counting 93 0 . 4 predict II reaioo 0.2 o 0---0.-2--0-.-4 --0-.-6 --0-.8--1 802 803 1 I I , I , 0.8 I I 0.6 I 0.4 , I I I I o .2 predict eDdrepOIl " I I I o I 0~-0~.~2--0~.74-~0~.~6-~0~.8~~lmn Figure 3: 3a) Trajectory of a 13 projected onto HU2xHU3. The output node a is> .5 below and right of dividing line. The projection for HU2xHU5 is very similar. 3b) Trajectory of A 13 (after a 13) projected onto HU2xHU3. The output node for the end symbol is > .5 on the 13th trajectory step left of the solid dividing line, and it is > .5 on the 11th step left of the dashed dividing line (the hyperplane projection must use values at the appropriate time steps), hence the system predicts the A-to-end transition. The graph for HU2xHU5 and HU2xHU4 is very similar. IIU4 " II J/ " 41 fJIJ fJIJ fJIJ fJIJ lib u 1.0 fJIJ 0.9 0.8 0. 7 z •• fJIJ. , • 0.1 L.-__ --::---:--_:-=-~----:---'':-.:....:-::--_:_::___:_" HU3 0.2 0.3 0 . 4 0. 5 0 . 6 0.7 0.8 0.9 1.0 Figure4: Clusters of lasts tate values anbm Bm, m > 1, projected onto HU3xHU4. Notice that for increasing n the system oscillates toward an attracting point of the system F'B 0 Ft:0F:;.
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Modeling acoustic correlations by factor analysis Lawrence Saul and Mazin Rahim {lsaul.mazin}~research.att.com AT&T Labs Research 180 Park Ave, D-130 Florham Park, NJ 07932 Abstract Hidden Markov models (HMMs) for automatic speech recognition rely on high dimensional feature vectors to summarize the shorttime properties of speech. Correlations between features can arise when the speech signal is non-stationary or corrupted by noise. We investigate how to model these correlations using factor analysis, a statistical method for dimensionality reduction. Factor analysis uses a small number of parameters to model the covariance structure of high dimensional data. These parameters are estimated by an Expectation-Maximization (EM) algorithm that can be embedded in the training procedures for HMMs. We evaluate the combined use of mixture densities and factor analysis in HMMs that recognize alphanumeric strings. Holding the total number of parameters fixed, we find that these methods, properly combined, yield better models than either method on its own. 1 Introduction Hidden Markov models (HMMs) for automatic speech recognition[l] rely on high dimensional feature vectors to summarize the short-time, acoustic properties of speech. Though front-ends vary from recognizer to recognizer, the spectral information in each frame of speech is typically codified in a feature vector with thirty or more dimensions. In most systems, these vectors are conditionally modeled by mixtures of Gaussian probability density functions (PDFs). In this case, the correlations between different features are represented in two ways[2]: implicitly by the use of two or more mixture components, and explicitly by the non-diagonal elements in each covariance matrix. Naturally, these strategies for modeling correlationsimplicit versus explicit-involve tradeoffs in accuracy, speed, and memory. This paper examines these tradeoffs using the statistical method of factor analysis. 750 L Saul and M Rahim The present work is motivated by the following observation. Currently, most HMMbased recognizers do not include any explicit modeling of correlations; that is to say-conditioned on the hidden states, acoustic features are modeled by mixtures of Gaussian PDFs with diagonal covariance matrices. The reasons for this practice are well known. The use offull covariance matrices imposes a heavy computational burden, making it difficult to achieve real-time recognition. Moreover, one rarely has enough data to (reliably) estimate full covariance matrices. Some of these disadvantages can be overcome by parameter-tying[3]-e.g., sharing the covariance matrices across different states or models. But parameter-tying has its own drawbacks: it considerably complicates the training procedure, and it requires some artistry to know which states should and should not be tied. Unconstrained and diagonal covariance matrices clearly represent two extreme choices for the hidden Markov modeling of speech. The statistical method of factor analysis[4,5] represents a compromise between these two extremes. The idea behind factor analysis is to map systematic variations of the data into a lower dimensional subspace. This enables one to represent, in a very compact way, the covariance matrices for high dimensional data. These matrices are expressed in terms of a small number of parameters that model the most significant correlations without incurring much overhead in time or memory. Maximum likelihood estimates of these parameters are obtained by an Expectation-Maximization (EM) algorithm that can be embedded in the training procedures for HMMs. In this paper we investigate the use of factor analysis in continuous density HMMs. Applying factor analysis at the state and mixture component level[6, 7] results in a powerful form of dimensionality reduction, one tailored to the local properties of speech. Briefly, the organization of this paper is as follows. In section 2, we review the method of factor analysis and describe what makes it attractive for large problems in speech recognition. In section 3, we report experiments on the speakerindependent recognition of connected alpha-digits. Finally, in section 4, we present our conclusions as well as ideas for future research. 2 Factor analysis Factor analysis is a linear method for dimensionality reduction of Gaussian random variables[4, 5]. Many forms of dimensionality reduction (including those implemented as neural networks) can be understood as variants of factor analysis. There are particularly close ties to methods based on principal components analysis (PCA) and the notion of tangent distance[8]. The combined use of mixture densities and factor analysis-resulting in a non-linear form of dimensionality reduction-was first applied by Hinton et al[6] to the modeling of handwritten digits. The EM procedure for mixtures of factor analyzers was subsequently derived by Ghahramani et al[7]. Below we describe the method offactor analysis for Gaussian random variables, then show how it can be applied to the hidden Markov modeling of speech. 2.1 Gaussian model Let x E nP denote a high dimensional Gaussian random variable. For simplicity, we will assume that x has zero mean. If the number of dimensions, D, is very large, it may be prohibitively expensive to estimate, store, multiply, or invert a full covariance matrix. The idea behind factor analysis is to find a subspace of much lower dimension, f « D, that captures most of the variations in x. To this end, let z E 'RJ denote a low dimensional Gaussian random variable with zero mean and Modeling Acoustic Correlations by Factor Analysis 751 identity covariance matrix: (1) We now imagine that the variable x is generated by a random process in which z is a latent (or hidden) variable; the elements of z are known as the factors. Let A denote an arbitrary D x f matrix, and let '11 denote a diagonal, positive-definite D x D matrix. We imagine that x is generated by sampling z from eq. (1), computing the D-dime.nsional vector Az, then adding independent Gaussian noise (with variances Wii) to each component of this vector. The matrix A is known as the factor loading matrix. The relation between x and z is captured by the conditional distribution: P(xlz) = 1'111- 1/ 2 e- HX-AZ)TI)-l(X-AZ) (211")D/2 (2) The marginal distribution for x is found by integrating out the hidden variable z. The calculation is straightforward because both P(z) and P(xlz) are Gaussian: P(x) = J dz P(xlz)P(z) (3) I'll + AAT I- 1/ 2 -!XT(I)+AATf1x (211")D/2 e (4) From eq. (4), we see that x is normally distributed with mean zero and covariance matrix '11 + AAT. It follows that when the diagonal elements ofw are small, most of the variation in x occurs in the subspace spanned by the columns of A. The variances Wii measure the typical size of componentwise ftuctations outside this subspace. Covariance matrices of the form '11 + AAT have a number of useful properties. Most importantly, they are expressed in terms of a small number of parameters, namely the D(f + 1) non-zero elements of A and W. If f ~ D, then storing A and '11 requires much less memory than storing a full covariance matrix. Likewise, estimating A and '11 also requires much less data than estimating a full covariance matrix. Covariance matrices of this form can be efficiently inverted using the matrix inversion lemma[9], (5) where I is the f x f identity matrix. This decomposition also allows one to compute the probability P(x) with only O(fD) multiplies, as opposed to the O(D2) multiplies that are normally required when the covariance matrix is non-diagonal. Maximum likelihood estimates of the parameters A and '11 are obtained by an EM procedure[4]. Let {xt} denote a sample of data points (with mean zero). The EM procedure is an iterative procedure for maximizing the log-likelihood, Lt In P(xt}, with P(Xt) given by eq. (4). The E-step of this procedure is to compute: Q(A', '11'; A, '11) = 'LJdz P(zIXt,A, w)lnP(z,xtIA', '11'). (6) t The right hand side of eq. (6) depends on A and '11 through the statistics[7]: E[zlxtl [I + AT w- 1 A]-lATw-1Xt, (7) E[zzT lxtl = [I + AT W- 1 A]-l + E[zlxtlE[zTlxtl. (8) Here, E['lxtl denotes an average with respect to the posterior distribution, P(zlxt, A, '11). The M-step of the EM algorithm is to maximize the right hand 752 L. Saul and M. Rahim side of eq. (6) with respect to'll' and A'. This leads to the iterative updates[7]: A' (~X'E[ZT IX,]) (~E[zzTIX,]) -1 (9) '11' diag { ~ ~ [x,x; - A'E[zlx,]xiJ }, (10) where N is the number of data points, and'll' is constrained to be purely diagonal. These updates are guaranteed to converge monotonically to a (possibly local) maximum of the log-likelihood. 2.2 Hidden Markov modeling of speech Consider a continuous density HMM whose feature vectors, conditioned on the hidden states, are modeled by mixtures of Gaussian PDFs. If the dimensionality of the feature space is very large, we can make use of the parameterization in eq. (4). Each mixture component thus obtains its own means, variances, and factor loading matrix. Taken together, these amount to a total of C(f + 2)D parameters per mixture model, where C is the number of mixture components, f the number of factors, and D the dimensionality of the feature space. Note that these models capture feature correlations in two ways: implicitly, by using two or more mixture components, and explicitly, by using one or more factors. Intuitively, one expects the mixture components to model discrete types of variability (e.g., whether the speaker is male or female), and the factors to model continuous types of variability (e.g., due to coarticulation or noise). Both types of variability are important for building accurate models of speech. It is straightforward to integrate the EM algorithm for factor analysis into the training of HMMs. Suppose that S = {xtl represents a sequence of acoustic vectors. The forward-backward procedure enables one to compute the posterior probability, ,tC = P(St = s, Ct = ciS), that the HMM used state s and mixture component cat time t. The updates for the matrices A $C and w3C (within each state and mixture component) have essentially the same form as eqs. (9-10), except that now each observation Xt is weighted by the posterior probability, ,tc . Additionally, one must take into account that the mixture components have non-zero means[7]. A complete derivation of these updates (along with many additional details) will be given in a longer version of this paper. Clearly, an important consideration when applying factor analysis to speech is the choice of acoustic features. A standard choice--and the one we use in our experiments-is a thirty-nine dimensional feature vector that consists of twelve cepstral coefficients (with first and second derivatives) and the normalized log-energy (with first and second derivatives). There are known to be correlations[2] between these features, especially between the different types of coefficients (e.g., cepstrum and delta-cepstrum). While these correlations have motivated our use of factor analysis, it is worth emphasizing that the method applies to arbitrary feature vectors. Indeed, whatever features are used to summarize the short-time properties of speech, one expects correlations to arise from coarticulation, background noise, speaker idiosynchrasies, etc. 3 Experiments Continuous density HMMs with diagonal and factored covariance matrices were trained to recognize alphanumeric strings (e.g., N Z 3 V J 4 E 3 U 2). Highly Modeling Acoustic Correlations by FactorAnalysis alpha-digits (ML) 37 3~~--5~--1~0---1~5---2~0--~25~~30--~ parameters 17 16 15 #: ;14 !! g13 II> 'E 12 ~ 11 10 90 753 alpha-dig~s (ML) \ . , . '0. 5 10 15 20 25 30 parameters Figure 1: Plots of log-likelihood scores and word error rates on the test set versus the number of parameters per mixture model (divided by the number of features). The stars indicate models with diagonal covariance matrices; the circles indicate models with factor analysis. The dashed lines connect the recognizers in table 2. confusable letters such as BjV, CjZ, and MjN make this a challenging problem in speech recognition. The training and test data were recorded over a telephone network and consisted of 14622 and 7255 utterances, respectively. Recognizers were built from 285 left-to-right HMMs trained by maximum likelihood estimation; each HMM modeled a context-dependent sub-word unit. Testing was done with a free grammar network (i.e., no grammar constraints). We ran several experiments, varying both the number of mixture components and the number of factors. The goal was to determine the best model of acoustic feature correlations. Table 1 summarizes the results of these experiments. The columns from left to right show the number of mixture components, the number of factors, the number of parameters per mixture model (divided by the feature dimension), the word error rates (including insertion, deletion, and substition errors) on the test set, the average log-likelihood per frame of speech on the test set, and the CPU time to recognize twenty test utterances (on an SGI R4000). Not surprisingly, the word accuracies and likelihood scores increase with the number of modeling parameters; likewise, so do the CPU times. The most interesting comparisons are between models with the same number of parameters-e.g., four mixture components with no factors versus two mixture components with two factors. The left graph in figure 1 shows a plot of the average log-likelihood versus the number of parameters per mixture model; the stars and circles in this plot indicate models with and without diagonal covariance matrices. One sees quite clearly from this plot that given a fixed number of parameters, models with non-diagonal (factored) covariance matrices tend to have higher likelihoods. The right graph in figure 1 shows a similar plot of the word error rates versus the number of parameters. Here one does not see much difference; presumably, because HMMs are such poor models of speech to begin with, higher likelihoods do not necessarily translate into lower error rates. We will return to this point later. It is worth noting that the above experiments used a fixed number of factors per mixture component. In fact , because the variability of speech is highly contextdependent, it makes sense to vary the number of factors, even across states within the same HMM. A simple heuristic is to adjust the number of factors depending on the amount of training data for each state (as determined by an initial segmentation of the training utterances). We found that this heuristic led to more pronounced 754 L Saul and M Rahim C f C(f + 2) word error (%) log-likelihood CPU time (sec) 1 0 2 16.2 32.9 25 1 1 3 14.6 34.2 30 1 2 4 13.7 34.9 30 1 3 5 13.0 35.3 38 1 4 6 12.5 35.8 39 2 0 4 13.4 34.0 30 2 1 6 12.0 35.1 44 2 2 8 11.4 35.8 48 2 3 10 10.9 36.2 61 2 4 12 10.8 36.6 67 4 0 8 11.5 34.9 46 4 1 12 10.4 35.9 80 4 2 16 10.1 36.5 93 4 3 20 10.0 36.9 132 4 4 24 9.8 37.3 153 8 0 16 10.2 35.6 93 8 1 24 9.7 36.5 179 8 2 32 9.6 37.0 226 16 0 32 9.5 36.2 222 Table 1: Results for different recogmzers. The columns indicate the number of mixture components, the number of factors, the number of parameters per mixture model (divided by the number of features), the word error rates and average loglikelihood scores on the test set, and the CPU time to recognize twenty utterances. C f C(f + 2) word error J % J log-likelihood CPU time Jse<J 1 2 4 12.3 35.4 32 2 2 8 10.5 36.3 53 4 2 16 9.6 37.0 108 Table 2: Results for recognizers with variable numbers of factors; f denotes the average number of factors per mixture component. differences in likelihood scores and error rates. In particular, substantial improvements were observed for three recognizers whose HMMs employed an average of two factors per mixture component; see the dashed lines in figure 1. Table 2 summarizes these results. The reader will notice that these recognizers are extremely competitive in all aspects of performance--accuracy, memory, and speed-with the baseline (zero factor) models in table 1. 4 Discussion In this paper we have studied the combined use of mixture densities and factor analysis for speech recognition. This was done in the framework of hidden Markov modeling, where acoustic features are conditionally modeled by mixtures of Gaussian PDFs. We have shown that mixture densities and factor analysis are complementary means of modeling acoustic correlations. Moreover, when used together, they can lead to smaller, faster, and more accurate recognizers than either method on its own. (Compare the last lines of tables 1 and 2.) Modeling Acoustic Correlations by Factor Analysis 755 Several issues deserve further investigation. First, we have seen that increases in likelihood scores do not always correspond to reductions in error rates. (This is a common occurrence in automatic speech recognition.) We are currently investigating discriminative methods[lO] for training HMMs with factor analysis; the idea here is to optimize an objective function that more directly relates to the goal of minimizing classification errors. Second, it is important to extend our results to large vocabulary tasks in speech recognition. The extreme sparseness of data in these tasks makes factor analysis an appealing strategy for dimensionality reduction. Finally, there are other questions that need to be answered. Given a limited number of parameters, what is the best way to allocate them among factors and mixture components? Do the cepstral features used by HMMs throwaway informative correlations in the speech signal? Could such correlations be better modeled by factor analysis? Answers to these questions can only lead to further improvements in overall performance. Acknowledgements We are grateful to A. Ljolje (AT&T Labs), Z. Ghahramani (University of Toronto) and H. Seung (Bell Labs) for useful discussions. We also thank P. Modi (AT&T Labs) for providing an initial segmentation of the training utterances. References [1] Rabiner, L., and Juang, B. (1993) Fundamentals of Speech Recognition. Englewood Cliffs: Prentice Hall. [2] Ljolje, A. (1994) The importance of cepstral parameter correlations in speech recognition. Computer Speech and Language 8:223-232. [3] Bellegarda, J., and Nahamoo, D. (1990) Tied mixture continuous parameter modeling for speech recognition. IEEE Transactions on Acoustics, Speech, and Signal Processing 38:2033-2045. [4] Rubin, D., and Thayer, D. (1982) EM algorithms for factor analysis. Psychometrika 47:69-76. [5] Everitt, B. (1984) An introduction to latent variable models. London: Chapman and Hall. [6] Hinton, G., Dayan, P., and Revow, M. (1996) Modeling the manifolds of images of handwritten digits. To appear in IEEE Transactions on Neural Networks. [7] Ghahramani, Z. and Hinton, G. (1996) The EM algorithm for mixtures of factor analyzers. University of Toronto Technical Report CRG-TR-96-1. [8] Simard, P., LeCun, Y., and Denker, J. (1993) Efficient pattern recognition using a new transformation distance. In J. Cowan, S. Hanson, and C. Giles, eds. Advances in Neural Information Processing Systems 5:50-58. Cambridge: MIT Press. [9] Press, W., Teukolsky, S., Vetterling, W., and Flannery, B. (1992) Numerical Recipes in C: The Art of Scientific Computing. Cambridge: Cambridge University Press. [10] Bahl, L., Brown, P., deSouza, P., and Mercer, 1. (1986) Maximum mutual information estimation of hidden Markov model parameters for speech recognition. In Proceedings of ICASSP 86: 49-52.
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MELONET I: Neural Nets for Inventing Baroque-Style Chorale Variations Dominik Hornel dominik@ira.uka.de Institut fur Logik, Komplexitat und Deduktionssysteme Universitat Fridericiana Karlsruhe (TH) Am Fasanengarten 5 D-76128 Karlsruhe, Germany Abstract MELONET I is a multi-scale neural network system producing baroque-style melodic variations. Given a melody, the system invents a four-part chorale harmonization and a variation of any chorale voice, after being trained on music pieces of composers like J. S. Bach and J . Pachelbel. Unlike earlier approaches to the learning of melodic structure, the system is able to learn and reproduce high-order structure like harmonic, motif and phrase structure in melodic sequences. This is achieved by using mutually interacting feedforward networks operating at different time scales, in combination with Kohonen networks to classify and recognize musical structure. The results are chorale partitas in the style of J. Pachelbel. Their quality has been judged by experts to be comparable to improvisations invented by an experienced human organist. 1 INTRODUCTION The investigation of neural information structures in music is a rather new, exciting research area bringing together different disciplines such as computer science, mathematics, musicology and cognitive science. One of its aims is to find out what determines the personal style of a composer. It has been shown that neural network models - better than other AI approaches - are able to learn and reproduce styledependent features from given examples, e.g., chorale harmonizations in the style of Johann Sebastian Bach (Hild et al., 1992). However when dealing with melodic sequences, e.g., folk-song style melodies, all of these models have considerable difficulties to learn even simple structures. The reason is that they are unable to capture high-order structure such as harmonies, motifs and phrases simultaneously occurring at multiple time scales. To overcome this problem, Mozer (Mozer, 1994) 888 D. Hamel proposes context units that learn reduced descriptions of a sequence of individual notes. A similar approach in MELONET (Feulner et Hornel, 1994) uses delayed update units that do not fire each time their input changes but rather at discrete time intervals. Although these models perform well on artificial sequences, they produce melodies that suffer from a lack of global coherence. The art of melodic variation has a long tradition in Western music. Almost every great composer has written music pieces inventing variations of a given melody, e.g., Mozart's famous variations KV 265 on the melody "Ah! Vous dirai-je, Maman", also known as "Twinkle twinkle little star". At the beginning of this tradition there is the baroque type of chorale variations. These are organ or harpsichord variations of a chorale melody composed for use in the Protestant church. A prominent representative of this kind of composition is J. Pachelbel (1653 - 1706) who wrote about 50 chorale variations or partitas on various chorale melodies. 2 TASK DESCRIPTION Given a chorale melody, the learning task is achieved in two steps: 1. A chorale harmonization of the melody is invented. 2. One of the voices of the resulting chorale is chosen and provided with melodic variations. Both subtasks are directly learned from music examples composed by J. Pachelbel and performed in an interactive composition process which results in a chorale variation of the given melody. The first task is performed by HARMONET, a neural network system which is able to harmonize melodies in the style of various composers like J. S. Bach. The second task is performed by the neural network system MELONET I, presented in the following. For simplicity we have considered melodic variations consisting of 4 sixteenth notes for each melody quarter note. This is the most common variation type used by baroque composers and presents a good starting point for even more complex variation types, since there are enough music examples for training and testing the networks, and because it allows the representation of higher-scale elements in a rather straightforward way. HARMONET is a system producing four-part chorales in various harmonization styles, given a one-part melody. It solves a musical real-world problem on a performance level appropriate for musical practice. Its power is based on a coding scheme capturing musically relevant information. and on the integration of neural networks and symbolic algorithms in a hierarchical system, combining the advantages of both. The details are not discussed in this paper. See (Hild et aI., 1992) or (Hornel et Ragg, 1996a) for a detailed account. 3 A MULTI-SCALE NEURAL NETWORK MODEL The learning goal is twofold. On the one hand, the results produced by the system should conform to musical rules. These are melodic and harmonic constraints such as the correct resolving of dissonances or the appropriate use of successive interval leaps. On the other hand, the system should be able to capture stilistic features from the learning examples, e.g., melodic shapes preferred by J. Pachelbel. The observation of musical rules and the aesthetic conformance to the learning set can be achieved by a multi-scale neural network model. The complexity of the learning task is reduced by decomposition in three subtasks (see Figure 1): MELONEI' I: Neural Netsfor Inventing Baroque-Style Chorale Variations 889 Harmony T T D, T S, T, D T k=tmod4 : MelodIc Vanallon Figure 1: Structure of the system and process of composing a new melodic variation. A melody (previously harmonized by HARMONET) is passed to the supernet which predicts the current motif class MGT from a local window given by melody notes MT to MT+2 and preceding motif class MGT-I. A similar procedure is performed at a lower time scale by the su bnet which predicts the next motif note Nt based on M CT, current harmony HT and preceding motif note Nt-I. The result is then returned to the supernet through the motif classifier to be considered when computing the next motif class MCT +1 . 1. A melody variation is considered at a higher time scale as a sequence of melodic groups, so-called motifs. Each quarter note of the given melody is varied by one motif. Before training the networks, motifs are classified according to their similarity. 2. One neural network is used to learn the abstract sequence of motif classes. Motif classes are represented in a l-of-n coding form where n is a fixed number of classes. The question it solves is: What kind of motif 'fits' a melody note depending on melodic context and the motif that has occurred before? No concrete notes are fixed by this network. It works at a higher scale and will therefore be called stlpernet in the following. 3. Another neural network learns the implementation of abstract motif classes into concrete notes depending on a given harmonic context. It produces a sequence of sixteenth notes - four notes per motif - that result in a melodic variation of the given melody. Because it works one scale below the supernet, it is called stlbnet. 4. The subnet sometimes invents a sequence of notes that does not coincide 890 D. Homel with the motif class determined by the supernet. This motif will be considered when computing the next motif class, however. and should therefore match the notes previously formed by the subnet. It is therefore reclassified by the motif classifier before the supernet determines the next motif class. The motivation of this separation into supernet and subnet arised from the following consideration: Having a neural network that learns sequences of sixteenth notes, it. would be easier for this network to predict notes given a contour of each motif. i.e. a sequence of interval directions to be produced for each quarter note. Consider a human organist who improvises a melodic variation of a given melody in real time. Because he has to take his decisions in a fraction of a second, he must at least have some rough idea in mind about what kind of melodic variation should be applied to the next melody note to obtain a meaningful continuation of the variation. Therefore, a neural network was introduced at a higher time scale, the training of which really improved the overall behavior of the system and not just shifted the learning problem to another time scale. 4 MOTIF CLASSIFICATION AND RECOGNITION In order to realize learning at different time scales as described above, we need a recognition component to find a suitable classification of motifs. This can be achieved using unsupervised learning, e.g. , agglomerative hierarchical clustering or Kohonen's topological feature maps (Kohonen, 1990). The former has the disadvantage however that an appropriate distance measure is needed which determines the similarity between small sequences of notes respectively intervals, whereas the latter allows to obtain appropriate motif classes through self-organization within a twodimensional surface. Figure 2 displays the motif representation and distribution of motif contours over a 10xlO Kohonen feature map. In MELONET I, the Kohonen algorithm is applied to all motifs contained in the training set. Afterwards a corresponding motif classification tree is recursively built from the Kohonen map. While cutting this classification tree at lower levels we can get more and more classes. One important problem remains to find an appropriate number of classes for the given learning task. This will be discussed in section 6. ~ ... -" .............................................. ', , jJ 3jl Winner 1 st interval 2nd interval 3rd interval -1 Figure 2: Motifrepresentation example (left) and motif contour distribution (right) over a 10xlO Kohonen feature map developed from one Pachelbel chorale variation (initial update area 6x6, initial adaptation height 0.95, decrease factor 0.995). Each cell corresponds to one unit in the KFM. One can see the arrangement of regions responding to motifs having different motif contours. MELONEr I: Neural Nets for Inventing Baroque-Style Chorale Variations 891 5 REPRESENTATION In general one can distinguish two groups of motifs: Melodic motifs prefer small intervals, mainly seconds, harmonic motifs prefer leaps and harmonizing notes (chord notes) . Both motif groups heavily rely on harmonic information. In melodic motifs dissonances should be correctly resolved, in harmonic motifs notes must fit the given harmony. Small deviations may have a significant effect on the quality of musical results. Thus our idea was to integrate musical knowledge about interval and harmonic relationships into an appropriate interval representation. Each note is represented by its interval to the first motif note, the so-called reference note. This is an important element contributing to the success of MELONET I. A similar idea for Jazz improvisation was followed in (Baggi, 1992) . The interval coding shown in Table 1 considers several important relationships: neighboring intervals are realized by overlapping bits, octave invariance is represented using a special octave bit. The activation of the overlapping bit was reduced from 1 to 0.5 in order to allow a better distinction of the intervals. 3 bits are used to distinguish the direction of the interval, 7 bits represent interval size. Complementary intervals such as ascending thirds and descending sixths have similar representations because they lead to the same note and can therefore be regarded as harmonically equivalent. A simple rhythmic element was then added using a tenuto bit (not shown -in Table 1) which is set when a note is tied to its predecessor. This final 3+1+7+1=12 bit coding gave the best results in our simulations. Table 1: Complementary Interval Coding direction octave interval size ninth \. 1 o 0 1 0 0 0 0 0 0.5 1 octave \. 1 o 0 1 1 0 0 0 0 0 0.5 seventh \. 1 o 0 0 0.5 1 0 0 0 0 0 sixth \. 100 0 0 0.5 1 0 0 0 0 fifth \. 100 0 0 0 0.5 1 0 0 0 fourth \. 1 0 0 0 0 0 0 0.5 1 0 0 third \. 1 o 0 0 0 0 0 0 0.5 1 0 second \. 1 o 0 0 0 0 0 0 0 0.5 1 pnme -+ 010 0 1 0 0 0 0 0 0.5 second /' o 0 1 0 0.5 1 0 0 0 0 0 third /' 0 0 1 0 0 0.5 1 0 0 0 0 fourth /' 0 0 1 0 0 0 0.5 1 0 0 0 fifth /' o 0 1 0 0 0 0 0.5 1 0 0 sixth /' o 0 1 0 0 0 0 0 0 . 5 1 0 seventh /' o 0 1 0 0 0 0 0 0 0.5 1 octave /' 0 0 1 1 1 0 0 0 0 0 0.5 ninth /' o 0 1 1 0.5 1 0 0 0 0 0 Now we still need a representation for harmony. It can be encoded as a harmonic field which is a vector of chord notes of the diatonic scale. The tonic T in C major for example contains 3 chord notes - C, E and G - which correspond to the first, third and fifth degree of the C major scale (1010100). This representation may be further improved. We have already mentioned that each note is represented by the interval to the first motif note (reference note). We can now encode the harmonic field starting with the first motif note instead of the first degree of the scale. This is equivalent to rotating the bits of the harmonic field vector. An example is displayed in Figure 3. The harmony of the motif is the dominant D, the first motif note is B which corresponds to the seventh degree of the C major scale. Therefore the 892 D.Homel harmonic field for D (0100101) is rotated by one position to the right resulting in (1010010). Starting with the first note B. the harmonic field indicates the intervals that lead to harmonizing notes B, D and G. In the right part of Figure 3 one can see a correspondance between bits activated in the harmonic field and bits set to 1 in the three interval codings. This kind of representation helps the neural network to directly establish a relationship between intervals and given harmony. third up o 0 1 0 0 0.5 1 0 0 0 0 , J J 3d I sixth up o 0 1 0 0 0 0 0 0.5 1 0 pnme 010 0 1 0 0 0 0 0 0.5 D harmonic field 1 0 1 0 0 1 0 Figure 3: Example illustrating the relationship between interval coding and rotat.ed harmonic field. Each note is represented by its interval to the first note. 6 PERFORMANCE We carried out several simulations to evaluate the performance of the system. Many improvements could be found however by just listening to the improvisations produced by the neural organist. One important problem was to find an appropriate number of classes for the given learning task. The following table lists the classification rate on the learning and validation set of the supernet and the subnet using 5, 12 and 20 motif classes. The learning set was automatically built from 12 Pachelbel chorale variations corresponding to 2220 patterns for the subnet and 555 for the supernet. The validation set includes 6 Pachelbel variations corresponding to 1396 patterns for the subnet and 349 for the supernet. Supernet and subnet were then trained independently with the RPROP learning algorithm. s'Upernet s'Ubnet 5 classes 12 classes 20 classes 5 classes 12 classes 20 classes learning set 91.17% 86.85% 87.57% 86.31% 93.92% 95.68% validation set 49.85% 40.69% 37.54% 79.15% 83.38% 86.96% The classification rate of both networks strongly depends on the number of classes, esp. on the validation set of the supernet. The smaller the number of classes, the better is the classification of the supernet because there are less alternatives to choose from. We can also notice an opposite development of the classification behavior for the subnet. The bigger the number of classes. the easier the subnet will be able to determine concrete motif notes for a given motif class. One can imagine that the optimal number of classes lies somewhere in the middle. Another idea is to form a committee of networks each of which is trained with different number of classes. We have also tested MELONET I on melodies that do not belong to the baroque era. Figure 4 shows a harmonization and variation of the melody "Twinkle twinkle little star" used by Mozart in his famous piano variations. It was produced by a network committee formed by 3*2=6 networks trained with 5, 12 and 20 classes. 7 CONCLUSION We have presented a neural network system inventing baroque-style variations on given melodies whose qualities are similar to those of an experienced human organMELONEI'L· Neural Nets for Inventing Baroque-Style Chorale Variations 893 ! . Figure 4: Melodic variation on "Twinkle twinkle little star" ist. The complex musical task could be learned introducing a multi-scale network model with two neural networks cooperating at different time scales, together with an unsupervised learning mechanism able to classify and recognize relevant musical structure. We are about to test this multi-scale approach on learning examples of other epochs, e.g., on compositions of classical composers like Haydn and Mozart or on Jazz improvisations. First results confirm that the system is able to reproduce stylespecific elements of other kinds of melodic variation as well. Another interesting question is whether the global coherence of the musical results may be further improved adding another network working at a higher level of abstraction, e.g., at. a phrase level. In summary, we believe that this approach presents an important step towards the learning of complete melodies. References Denis L. Baggi. NeurSwing: An Intelligent Workbench for the Investigation of Swing in Jazz. In: Readings in Computer-Generated Music, IEEE Computer Society Press, pp. 79-94, 1992. Johannes Feulner, Dominik Hornel. MELONET: Neural networks that learn harmony-based melodic variations. In: Proceedings of the 1994 International Computer Music Conference. ICMA Arhus, pp. 121-124, 1994. Hermann Hild, Johannes Feulner, Wolfram Menzel. HARMONET: A Neural Net for Harmonizing Chorales in the Style of J. S. Bach. In: Advances in Neural Information Processing 4 (NIPS 4), pp. 267-274. 1992. Dominik Hornel, Thomas Ragg. Learning Musical Structure and Style by Recognition, Prediction and Evolution. In: Proceedings of the 1996 International Computer Music Conference. ICMA Hong Kong, pp. 59-62, 1996. Dominik Hornel, Thomas Ragg. A Connectionist Model for the Evolution of Styles of Harmonization. In: Proceedings of the 1996 International Conference on Music Perception and Cognition. Montreal, 1996. Teuvo Kohonen. The Self-Organizing Map. In: Proceedings of the IEEE, Vol. 78, no. 9, pp. 1464-1480, 1990. Michael C. Mozer. Neural Network music composition by prediction. In: Connection Science 6(2,3), pp. 247-280, 1994.
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Hippocampal Model of Rat Spatial Abilities Using Temporal Difference Learning David J Foster* Centre for Neuroscience Edinburgh University Richard GM Morris Centre for Neuroscience Edinburgh University Abstract Peter Dayan E25-210, MIT Cambridge, MA 02139 We provide a model of the standard watermaze task, and of a more challenging task involving novel platform locations, in which rats exhibit one-trial learning after a few days of training. The model uses hippocampal place cells to support reinforcement learning, and also, in an integrated manner, to build and use allocentric coordinates. 1 INTRODUCTION Whilst it has long been known both that the hippocampus of the rat is needed for normal performance on spatial tasksl3, 11 and that certain cells in the hippocampus exhibit place-related firing,12 it has not been clear how place cells are actually used for navigation. One of the principal conceptual problems has been understanding how the hippocampus could specify or learn paths to goals when spatially tuned cells in the hippocampus respond only on the basis of the rat's current location. This work uses recent ideas from reinforcement learning to solve this problem in the context of two rodent spatial learning results. Reference memory in the watermazell (RMW) has been a key task demonstrating the importance of the hippocampus for spatial learning. On each trial, the rat is placed in a circular pool of cloudy water, the only escape from which is a platform which is hidden (below the water surface) but which remains in a constant position. A random choice of starting pOSition is used for each trial. Rats take asymptotically short paths after approximately 10 trials (see figure 1 a). Delayed match-to-place (DMP) learning is a refined version in which the platform'S location is changed on each day. Figure 1 b shows escape latencies for rats given four trials per day for nine days, with the platform in a novel position on each day. On early days, acquisition ·Crichton Street, Edinburgh EH8 9LE, United Kingdom. Funded by Edin. Univ. Holdsworth Scholarship, the McDonnell-Pew foundation and NSF grant IBN-9634339. Email: djf@cfn.ed.ac.uk 146 D. J Foster; R. G. M. Mo"is and P. Dayan 100 100 90 b 90 80 a _ 70 '" i oo '" 50 -' ~40 ~30 20 10 13 17 21 2S Figure 1: a) Latencies for rats on the reference memory in the watermaze (RMW) task (N=8). b) Latencies for rats on the Delayed Match-to-Place (DMP) task (N=62). is gradual but on later days, rats show one-trial learning, that is, near asymptotic performance on the second trial to a novel platform position. The RMW task has been extensively modelled. 6,4,5,20 By contrast, the DMP task is new and computationally more challenging. It is solved here by integrating a standard actor-critic reinforcement learning system2,7 which guarantees that the rat will be competent to perform well in arbitrary mazes, with a system that learns spatial coordinates in the maze. Temporal difference learning 1 7 (TO) is used for actor, critic and coordinate learning. TO learning is attractive because of its generality for arbitrary Markov decision problems and the fact that reward systems in vertebrates appear to instantiate it. 14 2 THEMODEL The model comprises two distinct networks (figure 2): the actor-critic network and a coordinate learning network. The contribution of the hippocampus, for both networks, is to provide a state-space representation in the form of place cell basis functions. Note that only the activities of place cells are required, by contrast with decoding schemes which require detailed information about each place cell.4 ACTOR-CRITIC SYSTEM COORDINATE SYSTEM r------------1 Remembered 1 Goal coordinates 1 1 VECTOR COMPUTA nONI ~ Coordinate Representation 1 1 ______ -------1 Figure 2: Model diagram showing the interaction between actor-critic and coordinate system components. Hippocampal Model of Rat Spatial Abilities Using TD Learning 147 2.1 Actor-Critic Learning Place cells are modelled as being tuned to location. At position p, place cell i has an output given by h(p) = exp{ -lip - sdI2/2(12}, where Si is the place field centre, and (1 = 0.1 for all place fields. The critic learns a value function V(p) = L:i wih(p) which comes to represent the distance of p from the goal, using the TO rule 6.w~ ex: 8t h(pt), where (1) is the TD error, pt is position at time t, and the reward r(pt, pt+I) is 1 for any move onto the platform, and 0 otherwise. In a slight alteration of the original rule, the value V (p) is set to zero when p is at the goal, thus ensuring that the total future rewards for moving onto the goal will be exactly 1. Such a modification improves stability in the case of TD learning with overlapping basis functions. The discount factor, I' was set to 0.99. Simultaneously the rat refines a policy, which is represented by eight action cells. Each action cell (aj in figure 2) receives a parameterised input at any position p: aj (p) = L:i qjdi (p). An action is chosen stochastically with probabilities given by P(aj) = exp{2aj}/ L:k exp{2ak}. Action weights are reinforced according to:2 (2) where 9j((Jt) is a gaussian function of the difference between the head direction (Jt at time t and the preferred direction of the jth action cell. Figure 3 shows the development of a policy over a few trials. V(p)l Triall V(p) 1 TrialS V(P)l Triall3 0.5 0.5 0.5 I 0. 01 0: 0.5 0.5 0.5 0.5 0.5 .---'---0.5 0 ------0 -0.5 -0.5 Figure 3: The RMW task: the value function gradually disseminates information about reward proximity to all regions of the environment. Policies and paths are also shown. There is no analytical guarantee for the convergence of TD learning with policy adaptation. However our simulations show that the algorithm always converges for the RMW task. In a simulated arena of diameter 1m and with swimming speeds of 20cm/s, the simulation matched the performance of the real rats very closely (see figure S). This demonstrates that TD-based reinforcement learning is adequately fast to account for the learning performance of real animals. 148 D. 1. Foster, R. G. M Morris and P. Dayan 2.2 Coordinate Learning Although the learning of a value function and policy is appropriate for finding a fixed platform, the actor-critic model does not allow the transfer of knowledge from the task defined by one goal position to that defined by any other; thus it could not generate the sort of one-trial learning that is shown by rats on the DMP task (see figure 1 b). This requires acquisition of some goal-independent know ledge about s~ace. A natural mechanism for this is the path integration or self-motion system. 0,10 However, path integration presents two problems. First, since the rat is put into the maze in a different position for each trial, how can it learn consistent coordinates across the whole maze? Second, how can a general, powerful, but slow, behavioral learning mechanism such as TO be integrated with a specific, limited, but fast learning mechanism involving spatial coordinates? Since TO critic learning is based on enforcing consistency in estimates of future reward, we can also use it to learn spatially consistent coordinates on the basis of samples of self-motion. It is assumed that the rat has an allocentric frame of reference.1s The model learns parameterised estimates of the x and y coordinates of all positions p: x(p) = Li w[ fi(P) and y(p) = Li wY h(p), Importantly, while place cells were again critical in supporting spatial representation, they do not embody a map of space. The coordinate functions, like the value function previously, have to be learned. As the simulated rat moves around, the coordinate weights {w[} are adjusted according to: t Llwi ()( (Llxt + X (pt+l ) - X(pt)) L At - k h(pk) (3) k=1 where Llxt is the self-motion estimate in the x direction. A similar update is applied to {wn. In this case, the full TO(A) algorithm was used (with A = 0.9); however TD(O) could also have been used, taking slightly longer. Figure 4a shows the x and y coordinates at early and late phases of learning. It is apparent that they rapidly become quite accurate - this is an extremely easy task in an open field maze. An important issue in the learning of coordinates is drift, since the coordinate system receives no direct information about the location of the origin. It turns out that the three controlling factors over the implicit origin are: the boundary of the arena, the prior setting of the coordinate weights (in this case all were zero) and the position and prior value of any absorbing area (in this case the platform). If the coordinate system as a whole were to drift once coordinates have been established, this would invalidate coordinates that have been remembered by the rat over long periods. However, since the expected value of the prediction error at time steps should be zero for any self-consistent coordinate mapping, such a mapping should remain stable. This is demonstrated for a single run: figure 4b shows the mean value of coordinates x evolving over trials, with little drift after the first few trials. We modeled the coordinate system as influencing the choice of swimming direction in the manner of an abstract action. I5 The (internally specified) coordinates of the most recent goal position are stored in short term memory and used, along with the current coordinates, to calculate a vector heading. This vector heading is thrown into the stochastic competition with the other possible actions, governed by a single weight which changes in a similar manner to the other action weights (as in equation 2, see also fig 4d), depending on the TO error, and on the angular proximity of the current head direction to the coordinate direction. Thus, whether the the coordinate-based direction is likely to be used depends upon its past performance. One simplification in the model is the treatment of extinction. In the DMP task, Hippocampal Model of Rat Spatial Abilities Using 1D Learning " TJUAL d i: ~Ol ~o !" ~o ,. 149 III .1 26 16 TRIAL Figure 4: The evolution of the coordinate system for a typical simulation run: a.) coordinate outputs at early and late phases of learning, b.) the extent of drift in the coordinates, as shown by the mean coordinate value for a single run, c.) a measure f d· A2 ~ ~ {X r (Pr.)-Xr -X(pr.)}2 o coor mate error for the same run (7E = r r. (Np-l)Nr ' where k indexes measurement points (max Np ) and r indexes runs (max Nr), Xr(Pk) is the model estimate of X at position Pk, X(Pk) is the ideal estimate for a coordinate system centred on zero, and Xr is the mean value over all the model coordinates, d.) the increase during training of the probability of choosing the abstract action. This demonstrates the integration of the coordinates into the control system. real rats extinguish to a platform that has moved fairly quickly whereas the actorcritic model extinguishes far more slowly. To get around this, when a simulated rat reaches a goal that has just been moved, the value and action weights are reinitialised, but the coordinate weights wf and wf, and the weights for the abstract action, are not. 3 RESULTS The main results of this paper are the replication by simulation of rat performance on the RMW and DMP tasks. Figures la and b show the course of learning for the rats; figures Sa and b for the model. For the DMP task, one-shot acquisition is apparent by the end of training. 4 DISCUSSION We have built a model for one-trial spatial learning in the watermaze which uses a single TD learning algorithm in two separate systems. One system is based on a reinforcement learning that can solve general Markovian decision problems, and the other is based on coordinate learning and is specialised for an open-field water maze. Place cells in the hippocampus offer an excellent substrate for learning the actor, the critic and the coordinates. The model is explicit about the relationship between the general and specific learning systems, and the learning behavior shows that they integrate seamlessly. As currently constituted, the coordinate system would fail if there were a barrier in the maze. We plan to extend the model to allow the coordinate system to specify abstract targets other than the most recent platform position - this could allow it fast navigation around a larger class of environments. It is also important to improve the model of learning 'set' behavior - the information about the nature of 150 D. 1. Foster; R. G. M. Mo"is and P. Dayan 14 a 12 b 12 10 §. z ~ S> .. j:10\ ~ . ~ ~ . '"~ ............................................ 0~D.~yl~~y~2~D~.y~3~D~.y~47~~yS~~~.~~~y~7~~~y~.7D.~y9~ Figure 5: a.) Performance of the actor-critic model on the RMW task, and b.) performance of the full model on the DMP task. The data for comparison is shown in figures la and b. the DMP task that the rats acquire over the course of the first few days of training. Interestingly, learning set is incomplete - on the first trial of each day, the rats still aim for the platform position on the previous day, even though this is never correct.16 The significant differences in the path lengths on the first trial of each day (evidence in figure Ib and figure 5b) come from the relative placements of the platforms. However, the model did not use the same positions as the empirical data, and, in any case, the model of exploration behavior is rather simplistic. The model demonstrates that reinforcement learning methods are perfectly fast enough to match empirical learning curves. This is fortunate, since, unlike most models specifically designed for open-field navigation,6,4,5,2o RL methods can provably cope with substantially more complicated tasks with arbitrary barriers, etc, since they solve the temporal credit assignment problem in its full generality. The model also addresses the problem that coordinates in different parts of the same environment need to be mutually consistent, even if the animal only experiences some parts on separate trials. An important property of the model is that there is no requirement for the animal to have any explicit knowledge of the relationship between different place cells or place field position, size or shape. Such a requirement is imposed in various models.9,4,6,2o Experiments that are suggested by this model (as well as by certain others) concern the relationship between hippocampally dependent and independent spatial learning. First, once the coordinate system has been acquired, we predict that merely placing the rat at a new location would be enough to let it find the platform in one shot, though it might be necessary to reinforce the placement e.g. by first placing the rat in a bucket of cold water. Second, we know that the establishment of place fields in an environment happens substantiallr faster than establishment of one-shot or even ordinary learning to a platform.2 We predict that blocking plasticity in the hippocampus following the establishment of place cells (possibly achieved without a platform) would not block learning of a platform. In fact, new experiments show that after extensive pre-training, rats can perform one-trial learning in the same environment to new platform positions on the DMP task without hippocampal synaptic plasticity. 16 This is in contrast to the effects of hippocampal lesion, which completely disrupts performance. According to the model, coordinates will have been learned during pre-training. The full prediction remains untested: that once place fields have been established, coordinates could be learned in the absence of hippocampal synaptic plasticity. A third prediction follows from evidence that rats with restricted hippocampal lesions can learn the fixed platform Hippocampal Model of Rat Spatial Abilities Using TD Learning 151 task, but much more slowly, based on a gradual "shaping" procedure.22 In our model, they may also be able to learn coordinates. However, a lengthy training procedure could be required, and testing might be complicated if expressing the knowledge required the use of hippocampus dependent short-term memory for the last platform location.I6 One way of expressing the contribution of the hippocampus in the model is to say that its function is to provide a behavioural state space for the solution of complex tasks. Hence the contribution of the hippocampus to navigation is to provide place cells whose firing properties remain consistent in a given environment. It follows that in different behavioural situations, hippocampal cells should provide a representation based on something other than locations and, indeed, there is evidence for this.8 With regard to the role of the hippocampus in spatial tasks, the model demonstrates that the hippocampus may be fundamentally necessary without embodying a map. References [1] Barto, AG & Sutton, RS (1981) BioI. Cyber., 43:1-8. [2] Barto, AG, Sutton, RS & Anderson, CW (1983) IEEE Trans. on Systems, Man and Cybernetics 13:834-846. [3] Barto, AG, Sutton, RS & Watkins, CJCH (1989) Tech Report 89-95, CAIS, Univ. Mass., Amherst, MA. [4] Blum, KI & Abbott, LF (1996) Neural Computation, 8:85-93. [5] Brown, MA & Sharp, PE (1995) Hippocampus 5:171-188. [6] Burgess, N, Reece, M & O'Keefe, J (1994) Neural Networks, 7:1065-1081. [7] Dayan, P (1991) NIPS 3, RP Lippmann et aI, eds., 464-470. [8] Eichenbaum, HB (1996) Curro Opin. Neurobiol., 6:187-195. [9] Gerstner, W & Abbott, LF (1996) J. Computational Neurosci. 4:79-94. [10] McNaughton, BL et a1 (1996) J. Exp. BioI., 199:173-185. [11] Morris, RGM et al (1982) Nature, 297:681-683. [12] O'Keefe, J & Dostrovsky, J (1971) Brain Res., 34(171). [13] Olton, OS & Samuelson, RJ (1976) J. Exp. Psych: A.B.P., 2:97-116. Rudy, JW & Sutherland, RW (1995) Hippocampus, 5:375-389. [14] SchUltz, W, Dayan, P & Montague, PR (1997) Science, 275, 1593-1599. [15] Singh, SP Reinforcement learning with a hierarchy of abstract models. [16] Steele, RJ & Morris, RGM in preparation. [17] Sutton, RS (1988) Machine Learning, 3:9-44. [18] Taube, JS (1995) J. Neurosci. 15(1):70-86. [19] Tsitsiklis, IN & Van Roy, B (1996) Tech Report LIDS-P-2322, M.LT. [20] Wan, HS, Touretzky, OS & Redish, AD (1993) Proc. 1993 Connectionist Models Summer School, Lawrence Erlbaum, 11-19. [21] Watkins, CJCH (1989) PhD Thesis, Cambridge. [22] Whishaw, IQ & Jarrard, LF (1996) Hippocampus [23] Wilson, MA & McNaughton, BL (1993) Science 261:1055-1058.
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EM Algorithms for PCA and SPCA Sam Roweis· Abstract I present an expectation-maximization (EM) algorithm for principal component analysis (PCA). The algorithm allows a few eigenvectors and eigenvalues to be extracted from large collections of high dimensional data. It is computationally very efficient in space and time. It also naturally accommodates missing infonnation. I also introduce a new variant of PC A called sensible principal component analysis (SPCA) which defines a proper density model in the data space. Learning for SPCA is also done with an EM algorithm. I report results on synthetic and real data showing that these EM algorithms correctly and efficiently find the leading eigenvectors of the covariance of datasets in a few iterations using up to hundreds of thousands of datapoints in thousands of dimensions. 1 Why EM for peA? Principal component analysis (PCA) is a widely used dimensionality reduction technique in data analysis. Its popularity comes from three important properties. First, it is the optimal (in tenns of mean squared error) linear scheme for compressing a set of high dimensional vectors into a set of lower dimensional vectors and then reconstructing. Second, the model parameters can be computed directly from the data - for example by diagonalizing the sample covariance. Third, compression and decompression are easy operations to perfonn given the model parameters - they require only matrix multiplications. Despite these attractive features however, PCA models have several shortcomings. One is that naive methods for finding the principal component directions have trouble with high dimensional data or large numbers of datapoints. Consider attempting to diagonalize the sample covariance matrix of n vectors in a space of p dimensions when n and p are several hundred or several thousand. Difficulties can arise both in the fonn of computational complexity and also data scarcity. I Even computing the sample covariance itself is very costly, requiring 0 (np2) operations. In general it is best to avoid altogether computing the sample • rowei s@cns . cal tech. edu; Computation & Neural Systems, California Institute of Tech. IOn the data scarcity front, we often do not have enough data in high dimensions for the sample covariance to be of full rank and so we must be careful to employ techniques which do not require full rank matrices. On the complexity front, direct diagonalization of a symmetric matrix thousands of rows in size can be extremely costly since this operation is O(P3) for p x p inputs. Fortunately, several techniques exist for efficient matrix diagonalization when only the first few leading eigenvectors and eigerivalues are required (for example the power method [10] which is only O(p2». EM Algorithms for PCA and SPCA 627 covariance explicitly. Methods such as the snap-shot algorithm [7] do this by assuming that the eigenvectors being searched for are linear combinations of the datapoints; their complexity is O(n3 ). In this note, I present a version of the expectation-maximization (EM) algorithm [1] for learning the principal components of a dataset. The algorithm does not require computing the sample covariance and has a complexity limited by 0 (knp) operations where k is the number of leading eigenvectors to be learned. Another shortcoming of standard approaches to PCA is that it is not obvious how to deal properly with missing data. Most of the methods discussed above cannot accommodate missing values and so incomplete points must either be discarded or completed using a variety of ad-hoc interpolation methods. On the other hand, the EM algorithm for PCA enjoys all the benefits [4] of other EM algorithms in tenns of estimating the maximum likelihood values for missing infonnation directly at each iteration. Finally, the PCA model itself suffers from a critical flaw which is independent of the technique used to compute its parameters: it does not define a proper probability model in the space of inputs. This is because the density is not nonnalized within the principal subspace. In other words, if we perfonn PCA on some data and then ask how well new data are fit by the model, the only criterion used is the squared distance of the new data from their projections into the principal subspace. A datapoint far away from the training data but nonetheless near the principal subspace will be assigned a high "pseudo-likelihood" or low error. Similarly, it is not possible to generate "fantasy" data from a PCA model. In this note I introduce a new model called sensible principal component analysis (SPCA), an obvious modification of PC A, which does define a proper covariance structure in the data space. Its parameters can also be learned with an EM algorithm, given below. In summary, the methods developed in this paper provide three advantages. They allow simple and efficient computation of a few eigenvectors and eigenvalues when working with many datapoints in high dimensions. They permit this computation even in the presence of missing data. On a real vision problem with missing infonnation, I have computed the 10 leading eigenvectors and eigenvalues of 217 points in 212 dimensions in a few hours using MATLAB on a modest workstation. Finally, through a small variation, these methods allow the computation not only of the principal subspace but of a complete Gaussian probabilistic model which allows one to generate data and compute true likelihoods. 2 Whence EM for peA? Principal component analysis can be viewed as a limiting case of a particular class of linearGaussian models. The goal of such models is to capture the covariance structure of an observed p-dimensional variable y using fewer than the p{p+ 1) /2 free parameters required in a full covariance matrix. Linear-Gaussian models do this by assuming that y was produced as a linear transfonnation of some k-dimensionallatent variable x plus additive Gaussian noise. Denoting the transfonnation by the p x k matrix C, and the ~dimensional) noise by v (with covariance matrix R) the generative model can be written as y = Cx+v x-N{O,I) v-N(O,R) (la) The latent or cause variables x are assumed to be independent and identically distributed according to a unit variance spherical Gaussian. Since v are also independent and nonnal distributed (and assumed independent of x), the model reduces to a single Gaussian model 2 All vectors are column vectors. To denote the transpose of a vector or matrix I use the notation x T . The determinant of a matrix is denoted by IAI and matrix inversion by A -1. The zero matrix is 0 and the identity matrix is I. The symbol", means "distributed according to". A multivariate normal (Gaussian) distribution with mean JL and covariance matrix 1:: is written as N (JL, 1::). The same Gaussian evaluated at the point x is denoted N (JL, 1::) Ix628 S. Roweis for y which we can write explicitly: y ",N (O,CCT + R) (lb) In order to save parameters over the direct covariance representation in p-space, it is necessary to choose k < p and also to restrict the covariance structure of the Gaussian noise v by constraining the matrix R.3 For example, if the shape of the noise distribution is restricted to be axis aligned (its covariance matrix is diagonal) the model is known asfactor analysis. 2.1 Inference and learning There are two central problems of interest when working with the linear-Gaussian models described above. The first problem is that of state inference or compression which asks: given fixed model parameters C and R, what can be said about the unknown hidden states x given some observations y? Since the datapoints are independent, we are interested in the posterior probability P (xly) over a single hidden state given the corresponding single observation. This can be easily computed by linear matrix projection and the resulting density is itself Gaussian: P( I ) = P(Ylx)P(x) = N(Cx,R)lyN(O,I)lx xy P(y) N(O,CCT+R)ly (2a) P (xly) = N ((3y,I - (3C) Ix , (3 = CT(CCT + R)-l (2b) from which we obtain not only the expected value (3y of the unknown state but also an estimate of the uncertainty in this value in the form of the covariance 1- (3C. Computing y from x (reconstruction) is also straightforward: P (ylx) = N (Cx, R) Iy. Finally, computing the likelihood of any datapoint y is merely an evaluation under (1 b). The second problem is that of learning, or parameter fitting which consists of identifying the matrices C and R that make the model assign the highest likelihood to the observed data. There are a family of EM algorithms to do this for the various cases of restrictions to R but all follow a similar structure: they use the inference formula (2b) above in the e-step to estimate the unknown state and then choose C and the restricted R in the m-step so as to maximize the expected joint likelihood of the estimated x and the observed y. 2.2 Zero noise limit Principal component analysis is a limiting case of the linear-Gaussian model as the covariance of the noise v becomes infinitesimally small and equal in all directions. Mathematically, PCA is obtained by taking the limit R = limf~O d. This has the effect of making the likelihood of a point y dominated solely by the squared distance between it and its reconstruction Cx. The directions of the columns of C which minimize this error are known as the principal components. Inference now reduces t04 simple least squares projection: P (xIY) = N ((3y,I - (3C) Ix , (3 = lim CT (CCT + d)-l (3a) f~O P (xly) = N ((CTC)-lCT y, 0) Ix = 6(x - (CTC)-lCT y) (3b) Since the noise has become infinitesimal, the posterior over states collapses to a single point and the covariance becomes zero. 3This restriction on R is not merely to save on parameters: the covariance of the observation noise must be restricted in some way for the model to capture any interesting or informative projections in the state x. If R were not restricted, the learning algorithm could simply choose C = 0 and then set R to be the covariance of the data thus trivially achieving the maximum likelihood model by explaining all of the structure in the data as noise. (Remember that since the model has reduced to a single Gaussian distribution for y we can do no better than having the covariance of our model equal the sample covariance of our data.) 4Recall that if C is p x k with p > k and is rank k then left multiplication by C T (CCT)-l (which appears not to be well defined because (CCT ) is not invertible) is exactly eqUivalent to left multiplication by (CT C) -1 CT. The intuition is that even though CCT truly is not invertible, the directions along which it is not invertible are exactly those which C T is about to project out. EM Algorithms for PCA and SPCA 629 3 An EM algorithm for peA The key observation of this note is that even though the principal components can be computed explicitly, there is still an EM algorithm for learning them. It can be easily derived as the zero noise limit of the standard algorithms (see for example [3, 2] and section 4 below) by replacing the usual e-step with the projection above. The algorithm is: • e-step: • m-step: x = (CTC)-lCTy cnew = YXT(XXT)-l where Y is a p x n matrix of all the observed data and X is a k x n matrix of the unknown states. The columns of C will span the space of the first k principal components. (To compute the corresponding eigenvectors and eigenvalues explicitly, the data can be projected into this k-dimensional subspace and an ordered orthogonal basis for the covariance in the subspace can be constructed.) Notice that the algorithm can be performed online using only a single datapoint at a time and so its storage requirements are only O(kp) + O(k2). The workings of the algorithm are illustrated graphically in figure 1 below. ~ 0 -I - 2 ,'I" Gaussian Input Data '. ';' : . ~ , ": ( . , . ' .. ' ,.'" -~3L --_'7'2-'-' --_~ I -~o----c~--:------: xl ~ 0 -I -2 Non-Gaussian Input Data " ~ '. l.·.· , I ', . '.' . " ,,' I . , . ',:', ~3~---~2-----~I--~O~--~--~--xl Figure 1: Examples of iterations of the algorithm. The left panel shows the learning of the first principal component of data drawn from a Gaussian distribution, while the right panel shows learning on data from a non-Gaussian distribution. The dashed lines indicate the direction of the leading eigenvector of the sample covariance. The dashed ellipse is the one standard deviation contour of the sample covariance. The progress of the algorithm is indicated by the solid lines whose directions indicate the guess of the eigenvector and whose lengths indicate the guess of the eigenvalue at each iteration. The iterations are numbered; number 0 is the initial condition. Notice that the difficult learning on the right does not get stuck in a local minimum, although it does take more than 20 iterations to converge which is unusual for Gaussian data (see figure 2). The intuition behind the algorithm is as follows: guess an orientation for the principal subspace. Fix the guessed subspace and project the data y into it to give the values of the hidden states x. Now fix the values ofthe hidden states and choose the subspace orientation which minimizes the squared reconstruction errors of the datapoints. For the simple twodimensional example above, I can give a physical analogy. Imagine that we have a rod pinned at the origin which is free to rotate. Pick an orientation for the rod. Holding the rod still, project every datapoint onto the rod, and attach each projected point to its original point with a spring. Now release the rod. Repeat. The direction of the rod represents our guess of the principal component of the dataset. The energy stored in the springs is the reconstruction error we are trying to minimize. 3.1 Convergence and Complexity The EM learning algorithm for peA amounts to an iterative procedure for finding the subspace spanned by the k leading eigenvectors without explicit computation of the sample 630 S. Roweis covariance. It is attractive for small k because its complexity is limited by 0 (knp) per iteration and so depends only linearly on both the dimensionality of the data and the number of points. Methods that explicitly compute the sample covariance matrix have complexities limited by 0 (np2), while methods like the snap-shot method that form linear combinations of the data must compute and diagonalize a matrix of all possible inner products between points and thus are limited by O(n2p) complexity. The complexity scaling of the algorithm compared to these methods is shown in figure 2 below. For each dimensionality, a random covariance matrix E was generated5 and then lOp points were drawn from N (0, E). The number of floating point operations required to find the first principal component was recorded using MATLAB'S flops function. As expected, the EM algorithm scales more favourably in cases where k is small and both p and n are large. If k ~ p ~ n (we want all the eigenvectors) then all methods are O(p3). The standard convergence proofs for EM [I] apply to this algorithm as well, so we can be sure that it will always reach a local maximum of likelihood. Furthennore, Tipping and Bishop have shown [8, 9] that the only stable local extremum is the global maximum at which the true principal subspace is found; so it converges to the correct result. Another possible concern is that the number of iterations required for convergence may scale with p or n. To investigate this question, I have explicitly computed the leading eigenvector for synthetic data sets (as above, with n = lOp) of varying dimension and recorded the number of iterations of the EM algorithm required for the inner product of the eigendirection with the current guess of the algorithm to be 0.999 or greater. Up to 450 dimensions (4500 datapoints), the number of iterations remains roughly constant with a mean of 3.6. The ratios of the first k eigenvalues seem to be the critical parameters controlling the number of iterations until convergence (For example, in figure I b this ratio was 1.0001.) ~~metbod Sompli Covariance + Dill· Smtple Covariance only Convergence Behaviour Figure 2: Time complexity and convergence behaviour of the algorithm. In all cases, the number of datapoints n is 10 times the dimensionality p. For the left panel, the number of floating point operations to find the leading eigenvector and eigenvalue were recorded. The EM algorithm was always run for exactly 20 iterations. The cost shown for diagonalization of the sample covariance uses the MATLAB functions cov and eigs. The snap-shot method is show to indicate scaling only; one would not normally use it when n > p . In the right hand panel, convergence was investigated by explicitly computing the leading eigenvector and then running the EM algorithm until the dot product of its guess and the true eigendirection was 0.999 or more. The error bars show ± one standard deviation across many runs. The dashed line shows the number of iterations used to produce the EM algorithm curve ('+') in the left panel. 5First, an axis-aligned covariance is created with the p eigenvalues drawn at random from a uniform distribution in some positive range. Then (p - 1) points are drawn from a p-dimensional zero mean spherical Gaussian and the axes are aligned in space using these points. EM Algorithms for PCA and SPCA 631 3.2 Missing data In the complete data setting, the values of the projections or hidden states x are viewed as the "missing information" for EM. During the e-step we compute these values by projecting the observed data into the current subspace. This minimizes the model error given the observed data and the model parameters. However, if some of the input points are missing certain coordinate values, we can easily estimate those values in the same fashion. Instead of estimating only x as the value which minimizes the squared distance between the point and its reconstruction we can generalize the e-step to: • generalized e-step: For each (possibly incomplete) point y find the unique pair of points x· and y. (such that x· lies in the current principal subspace and y. lies in the subspace defined by the known information about y) which minimize the norm IICx· - y·lI. Set the corresponding column of X to x* and the corresponding column ofY to y •. If y is complete, then y* = y and x* is found exactly as before. If not, then x* and y* are the solution to a least squares problem and can be found by, for example, Q R factorization of a particular constraint matrix. Using this generalized e-step I have found the leading principal components for datasets in which every point is missing some coordinates. 4 Sensible Principal Component Analysis If we require R to be a multiple €I of the identity matrix (in other words the covariance ellipsoid of v is spherical) but do not take the limit as E --t 0 then we have a model which I shall call sensible principal component analysis or SPCA. The columns of C are still known as the principal components (it can be shown that they are the same as in regular PC A) and we will call the scalar value E on the diagonal of R the global noise level. Note that SPCA uses 1 + pk - k(k - 1)/2 free parameters to model the covariance. Once again, inference is done with equation (2b). Notice however, that even though the principal components found by SPCA are the same as those for PCA, the mean of the posterior is not in general the same as the point given by the PCA projection (3b). Learning for SPCA also uses an EM algorithm (given below). Because it has afinite noise level E, SPCA defines a proper generative model and probability distribution in the data space: (4) which makes it possible to generate data from or to evaluate the actual likelihood of new test data under an SPCA model. Furthermore, this likelihood will be much lower for data far from the training set even if they are near the principal subspace, unlike the reconstruction error reported by a PCA model. The EM algorithm for learning an SPCA model is: • e-step: {3 = CT(CCT + d)-l J..Lx = (3Y :Ex = nI - n{3C + J..LxJ..L~ • m-step: cnew = Y J..L~:E-l Enew = trace[XXT CJ..Lx yT]/n2 Two subtle points about complexity6 are important to notice; they show that learning for SPCA also enjoys a complexity limited by 0 (knp) and not worse. 6 First, since d is diagonal, the inversion in the e-step can be performed efficiently using the matrix inversion lemma: {CCT + d)-l = (I/f - C(I + CTC/f)-ICT /f2 ). Second, since we are only taking the trace of the matrix in the m-step, we do not need to compute the fu\1 sample covariance XXT but instead can compute only the variance along each coordinate. 632 S. Roweis 5 Relationships to previous methods The EM algorithm for PCA, derived above using probabilistic arguments, is closely related to two well know sets of algorithms. The first are power iteration methods for solving matrix eigenvalue problems. Roughly speaking, these methods iteratively update their eigenvector estimates through repeated mUltiplication by the matrix to be diagonalized. In the case of PCA, explicitly forming the sample covariance and multiplying by it to perform such power iterations would be disastrous. However since the sample covariance is in fact a sum of outer products of individual vectors, we can multiply by it efficiently without ever computing it. In fact, the EM algorithm is exactly equivalent to performing power iterations for finding C using this trick. Iterative methods for partial least squares (e.g. the NIPALS algorithm) are doing the same trick for regression. Taking the singular value decomposition (SVD) of the data matrix directly is a related way to find the principal subspace. If Lanczos or Arnoldi methods are used to compute this SVD, the resulting iterations are similar to those of the EM algorithm. Space prohibits detailed discussion of these sophisticated methods, but two excellent general references are [5, 6]. The second class of methods are the competitive learning methods for finding the principal subspace such as Sanger's and Oja's rules. These methods enjoy the same storage and time complexities as the EM algorithm; however their update steps reduce but do not minimize the cost and so they typically need more iterations and require a learning rate parameter to be set by hand. Acknowledgements I would like to thank John Hopfield and my fellow graduate students for constant and excellent feedback on these ideas. In particular I am grateful to Erik Winfree for significant contributions to the missing data portion of this work, to Dawei Dong who provided image data to try as a real problem, as well as to Carlos Brody, San joy Mahajan, and Maneesh Sahani. The work of Zoubin Ghahrarnani and Geoff Hinton was an important motivation for this study. Chris Bishop and Mike Tipping are pursuing independent but yet unpublished work on a virtually identical model. The comments of three anonymous reviewers and many visitors to my poster improved this manuscript greatly. References [I] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society series B, 39: 1-38, 1977. [2] B. S. Everitt. An Introducction to Latent Variable Models. Chapman and Hill, London, 1984. [3] Zoubin Ghahramani and Geoffrey Hinton. The EM algorithm for mixtures of factor analyzers. Technical Report CRG-TR -96-1 , Dept. of Computer Science, University of Toronto, Feb. 1997. [4] Zoubin Ghahramani and Michael I. Jordan. Supervised learning from incomplete data via an EM approach. In Jack D. Cowan, Gerald Tesauro, and Joshua Alspector, editors, Advances in Neural Information Processing Systems, volume 6, pages 120-127. Morgan Kaufmann, 1994. [5] Gene H. Golub and Charles F. Van Loan. Matrix Computations. The Johns Hopkins University Press, Baltimore, MD, USA, second edition, 1989. [6] R. B. Lehoucq, D. C. Sorensen, and C. Yang. Arpack users' guide: Solution of large scale eigenvalue problems with implicitly restarted Arnoldi methods. Technical Report from http://www.caam.rice.edu/software/ARPACK/, Computational and Applied Mathematics, Rice University, October 1997. [7] L. Sirovich. Turbulence and the dynamics of coherent structures. Quarterly Applied Mathematics, 45(3):561-590, 1987. [8] Michael Tipping and Christopher Bishop. Mixtures of probabilistic principal component analyzers. Technical Report NCRG/97/003, Neural Computing Research Group, Aston University, June 1997. [9] Michael Tipping and Christopher Bishop. Probabilistic principal component analysis. Technical Report NCRG/97/010, Neural Computing Research Group, Aston University, September 1997. [10] J. H. Wilkinson. The AlgebraiC Eigenvalue Problem. Claredon Press, Oxford, England, 1965.
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Reinforcement Learning for Continuous Stochastic Control Problems Remi Munos CEMAGREF, LISC, Pare de Tourvoie, BP 121, 92185 Antony Cedex, FRANCE. Rerni.Munos@cemagref.fr Paul Bourgine Ecole Polyteclmique, CREA, 91128 Palaiseau Cedex, FRANCE. Bourgine@poly.polytechnique.fr Abstract This paper is concerned with the problem of Reinforcement Learning (RL) for continuous state space and time stocha.stic control problems. We state the Harnilton-Jacobi-Bellman equation satisfied by the value function and use a Finite-Difference method for designing a convergent approximation scheme. Then we propose a RL algorithm based on this scheme and prove its convergence to the optimal solution. 1 Introduction to RL in the continuous, stochastic case The objective of RL is to find -thanks to a reinforcement signal- an optimal strategy for solving a dynamical control problem. Here we sudy the continuous time, continuous state-space stochastic case, which covers a wide variety of control problems including target, viability, optimization problems (see [FS93], [KP95])}or which a formalism is the following. The evolution of the current state x(t) E 0 (the statespace, with 0 open subset of IRd), depends on the control u(t) E U (compact subset) by a stochastic differential equation, called the state dynamics: dx = f(x(t), u(t))dt + a(x(t), u(t))dw (1) where f is the local drift and a .dw (with w a brownian motion of dimension rand (j a d x r-matrix) the stochastic part (which appears for several reasons such as lake of precision, noisy influence, random fluctuations) of the diffusion process. For initial state x and control u(t), (1) leads to an infinity of possible traj~tories x(t). For some trajectory x(t) (see figure I)., let T be its exit time from 0 (with the convention that if x(t) always stays in 0, then T = 00). Then, we define the functional J of initial state x and control u(.) as the expectation for all trajectories of the discounted cumulative reinforcement : J(x; u(.)) = Ex,u(.) {loT '/r(x(t), u(t))dt +,,{ R(X(T))} 1030 R. Munos and P. Bourgine where rex, u) is the running reinforcement and R(x) the boundary reinforcement. 'Y is the discount factor (0 :S 'Y < 1). In the following, we assume that J, a are of class C2 , rand Rare Lipschitzian (with constants Lr and LR) and the boundary 80 is C2 . • · all · II • • xirJ • • • • • • • • Figure 1: The state space, the discretized ~6 (the square dots) and its frontier 8~6 (the round ones). A trajectory Xk(t) goes through the neighbourhood of state ~. RL uses the method of Dynamic Program~ing (DP) which generates an optimal (feed-back) control u*(x) by estimating the value function (VF), defined as the maximal value of the functional J as a function of initial state x : Vex) = sup J(x; u(.). u(.) (2) In the RL approach, the state dynamics is unknown from the system ; the only available information for learning the optimal control is the reinforcement obtained at the current state. Here we propose a model-based algorithm, i.e. that learns on-line a model of the dynamics and approximates the value function by successive iterations. Section 2 states the Hamilton-Jacobi-Bellman equation and use a Finite-Difference (FD) method derived from Kushner [Kus90] for generating a convergent approximation scheme. In section 3, we propose a RL algorithm based on this scheme and prove its convergence to the VF in appendix A. 2 A Finite Difference scheme Here, we state a second-order nonlinear differential equation (obtained from the DP principle, see [FS93J) satisfied by the value function, called the Hamilton-JacobiBellman equation. Let the d x d matrix a = a.a' (with' the transpose of the matrix). We consider the uniformly pambolic case, Le. we assume that there exists c > 0 such that V$ E 0, Vu E U, Vy E IRd ,2:t,j=l aij(x, U)YiYj 2: c1lY112. Then V is C2 (see [Kry80J). Let Vx be the gradient of V and VXiXj its second-order partial derivatives. Theorem 1 (Hamilton-Jacohi-Bellman) The following HJB equation holds : Vex) In 'Y + sup [rex, u) + Vx(x).J(x, u) + ! 2:~j=l aij VXiXj (x)] = 0 for x E 0 uEU Besides, V satisfies the following boundary condition: Vex) = R(x) for x E 80. Reinforcement Learningfor Continuous Stochastic Control Problems 1031 Remark 1 The challenge of learning the VF is motivated by the fact that from V, we can deduce the following optimal feed-back control policy: u*(x) E arg sup [r(x, u) + Vx(x).f(x, u) + ! L:7,j=l aij VXiXj (x)] uEU In the following, we assume that 0 is bounded. Let eI, ... , ed be a basis for JRd. Let the positive and negative parts of a function 4> be : 4>+ = ma.x(4),O) and 4>- = ma.x( -4>,0). For any discretization step 8, let us consider the lattices: 8Zd = {8. L:~=1 jiei} where j}, ... ,jd are any integers, and ~6 = 8Zd n O. Let 8~6, the frontier of ~6 denote the set of points {~ E 8Zd \ 0 such that at least one adjacent point ~ ± 8ei E ~6} (see figure 1). Let U6 cUbe a finite control set that approximates U in the sense: 8 ~ 8' => U6' C U6 and U6U6 = U. Besides, we assume that: Vi = l..d, (3) By replacing the gradient Vx(~) by the forward and backward first-order finitedifference quotients: ~;, V(~) = l [V(~ ± 8ei) V(~)l and VXiXj (~) by the secondorder finite-difference quotients: ~XiXi V(~) -b [V(~ + 8ei) + V(,' - 8ei) - 2V(O] ~;iXj V(~) = 2P[V(~ + 8ei ± 8ej) + V(~ - 8ei =F 8ej) -V(~ + 8ei) V(~ - 8ei) V(~ + 8ej) V(~ - 8ej) + 2V(~)] in the HJB equation, we obtain the following : for ~ E :£6, V6(~)In,+SUPUEUh {r(~,u) + L:~=1 [f:(~,u)'~~iV6(~) fi-(~,U)'~;iV6(~) + aii (~.u) ~ . . V(C) + " . . (at; (~,'U) ~ + . V(C) _ a~ (~,'U) ~ - . . V(C))] } = 0 2 X,X,'" wJ'l=~ 2 x,x.1'" 2 x,xJ '" Knowing that (~t In,) is an approximation of ( ,l:l.t -1) as ~t tends to 0, we deduce: V6(~) SUPuEUh [,"'(~'U)L(EEbP(~,U,()V6«()+T(~,u)r(~,u)] (4) with T(~, u) (5) which appears as a DP equation for some finite Markovian Decision Process (see [Ber87]) whose state space is ~6 and probabilities of transition: p(~,u,~ ± 8ei) p(~, u, ~ + 8ei ± 8ej) p(~,u,~ - 8ei ± 8ej) p(~,u,() "'~~r) [28Ift(~, u)1 + aii(~' u) - Lj=l=i laij(~, u)l] , "'~~r)a~(~,u)fori=f:j, (6) "'~~r)a~(~,u) for i =f: j, o otherwise. Thanks to a contraction property due to the discount factor" there exists a unique solution (the fixed-point) V to equation (4) for ~ E :£6 with the boundary condition V6(~) = R(~) for ~ E 8:£6. The following theorem (see [Kus90] or [FS93]) insures that V 6 is a convergent approximation scheme. 1032 R. Munos and P. Bourgine Theorem 2 (Convergence of the FD scheme) V D converges to V as 8 1 0 : lim /)10 VD(~) = Vex) un~formly on 0 ~-x Remark 2 Condition (3) insures that the p(~, u, () are positive. If this condition does not hold, several possibilities to overcome this are described in [Kus90j. 3 The reinforcement learning algorithm Here we assume that f is bounded from below. As the state dynami,:s (J and a) is unknown from the system, we approximate it by building a model f and a from samples of trajectories Xk(t) : we consider series of successive states Xk = Xk(tk) and Yk = Xk(tk + Tk) such that: - "It E [tk, tk + Tk], x(t) E N(~) neighbourhood of ~ whose diameter is inferior to kN.8 for some positive constant kN, - the control u is constant for t E [tk, tk + Tk], - T k satisfies for some positive kl and k2, (7) Then incrementally update the model : .1 ",n Yk - Xk n ~k=l Tk an(~,u) 1 n (Yk - Xk - Tk.fn(~, u)) (Yk - Xk - Tk·fn(~, u))' -;;; Lk=l Tk (8) and compute the approximated time T( x, u) ~d the approximated probabilities of transition p(~, u, () by replacing f and a by f and a in (5) and (6). We obtain the following updating rule of the V D -value of state ~ : V~+l (~) = sUPuEU/) [,~/:(x,u) L( p(~, u, ()V~(() + T(x, u)r(~, u)] (9) which can be used as an off-line (synchronous, Gauss-Seidel, asynchronous) or ontime (for example by updating V~(~) as soon as a trajectory exits from the neighbourood of ~) DP algorithm (see [BBS95]). Besides, when a trajectory hits the boundary [JO at some exit point Xk(T) then update the closest state ~ E [JED with: (10) Theorem 3 (Convergence of the algorithm) Suppose that the model as well as the V D -value of every state ~ E :ED and control u E UD are regularly updated (respectively with (8) and (9)) and that every state ~ E [JED are updated with (10) at least once. Then "Ie> 0, :3~ such that "18 ~ ~, :3N, "In 2: N, sUP~EE/) IV~(~) V(~)I ~ e with probability 1 Reinforcement Learningfor Continuous Stochastic Control Problems 1033 4 Conclusion This paper presents a model-based RL algorithm for continuous stochastic control problems. A model of the dynamics is approximated by the mean and the covariance of successive states. Then, a RL updating rule based on a convergent FD scheme is deduced and in the hypothesis of an adequate exploration, the convergence to the optimal solution is proved as the discretization step 8 tends to 0 and the number of iteration tends to infinity. This result is to be compared to the model-free RL algorithm for the deterministic case in [Mun97]. An interesting possible future work should be to consider model-free algorithms in the stochastic case for which a Q-Iearning rule (see [Wat89]) could be relevant. A Appendix: proof of the convergence Let M f ' Ma, M fr. and Ma .• be the upper bounds of j, a, f x and 0' x and m f the lower bound of f. Let EO = SUP€EI:h !V0(';) - V(';)I and E! = SUP€EI:b \V~(';) - VO(.;)\. A.I Estimation error of the model fn and an and the probabilities Pn Suppose that the trajectory Xk(t) occured for some occurence Wk(t) of the brownian motion: Xk(t) = Xk + f!k f(Xk(t),u)dt + f!" a(xk(t),U)dwk. Then we consider a trajectory Zk (t) starting from .; at tk and following the same brownian motion: Zk(t) ='; + fttk. f(Zk(t), u)dt + fttk a(zk(t), U)dWk' Let Zk = Zk(tk + Tk). Then (Yk - Xk) - (Zk -.;) = ftk [f(Xk(t), u) - f(Zk(t), u)] dt + ftt:.+Tk [a(xk(t), u) - a(zk(t), u)J dWk. Thus, from the C1 property of f and a, II(Yk - Xk) - (Zk - ';)11 ~ (Mf'" + M aJ.kN.Tk.8. (11) The diffusion processes has the following property ~ee for example the ItO-Taylor majoration in [KP95j) : Ex [ZkJ = ';+Tk.f(';, U)+O(Tk) which, from (7), is equivalent to: Ex [z~:g] = j(';,u) + 0(8). Thus from the law of large numbers and (11): li~-!~p Ilfn(';, u) - f(';, u)11 li;;:s~p II~ L~=l [Yk;kX& - ¥.] II + 0(8) (Mf:r: + M aJ·kN·8 + 0(8) = 0(8) w.p. 1 (12) Besides, diffusion processes have the following property (again see [KP95J): Ex [(Zk -.;) (Zk - .;)'] = a(';, U)Tk + f(';, u).f(';, U)'.T~ + 0(T2) which, from (7), is equivalent to: Ex [(Zk-€-Tkf(S'U)~(kZk-S-Tkf(S'U»/] = a(';, u) + 0(82). Let rk = Zk -.; - Tkf(';, u) and ik = Yk - Xk - Tkfn(';, u) which satisfy (from (11) and (12» : Ilrk - ikll = (Mf:r: + M aJ.Tk.kN.8 + Tk.o(8) (13) From the definition of Ci;;(';,u), we have: Ci;;(';,u) - a(';,u) = ~L~=l '\:1.' Ex [r~':k] + 0(82 ) and from the law of large numbers, (12) and (13), we have: li~~~p 11~(';,u) - a(';,u)11 = li~-!~p II~ L~=l rJ./Y - r~':k II + 0(82 ) Ilik -rkllli:!s!p~ fl (II~II + II~II) +0(82 ) = 0(82 ) 1034 R. Munos and P. Bourgine "In(';, u) - I(';, u)" ~ kf·8 w.p. 1 1Ill;;(';, u) - a(';, u)1I ~ ka .82 w.p. 1 (14) Besides, from (5) and (14), we have: 1 (c ) _ (c )1 < d.(k[.6 2+d.k,,62 ) J:2 < k J:2 T r.",U Tn r.",U _ (d.m,.6)2 U _ T'U (15) and from a property of exponential function, I,T(~.u) _ ,7' .. (€ .1£) I = kT.In ~ .82. (16) We can deduce from (14) that: . 1 ( ) -( )1 (2.6.Mt+d.Ma)(2.kt+d.k,,)62 k J: limsupp';,u,( -Pn';,u,( ~ 6mr(2.k,+d.ka)62 S; puw.p.l n-+oo (17) A.2 Estimation of IV~+l(';) - V6(.;) 1 Mter having updated V~(';) with rule (9), let A denote the difference IV~+l(';) - V6(.;) I. From (4), (9) and (8), A < ,T(€.U) L: [P(';, u, () - p(.;, u, ()] V6 (() + ( ,T(€.1£) ,7'(~'1£») L p(.;, u, () V 6 (() ( ( +,7' (€.u) . L:p(.;, u, () [V6(() V~(()] + L:p(.;, u, ().T(';, u) [r(';, u) - F(';, u)] ( ( + L:( p(.;, u, () [T(';, u) - T(';, u)] r(';, u) for all u E U6 As V is differentiable we have : Vee) = V(';) + VX ' (( -.;) + 0(1I( - ';11). Let us define a linear function V such that: Vex) = V(';) + VX ' (x - ';). Then we have: [P(';, u, () - p(.;, u, ()] V6(() = [P(';, u, () - p(.;, u, ()] . [V6(() - V(()] + [P(';,u,()-p(';,u,()]V((), thus: L:([p(';,u,()-p(';,u,()]V6(() = kp .E6.8 + L([P(';,U,()-p(.;,u,()] [V(() +0(8)] = [V(7J)-VUD] + kp .E6.8 + 0(8) = [V(7J) - V(1j)] + 0(8) with: 7J = L:( p(';, u, () (( -.;) and 1j = L:( p(.;, u, () (( - .;). Besides, from the convergence of the scheme (theorem 2), we have E6.8 = 0(8). From the linearity of V, IV(() - V(Z) I ~ II( -ZII·Mv", S; 2kp 82 . Thus IL( [P(';, u, () - p(.;, u, ()] V6 (() I = 0(8) and from (15), (16) and the Lipschitz property of r, A = 1'l'(€'U), L:( p(.;, u, () [V6(() - V~ (()] 1+ 0(8). As ,..,.7'(€.u) < 1 7'(€.U) In 1 < 1 _ T(€.u)-k.,.62 In 1 < 1 _ ( 6 _ !ix..82) In 1 I 2 'Y 2 'Y 2d(M[+d.M,,) 2 'Y ' we have: A = (1 k.8)E~ + 0(8) (18) with k = 2d(M[~d.M,,). Reinforcement Learning for Continuous Stochastic Control Problems 1035 A.3 A sufficient condition for sUP€EE~ IV~(~) V6(~)1 :S C2 Let us suppose that for all ~ E ~6, the following conditions hold for some a > 0 E~ > C2 =} IV~+I(O V6(~)1 :S E~ - a (19) E~ :S c2=}IV~+I(~)_V6(~)I:Sc2 (20) From the hypothesis that all states ~ E ~6 are regularly updated, there exists an integer m such that at stage n + m all the ~ E ~6 have been updated at least once since stage n. Besides, since all ~ E 8C6 are updated at least once with rule (10), V~ E 8C6, IV~(~) V6(~)1 = IR(Xk(T)) R(~)I :S 2.LR.8 :S C2 for any 8 :S ~3 = 2~lR' Thus, from (19) and (20) we have: E! > C2 =} E!+m :S E! - a E! :S C2 =} E!+m :S C2 Thus there exists N such that: Vn ~ N, E~ :S C2. A.4 Convergence of the algorithm Let us prove theorem 3. For any c > 0, let us consider Cl > 0 and C2 > 0 such that Cl +C2 = c. Assume E~ > £2, then from (18), A = E! - k.8'£2+0(8) :S E~ -k.8.~ for 8 :S ~3. Thus (19) holds for a = k.8.~. Suppose now that E~ :S £2. From (18), A :S (1 - k.8)£2 + 0(8) :S £2 for 8 :S ~3 and condition (20) is true. Thus for 8 :S min { ~1, ~2, ~3}, the sufficient conditions (19) and (20) are satisfied. So there exists N, for all n ~ N, E~ :S £2. Besides, from the convergence of the scheme (theorem 2), there exists ~o st. V8:S ~o, sUP€EE~ 1V6(~) V(~)I :S £1· Thus for 8 :S min{~o, ~1, ~2, ~3}, "3N, Vn ~ N, sup IV~(~) V(~)I :S sup IV~(~) V6(~)1 + sup 1V6(~) V(~)I :S £1 + c2 = £. €EE6 €EEh €EE6 References [BBS95j Andrew G. Barto, Steven J. Bradtke, and Satinder P. Singh. Learning to act using real-time dynamic programming. Artificial Intelligence, (72):81138, 1995. [Ber87j Dimitri P. Bertsekas. Dynamic Programming: Deterministic and Stochastic Models. Prentice Hall, 1987. [FS93j Wendell H. Fleming and H. Mete Soner. Controlled Markov Processes and Viscosity Solutions. Applications of Mathematics. Springer-Verlag, 1993. [KP95j Peter E. Kloeden and Eckhard Platen. Numerical Solutions of Stochastic Differential Equations. Springer-Verlag, 1995. [Kry80j N.V. Krylov. Controlled Diffusion Processes. Springer-Verlag, New York, 1980. [Kus90j Harold J. Kushner. Numerical methods for stochastic control problems in continuous time. SIAM J. Control and Optimization, 28:999-1048, 1990. [Mun97j [Wat89j Remi Munos. A convergent reinforcement learning algorithm in the continuous case based on a finite difference method. International Joint Conference on Art~ficial Intelligence, 1997. Christopher J.C.H. Watkins. Learning from delayed reward. PhD thesis, Cambridge University, 1989.
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Hierarchical Non-linear Factor Analysis and Topographic Maps Zoubin Ghahramani and Geoffrey E. Hinton Dept. of Computer Science, University of Toronto Toronto, Ontario, M5S 3H5, Canada http://www.cs.toronto.edu/neuron/ {zoubin,hinton}Ocs.toronto.edu Abstract We first describe a hierarchical, generative model that can be viewed as a non-linear generalisation of factor analysis and can be implemented in a neural network. The model performs perceptual inference in a probabilistically consistent manner by using top-down, bottom-up and lateral connections. These connections can be learned using simple rules that require only locally available information. We then show how to incorporate lateral connections into the generative model. The model extracts a sparse, distributed, hierarchical representation of depth from simplified random-dot stereograms and the localised disparity detectors in the first hidden layer form a topographic map. When presented with image patches from natural scenes, the model develops topographically organised local feature detectors. 1 Introduction Factor analysis is a probabilistic model for real-valued data which assumes that the data is a linear combination of real-valued uncorrelated Gaussian sources (the factors). After the linear combination, each component of the data vector is also assumed to be corrupted by additional Gaussian noise. A major advantage of this generative model is that, given a data vector, the probability distribution in the space of factors is a multivariate Gaussian whose mean is a linear function of the data. It is therefore tractable to compute the posterior distribution exactly and to use it when learning the parameters of the model (the linear combination matrix and noise variances). A major disadvantage is that factor analysis is a linear model that is insensitive to higher order statistical structure of the observed data vectors. One way to make factor analysis non-linear is to use a mixture of factor analyser modules, each of which captures a different linear regime in the data [3]. We can view the factors of all of the modules as a large set of basis functions for describing the data and the process of selecting one module then corresponds to selecting an appropriate subset of the basis functions. Since the number of subsets under consideration is only linear in the number of modules, it is still tractable to compute Hierarchical Non-linear Factor Analysis and Topographic Maps 487 the full posterior distribution when given a data point. Unfortunately, this mixture model is often inadequate. Consider, for example, a typical image that contains multiple objects. To represent the pose and deformation of each object we want a componential representation of the object's parameters which could be obtained from an appropriate factor analyser. But to represent the multiple objects we need several of these componential representations at once, so the pure mixture idea is not tenable. A more powerful non-linear generalisation of factor analysis iF to have a large set of factors and to allow any subset of the factors to be selected. This can be achieved by using a generative model in which there is a high probability of generating factor activations of exactly zero. 2 Rectified Gaussian Belief Nets The Rectified Gaussian Belief Net (RGBN) uses multiple layers of units with states that are either positive real values or zero [5]. Its main disadvantage is that computing the posterior distribution over the factors given a data vector involves Gibbs sampling. In general, Gibbs sampling can be very time consuming, but in practice 10 to 20 samples per unit have proved adequate and there are theoretical reasons for believing that learning can work well even when the Gibbs sampling fails to reach equilibrium [10]. We first describe the RGBN without considering neural plausibility. Then we show how lateral interactions within a layer can be used to perform probabilistic inference correctly using locally available information. This makes the RGBN far more plausible as a neural model than a sigmoid belief net [9, 8] because it means that Gibbs sampling can be performed without requiring units in one layer to see the total top-down input to units in the layer below. The generative model for RGBN's consists of multiple layers of units each of which has a real-valued unrectified state, Yj, and a rectified state, [Yj]+, which is zero if Yj is negative and equal to Yj otherwise. This rectification is the only non-linearity in the network. 1 The value of Yj is Gaussian distributed with a standard deviation (Jj and a mean, ih that is determined by the generative bias, gOj, and the combined effects of the rectified states of units, k, in the layer above: Yj = gOj + Lgkj[Yk]+ k (1) The rectified state [Yj]+ therefore has a Gaussian distribution above zero, but all of the mass of the Gaussian that falls below zero is concentrated in an infinitely dense spike at zero as shown in Fig. la. This infinite density creates problems if we attempt to use Gibbs sampling over the rectified states, so, following a suggestion by Radford Neal, we perform Gibbs sampling on the unrectified states. Consider a unit, j, in some intermediate layer of a multilayer RGBN. Suppose that we fix the unrectified states of all the other units in the net. To perform Gibbs sampling, we need to stochastically select a value for Yj according to its distribution given the unrectified states of all the other units. If we think in terms of energy functions, which are equal to negative log probabilities (up to a constant), the rectified states of the units in the layer above contribute a quadratic energy term by determining Yj. The unrectified states of units, i, in the layer below contribute a constant if [Yj]+ is 0, and if [Yj]+ is positive they each contribute a quadratic term 1 The key arguments presented in this paper hold for general nonlinear belief networks as long as the noise is Gaussian; they are not specific to the rectification nonlinearity. 488 a b c W / I I I ~----..J,' Top-down , , '-_ .. " -3-2-1 0 1 2 3 -3-2-1 0 1 2 3 Y Y because of the effect of [Yj] + on Yi. Z Ghahramani and G. E. Hinton Figure 1: a) Probability density in which all the mass of a Gaussian below zero has been replaced by an infinitely dense spike at zero. b) Schematic of the density of a unit's unrectified state. c) Bottomup and top-down energy functions corresponding to b. (2) where h is an index over all the units in the same layer as j including j itself. Terms that do not depend on Yj have been omitted from Eq. 2. For values of Yj below zero there is a quadratic energy function which leads to a Gaussian distribution. The same is true for values of Yj above zero, but it is a different quadratic (Fig. Ic) . The Gaussian distributions corresponding to the two quadratics must agree at Yj = 0 (Fig. Ib). Because this distribution is piecewise Gaussian it is possible to perform Gibbs sampling exactly. Given samples from the posterior, the generative weights of a RGBN can be learned by using the online delta rule to maximise the log probability of the data. 2 (3) The variance of the local Gaussian noise of each unit, o}, can also be learned by an online rule, D-.o} = f [(Yj - Yj)2 - o}]. Alternatively, o} can be fixed at I for all hidden units and the effective local noise level can be controlled by scaling the generative weights. 3 The Role of Lateral Connections in Perceptual Inference In RGBNs and other layered belief networks, fixing the value of a unit in one layer causes correlations between the parents of that unit in the layer above. One of the main reasons why purely bottom-up approaches to perceptual inference have proven inadequate for learning in layered belief networks is that they fail to take into account this phenomenon, which is known as "explaining away." Lee and Seung (1997) introduced a clever way of using lateral connections to handle explaining away effects during perceptual inference. Consider the network shown in Fig. 2. One contribution, Ebelow, to the energy of the state of the network is the squared difference between the unrectified states of the units in one layer, Yj, a.nd the top-down expectations generated by the states of units in the layer above. Assuming the local noise models for the lower layer units all have unit variance, and 2 If Gibbs sampling has not been run long enough to reach equilibrium, the delta rule follows the gradient of the penalized log probability of the data [10]. The penalty term is the Kullback-Liebler divergence between the equilibrium distribution and the distribution produced by Gibbs sampling. Other things being equal, the delta rule therefore adjusts the parameters that determine the equilibrium distribution to reduce this penalty, thus favouring models for which Gibbs sampling works quickly. Hierarchical Non-linear Factor Analysis and Topographic Maps 489 ignoring biases and constant terms that are unaffected by the states of the units Ebe\ow = ~ l:)Yj - Yj)2 = ~ I)Yj - 2:k[Yk]+9kj)2. j j (4) Rearranging this expression and setting rjk = gkj and mkl = - Lj gkjglj we get Ebe\ow = ~ LyJ - L[Yk]+ LYjrjk ~ L[Yk]+ L[y!l+mkl . (5) j k j k I This energy function can be exactly implemented in a network with recognition weights, rjk, and symmetric lateral interactions, mkl. The lateral and recognition connections allow a unit, k, to compute how Ebe\ow for the layer below depends on its own state and therefore they allow it to follow the gradient of E or to perform Gibbs sampling in E . Figure 2: A small segment of a network, showing the generative weights (dashed) and the recognition and lateral weights (solid) which implement perceptual inference and correctly handle explaining away effects. Seung's trick can be used in an RGBN and it eliminates the most neurally implausible aspect of this model which is that a unit in one layer appears to need to send both its state Y and the top-down prediction of its state Y to units in the layer above. Using the lateral connections, the units in the layer above can, in effect, compute all they need to know about the top-down predictions. In computer simulations, we can simply set each lateral connection mkl to be the dot product - 2:j gkjglj. It is also possible to learn these lateral connections in a more biologically plausible way by driving units in the layer below with unit-variance independent Gaussian noise and using a simple anti-Hebbian learning rule. Similarly, a purely local learning rule can learn recognition weights equal to the generative weights . . If units at one layer are driven by unit-variance, independent Gaussian noise, and these in turn drive units in the layer below using the generative weights, then Hebbian learning between the two layers will learn the correct recognition weights [5]. 4 Lateral Connections in the Generative Model When the generative model contains only top-down connections, lateral connections make it possible to do perceptual inference using locally available information. But it is also possible, and often desirable, to have lateral connections in the generative model. Such connections can cause nearby units in a layer to have a priori correlated activities, which in turn can lead to the formation of redundant codes and, as we will see, topographic maps. Symmetric lateral interactions between the unrectified states of units within a layer have the effect of adding a quadratic term to the energy function EMRF = ~ L: L Mkl YkYI, (6) k I which corresponds to a Gaussian Markov Random Field (MRF). During sampling, this term is simply added to the top-down energy contribution. Learning is more difficult. The difficulty sterns from the need to know the derivatives of the partition function of the MRF for each data vector. This partition function depends on the 490 Z Ghahramani and G. E. Hinton top-down inputs to a layer so it varies from one data vector to the next, even if the lateral connections themselves are non-adaptive. Fortunately, since both the MRF and the top-down prediction define Gaussians over the states of the units in a layer, these derivatives can be easily calculated. Assuming unit variances, tlYj; = , ([Yj]+(Y; - ii;) + [Yj]+ ~ [M(I + M)-ll;. ii.) (7) where M is the MRF matrix for the layer including units i and k, and I is the identity matrix. The first term is the delta rule (Eq. 3); the second term is the derivative of the partition function which unfortunately involves a matrix inversion. Since the partition function for a multivariate Gaussian is analytical it is also possible to learn the lateral connections in the MRF. Lateral interactions between the rectified states of units add the quadratic term ~ Lk Ll Mkl [Yk]+[YzJ+· The partition function is no longer analytical, so computing the gradient of the likelihood involves a two-phase Boltzmann-like procedure: !19ji = f ([Yj]+Yi) * - ([Yj]+Yi r) , (8) where 0* averages with respect to the posterior distribution of Yi and Yj, and 0averages with respect to the posterior distribution of Yj and the prior of Yi given units in the same layer as j. This learning rule suffers from all the problems of the Boltzmann machine, namely it is slow and requires two-phases. However, there is an approximation which results in the familiar one-phase delta rule that can be described in three equivalent ways: (1) it treats the lateral connections in the generative model as if they were additional lateral connections in the recognition model; (2) instead of lateral connections in the generative model it assumes some fictitious children with clamped values which affect inference but whose likelihood is not maximised during learning; (3) it maximises a penalized likelihood of the model without the lateral connections in the generative model. 5 Discovering depth in simplified stereograms Consider the following generative process for stereo pairs. Random dots of uniformly distributed intensities are scattered sparsely on a one-dimensional surface, and the image is blurred with a Gaussian filter. This surface is then randomly placed at one of two different depths, giving rise to two possible left-to-right disparities between the images seen by each eye. Separate Gaussian noise is then added to the image seen by each eye. Some images generated in this manner are shown in Fig. 3a. Figure 3: a) Sample data from the stereo disparity problem. The left and right column of each 2 x 32 image are the inputs to the left and right eye, respectively. Periodic boundary conditions were used. The value of a pixel is represented by the size of the square, with white being positive and black being negative. Notice that pixel noise makes it difficult to infer the disparity, i.e. the vertical shift between the left and right columns, in some images. b) Sample images generated by the model after learning. We trained a three-layer RGBN consisting of 64 visible units, 64 units in the first hidden layer and 1 unit in the second hidden layer on the 32-pixel wide stereo Hierarchical Non-linear Factor Analysis and Topographic Maps 491 disparity problem. Each of the hidden units in the first hidden layer was connected to the entire array of visible units, i.e. it had inputs from both eyes. The hidden units in this layer were also laterally connected in an MRF over the unrectified units. Nearby units excited each other and more distant units inhibited each other, with the net pattern of excitation/inhibition being a difference of two Gaussians. This MRF was initialised with large weights which decayed exponentially to zero over the course of training. The network was trained for 30 passes through a data set of 2000 images. For each image we used 16 iterations of Gibbs sampling to approximate the posterior distribution over hidden states. Each iteration consisted of sampling every hidden unit once in a random order. The states after the fourth iteration of Gibbs sampling were used for learning, with a learning rate of 0.05 and a weight decay parameter of 0.001. Since the top level of the generative process makes a discrete decision between left and right global disparity we used a trivial extension of the RGBN in which the top level unit saturates both at 0 and 1. a ._--="TI:~£:I=-[J __ IEI[I:I _1II_-=-_.-:rr::JI...___I:IIUI::JI-L1D-.--:tIl::Jl-=-::l .-'-' _______ OW''--o--.,u'-'-''__=_-..._.-'-"._ b c Figure 4: Generative weights of a three-layered RGBN after being trained on the stereo disparity problem. a) Weights from the top layer hidden unit to the 64 middle-layer hidden units. b) Biases of the middle-layer hidden units, and c) weights from the hidden units to the 2 x 32 visible array. Thirty-two of the hidden units learned to become local left-disparity detectors, while the other 32 became local right-disparity detectors (Fig. 4c). The unit in the second hidden layer learned positive weights to the left-disparity detectors in the layer below, and negative weights to the right detectors (Fig. 4a). In fact, the activity of this top unit discriminated the true global disparity of the input images with 99% accuracy. A random sample of images generated by the model after learning is shown in Fig. 3b. In addition to forming a hierarchical distributed representation of disparity, units in the hidden layer self-organised into a topographic map. The MRF caused high correlations between nearby units early in learning, which in turn resulted in nearby units learning similar weight vectors. The emergence of topography depended on the strength of the MRF and on the speed with which it decayed. Results were relatively insensitive to other parametric changes. We also presented image patches taken from natural images [1] to a network with units in the first hidden layer arranged in laterally-connected 2D grid. The network developed local feature detectors, with nearby units responding to similar features (Fig. 5). Not all units were used, but the unused units all clustered into one area. 6 Discussion Classical models of topography formation such as Kohonen's self-organising map [6] and the elastic net [2, 4] can be thought of as variations on mixture models where additional constraints have been placed to encourage neighboring hidden units to have similar generative weights. The problem with a mixture model is that it cannot handle images in which there are several things going on at once. In contrast, we 492 Z. Ghahramani and G. E. Hinton Figure 5: Generative weights of an RGBN trained on 12 x 12 natural image patches: weights from each of the 100 hidden units which were arranged in a 10 x 10 sheet with toroidal boundary conclitions. have shown that topography can arise in much richer hierarchical and componential generative models by inducing correlations between neighboring units. There is a sense in which topography is a necessary consequence of the lateral connection trick used for perceptual inference. It is infeasible to interconnect all pairs of units in a cortical area. If we assume that direct lateral interactions (or interactions mediated by interneurons) are primarily local, then widely separated units will not have the apparatus required for explaining away. Consequently the computation of the posterior distribution will be incorrect unless the generative weight vectors of widely separated units are orthogonal. If the generative weights are constrained to be positive, the only way two vectors can be orthogonal is for each to have zeros wherever the other has non-zeros. Since the redundancies that the hidden units are trying to model are typically spatially localised, it follows that widely separated units must attend to different parts of the image and units can only attend to overlapping patches if they are laterally interconnected. The lateral connections in the generative model assist in the formation of the topography required for correct perceptual inference. Acknowledgements. We thank P. Dayan, B. Frey, G. Goodhill, D. MacKay, R. Neal and M. Revow. The research was funded by NSERC and ITRC. GEH is the Nesbitt-Burns fellow of CIAR. References [1] A. Bell & T. J. Sejnowski. The 'Independent components' of natural scenes are edge filters. Vision Research, In Press. [2] R. Durbin & D. Willshaw. An analogue approach to the travelling salesman problem using an elastic net method. Nature, 326(16):689-691, 1987. [3] Z. Ghahramani & G. E. Hinton. The EM algorithm for mixtures of factor analyzers. Univ. Toronto Technical Report CRG-TR-96-1, 1996. [4] G. J. Goodhill & D. J. Willshaw. Application of the elatic net algorithm to the formation of ocular dominance stripes. Network: Compo in Neur. Sys., 1:41-59, 1990. [5] G. E. Hinton & Z. Ghahramani. Generative models for cliscovering sparse clistributed representations. Philos. Trans. Roy. Soc. B, 352:1177-1190, 1997. [6] T. Kohonen. Self-organized formation of topologically correct feature maps. Biological Cybernetics, 43:59-69, 1982. [7] D. D. Lee & H. S. Seung. Unsupervised learning by convex and conic cocling. In M. Mozer, M. Jordan, & T. Petsche, eds., NIPS 9. MIT Press, Cambridge, MA, 1997. [8] M. S. Lewicki & T. J. Sejnowski. Bayesian unsupervised learning of higher order structure. In NIPS 9. MIT Press, Cambridge, MA, 1997. [9] R. M. Neal. Connectionist learning of belief networks. Arti/. Intell., 56:71-113, 1992. [10] R. M. Neal & G. E. Hinton. A new view of the EM algorithm that justifies incremental and other variants. Unpublished Manuscript, 1993.
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Learning Path Distributions using Nonequilibrium Diffusion Networks Paul Mineiro * pmineiro~cogsci.ucsd.edu Department of Cognitive Science University of California, San Diego La Jolla, CA 92093-0515 Javier Movellan movellan~cogsci.ucsd.edu Department of Cognitive Science University of California, San Diego La Jolla, CA 92093-0515 Ruth J. Williams williams~math.ucsd.edu Department of Mathematics University of California, San Diego La Jolla, CA 92093-0112 Abstract We propose diffusion networks, a type of recurrent neural network with probabilistic dynamics, as models for learning natural signals that are continuous in time and space. We give a formula for the gradient of the log-likelihood of a path with respect to the drift parameters for a diffusion network. This gradient can be used to optimize diffusion networks in the nonequilibrium regime for a wide variety of problems paralleling techniques which have succeeded in engineering fields such as system identification, state estimation and signal filtering. An aspect of this work which is of particular interest to computational neuroscience and hardware design is that with a suitable choice of activation function, e.g., quasi-linear sigmoidal, the gradient formula is local in space and time. 1 Introduction Many natural signals, like pixel gray-levels, line orientations, object position, velocity and shape parameters, are well described as continuous-time continuous-valued stochastic processes; however, the neural network literature has seldom explored the continuous stochastic case. Since the solutions to many decision theoretic problems of interest are naturally formulated using probability distributions, it is desirable to have a flexible framework for approximating probability distributions on continuous path spaces. Such a framework could prove as useful for problems involving continuous-time continuous-valued processes as conventional hidden Markov models have proven for problems involving discrete-time sequences. Diffusion networks are similar to recurrent neural networks, but have probabilistic dynamics. Instead of a set of ordinary differential equations (ODEs), diffusion networks are described by a set of stochastic differential equations (SDEs). SDEs provide a rich language for expressing stochastic temporal dynamics and have proven • To whom correspondence should be addressed. Learning Path Distributions Using Nonequilibriwn Diffusion Networks 599 Figure 1: An example where the average of desirable paths yields an undesirable path, namely one that collides with the tree. useful in formulating continuous-time statistical inference problems, resulting in such solutions as the continuous Kalman filter and generalizations of it like the condensation algorithm (Isard & Blake, 1996). A formula is given here for the gradient of the log-likelihood of a path with respect to the drift parameters for a diffusion network. Using this gradient we can potentially optimize the model to approximate an entire probability distribution of continuous paths, not just average paths or equilibrium points. Figure 1 illustrates the importance of this kind of learning by showing a case in which learning average paths would have undesirable results, namely collision with a tree. Experience has shown that learning distributions of paths, not just averages, is crucial for dynamic perceptual tasks in realistic environments, e.g., visual contour tracking (Isard & Blake, 1996). Interestingly, with a suitable choice of activation function, e.g., quasilinear sigmoidal, the gradient formula depends only upon local computations, i.e., no time unfolding or explicit backpropagation of error is needed. The fact that noise localizes the gradient is of potential interest for domains such as theoretical neuroscience, cognitive modeling and hardware design. 2 Diffusion Networks Hereafter Cn refers to the space of continuous Rn-valued functions over the time interval [0, TJ, with T E R, T > ° fixed throughout this discussion. A diffusion network with parameter A E RP is a random process defined via an Ito SD E of the form dX(t) = JL(t, X(t), A)dt + adB(t), (1) X(O) '" 1I, where X is a Cn-valued process that represents the temporal dynamics of the n nodes in the network; JL: [0, T] x R n x RP -+ Rn is a deterministic function called the drift; A E RP is the vector of drift parameters, e.g., synaptic weights, which are to be optimized; B is a Brownian motion process which provides the random driving term for the dynamics; 1I is the initial distribution of the solution; and a E R, a > 0, is a fixed constant called the dispersion coefficient, which determines the strength of the noise term. In this paper we do not address the problem of optimizing the dispersion or the initial distribution of X. For the existence and uniqueness in law of the solution to (1) JL(',', A) must satisfy some conditions. For example, it is sufficient that it is Borel measurable and satisfies a linear growth condition: IJL(t, x, A)I S; K-\(l + Ixl) for some K-\ > ° and all t E [0, T], x ERn; see 600 P. Mineiro, 1. Movellan and R. 1. Williams (Karatzas & Shreve, 1991, page 303) for details. It is typically the case that the n-dimensional diffusion network will be used to model d-dimensional observations with n > d. In this case we divide X into hidden and observablel components, denoted Hand 0 respectively, so that X = (H,O). Note that with a = 0 in equation (1), the model becomes equivalent to a continuoustime deterministic recurrent neural network. Diffusion networks can therefore be thought of as neural networks with "synaptic noise" represented by a Brownian motion process. In addition, diffusion networks have Markovian dynamics, and hidden states if n > d; therefore, they are also continuous-time continuous-state hidden Markov models. As with conventional hidden Markov models, the probability density of an observable state sequence plays an important role in the optimization of diffusion networks. However, because X is a continuous-time process, care must taken in defining a probability density. 2.1 Density of a continuous observable path Let (XA, B>') defined on some filtered probability space (0, F, {Fd, p) be a (weak) solution of (1) with fixed parameter A. Here X>' = (HA,O>') represents the states of the network and is adapted to the filtration {Fd, B>' is an n-dimensional {Fdmartingale Brownian motion and the filtration {Ft } satisfies the usual conditions (Karatzas and Shreve, 1991, page 300). Let Q>' be the unique probability law generated by any weak solution of (1) with fixed parameter A Q>'(A) = p(X>' E A) for all A E F, (2) where F is the Borel sigma algebra generated by the open sets of Cn. Setting n = Cn, nh = Cn- d, and no = Cd with associated Borel a-algebras F, Fh and Fo, respectively, we have n = nh x no, F = Fh ® Fo, and we can define the marginal laws for the hidden and observable components of the network by Q~(Ah) = QA(Ah x Cd) ~ P(HA E Ah) for all Ah E Fh, (3) Q~(Ao) = Q>'(Cn-d X Ao) ~ p(O>' E Ao) for all Ao E Fo. (4) For our purposes the appropriate generalization of the notion of a probability density on Rm to the general probability spaces considered here is the Radon-Nikodym derivative with respect to a reference measure that dominates all members of the family {Q>'} >'ERp (Poor, 1994, p.264ff). A suitable reference measure P is the law of the solution to (1) with zero drift (IJ = 0). The measures induced by this reference measure over Fh and Fo are denoted by Ph and Po, respectively. Since in the reference model there are no couplings between any of the nodes in the network, the hidden and observable processes are independent and it follows that P(Ah x Ao) = Ph (Ah)Po(Ao) for all Ah E Fh,Ao E .1'0' (5) The conditions on IJ mentioned above are sufficient to ensure a Radon-Nikodym derivative for each QA with respect to the reference measure. Using Girsanov's Theorem (Karatzas & Shreve, 1991, p.190ff) its form can be shown to be Z'(w) = ~~ (w) = exp { :' lT I'(t,w(t), A) . dw(t) - 2~' lT II'(t, w(t), A)I'dt} , wE \1, (6) lIn our treatment we make no distinction between observables which are inputs and those which are outputs. Inputs can be conceptualized as observables under "environmental control," i.e., whose drifts are independent of both A and the hidden and output processes. Learning Path Distributions Using Nonequilibriwn Diffusion Networks 601 where the first integral is an Ito stochastic integral. The random variable Z>. can be interpreted as a likelihood or probability density with respect to the reference mode12 . However equation (6) defines the density of Rn-valued paths of the entire network, whereas our real concern is the density of Rd-valued observable paths. Denoting wE 0 as w = (Wh,wo) where Wh E Oh and Wo E 0 0 , note that (7) (8) and therefore the Radon-Nikodym derivative of Q~ with respect to Po, the density of interest, is given by Z;(wo) = ~~: (wo) = EPh[Z>'(., wo»)' Wo E 0 0 , (9) 2.2 Gradient of the density of an observable path The gradient of Z; with respect to A is an important quantity for iterative optimization of cost functionals corresponding to a variety of problems of interest, e.g., maximum likelihood estimation of diffusion parameters for continuous path density estimation. Formal differentiation3 of (9) yields where \7>.logZ~(wo) = EPh[Z~lo( · ,wo)\7),logZ'\(.,wo )), (10) >. 6 Z)'(w) Zh lo(w) = Z;(wo) ' \7),logZ)'(w) = -\- rT J(t,W(t),A)' dl(w,t), a Jo J. ( A) ~ 8J.lk(t, x , A) Jk t,x, 8A ' ' J lew, t) ~ wet) - w(O) -lot J.l(s,w(s), A)ds. (11) (12) (13) (14) Equation (10) states that the gradient of the density of an observable path can be found by clamping the observable nodes to that path and performing an average of Z~lo \7.x log Z)' with respect to Ph, i.e., average with respect to the hidden paths distributed as a scaled Brownian motion. This makes intuitive sense: the output gradient of the log density is a weighted average of the total gradient of the log density, where each hidden path contributes according to its likelihood Z~lo given the output. In practice to evaluate the gradient, equation (10) must be approximated. Here we use Monte Carlo techniques, the efficiency of which can be improved by sampling according to a density which reduces the variance of the integrand. Such a density 2To ease interpretation of (6) consider the simpler case of a one-dimensional Gaussian random variable with mean JL and variance (J"2. The ratio of the density of such a model with respect to an equivalent model with zero mean is exp(-f<JLx ~JL2). Equation (6) can be viewed as a generalization of this same idea to Brownian motion. 3See (Levanony et aL, 1990) for sufficient conditions for the differentiation in equation (10) to be valid. 602 P. Mineiro, J. Movellan and R. 1. Williams is available for models with hidden dynamics which do not explicitly depend upon the observables, i.e., the observable nodes do not send feedback connections to the hidden states. Models which obey this constraint are henceforth denoted factorial. Denoting J-Lh and J-Lo as the hidden and observable components, respectively, of the drift vector, and Bh and Bo as the hidden and observable components, respectively, of the Brownian motion, for a factorial network we have dH(t) = J-Lh(t, H(t), >')dt + adBh(t), dO(t) = J-Lo(t, H(t), O(t), >')dt + adBo(t). (15) (16) The drift for the hidden variables does not depend on the observables, and Girsanov's theorem gives us an explicit formula for the density of the hidden process. dQ>' { 1 iT Z~(Wh) = dP: (Wh) = exp a2 0 J-Lh(t, Wh(t), >.) . dwh(t) - 2~2 /,T Il'h(t,wh(t), ,\)I'dt} . Equations (9) and (10) can then be written in the form Z;(wo) = EQ~ [Zolh(', wo)], >. Q). [Z;lh("WO) >. 1 v\ log Zo (wo) = E h Z;(wo) yo >.log Z (', wo) , where II Z>'(w) { 1 iT Z;lh(W) = Z~(Wh) = exp a2 0 J-Lo(t,w(t), >.) . dwo(t) - 2~' /,T lI'o( t, w( t), ,\) I' dt } . (17) (18) (19) (20) Note the expectations are now performed using the measure Q~. We can easily generate samples according to Q~ by numerically integrating equation (15), and in practice this leads to more efficient Monte Carlo approximations of the likelihood and gradient. 3 Example: Noisy Sinusoidal Detection This problem is a simple example of using diffusion networks for signal detection. The task was to detect a sinusoid in the presence of additive Gaussian noise. Stimuli were generated according to the following process Y(t,w) = 1A(w).!.sin(47l't) + B(t,w), (21) 7l' where t E [0,1/2]. Here Y is assumed anchored in a probability space (0, F, P) large enough to accommodate the event A which indicates a signal or noise trial. Note that under P, B is a Brownian motion on Cd independent of A. A model was optimized using 100 samples of equation (21) given W E A, i.e., 100 stimuli containing a signal. The model had four hidden units and one observable unit (n = 5, d = 1). The drift of the model was given by J-L(t, x, >.) = () + W . g(x), (22) 1 gj(x) = 1 + e-Zj , j E {I, 2, 3, 4, 5}, Learning Path Distributions Using Nonequilibriwn Diffusion Networks 0.8 l" 0.6 ! ,I t 0.4 rf ~ 0.2 ROC Curve, Sinewave Detection Problem ...•. ---A----~----~- --·~ ·· ··-·T-.-.. -. d'=1.89 ---.---1" ....... o L-____ ~ __ ~~ __ ~ ____ ~ ____ ~ o 0.2 0.4 0.6 0.8 hit rate 603 Figure 2: Receiver operating characteristic (ROC) curve for a diffusion network performing a signal detection task involving noisy sinusoids, Dotted line: Detection performance estimated numerically using 10000 novel stimuli. Solid line: Best fit curve corresponding to d' = 1.89. This value of d' corresponds to performance within 1.5% of the Bayesian limit. where 0 E R5 and W is a 5x5 real-valued connection matrix. In this case ~ = {{Oi}, {Wij }, i,j = 1, ... ,5}. The connections from output to hidden units were set to zero, allowing use of the more efficient techniques for factorial networks described above. The initial distribution for the model was a 8-function at (1, -1, 1, -1,0). The model was numerically simulated with ilt = 0.01, and 100 hidden samples were used to approximate the likelihood and gradient of the log- likelihood, according to equations (18) and (19). The conjugate gradient algorithm was used for training, with the log-likelihood of the data as the cost function. Once training was complete, the parameter estimation was tested using 10000 novel stimuli and the following procedure. Given a new stimuli y we used the model to estimate the likelihood Zo(Y I A) ~ Z~(Y), where ~ is the parameter vector at the end of training. The decision rule employed was D(Y) = {sig.nal if Zo(~ I A) > b, nOIse otherwIse, (23) where b E R is a bias term representing assumptions about the apriori probability of a signal trial. By sweeping across different values of b the receiver-operator characteristic (ROC) curve is generated. This curve shows how the probability of a hit, P(D = signal I A), and the probability of a false alarm, P(D = signal I AC), are related. From this curve the parameter d', a measure of sensitivity independent of apriori assumptions, can be estimated. Figure 2 shows the ROC curve as found by numerical simulation, and the curve obtained by the best fit value d' = 1.89. This value of d' corresponds to a 82.7% correct detection rate for equal prior signal probabilities. The theoretically ideal observer can be derived for this problem, since the profile of the unperturbed signal is known exactly (Poor, 1994, p. 278ff). For this problem the optimal observer achieves d'max = 2, which implies at equal probabilities for signal and noise trials, the Bayesian limit corresponds to a 84.1 % correct detection rate. The detection system based upon the diffusion network is therefore operating close to the Bayesian limit, but was designed using only implicit information, i.e., 100 training examples, about the structure of the signal to be detected, in contrast to the explicit information required to design the optimal Bayesian classifier.
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Unsupervised On-Line Learning of Decision Trees for Hierarchical Data Analysis Marcus Held and Joachim M. Buhmann Rheinische Friedrich-Wilhelms-U niversitat Institut fUr Informatik III, ROmerstraBe 164 D-53117 Bonn, Germany email: {held.jb}.cs.uni-bonn.de WWW: http://www-dbv.cs.uni-bonn.de Abstract An adaptive on-line algorithm is proposed to estimate hierarchical data structures for non-stationary data sources. The approach is based on the principle of minimum cross entropy to derive a decision tree for data clustering and it employs a metalearning idea (learning to learn) to adapt to changes in data characteristics. Its efficiency is demonstrated by grouping non-stationary artifical data and by hierarchical segmentation of LANDSAT images. 1 Introduction Unsupervised learning addresses the problem to detect structure inherent in unlabeled and unclassified data. The simplest, but not necessarily the best approach for extracting a grouping structure is to represent a set of data samples X = {Xi E Rdli = 1, ... ,N} by a set of prototypes y = {Ya E Rdlo = 1, .. . ,K}, K « N. The encoding usually is represented by an assignment matrix M = (Mia), where Mia = 1 if and only if Xi belongs to cluster 0, and Mia = 0 otherwise. According to this encoding scheme, the cost function 1i (M, Y) = ~ L:~1 MiaV (Xi, Ya) measures the quality of a data partition, Le., optimal assignments and prototypes (M,y)OPt = argminM,y1i (M,Y) minimize the inhomogeneity of clusters w.r.t. a given distance measure V. For reasons of simplicity we restrict the presentation to the ' sum-of-squared-error criterion V(x, y) = !Ix - YI12 in this paper. To facilitate this minimization a deterministic annealing approach was proposed in [5] which maps the discrete optimization problem, i.e. how to determine the data assignments, via the Maximum Entropy Principle [2] to a continuous parameter esUnsupervised On-line Learning of Decision Trees for Data Analysis 515 timation problem. Deterministic annealing introduces a Lagrange multiplier {3 to control the approximation of 11. (M, Y) in a probabilistic sense. Equivalently to maximize the entropy at fixed expected K-means costs we minimize the free energy :F = ~ 2:f::1ln (2::=1 exp (-{3V (Xi, Ya:))) w.r.t. the prototypes Ya:. The assignments Mia: are treated as random variables yielding a fuzzy centroid rule N N Ya: = L i=l (Mia:)xdLi=l (Mia:) , (1) where the expected assignments (Mia:) are given by Gibbs distributions (Mia:) = :xp (-{3V (Xi,Ya:)) . (2) 2:/l=1 exp ( -{3V (Xi, Ya:)) For a more detailed discussion of the DA approach to data clustering cf. [1, 3, 5]. In addition to assigning data to clusters (1,2), hierarchical clustering provides the partitioning of data space with a tree structure. Each data sample X is sequentially assigned to a nested structure of partitions which hierarchically cover the data space Rd. This sequence of special decisions is encoded by decision rules which are attached to nodes along a path in the tree (see also fig. 1). Therefore, learning a decision tree requires to determine a tree topology, the accompanying assignments, the inner node labels S and the prototypes y at the leaves. The search of such a hierarchical partition of the data space should be guided by an optimization criterion, i.e., minimal distortion costs. This problem is solvable by a two-stage approach, which on the one hand minimizes the distortion costs at the leaves given the tree structure and on the other hand optimizes the tree structure given the leaf induced partition of Rd. This approach, due to Miller & Rose [3], is summarized in section 2. The extensions for adaptive online learning and experimental results are described in sections 3 and 4, respectively. x ~ S S partition /'\ of data space • S S j'\ /4\ a a b c d e f Figure 1: Right: Topology of a decision tree. Left: Induced partitioning of the data space (positions of the letters also indicate the positions of the prototypes). Decisions are made according to the nearest neighbor rule. 2 Unsupervised Learning of Decision Trees Deterministic annealing of hierarchical clustering treats the assignments of data to inner nodes of the tree in a probabilistic way analogous to the expected assignments of data to leaf prototypes. Based on the maximum entropy principle, the probability ~~j that data point Xi reaches inner node Sj is recursively defined by (see [3]): exp (-,V(Xi,Sj)) ~~root:= 1, ~~j = ~~parent(j)1ri,j, 1ri,j = 2: exp(-,V(Xi,Sk)) , (3) kEsiblings(j) 516 M. Held and 1. M. Buhmann where the Lagrange multiplier, controls the fuzziness of all the transitions 1fi,j' On the other hand, given the tree topology and the prototypes at the leaves, the maximum entropy principle naturally recommends an ideal probability cpLl< at leaf Yet, resp. at an inner node sj> cp~ = exp(-j1V(Xi,Yet)) and cpl. = :E CP!k' (4) ',et L: exp(-j1V(Xi'Y/L)) l,J kEdescendants(j) , /LEY We apply the principle of minimum cross entropy for the calculation of the prototypes at the leaves given a priori the probabilities for the parents of the leaves. Minimization of the cross entropy with fixed expected costs (HXi) = L:et (Miet)V (Xi, Yet) for the data point Xi yields the expression m~n I({(Miet)}II{Cp~parent(et)/K}) = min Let (Miet) In cpJMiet) , (5) {(M.e»} {(M.e>)} i,parent(et) where I denotes the Kullback-Leibler divergence and K defines the degree of the inner nodes. The tilted distribution ( Cp~parent(et) exp (-j1V (Xi, Yet)) Miet) = H . L: /L cp i,parent(/L) exp ( - j1V (Xi, Y /L)) (6) generalizes the probabilistic assignments (2). In the case of Euclidian distances we again obtain the centroid formula (1) as the minimum of the free energy F = - h L::l ln [L:etEY cprparent(et) exp (-j1V (Xi, Yet))]. Constraints induced by the tree structure are incorporated in the assignments (6). For the optimization of the hierarchy, Miller and Rose in a second step propose the minimization of the distance between the hierarchical probabilities CP~. and the ideal probabilities Cp~ ., the distance being measured by the Kullback-Leibler divergence ' N cp~ . ~T L I ({ Cp~,j }II{ Cp~j}) == ~,W L :E cpL In cp~J . (7) , BjEparent(Y) BjEparent(Y)i=l t,J Equation (7) describes the minimization of the sum of cross entropies between the probability densities CP~. and CP~. over the parents of the leaves. Calculating the gradients for the inner ;'odes S j ~d the Lagrange multiplier, we receive N N -2, L (Xi - Sj) {cpL cp!,parent(j)1fi,j} := -2,:E ~1 (Xi, Sj), (8) i=l i=l N N L:E V (Xi, Sj) {cpL cpLparent(j)1fi,j} := L:E ~2 (Xi, Sj). (9) i=l jES i=l jES The first gradient is a weighted average of the difference vectors (Xi - Sj), where the weights measure the mismatch between the probability CPtj and the probability induced by the transition 1fi,j' The second gradient (9) measures the scale - V (Xi, Sj) - on which the transition probabilities are defined, and weights them with the mismatch between the ideal probabilities. This procedure yields an algorithm which starts at a small value j1 with a complete tree and identical test vectors attached to all nodes. The prototypes at the leaves are optimized according to (6) and the centroid rule (1), and the hierarchy is optimized by (8) and (9). After convergence one increases j1 and optimizes the hierarchy and the prototypes at the leaves again. The increment of j1leads to phase transitions where test vectors separate from each other and the formerly completely degenerated tree evolves its structure. For a detailed description of this algorithm see [3]. Unsupervised On-line Learning of Decision Trees for Data Analysis 517 3 On-Line Learning of Decision Trees Learning of decision trees is refined in this paper to deal with unbalanced trees and on-line learning of trees. Updating identical nodes according to the gradients (9) with assignments (6) weighs parameters of unbalanced tree structures in an unsatisfactory way. A detailed analysis reveals that degenerated test vectors, i.e., test vectors with identical components, still contribute to the assignments and to the evolution of /. This artefact is overcome by using dynamic tree topologies instead of a predefined topology with indistinguishable test vectors. On the other hand, the development of an on-line algorithm makes it possible to process huge data sets and non-stationary data. For this setting there exists the need of on-line learning rules for the prototypes at the leaves, the test vectors at the inner nodes and the parameters / and (3. Unbalanced trees also require rules for splitting and merging nodes. Following Buhmann and Kuhnel [1] we use an expansion of order O(I/n) of (1) to estimate the prototypes for the Nth datapoint N '" N-l (M:;;I) ( N-I) Ya '" Ya + 'TJa N-1M XN - Ya , POI (10) where P~ ~ p~-1 +1/M ((M:;;I) - p~-l) denotes the probability of the occurence of class o. The parameters M and'TJa are introduced in order to take the possible non-stationarity of the data source into account. M denotes the size of the data window, and 'TJa is a node specific learning rate. Adaptation of the inner nodes and of the parameter / is performed by stochastic approximation using the gradients (8) and (9) (11) (12) For an appropriate choice of the learning rates 'TJ, the learning to learn approach of Murata et al. [4] suggests the learning algorithm (13) The flow 1 in parameter space determines the change of w N -1 given a new datapoint XN. Murata et al. derive the following update scheme for the learning rate: rN (1 - 8)rN- 1 + 81 (XN, W N- 1 ) , 'TJN _ 'TJN- I + Vl",N-l (v2I1rNII- 'TJN-l) , (14) (15) where VI, v2 and 8 are control parameters to balance the tradeoff between accuracy and convergence rate. rN denotes the leaky average of the flow at time N. The adaptation of (3 has to observe the necessary condition for a phase transition (3 > (3erit == 1/28rnax , 8rnax being the largest eigenvalue of the covariance matrix [3] M M ~a = L (Xi - Ya) (Xi - Ya)t (Mia)/L(Mia ). (16) i=l i=l Rules for splitting and merging nodes of the tree are introduced to deal with unbalanced trees and non-stationary data. Simple rules measure the distortion costs at the prototypes of the leaves. According to these costs the leaf with highest 518 M Held and 1. M Buhmann distortion costs is split. The merging criterion combines neighboring leaves with minimal distance in a greedy fashion. The parameter M (10), the typical time scale for changes in the data distribution is used to fix the time between splitting resp. merging nodes and the update of (3. Therefore, M controls the time scale for changes of the tree topology. The learning parameters for the learning to learn rules (13)-(15) are chosen empirically and are kept constant for all experiments. 4 Experiments The first experiment demonstrates how a drifting two dimensional data source can be tracked. This data source is generated by a fixed tree augmented with transition probabilities at the edges and with Gaussians at the leaves. By descending the tree structure this ~enerates an Li.d. random variable X E R2, which is rotated around the origin of R to obtain a random variable T(N) = R(w, N)X. R is an orthogonal matrix, N denotes the number of the actual data point and w denotes the angular velocity, M = 500. Figure 2 shows 45 degree snapshots of the learning of this nonstationary data source. We start to take these snapshots after the algorithm has developed its final tree topology (after ~ 8000 datapoints). Apart from fluctuations of the test vectors at the leaves, the whole tree structure is stable while tracking the rotating data source. Additional experiments with higher dimensional data sources confirm the robustness of the algorithm w.r.t. the dimension of the data space, i. e. similiar tracking performances for different dimensions are observed, where differences are explained as differences in the data sources (figure 3). This performance is measured by the variance of the mean of the distances between the data source trajectory and the trajectories of the test vectors at the nodes of the tree. 7 ) 8) Figure 2: 45 degree snapshots of the learning of a data source which rotates with a velocity w = 271"/30000 (360 degree per 30000 data samples:. A second experiment demonstrates the learning of a switching data source. The results confirm a good performance concerning the restructuring of the tree (see figure 4). In this experiment the algorithm learns a given data source and after 10000 data points we switch to a different source. As a real-world example of on-line learning of huge data sources the algorithm is applied to the hierarchical clustering of 6- dimensional LANDSAT data. The heat Unsupervised On-line Learning of Decision Trees for Data Analysis -0.5 ·1 ' 1.5 I ~ . ~ -3.5 ~ -4 -4.5 ·5 ·5.5 0 2dim4dim ~ .-12dim ..... 18dim . 10000 20000 30000 40000 50000 60000 70000 80000 90000 N 519 Figure 3: Tracking performance for different dimensions. As data sources we use d-dimensional Gaussians which are attached to a unit sphere. To the components of every random sample X we add sin(wN) in order to introduce non stationarity. The first 8000 samples are used for the development of the tree topology. A ) \ ~, k II B) \ ~"" k / a \ b 1\ "'-, . 1 \ h \ 1 ""., . / 1 \ \ J " "\" \ 9" , ~, .>,\" . L ' .. __ \ .• '( ..____. \ _ , ___ __. '" ._________ c . ._" ,-_.-" -'-----?/l i >I--."/_-J--; v' j ''-.--~.~ " ".( - #-\01f;;<1 \ __ }-;- I g f <' J \ ;r" " n'/ .'6> 1• -\--<I!..~ tf '\' ~ ____ C e I:' \\Q \ h \,m -----b~~<\ :' 1 :1'" e ', \' \ ' I l/ / \ \ \" \ , / \ \' \ \ I \ \ I / \1 ~ a b m J k 1 defg nohi Figure 4: Learning a switching data source: top: a) the partition of the data space after 10000 data samples given the first source, b) the restructured partition after additional 2500 samples. Below: accompanying tree topologies. channel has been discarded because of its reduced resolution. In a preprocessing step all channels are rescaled to unit variance, which alternatively could be established by using a Mahalanobis distance. Note that the decision tree which clusters this data supplies us with a hierarchical segmentation of the corresponding LANDSAT image. A tree of 16 leaves has been learned on a training set of 128 x 128 data samples, and it has been applied to a test set of 128 x 128 LANDSAT pixels. The training is established by 15 sequential runs through the test set, where after each M = 16384 run a split of one node is carried out. The resulting empirical errors (0.49 training distortion and 0.55 test distortion) differ only slightly from the errors obtained by the LBG algorithm applied to the whole training set (0.42 training distortion and 0.52 test distortion). This difference is due to the fact that not every data point is assigned to the nearest leaf prototype by a decision tree induced partition. The segmentation of the test image is depicted in figure 5. 5 Conclusion This paper presents a method for unsupervised on-line learning of decision trees. We overcome the shortcomings of the original decision tree approach and extend 520 M Held and J. M. Buhmann Figure 5: Hierarchical segmentation of the test image. The root represents the original image, i.e., the gray scale version of the three color channels. it to the realm of on-line learning of huge data sets and of adaptive learning of non-stationary data. Our experiments demonstrate that the approach is capable of tracking gradually changing Or switching environments. Furthermore, the method has been successfully applied to the hierarchical segmentation of LANDSAT images. Future work will address active data selection issues to significantly reduce the uncertainty of the most likely tree parameters and the learning questions related to different tree topologies. Acknowledgement: This work has been supported by the German Israel Foundation for Science and Research Development (GIF) under grant #1-0403-001.06/95 and by the Federal Ministry for Education, Science and Technology (BMBF #01 M 3021 A/4). References [1] J. M. Buhmann and H. Kuhnel. Vector quantization with complexity costs. IEEE Transactions on Information Theory, 39(4):1133-1145, July 1993. [2] T.M. Cover and J. Thomas. Elements of Information Theory. Wiley & Sons, 1991. [3] D. Miller and K Rose. Hierarchical unsupervised learning with growing via phase transitions. Neural Computation, 8:425-450, February 1996. [4] N. Murata, K-R. Muller, A. Ziehe, and S. Amari. Adaptive on-line learning in changing environments. In M.C. Mozer, M.I. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems, number 9, pages 599-605. MIT Press, 1997. [5] K Rose, E. Gurewitz, and G.C. Fox. A deterministic annealing approach to clustering. Pattern Recognition Letters, 11(9):589-594, September 1990.
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Recovering Perspective Pose with a Dual Step EM Algorithm Andrew D.J. Cross and Edwin R. Hancock, Department of Computer Science, University of York, York, YOl 5DD, UK. Abstract This paper describes a new approach to extracting 3D perspective structure from 2D point-sets. The novel feature is to unify the tasks of estimating transformation geometry and identifying pointcorrespondence matches. Unification is realised by constructing a mixture model over the bi-partite graph representing the correspondence match and by effecting optimisation using the EM algorithm. According to our EM framework the probabilities of structural correspondence gate contributions to the expected likelihood function used to estimate maximum likelihood perspective pose parameters. This provides a means of rejecting structural outliers. 1 Introduction The estimation of transformational geometry is key to many problems of computer vision and robotics [10]. Broadly speaking the aim is to recover a matrix representation of the transformation between image and world co-ordinate systems. In order to estimate the matrix requires a set of correspondence matches between features in the two co-ordinate systems [11]. Posed in this way there is a basic chickenand-egg problem. Before good correspondences can be estimated, there need to be reasonable bounds on the transformational geometry. Yet this geometry is, after all, the ultimate goal of computation. This problem is usually overcome by invoking constraints to bootstrap the estimation of feasible correspondence matches [5, 8]. One of the most popular ideas is to use the epipolar constraint to prune the space of potential correspondences [5]. One of the drawbacks of this pruning strategy is that residual outliers may lead to ill-conditioned or singular parameter matrices [11]. Recovering Perspective Pose with a Dual Step EM Algorithm 781 The aim in this paper is to pose the two problems of estimating transformation geometry and locating correspondence matches using an architecture that is reminiscent of the hierarchical mixture of experts algorithm [6]. Specifically, we use a bi-partite graph to represent the current configuration of correspondence match. This graphical structure provides an architecture that can be used to gate contributions to the likelihood function for the geometric parameters using structural constraints. Correspondence matches and transformation parameters are estimated by applying the EM algorithm to the gated likelihood function. In this way we arrive at dual maximisation steps. Maximum likelihood parameters are found by minimising the structurally gated squared residuals between features in the two images being matched. Correspondence matches are updated so as to maximise the a posteriori probability of the observed structural configuration on the bi-partite association graph. We provide a practical illustration in the domain of computer vision which is aimed at matching images of floppy discs under severe perspective foreshortening. However, it is important to stress that the idea of using a graphical model to provide structural constraints on parameter estimation is a task of generic importance. Although the EM algorithm has been used to extract affine and Euclidean parameters from point-sets [4] or line-sets [9], there has been no attempt to impose structural constraints of the correspondence matches. Viewed from the perspective of graphical template matching [1, 7] our EM algorithm allows an explicit deformational model to be imposed on a set of feature points. Since the method delivers statistical estimates for both the transformation parameters and their associated covariance matrix it offers significant advantages in terms of its adaptive capabilities. 2 Perspective Geometry Our basic aim is to recover the perspective transformation parameters which bring a set of model or fiducial points into correspondence with their counterparts in a set of image data. Each point in the image data is represented by an augmented vector of co-ordinates Wi = (Xi, Yi, l)T where i is the point index. The available set of image points is denoted by w = {Wi' Vi E 'D} where'D is the point index-set. The fiducial points constituting the model are similarly represented by the set of augmented co-ordinate vectors z = b j , Vj EM}. Here M is the index-set for the model feature-points and the 'l!j represent the corresponding image co-ordinates. Perspective geometry is distinguished from the simpler Euclidean (translation, rotation and scaling) and affine (the addition of shear) cases by the presence of significant foreshortening. We represent the perspective transformation by the parameter matrix ( ¢>(n) 1,1 ~(n) = ,hen) 'f'2,1 ¢>(n) 3,1 (1) Using homogeneous co-ordinates, the transformation between model and data is zen) = ( 1 )-l~(n)z. where \lI(n) = (,h(n) ,hen) 1)T is a column-vector formed -J ZT .'lI(n) -J' 'f'3,1 ''f'3,2 , -J from the elements in bottom row of the transformation matrix. 782 A. D. J. Cross and E. R. Hancock 3 Relational Constraints One of our goals in this paper is to exploit structural constraints to improve the recovery of perspective parameters from sets of feature points. We abstract the process as bi-partite graph matching. Because of its well documented robustness to noise and change of viewpoint, we adopt the Delaunay triangulation as our basic representation of image structure [3]. We establish Delaunay triangulations on the data and the model, by seeding Voronoi tessellations from the feature-points. Tlie process of Delaunay triangulation generates relational graphs from the two sets of point-features. More formally, the point-sets are the nodes of a data graph GD = {V,ED} and a model graph GM = {M,EM}. Here ED ~ V X V and EM ~ M x M are the edge-sets of the data and model graphs. Key to our matching process is the idea of using the edge-structure of Delaunay graphs to constrain the correspondence matches between the two point-sets. This correspondence matching is denoted by the function j : M -+ V from the nodes of the data-graph to those of the model graph. According to this notation the statement j(n)(i) = j indicates that there is a match between the node i E V of the model-graph to the node j E M of the model graph at iteration n of the algorithm. We use the binary indicator s~n) = {I if j(n)(i) = j (2) t,) 0 otherwise to represent the configuration of correspondence matches. We exploit the structure of the Delaunay graphs to compute the consistency of match using the Bayesian framework for relational graph-matching recently reported by Wilson and Hancock [12]. Suffice to say that consistency of a configuration of matches residing on the neighbourhood Ri = i U {k ; (i, k) E ED} of the node i in the data-graph and its counterpart Sj = j U {I ; (j,l) E Em} for the node j in the model-graph is gauged by Hamming distance. The Hamming distance H(i,j) counts the number of matches on the data-graph neighbourhood Ri that are inconsistently matched onto the model-graph neighbourhood Sj. According to Wilson and Hancock [12] the structural probability for the correspondence match j(i) = j at iteration n of the algorithm is given by exp [-/3H(i,j)] ( ~) = ----=----;:-----"-----:;t,) LjEM exp [-/3H(i,j)] (3) In the above expression, the Hamming distance is given by H(i,j) L(k,I)ER;eSj (l-si~h where the symbol- denotes the composition of the data-graph relation Ri and the model-graph relation Sj. The exponential constant /3 = In 1 Ft, is related to the uniform probability of structural matching errors Pe . This probability is set to reflect the overlap of the two point-sets. In the work reported here p. 2I1MI-ID\1 we set e I1MI+IDI . 4 The EM Algorithm Our aim is to extract perspective pose parameters and correspondences matches from the two point-sets using the EM algorithm. According to the original work Recovering Perspective Pose with a Dual Step EM Algorithm 783 of Dempster, Laird and Rubin [2] the expected likelihood function is computed by weighting the current log-probability density by the a posteriori measurement probabilities computed from the preceding maximum likelihood parameters. Jordan and Jacobs [6] augment the process with a graphical model which effectively gates contributions to the expected log-likelihood function. Here we provide a variant of this idea in which the bi-partite graph representing the correspondences matches gate the log-likelihood function for the perspective pose parameters. 4.1 Mixture Model Our basic aim is to jointly maximize the data-likelihood p(wlz, f,~) over the space of correspondence matches f and the matrix of perspective parameters ~. To commence our development, we assume observational independence and factorise the conditional measurement density over the set of data-items p(wlz, f,~) = II p(wil z , f,~) (4) iE'D In order to apply the apparatus of the EM algorithm to maximising p(wlz,f,~) with respect to f and ~, we must establish a mixture model over the space of correspondence matches. Accordingly, we apply Bayes theorem to expand over the space of match indicator variables. In other words, p(wilz,j,~) = L P(Wi,Si,jIZ,f,~) (5) Si ,jE! In order to develop a tractable likelihood function, we apply the chain rule of conditional probability. In addition, we use the indicator variables to control the switching of the conditional measurement densities via exponentiation. In other words we assume p(wilsi,j'~j,~) = p(wil~j, ~)Si.j. With this simplification, the mixture model for the correspondence matching process leads to the following expression for the expected likelihood function Q(f(n+l), ~(n+l)lf(n), ~(n») = L L P(si,jIW, z, f(n), ~(n»)s~~) Inp(WiIZj' ~(n+1») iE'D iEM (6) To further simplify matters we make a mean-field approximation and replace s~~) by its average value, i.e. we make use of the fact that E(s~~») = (i,'j). In this way the structural matching probabilities gate contributions to the expected likelihood function. This mean-field approximation alleviates problems associated with local optima which are likely to occur if the likelihood function is discretised by gating with Si,j' 4.2 Expectation Using the Bayes rule, we can re-write the a posteriori measurement probabilities in terms of the components of the conditional measurement densities appearing in the mixture model in equation (5) r(n)p(w Iz ~(n») P( .. 1 fen) ~(n+1») = ':.i,j -i -j, St,} W, z, , ~ (n) Lj/EM (i,j' p(wil?;jl, ~(n») (7) 784 A. D. J. Cross and E. R. Hancock In order to proceed with the development of a point registration process we require a model for the conditional measurement densities, i.e. p(wil?;j, cf>(n»). Here we assume that the required model can be specified in terms of a multivariate Gaussian distribution. The random variables appearing in these distributions are the error residuals for the position predictions of the jth model line delivered by the current estimated transformation parameters. Accordingly we write (n») _ 1 [ 1 ( (n»)T~-l ( _ (n»)] ) p(wil?;j,cf> (27l")~Mexp -2 Wi -?;j L..J Wi 'l.j (8 In the above expression ~ is the variance-covariance matrix for the vector of errorresiduals fi,j(cf>(n») = Wi - 'l.;n) between the components of the predicted measurement vectors 'l.j and their counterparts in the data, i.e. Wi. Formally, the matrix is related to the expectation of the outer-product of the error-residuals i.e. ~ = E[fi,j(cf>(n»)fi,j(cf>(n»)T]. 4.3 Maximisation The maximisation step of our matching algorithm is based on two coupled update processes. The first of these aims to locate maximum a posteriori probability correspondence matches. The second class of update operation is concerned with locating maximum likelihood transformation parameters. We effect the coupling by allowing information flow between the two processes. Correspondences located by maximum a posteriori graph-matching are used to constrain the recovery of maximum likelihood transformation parameters. A posteriori measurement probabilities computed from the updated transformation parameters are used to refine the correspondence matches. In terms of the indicator variables matches the configuration of maximum a posteriori probability correspondence matches is updated as follows exp [-f3 LCk,I)ER,.Sj (1 - s~~l)] j(n+1)(i) = argmaxP(?; .IWi,cf>(n») (9) JEM J [ (n) ] LjEM exp -f3 L(k ,I)ER,.Sj (1 - sk ,l ) The maximum likelihood transformation parameters satisfy the condition cf>(n+l) = argmin '""' '""' P(z.lw . cf>(n»);~~)(w . z(n»)T~-l(w . - zen») (10) ~ L...J L...J -J -P "1 ,J -1 -J -1-J iE'DiEM In the case of perspective geometry where we have used homogeneous co-ordinates the saddle-point equations are not readily amenable in a closed-form linear fashion. Instead, we solve the non-linear maximisation problem using the LevenbergMarquardt technique. This non-linear optimisation technique offers a compromise between the steepest gradient and inverse Hessian methods. The former is used when close to the optimum while the latter is used far from it. 5 Experiments The real-world evaluation of our matching method is concerned with recognising planer objects in different 3D poses. The object used in this study is a 3.5 inch Recovering Perspective Pose with a Dual Step EM Algorithm 785 floppy disk which is placed on a desktop. The scene is viewed with a low-quality SGI IndyCam. The feature points used to triangulate the object are corners. Since the imaging process is not accurately modelled by a perspective transformation under pin-hole optics, the example provides a challenging test of our matching process. Our experiments are illustrated in Figure 1. The first two columns show the views under match. In the first example (the upper row of Figure 1) we are concerned with matching when there is a significant difference in perspective forshortening. In the example shown in the lower row of Figure 1, there is a rotation of the object in addition to the foreshortening. The images in the third column are the initial matching configurations. Here the perspective parameter matrix has been selected at random. The fourth column in Figure 1 shows the final matching configuration after the EM algorithm has converged. In both cases the final registration is accurate. The algorithm appears to be capable of recovering good matches even when the initial pose estimate is poor. Figure 1: Images Under Match, Initial and Final Configurations. We now turn to measuring the sensitivity of our method. In order to illustrate the benefits offered by the structural gating process, we compare its performance with a conventional least-squares parameter estimation process. Figure 2 shows a comparison of the two algorithms for a problem involving a point-set of 20 nodes. Here we show the RMS error as a function of the number of points which have correct correspondence matches. The break-even point occurs when 8 nodes are initially matched correctly and there are 12 errors. Once the number of initially correct correspondences exceeds 8 then the EM method consistently outperforms the least-squares estimation. 6 Conclusions Our main contributions in this paper are twofold. The theoretical contribution has been to develop a mixture model that allows a graphical structure to to constrain the estimation of maximum likelihood model parameters. The second contribution is a practical one, and involves the application of the mixture model to the estimation of perspective pose parameters. There are a number of ways in which the ideas developed in this paper can be extended. For instance, the framework is readily extensible to the recognition of more complex non-planar objects. 786 ,. 12 ~ c!! 08 f f os .. o. References ! I ", 1 l't- " i. A. D. J. Cross and E. R. Hancock LSF a1andardLSF strucknI ····· Figure 2: Structural Sensitivity. [1] Y. Amit and A. Kong, "Graphical Templates for Model Registration", IEEE PAMI, 18, pp. 225-236, 1996. (2] A.P. Dempster, Laird N.M. and Rubin D.B., "Maximum-likelihood from incomplete data via the EM algorithm", J. Royal Statistical Soc. Ser. B (methodological},39, pp 1-38, 1977. (3] O.D. Faugeras, E. Le Bras-Mehlman and J-D. Boissonnat, "Representing Stereo Data with the Delaunay Triangulation", Artificial Intelligence, 44, pp. 41-87, 1990. (4] S. Gold, Rangarajan A. and Mjolsness E., "Learning with pre-knowledge: Clustering with point and graph-matching distance measures", Neural Computation, 8, pp. 787804, 1996. (5] R.I. Hartley, "Projective Reconstruction and Invariants from Multiple Images", IEEE PAMI, 16, pp. 1036-1041, 1994. (6] M.I. Jordan and R.A. Jacobs, "Hierarchical Mixtures of Experts and the EM Algorithm" , Neural Computation, 6, pp. 181-214, 1994. [7] M. Lades, J .C. Vorbruggen, J. Buhmann, J. Lange, C. von der Maalsburg, R.P. Wurtz and W.Konen, "Distortion-invariant object-recognition in a dynamic link architecture", IEEE Transactions on Computers, 42, pp. 300-311, 1993 [8] D.P. McReynolds and D.G. Lowe, "Rigidity Checking of 3D Point Correspondences under Perspective Projection", IEEE PAMI, 18 , pp. 1174-1185, 1996. (9] S. Moss and E.R. Hancock, "Registering Incomplete Radar Images with the EM Algorithm", Image and Vision Computing, 15, 637-648, 1997. [10] D. Oberkampf, D.F. DeMenthon and L.S. Davis, "Iterative Pose Estimation using Coplanar Feature Points", Computer Vision and Image Understanding, 63, pp. 495511, 1996. (11] P. Torr, A. Zisserman and S.J. Maybank, "Robust Detection of Degenerate Configurations for the Fundamental Matrix", Proceedings of the Fifth International Conference on Computer Vision, pp. 1037-1042, 1995. (12] R.C. Wilson and E.R. Hancock, "Structural Matching by Discrete Relaxation", IEEE PAMI, 19, pp.634-648 , 1997.
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Learning to Order Things William W. Cohen Robert E. Schapire Yoram Singer AT&T Labs, 180 Park Ave., Florham Park, NJ 07932 { wcohen,schapire,singer} @research.att.com Abstract There are many applications in which it is desirable to order rather than classify instances. Here we consider the problem of learning how to order, given feedback in the form of preference judgments, i.e., statements to the effect that one instance should be ranked ahead of another. We outline a two-stage approach in which one first learns by conventional means a preference Junction, of the form PREF( u, v), which indicates whether it is advisable to rank u before v. New instances are then ordered so as to maximize agreements with the learned preference function. We show that the problem of finding the ordering that agrees best with a preference function is NP-complete, even under very restrictive assumptions. Nevertheless, we describe a simple greedy algorithm that is guaranteed to find a good approximation. We then discuss an on-line learning algorithm, based on the "Hedge" algorithm, for finding a good linear combination of ranking "experts." We use the ordering algorithm combined with the on-line learning algorithm to find a combination of "search experts," each of which is a domain-specific query expansion strategy for a WWW search engine, and present experimental results that demonstrate the merits of our approach. 1 Introduction Most previous work in inductive learning has concentrated on learning to classify. However, there are many applications in which it is desirable to order rather than classify instances. An example might be a personalized email filter that gives a priority ordering to unread mail. Here we will consider the problem of learning how to construct such orderings, given feedback in the form of preference judgments, i.e., statements that one instance should be ranked ahead of another. Such orderings could be constructed based on a learned classifier or regression model, and in fact often are. For instance, it is common practice in information retrieval to rank documents according to their estimated probability of relevance to a query based on a learned classifier for the concept "relevant document." An advantage of learning orderings directly is that preference judgments can be much easier to obtain than the labels required for classification learning. For instance, in the email application mentioned above, one approach might be to rank messages according to their estimated probability of membership in the class of "urgent" messages, or by some numerical estimate of urgency obtained by regression. Suppose, however, that a user is presented with an ordered list of email messages, and elects to read the third message first. Given this election, it is not necessarily the case that message three is urgent, nor is there sufficient information to estimate any numerical urgency measures; however, it seems quite reasonable to infer that message three should have been ranked ahead of the others. Thus, in this setting, obtaining preference information may be easier and more natural than obtaining the information needed for classification or regression. 452 w. W. Cohen, R. E. Schapire and Y. Singer In the remainder of this paper, we will investigate the following two-stage approach to learning how to order. In stage one, we learn a preference junction, a two-argument function PREF( u, v) which returns a numerical measure of how certain it is that u should be ranked before v. In stage two, we use the learned preference function to order a set of new instances U; to accomplish this, we evaluate the learned function PREF( u, v) on all pairs of instances u, v E U, and choose an ordering of U that agrees, as much as possible, with these pairwise preference judgments. This general approach is novel; for related work in various fields see, for instance, references [2, 3, 1, 7, 10]. As we will see, given an appropriate feature set, learning a preference function can be reduced to a fairly conventional classification learning problem. On the other hand, finding a total order that agrees best with a preference function is NP-complete. Nevertheless, we show that there is an efficient greedy algorithm that always finds a good approximation to the best ordering. After presenting these results on the complexity of ordering instances using a preference function, we then describe a specific algorithm for learning a preference function. The algorithm is an on-line weight allocation algorithm, much like the weighted majority algorithm [9] and Winnow [8], and, more directly, Freund and Schapire's [4] "Hedge" algorithm. We then present some experimental results in which this algorithm is used to combine the results of several "search experts," each of which is a domain-specific query expansion strategy for a WWW search engine. 2 Preliminaries Let X be a set of instances (possibly infinite). A preference junction PREF is a binary function PREF : X x X ~ [0,1]. A value of PREF(u, v) which is close to 1 or a is interpreted as a strong recommendation that u should be ranked before v. A value close to 1/2 is interpreted as an abstention from making a recommendation. As noted above, the hypothesis of our learning system will be a preference function, and new instances will be ranked so as to agree as much as possible with the preferences predicted by this hypothesis. In standard classification learning, a hypothesis is constructed by combining primitive features. Similarly, in this paper, a preference function will be a combination of other preference functions. In particular, we will typically assume the availability of a set of N primitive preference functions RI , ... , RN. These can then be combined in the usual ways, e.g., with a boolean or linear combination of their values; we will be especially interested in the latter combination method. It is convenient to assume that the Ri'S are well-formed in certain ways. To this end, we introduce a special kind of preference function called a rank ordering. Let S be a totally ordered setl with' >' as the comparison operator. An ordering function into S is a function f : X ~ S. The function f induces the preference function Rj, defined as { I if f (u) > f ( v) Rj(u,v) ~ 0 21 if f(u) < f(v) otherwise. We call Rf a rank ordering for X into S. If Rf(u, v) = I, then we say that u is preferred to v, or u is ranked higher than v. It is sometimes convenient to allow an ordering function to "abstain" and not give a preference for a pair u, v. Let ¢> be a special symbol not in S, and let f be a function into S U {¢>}. We will interpret the mapping f (u) = ¢> to mean that u is "unranked," and let Rf (u, v) = ! if either u or v is unranked. To give concrete examples of rank ordering, imagine learning to order documents based on the words that they contain. To model this, let X be the set of all documents in a repository, )That is, for all pairs of distinct elements 8J, 82 E S, either 8) < 82 or 8) > 82 . Learning to Order Things 453 and for N words WI, ... , W N. let Ii (u) be the number of occurrences of Wi in u. Then Rf; will prefer u to v whenever Wi occurs more often in u than v. As a second example. consider a meta-search application in which the goal is to combine the rankings of several WWW search engines. For N search engines el, ... , eN. one might define h so that R'i prefers u to v whenever u is ranked ahead of v in the list Li produced by the corresponding search engine. To do this, one could let Ii(u) = -k for the document u appearing in the k-th position in the list L i • and let Ii( u) = </> for any document not appearing in L i . 3 Ordering instances with a preference function We now consider the complexity of finding the total order that agrees best with a learned preference function. To analyze this. we must first quantify the notion of agreement between a preference function PREF and an ordering. One natural notion is the following: Let X be a set. PREF be a preference function. and let p be a total ordering of X. expressed again as an ordering function (i.e .• p( u) > p( v) iff u precedes v in the order). We define AGREE(p, PREF) to be the sum of PREF( u, v) over all pairs u, v such that u is ranked ahead of v by p: AGREE(p, PREF) = PREF(u, v). (1) u.v:p{ul>p{v) Ideally. one would like to find a p that maximizes AGREE(p, PREF). This general optimization problem is of little interest since in practice, there are many constraints imposed by learning: for instance PREF must be in some restricted class of functions. and will generally be a combination of relatively well-behaved preference functions R i . A more interesting question is whether the problem remains hard under such constraints. The theorem below gives such a result. showing that the problem is NP-complete even if PREF is restricted to be a linear combination of rank orderings. This holds even if all the rank orderings map into a set S with only three elements. one of which mayor may not be </>. (Clearly. if S consists of more than three elements then the problem is still hard.) Theorem 1 The following decision problem is NP-complete: Input: A rational number 1\,; a set X; a set S with lSI ~ 3; a collection of N ordering functions Ii : X -t S; and a preference function PREF defined as PREF(u, v) = L~I wiR'i (u, v) where w = (WI, ... ,WN) is a weight vector in [0, l]N with L~I Wi = 1. Question: Does there exist a total order p such that AGREE(p, PREF) ~ I\,? The proof (omitted) is by reduction from CYCLIC-ORDERING [5. 6]. Although this problem is hard when lSI ~ 3. it becomes tractable for linear combinations of rank orderings into a set S of size two. In brief. suppose one is given X, Sand PREF as in Theorem 1, save that S is a two-element set. which we assume without loss of generality to be S = {O, I}. Now define p(u) = Li Wdi(U). It can be shown that the total order defined by p maximizes AGREE(p, PREF). (In case of a tie, p( u) = p( v) for distinct u and v. p defines only a partial order. The claim still holds in this case for any total order which is consistent with this partial order.) Of course, when lSI = 2, the rank orderings are really only binary classifiers. The fact that this special case is tractable underscores the fact that manipulating orderings can be computationally more difficult than performing the corresponding operations on binary classifiers. Theorem 1 implies that we are unlikely to find an efficient algorithm that finds the optimal total order for a weighted combination of rank orderings. Fortunately. there do exist efficient algorithms for finding an approximately optimal total order. Figure 1 summarizes a greedy 454 w. W. Cohen, R. E. Schapire and Y. Singer Algorithm Order-By-Preferences Inputs: an instance set X; a preference function PREF Output: an approximately optimal ordering function p let V = X for each v E V do7l'(v) = LUEVPREF(v,u) - LUEVPREF(u,v) while V is non-empty do let t = argmaxuEv 71'(u) let pet) = IVI V=V-{t} for each v E V do 71'(v) = 71'(v) + PREF(t, v) - PREF(v, t) endwhile Figure 1: A greedy ordering algorithm algorithm that produces a good approximation to the best total order, as we will shortly demonstrate. The algorithm is easiest to describe by thinking of PREF as a directed weighted graph where, initially, the set of vertices V is equal to the set of instances X, and each edge u -t v has weight PREF( u, v). We assign to each vertex v E V a potential value 71'( v), which is the weighted sum of the outgoing edges minus the weighted sum of the ingoing edges. That is, 71'(v) = LUEV PREF(v,u) - LUEV PREF(u, v) . The greedy algorithm then picks some node t that has maximum potential, and assigns it a rank by setting pet) = lVI, effectively ordering it ahead of all the remaining nodes. This node, together with all incident edges, is then deleted from the graph, and the potential values 71' of the remaining vertices are updated appropriately: This process is repeated until the graph is empty; notice that nodes removed in subsequent iterations will have progressively smaller and smaller ranks. The next theorem shows that this greedy algorithm comes within a factor of two of optimal. Furthermore, it is relatively simple to show that the approximation factor of 2 is tight. Theorem 2 Let OPT(PREF) be the weighted agreement achieved by an optimal total orderfor the preference junction PREF and let APPROX(PREF) be the weighted agreement achieved by the greedy algorithm. Then APPROX(PREF) ;::: !OPT(PREF). 4 Learning a good weight vector In this section, we look at the problem of learning a good linear combination of a set of preference functions. Specifically, we assume access to a set of ranking experts which provide us with preference functions Ri of a set of instances. The problem, then, is to learn a preference function of the form PREF(u,v) = L~I wiRi(U,V). We adopt the on-line learning framework first studied by Littlestone [8J in which the weight Wi assigned to each ranking expert Ri is updated incrementally. Learning is assumed to take place in a sequence of rounds. On the t-th round, the learning algorithm is provided with a set X t of instances to be ranked and to a set of N preference functions R~ of these instances. The learner may compute R!( u, v) for any and all preference functions R~ and pairs u, v E X t before producing a final ordering Pt of xt. Finally, the learner receives feedback from the environment. We assume that the feedback is an arbitrary set of assertions of the form "u should be preferred to v." That is, formally we regard the feedback on the t-th round as a set Ft of pairs (u, v) indicating such preferences. The algorithm we propose for this problem is based on the "weighted majority algorithm" [9J and, more directly, on the "Hedge" algorithm [4]. We define the loss of a preference function Learning to Order Things Allocate Weights for Ranking Experts Parameters: (3 E [0,1], initial weight vector WI E [0, I]N with l:~1 wl = 1 N ranking experts, number of rounds T Do fort = 1,2, ... ,T 1. Receive a set of elements X t and preference functions R~, ... , R'N. 455 2. Use algorithm Order-By-Preferences to compute ordering function Pt which approximatesPREFt(u,v) = E~I wiRHu,v). 3. Order X t using Pt. 4. Receive feedback Ft from the user. 5. Evaluate losses Loss(RL Ft) as defined in Eq. (2). 6. Set the new weight vector w!+ 1 = w! . (3Loss(R: ,Ft) / Zt where Zt is a normalization constant, chosen so that E~I w!+1 = 1. Figure 2: The on-line weight allocation algorithm. R with respect to the user's feedback F as L (R F) ~ E(U,V)EF(1 - R(u,v)) oss , IFI' (2) This loss has a natural probabilistic interpretation. If R is viewed as a randomized prediction algorithm that predicts that u will precede v with probability R(u, v), then Loss(R, F) is the probability of R disagreeing with the feedback on a pair (u, v) chosen uniformly at random from F. We now can use the Hedge algorithm almost verbatim, as shown in Figure 2. The algorithm maintains a positive weight vector whose value at time t is denoted by w t = (wf, . . . , w'N). If there is no prior knowledge about the ranking experts, we set all initial weights to be equal so that wI = 1/ N. The weight vector w t is used to combine the preference functions of the different experts to obtain the preference function PREFt = E~ I w~ R~. This, in tum, is converted into an ordering Pt on the current set of elements Xl using the method described in Section 3. After receiving feedback pt, the loss for each preference function Loss(RL Ft) is evaluated as in Eq. (2) and the weight vector w t is updated using the mUltiplicative rule W!+I = w~ . (3LQss(R: ,Ft) / Zt where (3 E [0, 1] is a parameter, and Zt is a normalization constant, chosen so that the weights sum to one after the update. Thus, based on the feedback, the weights of the ranking experts are adjusted so that experts producing preference functions with relatively large agreement with the feedback are promoted. We will briefly sketch the theoretical rationale behind this algorithm. Freund and Schapire [4] prove general results about Hedge which can be applied directly to this loss function. Their results imply almost immediately a bound on the cumulative loss of the preference function PREFt in terms of the loss of the best ranking expert, specifically T T LLoss(PREFt,Ft ) ~ a,Bm~n LLoss(RLFt) +c,BlnN l t=1 t=1 where a,B = InO / (3) / (1 - (3) and C,B = 1/( I - (3). Thus, if one of the ranking experts has low loss, then so will the combined preference function PREFt . However, we are not interested in the loss ofPREFt (since it is not an ordering), but rather in the performance of the actual ordering Pt computed by the learning algorithm. Fortunately, 456 w. W. Cohen, R. E. Schapire and y. Singer the losses of these can be related using a kind of triangle inequality. It can be shown that, for any PREF, F and p: OISAGREE(p PREF) Loss(Rp, F) ~ IFI ' + Loss(PREF, F) (3) where, similar to Eq. (1), OISAGREE(p, PREF) = Lu,v:p(u»p(v)(l - PREF(u, v)). Not surprisingly, maximizing AGREE is equivalent to minimizing DISAGREE. So, in sum, we use the greedy algorithm of Section 3 to minimize (approximately) the first term on the right hand side ofEq. (3), and we use the learning algorithm Hedge to minimize the second term. 5 Experimental results for metasearch We now present some experiments in learning to combine the results of several WWW searches. We note that this problem exhibits many facets that require a general approach such as ours. For instance, approaches that learn to combine similarity scores are not applicable since the similarity scores of WWW search engines are often unavailable. We chose to simulate the problem of learning a domain-specific search engine. As test cases we picked two fairly narrow classes of queries-retrieving the home pages of machine learning researchers (ML), and retrieving the home pages of universities (UNIV). We obtained a listing of machine learning researchers, identified by name and affiliated institution, together with their home pages, and a similar list for universities, identified by name and (sometimes) geographical location. Each entry on a list was viewed as a query, with the associated URL the sole relevant document. We then constructed a series of special-purpose "search experts" for each domain. These were implemented as query expansion methods which converted a name, affiliation pair (or a name, location pair) to a likely-seeming Altavista query. For example, one expert for the ML domain was to search for all the words in the person's name plus the words "machine" and "learning," and to further enforce a strict requirement that the person's last name appear. Overall we defined 16 search experts for the ML domain and 22 for the UN IV domain. Each search expert returned the top 30 ranked documents. In the ML domain there were 210 searches for which at least one search expert returned the named home page; for the UNIV domain, there were 290 such searches. For each query t, we first constructed the set X t consisting of all documents returned by all of the expanded queries defined by the search experts. Next, each search expert i computed a preference function R~. We chose these to be rank orderings defined with respect to an ordering function If in the natural way: We assigned a rank of if = 30 to the first listed document, Ii = 29 to the second-listed document, and so on, finally assigning a rank of Ii = 0 to every document not retrieved by the expanded query associated with expert i. To encode feedback, we considered two schemes. In the first we simulated complete relevance feedback-that is, for each query, we constructed feedback in which the sole relevant document was preferred to all other documents. In the second, we simulated the sort of feedback that could be collected from "click data," i.e., from observing a user's interactions with a metasearch system. For each query, after presenting a ranked list of documents, we noted the rank of the one relevant document. We then constructed a feedback ranking in which the relevant document is preferred to all preceding documents. This would correspond to observing which link the user actually followed, and making the assumption that this link was preferred to previous links. To evaluate the expected performance of a fully-trained system on novel queries in this domain, we employed leave-one-out testing. For each query q, we removed q from the Learning to Order Things 457 ML Domain University Domain Top 1 Top lO Top 30 Av. rank Top 1 Top lO Top 30 Av. rank Learned System (Full Feedback) 114 185 198 4.9 111 225 253 7.8 Learned System ("Click Data") 93 185 198 4.9 87 229 259 7.8 Naive 89 165 176 7.7 79 157 191 14.4 Best (Top 1) 119 170 184 6.7 112 221 247 8.2 Best (Top 10) 114 182 190 5.3 111 223 249 8.0 Best (Top 30) 97 181 194 5.6 111 223 249 8.0 Best (Av. Rank) 114 182 190 5.3 111 223 249 8.0 Table 1: Comparison of learned systems and individual search queries query set, and recorded the rank of q after training (with (3 = 0.5) on the remaining queries. For click data feedback, we recorded the median rank over 100 randomly chosen permutations of the training queries. We the computed an approximation to average rank by artificially assigning a rank of 31 to every document that was either unranked, or ranked above rank 30. (The latter case is to be fair to the learned system, which is the only one for which a rank greater than 30 is possible.) A summary of these results is given in Table 1, together with some additional data on "top-k performance"-the number of times the correct homepage appears at rank no higher than k. In the table we give the top-k performance (for three values of k) and average rank for several ranking systems: the two learned systems, the naive query (the person or university's name), and the single search expert that performed best with respect to each performance measure. The table illustrates the robustness of the learned systems, which are nearly always competitive with the best expert for every performance measure listed; the only exception is that the system trained on click data trails the best expert in top-k performance for small values of k. It is also worth noting that in both domains, the naive query (simply the person or university's name) is not very effective. Even with the weaker click data feedback, the learned system achieves a 36% decrease in average rank over the naive query in the ML domain, and a 46% decrease in the UNIV domain. To summarize the experiments, on these domains, the learned system not only performs much better than naive search strategies; it also consistently performs at least as well as, and perhaps slightly better than, any single domain-specific search expert. Furthermore, the performance of the learned system is almost as good with the weaker "click data" training as with complete relevance feedback. References [1] D.S. Hochbaum (Ed.). Approximation Algorithms for NP-hard problems. PWS Publishing Company, 1997. [2] O. Etzioni, S. Hanks, T. Jiang, R M. Karp, O. Madani, and O. Waarts. Efficient information gathering on the internet. In 37th Ann. Symp. on Foundations of Computer Science, 1996. [3] P.C Fishburn. The Theory of Social Choice. Princeton University Press, Princeton, NJ, 1973. [4] Y. Freund and RE. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. Journal of Computer and System Sciences, 1997. [5] Z. Galil and N. Megido. Cyclic ordering is NP-complete. Theor. Compo Sci. , 5:179-182, 1977. [6] M.R Gary and D.S. Johnson. Computers and Intractibility: A Guide to the Theory of NPcompleteness. W. H. Freeman and Company, New York, 1979. [7j P.B. Kantor. Decision level data fusion for routing of documents in the TREC3 context: a best case analysis of worste case results. In TREC-3, 1994. [8] N. Littlestone. Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning, 2(4), 1988. [9) N. Littlestone and M.K. Warmuth. The weighted majority algorithm. Infonnation and Computation, 108(2):212-261, 1994. [10] K.E. Lochbaum and L.A. Streeter. Comparing and combining the effectiveness of latent semantic indexing and the ordinary vector space model for information retrieval. 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Analog VLSI Model of Intersegmental Coordination With Nearest-Neighbor Coupling Girish N. Patel girish@ece.gatech.edu Jeremy H. Holleman jeremy@ece.gatech.edu Stephen P. De Weerth steved@ece.gatech.edu School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, Ga. 30332-0250 Abstract We have a developed an analog VLSI system that models the coordination of neurobiological segmental oscillators. We have implemented and tested a system that consists of a chain of eleven pattern generating circuits that are synaptically coupled to their nearest neighbors. Each pattern generating circuit is implemented with two silicon Morris-Lecar neurons that are connected in a reciprocally inhibitory network. We discuss the mechanisms of oscillations in the two-cell network and explore system behavior based on isotropic and anisotropic coupling, and frequency gradients along the chain of oscillators. 1 INTRODUCTION In recent years, neuroscientists and modelers have made great strides towards illuminating structure and computational properties in biological motor systems. For example, much progress has been made toward understanding the neural networks that elicit rhythmic motor behaviors, including leech heartbeat (Calabrese and De Schutter, 1992), crustacean stomatogastric mill (Selverston, 1989) and tritonia swimming (Getting, 1989). In particular, segmented locomotory systems, such as those that underlie swimming in the lamprey (Cohen and Kiemel, 1993, Sigvardt, 1993, Grillner et aI, 1991) and in the leech (Friesen and Pearce, 1993), are interesting from an quantitative perspective. In these systems, it is clear that coordinated motor behaviors are a result of complex interactions among membrane, synaptic, circuit, and system properties. However, because of the lack of sufficient neural underpinnings, a complete understanding of the computational principles in these systems is still lacking. Abstracting the biophysical complexity by modeling segmented systems as coupled nonlinear oscillators is one approach that has provided much insight into the operation of these systems (Cohen et ai, 1982). More specifically, this type of modeling work has illuminated computational properties that give rise to phase constancy, a motor behavior that is characterized by intersegmental phase lags that are maintained at constant values independent of swimming frequency. For example, it has been shown that frequency gradients and asymmetrical coupling play an important role in establishing phase lags of correct sign and amplitude (Kopell and Ennentrout, 1988) as well as appropriate boundary conditions (Williams and Sigvardt, 1994). Although theoretical modeling has provided much insight into the operation of interseg720 G. N. Patel, 1. H. Holleman and S. P. DeWeerth mental systems. these models have limited capacity for incorporating biophysical properties and complex interconnectivity. Software and/or hardware emulation provides the potential to add such complexity to system models. Additionally, the modularity and regularity in the anatomical and computational structures of intersegmental systems facilitate scalable representations. These factors make segmented systems particularly viable for modeling using neuromorphic analog very large-scale integrated (aVLSI) technology. In general, biological motor systems have a number of properties that make their realtime modeling using aVLSI circuits interesting and approachable. Like their sensory counterparts, they exhibit rich emergent properties that are generated by collective architectures that are regular and modular. Additionally, the fact that motor processing is at the periphery of the nervous system makes the analysis of the system behavior accessible due to the fact that output of the system (embodied in the motor actions) is observable and facilitates functional analysis. The goals in this research are i) to study how the properties of individual neurons in a network affect the overall system behavior; (ii) to facilitate the validation of the principles underlying intersegmental coordination; and (iii) to develop a real-time, low power, motion control system. We want to exploit these principles and architectures both to improve our understanding of the biology and to design artificial systems that perfonn autonomously in various environments. In this paper we present an analog VLSI model of intersegmental coordination that addresses the role of frequency gradients and asymmetrical coupling. Each segment in our system is implemented with two silicon model neurons that are connected in a reciprocally inhibitory network. A model of intersegmental coordination is implemented by connecting eleven such oscillators, with nearest neighbor coupling. We present the neuron model, and we investigate the role of frequency gradients and asymmetrical coupling in the establishment of phase lags along a chain these neural oscillators. 2 NEURON MODEL In order to produce bursting activity, a neuron must possess "slow" intrinsic time constants in addition to the "fast" time constants that are necessary for the generation of spikes. Hardware models of neurons with both slow ana fast time constants have been designed based upon previously described Hodgkin-Huxley neuron models (Mahowald and Douglas, 1991). Although these circuits are good models of their biological counterparts, they are relatively complex. with a large parameter space and transistor count, limiting their usefulness in the development of large-scale systems. It has been shown (Skinner. 1994). however, that pattern generation can be represented with only the slow time constants, creating a system that represents the envelope of the bursting oscillations without the individual spikes. Model neurons with only slow time constants have been proposed by Morris and Lecar (1981). We have implemented an analog VLSI model of the Morris-Lecar Neuron (Patel and De Weerth. 1997). Figure I shows the circuit diagram of this neuron. The model consists of two state variables: one corresponding to the membrane potential (V) and One corresponding to a slow variable (N). The slow variable is obtained by delaying the mem., I p.Vprel IN Ie '-----+--~~---+--_v LoJ T L _____ ...J Figure 1: Circuit diagram of silicon Morris-Lecar Neuron v A VISI Model of Intersegmental Coordination brane potential by way of an operational transconductance amplifier (OTA) connected in unity gain configuration with load capacitor C2 . The membrane potential is obtained by injecting two positive currents (lext and iH) and two negative currents (iL and isyn) into capacitor C I . Current iH raises the membrane potential towards V High when the membrane potential increases above V H' whereas current iL lowers the membrane potential towards V Low when the delayed membrane potential increases above V L' The synaptic current, isyn ' activates when the presynaptic input, V Pre' increases above V thresh . Assuming operation of transistors in weak inversion and synaptic coupling turned off (isyn = 0) the equations of motion for the system are. . exp(K(V - VH)/UT) exp(K(N - VL)/UT) C I V = II (V, N) = 'extap + IH I + exp(IC(V _ VH)/UT) a p -Il.l + exp(K(N _ VL)/UT) aN C2'" = 12(V, N) = 'ttanh(K(V - N)/(2UT »( 1- exp«N - V dd )/UT ») The terms (Xp and (XN' where (Xp = 1 - exp(V - VHigh)/UT and (XN = 1 - exp(V Low - V)/UT ,correspond to the ohmic effect of transistor Ml and M2 respectively. K' corresponds to the back-gate effect of a MOS transistor operated in weak inversion, and UT corresponds to the thermal voltage. We can understand the behavior of this circuit by analyzing the geometry of the curves that yield zero motion (i.e., when II (V, N) = 12( V, N) = 0). These curves, referred to as nullclines, are shown in Figure 2 for various values of external current. The externally applied constant current (lext)' which has the effect of shifting the V nullcline in the positive vertical direction (see Figure 2), controls the mode of operation of the neuron. When the V - and N nullclines intersect between the local minimum and local maximum of the V nullcline (P2 in Figure 2), the resulting fixed point is unstable and the trajectories of the system approach a stable limit-cycle (an endogenous bursting mode). Fixed points to the left of the local minimum (PI in Figure 2) or to the right of the local maximum (P3 in Figure 2) are stable and correspond to a silent mode and a tonic mode of the neuron respectively. An inhibitory synaptic current (isyn ) has the effect of shifting the V nullcline in the negative vertical direction; depending on the state of a presynaptic cell, i Syn can dynamically change the mode of operation of the neuron. 3 TWO-CELL NETWORK When two cells are connected in a reciprocally inhibitory network, the two cells will oscillate in antiphase depending on the conditions of the free and inhibited cells and the value of the synaptic threshold (Skinner et. aI, 1994). We assume that the turn-on characteristics of the synaptic current is sharp (valid for large V High - V Low) such that when the membrane potential of a presynaptic cell reaches above V thresh' the postsynaptic cell is immediately inhibited by application of negative current Isyn to its membrane potential. 26 255 z 2.5 2.45 24 I , P1 ___ 24 ... -- - 2.45 2.5 V P3 255 , I \, , ' " , 11.0 2.6 - lext = 5 nA lext = 2.5 nA lext = 0 nA trajectories Figure 2: Nullcline and corresponding trajectories of silicon Morris-Lecar neuron. 721 722 G. N. Patel, J H Holleman and S. P. DeWeerth If the free cell is an endogenous burster, the inhibited cell is silent, and the synaptic threshold is between the local maximum of the free cell and the local minimum in the inhibited cell. the mechanism for oscillation is due to intrinsic release. This mechanism can be understood by observing that the free cell undergoes rapid depolarization when its state approaches the local maximum thus facilitating the release of the inhibited cell. If the free cell is tonic and the inhibited cell is an endogenous burster (and conditions on synaptic threshold are the same as in the intrinsic release case). then the oscillations are due to an intrinsic escape mechanism. This mechanism is understood by observing that the inhibited cell undergoes rapid hyperpolarization. thus escaping inhibition. when its state approaches the local minimum. Note. in both intrinsic release and intrinsic escape mechanisms. the synaptic threshold has no effect on oscillator period because rapid changes in membrane potential occur before the effect of synaptic threshold. When the free cell is an endogenous burster. the inhibited cell is silent. and the synaptic threshold is to the right of the local maximum of the free cell. then the oscillations are due to a synaptic release mechanism. This mechanism can be understood by observing that when the membrane potential of the free cell reaches below the synaptic threshold. the free cell ceases to inhibit the other cell which causes the release of the inhibited cell. When the free cell is tonic. and the inhibited cell is an endogenous burster. and the synaptic threshold is to the left of the local minimum of the inhibited cell, then the oscillations are due to a synaptic escape mechanism. This mechanism can be understood by observing that when the membrane potential of the inhibited cell crosses above the synaptic threshold, then the membrane potential of the inhibited cell is large enough to inhibit the free cell. Note, increasing the synaptic threshold h~ the effect of increasing oscillator frequency for the synaptic release mechanism, however, oscillator frequency under the synaptic escape mechanism will decrease with an increase in the synaptic threshold. By setting V High - V Low to a large value. the synaptic currents appear to have a sharp cutoff. However, because transistor currents saturate within a few thermal voltages, the null clines due to the membrane potential appear less cubic-like and more square-like. This does not effect the qualitative behavior of the circuit. as we are able to produce antiphasic oscillations due to all four mechanisms. Figure 3 illustrates the four modes of oscillations under various parameter regimes. Figure 31\ show typical waveforms from two silicon neurons when they are configured in a reciprocally inhibitory network. The oscillations in this case are due to intrinsic release mechanism and the frequency of oscillations are insensitive to the synaptic threshold. When the synaptic threshold is increased above 2.5 volts, the oscillations are due to the synaptic release mechanism and the oscillator frequency will increase as the synaptic threshold is increased. as shown in Figure 3C. By adjusting lex! such that the free cell is tonic and the inhibited cell bursts endogenously. we are able to produce oscillations due to the intrinsic escape mechanism, as -E..,..,. t.Ioc:horOsm II V1IU B lIlJ1J JUlf1 nJU\ c D 1 3.: (25 '" i 2.5 i 2 l.: .:! j ' .5 '-'0 2 , ~ 0 2 ~ Figure 3: Experimental results from two neurons connected in a reciprocally inhibitory network. Antiphasic oscillations due to intrinsic release mechanism (A), and intrinsic escape mechanism (B). Dependence of oscillator frequency on synaptic threshold for the synaptic release mechanism (C) and synaptic escape mechanism (D) A VISI Model of Intersegmental Coordination shown in Figure 3B. As the synaptic threshold is decreased below 0.3 volts, the oscillations are caused by the synaptic escape mechanism and oscillator frequency increases as the synaptic threshold is decreased. The sharp transition between intrinsic and synaptic mechanisms is due to nullclines that appear square-like. 4 CHAIN OF COUPLED NEURAL OSCILLATORS In order to build a chain of pattern generating circuits with nearest neighbor coupling, we designed our silicon neurons with five synaptic connections. The connections are made using the synaptic spread rule proposed by Williams (1990). The rule states that a neuron in any given segment can only connect to neurons in other segments that are homologues to the neurons it connects to in the local segment. Therefore, each neuron makes two inhibitory, contralateral connections and two excitatory, ipsilateral connections (as well a single inhibitory connection in the local segment). The synaptic circuit, shown in the dashed box in Figure I, is repeated for each inhibitory synapse and its complementary version is repeated for the excitatory synapses. In order to investigate the role of frequency gradients, each neural oscillator has an independent parameter, lex!' for setting the intrinsic oscillator period. A set of global parameters, IL , IH , It' V H' V L' V Hi!\h ' and V Low control the mechanism of oscilJation. These parameters are set such that the mechanism of oscillation is intrinsic release. Because of inherent mismatch of devices in CMOS technology, a consequence in our model is that neurons with equal parameters do not necessarily behave with similar performance. Figure 4A illustrates the)ntrinsic oscillator period along the length of system when all neurons receive the same parameters. When the oscillators are symmetrically coupled, the resulting phase differences along the chain are nonzero, as shown in Figure 4B. The phase lags are negative with respect to the head position, thus the default swim direction is backward. As the coupling strength is increased, indicated by the lowermost curves in Figure 4B, the phase lags become smaller, as expected, but do not diminish to produce synchronous oscillations. When the oscillators are locked to one common frequency, n, theory predicts (Kopell and Ermentrout, 1988) that the common frequency is dependent on intrinsic oscillator frequencies, and coupling from neighboring oscillators. In addition, under the condition of weak coupling, the effect of coupling can be quantified with coupling functions that depends on the phase difference between neighboring oscillators: n = CJ)i + H~($;) + H~(~i_l) where, CJ)i is the intrinsic frequency of a given oscillator, H A and H D are coupling functions in the ascending and descending directions respectively, and $i is the phase difference between the (i+ 1 )th and ith oscillator. This equation suggests that the phase lags must be large in order to compensate for large variations in the intrinsic oscillator freA 1.5 f ·4 § 1.3 ~, 2 i 1: 0.9 B 0 0 0 o 0 0 c ! 0 0 0 4 6 10 15,--.--------, 10 , I, __ .. --............ -:.. -; "'-.. ~ ... I 0 ~ I -5 -10 _15L----------' 6 8 10 '-~ Figure 4: Experimental data obtained from system of coupled oscillators. 723 724 C. N Patel, J H. Holleman and S. P. DeWeerth quencies. Another factor that can effect the intersegmental phase lag is the degree of anisotropic coupling. To investigate the effect of asymmetrical coupling. we adjusted lex! in each segment so to produce uniform intrinsic oscillator periods (to within ten percent of 115 ms) along the length of the system. Asymmetrical coupling is established by maintaining Vavg == (VAse - V DEs)/2 at 0.7 volts and varying V delta == V ASC - V DES from 0.4 to - 0.4 volts. VAse and V DFS correspond to the bias voltage that sets the synaptic conductance of presynaptic inputs arriving from the ascending and descending directions respectively. Throughout the experiment. the average of inhibitory (contralateral) and excitatory (ipsilateral) connections from one direction are maintained at equal levels. Figure 4C shows the intersegmental phase lags at different levels of anisotropic coupling. Stronger ascending weights (V delta = 0.4. 0.2 volts) produced negative phase lags. corresponding to backward swimming. while stronger descending connections (V delta = -0.4. -0.2 volts) produce positive phase lags. corresponding to backward swimming. Although mathematical models suggest that stronger ascending coupling should produce forward swimming. we feel that the type of coupling (inhibitory contralateral and excitatory ipsilateral connections) and the oscillatory mode (intrinsic release) of the segmental oscillators may account for this discrepancy. To study the effects of frequency gradients. we adjusted lexl at each segment such that the that the oscillator period from the head to the tail (from segment I to segment II) varied from 300 ms to lOOms in 20 ms increments. In addition. to minimize the effect of asymmetrical coupling. we set Vavg = 0.8 volts and V delta = 0 volts. The absolute phase under these conditions are shown in Figure SA. The phase lags are negative with respect to the head position. which corresponds to backward swimming. With a positive frequency gradient. head oscillator at lOOms and tail oscillator at 300 ms. the resulting phases are in the opposite direction, as shown in Figure 5B. These results are consistent with mathematical models and the trailing oscillator hypothesis as expounded by Grinner et. al. (1991). 5 CONCLUSIONS AND FUTURE WORK We have implemented and tested an analog VLSI model of intersegmental coordination with nearest neighbor coupling. We have explored the effects of anisotropic coupling and frequency gradients on system behavior. One of our results-stronger ascending connections produced backward swimming instead of forward swimming-is contrary to theory. There are two factors that may account for this discrepancy: i) our system exhibits inherent spatial disorder in the parameter space due to device mismatch. and ii) the operating point at which we performed the experiments retains high sensitivity to neuron parameter variations and oscillatory modes. We are continuing to explore the parameter space to determine if there are more robust operating points. We expect that the limitation of our system to only including nearest-neighbor connections is a major factor in the large phase-lag variations that we observed. The importance of both short and long distance connections in the regulation of constant phase under conditions of large variability in the parameter space has been shown by Cohen and Kiemel (1993). To address these issues, we are currently designing a system that facilitates both short and long distance connections (DeWeerth et al. 1997). Additionally. to study the A B eo .. -10 r l~: .. "" t: I .. 10 -40 -70 -40 • 10 _ .... Figure 5: Absolute phase with negative (A) and positive (B) frequency gradients. A VUI Model of Intersegmental Coordination role of sensory feedback and to close the loop between neural control and motor behavior, we are also building a mechanical segmented system into which we will incorporate our a VLSI models. Acknowledgments This research is supported by NSF grant IBN-95 I 1721. We would like to thank Avis Cohen for discussion on computational properties that underlie coordinated motor behavior in the lamprey swim system. We would also like to thank Mario Simoni for discussions on pattern generating circuits. We thank the Georgia Tech Analog Consortium for supporting students with travel funds. References Calabrese, R. and De Schutter, E. (1992). Motor-pattern-generating networks in Invertebrates: Modeling Our Way Toward Understanding. TINS, 15.11 :439-445. Cohen, A, Holms, P. and Rand R. (1982). The nature of coupling between segmental oscillators of the lamprey spinal generator for locomotion: A mathematical model. J. Math BioI. 13:345-369. Cohen, A. and Kiemel, T. (1993). Intersegmental coordination: lessons from modeling systems of coupled non-linear oscillators. Amer. Zool., 33:54-65. DeWeerth, S., Patel, G., Schimmel, D., Simoni, M. and Calabrese, R. (1997). In Proceedings of the Seventeenth Conference on Advanced Research in VLSI, R.B. Brown and A.T. Ishii (eds), Los Alamitos, CA: IEEE Computer Society, 182-200. Friesen, 0 and Pearce, R. (1993). Mechanisms of intersegmental coordination in leech locomotion. SINS 5:41-47. Getting, P. (1989). A network oscillator underlying swimming in Tritonia. In Cellular and Neuronal Oscillators, J.w. Jacklet (ed), New York: Marcel Dekker, 101-128. Grillner, S. Wallen, P., Brodin, L. and Lansner, A. (1991). Neuronal network generating locomotor behavior In lamprey: circuitry, transmitter, membrane properties, and simulation. Ann. Rev. Neurosci., 14:169-169. Kopell, N. and Ennentrout, B. (1988). Coupled oscillators and the design of central pattern generators. Math Biosci. 90:87-109. Mahowald, M. and Douglas, R. (1991) A silicon neuron. Nature, 354:515-518. Morris, C. and Lecar, H. (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys. 1, 35: 193-213. Patel, G., DeWeerth, S. (1997). Analogue VLSI Morris-Lecar neuron. Electronic LeUers, lEE. 33.12: 997-998. Sigvardt, K.(l993). Intersegmental coordination in the lamprey central pattern generator for locomotion. SINS 5:3-15. Selverston, A. (1989) The Lobster Gastric Mill Oscillator, In Cellular and Neuronal Oscillators, J.W. Jacklet (ed), New York: Marcel Dekker, 338-370. Skinner, F., Kopell, N., and Marder E. (1994) Mechanisms for Oscillation and Frequency Control' in Reciprocally Inhibitory Model Neural Networks., 1. of Compo Neuroscience, ] :69-87. Wiliams, T. (1992). Phase Coupling and Synaptic Spread in Chains of Coupled Neuronal Oscillators. Science, vol. 258. Williams, T., Sigvardt, K. (1994) intersegmental phase lags in the lamprey spinal cord: experimental confirmation ofthe existence of a boundary region. 1. ofComp. Neuroscience, 1:61-67. 725
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Globally Optimal On-line Learning Rules Magnus Rattray*and David Saadt Department of Computer Science & Applied Mathematics, Aston University, Birmingham B4 7ET, UK. Abstract We present a method for determining the globally optimal on-line learning rule for a soft committee machine under a statistical mechanics framework. This work complements previous results on locally optimal rules, where only the rate of change in generalization error was considered. We maximize the total reduction in generalization error over the whole learning process and show how the resulting rule can significantly outperform the locally optimal rule. 1 Introduction We consider a learning scenario in which a feed-forward neural network model (the student) emulates an unknown mapping (the teacher), given a set of training examples produced by the teacher. The performance of the student network is typically measured by its generalization error, which is the expected error on an unseen example. The aim of training is to reduce the generalization error by adapting the student network's parameters appropriately. A common form of training is on-line learning, where training patterns are presented sequentially and independently to the network at each learning step. This form of training can be beneficial in terms of both storage and computation time, especially for large systems. A frequently used on-line training method for networks with continuous nodes is that of stochastic gradient descent, since a differentiable error measure can be defined in this case. The stochasticity is a consequence of the training error being determined according to only the latest, randomly chosen, training example. This is to be contrasted with batch learning, where all the training examples would be used to determine the training error leading to a deterministic algorithm. Finding an effective algorithm for discrete networks is less straightforward as the error measure is not differentiable. • rattraym@aston.ac.uk t saadd@aston.ac.uk Globally Optimal On-line Learning Rules 323 Often, it is possible to improve on the basic stochastic gradient descent algorithm and a number of modifications have been suggested in the literature. At late times one can use on-line estimates of second order information (the Hessian or its eigenvalues) to ensure asymptotically optimal performance (e.g., [1, 2]). A number of heuristics also exist which attempt to improve performance during the transient phase of learning (for a review, see [3]). However, these heuristics all require the careful setting of parameters which can be critical to their performance. Moreover, it would be desirable to have principled and theoretically well motivated algorithms which do not rely on heuristic arguments. Statistical mechanics allows a compact description for a number of on-line learning scenarios in the limit of large input dimension, which we have recently employed to propose a method for determining globally optimal learning rates for on-line gradient descent [4]. This method will be generalized here to determine globally optimal on-line learning rules for both discrete and continuous machines. That is, rules which provide the maximum reduction in generalization error over the whole learning process. This provides a natural extension to work on locally optimal learning rules [5, 6], where only the rate of change in generalization error is optimized. In fact, for simple systems we sometimes find that the locally optimal rule is also globally optimal. However, global optimization seems to be rather important in more complex systems which are characterized by more degrees of freedom and often require broken permutation symmetries to learn perfectly. We will outline our general formalism and consider two simple and tractable learning scenarios to demonstrate the method. It should be pointed out that the optimal rules derived here will often require knowledge of macroscopic properties related to the teacher's structure which would not be known in general. In this sense these rules do not provide practical algorithms as they stand, although some of the required macroscopic properties may be evaluated or estimated on the basis of data gathered as the learning progresses. In any case these rules provide an upper bound on the performance one could expect from a real algorithm and may be instrumental in designing practical training algorithms. 2 The statistical mechanics framework For calculating the optimal on-line learning rule we employ the statistical mechanics description of the learning process. Under this framework, which may be employed for both smooth [7,8] and discrete sy.stems (e.g. [9]), the learning process is captured by a small number of self-averaging statistics whose trajectory is deterministic in the limit of large input dimension. In this analysis the relevant statistics are overlaps between weight vectors associated with different nodes of the student and teacher networks. The equations of motion for the evolution of these overlaps can be written in closed form and can be integrated numerically to describe the dynamics. We will consider a general two-layer soft committee machinel . The desired teacher mapping is from an N-dimensional input space e E RN onto a scalar ( E R, which the student models through a map a(J,e) = 2:~l g(Ji ·e), where g(x) is the activation function for the hidden layer, J == {Jih<i<K is the set of input-to-hidden adaptive weights for the K hidden nodes and the hidden-to-output weights are set to 1. The activation of hidden node i under presentation of the input pattern eJ' is denoted xr = J i . eJ'· IThe general result presented here also applies to the discrete committee machine, but we will limit our discussion to the soft-committee machine. 324 M. Rattray and D. Saad Training examples are of the form (el', (I') where J.L = 1,2, ... , P. The components of the independently drawn input vectors el' are uncorrelated random variables with zero mean and unit variance. The corresponding output (I' is given by a deterministic teacher of a similar configuration to the student except for a possible difference in the number M of hidden units and is of the form (I' = l:!1 g(Bn . el'), where B == {Bnh<n<M is the set of input-to-hidden adaptive weights. The activation of hidden node n under presentation of the input pattern el' is denoted Y~ = Bn . el'. We will use indices i, j, k, I ... to refer to units in the student network and n, m, ... for units in the teacher network. We will use the commonly used quadratic deviation E(J,e) == ~ [ a(J,e) - (]2, as the measure of disagreement between teacher and student. The most basic learning rule is to perform gradient descent on this quantity. Performance on a typical input defines the generalization error Eg(J) == (E(J,e»{(} through an average over all possible input vectors e. The general form of learning rule we will consider is, J~+l = J~ + ~FfJ(xl' ~I') el' , 'N' ,." ~ (1) where F == {Fi} depends only on the student activations and the teacher's output, and not on the teacher activations which are unobservable. Note that gradient descent on the error takes this general form, as does Hebbian learning and other training algorithms commonly used in discrete machines. The optimal F can also depend on the self-averaging statistics which describe the dynamics, since we know how they evolve in time. Some of these would not be available in a practical application, although for some simple cases the unobservable statistics can be deduced from observable quantities. This is therefore an idealization rather than a practical algorithm and provides a bound on the performance of a real algorithm. The activations are distributed according to a multivariate Gaussian with covariances: (XiXk) = Ji·Jk == Qik, (XiYn) = Ji·Bn == Rin , and (YnYm) = Bn·Bm == Tnm, measuring overlaps between student and teacher vectors. Angled brackets denote averages over input patterns. The covariance matrix completely describes the state of the system and in the limit of large N we can write equations of motion for each macroscopic (the Tnm are fixed and define the teacher): dRin ( ) dQik ( dO! = FiYn do: = Fixk + FkXi + FiFk) , (2) where angled brackets now denote the averages over activations, replacing the averages over inputs, and 0: = J.LIN plays the role of a continuous time variable. 3 The globally optimal rule Carrying out the averaging over input patterns one obtains an expression for the generalization error which depends exclusively on the overlaps R,Q and T. Using the dependence of their dynamics (Eq. 2) on F one can easily calculate the locally optimal learning rule [5] by taking the functional derivative of dEg(F)/do: to zero, looking for the rule that will maximize the reduction in generalization error at the present time step. This approach has been shown to be successful in some training scenarios but is likely to fail where the learning process is characterized by several phases of a different natures (e.g., multilayer networks). The globally optimal learning rule is found by minimizing the total change in generalization error over a fixed time window, la l dE la l ~fg(F)= dgdo: = £(F, 0:) do:. ao 0: ao (3) Globally Optimal On-line Learning Rules 325 This is a functional of the learning rule which we minimize by a variational approach. First we can rewrite the integrand by expanding in terms of the equations of motion, each constrained by a Lagrange multiplier, £(F ) = ~ 8fg dRin ~ 8fg dQik ~..\ . (d~n _ (F.- ») ,a ~ 8R. da + ~ 8Q. da + ~ In d IYn in In ik Ik in a + L Vik (d~ik - (FiXk + FkXi + FiFk») . ik a (4) The expression for £ still involve two multidimensional integrations over x and y, so taking variations in F, which may depend on x and ( but not on y, we find an expression for the optimal rule in terms of the Lagrange multipliers: 1 -1 _ F = -x -"2v .\y (5) where v = [Vij] and .\ = [..\in]. We define y to be the teacher's expected field given the teacher's output and the student activations, which are observable quantities: y = / dyyp(ylx,() . (6) Now taking variations in the overlaps w.r.t. the integral in Eq. (3) we find a set of differential equations for the Lagrange multipliers1 d..\km _ L,,\ . 8{FiYn) _ LV .. 8{FiXj + FjXi + FiFj) da . In 8Rkm .. I} 8Rkm In IJ (7) where F takes its optimal value defined in Eq. (5). The boundary conditions for the Lagrange multipliers are, and (8) which are found by minimizing the rate of change in generalization error at ai, so that the globally optimal solution reduces to the locally optimal solution at this point, reflecting the fact that changes at al have no affect at other times. IT the above expressions do not yield an explicit formula for the optimal rule then the rule can be determined iteratively by gradient descent on the functional Llfg(F). To determine all the quantities necessary for this procedure requires that we first integrate the equations for the overlaps forward and then integrate the equations for the Lagrange multipliers backwards from the boundary conditions in Eq. (8). 4 Two tractable examples In order to apply the above results we must be able to carry out the average in Eq. (6) and then in Eq. (7). These averages are also required to determine the locally optimal learning rule, so that the present method can be extended to any of the problems which have already been considered under the criteria of local optimality. Here we present two examples where the averages can be computed in closed form. The first problem we consider is a boolean perceptron learning a 326 M. Rattray and D. Saad linearly separable task where we retrieve the locally optimal rule [5]. The second problem is an over-realizable task, where a soft committee machine student learns a perceptron with a sigmoidal response. In this example the globally optimal rule significantly outperforms the locally optimal rule and exhibits a faster asymptotic decay. Boolean perceptron: For the boolean perceptron we choose the activation function g(x) = sgn(x) and both teacher and student have a single hidden node (M = K = 1). The locally optimal rule was determined by Kinouchi and Caticha [5] and they supply the expected teacher field given the teacher output ( = sgn(y) and the student field x (we take the teacher length T = 1 without loss of generality), _ R ( (~exp(-if-») y = x + ---'-----,,..-----:Q ,erfc (-5?) (9) Substituting this expression into the Lagrange multiplier dynamics in Eq. (7) shows that the ratio of .x to v is given by .x/v = -2Q / R, and Eq. (5) then returns the locally optimal value for the optimal rule: ( fi.. exp( _,2z 2 ) F = V i 2. (10) ,erfc (-~,) This rule leads to modulated Hebbian learning and the resulting dynamics are discussed in [5]. We also find that the locally optimal rule is retrieved when the teacher is corrupted by output or weight noise [9]. Soft committee machine learning a continuous perceptron: In this example the teacher is an invertible perceptron (M = 1) while the student is a soft committee machine with an arbitrary number (K) of hidden nodes. We choose the activation function g(x) = erf(x/v'2) for both the student and teacher since this allows the generalization error to be determined in closed form [7]. This is an example of an over-realizable task, since the student has greater complexity than is required to learn the teacher's mapping. The locally optimal rule for this scenario was determined recently [6]. Since the teacher is invertible, the expected teacher activation fi is trivially equal to the true activation y. This leads to a particularly simple form for the dynamics (the n suffix is dropped since there is only one teacher node), dRi dQik do: = biT - Ri do: = bibkT - Qik , (11) where we have defined bi = - Ej vi/ .xj /2 and the optimal rule is given by Fi = biy Xi. The Lagrange multiplier dynamics in Eq. (7) then show that the relative ratios of each Lagrange multiplier remain fixed over time, so that bi is determined by its boundary value (see Eq. (8». It is straightforward to find solutions for long times, since the bi approach limiting values for very small generalization error (there are a number of possible solutions because of symmetries in the problem but any such solution will have the same performance for long times). For example, one possible solution is to have b1 = 1 and bi = 0 for all i f:. 1, which leads to an exponential decay of weights associated with all but a single node. This shows how the optimal performance is achieved when the complexity of the student matches that of the teacher. Figure 1 shows results for a three node student learning a continuous perceptron. Clearly, the locally optimal rule performs poorly in comparison to the globally Globally Optimal On-line Learning Rules 327 100 0.8 10-2 --. __ .€g ~n 0.6 ..... '" .... 10-4 '" 0.4 .... .... (" .... ......... 0.2 i ,. ......... . " 10'" 0 , -0.2 '. 10'" , -0.4 , .... 10-'0 -0.6 " -0.8 10-'2 -1 0 5 10 15 20 25 0 5 10 15 20 25 a a Figure 1: A three node soft committee machine student learns from an continuous percept ron teacher. The figure on the left shows a log plot of the generalization error for the globally optimal (solid line) and locally optimal (dashed line) algorithms. The figure on the right shows the student-teacher overlaps for the locally optimal rule, which exhibit a symmetric plateau before specialization occurs. The overlaps where initialized randomly and uniformly with Qii E [0,0.5] and ~,Qi*j E [0,10-6]. optimal rule. In this example the globally optimal r.ule arrived at was one in which two nodes became correlated with the teacher while a third became anti-correlated, showing another possible variation on the optimal rule (we determined this rule iteratively by gradient descent in order to justify our general approach, although the observations above show how one can predict the final result for long times). The locally optimal rule gets caught in a symmetric plateau, characterized by a lack of differentiation between student vectors associated with different nodes, and also displays a slower asymptotic decay. 5 Conclusion and future work We have presented a method for determining the optimal on-line learning rule for a soft committee machine under a statistical mechanics framework. This result complements previous work on locally optimal rules which sought only to optimize the rate of change in generalization error. In this work we considered the global optimization problem of minimizing the total change in generalization error over the whole learning process. We gave two simple examples for which the rule could be determined in closed form, for one of which, an over-realizable learning scenario, it was shown how the locally optimal rule performed poorly in comparison to the globally optimal rule. It is expected that more involved systems will show even greater difference in performance between local and global optimization and we are currently applying the method to more general teacher mappings. The main technical difficulty is in computing the expected teacher activation in Eq. (6) and this may require the use of approximate methods in some cases. It would be interesting to compare the training dynamics obtained by the globally optimal rules to other approaches, heuristic and principled, aimed at incorporating information about the curvature of the error surface into the parameter modification rule. In particular we would like to examine rules which are known to be optimal asymptotically (e.g. [10]). Another important issue is whether one can apply these results to facilitate the design of a practical learning algorithm. 328 M. Rattray and D. Saad Acknowledgement This work was supported by the EPSRC grant GR/L19232. References [1] G. B. Orr and T. K. Leen in Advances in Neural Information Processing Systems, vol 9, eds M. C. Mozer, M. I. Jordan and T. Petsche (MIT Press, Cambridge MA, 1997) p 606. [2] Y. LeCun, P. Y. Simard and B. Pearlmutter in Advances in Neural Information Processing Systems, vol 5, eds S. J. Hanson, J. D. Cowan and C. 1. Giles (Morgan Kaufman, San Mateo, CA, 1993) P 156. [3] C. M. Bishop, Neural networks for pattern recognition, (Oxford University Press, Oxford, 1995). [4] D. Saad and M. Rattray, Phys. Rev. Lett. 79, 2578 (1997). [5] O. Kinouchi and N. Caticha J. Phys. A 25, 6243 (1992). [6] R. Vicente and N. Caticha J. Phys. A 30, L599 (1997). [7] D. Saad and S. A. Solla, Phys. Rev. Lett. 74, 4337 (1995) and Phys. Rev. E 52 4225 {1995}. [8] M. Biehl and H. Schwarze, J. Phys. A 28, 643 (1995). [9] M. Biehl, P. Riegler and M. Stechert, Phys. Rev. E 52, R4624 (1995). [10] S. Amari in Advances in Neural Information Processing Systems, vol 9, eds M. C. Mozer, M. I. Jordan and T. Petsche (MIT Press, Cambridge MA, 1997).
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Ensemble Learning for Multi-Layer Networks David Barber· Christopher M. Bishopt Neural Computing Research Group Department of Applied Mathematics and Computer Science Aston University, Birmingham B4 7ET, U.K. http://www.ncrg.aston.ac.uk/ Abstract Bayesian treatments of learning in neural networks are typically based either on local Gaussian approximations to a mode of the posterior weight distribution, or on Markov chain Monte Carlo simulations. A third approach, called ensemble learning, was introduced by Hinton and van Camp (1993). It aims to approximate the posterior distribution by minimizing the Kullback-Leibler divergence between the true posterior and a parametric approximating distribution. However, the derivation of a deterministic algorithm relied on the use of a Gaussian approximating distribution with a diagonal covariance matrix and so was unable to capture the posterior correlations between parameters. In this paper, we show how the ensemble learning approach can be extended to fullcovariance Gaussian distributions while remaining computationally tractable. We also extend the framework to deal with hyperparameters, leading to a simple re-estimation procedure. Initial results from a standard benchmark problem are encouraging. 1 Introduction Bayesian techniques have been successfully applied to neural networks in the context of both regression and classification problems (MacKay 1992; Neal 1996). In contrast to the maximum likelihood approach which finds only a single estimate for the regression parameters, the Bayesian approach yields a distribution of weight parameters, p(wID), conditional on the training data D, and predictions are ex·Present address: SNN, University of Nijmegen, Geert Grooteplein 21, Nijmegen, The Netherlands. http://wvw.mbfys.kun.n1/snn/ email: davidbbbfys.kun.n1 tpresent address: Microsoft Research Limited, St George House, Cambridge CB2 3NH, UK. http://vvv.research.microsoft . com email: cmbishopbicrosoft.com 396 D. Barber and C. M. Bishop pressed in terms of expectations with respect to the posterior distribution (Bishop 1995). However, the corresponding integrals over weight space are analytically intractable. One well-established procedure for approximating these integrals, known as Laplace's method, is to approximate the posterior distribution by a Gaussian, centred at a mode of p(wID), in which the covariance of the Gaussian is determined by the local curvature of the posterior distribution (MacKay 1995). The required integrations can then be performed analytically. More recent approaches involve Markov chain Monte Carlo simulations to generate samples from the posterior (Neal 1996}. However, such techniques can be computationally expensive, and they also suffer from the lack of a suitable convergence criterion. A third approach, called ensemble learning, was introduced by Hinton and van Camp (1993) and again involves finding a simple, analytically tractable, approximation to the true posterior distribution. Unlike Laplace's method, however, the approximating distribution is fitted globally, rather than locally, by minimizing a Kullback-Leibler divergence. Hinton and van Camp (1993) showed that, in the case of a Gaussian approximating distribution with a diagonal covariance, a deterministic learning algorithm could be derived. Although the approximating distribution is no longer constrained to coincide with a mode of the posterior, the assumption of a diagonal covariance prevents the model from capturing the (often very strong) posterior correlations between the parameters. MacKay (1995) suggested a modification to the algorithm by including linear preprocessing of the inputs to achieve a somewhat richer class of approximating distributions, although this was not implemented. In this paper we show that the ensemble learning approach can be extended to allow a Gaussian approximating distribution with an general covariance matrix, while still leading to a tractable algorithm. 1.1 The Network Model We consider a two-layer feed-forward network having a single output whose value is given by H /(x,w) = LViU(Ui'X) (1) i=1 where w is a k-dimensional vector representing all of the adaptive parameters in the model, x is the input vector, {ud, i = 1, ... , H are the input-to-hidden weights, and {Vi}, i = 1, ... ,H are the hidden-to-output weights. The extension to multiple outputs is straightforward. For reasons of analytic tractability, we choose the sigmoidal hidden-unit activation function u(a) to be given by the error function u(a} = f! loa exp (-82/2) d8 (2) which (when appropriately scaled) is quantitatively very similar to the standard logistic sigmoid. Hidden unit biases are accounted for by appending the input vector with a node that is always unity. In the current implementation there are no output biases (and the output data is shifted to give zero mean), although the formalism is easily extended to include adaptive output biases (Barber and Bishop 1997) . . The data set consists of N pairs of input vectors and corresponding target output values D = {x~, t~} ,It = 1, ... , N. We make the standard assumption of Gaussian noise on the target values, with variance (3-1. The likelihood of the training data is then proportional to exp(-(3ED ), where the training error ED is 1", 2 ED{w) = 2 ~ (J{x~, w) t~) . (3) ~ Ensemble Leamingfor Multi-Layer Networks 397 The prior distribution over weights is chosen to be a Gaussian of the form p(w) (X exp (-Ew(w)) (4) where Ew(w) = !wT Aw, and A is a matrix of hyper parameters. The treatment of (3 and A is dealt with in Section 2.1. From Bayes' theorem, the posterior distribution over weights can then be written 1 p(wID) = z exp (-(3ED(w) - Ew(w)) (5) where Z is a normalizing constant. Network predictions on a novel example are given by the posterior average of the network output (f(x)) = J f(x, w)p(wID) dw. (6) This represents an integration over a high-dimensional space, weighted by a posterior distribution p(wID) which is exponentially small except in narrow regions whose locations are unknown a-priori. The accurate evaluation of such integrals is thus very difficult. 2 . Ensemble Learning Integrals of the form (6) may be tackled by approximating p(wID) by a simpler distribution Q(w). In this paper we choose this approximating distribution to be a Gaussian with mean wand covariance C. We determine the values of w and C by minimizing the Kullback-Leibler divergence between the network posterior and approximating Gaussian, given by J { Q(w) } F [Q] = Q(w) In p(wID) dw (7) J Q(w) In Q(w)dw - J Q(w) Inp(wID) dw. (8) The first term in (8) is the negative entropy of a Gaussian distribution, and is easily evaluated to give ! In det (C) + const. From (5) we see that the posterior dependent term in (8) contains two parts that depend on the prior and likelihood J Q(w)Ew(w)dw + J Q(w)ED(w)dw. (9) Note that the normalization coefficient Z-l in (5) gives rise to a constant additive term in the KL divergence and so can be neglected. The prior term Ew (w) is quadratic in w, and integrates to give Tr(CA) + ~wT Aw. This leaves the data dependent term in (9) which we write as J (3N L = Q(W)ED(W)dw = "2 I: l(xl!, tl!) I!=l (10) where l(x, t) = J Q(w) (J(x, W))2 dw - 2t J Q(w)f(x, w) dw + t2• (11) 398 D. Barber and C. M. BisJwp For clarity, we concentrate only on the first term in (11), as the calculation of the term linear in I(x, w) is similar, though simpler. Writing the Gaussian integral over Q as an average, ( ), the first term of (11) becomes H ((I(x, w»2) = L (vivju(uTx)u(uJx»). (12) i,j=I To simplify the notation, we denote the set of input-to-hidden weights (Ul' ... , UH) by u and the set of hidden-to-output weights, (VI' ... ' V H) by v. Similarly, we partition the covariance matrix C into blocks, C uu , C vu , Cvv , and C vu = C~v. As the components of v do not enter the non-linear sigmoid functions, we can directly integrate over v, so that each term in the summation (12) gives ((Oij + (u - IT)T \Ilij (u - IT) + n~ (u - IT») u (uTxi) u (uTxj)) (13) where Oij (Cvv - CvuCuu -lCuV)ij + "hvj (14) \Ilij Cuu -ICu,v=:iC lI=:j,uCuu -1, (15) nij 2Cuu -ICu,lI=:jVi. (16) Although the remaining integration in (13) over u is not analytically tractable, we can make use of the following result to reduce it to a one-dimensional integration (u (z·a + ao) u (z·b + bo»)z = (u (zlal + 0.0) u ( za Tb + bolal )) Vlal2 (1 + Ib 12) - (aT b)2 z (17) where a and b are vectors and 0.0, bo are scalar offsets. The avera~e on the left of (17) is over an isotropic multi-dimensional Gaussian, p(z) ex: exp( -z z/2), while the average on the right is over the one-dimensional Gaussian p(z) ex: exp( -z2 /2). This result follows from the fact that the vector z only occurs through the scalar product with a and b, and so we can choose a coordinate system in which the first two components of z lie in the plane spanned by a and b. All orthogonal components do not appear elsewhere in the integrand, and therefore integrate to unity. The integral we desire, (13) is only a little more complicated than (17) and can be evaluated by first transforming the coordinate system to an isotopic basis z, and then differentiating with respect to elements of the covariance matrix to 'pull down' the required linear and quadratic terms in the u-independent pre-factor of (13). These derivatives can then be reduced to a form which requires only the numerical evaluation of (17). We have therefore succeeded in reducing the calculation of the KL divergence to analytic terms together with a single one-dimensional numerical integration of the form (17), which we compute using Gaussian quadrature1 . Similar techniques can be used to evaluate the derivatives of the KL divergence with respect to the mean and covariance matrix (Barber and Bishop 1997). Together with the KL divergence, these derivatives are then used in a scaled conjugate gradient optimizer to find the parameters w and C that represent the best Gaussian fit. The number of parameters in the covariance matrix scales quadratically with the number of weight parameters. We therefore have also implemented a version with 1 Although (17) appears to depend on 4 parameters, it can be expressed in terms of 3 independent parameters. An alternative to performing quadrature during training would therefore be to compute a 3-dimensionallook-up table in advance. Ensemble Learning for Multi-Layer Networks 399 Posterior laplace fit Minimum KLD fit Minimum KL fit Figure 1: Laplace and minimum Kullback-Leibler Gaussian fits to the posterior. The Laplace method underestimates the local posterior mass by basing the covariance matrix on the mode alone, and has KL value 41. The minimum KullbackLeibler Gaussian fit with a diagonal covariance matrix (KLD) gives a KL value of 4.6, while the minimum Kullback-Leibler Gaussian with full covariance matrix achieves a value of 3.9. a constrained covariance matrix s C = diag(di,· · ·, d~) + L sisT (18) i=l which is the form of covariance used in factor analysis (Bishop 1997). This reduces the number offree parameters in the covariance matrix from k(k + 1)/2 to k(s + 1) (representing k(s + 1) - s(s - 1)/2 independent degrees of freedom) which is now linear in k. Thus, the number of parameters can be controlled by changing sand, unlike a diagonal covariance matrix, this model can still capture the strongest of the posterior correlations. The value of s should be as large as possible, subject only to computational cost limitations. There is no 'over-fitting' as s is increased since more flexible distributions Q(w) simply better approximate the true posterior. We illustrate the optimization of the KL divergence using a toy problem involving the posterior distribution for a two-parameter regression problem. Figure 1 shows the true posterior together with approximations obtained from Laplace's method, ensemble learning with a diagonal covariance Gaussian, and ensemble learning using an unconstrained Gaussian. 2.1 Hyperparameter Adaptation So far, we have treated the hyperparameters as fixed. We now extend the ensemble learning formalism to include hyperparameters within the Bayesian framework. For simplicity, we consider a standard isotropic prior covariance matrix of the form A = aI, and introduce hyperpriors given by Gamma distributions lnp (a) In {aa-l exp ( -~) } + const (19) lnp (f3) = In {f3 C- 1 exp ( -~) } + const (20) 400 D. Barber and C. M. BisJwp where a, b, c, d are constants. The joint posterior distribution of the weights and hyperparameters is given by p (w, a, ,BID) <X P (Dlw, j3) p (wla) p (a) p (,B) in which lnp (Dlw,,B) N - ,BED + "2 In,B + const lnp (wla) k -alwl2 + '2 In a + const (21) (22) (23) We follow MacKay (1995) by modelling the joint posterior p (w, a, ,BID) by a factorized approximating distribution of the form Q(w)R(a)S(,B) (24) where Q(w) is a Gaussian distribution as before, and the functional forms of Rand S are left unspecified. We then minimize the KL divergence J {Q(W)R(a)S(,B) } F[Q,R,S] = Q(w)R(a)S(,B) In p(w,a,,BID) dwdad,B. (25) Consider first the dependence of (25) on Q(w) F [QJ - J Q(w)R(a)S(j3) { -,BED(W) ~lwl2 -In Q(w) } + const (26) - -J Q(w) { -73ED(W) - ~lwl2 -In Q(W)} + const (27) where a = J R(a)ada and 73 = J S(,B)j3d,B. We see that (27) has the form of (8), except that the fixed hyperparameters are now replaced with their average values. To calculate these averages, consider the dependence of the functional F on R(a) F[R] - J Q(W)R(a)S(j3){-~lwI2+~lna+(a-1)lna-i} dwdad,B -J R(a) {; + (r - 1) Ina -In R(a)} da + const (28) where r = ~ +a and lis = ~lwl2 + ~TrC+ lib. We recognise (28) as the KullbackLeibler divergence between R(a) and a Gamma distribution. Thus the optimum R(a) is also Gamma distributed R(a) <X a r - 1 exp (-;) . (29) We therefore obtain a = rs. A similar procedure for S(,B) gives 73 = uv, where u = ~ + c and 11v = (ED) + lid, in which (ED) has already been calculated during the optimization of Q(w). This defines an iterative procedure in which we start by initializing the hyperparameters (using the mean of the hyperprior distributions) and then alternately optimize the KL divergence over Q(w) and re-estimate a and 73. 3 Results and Discussion As a preliminary test of our method on a standard benchmark problem, we applied the minimum KL procedure to the Boston Housing dataset. This is a one Ensemble Learning for Multi-Layer Networks 401 I Method Test Error Ensemble (s == 1) 0.22 Ensemble (diagonal) 0.28 Laplace 0.33 Table 1: Comparison of ensemble learning with Laplace's method. The test error is defined to be the mean squared error over the test set of 378 examples. dimensional regression problem, with 13 inputs, in which the data for 128 training examples was obtained from the DELVE archive2 • We trained a network of four hidden units, with covariance matrix given by (18) with s = 1, and specified broad hyperpriors on a and (3 (a = 0.25, b = 400, c = 0.05, and d = 2000). Predictions are made by evaluating the integral in (6). This integration can be done analytically as a consequence of the form of the sigmoid function given in (2). We compared the performance of the KL method against the Laplace framework of MacKay (1995) which also treats hyperparameters through a re-estimation procedure. In addition we also evaluated the performance of the ensemble method using a diagonal covariance matrix. Our results are summarized in Table 1. Acknowledgements We would like to thank Chris Williams for helpful discussions. Supported by EPSRC grant GR/J75425: Novel Developments in Learning Theory for Neural Networks. References Barber, D. and C. M. Bishop (1997). On computing the KL divergence for Bayesian neural networks. Technical report, Neural Computing Research Group, Aston University, Birmingham, {;.K. Bishop, C. M. (1995). Neural Networks for Pattern Recognition. Oxford University Press. Bishop, C. M. (1997). Latent variables, mixture distributions and topographic mappings. Technical report, Aston University. To appear in Proceedings of the NATO Advanced Study Institute on Learning in Graphical Models, Erice. Hinton, G. E. and D. van Camp (1993). Keeping neural networks simple by minimizing the description length of the weights. In Proceedings of the Sixth Annual Conference on Computational Learning Theory, pp. 5-13. MacKay, D. J. C. (1992). A practical Bayesian framework for back-propagation networks. Neural Computation 4 (3), 448-472. MacKay, D. J. C. (1995). Developments in probabilistic modelling with neural networks--ensemble learning. In Neural Networks: Artificial Intelligence and Industrial Applications. Proceedings of the 3rd Annual Symposium on Neural Networks, Nijmegen, Netherlands, 14-15 September 1995, Berlin, pp. 191-198. Springer. MacKay, D. J. C. (1995). Probable networks and plausible predictions - a review of practical Bayesian methods for supervised neural networks. Network: Computation in Neural Systems 6(3), 469-505. Neal, R. M. (1996). Bayesian Learning for Neural Networks. Springer. Lecture Notes in Statistics 118. 2See http://wvv . cs. utoronto. cal "-'delve I
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Estimating Dependency Structure as a Hidden Variable Marina Meill and Michael I. Jordan {mmp, jordan}@ai.mit.edu Center for Biological & Computational Learning Massachusetts Institute of Technology 45 Carleton St. E25-201 Cambridge, MA 02142 Abstract This paper introduces a probability model, the mixture of trees that can account for sparse, dynamically changing dependence relationships. We present a family of efficient algorithms that use EM and the Minimum Spanning Tree algorithm to find the ML and MAP mixture of trees for a variety of priors, including the Dirichlet and the MDL priors. 1 INTRODUCTION A fundamental feature of a good model is the ability to uncover and exploit independencies in the data it is presented with. For many commonly used models, such as neural nets and belief networks, the dependency structure encoded in the model is fixed, in the sense that it is not allowed to vary depending on actual values of the variables or with the current case. However, dependency structures that are conditional on values of variables abound in the world around us. Consider for example bitmaps of handwritten digits. They obviously contain many dependencies between pixels; however, the pattern of these dependencies will vary across digits. Imagine a medical database recording the body weight and other data for each patient. The body weight could be a function of age and height for a healthy person, but it would depend on other conditions if the patient suffered from a disease or was an athlete. Models that are able to represent data conditioned dependencies are decision trees and mixture models, including the soft counterpart of the decision tree, the mixture of experts. Decision trees however can only represent certain patterns of dependecy, and in particular are designed to represent a set of conditional probability tables and not a joint probability distribution. Mixtures are more flexible and the rest of this paper will be focusing on one special case called the mixtures of trees. We will consider domains where the observed variables are related by pairwise dependencies only and these dependencies are sparse enough to contain no cycles. Therefore they can Estimating Dependency Structure as a Hidden Variable 585 be represented graphically as a tree. The structure of the dependencies may vary from one instance to the next. We index the set of possible dependecy structures by a discrete structure variable z (that can be observed or hidden) thereby obtaining a mixture. In the framework of graphical probability models, tree distributions enjoy many properties that make them attractive as modelling tools: they have a flexible topology, are intuitively appealing, sampling and computing likelihoods are linear time, simple efficient algorithms for marginalizing and conditioning (O(1V12) or less) exist. Fitting the best tree to a given distribution can be done exactly and efficiently (Chow and Liu, 1968). Trees can capture simple pairwise interactions between variables but they can prove insufficient for more complex distributions. Mixtures of trees enjoy most of the computational advantages of trees and, in addition, they are universal approximators over the space of all distributions. Therefore, they are fit for domains where the dependency patterns become tree like when a possibly hidden variable is instantiated. Mixture models have been extensively used in the statistics and neural network literature. Of relevance to the present work are the mixtures of Gaussians, whose distribution space, in the case of continuous variables overlaps with the space of mixtures of trees. Work on fitting a tree to a distribution in a Maximum-Likelihood (ML) framework has been pioneered by (Chow and Liu, 1968) and was extended to poly trees by (Pearl, 1988) and to mixtures of trees with observed structure variable by (Geiger, 1992; Friedman and Goldszmidt, 1996). Mixtures of factorial distributions were studied by (Kontkanen et al., 1996) whereas (Thiesson et aI., 1997) discusses mixtures of general belief nets. Multinets (Geiger, 1996) which are essentially mixtures of Bayes nets include mixtures of trees as a special case. It is however worth studying mixtures of trees separately for their special computational advantages. This work presents efficient algorithms for learning mixture of trees models with unknown or hidden structure variable. The following section introduces the model; section 3 develops the basic algorithm for its estimation from data in the ML framework. Section 4 discusses the introduction of priors over mixtures of trees models and presents several realistic factorized priors for which the MAP estimate can be computed by a modified versions of the basic algorithm. The properties of the model are verified by simulation in section 5 and section 6 concludes the paper. 2 THE MIXTURE OF TREES MODEL In this section we will introduce the mixture of trees model and the notation that will be used throughout the paper. Let V denote the set of variables of interest. According to the graphical model paradigm, each variable is viewed as a vertex of a graph. Let Tv denote the number of values of variable v E V, XV a particular value of V, XA an assignment to the variables in the subset A of V. To simplify notation Xv will be denoted by x. We use trees as graphical representations for families of probability distributions over V that satisfy a common set of independence relationships encoded in the tree topology. In this representation, an edge of the tree shows a direct dependence, or, more precisely, the absence of an edge between two variables signifies that they are independent, conditioned on all the other variables in V. We shall call a graph that has no cycles a tree I and shall denote by E the set of its (undirected) edges. A probability distribution T that is conformal with the tree (V, E) is a distribution that can be factorized as: () IT(u,v)EE Tuv (xu, xv) T X = IT T, (x )degv-l vEV v v (1) Here deg v denotes the degree of v, e.g. the number of edges incident to node v E V. The l In the graph theory literature, our definition corresponds to a forest. The connected components of a forest are called trees. 586 M. MeillJ and M. I. Jordan factors Tuv and Tv are the marginal distributions under T: Tuv(xu,xv) = 2: T(xu , xv,XV-{u,v}), Tv(xv) = 2: T(xv,xv-{v})' (2) XV-{u.v} Xv-tv} The distribution itself will be called a tree when no confusion is possible. Note that a tree distribution has for each edge (u, v) E E a factor depending on xu, Xv onlyl If the tree is connected, e.g. it spans all the nodes in V , it is often called a spanning tree. An equivalent representation for T in terms of conditional probabilities is T(x) = II Tvlpa(v)(xvlxpa(v») vEV (3) The form (3) can be obtained from (1) by choosing an arbitrary root in each connected component and recursively substituting Tvt';V) by Tvlpa(v) starting from the root. pa(v) represents the parent of v in the thus directed tree or the empty set if v is the root of a connected component. The directed tree representation has the advantage of having independent parameters. The total number of free parameters in either representation is E(u,v)EET rurv - EVEv(degv - l)rv . Now we define a mixture of trees to be a distribution of the form m Q(X) = 2: AkTk(x); Ak 2: 0, k = 1, . .. , m; (4) k=1 From the graphical models perspecti ve, a mixture of trees can be viewed as a containing an unobserved choice variable z, taking value k E {I, ... m} with probability Ak. Conditioned on the value of z the distribution of the visible variables X is a tree. The m trees may have different structures and different parameters. Note that because of the structure variable, a mixture of trees is not properly a belief network, but most of the results here owe to the belief network perspective. 3 THE BASIC ALGORITHM: ML FITIING OF MIXTURES OF TREES This section will show how a mixture of trees can be fit to an observed dataset in the Maximum Likelihood paradigm via the EM algorithm (Dempster et al., 1977). The observations are denoted by {xl , x2 , ... , xN}; the corresponding values of the structure variable are {zi,i=I, ... N}. Following a usual EM procedure for mixtures, the Expectation (E) step consists in estimating the posterior probability of each tree to generate datapoint xi Pr[zi = klxl , ... ,N, model] = 'Yk(i) = AkTk(x:). Lkl AklTk (x') (5) Then the expected complete log-likelihood to be maximized by the M step of the algorithm is m N E[Ic Ixl , ... N , model] L rk[log Ak + L pk(xi) 10gTk(xi)] (6) k=1 i=1 N rk = 2: 'Yk(X i ), (7) i=1 The maximizing values for the parameters A are Akew = rk/ N. To obtain the new distributions Tk, we have to maximize for each k the expression that is the negative of the Estimating Dependency Structure as a Hidden Variable Figure 1: The Basic Algorithm: ML Fitting of a Mixture of Trees Input:Dataset {xl, ... xN} Initial model m, Tk, ,\k, k = I, . .. m Procedure MST( weights) that fits a maximum weight spanning tree over V Iterate until convergence Estep: compute'Y~, pk(X') fork = I, . .. m, i= 1, . . . Nby(5),(7) Mstep: Ml. M2. MJ. M4. M5. '\k +- rk/N, k = I, ... m compute marginals P:, p!v, U, v E V, k = I, ... m compute mutual information I!v u, v E V, k = I, ... m call MST( { I!v }) to generate ETk for k = I, ... m T!v +- p!v, ; T: +- P: for (u, v) E ETk, k = I, ... m crossentropy between pk and Tk. N L pk(xi ) 10gTk(xi) i=l 587 (8) This problem can be solved exactly as shown in (Chow and Liu, 1968). Here we will give a brief description of the procedure. First, one has to compute the mutual information between each pair of variables in V under the target distribution P '"" ( ) PUtl(xu, Xtl) JUti = Jvu = L.J Puv Xu, Xv log Pu(xu)PtI(xv)' u, v E V, u f=v. X",X v (9) Second, the optimal tree structure is found by a Maximum Spanning Tree (MST) algorithm using JUti as the weight for edge (u, v), \lu, v E V.Once the tree is found, its marginals Tutl (or Tul v), (u, v) E ET are exactly equal to the corresponding marginals PUti of the target distribution P. They are already computed as an intermediate step in the computation of the mutual informations JUti (9). In our case, the target distribution for Tk is represented by the posterior sample distribution pk. Note that although each tree fit to pk is optimal, for the encompassing problem of fitting a mixture of trees to a sample distribution only a local optimum is guaranteed to be reached. The algorithm is summarized in figure 1. This procedure is based on one important assumption that should be made explicit now. It is the Parameter independence assumption: The distribution T:1pa( tI) for any k, v and value of pa( v) is a multinomial with rv - 1 free parameters that are independent of any other parameters of the mixture. It is possible to constrain the m trees to share the same structure, thus constructing a truly Bayesian network. To achieve this, it is sufficient to replace the weights in step M4 by Lk J~tI and run the MST algorithm only once to obtain the common structure ET. The tree stuctures obtained by the basic algorithm are connected. The following section will give reasons and ways to obtain disconnected tree structures. 4 MAP MIXTURES OF TREES In this section we extend the basic algorithm to the problem of finding the Maximum a Posteriori (MAP) probability mixture of trees for a given dataset. In other words, we will consider a nonuniform prior P[mode/] and will be searching for the mixture of trees that maximizes log P[model\x1 , .. . N] = 10gP[xl, ... N\model] + log P[model] + constant. (10) Factorized priors The present maximization problem differs from the ML problem solved in the previous section only by the addition of the term log P[model]. We can as well 588 M. Meilii and M. l. Jordan approach it from the EM point of view, by iteratively maximizing E [logP[modelJxl , ... N, ZI, ... NJ] = E[lc{xl , ... N, zl , ... NJmodel)] + 10gP[model] (11) It is easy to see that the added term does not have any influence on the E step,which will proceed exactly as before. However, in the M step, we must be able to successfully maximize the r.h.s. of (11). Therefore, we look for priors of the form m P[model] = P[AI, ... m] II P[Tkl (12) k=1 This class of priors is in agreement with the parameter independence assumption and includes the conjugate prior for the multinomial distribution which is the Dirichlet prior. A Dirichlet prior over a tree can be represented as a table of fictitious marginal probabilities P~~ for each pair u, v of variables plus an equivalent sample size Nt that gives the strength of the prior (Heckerman et al., 1995). However, for Dirichlet priors, the maximization over tree structures (corresponding to step M4) can only be performed iteratively (Meilli et al., 1997). MDL (Minimum Description Length) priors are less informative priors. They attempt to balance the number of parameters that are estimated with the amount of data available, usually by introducing a penalty on model complexity. For the experiments in section 5 we used edge pruning. More smoothing methods are presented in (Meilli et al., 1997). To penalize the number of parameters in each component we introduce a prior that penalizes each edge that is added to a tree, thus encouraging the algorithm to produce disconnected trees. The edge pruning prior is P [T] <X exp [-(3 L:utJ E ET L\utJ] . We choose a uniform penalty L\utJ = 1. Another possible choice is L\utJ = (ru - 1)( rtJ - 1) which is the number of parameters introduced by the presence of edge (u , v) w.r.t. a factorized distribution. Using this prior is equivalent to maximizing the following expression in step M4 of the Basic Algorithm (the index k being dropped for simplicity) argmax L: max[O, r1utJ (3~tJ] = argmax L: WutJ ET utJEET ET utJEET (13) 5 EMPIRICAL RESULTS We have tested our model and algorithms for their ability to retrieve the dependency structure in the data, as classifiers and as density estimators. For the first objective, we sampled 30,000 datapoints from a mixture of 5 trees over 30 variables with rtJ = 4 for all vertices. All the other parameters of the generating model and the initial points for the algorithm were picked at random. The results on retrieving the original trees were excellent: out of 10 trials, the algorithm failed to retrieve correctly only 1 tree in 1 trial. This bad result can be accounted for by sampling noise. The tree that wasn't recovered had a A of only 0.02. The difference between the log likelihood of the samples of the generating tree and the approximating tree was 0.41 bits per example. For classification, we investigated the performance of mixtures of trees on a the Australian Credit dataset from the UCI repository2. The data set has 690 instances of 14-dimensional attribute vectors. Nine attributes are discrete ( 2 - 14 values) and 5 are continuous. The class variable has 6 values. The continuous variables were discretized in 3 - 5 uniform bins each. We tested mixtures with different values for m and for the edge pruning parameter (3. For comparison we tried also mixtures of factorial distributions of different sizes. One tenth of the data, picked randomly at each trial, was used for testing and the rest for training. In the training phase, we learned a MT model of the joint distribution of all the 15 variables. 2 http://www.ics . uci.edu/-mlearn/MLRepository.html Estimating Dependency Structure as a Hidden Variable 589 Figure 2: Performance of different algorithms on the Australian Credit dataset. - is mixture of trees with j3 = 10, - - is mixture of trees with beta = 11m, -.- is mixture of factorial distributions. 88 87 86 (385 C1) 584 u ~83 '? 82 I 81 800 I I' I ' .. I 5 10 ... 15 m , , , -, , , ~'i t7 ~ .., 20 25 30 Table 1: a) Mixture of trees compression rates [log lte&t! Nte&t1. b) Compression rates (bits/digit) for the single digit (Digit) and double digit (Pairs) datasets. MST is mixtures of trees, MF is a mixture of factorial distributions, BR is base rate model, H-WS is Helmholtz Machine trained with the wake-sleep algorithm (Frey et aI., 1996), H-MF is Helmholtz Machine trained with the Mean Field approximation, FV is a fully visible bayes net. (*=best) (a) (b) Algorithm Digits Pairs m Digits Pairs 16 *34.72 79.25 32 34.48 *78.99 64 34.84 79.70 128 34.88 81 .26 gzip 44.3 89.2 BR 59.2 118.4 MF 37.5 92.7 H-MF 39.5 80.7 H-WS 39.1 80.4 FV 35.9 *72.9 MT *34.7 79.0 In the testing phase, the output of our classifier was chosen to be the class value with the largest posterior probability given the inputs. Figure 2 shows that the results obtained for mixtures of trees are superior to those obtained for mixtures of factorial distributions.For comparison, correct classification rates obtained and cited in (Kontkanen et aI., 1996) on training/test sets of the same size are: 87.2next best model (a decision tree called CaI50). We also tested the basic algorithm as a density estimator by running it on a subset of binary vector representations of handwritten digits and measuring the compression rate. One dataset contained images of single digits in 64 dimensions, the second contained 128 dimensional vectors representing randomly paired digit images. The training, validation and test set contained 6000, 2000, and 5000 exemplars respectively. The data sets, the training conditions and the algorithms we compared with are described in (Frey et aI., 1996). We tried mixtures of 16, 32, 64 and 128 trees, fitted by the basic algorithm. The results (shown in table1 averaged over 3 runs) are very encouraging: the mixture of trees is the absolute winner for compressing the simple digits and comes in second as a model for pairs of digits. This suggests that our model (just like the mixture of factorized distributions) is able to perform good compression of the digit data but is unable to discover the independency in the double digit set. 590 M. Meilli and M. I. Jordan 6 CONCLUSIONS This paper has shown a method of modeling and exploiting sparse dependency structure that is conditioned on values of the data. By using trees, our method avoids the exponential computation demands that plague both inference and structure finding in wider classes of belief nets. The algorithms presented here are linear in m and N and quadratic in I V I. Each M step is performing exact maximization over the space of all the tree structures and parameters. The possibility of pruning the edges of the components of a mixture of trees can playa role in classification, as a means of automatically selecting the variables that are relevant for the task. The importance of using the right priors in constructing models for real-world problems can hardly be understated. In this context, the present paper has presented a broad class of priors that are efficiently handled in the framework of our algorithm and it has shown that this class includes important priors like the MDL prior and the Dirichlet prior. Acknowledgements Thanks to Quaid Morris for running the digits and structure finding experiments and to Brendan Frey for providing the digits datasets. References Chow, C. K. and Liu, C. N. (1968). Approximating discrete probability distributions with dependence trees. "IEEE Transactions on Information Theory ", IT-14(3 ):462-467. Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, B, 39: 1-38. Frey, B. J., Hinton, G. E., and Dayan, P. (1996). Does the wake-sleep algorithm produce good density estimators? In Touretsky, D., Mozer, M., and Hasselmo, M., editors, Neural Information Processing Systems, number 8, pages 661-667. MIT Press. Friedman, N. and Goldszmidt, M. (1996). Building classifiers using Bayesian networks. In Proceedings of the National Conference on Artificial Intelligence (AAAI 96), pages 1277-1284, Menlo Park, CA. AAAI Press. Geiger, D. (1992). An entropy-based learning algorithm of bayesian conditional trees. In Proceedings of the 8th Conferenceon Uncertainty in AI, pages 92-97. Morgan Kaufmann Publishers. Geiger, D. (1996). Knowledge representation and inference in similarity networks and bayesian multinets. "Artificial Intelligence", 82:45-74. Heckerman, D., Geiger, D., and Chickering, D. M. (1995). Learning Bayesian networks: the combination of knowledge and statistical data. Machine Learining, 20(3): 197-243. Kontkanen, P., Myllymaki, P., and Tirri, H. (1996). Constructing bayesian finite mixture models by the EM algorithm. Technical Report C-1996-9, Univeristy of Helsinky. Department of Computer Science. Meilli, M., Jordan, M. I., and Morris, Q. D. (1997). Estimating dependency structure as a hidden variable. Technical Report AIM-1611 ,CBCL-151, Massachusetts Institute of Technology, Artificial Intelligence Laboratory. Pearl, 1. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufman Publishers, San Mateo, CA. Thiesson, B., Meek, C., Chickering, D. M., and Heckerman, D. (1997). Learning mixtures of Bayes networks. Technical Report MSR-POR-97-30, Microsoft Research.
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Ensemble and Modular Approaches for Face Detection: a Comparison Raphael Feraud ·and Olivier Bernier t France-Telecom CNET DTLjDLI Technopole Anticipa, 2 avenue Pierre Marzin, 22307 Lannion cedex, FRANCE Abstract A new learning model based on autoassociative neural networks is developped and applied to face detection. To extend the detection ability in orientation and to decrease the number of false alarms, different combinations of networks are tested: ensemble, conditional ensemble and conditional mixture of networks. The use of a conditional mixture of networks allows to obtain state of the art results on different benchmark face databases. 1 A constrained generative model Our purpose is to classify an extracted window x from an image as a face (x E V) or non-face (x EN). The set of all possible windows is E = V uN, with V n N = 0. Since collecting a representative set of non-face examples is impossible, face detection by a statistical model is a difficult task. An autoassociative network, using five layers of neurons, is able to perform a non-linear dimensionnality reduction [Kramer, 1991]. However, its use as an estimator, to classify an extracted window as face or non-face, raises two problems: 1. V', the obtained sub-manifold can contain non-face examples (V C V'), 2. owing to local minima, the obtained solution can be close to the linear solution: the principal components analysis. Our approach is to use counter-examples in order to find a sub-manifold as close as possible to V and to constrain the algorithm to converge to a non-linear solution [Feraud, R. et al., 1997]. Each non-face example is constrained to be reconstructed as its projection on V. The projection P of a point x of the input space E on V, is defined by: ·email: feraud@lannion.cnet.fr t email: bernier@lannion.cnet.fr Ensemble and Modular Approaches for Face Detection: A Comparison 473 • if x E V, then P{ x) = x, • if x rJ. V: P{x) = argminYEv{d(x, V)), where d is the Euclidian distance. During the learning process, the projection P of x on V is approximated by: P(x) ,...., ~ 2:7=1 Vi, where VI, V2, •.. , Vn , are the n nearest neighbours, in the training set of faces, of v, the nearest face example of x. The goal of the learning process is to approximate the distance V of an input space element x to the set of faces V: • V{x, V) = Ilx - P(x)11 ,...., it (x - £)2, where M is the size of input image x and £ the image reconstructed by the neural network, • let x E £, then x E V if and only if V{x, V) S T, with T E IR, where T is a threshold used to adjust the sensitivity of the model. 15 x 20 outputs 50 neurons 35 neurons 15 x 20 inputs Figure 1: The use of two hidden layers and counter-examples in a compression neural network allows to realize a non-linear dimensionality reduction. In the case of non-linear dimensionnality reduction, the reconstruction error is related to the position of a point to the non-linear principal components in the input space. Nevertheless, a point can be near to a principal component and far from the set of faces. With the algorithm proposed, the reconstruction error is related to the distance between a point to the set of faces. As a consequence, if we assume that the learning process is consistent [Vapnik, 1995], our algorithm is able to evaluate the probability that a point belongs to the set of faces. Let y be a binary random variable: y = 1 corresponds to a face example and y = 0 to a non-face example, we use: (,,_r):l P(y = llx) = e0'2 ,where (j depends on the threshold T The size of the training windows is 15x20 pixels. The faces are normalized in position and scale. The windows are enhanced by histogram equalization to obtain a relative independence to lighting conditions, smoothed to remove the noise and normalized by the average face, evaluated on the training set. Three face databases are used: after vertical mirroring, B fl is composed of 3600 different faces with orientation between 0 degree and 20 degree, Bf2 is composed of 1600 different faces with orientation between 20 degree and 60 degree and B f3 is the concatenation of Bfl and Bf2, giving a total of 5200 faces. All of the training faces are extracted 474 R. Feraud and O. Bernier from the usenix face database(**), from the test set B of CMU(**), and from 100 images containing faces and complex backgrounds . • Figure 2: Left to right: the counter-examples successively chosen by the algorithm are increasingly similar to real faces (iteration 1 to 8) . The non-face databases (Bn!! ,Bn!2,Bn!3), corresponding to each face database, are collected by an iterative algorithm similar to the one used in [Sung, K. and Poggio, T ., 1994] or in [Rowley, H. et al., 1995]: • 1) Bn! = 0, T = Tmin , • 2) the neural network is trained with B! + Bn!, • 3) the face detection system is tested on a set of background images, • 4) a maximum of 100 su bimages Xi are collected with V (Xi , V) ~ T , • 5) Bn! = Bn! + {xo, .. . , xn } , T = T + Jl , with Jl > 0, • 6) while T < Tmax go back to step 2. After vertical mirroring, the size of the obtained set of non-face examples is respectively 1500 for Bn!! , 600 for Bn!2 and 2600 for Bn!3. Since the non-face set (N) is too large, it is not possible to prove that this algorithm converge in a finite time. Nevertheless, in only 8 iterations, collected counter-examples are close to the set of faces (Figure 2) . Using this algorithm, three elementary face detectors are constructed: the front view face detector trained on Bfl and Bnfl (CGM1), the turned face detector trained on Bf2 and Bn!2 (CGM2) and the general face detector trained on B!3 and Bn!3 (CGM3). To obtain a non-linear dimensionnality reduction, five layers are necessary. However, our experiments show that four layers are sufficient. Consequently, each CGM has four layers (Figure 1) . The first and last layers consist each of 300 neurons, corresponding to the image size 15x20. The first hidden layer has 35 neurons and the second hidden layer 50 neurons. In order to reduce the false alarm rate and to extend the face detection ability in orientation, different combinations of networks are tested. The use of ensemble of networks to reduce false alarm rate was shown by [Rowley, H. et al., 1995]. However, considering that to detect a face in an image, there are two subproblems to solve, detection of front view faces and turned faces, a modular architecture can also be used. 2 Ensemble of CGMs Generalization error of an estimator can be decomposed in two terms: the bias and the variance [Geman, S. et al., 1992]. The bias is reduced with prior knowledge. The use of an ensemble of estimators can reduce the variance when these estimators are independently and identically distributed [Raviv, Y. and Intrator, N., 1996]. Each face detector i produces: Assuming that the three face detectors (CGM1,CGM2,CGM3) are independently and identically distributed (iid), the ouput of the ensemble is: Ensemble and Modular Approaches for Face Detection: A Comparison 475 3 Conditional mixture of CGMs To extend the detection ability in orientation, a conditional mixture of CGMs is tested. The training set is separated in two subsets: front view faces and the corresponding counter-examples (B = 1) and turned faces and the corresponding counter-examples (B = 0). The first subnetwork (CGMl) evaluates the probability of the tested image to be a front view face, knowing the label equals 1 (P(y = llx, B = 1)). The second (CGM2) evaluates the probability of the tested image to be a turned face, knowing the label equals 0 (P(y = llx, B = 0)). A gating network is trained to evaluate P(B = llx), supposing that the partition B = 1, B = 0 can be generalized to every input: E[Ylx] = E[yIB = 1, x]f(x) + E[yIB = 0, x](l- f(x)) Where f(x) is the estimated value of P{B = llx) . This system is different from a mixture of experts introduced by [Jacobs, R. A. et aI., 1991]: each module is trained separately on a subset of the training set and then the gating network learns to combine the outputs. 4 Conditional ensemble of CGMs To reduce the false alarm rate and to detect front view and turned faces, an original combination, using (CGMl,CGM2) and a gate network, is proposed. Four sets are defined: • F is the front view face set, • P is the turned face set, with F n p = 0, • V = F U P is the face set, • N is the non-face set, with V n N = 0, Our goal is to evaluate P{x E Vlx). Each estimator computes respectively: • P(x E Fix E FuN, x) (CGMl(x)), • P(x E Pix E puN, x) (CGM2(x)), Using the Bayes theorem, we obtain: P(x E Fix) P(x E Fix E FUN,x) = P(x E FUNlx)P(x E FuNlx E F,x) Since x E F => x E FuN, then: P(x E Fix) P(x E Flx,x E FuN) = P(x E FUNlx) 476 R. Feraud and o. Bernier ~ P(x E Fix) = P(x E Fix E FUN,x)P(x E FUNlx) ~ P(x E Fix) = P(x E Fix E FUN,x)[P(x E Fix) + P(x E Nix)] In the same way, we have: P(x E Pix) = P(x E Pix E puN, x)[P(x E Pix) + P(x E Nix)] Then: P(x E Vlx) = P(x E Nlx)[P(x E pix E puN, x) + P(x E Fix E FuN, x)] +P(x E Plx)P(x E Pix E puN, x) + P(x E Flx)P(x E Fix E FuN, x) Rewriting the previous equation using the following notation, CGMl(x) for P(x E Fix E FuN, x) and CGMl(x) for P(x E Pix E puN, x), we have: P(x E Vlx) = P(x E Nlx)[CGMl(x) + CGM2(x)] (1) +P(x E Plx)CGM2(x) + P(x E Flx)CGMl(x) (2) Then, we can deduce the behaviour of the conditional ensemble: • in N, if the output of the gate network is 0.5, as in the case of ensembles, the conditional ensemble reduces the variance of the error (first term of the right side of the equation (1)), • in V, as in the case of the conditional mixture, the conditional ensemble permits to combine two different tasks (second term of the right side of the equation (2)) : detection of turned faces and detection of front view faces. The gate network f(x) is trained to calculate the probability that the tested image is a face (P(x E Vlx)), using the following cost function: C = ~ ([f(xi)MGCl(x) + (1- f(xd)]MGC2(x) - Yi)2 + ~ (J(xd - 0.5)2 x.EV xiEN 5 Discussion Each 15x20 subimage is extracted and normalized by enhancing, smoothing and substracting the average face, before being processed by the network. The detection threshold T is fixed for all the tested images. To detect a face at different scales, the image is subsampled. The first test allows to evaluate the limits in orientation of the face detectors. The sussex face database(**), containing different faces with ten orientations betwen 0 degree and 90 degrees, is used (Table 1). The general face detector (CGM3) uses the same learning face database than the different mixtures of CGMs. Nevertheless, CGM3 has a smaller orientation range than the conditional mixtures of CGMs, and the conditional ensemble of CGMs. Since the performances of the ensemble of CGMS are low, the corresponding hypothesis (the CGMs are iid) is invalid. Moreover, this test shows that the combination by a gating neural network ofCGMs, Ensemble and Modular Approaches for Face Detection: A Comparison 477 Table l :Results on Sussex face database orientation CGM1 CGM2 CGM3 Ensemble Conditional Conditional (degree) (1,2,3) ensemble 0 100.0 0 10 62.5 % 20 50.0 % 30 12.5 % 40 0.0 % 50 0.0 % 60 0.0 % 70 0.0 % trained on different training set, allows to extend the detection ability to both front view and turned faces. The conditional mixture of CGMs obtains results in term of orientation and false alarm rate close to the best CGMs used to contruct it (see Table 1 and Table 2). The second test allows to evaluate the false alarms rate and to compare our results with the best results published so far on the test set A [Rowley, H. et al., 1995] of the CMU (**), containing 42 images of various quality. First, these results show that the model, trained without counter-examples (GM), overestimates the distribution offaces and its false alarm rate is too important to use it as a face detector. Second, the estimation of the probability distribution of the face performed by one CGM (CGM3) is more precise than the one obtained by [Rowley, H. et al., 1995] with one SWN (see Table 2). The conditional ensemble of CGMs and the conditional mixture of CG Ms obtained a similar detection rate than an ensemble of SWN s [Rowley, H. et al. , 1995], but with a false alarm rate two or three times lower. Since the results of the conditional ensemble of CG Ms and the conditional mixture of CGMs are close on this test, the detection rate versus the number of false alarms is plotted (Figure 3), for different thresholds. The conditional mixture of CGMs curve is above the one for the conditional ensemble of CG Ms. l00~--~----~----~----~----~--~ ~ ........ __ ........ ,,............. ..... 10 3D Cl so 110 70 10 Numt.rolt .. .wnw Figure 3: Detection rate versus number of false alarms on the CMU test set A. In dashed line conditional ensemble and in solid line conditional mixture. The conditionnal mixture of CG Ms is used in an application called LISTEN [Collobert, M. et al. , 1996]: a camera detects, tracks a face and controls a microphone array towards a speaker, in real time. The small size of the subimages (15x20) processed allows to detect a face far from the camera (with an aperture of 60 degrees, the maximum distance to the camera is 6 meters). To detect a face in real time, the number of tested hypothesis is reduced by motion and color analysis. 478 R. Feraud and o. Bernier Table 2:Results on the CMU test set A GM: the model trained without counter-examples, CGM1: face detector, CGM2: turned face detector, CGM3: general face detector. SWN: shared weight network. (*) Considering that our goal is to detect human faces, non-human faces and rough face drawings have not been taken into account. Model Detection rate GM 84% CGMI 77 % CGM2 CGM3 [Rowley, 1995] (one SWN) Ensemble (CGM1,CGM2,CGM3) Conditional ensemble (CGM1,CGM2,gate) [Rowley, 1995] (three SWNs) Conditional mixture (CGM1,CGM2,gate) 127/164 85 % 139/164 85 % 139/164 84 % 142/169* 74 % 121/164 82 % 134/164 85 % 144/169* 87 % 142/164 +/- 5 % +/- 5 % +/- 5 % +/- 5 % +/- 5 % +/- 5 % +/- 5 % +/- 5 % +/- 5 % False alarms rate 1/1,000 5.43/1,000,000 47/33,700,000 6.3/1,000,000 212/33,700,000 1.36/1,000,000 46/33,700,000 8.13/1,000,000 179/22,000,000 0.71/1,000,000 24/33,700,000 0.77/1,000,000 26/33,700,000 2.13/1,000,000 47/22,000,000 1.15/1,000,000 39/33,700,000 +/- 0.1/1,000,00( +/- 0.38/1,000,00 + /- 0.37/1,000,00 +/- 0.41/1,000,00 +/- 0.4/1,000,00( +/- 0.43/1,000,00 +/- 0.38/1,000,00 +/- 0.42/1,000,00 +/- 0.35/1,000,001 (**) usenix face database, sussex face database and CMU test sets can be retrieved at www.cs.rug.nl/peterkr/FACE/face.html. References [Collobert, M. et al., 1996] Collobert, M., Feraud, R, Le Tourneur, G., Bernier, 0., Viallet, J.E, Mahieux, Y., and Collobert, D. (1996). Listen: a system for locating and tracking individual speaker. In Second International Conference On Automatic Face and Gesture Recognition. [Feraud, R et al., 1997] Feraud, R, Bernier, 0., and Collobert, D. (1997). A constrained generative model applied to face detection. Neural Processing Letters. [Geman, S. et al., 1992] Geman, S., Bienenstock, E., and Doursat, R (1992). Neural networks and the bias-variance dilemma. Neural Computation, 4:1-58. [Jacobs, R A. et al., 1991] Jacobs, R. A., Jordan, M. I., Nowlan, S. J., and Hinton, G. E. (1991). Adaptative mixtures of local experts. Neural Computation, 3:79-87. [Kramer, 1991] Kramer, M. (1991). Nonlinear principal component analysis using autoassociative neural networks. AIChE Journal, 37:233-243. [Raviv, Y. and Intrator, N., 1996] Raviv, Y. and Intrator, N. (1996). Bootstrapping with noise: An effective regularization technique. Connection Science, 8:355-372. [Rowley, H. et al., 1995] Rowley, H., Baluja, S., and Kanade, T. (1995). Human face detection in visual scenes. In Neural Information Processing Systems 8. [Sung, K. and Poggio, T ., 1994] Sung, K. and Poggio, T. (1994). Example-based learning for view-based human face detection. Technical report, M.I.T. [Vapnik, 1995] Vapnik, V. (1995). The Nature of Statistical Learning Theory. SpringerVerlag New York Heidelberg Berlin.
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Intrusion Detection with Neural Networks Jake Ryan* Department of Computer Sciences The University of Texas at Austin Austin, TX 78712 raven@cs.utexas.edu Meng-Jang Lin Department of Electrical and Computer Engineering The University of Texas at Austin Austin, TX 78712 mj@orac.ece .utexas.edu Risto Miikkulainen Department of Computer Sciences The University of Texas at Austin Austin, TX 78712 risto@cs.utexas.edu Abstract With the rapid expansion of computer networks during the past few years, security has become a crucial issue for modern computer systems. A good way to detect illegitimate use is through monitoring unusual user activity. Methods of intrusion detection based on hand-coded rule sets or predicting commands on-line are laborous to build or not very reliable. This paper proposes a new way of applying neural networks to detect intrusions. We believe that a user leaves a 'print' when using the system; a neural network can be used to learn this print and identify each user much like detectives use thumbprints to place people at crime scenes. If a user's behavior does not match hislher print, the system administrator can be alerted of a possible security breech. A backpropagation neural network called NNID (Neural Network Intrusion Detector) was trained in the identification task and tested experimentally on a system of 10 users. The system was 96% accurate in detecting unusual activity, with 7% false alarm rate. These results suggest that learning user profiles is an effective way for detecting intrusions. 1 INTRODUCTION Intrusion detection schemes can be classified into two categories: misuse and anomaly intrusion detection. Misuse refers to known attacks that exploit the known vulnerabilities of the system. Anomaly means unusual activity in general that could indicate an intrusion. ·Currently: MCI Communications Corp., 9001 N. IH 35, Austin, TX 78753; jake.ryan@mci.com. 944 1. Ryan, M-J. Lin and R. Miikkulainen If the observed activity of a user deviates from the expected behavior, an anomaly is said to occur. Misuse detection can be very powerful on those attacks that have been programmed in to the detection system. However, it is not possible to anticipate all the different attacks that could occur, and even the attempt is laborous. Some kind of anomaly detection is ultimately necessary. One problem with anomaly detection is that it is likely to raise many false alarms. Unusual but legitimate use may sometimes be considered anomalous. The challenge is to develop a model of legitimate behavior that would accept novel legitimate use. It is difficult to build such a model for the same reason that it is hard to build a comprehensive misuse detection system: it is not possible to anticipate aU possible variations of such behavior. The task can be made tractable in three ways: (1) Instead of general legitimate use, the behavior of individual users in a particular system can be modeled. The task of characterizing regular patterns in the behavior of an individual user is an easier task than trying to do it for aU users simultaneously. (2) The patterns of behavior can be learned for examples of legitimate use, instead of having to describe them by hand-COding possible behaviors. (3) Detecting an intrusion real-time, as the user is typing commands, is very difficult because the order of commands can vary a lot. In many cases it is enough to recognize that the distribution of commands over the entire login session, or even the entire day, differs from the usual. The system presented in this paper, NNID (Neural Network Intrusion Detector), is based on these three ideas. NNID is a backpropagation neural network trained to identify users based on what commands they use during a day. The system administrator runs NNID at the end of each day to see if the users' sessions match their normal pattern. If not, an investigation can be launched. The NNID model is implemented in a UNIX environment and consists of keeping logs of the commands executed, forming command histograms for each user, and learning the users' profiles from these histograms. NNID provides an elegant solution to off-line monitoring utilizing these user profiles. In a system of 10 users, NNID was 96% accurate in detecting anomalous behavior (i.e. random usage patterns), with a false alarm rate of 7%. These results show that a learning offline monitoring system such as NNID can achieve better performance than systems that attempt to detect anomalies on-line in the command sequences, and with computationally much less effort. The rest of the paper outlines other approaches to intrusion detection and motivates the NNID approach in more detail (sections 2 and 3), presents the implementation and an evaluation on a real-world computer system (sections 4 and 5), and outlines some open issues and avenues for future work (section 6). 2 INTRUSION DETECTION SYSTEMS Many misuse and anomaly intrusion detection systems (lDSs) are based on the general model proposed by Denning (1987). This model is independent of the platform, system vulnerability, and type of intrusion. It maintains a set of historical profiles for users, matches an audit record with the appropriate profile, updates the profile whenever necessary, and reports any anomalies detected. Another component, a rule set, is used for detecting misuse. Actual systems implement the general model with different techniques (see Frank 1994; Mukherjee et al. 1994, for an overview). Often statistical methods are used to measure how anomalous the behavior is, that is, how different e.g. the commands used are from normal behavior. Such approaches require that the distribution of subjects' behavior is known. The behavior can be represented as a rule-based model (Garvey and Lunt 1991), in terms of predictive pattern generation (Teng et al. 1990), or using state transition analysis (Porras Intrusion Detection with Neural Networks 945 et al. 1995). Pattern matching techniques are then used to detennine whether the sequence of events is part of normal behavior, constitutes an anomaly, or fits the description of a known attack. IDSs also differ in whether they are on-line or off-line. Off-line IDSs are run periodically and they detect intrusions after-the-fact based on system logs. On-line systems are designed to detect intrusions while they are happening, thereby allowing for quicker intervention. On-line IDSs are computationally very expensive because they require continuous monitoring. Decisions need to be made quickly with less data and therefore they are not as reliable. Several IDSs that employ neural networks for on-line intrusion detection have been proposed (Debar et al. 1992; Fox et al. 1990). These systems learn to predict the next command based on a sequence of previous commands by a specific user. Through a shifting window, the network receives the w most recent commands as its input. The network is recurrent, that is, part of the output is fed back as the input for the next step; thus, the network is constantly observing the new trend and "forgets" old behavior over time. The size of the window is an important parameter: If w is too small, there will be many false positives; if it is too big, the network may not generalize well to novel sequences. The most recent of such systems (Debar et al. 1992) can predict the next command correctly around 80% of the time, and accept a command as predictable (among the three most likely next commands) 90% of the time. One problem with the on-line approach is that most of the effort goes into predicting the order of commands. In many cases, the order does not matter much, but the distribution of commands that are used is revealing. A possibly effective approach could therefore be to collect statistics about the users' command usage over a period of time, such as a day, and try to recognize the distribution of commands as legitimate or anomalous off-line. This is the idea behind the NNID system. 3 THE NNID SYSTEM The NNID anomaly intrusion detection system is based on identifying a legitimate user based on the distribution of commands she or he executes. This is justifiable because different users tend to exhibit different behavior, depending on their needs of the system. Some use the system to send and receive e-mail only, and do not require services such as programming and compilation. Some engage in all kinds of activities including editing, programming, e-mail, Web browsing, and so on. However, even two users that do the same thing may not use the same application program. For example, some may prefer the "vi" editor to "emacs", favor "pine" over "elm" as their mail utility program, or use "gcc" more often than "cc" to compile C programs. Also, the frequency with which a command is used varies from user to user. The set of commands used and their frequency, therefore, constitutes a 'print' of the user, reflecting the task performed and the choice of application programs, and it should be possible to identify the user based on this information. It should be noted that this approach works even if some users have aliases set up as shorthands for long commands they use frequently, because the audit log records the actual commands executed by the system. Users' privacy is not violated, since the arguments to a command do not need to be recorded. That is, we may know that a user sends e-mail five times a day, but we do not need to know to whom the mail is addressed. Building NNID for a particular computer system consists of the following three phases: 1. Collecting training data: Obtain the audit logs for each user for a period of several days. For each day and user, form a vector that represents how often the user executed each command. 946 1 Ryan, M-J. Un and R. Miikkulainen as awk be 61btex calendar cat chmOd comsat cp cpp cut cvs date df diff du dvips egrep elm emacs expr fgrep filter find finger fmt from ftp gcc gdb ghostview gmake grep gs gzip hostname id ifConfig Ispell fast Id fess look Ipq Ipr Iprm Is machine mail make man mesg metamail rillCdir more movemail mpage mt mv netscape netstat nm objdump perl pgp ping ps pwd rcp resize rm rsh sed sendmail sh sort strip stty tail tar tcsh tee test tgif top tput tr tty uname vacation vi virtex w wc whereis xbiff++ xca1c xdvi xhost xterm Table 1: The 100 commands used to describe user behavior. The number of times the user executed each of these commands during the day was recorded, mapped into a nonlinear scale of 11 intervals, and concatenated into a l00-dimensional input vector, representing the usage pattern for that user for that day. 2. Training: Train the neural network to identify the user based on these command distribution vectors. 3. Perfonnance: Let the network identify the user for each new command distribution vector. If the network's suggestion is different from the actual user, or if the network does not have a clear suggestion, signal an anomaly. The particular implementation of NNID and the environment where it was tested is described in the next section. 4 EXPERIMENTS The NNID system was built and tested on a machine that serves a particular research group at the Department of Electrical and Computer Engineering at the University of Texas at Austin. This machine has 10 total users; some are regular users, with several other users logging in intennittently. This platfonn was chosen for three reasons: 1. The operating system (NetBSD) provides audit trail logging for accounting purposes and this option had been enabled on this system. 2. The number of users and the total number of commands executed per day are on an order of magnitude that is manageable. Thus, the feasibility of the approach could be tested with real-world data without getting into scalability issues. 3. The system is relatively unknown to outsiders and the users are all known to us, so that it is likely that the data collected on it consists of nonnal user behavior (free of intrusions). Data was collected on this system for 12 days, resulting in 89 user-days. Instead of trying to optimize the selection of features (commands) for the input, we decided to simply use a set of 100 most common commands in the logs (listed in Table 1), and let the network figure out what infonnation was important and what superfluous. Intelligent selection of features might improve the results some but the current approach is easy to implement and proves the point. In order to introduce more overlap between input vectors, and therefore better generalization, the number of times a command was used was divided into intervals. There were 11 intervals, non-linearly spaced, so that the representation is more accurate at lower frequencies where it is most important. The first interval meant the command was never used; the second that it was used once or twice, and so on until the last interval where the command was used more than 500 times. The intervals were represented by values from 0.0 to 1.0 in 0.1 increments. These values, one for each command, were then concatenated into a 100-dimensional command distribution vector (also called user vector below) to be used as input to the neural network. Intrusion Detection with Neural Networks 947 The standard three-layer backpropagation architecture was chosen for the neural network. The idea was to get results on the most standard and general architecture so that the feasibility of the approach could be demonstrated and the results would be easily replicable. More sophisticated architectures could be used and they would probably lead to slightly better results. The input layer consisted of 100 units, representing the user vector; the hidden layer had 30 units and the output layer 10 units, one for each user. The network was implemented in the PlaNet Neural Network simulator (Miyata 1991). 5 RESULTS To avoid overtraining, several training sessions were run prior to the actual experiments to see how many training cycles would give the highest performance. The network was trained on 8 randomly chosen days of data (65 user vectors), and its performance was tested on the remaining 4 days (24 vectors) after epochs 30, 50, 100,200, and 300, of which 100 gave the best performance. Four splits of the data into training and testing sets were created by randomly picking 8 days for training. The reSUlting four networks were tested in two tasks: 1. Identifying the user vectors of the remaining 4 days. If the activation of the output unit representing the correct user was higher than those of all other units, and also higher than 0.5, the identification was counted as correct. Otherwise, a false positive was counted. 2. Identifying 100 randomly-generated user vectors. If all output units had an activation less than 0.5, the network was taken to correctly identify the vector as an anomaly (i.e. not any of the known users in the system). Otherwise, the most highly active output unit identifies the network's suggestion. Since all intrusions occur under one of the 10 user accounts, there is a 111 0 chance that the suggestion would accidentally match the compromised user account and the intrusion would not be detected. Therefore, 1/10 of all such cases were counted as false negatives. The second test is a suggestive measure of the accuracy of the system. It is not possible to come up with vectors that would represent a good sampling of actual intrusions; the idea here was to generate vectors where the values for each command were randomly drawn from the distribution of values for that command in the entire data set. In other words, the random test vectors had the same first-order statistics as the legitimate user vectors, but had no higher-order correlations. Therefore they constitute a neutral but realistic sample of unusual behavior. All four splits led to similar results. On average, the networks rejected 63% of the random user vectors, leading to an anomaly detection rate of 96%. They correctly identified the legitimate user vectors 93% of the time, giving a false alarm rate of 7%. Figure 1 shows the output of the network for one of the splits. Out of 24 legitimate user vectors, the network identified 22. Most of the time the correct output unit is very highly activated, indicating high certainty of identification. However, the activation of the highest unit was below 0.5 for two of the inputs, resulting in a false alarm. Interestingly, in all false alarms in all splits, the falsely-accused user was always the same. A closer look at the data set revealed that there were only 3 days of data on this user. He used the system very infrequently, and the network could not learn a proper profile for him. While it would be easy to fix this problem by collecting more data in this case, we believe this is a problem that would be difficult to rule out in general. No matter how much data one collects, there may still not be enough for some extremely infrequent user. Therefore, we believe the results obtained in this rather small data set give a realistic picture of the performance of the NNID system. 948 1. Ryan, M-l. lin and R. Miikkulainen xr~ct , • •• .. ' . •• . O~t.,ut;': .> 4 • • tI :l ~t· 4 , • tI " ~t<'4'" t " ~~,.,,, · " • . ... . • • t ) 2 3 • 5 6 7 B ~ 0 L Z ~ • 5 6 7 B 9 0 1 2 3 • 5 6 • e 9 1 2 ~ , 5 6 7 8 9 O.lt('t-rt D.itrut 00. ....... Oo.t"", ... • . • • • D.~t '" •• . ~ . ~~- . . ~ . ~ ~ ~< , • ~ 0 < , Out.~, ~ " ~ 0 .. •• . .. • • • • D.np;;.. '" 6 . . , ~ . n..,.!. - o • ~ ~ ~t . g • . ~ Out.~, ~"O' ti ... • • • D~~L "" ~ . . " u " ~"":;l • -. ~ " u " o::.+~" Eo .. ~ ~ . " Ou<.~. L , , ~ ~ · " •• ... . .. ~ D~t~2~'56 e ~~t _ • , 6 " 9 ~Zl.'5 t " o::...~"'.'~ • s Figure 1: User identification with the NNID Network. The output layer of NNID is shown for each of the 24 test vectors in one of the 4 splits tested. The output units are lined up from left to right, and their activations are represented by the size of the squares. In this split there were two false alarms: one is displayed in the top right with activation 0.01, and one in the second row from the bottom, second column from the left with 0.35. All the other test vectors are identified correctly with activation higher than 0.5. 6 DISCUSSION AND FUTURE WORK An important question is, how well does the performance of NNID scale with the number of users? Although there are many computer systems that have no more than a dozen users, most intrusions occur in larger systems with hundreds of users. With more users, the network would have to make finer distinctions, and it would be difficult to maintain the same low level of false alarms. However, the rate of detecting anomalies may not change much, as long as the network can learn the user patterns well. Any activity that differs from the user's normal behavior would still be detected as an anomaly. Training the network to represent many more users may take longer and require a larger network, but it should be possible because the user profiles share a lot of common structure, and neural networks in general are good at learning such data. Optimizing the set of commands included in the user vector, and the size of the value intervals, might also have a large impact on performance. It would be interesting to determine the curve of performance Intrusion Detection with Neural Networks 949 versus the number of users, and also see how the size of the input vector and the granularity of the value intervals affect that curve. This is the most important direction of future work. Another important issue is, how much does a user's behavior change over time? If behavior changes dramatically, NNID must be recalibrated often or the number of false positives would increase. Fortunately such retraining is easy to do. Since NNID parses daily activity of each user into a user-vector, the user profile can be updated daily. NNID could then be retrained periodically. In the current system it takes only about 90 seconds and would not be a great burden on the system. 7 CONCLUSION Experimental evaluation on real-world data shows that NNID can learn to identify users simply by what commands they use and how often, and such an identification can be used to detect intrusions in a network computer system. The order of commands does not need to be taken into account. NNID is easy to train and inexpensive to run because it operates off-line on daily logs. As long as real-time detection is not required, NNID constitutes a promising, practical approach to anomaly intrusion detection. Acknowledgements Special thanks to Mike Dahlin and Tom Ziaja for feedback on an earlier version of this paper, and to Jim Bednar for help with the PlaNet simulator. This research was supported in part by DOD-ARPA contract F30602-96-1-0313, NSF grant IRI-9504317, and the Texas Higher Education Coordinating board grant ARP-444. References Debar, H., Becker, M., and Siboni, D. (1992). A neural network component for an intrusion detection system. In Proceedings of the 1992 IEEE Computer Society Symposium on Research in Computer Security and Privacy, 240-250. Denning, D. E. (1987). An intrusion detection model. IEEE Transactions on Software Engineering, SE-13:222-232. Fox, K. L., Henning, R. R., Reed, J. H., and Simonian, R. (1990). A neural network approach towards intrusion detection. In Proceedings of the 13th National Computer Security Conference, 125-134. Frank, J. (1994). Artificial intelligence and intrusion detection: Current and future directions. In Proceedings of the National 17th Computer Security Conference. Garvey, T. D., and Lunt, T. F. (1991). Model-based intrusion detection. In Proceedings of the 14th National Computer Security Conference. Miyata, Y. (1991). A User's Guide to PlaNet Version 5.6 -A Toolfor Constructing, Running, and Looking in to a PDP Network. Computer Science Department, University of Colorado, Boulder, Boulder, CO. Mukherjee, B., Heberlein, L. T., and Levitt, K. N. (1994). Network intrusion detection. IEEE Network, 26-41. Porras, P. A., IIgun, K., and Kemmerer, R. A. (1995). State transition analysis: A rulebased intrusion detection approach. IEEE Transactions on Software Engineering, SE21 : 181-199. Teng, H. S., Chen, K., and Lu, S. C. (1990). Adaptive real-time anomaly detection using inductively generated sequential patterns. In Proceedings of the 1990 IEEE Symposium on Research in Computer Security and Privacy, 278-284.
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Stacked Density Estimation Padhraic Smyth * Information and Computer Science University of California, Irvine CA 92697-3425 smythCics.uci.edu David Wolpert NASA Ames Research Center Caelum Research MS 269-2, Mountain View, CA 94035 dhwCptolemy.arc.nasa.gov Abstract In this paper, the technique of stacking, previously only used for supervised learning, is applied to unsupervised learning. Specifically, it is used for non-parametric multivariate density estimation, to combine finite mixture model and kernel density estimators. Experimental results on both simulated data and real world data sets clearly demonstrate that stacked density estimation outperforms other strategies such as choosing the single best model based on cross-validation, combining with uniform weights, and even the single best model chosen by "cheating" by looking at the data used for independent testing. 1 Introduction Multivariate probability density estimation is a fundamental problem in exploratory data analysis, statistical pattern recognition and machine learning. One frequently estimates density functions for which there is little prior knowledge on the shape of the density and for which one wants a flexible and robust estimator (allowing multimodality if it exists). In this context, the methods of choice tend to be finite mixture models and kernel density estimation methods. For mixture modeling, mixtures of Gaussian components are frequently assumed and model choice reduces to the problem of choosing the number k of Gaussian components in the model (Titterington, Smith and Makov, 1986) . For kernel density estimation, kernel shapes are typically chosen from a selection of simple unimodal densities such as Gaussian, triangular, or Cauchy densities, and kernel bandwidths are selected in a data-driven manner (Silverman 1986; Scott 1994). As argued by Draper (1996), model uncertainty can contribute significantly to pre• Also with the Jet Propulsion Laboratory 525-3660, California Institute of Technology, Pasadena, CA 91109 Stacked Density Estimation 669 dictive error in estimation. While usually considered in the context of supervised learning, model uncertainty is also important in unsupervised learning applications such as density estimation. Even when the model class under consideration contains the true density, if we are only given a finite data set, then there is always a chance of selecting the wrong model. Moreover, even if the correct model is selected, there will typically be estimation error in the parameters of that model. These difficulties are summarized by wri ting P(f I D) = L J dOMP(OM I D,M) x P(M I D) x fM,9M' (1) M where f is a density, D is the data set, M is a model, and OM is a set of values for the parameters for model M. The posterior probability P( M I D) reflects model uncertainty, and the posterior P(OM I D, M) reflects uncertainty in setting the parameters even once one knows the model. Note that if one is privy to P(M, OM), then Bayes' theorem allows us to write out both of our posteriors explicitly, so that we explicitly have P(f I D) (and therefore the Bayes-optimal density) given by a weighted average of the fM,9M" (See also Escobar and West (1995)). However even when we know P(M, OM), calculating the combining weights can be difficult. Thus, various approximations and sampling techniques are often used, a process that necessarily introduces extra error (Chickering and Heckerman 1997). More generally, consider the case of mis-specified models where the model class does not include the true model, so our presumption for P(M, OM) is erroneous. In this case often one should again average. Thus, a natural approach to improving density estimators is to consider empiricallydriven combinations of multiple density models. There are several ways to do this, especially if one exploits previous combining work in supervised learning. For example, Ormontreit and Tresp (1996) have shown that "bagging" (uniformly weighting different parametrizations of the same model trained on different bootstrap samples) , originally introduced for supervised learning (Breiman 1996a), can improve accuracy for mixtures of Gaussians with a fixed number of components. Another supervised learning technique for combining different types of models is "stacking" (Wolpert 1992), which has been found to be very effective for both regression and classification (e .g., Breiman (1996b)) . This paper applies stacking to density estimation, in particular to combinations involving kernel density estimators together with finite mixture model estimators. 2 Stacked Density Estimation 2.1 Background on Density Estimation with Mixtures and Kernels Consider a set of d real-valued random variables X = {Xl, . . . , xd} Upper case symbols denote variable name.s (such as Xi) and lower-case symbols a particular value of a variable (such as xJ). ~ is a realization of the vector variable X. J(~) is shorthand for f(X = ~) and represents the joint probability distribution of X. D = {~1 ' .. . ' ~N} is a training data set where each sample ~i' 1 :::; i :::; N is an independently drawn sample from the underlying density function J(~) . A commonly used model for density estimation is the finite mixture model with k components, defined as: k fk(~J = L aigi(~), (2) i=l 670 P. Smyth and D. Wolpert where I:~=1 Ctj = 1. The component gj's are usually relatively simple unimodal densities such as Gaussians. Density estimation with mixtures involves finding the locations, shapes, and weights of the component densities from the data (using for example the Expectation-Maximization (EM) procedure). Kernel density estimation can be viewed as a special case of mixture modeling where a component is centered at each data point, given a weight of 1/ N, and a common covariance structure (kernel shape) is estimated from the data. The quality of a particular probabilistic model can be evaluated by an appropriate scoring rule on independent out-of-sample data, such as the test set log-likelihood (also referred to as the log-scoring rule in the Bayesian literature). Given a test data set Dte3t , the test log likelihood is defined as logf(Dte3tlfk(~)) = l: logfk(~i) (3) Dteof This quantity can play the role played by classification error in classification or squared error in regression. For example, cross-validated estimates of it can be used to find the best number of clusters to fit to a given data set (Smyth, 1996). 2.2 Background on Stacking Stacking can be used either to combine models or to improve a single model. In the former guise it proceeds as follows. First, subsamples of the training set are formed. Next the models are all trained on one subsample and resultant joint predictive behavior on another subs ample is observed, together with information concerning the optimal predictions on the elements in that other subsample. This is repeated for other pairs of subsamples of the training set. Then an additional ("stacked") model is trained to learn, from the subsample-based observations, the relationship between the observed joint predictive behavior of the models and the optimal predictions. Finally, this learned relationship is used in conjunction with the predictions of the individual models being combined (now trained on the entire data set) to determine the full system's predictions. 2.3 Applying Stacking to Density Estimation Consider a set of M different density models, fm(~), 1 ~ m ~ M. In this paper each of these models will be either a finite mixture with a fixed number of component densities or a kernel density estimate with a fixed kernel and a single fixed global bandwidth in each dimension. (In general though no such restrictions are needed.) The procedure for stacking the M density models is as follows: 1. Partition the training data set D v times, exactly as in v-fold cross validation (we use v = 10 throughout this paper), and for each fold: (a) Fit each of the M models to the training portion ofthe partition of D . (b) Evaluate the likelihood of each data point in the test partition of D, for each of the M fitted models. 2. After doing this one has M density estimates for each of N data points, and therefore a matrix of size N x M, where each entry is fm(~) , the out-of-sample likelihood of the mth model on the ith data point. 3. Use that matrix to estimate the combination coefficients {Pl, ... , PM} that maximize the log-likelihood at the points ~i of a stacked density model of Stacked Density Estimation 671 the form: M fstacked (.~) = I': f3m f m (~J. m=l Since this is itself a mixture model, but where the fm(~i) are fixed, the EM algorithm can be used to (easily) estimate the f3m. 4. Finally, re-estimate the parameters of each of the m component density models using all of the training data D. The stacked density model is then the linear combination of those density models, with combining coefficients given by the f3m. 3 Experimental Results In our stacking experiments M = 6: three triangular kernels with bandwidths of 0.1,0.4, and 1.5 of the standard deviation (of the full data set) in each dimension, and three Gaussian mixture models with k = 2,4, and 8 components. This set of models was chosen to provide a reasonably diverse representational basis for stacking. We follow roughly the same experimental procedure as described in Breiman (1996b) for stacked regression: • Each data set is randomly split into training and test partitions 50 times, where the test partition is chosen to be large enough to provide reasonable estimates of out-of-sample log-likelihood. • The following techniques are run on each training partition: 1. Stacking: The stacked combination of the six constituent models. 2. Cross-Validation: The single best model as indicated by the maximum likelihood score of the M = 6 single models in the N x M cross-validated table of likelihood scores. 3. Uniform Weighting: A uniform average of the six models. 4. "Cheating:" The best single model, i.e., the model having the largest likelihood on the test data partition, 5. Truth: The true model structure, if the true model is one of the six generating the data (only valid for simulated data). • The log-likelihoods of the models resulting from these techniques are calculated on the test data partition. The log-likelihood of a single Gaussian model (parameters determined on the training data) is subtracted from each model's log-likelihood to provide some normalization of scale. 3.1 Results on Real Data Sets Four real data sets were chosen for experimental evaluation. The diabetes data set consists of 145 data points used in Gaussian clustering studies by Banfield and Raftery (1991) and others. Fisher's iris data set is a classic data set in 4 dimensions with 150 data points. Both of these data sets are thought to consist roughly of 3 clusters which can be reasonably approximated by 3 Gaussians. The Barney and Peterson vowel data (2 dimensions, 639 data points) contains 10 distinct vowel sounds and so is highly multi-modal. The star-galaxy data (7 dimensions, 499 data points) contains non-Gaussian looking structure in various 2d projections. Table 1 summarizes the results. In all cases stacking had the highest average loglikelihood, even out-performing "cheating" (the single best model chosen from the test data). (Breiman (1996b) also found for regression that stacking outperformed 672 P. Smyth and D. Wolperl Table 1: Relative performance of stacking multiple mixture models, for various data sets, measured (relative to the performance of a single Gaussian model) by mean log-likelihood on test data partitions. The maximum for each data set is underlined. II Data Set I Gaussian I Cross-Validation I "Cheating" I Uniform I Stacking II Diabetes -352.9 27.8 30.4 29.2 31.8 Fisher's Iris -52.6 18.3 21.2 18.3 22.5 Vowel 128.9 53.5 54.6 40.2 55.8 Star-Galaxy -257.0 678.9 721.6 789.1 888.9 Table 2: Average across 20 runs of the stacked weights found for each constituent model. The columns with h = .. . are for the triangular kernels and the columns with k = . .. are for the Gaussian mixtures. II Data Set I h=O.1 I h=O.4 I h=1.5 I k = 2 I k = 4 I k = 8 1/ DIabetes 0.01 0.09 0.03 0.13 0.41 0.32 Fisher's Iris 0.02 0.16 0.00 0.26 0.40 0.16 Vowel 0.00 0.25 0.00 0.02 0.20 0.53 Star-Galaxy 0.00 0.04 0.03 0.03 0.27 0.62 the "cheating" method.) We considered two null hypotheses: stacking has the same predictive accuracy as cross-validation, and it has the same accuracy as uniform weighting. Each hypothesis can be rejected with a chance ofless than 0.01% of being incorrect, according to the Wilcoxon signed-rank test i.e., the observed differences in performance are extremely strong even given the fact that this particular test is not strictly applicable in this situation. On the vowel data set uniform weighting performs much worse than the other methods: it is closer in performance to stacking on the other 3 data sets. On three of the data sets, using cross-validation to select a single model is the worst method. "Cheating" is second-best to stacking except on the star-galaxy data, where it is worse than uniform weighting also: this may be because the star-galaxy data probably induces the greatest degree of mis-specification relative to this 6-model class (based on visual inspection). Table 2 shows the averages of the stacked weight vectors for each data set. The mixture components generally got higher weight than the triangular kernels. The vowel and star-galaxy data sets have more structure than can be represented by any of the component models and this is reflected in the fact that for each most weight is placed on the most complex mixture model with k = 8. 3.2 Results on Simulated Data with no Model Mis-Specification We simulated data from a 2-dimensional 4-Gaussian mixture model with a reasonable degree of overlap (this is the data set used in Ripley (1994) with the class labels removed) and compared the same models and combining/selection schemes as before, except that "truth" is also included, i.e., the scheme which always selects the true model structure with k = 4 Gaussians. For each training sample size, 20 different training data sets were simulated, and the mean likelihood on an independent test data set of size 1000 was reported. Stacked Density Estimation 673 250 Slacking l..... . -. w Ul .+ .. . ' .' 0 I0 . ~ . , fl]200 I~ ~ ----~ lJnIfonn ~lSO .~ ~ , <' ..J ~ ~ ~~-: . • i ,-8 I ..J 100 I ~ I • I ". ~ Ch.-ung )Y so / / TrueK + I I ~ I 0 20 .0 60 80 100 120 1.0 160 180 200 TRAINING SAMPLE SIZE Figure 1: Plot of mean log-likelihood (relative to a single Gaussian model) for various density estimation schemes on data simulated from a 4-component Gaussian mixture. Note that here we are assured of having the true model in the set of models being considered, something that is presumably never exactly the case in the real world (and presumably was not the case for the experiments recounted in Table 1.) Nonetheless, as indicated in (Figure 1), stacking performed about the same as the "cheating" method and significantly outperformed the other methods, including "truth." (Results where some of the methods had log-likelihoods lower than the single Gaussian are not shown for clarity). The fact that "truth" performed poorly on the smaller sample sizes is due to the fact that with smaller sample sizes it was often better to fit a simpler model with reliable parameter estimates (which is what "cheating" typically would do) than a more complex model which may overfit (even when it is the true model structure). As the sample size increases, both "truth" and cross-validation approach the performance of "cheating" and stacking: uniform weighting is universally poorer as one would expect when the true model is within the model class. The stacked weights at the different sample sizes (not shown) start out with significant weight on the triangular kernel model and gradually shift to the k = 2 Gaussian mixture model and finally to the (true) k = 4 Gaussian model as sample size grows. Thus, stacking is seen to incur no penalty when the true model is within the model class being fit. In fact the opposite is true; for small sample sizes stacking outperforms other density estimation techniques which place full weight on a single (but poorly parametrized) model. 4 Discussion and Conclusions Selecting a global bandwidth for kernel density estimation is still a topic of debate among statisticians. Stacking allows the possibility of side-stepping the issue of a single bandwidth by combining kernels with different bandwidths and different kernel shapes. A stacked combination of such kernel estimators is equivalent to using 674 P. Smyth and D. Wolpert a single composite kernel that is a convex combination of the underlying kernels. For example, kernel estimators based on finite support kernels can be regularized in a data-driven manner by combining them with infinite support kernels. The key point is that the shape and width of the resulting "effective" kernel i8 driven by the data. It is also worth noting that by combining Gaussian mixture models with different k values one gets a hierarchical "mixture of mixtures" model. This hierarchical model can provide a natural multi-scale representation of the data, which is clearly similar in spirit to wavelet density estimators, although the functional forms and estimation methodologies for each technique can be quite different. There is also a representational similarity to Jordan and Jacob's (1994) "mixture of experts" model where the weights are allowed to depend directly on the inputs. Exploiting that similarity, one direction for further work is to investigate adaptive weight parametrizations in the stacked density estimation context. Acknowledgements The work of P.S. was supported in part by NSF Grant IRI-9703120 and in part by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. References Banfield, J. D., and Raftery, A. E., 'Model-based Gaussian and non-Gaussian clustering,' Biometrics, 49, 803-821, 1993. Breiman, L., 'Bagging predictors,' Machine Learning, 26(2), 123-140, 1996a. Breiman, L., 'Stacked regressions,' Machine Learning, 24, 49-64, 1996b. Chickering, D. M., and Heckerman, D., 'Efficient approximations for the marginal likelihood of Bayesian networks with hidden variables,' Machine Learning, In press. Draper, D, 'Assessment and propagation of m0del uncertainty (with discussion),' Journal of the Royal Statistical Society B, 57, 45-97, 1995. Escobar, M. D., and West, M., 'Bayesian density estimation and inference with mixtures,' J. Am. Stat. Assoc., 90, 577-588, 1995. Jordan, M. 1. and Jacobs, R. A., 'Hierarchical mixtures of experts and the EM algorithm,' Neural Computation, 6, 181-214, 1994. Madigan, D., and Raftery, A. E., 'Model selection and accounting for model uncertainty in graphical models using Occam's window,' J. Am. Stat. Assoc., 89, 1535-1546, 1994. Ormeneit, D., and Tresp, V., 'Improved Gaussian mixture density estimates using Bayesian penalty terms and network averaging,' in Advances in Neural Information Processing 8, 542-548, MIT Press, 1996. Ripley, B. D. 1994. 'Neural networks and related methods for classification (with discussion),' J. Roy. Stat. Soc. B, 56,409-456. Smyth, P.,'Clustering using Monte-Carlo cross-validation,' in Proceedings of the Second International Conference on Knowledge Discovery and Data Mining, Menlo Park, CA: AAAI Press, pp.126-133, 1996. Titterington, D. M., A. F. M. Smith, U. E. Makov, Statistical Analysis of Finite Mixture Distributions, Chichester, UK: John Wiley and Sons, 1985 Wolpert, D. 1992. 'Stacked generalization,' Neural Networks, 5, 241-259,
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An Analog VLSI Neural Network for Phasebased Machine Vision KwokFaiHui Bertram E. Shi Department of Electrical and Electronic Engineering Fujitsu Microelectronics Pacific Asia Ltd. Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong Suite 1015-20, Tower 1 Grand Century Place 193 Prince Edward Road West Mongkok, Kowloon, Hong Kong. Abstract We describe the design, fabrication and test results of an analog CMOS VLSI neural network prototype chip intended for phase-based machine vision algorithms. The chip implements an image filtering operation similar to Gabor-filtering. Because a Gabor filter's output is complex valued, it can be used to define a phase at every pixel in an image. This phase can be used in robust algorithms for disparity estimation and binocular stereo vergence control in stereo vision and for image motion analysis. The chip reported here takes an input image and generates two outputs at every pixel corresponding to the real and imaginary parts of the output. 1 INTRODUCTION Gabor filters are used as preprocessing stages for different tasks in machine vision and image processing. Their use has been partially motivated by findings that two dimensional Gabor filters can be used to model receptive fields of orientation selective neurons in the visual cortex (Daugman, 1980) and three dimensional spatio-temporal Gabor filters can be used to model biological image motion analysis (Adelson, 1985). A Gabor filter has a complex valued impulse response which is a complex exponential modulated by a Gaussian function. In one dimension, x2 x2 g(x) = _1_e-202/OOxox = _1_e -202 (cos (00 x) + jsin (00 x» ./2itcr ./2itcr xo xo where OOxo and cr are real constants corresponding to the angular frequency of the complex exponential and the standard deviation of the Gaussian. An Analog VLSI Neural Network/or Phase-based Machine Vision 727 The phase of the complex valued filter output at a given pixel is related to the location of edges and other features in the input image near that pixel. Because translating the image input results in a phase shift in the Gabor output, several authors have developed "phasebased" approaches to disparity estimation (Westelius, 1995) and binocular vergence control (Theimer, 1994) in stereo vision and image motion analysis (Fleet, 1992). Barron et. al.'s comparison (Barron, 1992) of algorithms for optical flow estimation indicates that Fleet's algorithm is the most accurate among those tested. The remainder of this paper describes the design, fabrication and test results of a prototype analog VLSI continuous time neural network which implements a complex valued filter similar to the Gabor. 2 NETWORK AND CIRCmT ARCmTECTURE The prototype implements a Cellular Neural Network (CNN) architecture for Gabor-type image filtering (Shi, 1996). It consists of an array of neurons, called "cells," each corresponding to one pixel in the image to be processed. Each cell has two outputs v,(n) and vi(n) which evolve over time according to the equation rv,(n)l [coS(o -sinw l [v (n - l~ [2 +).2 0 1 [v (n)l [cosw sinw] [v (n + 1)1 [).2u(n)1 lvj(n)j = sinw:: cosw:: v~(n - l)j 0 2 + ).2J v~n)j + -sin::o cosW:: v~n + 1)j + 0 J where A. > 0 and 0)0 E [0,21t] are real constants and u(n) is the input image. The feedback from neighbouring cells' outputs enables information to be spread globally throughout the array. This network has a unique equilibrium point where the outputs correspond to the real and imaginary parts of the result of filtering the image with a complex valued discrete space convolution kernel which can be approximated by g(n) = ~e-A.lnli!ll .. o(n). 2 The Gaussian function of the Gabor filter has been replaced by (A./2) e-A.1xt . The larger A. is, the narrower the impulse response and the larger the bandwidth. Figure 1 shows the real (a) and imaginary (b) parts of g(n) for A. = 0.3 and O)xo = 0.93. The dotted lines show the function which modulates the complex exponential. -4:10 _1~ _10 -I (a) '. '. , , , . , . , . (b) Figure 1: The Real and Imaginary Parts of the Impulse Response. In the circuit implementation of this CNN, each output corresponds to the voltage across a capacitor. We selected the circuit architecture in Figure 2 because it was the least sensitive to the effects of random parameter variations among those we considered (Hui, 1996). In the figure, resistor labels denote conductances and trapezoidal blocks represent transconductance amplifiers labelled by their gains. 728 B. E. Shi and K. F. Hui Figure 2: Circuit Implementation of One Neuron. The circuit implementation also gives good intuitive understanding of the CNN's operation. Assume that the input image is an impulse at pixel n. In the circuit, this corresponds to setting the current source A.2u(n) to 1..2 amps and setting the remaining current sources to zero. If the gains and conductances were chosen so that').. = 0.3 and w.w = 0.93. then the steady state voltages across the lower capacitors would follow the spatial distribution shown in Figure l(a) where the center peak occurs at cell n and the voltages across the upper capacitors would follow the distribution shown in Figure l(b). To see how this would arise in the circuit, consider the current supplied by the source ')..2u(n) . Part of the current flows through the conductance Go pushing the voltage v,(n) positive. As this voltage increases, the two resistors with conductance G1 cause a smoothing effect which pulls the voltages v,(n-l) and v,(n + 1) up towards v,(n). Current also flows through the diagonal resistor with conductance G2 pulling vj(n + 1) positive as well. At the same time, the transconductance amplifier with input v,(n) draws current from node vj(n - 1) pushing vj(n - 1) negative. The larger G2 , the more the voltages at nodes vj(n - 1) and vj(n + 1) are pushed negative and positive. On the other hand, the larger G1 ' the greater the smoothing between nodes. Thus, the larger the ratio sinwxo --- = tanwxo ' coswxo the higher the spatial frequency wxo at which the impulse response oscillates. 3 DESIGN OF CMOS BUILDING BLOCKS This section describes CMOS transistor circuits which implement the transconductance amplifiers and resistors in Figure 2. It is not necessary to implement the capacitors explicitly. Since the equilibrium point of the CNN is unique, the parasitic capacitances of the circuit are sufficient to ensure the circuit operates correctly. 3.1 ·TRANSCONDUCTANCE AMPLIFIER The transconductance amplifiers can be implemented using the circuit shown in Figure 3(a). For Yin == V GND' the output current is approximately lout = Jf3/ss Yin where f3 n = Iln Cox ( W / L) and (W / L) is the widthllength ratio of the differential pair. The transistors in the current mirrors are assumed to be matched. Using cascoded current mirAn Analog VLSI Neural Networkfor Phase-based Machine l-lsion 729 rors decreases static errors such as offsets caused by the finite output impedance of the MOS transistors in saturation. (a) (b) Figure 3: The CMOS Circuits Implementing OTAs and Resistors 3.2 RESISTORS Since the convolution kernels implemented are modulated sine and cosine functions, the nodal voltages ve(n) and v o(n) can be both positive and negative with respect to the ground potential. The resistors in the circuit must be floating and exhibit good linearity and invariance to common mode offsets for voltages around the ground potential. Many MOS resistor circuits require bias circuitry implemented at every resistor. Since for image processing tasks, we are interested in maximizing the number of pixels processed, eliminating the need for bias circuitry at each cell will decrease its area and in turn increase the number of cells implementable within a given area. Figure 3(b) shows a resistor circuit which satisfies the requirements above. This circuit is essentially a CMOS transmission gate with adjustable gate voltages. The global bias circuit which generates the gate voltages in the CMOS resistor is shown on the left. The gate bias voltages V GI and V G2 are distributed to each resistor designed with the same value. Both transistors Mn and Mp operate in the conduction region where (Enz, 1995) IDn = nn~n(Vpn- VD;VS)(VD_VS) andIDp = -np~p(vpp- VD;VS)(VD_VS) and V Pn and V P are nonlinear functions of the gate and threshold voltages. The sizing of the NMOS and PMOS transistors can be chosen to decrease the effect of the nonlinearity 730 B. E. Shi and K. F. Hui due to the (V D + Vs) 12 tenns. The conductance of the resistors can be adjusted using [bias . 3.3 LIMITATIONS Due to the physical constraints of the circuit realizations, not all values of A. and ooxo can be realized. Because the conductance values are non-negative and the OTA gains are nonpositive both G1 and G2 must be non-negative. This implies that ooxo must lie between 0 and 1t/2. Because the conductance Go is non-negative, 1..2 ~ - 2 + 2cosooxo + sinooxo ' Figure 4 shows the range of center frequencies ooxo (nonnalized by 1t) and relative bandwidths (2A./ooxo) achievable by this realization. Not all bandwidths are achievable for ooxo ~ 2atanO.5 == 0.31t . I I 07 0.1 01 Figure 4: The filter parameters implementable by the circuit realization. 4 TEST RESULTS The circuit architecture and CMOS building blocks described above were fabricated using the Orbit 2Jlm n-well process available through MOSIS. In this prototype, a 13 cell one dimensional array was fabricated on a 2.2mm square die. The value of ooxo is fixed at 2 atan 0.5 == 0.927 by transistor sizing. This is the smallest spatial frequency for which all bandwidths can be obtained. In addition, Go = 1..2 for this value of ooxo . The width of the impulse response is adjustable by changing the externally supplied bias current shown in Figure 3(b) controlling Go . The transconductance amplifiers and resistors are designed to operate between ±300m V . The currents representing the input image are provided by transconductance amplifiers internal to the chip which are controlled by externally applied voltages. Outputs are read off the chip in analog fonn through two common read-out amplifiers: one for the real part of the impulse response and one for the imaginary part. The outputs of the cells are connected in tum to the inputs of the read-out amplifier through transmission gates controlled by a shift register. The chip requires ±4 V supplies and dissipates 35m W. To measure the impulse response of the filters, we applied 150m V to the input corresponding to the middle cell of the array and OV to the remaining inputs. The output voltages from one chip as a function of cell number are shown as solid lines in Figure 5(a, b). To correct for DC offsets, we also measured the output voltages when all of the inputs were grounded, as shown by the dashed lines in the figure. The DC offsets can be separated into two components: a constant offset common to all cells in the array and a small offset which varies from cell to cell. For the chip shown, the constant offset is approximately An Analog VISI Neural Networkfor Phase-based Machine VISion 731 (a) (b) (c) (d) Figure 5: DC Measurements from the Prototype l00mV and the small variations have a standard deviation of 2OmY. These results are consistent with the other chips. The constant offset is primarily due to the offset voltage in the read-out amplifier. The small variations from cell to cell are the result of both parameter variations from cell to cell and offsets in the transconductance amplifiers of each cell. By subtracting the DC zero-input offsets at each cell from the outputs, we can observe that the impulse response closely matches that predicted by the theory. The dotted lines in Figure 5(c, d) show the offset corrected outputs for the same chip as shown in Figure 5(a, b). The solid lines shows the theoretical output of the chip using parameters A. and O)xo chosen to minimize the mean squared error between the theory and the data. The chip was designed for A. = 0.210 and O)xo = 0.927 . The parameters for the best fit are A. = 0.175 and O)xo = 0.941 . The signal to noise ratio, as defined by the energy in the theoretical output divided by the energy in the error between theory and data, is 19.3dB. Similar measurements from two other chips gave SIgnal to noise ratios of 29.OdB (A. = 0.265, O)xo = 0.928) and 30.6dB (A. = 0.200, O)xo = 0.938). To measure the speed of the chips, we grounded all of the inputs except that of the middle cell to which we attached a function generator generating a square wave switching between ±200mV. The rise times (10% to 90%) at the output of the chip for each cell were measured and ranged between 340 and 528 nanoseconds. The settling times will not increase if the number of cells increases since the outputs are computed in parallel. The settling time is primarily determined by the width of the impUlse response. The wider the impulse response, the farther information must propagate through the array and the slower the settling time. 732 B. E Shi and K. F. Hui 5 CONCLUSION We have described the architecture, design and test results from an analog VLSI prototype of a neural network which filters images with convolution kernels similar to those of the Gabor filter. Our future work on chip design includes fabricating chips with larger numbers of cells, two dimensional arrays and chips with integrated photosensors which acquire and process images simultaneously. We are also investigating the use of these neural network chips in binocular vergence control of an active stereo vision system. Acknowledgements This work was supported by the Hong Kong Research Grants Council (RGC) under grant number HKUST675/95E. References E. H. Adelson, and J. R. Bergen, "Spatiotemporal energy models for the perception of motion",1. Optical Society of America A, vol. 2, pp. 284-299, Feb. 1985. J. Barron, D. S. Fleet, S. S. Beauchemin, and T. A. Burkitt, "Performance of optical flow techniques," in Proc. ofCVPR, (Champaign, IL), pp. 236-242, IEEE, 1992. J. G. Daugman, "Two-dimensional spectral analysis of cortical receptive field profiles," Vision Research, vol. 20, pp. 847-856, 1980. C. C. Enz, F. Krummenacher, and E. A. Vittoz, "An analytical MaS transistor model valid in all regions of operation and dedicated to low-voltage and low-current applications," Analog Integrated Circuits and Signal Processing, vol.8, no.l, p83-114, Jut 1995. D. J. Fleet, Measurement of Image Velocity, Boston. MA: Kluwer Academic Publishers, 1992. K. F. Hui and B. E. Shi, "Robustness of CNN Implementations for Gabor-type Filtering," Proc. of Asia Pacific Conference on Circuits and Systems, pp. 105-108, Nov. 1996. B. E. Shi, "Gabor-type image filtering with cellular neural networks," Proceedings of the 1996 IEEE International Symposium on Circuits and Systems, vol. 3, pp. 558-561, May 1996. W. M. Theimer and H. A Mallot, "Phase-based binocular vergence control and depth reconstruction using active vision," CVGIP: Image Understanding, vol. 60, no. 3, pp. 343-358, Nov. 1994. C.-J. Weste1ius, H. Knutsson, J. Wiklund and c.-F. Westin, "Phase-based disparity estimation," in 1. L. Crowley and H. I. Christensen, eds., Vision as Process, chap. 11, pp. 157178, Springer-Verlag, Berlin, 1995.
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Combining Classifiers Using Correspondence Analysis Christopher J. Merz Dept. of Information and Computer Science University of California, Irvine, CA 92697-3425 U.S.A. cmerz@ics.uci.edu Category: Algorithms and Architectures. Abstract Several effective methods for improving the performance of a single learning algorithm have been developed recently. The general approach is to create a set of learned models by repeatedly applying the algorithm to different versions of the training data, and then combine the learned models' predictions according to a prescribed voting scheme. Little work has been done in combining the predictions of a collection of models generated by many learning algorithms having different representation and/or search strategies. This paper describes a method which uses the strategies of stacking and correspondence analysis to model the relationship between the learning examples and the way in which they are classified by a collection of learned models. A nearest neighbor method is then applied within the resulting representation to classify previously unseen examples. The new algorithm consistently performs as well or better than other combining techniques on a suite of data sets. 1 Introduction Combining the predictions of a set of learned models! to improve classification and regression estimates has been an area of much research in machine learning and neural networks [Wolpert, 1992, Merz and Pazzani, 1997, Perrone, 1994, Breiman, 1996, Meir, 1995]. The challenge of this problem is to decide which models to rely on for prediction and how much weight to give each. The goal of combining learned models is to obtain a more accurate prediction than can be obtained from any single source alone. 1 A learned model may be anything from a decision/regression tree to a neural network. 592 C. l Men Recently, several effective methods have been developed for improving the performance of a single learning algorithm by combining multiple learned models generated using the algorithm. Some examples include bagging [Breiman, 1996], boosting [Freund, 1995], and error correcting output codes [Kong and Dietterich, 1995]. The general approach is to use a particular learning algorithm and a model generation technique to create a set of learned models and then combine their predictions according to a prescribed voting scheme. The models are typically generated by varying the training data using resampling techniques such as bootstrapping [Efron and Tibshirani, 1993J or data partitioning [Meir, 1995]. Though these methods are effective, they are limited to a single learning algorithm by either their model generation technique or their method of combining. Little work has been done in combining the predictions of a collection of models generated by many learning algorithms each having different representation and/or search strategies. Existing approaches typically place more emphasis on the model generation phase rather than the combining phase [Opitz and Shavlik, 1996]. As a result, the combining method is rather limited. The focus of this work is to present a more elaborate combining scheme, called SCANN, capable of handling any set of learned models, and evaluate it on some real-world data sets. A more detailed analytical and empirical study of the SCANN algorithm is presented in [Merz, 1997]. This paper describes a combining method applicable to model sets that are homogeneous or heterogeneous in their representation and/or search techniques. Section 2 describes the problem and explains some of the caveats of solving it. The SCANN algorithm (Section 3), uses the strategies of stacking [Wolpert, 1992J and correspondence analysis (Greenacre, 1984] to model the relationship between the learning examples and the way in which they are classified by a collection of learned models. A nearest neighbor method is then applied to the resulting representation to classify previously unseen examples. In an empirical evaluation on a suite of data sets (Section 4), the naive approach of taking the plurality vote (PV) frequently exceeds the performance of the constituent learners. SCANN, in turn, matches or exceeds the performance of PV and several other stacking-based approaches. The analysis reveals that SCANN is not sensitive to having many poor constituent learned models, and it is not prone to overfit by reacting to insignificant fluctuations in the predictions of the learned models. 2 Problem Definition and Motivation The problem of generating a set of learned models is defined as follows. Suppose two sets of data are given: a learning set C = {(Xi, Yi), i = 1, .. . ,I} and a test set T = {(Xt, yd, t = 1, .. . , T}. Xi is a vector of input values which are either nominal or numeric values, and Yi E {Cl , ... , Cc} where C is the number of classes. Now suppose C is used to build a set of N functions, :F = {fn (x)}, each element of which approximates f(x) , the underlying function. The goal here is to combine the predictions of the members of :F so as to find the best approximation of f(x). Previous work [Perrone, 1994] has indicated that the ideal conditions for combining occur when the errors of the learned models are uncorrelated. The approaches taken thus far attempt to generate learned models which make uncorrelated errors by using the same algorithm and presenting different samples of the training data [Breiman, 1996, Meir, 1995], or by adjusting the search heuristic slightly [Opitz and Shavlik, 1996, Ali and Pazzani, 1996J. No single learning algorithm has the right bias for a broad selection of problems. Combining Classifiers Using Correspondence Analysis 593 Therefore, another way to achieve diversity in the errors of the learned models generated is to use completely different learning algorithms which vary in their method of search and/or representation. The intuition is that the learned models generated would be more likely to make errors in different ways. Though it is not a requirement of the combining method described in the next section, the group of learning algorithms used to generate :F will be heterogeneous in their search and/or representation methods (i.e., neural networks, decision lists, Bayesian classifiers, decision trees with and without pruning, etc.). In spite of efforts to diversify the errors committed, it is still likely that some of the errors will be correlated because the learning algorithms have the same goal of approximating f, and they may use similar search strategies and representations. A robust combining method must take this into consideration. 3 Approach The approach taken consists of three major components: Stacking, Correspondence Analysis, and Nearest Neighbor (SCANN) . Sections 3.1-3.3 give a detailed description of each component, and section 3.4 explains how they are integrated to form the SCANN algorithm. 3.1 Stacking Once a diverse set of models has been generated, the issue of how to combine them arises. Wolpert (Wolpert, 1992] provided a general framework for doing so called stacked genemiization or stacking. The goal of stacking is to combine the members of:F based on information learned about their particular biases with respect to £2. The basic premise of stacking is that this problem can be cast as another induction problem where the input space is the (approximated) outputs of the learned models, and the output space is the same as before, i.e., The approximated outputs of each learned model, represented as jn(Xi), are generated using the following in-sample/out-of-sample approach: 1. Divide the £0 data up into V partitions. 2. For each partition, v, • Train each algorithm on all but partition v to get {j;V}. • Test each learned model in {j;V} on partition v. • Pair the predictions on each example in partition v (i.e., the new input space) with the corresponding output, and append the new examples to £1 3. Return £1 3.2 Correspondence Analysis Correspondence Analysis (CA) (Greenacre, 1984] is a method for geometrically exploring the relationship between the rows and columns of a matrix whose entries are categorical. The goal here is to explore the relationship between the training 2Henceforth £ will be referred to as £0 for clarity. 594 c. 1. Men Table 1· Correspondence Analysis calculations. Stage Symbol Definition Description 1 N (I x J) indicator matrix Records votes of learned models. 2:1 J Grand total of table N. n i=1 2:j=1 nij r ri = ni+/n Row masses. c cj=n+j/n Column masses. P (1/n)N Correspondence matrix. Dc (J x J) diagonal matrix Masses c on diagonal. Dr (I X I) diagonal matrix Masses r on diagonal. A Dr -1/2(p _ rcT)Dc -1/2 Standardized residuals. 2 A urv'! SVD of A. 3 F Dr -1/2ur Principal coordinates of rows. G Dc -1/2vr Principal coordinates of columns. examples and how they are classified by the learned models. To do this, the prediction matrix, M, is explored where min = in (xd (1 ::; i ::; I, and 1 ::; n ::; N). It is also important to see how the predictions for the training examples relate to their true class labels, so the class labels are appended to form M', an (I x J) matrix (where J = N + 1). For proper application of correspondence analysis, M' must be converted to an (I x (J . C)) indicator matrix, N, where ni,(joJ+e) is a one exactly when mij = ee, and zero otherwise. The calculations of CA may be broken down into three stages (see Table 1). Stage one consists of some preprocessing calculations performed on N which lead to the standardized residual matrix, A . In the second stage, a singular value decomposition (SVD) is performed on A to redefine it in terms ofthree matrices: U(lXK), r(KxK) ' and V(KXJ), where K = min(I - 1, J - 1). These matrices are used in the third stage to determine F(lXK) and G(JxK) , the coordinates of the rows and columns of N, respectively, in the new space. It should be noted that not all K dimensions are necessary. Section 3.4, describes how the final number of dimensions, K *, is determined. Intuitively, in the new geometric representation, two rows, fp* and fq*, will lie close to one another when examples p and q receive similar predictions from the collection of learned models. Likewise, rows gr* and gu will lie close to to one another when the learned models corresponding to r and s make similar predictions for the set of examples. Finally, each column, r, has a learned model, j', and a class label, c', with which it is associated; fp* will lie closer to gr* when model j' predicts class c'. 3.3 Nearest Neighbor The nearest neighbor algorithm is used to classify points in a weighted Euclidean space. In this scenario, each possible class will be assigned coordinates in the space derived by correspondence analysis. Unclassified examples will be mapped into the new space (as described below) , and the class label corresponding to the closest class point is assigned to the example. Since the actual class assignments for each example reside in the last C columns of N, their coordinates in the new space can be found by looking in the last Crows of G. For convenience, these class points will be called Class!, . .. , Classc . To classify an unseen example, XTest, the predictions of the learned models on XTest must be converted to a row profile, rT , oflength J . C, where r& oJ+e) is 1/ J exactly Combining Classifiers Using Correspondence Analysis 595 Table 2: Experimental results. PV SCANN S-BP S-BAYES Best Ind. vs PV vs PV vs PV vs PV Data set error ratio ratio ratio ratio abalone 80.35 .490 .499 .487 .535111' bal 13.81 .900 .859 .992 .911BP breast 4.31 .886 .920 .881 .938BP credit 13.99 .999 1.012 1.001 1.054BP dementia 32.78 .989 1.037 .932 1.048c4.5 glass 31.44 1.008 1.158 1.215 1.1550C1 heart 18.17 .964 .998 .972 .962BP ionosphere 3.05 .691 1.289 1.299 2.175c4.5 .. 4.44 1.017 1.033 1.467 1.150oc1 1flS krk 6.67 1.030 1.080 1.149 1. 159NN liver 29.33 1.035 1.077 1.024 1.138cN2 lymphography 17.78 1.017 1.162 1.100 .983Pebl8 musk 13.51 .812 .889 .835 1. 113Peb13 retardation 32.64 .970 .960 .990 .936Baye3 sonar 23.02 .990 1.079 1.007 1.048BP vote 5.24 .903 .908 .893 .927c4.5 wave 21.94 1.008 1.109 1.008 1.200Pebl8 wdbc 4.27 1.000 1.103 1.007 1. 164NN when mij = ee, and zero otherwise. However, since the example is unclassified, XTe3t is of length (J - 1) and can only be used to fill the first (( J - 1) . C) entries in iT. For this reason, C different versions are generated, i.e., iT, . .. , i c , where each one "hypothesizes" that XTe3t belongs to one of the C classes (by putting 1/ J in t~e appropr~ate col~~) .. Loc~ting thes=l.rofiles in the scale~ sp~ce is a matter of s1mple matflx multIphcatIOn, 1.e., f'[ = re Gr-1. The f'[ wh1ch lies closest to a class point, say Classc') is considered the "correct" hypothesized class, and XTe3t is assigned the class label c' . 3.4 The SCANN Algorithm Now that the three main parts of the approach have been described, a summary of the SCANN algorithm can be given as a function of Co and the constituent learning algorithms, A. The first step is to use Co and A to generate the stacking data, C1 , capturing the approximated predictions of each learned model. Next, C1 is used to form the indicator matrix, N. A correspondence analysis is performed on N to derive the scaled space, A = urvT. The number of dimensions retained from this new representation, K *, is the value which optimizes classification on C 1 . The resulting scaled space is used to derive the row/column coordinates F and G, thus geometrically capturing the relationships between the examples, the way in which they are classified, and their position relative to the true class labels. Finally, the nearest neighbor strategy exploits the new representation by predicting which class is most likely according to the predictions made on a novel example. 596 C. J Merz 4 Experimental Results The constituent learning algorithms, A, spanned a variety of search and/or representation techniques: Backpropagation (BP) [Rumelhart et al., 1986], CN2 [Clark and Niblett, 1989], C4.5 [Quinlan, 1993], OC1 [Salzberg; and Beigel, 1993], PEBLS [Cost, 1993], nearest neighbor (NN), and naive Bayes. Depending on the data set, anywhere from five to eight instantiations of algorithms were applied. The combining strategies evaluated were PV, SCANN, and two other learners trained on £1: S-BP, and S-Bayes. The data sets used were taken from the UCI Machine Learning Database Repository [Merz and Murphy, 1996], except for the unreleased medical data sets: retardation and dementia. Thirty runs per data set were conducted using a training/test partition of 70/30 percent. The results are reported in Table 2. The first column gives the mean error rate over the 30 runs of the baseline combiner, PV. The next three columns ("SCANN vs PV", "S-BP vs PV", and "S-Bayes vs PV") report the ratio of the other combining strategies to the error rate of PV. The column labeled "Best Ind. vs PV" reports the ratio with respect to the model with the best average error rate. The superscript of each entry in this column denotes the winning algorithm. A value less than 1 in the "a vs b" columns represents an improvement by method a over method b. Ratios reported in boldface indicate the difference between method a and method b is significant at a level better than 1 percent using a two-tailed sign test. It is clear that, over the 18 data sets, SCANN holds a statistically significant advantage on 7 sets improving upon PV's classification error by 3-50 percent. Unlike the other combiners, SCANN posts no statistically significant losses to PV (i.e., there were 4 losses each for S-BP and S-Bayes). With the exception of the retardation data set, SCANN consistently performs as well or better than the best individual learned model. In the direct comparison of SCANN with the S-BP and S-Bayes, SCANN posts 5 and 4 significant wins, respectively, and no losses. The most dramatic improvement of the combiners over PV came in the abalone data set. A closer look at the results revealed that 7 of the 8 learned models were very poor classifiers with error rates around 80 percent, and the errors of the poor models were highly correlated. This empirically demonstrates PV's known sensitivity to learned models with highly correlated errors. On the other hand, PV performs well on the glass and wave data sets where the errors of the learned models are measured to be fairly uncorrelated. Here, SCANN performs similarly to PV, but S-BP and S-Bayes appear to be overfitting by making erroneous predictions based on insignificant variations on the predictions of the learned models. 5 Conclusion A novel method has been introduced for combining the predictions of heterogeneous or homogeneous classifiers. It draws upon the methods of stacking, correspondence analysis and nearest neighbor. In an empirical analysis, the method proves to be insensitive to poor learned models and matches the performance of plurality voting as the errors of the learned models become less correlated. References [Ali and Pazzani, 1996) Ali, K. and Pazzani, M. (1996). Error reduction through learning multiple descriptions. Machine Learning, 24:173. Combining Classifiers Using Correspondence Analysis 597 [Breiman, 1996] Breiman, L. (1996). Bagging predictors. Machine Learning, 24(2):123-40. [Clark and Niblett, 1989] Clark, P. and Niblett, T. (1989). The CN2 induction algorithm. Machine Learning, 3(4):261-283. [Cost, 1993] Cost, S.; Salzberg, S. (1993). A weighted nearest neighbor algorithm for learning with symbolic features. Machine Learning, 10(1):57-78. [Efron and Tibshirani, 1993] Efron, B. and Tibshirani, R. (1993). An Introduction to the Bootstrap. Chapman and Hall, London and New York. [Freund, 1995] Freund, Y. (1995). Boosting a weak learning algorithm by majority. Information and Computation, 121(2):256-285. Also appeared in COLT90. [Greenacre, 1984] Greenacre, M. J. (1984). Theory and Application of Correspondence Analysis. Academic Press, London. [Kong and Dietterich, 1995] Kong, E. B. and Dietterich, T. G. (1995). Errorcorrecting output coding corrects bias and variance. In Proceedings of the 12th International Conference on Machine Learning, pages 313-321. Morgan Kaufmann. [Meir, 1995] Meir, R. (1995). Bias, variance and the combination ofleast squares estimators. In Tesauro, G., Touretzky, D., and Leen, T., editors, Advances in Neural Information Processing Systems, volume 7, pages 295-302. The MIT Press. [Merz, 1997] Merz, C. (1997). Using correspondence analysis to combine classifiers. Submitted to Machine Learning. [Merz and Murphy, 1996] Merz, C. and Murphy, P. (1996). UCI repository of machine learning databases. [Merz and Pazzani, 1997] Merz, C. J. and Pazzani, M. J. (1997). Combining neural network regression estimates with regularized linear weights. In Mozer, M., Jordan, M., and Petsche, T., editors, Advances in Neural Information Processing Systems, volume 9. The MIT Press. [Opitz and Shavlik, 1996] Opitz, D. W. and Shavlik, J. W. (1996). Generating accurate and diverse members of a neural-network ensemble. In Touretzky, D. S., Mozer, M. C., and Hasselmo, M. E., editors, Advances in Neural Information Processing Systems, volume 8, pages 535-541. The MIT Press. [Perrone, 1994] Perrone, M. P. (1994). Putting it all together: Methods for combining neural networks. In Cowan, J. D., Tesauro, G., and Alspector, J., editors, Advances in Neural Information Processing Systems, volume 6, pages 1188-1189. Morgan Kaufmann Publishers, Inc. [Quinlan, 1993] Quinlan, R. (1993). G..4-5 Programs for Machine Learning. Morgan Kaufmann, San Mateo, CA. [Rumelhart et al., 1986] Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986). Learning internal representations by error propagation. In Rumelhart, D. E., McClelland, J. 1., and the PDP research group., editors, Parallel distributed processing: Explorations in the microstructure of cognition, Volume 1: Foundations. MIT Press. [Salzberg; and Beigel, 1993] Salzberg;, S. M. S. K. S. and Beigel, R. (1993). OC1: Randomized induction of oblique decision trees. In Proceedings of AAAI-93. AAAI Pres. [Wolpert, 1992] Wolpert, D. H. (1992). Stacked generalization. Neural Networks, 5:241-259.
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Synchronized Auditory and Cognitive 40 Hz Attentional Streams, and the Impact of Rhythmic Expectation on Auditory Scene Analysis Bill Baird Dept Mathematics, U.C.Berkeley, Berkeley, Ca. 94720. baird@math.berkeley.edu Abstract We have developed a neural network architecture that implements a theory of attention, learning, and trans-cortical communication based on adaptive synchronization of 5-15 Hz and 30-80 Hz oscillations between cortical areas. Here we present a specific higher order cortical model of attentional networks, rhythmic expectancy, and the interaction of hi~her order and primar¥, cortical levels of processing. It accounts for the' mismatch negativity' of the auditory ERP and the results of psychological experiments of Jones showing that auditory stream segregation depends on the rhythmic structure of inputs. The timing mechanisms of the model allow us to explain how relative timing information such as the relative order of events between streams is lost when streams are formed. The model suggests how the theories of auditory perception and attention of Jones and Bregman may be reconciled. 1 Introduction Amplitude patterns of synchronized "gamma band" (30 to 80 Hz) oscillation have been observed in ilie ensemble activity (local field potentials) of vertebrate olfactory, visual, auditory, motor, and somatosensory cortex, and in the retlna, thalamus, hippocampus, reticular formation, and EMG. Such activity has not only been found in primates, cats, rabbits and rats, but also insects, slugs, fish, amphibians, reptiles, and birds. This suggests that gamma oscillation may be as fundamental to neural processing at the network level as action potentials are at the cellular level. We have shown how oscillatory associative memories may be coupled to recognize and generate sequential behavior, and how a set of novel mechanisms utilizing these complex dynamics can be configured to solve attentional and perceptual processing problems. For pointers to full treatment wi th mathematics and complete references see [BaIrG et al., 1994]. An important element of intra-cortical communication in the brain, and between modules in this architecture, is the ability of a module to detect and respond to the proper input signal from a particular module, when inputs from other modules which are irrelevant to tlie present computation are contributing crosstalk noise. We have demonstrated that selective control of synchronization, which we hypothesize to be a model of "attention", can be used to solve this coding problem and control program flow in an architecture with dynamic attractors [Baird et al., 1994]. Using dynamical systems theory, the architecture is constructed from recurrently interconnected oscillatory associative memory modules that model higher order sensory and motor areas of cortex. The modules learn connection weights between themselves which cause the system to evolve under a 5-20 Hz clocked sensory-motor processing cycle by a sequence 4 B. Baird of transitions of synchronized 30-80 Hz oscillatory attractors within the modules. The architecture employ's selective"attentional" control of the synchronization of the 30-80 Hz gamma band OSCIllations between modules to direct the flow of computation to recognize and generate sequences. The 30-80 Hz attractor amplitude patterns code the information content of a cortIcal area, whereas phase and frequency are used to "softwire" the network, since only the synchronized areas communicate by exchanging amplitude information. The system works like a broadcast network where the unavoidable crosstalk to all areas from previous learned connections is overcome by frequency coding to allow the moment to moment operation of attentional communication only between selected task-relevant areas. The behavior of the time traces in different modules of the architecture models the temporary appearance and switching of the synchronization of 5-20 and 30-80 Hz oscillations between cortical areas that is observed during sensorimotor tasks in monkeys and numans. The architecture models the 5-20 Hz evoked potentials seen in the EEG as the control signals which determine the sensory-motor processing cycle. The 5-20 Hz clocks which drive these control signals in the archItecture model thalamic pacemakers which are thought to control the excitabili ty of neocortical tissue through similar nonspecific biasing currents that cause the cogni tive and sensory evoked potentials of the EEG. The 5-20 Hz cycles "quantize time" and form the basis of derived somato-motor rhythms with periods up to seconds that entrain to each other in motor coordination and to external rhythms in speech perception [Jones et al., 1981]. 1.1 Attentional Streams of Synchronized 40 Hz Activity There is extensive evidence for the claim of the model that the 30-80 Hz gamma band activity in the brain accomplishes attentional processing, since 40 Hz appears in cortex when and where attention is required. For example, it is found in somatosensory, motor and premotor cortex of monkeys when they must pick a rasin out of a small box, but not when a habitual lever press delivers the reward. In human attention experiments, 30-80 Hz activity goes up in the contralateral auditory areas when subjects are mstructed to pay attention to one ear and not the other. Gamma activity declines in the dominant hemisphere along with errors in a learnable target and distractors task, but not when the distractors and target vary at random on each trial. Anesthesiologists use the absence of 40 Hz activity as a reliable indicator of unconsciousness. Recent work has shown that cats with convergent and divergent strabismus who fail on tasks where perceptual binding is required also do not exhibit cortical synchrony. This is evidence that gamma synchronization IS perceptually functional and not epiphenomenal. The architecture illustrates the notion that synchronization of gamma band activity not only"binds" the features of inputs in primary sensory cortex into "objects", but further binds the activity of an attended object to oscillatory activity in associational and higher-order sensory and motor cortical areas to create an evolving attentional network of intercommunicating cortical areas that directs behavior. The binding of sequences of attractor transitions between modules of the architecture by synchronization of their activity models the physiological mechanism for the formation of perceptual and cognitive "streams" investigated by Bregman [Bregman, 1990], Jones [Jones et aI., 1981], and others. In audition, according to Bregman's work, successive events of a sound source are bound together into a distinct sequence or "stream" and segy:egated from other sequences so that one pays attention to onl¥, one sound source at a time (the cocktail party problem). Higher order cortical or "cognitive' streams are in evidence when subjects are unable to recall the relative order of the telling of events between two stories told in alternating segments. MEG tomographic observations show large scale rostral to caudal motor-sensory sweeps of coherent thalamo-cortical40Hz activity accross the entire brain, the phase of which is reset by sensory input in waking, but not in dream states [Llinas and Ribary, 1993]. This suggests an inner higher order "attentional stream" is constantly cycling between motor (rostral) and sensory (caudal) areas in the absence of input. It may be interrupted by input "pop out" from primary areas or it may reach down as a "searchlight" to synchromze with particular ensembles of primary activity to be attended. 2 Jones Theory of Dynamic Attention Jones [Jones et al., 1981] has developed a psychological theory of attention,perception, and motor timing based on the hypotheSIS that these processes are organized by neural rhythms in the range of 10 to .5 Hz - the range within which subjects perceive pen odic events as a rhythm. These rhythms provide a multi scale representation of time and selectively synchronize with the prominant periodiCities of an input to provide a -temporal expectation mechanism for attention to target particular points in time. 40 Hz Attentional Streams, Rhythmic Expectation, and Auditory Scene Analysis 5 For example, some work suggests that the accented parts of speech create a rhythm to which listeners entrain. Attention can then be focused on these expected locations as recognition anchor points for inference of less prominant parts of the speech stream. This is the temporal analog of the body centered spatial coordinate frame and multiscale covert attention window system in vision. Here the body centered temporal coordinates of the internal time base orient by entrainment to the external rhythm, and the window of covert temporal attention can then select a level of the multiscale temporal coordinates. In this view, just as two cortical areas must synchronize to communicate, so must two nervous systems. Work using frame by frame film analysis of human verbal interaction, shows evidence of "interactional synchrony" of gesture and body movement changes and EEG of both speaker and listener With the onsets of phonemes in speech at the level of a 10 Hz "microrhythm" - the base clock rate of our models. Normal infants synchronize their spontaneous body ftailings at this 10 Hz level to the mothers voice accents, while autistic and s~hitzophrenic children fail to show interactional synchrony. Autistics are unable to tap in orne t9 a metronome. Neural expectation rhythms that support Jones' theory have been found in the auditory EEG. In experiments where the arrival time of a target stimulus is regular enough to be learned by an experimental subject, it has been shown that the 10 Hz activity in advance of the stimulus becomes phase locked to that expected arrival time. This fits our model of rhythmic expectation where the 10Hz rhythm IS a fast base clock that is shifted in phase and frequency to produce a match in timmg between the stimulus arrival and the output of longer period cycles derived from this base clock. 2.1 Mismatch Negativity The "mismatch negativity" (MNN) [Naatanen, 1992] of the auditory evoked potential appears to be an important physiological indicator of the action of a neural expectancy system like that proposed by Jones. It has been localized to areas within primary auditory cortex by MEG studies [Naatanen, 1992] and it appears as an increased negativity of the ERP in the region of the N200 peak whenever a psycbologically discriminable deviation of a repetitive auditory stimulus occurs. Mismatch IS caused by deviations in onset or offset time, rise time, frequency, loudness, timbre, phonetic structure, or spatial location of a tone in the sequence. The mismatch is abolished by blockers of the action ofNMDA channels [Naatanen, 1992] which are important for the synaptic changes underlying the kind of Hebbian learning which is used in the model. MNN is not a direct function of echoic memory because it takes several repetitions for the expectancy to begin to develop, and it decays 10 2 - 4 seconds. It appears only for repetition periods greater that 50-100 msec and less than 2-4 seconds. Thus the time scale of its operation is 10 the appropriate range for Jones' expectancy system. Stream formation also takes several cycles of stimulus repetition to builo up over 2-4 seconds and decays away within 2-4 seconds in the absence of stimulation. Those auditory stimulus features which cause streaming are also features which cause mismatch. This supports the hypothesis in the model that these phenomena are functionally related. Finally, MNN can occur independent of attention - while a subject is reading or doing a visual discrimination task. ThIS implies that the auditory system at least must have its own timing system that can generate timmg and expectancies independent of other behavior. We can talk or do internal verbal thinking while doing other tasks. A further component of this negativity appears in prefrontal cortex and is thought by Nataanen to initiate attentional switchmg toward the deVIant event causing perceptual "pop out" [Naatanen, 1992]. Stream formation is known to affect rhythm perception. The galloping rhythm of high H and low L tones - HLH-HLH-HLH, for example becomes two separate Isochronous rhythmic streams of H-H-H-Hand L-L-L-L when the H and L tones are spread far enough apart [Bregman, 1990]. Evidence for the effect of in'putrhythms on stream formation, however, is more sparse, and we focus here on the simulatIOn of a particular set of experiments by Jones [J ones et al., 1981] and Bregman [Bregman, 1990] where this effect has been demonstrated. 2.2 Jones-Bregman Experiment Jones [Jones et al., 1981] replicated and altered a classic streaming experiment of Bregman and Rudnicky [Bregman, 1990], and found that their result depended on a specific choice of the rhythm of presentation. The experiment required human subjects to determine of the order of presentation of a pair of high target tones AB or BA of slIghtly different frequencies. Also presented before and after the target tones were a series of identical much lower frequency tones called the capture tones CCC and two identical tones of intermediate fre6 B. Baird quency before and after the target tones called the flanking tones F - CCCFABFCCC. Bregman and Rudnicky found that target order determination performance was best when tfie capture tones were near to the flanking tones in frequency, and deteriorated as the captor tones were moved away. Their explanation was that the flanking tones were captured by the background capture tone stream when close in frequency, leav10g the target tones to stand out by themselves in the attended stream. When the captor tones were absent or far away in frequency, the flanking tones were included in the attended stream and obscured the target tones. Jones noted that the flanking tones and the capture stream were presented at a stimulus onset rate of one per 240 ms and the targets appeared at 80 ms intervals. In her experiments, when the captor and flanking tones were given a rhythm in common with the targets, no effect of the distance of captor and flanking tones appeared. This suggested that rfiythmic distinction of targets and dlstractors was necessary 10 addition to the frequency diStinction to allow selective attention to segregate out the target stream. Because performance in the single rhythm case was worse than that for the control condition without captors, it appeared that no stream segregation of targets and captors and flanking tones was occurring until the rhythmic difference was added. From this evidence we malie the assumption in the model that the distance of a stimulus in time from a rhythmic expectancy acts like the distance between stimuli in pitch, loudness, timbre, or spatial location as/actor for theformation of separate streams. 3 Architecture and Simulation To implement Jones's theory in the model and account for her data, subsets of the oscillatory modules are dedicated to form a rhythmic temporal coordinate frame or time base by dividing down a thalamic 10 Hz base clock rate in steps from 10 to .5 Hz. Each derived clock is created by an associative memory module that has been specialized to act stereotypically as a counter or shift register by repeatedly cycling through all its attractors at the rate of one for each time step of its clock. Its overall cycle time is therefore determined by the number of attractors. Each cycle is guaranteed to be identical, as required for clocklike function, because of the strong attractors that correct the perturbing effect of noise. Only one step of the cycle can send output back to primary cortex - the one with the largest weight from receiving the most matco to incoming stimuli. Each clock derived in this manner from a thalamic base clock will therefore phase reset itself to get the best match to incoming rhythms. The match can be further refined by frequency and phase entrainment of the base clock itself. Three such counters are sufficient to model the rhythms in Jones' experiment as shown in the architecture of figure 1. The three counters divide the 12.5 Hz clock down to 6.25 and 4.16 Hz. The first contains one attractor at the base clock rate which has adapted to entrain to the 80 msec period of target stimulation (12.5 Hz). The second cycles at 12.5/2 = 6.25 Hz, alternating between two attractors, and the third steps through three attractors, to cycle at 12.5/3 = 4.16 Hz, which is the slow rhythm of the captor tones. The modules of the time base send theirinternal30-80 Hz activity to {>rimary auditory cortex in 100msec bursts at these different rhythmic rates through fast adapting connections (which would use NMDA channels in the brain) that continually attempt to match incoming stimulus patterns using an incremental Hebbian learning rule. The weights decay. to zero over 2-4 sec to simulate the data on the rise and fall of the mismatch negativity. These weights effectively compute a low frequency discrete Fourier transform over a sliding window of several seconds, and the basic periodic structure of rhythmic patterns is quickly matched. This serves to establish a quantized temporal grid of expectations against which expressive timing deviations in speech and music can be experienced. Following Jones [Jones et al., 1981], we hypothesize that this happens automatically as a constant adaptation to environmental rhythms, as suggested by the mismatch negativity. Retained in these weights of the timebase is a special k10d of short term memory of the activity which includes temporal information since the timebase will partially regenerate the prevIous activity in primary cortex at the expected recurrence time. This top-down input causes enchanced sensitivity in target units by increasing their gain. Those patterns which meet these established rhythmic expectancy signals in time are thereby boosted in amplitude and pulled into synchrony with the 30-80 Hz attentional searchlight stream to become part of the attentional network sending input to higher areas. In accordance with Jones' theory, voluntary top-down attention can probe input at different hierarchical levels of periodicity by selectively synchronizing a particular cortical column in the time base set to the 40 Hz frequency of the inner attention stream. Then the searchlight into primary cortex is synchro40 Hz Attentional Streams, Rhythmic Expectation, and Auditory Scene Analysis Dynamic Attention Architecture Higher Order AuditoryCortex Synchronization Timebase ~" ..... ~: : :" ::. '---...,. Pitch Attentionallnput Stream Rhythmic Searchlight 10 Hz Clock w=, :J'~:' : '0" .' ~ :' ~ ~ Cycle -.-.. -.-.-.-.-.-.. -.-.-.-.-.~ High Target Tones " -'1 _._._._._._._._._.~ B ~. • • . I _ . • • • • • • • • • • • • • • -~ II. ' @H Fast Weights :-. 1 '1 _ . ~F I '_._._._._ . _._._._._._._._._._.~ High Flanking Tones Low Captor Tones Input I 1':'."", \ • • ~ I .:::::::(~~ I f • I .............. I ' · " 1 r ' - ' . _.-._. C I._._._._._._._._ ... _. __ ~::-:-: __ .:-._. C ._._._._._._ . _. C ._._._._~ Primary Auditory Cortex ~----------------------------------------------------~Time 7 Figure 1: Horizontally arrayed units at the top model higher order auditory and motor cortical columns which are sequentially clocked by the (thalamic) base clock on the right to alternate attractor transitions between upper hidden (motor) and lower context (sensory) layers to act as an Elman net. Three cortical regions are shown - sequence representation memory, attentional synchronization control, and a rhythmic timebase of three counters. The hidden and context layers consist of binary "units" composed of two oscillatory attractors. Activity levels oscillate up and down through the plane of the paper. Dotted hnes show frequency shifting outputs from the synchromzation (attention) control modules. The lower vertical set of units IS a sample of primary auditory cortex frequency channels at the values used in the Jones-Bregman expenment. The dashed lines show the rhythmic pattern of the target, flanking, and captor tones moving in time from left to right to Impact on auditory cortex. nizing and reading in activity occuring at the peaks of that particular time base rhythm. 3.1 Cochlear and Primary Cortex Model At present, we have modeled only the minimal aspects of primary auditory conex sufficient to qualitatively simulate the Jones-Bregman experiment, but the principles at work allow expansion to larger scale models with more stimulus features. We simulate four sites in auditory conex corresponding to the four frequencies of stimuli used in the experiment, as shown in figure 1. There are two close high frequency target tones, one high flanking frequency location, and the low frequency location of the captor stream. These cortical locations are modeled as oscillators with the same equations used for associative memory modules [Baird et al., 1994], with full linear cross coupling weights. This lateral connectivity is sufficient to promote synchrony among simultaneously activated oscillators, but insufficient to activate them strongly in the absence of externallOput. This makes full synchrony of activated units the default condition in the model conex, as in Brown's model [Brown and Cooke, 1996]. so that the background activation is coherent, and can be read into higher order cortical levels which synchronize with it. The system assumes that all input is due to the same environmental source in the absence of evidence for segregation [Bregman, 1990]. Brown and Cooke [Brown and Cooke, 1996] model the cochlear and brain stem nuclear output as a set of overlapping bandpass ("gammatone") filters consistent with auditory nerve responses and psychophysical "critical bands". A tone can excite several filter outputs at once. We apprOXImate this effect of the gammatone filters as a lateral fan out of input activations with weights that spread the activation in the same way as the overlapping gammatone 8 B. Baird filters do. Experiments show that the intrinsic resonant or "natural" frequencies or "eigenfr~uencies" of cortical tissue within the 30-80 Hz gamma band vary within individuals on different trials of a task, and that neurotransmitters can quickly alter these resonant frequencies of neural clocks. Following the evidence that the oscillation frequency of binding in vision goes up with the speed of motion of an object, we assume that unattended activity in auditory cortex synchromzes at a default background fr~uency of 35 Hz, while the higher order attentional stream is at a higher frequency of 40 Hz. Just as fast motion in vision can cause stimulus driven capture of attention, we hypothesize that expectancy mismatch in audition causes the deviant activity to be boosted above the default background frequency to facilitate synchronization with the attentional stream at 40 Hz. This models the mechanism of involuntary stimulus driven attentional "pop out". Multiple streams of primary cortex activity synchronized at different eigenfrequencies can be selectively attended by unifonnly sweeping the eigenfrequencies of all primary ensembles through the passband of the 40 Hz higner order attentional stream to "tune in" each in turn as a radio reciever does. Following, but modifing the approach of Brown and Cooke [Brown and Cooke, 1996], the core of our primary cortex stream fonning model is a fast learning rule that reduces the lateral coupling and (in our model) spreads apart the intrinsic cortIcal frequencies of sound frequency cflannels that do not exhibit the same amplitude of activity at the same time. This coupling and eigenfrequency difference recovers between onsets. In the absence of lateral synchronizing connectlons or coherent top down driving, synchrony between cortical streams is rapidly lost because of their distant resonant frequencIes. Activity not satisfying the Gestalt prinCIple of "common fate" [Bregman, 1990] is thus decorrelated. The trade off of the effect of temporal and sound frequency proximity on stream segregation follows because close stimulus frequencies excite each other's channel filters. Each produces a similar output in the other, and their activitites are not decorrelated by coupling reduction and resonant frequency shifts. On the other hand, to the extent that they are distant enough in sound frequency, each tone onset weakens the weights and shifts the eigenfrequencies of the other channels that are not simultaneously active. This effect is greater, the faster the presentation rate, because the weight recovery rate is overcome. This recovery rate can then be adjusted to yield stream segregation at the rates reported by van Noorden [Bregman, 1990] for given sound frequency separations. 3.2 Sequential Grouping by Coupling and Resonant Frequency Labels In the absence of rhythmic structure in the input, the temporary weights and resonant frequency "labels" serve as a short tenn "stream memory" to brid~e time (up to 4 seconds) so that the next nearby input is "captured" or "sequentially bound into the same ensemble of synchronized activity. This pattern of synchrony in primary cortex has been made into a temporary attractor by the temporary weight and eigenfrequency changes from the previous stimulation. This explains the single tone capture expenments where a series of ioentical tones captures later nearby tones. For two points in tIme to be sequentially grouped by this mechanIsm, there is no need for activity to continue between onsets as in Browns model [Brown and Cooke, 1996], or to be held in multiple spatial locations as Wang [Wang, 1995] does. Since the gamma band response to a single auditory input onset lasts only 100 - 150 ms, there is no 40 Hz activity available in prim~ cortex (at most stimulus rates) for succesive inputs to synchronize with for sequential bmding by these mechanisms. Furthermore, the decorrelation rule, when added to the mechanism of timing expectancies, explains the loss of relative timing (order) between streams, since the lateral connections that normally broadcast actual and expected onsets accross auditory cortex, are cut between two streams by the decorrelating weight reduction. Expected and actual onset events in different streams can no longer be directly (locally) compared. Experimental evidence for the broadcast of expectancies comes from the fast generalization to other frequencies of a learned expectancy for the onset time of a tone of a particular frequency (Schreiner lab personal commumcation). When rhythmic structure is present, the expectancy system becomes engaged, and this becomes an additional feature dimension along which stimuli can be segregated. Distance from expected timing as well as sound quality is now an added factor causing stream formation by decoupling and eigenfrequency ShIft. Feedback of expected input can also partially"fill in" missing input for a cycle or two so that the expectancy protects the binding of features of a stimulus and stabilizes a perceptual stream accross seconds of time. 40 Hz AItentional Streams, Rhythmic Expectation, and Auditory Scene Analysis 9 3.3 Simulation or the Jones-Bregman Experiment Figure 2 shows the architecture used to simulate the Jones-Bregman experiment. The case shown is where the flanking tones are in the same stream as the targets because the captor stream is at the lower sound frequency channel. At the particular pomt in time shown here, the first flanking tone has just fimshed, and the first target tone has arrived. Both channels are therfore active, and synchronized with the attentional stream into the higher order sequence recognizer. Our mechanistic explanation of the Bregman result is that the early standard target tones arriving at the 80 msec rate first prime ttie dynamic attention system by setting the 80 msec clock to oscillate at 40 Hz and depressing the oscillation frequency of other auditory cortex background activi~y. Then the slow captor tones at the 240 msec period establish a background stream at 30 Hz with a rhythmic expectancy that is later violated by the appearance of the fast target tones. These now fall outside the correlation attractor basin of the background stream because the mismatch increases their cortical oscillation frequency. They are explicitly brought into the 40 Hz foreground frequency by the mismatch pop out mechanism. This allows the attentional stream into the Elman sequence recognition units to synchronize and read in activity due to the target tones for order determination. It is assisted by the timebase searchlight at the 80 msec period which synchronizes and enhances activity arriving at that rhythm. In the absence of a rhythmic distmction for the target tones, their sound frequency difference alone is insufficient to separate them from the background stream, and the targets cannot be reliably discriminated. In this simulation, the connections to the first two Elman associative memory units are hand wired to the A and B primary cortex oscillators to act as a latching, order determining switch. If sy-nchronized to the memory unit at the attentional stream frequency, the A target tone OSCillator will drive the first memory unit into the 1 attractor whicfi then inhibits the second unit from being driven to 1 by the B target tone. The second unit has similar wiring from the B tone oscillator, so that the particular higher order (intermediate term) memory unit which is left in the 1 state after a tnal indicates to the rest of the brain which tone came first. The flanking and high captor tone oscillator is connected equally to both memory units, so that a random attractor transition occurs before the targets amve, when it is interfering at the 40 Hz attentional frequency, and poor order determination results. If the flanking tone oscillator is in a separate stream along with the captor tones at the background eigenfrequency of 35 Hz, it is outside the recieving passband of the memory units and cannot cause a spurious attractor transition. This architecture demonstrates mechanisms that integrate the theories of Jones and Bregman about auditory perception. Stream formation is a preattentive process that works well on non-rhythmic inputs as Bregman asserts, but an equally primary and preattentive rhythmic expectancy process is also at work as Jones asserts and the mismatch negativity indicates. This becomes a factor in stream formation when rhythmic structure is present in stimuli as demonstrated by Jones. References [Baird et al., 1994] Baird, B., Troyer, T., and Eeckman, F. H. (1994). Gramatical inference by attentional control of synchronization in an oscillating elman network. In Hanson, S., Cowan, J., and Giles, C., editors, Advances in Neural InjormationProcessing Systems 6, pages 67-75. Morgan Kaufman. [Bregman, 1990] Bregman, A. S. (1990). Auditory Scene Analysis. MIT Press, Cambridge. [Brown and Cooke, 1996] Brown, G. and Cooke, M. (1996). A neural oscillator model of auditory stream segregation. In JJCAI Workshop on Computational Auditory Scene Analysis. to appear. [Jones et al., 1981] Jones, M., Kidd, G., and Wetzel, R. (1981). Evidence for rhythmic attention. Journal 0/ Experimental Psychology: Human Perception and Performance, 7: 1059-1073. [Llinas and Ribary, 1993] Llinas, R. and Ribary, U. (1993). Coherent 40-hz oscillation characterizes dream state in humans. Proc. Natl. Acad. Sci. USA,90:2078-2081. [Naatanen, 1992] Naatanen, R. (1992). Attention and Brain Function. Erlbaum, New Jersey. [Wang, 1~95] Wang, D. (1995). ~n oscillatory correlation theory of te~poral pattern segmentatIOn. In Covey, E., Hawkms, H., McMullen, T., and Port, R., edaors, Neural Representations o/Temporal Patterns. Plenum. to appear.
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From Regularization Operators to Support Vector Kernels Alexander J. Smola GMDFIRST Rudower Chaussee 5 12489 Berlin, Germany smola@first.gmd.de Bernhard Scholkopf Max-Planck-Institut fur biologische Kybernetik Spemannstra.Be 38 72076 Ttibingen, Germany bs-@mpik-tueb.mpg.de Abstract We derive the correspondence between regularization operators used in Regularization Networks and Hilbert Schmidt Kernels appearing in Support Vector Machines. More specifica1ly, we prove that the Green's Functions associated with regularization operators are suitable Support Vector Kernels with equivalent regularization properties. As a by-product we show that a large number of Radial Basis Functions namely conditionally positive definite functions may be used as Support Vector kernels. 1 INTRODUCTION Support Vector (SV) Machines for pattern recognition, regression estimation and operator inversion exploit the idea of transforming into a high dimensional feature space where they perform a linear algorithm. Instead of evaluating this map explicitly, one uses Hilbert Schmidt Kernels k(x, y) which correspond to dot products of the mapped data in high dimensional space, i.e. k(x, y) = (<I>(x) · <I>(y)) (I) with <I> : .!Rn --* :F denoting the map into feature space. Mostly, this map and many of its properties are unknown. Even worse, so far no general rule was available. which kernel should be used, or why mapping into a very high dimensional space often provides good results, seemingly defying the curse of dimensionality. We will show that each kernel k(x, y) corresponds to a regularization operator P, the link being that k is the Green's function of P* P (with F* denoting the adjoint operator of F). For the sake of simplicity we shall only discuss the case of regression our considerations, however, also hold true foi the other cases mentioned above. ~ We start by briefly reviewing the concept of SV Machines (section 2) and of Regularization Networks (section 3). Section 4 contains the main result stating the equivalence of both 344 A. J. Smola and B. Schollwpf methods. In section 5, we show some applications of this finding to known SV machines. Section 6 introduces a new class of possible SV kernels, and, finally, section 7 concludes the paper with a discussion. 2 SUPPORT VECTOR MACHINES The SV algorithm for regression estimation, as described in [Vapnik, 1995] and [Vapnik et al., 1997], exploit~ the idea of computing a linear function in high dimensional feature space F (furnished with a dot product) and thereby computing a nonlinear function in the space of the input data !Rn. The functions take the form f(x) = (w · <ll(x)) + b with ell : !Rn -+ :F and w E F. In order to infer f from a training set {(xi, Yi) I i = 1, ... , f., Xi E !Rn, Yi E IR}, one tries to minimize the empirical risk functional Remp[f] together with a complexity term l!wll 2, thereby enforcingflatness in feature space, i.e. to minimize 1 l Rreg[/] = Remp[!] + Allwll 2 =f. :L;c(f(xi),yi) + Allwll 2 i=l (2) with c(f(xi),yi) being the cost function determining how deviations of f(xi) from the target values Yi should be penalized, and A being a regularization constant. As shown in [Vapnik, 1995] for the case of €-insensitive cost functions, c(f(x) ) = { lf(x)- yl- € for lf(_x)- Yl ;::: e ' Y 0 otherwtse ' (3) (2) can be minimized by solving a quadratic programming problem formulated in terms of dot products in :F. It turns out that the solution can be expressed in terms of Support Vectors, w = :Ef=I Cti<ll(xi), and therefore l l f(x) = L ai(<ll(xi) · <ll(x)) + b = L aik(xi, x) + b, (4) i=l i=l where k(xi, x) is a kernel function computing a dot product in feature space (a concept introduced by Aizerrnan et al. [ 1964]). The coefficients ai can be found by solving a quadratic programming problem (with Kii := k(xi, Xj) and ai = f3i - {3i): l l minimize ! L (f3i- f3i)(f3j- f3J)Kij- "£, (!3i- f3i)Yi- (f3i + f3i)e i,j=l i=l f. (5) subject to "£, f3i - !3i = 0, f3i, !3i E [0, ft] · i=l Note that (3) is not the only possible choice of cost functions resulting in a quadratic programming problem (in fact quadratic parts and infinities are admissible, too). For a detailed discussion see [Smola and Scholkopf, 1998]. Also note that any continuous symmetric function k(x, y) E L2 ® L2 may be used as an admissible Hilbert-Schmidt kernel if it satisfies Mercer's condition // k(x,y)g(x)g(y)dxdy;::: 0 for all g E L2 (IRn)~ (6) 3 REGULARIZATION NETWORKS Here again we start with minimizing the empirical risk functional Remp[!] plus a regularization term liP /11 2 defined by a regularization operator Pin the sense of Arsenin and From Regularization Operators to Support Vector Kernels 345 Tikhonov [1977]. Similar to (2), we minimize l ~ 2 1"' ~ 2 Rreg[f] = Remp + .\IIPJII = f £_-c(f(xi),yi) + -XIIPJII · i=1 (7) Using an expansion off in terms of some symmetric function k(xi, Xj) (note here, that k need not fulfil Mercer's condition), f(x) = 2: aik(xi, x) + b, (8) and the cost function defined in (3 ), this leads to a quadratic programming problem similar to the one for SVs: by computing Wolfe's dual (for details of the calculations see [Smola and SchOlkopf, 1998]), and using Dij := ((Fk)(xi, .) · (Fk)(xj, .)) (9) ((f · g) denotes the dot product of the functions f and g in Hilbert Space, t.e. I !(x)g(x)dx), we get a= n-1 K(i]- /3*), with f3i, f3i being the solution of l l mm1m1Ze ! 2: (f3i- f3i)(f3j- {3j)(KD- 1 K)ij- L (f3i- f3i)Yi- (f3i + f3i)E i,j=1 i=l f_ subject to L f3i - f3i = 0, f3i, f3i E [0, A] i=1 (I 0) Unfortunately this setting of the problem does not preserve sparsity in terms of the coefficients, as a potentially sparse decomposition in terms of f3i and f3i is spoiled by n-1 K, which in general is not diagonal (the expansion (4) on the other hand does typically have many vanishing coefficients). 4 THE EQUIVALENCE OF~BOTH METHODS Comparing (5) with (10) leads to the question if and under which condition the two methods might be equivalent and therefore also under which conditions regularization networks might lead to sparse decompositions (i.e. only a few of the expansion coefficients in f would differ from zero). A sufficient condition is D = K (thus K n-1 K = K), i.e. (11) Our goal now is twofold: • Given a regularization operator P, find a kernel k such that a SV machine using k will not only enforce flatness in feature space, but also correspond to minimizing a regularized risk functional with P as regularization operator. • Given a Hilbert Schmidt kernel k, find a regularization operator P such that a SV machine using this kernel can be viewed as a Regularization Network using P. These two problems can be solved by employing the concept of Green's functions as described in [Girosi et al., 1993]. These functions had been introduced in the context of solving differential equations. For our purpose, it is sufficient to know that the Green's functions Gx, (x) ofF* P satisfy (12) Here, 8xi (x) is the 8-distribution (not to be confused with the Kronecker symbol8ij) which has the property that (f · 8x.) = f(xi). Moreover we require for all Xi the projection of Gx, (x) onto the null space of F* P to be zero. The relationship between kernels and regularization operators is formalized in the following proposition. 346 A. I. Smola and B. Schtilkopf Proposition 1 Let P be a regularization operator, and G be the Green's function of P* P. Then G is a Hilbert Schmidt-Kernel such that D = K. SV machines using G minimize risk functional (7) with Pas regularization operator. Proof: Substituting ( 12) into GxJ (xi) = ( GxJ (.) · 8x• (.)) yields Gxi (xi) = ( (PGx, ){.) · (PGxJ(.)) = Gx; {xj), (13) hence G(xi,Xj) := Gx,{xj) is symmetric and satisfies (11). Thus the SVoptimization problem (5) is equivalent to the regularization network counterpart ( 10). Furthermore G is an admissible positive kernel, as it can be written as a dot product in Hilbert Space, namely (14) In the following we will exploit this relationship in both ways: to compute Green's functions for a given regularization operator P and to infer the regularization operator from a given kernel k. 5 TRANSLATION INVARIANT KERNELS Let us now more specifically consider regularization operators P that may be written as multiplications in Fourier space [Girosi et al., 1993] (Pi· P ) = 1 f f{w)fJ(w) dw g (21l')n/2 Jn P(w) (15) with ](w) denoting the Fourier transform of j(x), and P(w) = P( -w) real valued, nonnegative and converging uniformly to 0 for lwl --+ oo and n = supp[P(w)]. Small values of P(w) correspond to a strong attenuation of the corresponding frequencies. For regularization operators defined in Fourier Space by (15) it can be shown by exploiting P(w) = P(-w) = P(w) that G(Xi x) = 1 f eiw(x;-x) P(w)dw l {27!' )n/2 }JR.n (16) is a corresponding Green's function satisfying translational invariance, i.e. G(xi,xj) = G (Xi - xi), and G ( w) = P ( w). For the proof, one only has to show that G satisfies (11 ). This provides us with an efficient tool for analyzing SV kernels and the types of capacity control they exhibit Example 1 (Bq-splines) Vapnik et al. [ 1997] propose to use Bq-splines as building blocks for kernels, i.e. (17) i=l with x E !Rn. For the sake of simplicity, we consider the case n 1. Recalling the definition . Bq = ®q+11[-o.5,o.5] (18) (® denotes the convolution and Ix the indicator function on X), we can utilize the above result and the Fourier-Plancherel identity to construct the Fourier representation of the corresponding regularization operator. Up to a multiplicative constant, it equals P(w) = k(w) = sinc(q+l)(Wi). (19) 2 From Regularization Operators to Support W?ctor Kernels 347 This shows that only B-splines of odd order are admissible, as the even ones have negative parts in the Fourier spectrum (which would result in an amplification of the corresponding frequency components). The zeros ink stem from the fact that B1 has only compact support [-(k+ 1)/2, (k+ 1)/2). By using this kernel we trade reduced computational complexity in calculatingf(we only have to take points with llxi- xi II :S cfrom some limited neighborhood determined by c into account)for a possibly worse performance of the regularization operator as it completely ~emovesfrequencies wp with k(wp) = 0. Example 2 (Dirichlet kernels) In [Vapnik et al., 1997], a class of kernels generating Fourier expansions was introduced, k(x) = sin(2N + 1)x/2. (20) sinx/2 (As in example 1 we consider x E ~1 to avoid tedious notation.) By construction, this kernel corresponds to P(w) = ~ L~-N 6(w- i). A regularization operator with these properties, however; may not be desirable as it only damps a finite number of frequencies and leaves all other frequencies unchanged which can lead to overjitting (Fig. 1 ). 05 -I -· _, -15 \ !\ 'i 'I I. I• ,\ I l I \ : \ .I 'I -.:.,,\ 1\ .'i \ I.\ r ""t-4 \I ~ -10 -· 10 15 Figure 1: Left: Interpolation with a Dirichlet Kernel of order N = 10. One can clearly observe the overfitting (dashed line: interpolation, solid line: original data points, connected by lines). Right: Interpolation of the same data with a Gaussian Kernel of width CT2 = 1. Example 3 (Gaussian kernels) Following the exposition of Yuille and Grzywacz [ 1988] as described in [Girosi et al., 1993], one can see that for I 2m 11P!II2 = dx L ~!2m com f(x)) 2 m (21) with 62m = 6. m and 62m+l = V' 6. m. 6. being the Laplacian and V' the Gradient operator; we get Gaussians kernels k(x) = exp ( -~~~~ 2 ). (22) Moreover; we can provide an equivalent representation of P in terms of its Fourier properties, i.e. P(w) = exp(- u2 \kxll 2 ) up to a multiplicative constant. Training a SV machine with Gaussian RBF kernels [Scholkopf et al., 1997] corresponds to minimizing the specific cost function with a regularization operator of type (21 ). This also explains the good performance of SV machines in this case, as it is by no means obvious that choosing a flat fum:;tion in high dimensional space will correspond to a simple function in low dimensional space, as showed in example 2. Gaussian kernels tend to yield good performance under g'eneral smoothness assumptions and should be considered especially if no additional knowledge of the data is available. 348 A. J. Smola and B. Scholkopf 6 A NEW CLASS OF SUPPORT VECTOR KERNELS We will follow the lines of Madych and Nelson [ 1990] as pointed out by Girosi et al. [ 1993]. Our main statement is that conditionally positive definite functions ( c.p.d.) generate admissible SV kernels. This is very useful as the property of being c.p.d. often is easier to verify than Mercer's condition, especially when combined with the results of Schoenberg and Micchelli on the connection between c.p.d. and completely monotonic functions [Schoenberg, 1938, Micchelli, 1986]. Moreover c.p.d. functions lead to a class of SV kernels that do not necessarily satisfy Mercer's condition. Definition 1 (Conditionally positive definite functions) A continuous function h, defined on [0, oo), is said to be conditionally positive definite · ( c.p.d.) of order m on m.n if for any distinct points x1, ... , Xt E m.n and scalars c1, ... , Ct the quadratic form Eri=l cicih(llxi- Xj II) is nonnegative provided that E~=l Cip(xi) = 0 for all polynomials p on m.n of degree lower than m. Proposition 2 (c.p.d. functions and admissible kernels) Define II~ the space of polynomials of degree lower than m on IRn. Every c.p.d. function h of order m generates an admissible Kernel for SV expansions on the space of functions f orthogonal to II~ by setting k(xi, Xj) := h(llxi- Xjll 2). Proof: In [Dyn, /991] and [Madych and Nelson, 1990] it was shown that c.p.d. functions h generate semi-norms 11-llh by (23) Provided that the projection off onto the space of polynomials of degree lower than m is zero. For these functions, this, however. also defines a dot product in some feature space. Hence they can be used as SV kernels. Only c.p.d. functions of order m up to 2 are of practical interest for SV methods (for details see [Smola and Scholkopf, 1998]). Consequently, we may use kernels like the ones proposed in [Girosi et al., 1993] as SV kernels: k(x,y) = e-.BIJx-yJJ2 Gaussian, (m = 0); (24) k(x,y) = -v'llx- Yll 2 + c2 multiquadric, (m = 1) (25) k(x,y) = 1 inverse multiquadric, (m = 0) (26) y'Jix-yJ12+c2 k(x,y) = llx- Yll 2 ln llx- Yll thin plate splines, ( m = 2) (27) 7 DISCUSSION We have pointed out a connection between SV kernels and regularization operators. As one of the possible implications of this result, we hope that it will deepen our understanding of SV machines and of why they have been found to exhibit high generalization ability. In Sec. 5, we have given examples where only the translation into the regularization framework provided insight in why certain kernels are preferable to others. Capacity control is one of the strengths of SV machines; however, this does not mean that the structure of the learning machine, i.e. the choice of a suitable kernel for a given task, should be disregarded. On the contrary, the rather general class of admissible SV kernels should be seen as another strength, provided that we have a means of choosing the right kernel. The newly established link to regularization theory can thus be seen as a tool for constructing the structure consisting of sets of functions in which the SV machine (approximately) performs structural From Regularization Operators to Support \iector Kernels 349 risk minimization (e.g. [Vapnik, 1995]). For a treatment of SV kernels in a Reproducing Kernel Hilbert Space context see [Girosi, 1997]. Finally one should leverage the theoretical results achieved for regularization operators for a better understanding of SVs (and vice versa). By doing so this theory might serve as a bridge for connecting two (so far) separate threads of machine learning. A trivial example for such a connection would be a Bayesian interpretation of SV machines. In this case the choice of a special kernel can be regarded as a prior on the hypothesis space with P[f] ex exp{ ->.IIF 1112). A more subtle reasoning probably will be necessary for understanding the capacity bounds [Vapnik, 1995] from a Regularization Network point of view. Future work will include an analysis of the family of polynomial kernels, which perform very well in Pattern Classification [SchOlkopf et al., 1995]. Acknowledgements AS is supported by a grant of the DFG (# Ja 379/51 ). BS is supported by the Studienstiftung des deutschen Volkes. The authors thank Chris Burges, Federico Girosi, Leo van Hemmen, Klaus-Robert Muller and Vladimir Vapnik for helpful discussions and comments. References M.A. Aizerman, E. M. Braverman, and L. I. Rozonoer. Theoretical foundations of the potential function method in pattern recognition learning. Automation and Remote Control, 25:821-837, 1964. N. Dyn. Interpolation and approximation by radial and related functions. In C.K. Chui, L.L. Schumaker, and D.J. Ward, editors, Approximation Theory, VI, pages 211-234. Academic Press, New York, 1991. F. Girosi. An equivalence between sparse approximation and suppm1 vector machines. A.I. Memo No. 1606, MIT, 1997. F. Girosi, M. Jones, and T. Poggio. Priors, stabilizers and basis functions: From regularization to radial, tensor and additive splines. A.I. Memo No. 1430, MIT, 1993. W.R. Madych and S.A. Nelson. Multivariate interpolation and conditionally positive definite functions. II. Mathematics of Computation, 54(189):211-230, 1990. C. A. Micchelli. Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constructive Approximation, 2:11-22, 1986. I.J. Schoenberg. Metric spaces and completely monotone functions. Ann. of Math., 39: 811-841, 1938. B. Scholkopf, C. Burges, and V. Vapnik. Extracting support data for a given task. In U. M. Fayyad and R. Uthurusamy, editors, Proc. KDD I, Menlo Park, 1995. AAAI Press. B. SchOlkopf, K. Sung, C. Burges, F. Girosi, P. Niyogi, T. Poggio, and V. Vapnik. Comparing support vector machines with gaussian kernels to radial basis function classifiers. IEEE Trans. Sign. Processing, 45:2758-2765, 1997. A. J. Smola and B. SchOlkopf. On a kernel-based method for pattern recognition, regression, approximation and operator inversion. Algorithmica, 1998. see also GMD Technical Report 1997- I 064, URL: http://svm.first.gmd.de/papers.html. V. Vapnik. The Nature of Statistical Learning Theory. Springer Verlag, New York, 1995. V. Vapnik, S. Golowich, and A. Smola: Support vector method for function approximation, regression estimation, and signal processing. In NIPS 9, San Mateo, CA, 1997. A. Yuille and N. Gr:z;ywacz. The motion coherence theory. In Proceedings of the International Conference on Computer Vision, pages 344-354, Washington, D.C., 1988. IEEE Computer Society Press.
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Two Approaches to Optimal Annealing Todd K. Leen Dept of Compo Sci. & Engineering Oregon Graduate Institute of Science and Technology P.O.Box 91000, Portland, Oregon 97291-1000 tleen@cse.ogi.edu Bernhard Schottky and David Saad Neural Computing Research Group Dept of Compo Sci. & Appl. Math. Aston University Birmingham, B4 7ET, UK schottba{ saadd}@aston.ac.uk Abstract We employ both master equation and order parameter approaches to analyze the asymptotic dynamics of on-line learning with different learning rate annealing schedules. We examine the relations between the results obtained by the two approaches and obtain new results on the optimal decay coefficients and their dependence on the number of hidden nodes in a two layer architecture. 1 Introduction The asymptotic dynamics of stochastic on-line learning and it's dependence on the annealing schedule adopted for the learning coefficients have been studied for some time in the stochastic approximation literature [1, 2] and more recently in the neural network literature [3, 4, 5]. The latter studies are based on examining the KramersMoyal expansion of the master equation for the weight space probability densities. A different approach, based on the deterministic dynamics of macroscopic quantities called order parameters, has been recently presented [6, 7]. This approach enables one to monitor the evolution of the order parameters and the system performance at all times. In this paper we examine the relation between the two approaches and contrast the results obtained for different learning rate annealing schedules in the asymptotic regime. We employ the order parameter approach to examine the dependence of the dynamics on the number of hidden nodes in a multilayer system. In addition, we report some lesser-known results on non-standard annealing schedules 302 T. K. Leen, B. Schottky and D. Saad 2 Master Equation Most on-line learning algorithms assume the form Wt+l = Wt + 1]o/tP H(wt,xt) where Wt is the weight at time t, Xt is the training example, and H(w,x) is the weight update. The description of the algorithm's dynamics in terms of weight space probability densities starts from the master equation P(w',t+1)= JdW (8(w'-w-~~H(w,x)))xP(w,t) (1) where ( .. '}x indicates averaging with respect to the measure on x, P(w,t) is the probability density on weights at time t, and 8( ... ) is the Dirac function. One may use the Kramers-Moyal expansion of Eq.(l) to derive a partial differential equation for the weight probability density (here in one dimension for simplicity) {3, 4] at P ( w, t) = t (~~) i (7~ r a~ [ (Hi ( w, x) > x P ( w, t)] . (2) t=l Following {3], we make a small noise expansion for (2) by decomposing the weight trajectory into a deterministic and stochastic pieces or (TJo)-"( ~ = tP (w-</>(t)) (3) where </>( t) is the deterministic trajectory, and ~ are the fluctuations. Apart from the factor (1]0/ tP)'Y that scales the fluctuations, this is identical to the formulation for constant learning in {3]. The proper value for the unspecified exponent, will emerge from homogeneity requirements. Next, the dependence of the jump moments (Hi (w, x) > on 1]0 is explicated by a Taylor series expansion about the deterministic path </>. T!{e coefficients in this series expansion are denoted a~i) == ai (Hi(w,x))x /awilw=tI> Finally one rewrites (2) in terms of </> and ~ and the expansion of the jump moments, taking care to transform the differential operators in accordance with (3). These transformations leave equations of motion for </> and the density I1(~, t) on the fluctuations d</> dt = = (4) For stochastic descent H ( w, x) = -V w E( w, x) and (4) describes the evolution of </> as descent on the average cost. The fluctuation equation (5) requires further manipulation whose form depends on the context. For the usual case of descent in a quadratic minimum (ail) = -G, minus the cost function curvature), we take ''i = 1/2 to insure that for any m, terms in the sum are homogeneous in T}o/tP For constant learning rate (p = 0), rescaling time as t ~ 1]ot allows (5) to be written in a form convenient for perturbative analysis in 1]0 Typically, the limit 1]0 ~ 0 is invoked and only the lowest order terms in 1]0 retained (e.g. [3]). These comprise a diffusion operator, which results in a Gaussian approximation for equilibrium densities. Higher order terms have been successfully used to calculate corrections to the equilibrium moments in powers of 1]0 [8]. Two Approaches to Optimal Annealing 303 Of primary interest here is the case of annealed learning, as required for convergence of the parameter estimates. Again assuming a quadratic bowl and 'Y = 1/2, the first few terms of (5) are GtII = - :t Gd~II) - all) i; Ge(~II) + ~a~O): Gl II + 0 (: r/2. (6) As t -+ 00 the right hand side of (6) is dominated by the first three terms (since o < p S 1). Precisely which terms dominate depends on p. . We will first review the classical case p = 1. Asymptotically 1> -+ w"', a local optimum. The first three leading terms on the right hand side of (6) are all of order lit. For t -+ 00, we discard the remaining terms. From the resulting equation we recover a Gaussian equilibrium distribution for ~, or equivalently for Vt ( w - w"') == Vtv where v is called the weight error. The asymptotically normal distribution for Vtv has variance 0'0v from which the asymptotic expected squared weight error can be derived 1 2 (0) 1· E[I 12] 2 7Jo a 2 1 1m v -0' -t->oo -,;tv t 27]0 G'" - 1 t (7) where G'" == G(w"') is the curvature at the local optimum. Positive O',;t v requires T]o > 11 (2G"'). If this condition is not met the expected squared weight offset converges as (1It)1-2'1oG', slower than lit [5, for example, and references therein]. The above confirms the classical results [1] on asymptotic normality and convergence rate for 1 I t annealing. For the case 0 < p < 1, the second and third terms on the right hand side of (6) will dominate as t -+ 00. Again, we have a Gaussian equilibrium density for ~. Consequently ViP v is asymptotically normal with variance O'~v leading to the expected squared weight error 2 1 0' r;-;; ytPv tP = T]o a~O) .!.. 2G tP (8) Notice that the convergence is slower than lit and that there is no critical value of the learning rate to obtain a sensible equilibrium distribution. (See [9] for earlier results on 11tP annealing.) The generalization error follows the same decay rate as the expected weight offset. In one dimension, the expected squared weight offset is directly related to excess generalization error (the generalization error minus the least generalization error achievable) Eg = G E[v2 ]. In multiple dimensions, the expected squared weight offset, together with the maximum and minimum eigenvalues of G'" provide upper and lower bounds on the excess generalization error proportional to E[lvI2 ], with the criticality condition on G'" (for p = 1 )replaced with an analogous condition on its eigenvalues. 3 Order parameters In the Master equation approach, one focuses attention on the weight space distribution P( w, t) and calculates quantities of interested by averaging over this density. An alternative approach is to choose a smaller set of macroscopic variables that are sufficient for describing principal properties of the system such as the generalization error (in contrast to the evolution of the weights w which are microscopic). 304 T. K. Leen, B. Schottky and D. Saad Formally, one can replace the parameter dynamics presented in Eq.(1) by the corresponding equation for macroscopic observables which can be easily derived from the corresponding expressions for w. By choosing an appropriate set of macroscopic variables and invoking the thermodynamic limit (i.e., looking at systems where the number of parameters is infinite), one obtains point distributions for the order parameters, rendering the dynamics deterministic. Several researchers [6, 7] have employed this approach for calculating the tr~ing dynamics of a soft committee machine (SCM) . The SCM maps inputs x E RN to a scalar, through a model p{w,x) = 2:~lg{Wi' x). The activation function of the hidden units is g{u) == erf{u/V2) and Wi is the set of input-to-hidden adaptive weights for the i = 1 ... K hidden nodes. The hidden-to-output weights are set to 1. This architecture preserves most of the properties of the learning dynamics and the evolution of the generalization error as a general two-layer network, and the formalism can be easily extended to accommodate adaptive hidden-to-output weights [10]. Input vectors x are independently drawn with zero mean and unit variance, and the corresponding targets y are generated by deterministic teacher network corrupted by additive Gaussian output noise of zero mean and variance O'~. The teacher network is also a SCM, with input-to-hidden weights wi. The order parameters sufficient to close the dynamics, and to describe the network generalization error are overlaps between various input-to-hidden vectors Wi . Wk == Qik, Wi' W~ _ Rin, and w~· w~ == Tnm . Network performance is measured in terms of the generalization error Eg{W) _ (1/2 [ p(w, x) - Y ]2)~. The generalization error can be expressed in closed form in terms of the order parameters in the thermodynamiclimit (N -+ 00). The dynamics of the latter are also obtained in closed form [7]. These dynamics are coupled nonlinear ordinary differential equations whose solution can only be obtained through numerical integration. However, the asymptotic behavior in the case of annealed learning is amenable to analysis, and this is one of the primary results of the paper. We assume an isotropic teacher Tnm = 8nm and use this symmetry to reduce the system to a vector of four order parameters uT = (r, q, s, c) related to the overlaps by Rin = 8in (1 + r) + (1- 8in)S and Qik = 8ik(1 + q) + {1- 8ik)C. With learning rate annealing and limt-+oo u = ° we describe the dynamics in this vicinity by a linearization of the equations of motion in [7]. The linearization is d 2 2 dt U = rJ1\d U + rJ 0' /I b , where O'~ is the noise variance, b T = ~ (0,1/,;3,0,1/2), rJ = rJo/tP, and M is 2 M = 3V31T -4 4 3 --V3 2 3V3 3 -~(K -lhI(3) 4 3 -(K - 1)V3 3 2 2 --(K-2)+2 J3 3V3(K - 2) + ~ 3 -(K -1)V3 ~ --(K - 1)V3 2 o -3V3(K - 2) + ~ V3 (9) (10) The asymptotic equations of motion (9) were derived by dropping terms of order O(rJlluI12) and higher, and terms of order O{rJ2 u). While the latter are linear in the order parameters, they are dominated by the rJu and rJ20'~b terms in (9) as t -+ 00. Two Approaches to Optimal Annealing 305 This choice of truncations sheds light on the approach to equilibrium that is not implicit in the master equation approach. In the latter, the dominant terms for the asymptotics of (6) were identified by time scale of the coefficients, there was no identification of system observables that signal when the asymptotic regime is entered. For the order parameter approach, the conditions for validity of the asymptotic approximations are cast in terms of system observables 1JU vs rlu VS 1J2 fI~. The solution to (9) is u(t) = -yet, to) Uo + fI~ j3(t, to) b where Uo == u(to) and -y(t, to) = exp {M lot dr 1J(r)} and j3(t, to) = t dr -yet, r) 1J2( r). lto (11) (12) The asymptotic order parameter dynamics allow us to compute the generalization error (to first order in u) K(l K-1 ) E/ = -:; J3(q- 2r) + -2-(C- 28) . (13) Using the solution of Eq.(l1), the generalization error consists of two pieces: a contribution depending on the actual initial conditions Uo and a contribution due to the second term on the r.h.s. of Eq.(l1), independent of Uo. The former decays more rapidly than the latter, and we ignore it in what follows. Asymptotically, the generalization error is of the form E/ = fI;(CI0l(t) + C202(t)), where Cj are K dependent coefficients, and OJ are eigenmodes that evolve as OJ = 1J5 [! _ to'it)Ot;;-(O'it)O+I)] . (14) 1 + (}:j1Jo t with eigenvalues (Fig. l(a)) (}:1 = -~ (~ -2) and (}:2 = -~ (~ +2(K -1)) (15) The critical learning rate 1J~rit, above which the generalization decays as lit is, for K> 2 , crit (1 1 ) 7r 110 = max - (}:1 ,- (}:2 = 41 J3 - 2 . (16) For 1Jo > 1J~rit both modes OJ, i = 1,2 decay as lit, and so 2 2 (Cl C2) 1 2 ( 1 E/ = -fI1I 11o 1 + (}:11JO + 1 + (}:21Jo t == fIll f 1Jo,K) t (17) Minimizing the prefactor f (1Jo, K) in (17) minimizes the asymptotic error. The values 1J~Pt (K) are shown in Fig. 1 (b), where the special case of K = 1 (see below) is also included: There is a significant difference between the values for K = 1 and K = 2 and a rather weak dependence on K for K ~ 2. The sensitivity of the generalization error decay factor on the choice of 1Jo is shown in Fig. 1 ( c). The influence of the noise strength on the generalization error can be seen directly from (17): the noise variance fI; is just a prefactor scaling the lit decay. Neither the value for the critical nor for the optimal1Jo is influenced by it. 306 T. K. Leen, B. Schottky and D. Saad The calculation above holds for the case K = 1 (where c and s and the mode 01 are absent). In this case opt(K _ 1) - 2 crit(R" - 1) __ 2. _ J311" 110 -1]0 . <l:2 2 (18) Finally, for the general annealing schedule of the form 1] = 1]oltP with 0 < p < 1 the equations of motion (11) can be investigated, and one again finds 11tP decay. 4 Discussion and summary We employed master equation and order parameter approaches to study the convergence of on-line learning under different annealing schedules. For the lit annealing schedule, the small noise expansion provides a critical value of 1]0 (7) in terms of the curvature, above which Vt v is asymptotically normal, and the generalization decays as lit. The approach is general, but requires knowledge of the first two jump moments in the asymptotic regime for calculating tl~e relevant properties. By restricting the order parameters approach to a symmetric task characterized by a set of isotropic teacher vectors, one can explicitly solve the dynamics in the asymptotic regime for any number of hidden nodes, and provide explicit expressions for the decaying generalization error and for the critical (16) and optimal learning rate prefactors for any number of hidden nodes K. Moreover, one can study the sensitivity of the generalization error decay to the choice of this prefactor. Similar results have been obtained for the critical learning rate prefactors using both methods, and both methods have been used to study general11tP annealing. However, the order parameters approach enables one to gain a complete description of the dynamics and additional insight by restricting the task examined. Finally the order parameters approach expresses the dynamics in terms of ordinary differential equations, rather than partial differential equations; a clear advantage for numerical investigations. The order parameter approach provides a potentially helpful insight on the passage into the asymptotic regime. Unlike the truncation of the small noise expansion, the truncation of the order parameter equations to obtain the asymptotic dynamics is couched in terms of system observables (c.f. the discussion following (10)). That is, one knows exactly which observables must be dominant for the system to be in the asymptotic regime. Equivalently, starting from the full equations, the order parameters approach can tell us when the system is close to the equilibrium distribution. Although we obtained a full description of the asymptotic dynamics, it is still unclear how relevant it is in the larger picture which includes all stages of the training process, as in many cases it takes a prohibitively long time for the system to reach the asymptotic regime. It would be interesting to find a way of extending this framework to gain insight into earlier stages of the learning process. Acknowledgements: DS and BS would like to thank the Leverhulme Trust for their support (F J250JK). TL thanks the International Human Frontier Science Program (SF 473-96), and the NSF (ECS-9704094) for their support. Two Approaches to Optimal Annealing -2 -4 -6 -8 " . " ~ . ..... " "''' .. -'." ". -'II;. "." .... ", .... ......... '. -10 I .... = ~~' ....... . -12 L-__ ....i.---'--....L.~"'--'-.....&--..L.---'''-'•• ........J 2 4 6 8 10 12 14 16 18 20 (a) K 20 18 -K=I ...... K=2 16 -- K=3 .. .. K=5 14 12 ,. f ., . ,-, ::.:: 10 0 ~ 8 ;;::: I\ .... ~\ .... .. .. ... ... . . \\ 6 4 . \ \ .. ~'----------.................... 2 0 2 4 6 8 10 12 14 16 18 (c) t'lo References wr----~~~-~~~~ -T/"'" I 18 0 .. .... f(l1O"'. K) 16 14 ::v ................... . 8 ..• 6 • ......• .... 4 ..... 2 ....•. ..... . ..•. OL-~~--~--'-~--~~~~ I 2 3 4 5 6 7 8 9 10 (b) K Figure 1: (a) The dependence of the eigenvalues of M on the the number of hidden units K. Note that the constant eigenvalue al dominates the convergence for K ~ 2_ (b) 1]gpt and the resulting generalization prefactor / (1]gpt ,K) of (17) as a function of K_ (c) The dependence of the generalization error decay prefactor / (1]0, K) on the choice of 1]0. [1] V. Fabian. Ann. Math. Statist., 39, 1327 1968. 307 [2] L. Goldstein. Technical Report DRB-306, Dept. of Mathematics, University of Southern California, LA, 1987. [3] T. M. Heskes and B. Kappen, Phys. Rev. A 44, 2718 (1991)_ [4] T. K. Leen and J. E. Moody. In Giles, Hanson, and Cowan, editors, Advances in Neural In/ormation Processing Systems, 5, 451, San Mateo, CA, 1993. Morgan Kaufmann. [5] T. K. Leen and G. B. Orr. In J.D. Cowan, G. Tesauro, and J. Alspector, editors, Advances in Neural Information Processing Systems 6, 477 ,San Francisco, CA., 1994. Morgan Kaufmann Publishers. [6] M. Biehl and H. Schwarze, J. Phys. A 28, 643 (1995). [7] D. Saad and S_A. Solla Phys. Rev. Lett. 74, 4337 (1995) and Phys. Rev. E 52 4225 (1995). [8] G. B. Orr. Dynamics and Algorithms for Stochastic Search. PhD thesis, Oregon Graduate Institute, October 1996. [9] Naama Barkai. Statistical Mechanics of Learning_ PhD thesis, Hebrew University of Jerusalem, August 1995. [10] P. Riegler and M. Biehl J_ Phys. A 28, L507 (1995).
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Bidirectional Retrieval from Associative Memory Friedrich T. Sommer and Gunther Palm Department of Neural Information Processing University of Ulm, 89069 Ulm, Germany {sommer,palm}~informatik.uni-ulm.de Abstract Similarity based fault tolerant retrieval in neural associative memories (N AM) has not lead to wiedespread applications. A drawback of the efficient Willshaw model for sparse patterns [Ste61, WBLH69], is that the high asymptotic information capacity is of little practical use because of high cross talk noise arising in the retrieval for finite sizes. Here a new bidirectional iterative retrieval method for the Willshaw model is presented, called crosswise bidirectional (CB) retrieval, providing enhanced performance. We discuss its asymptotic capacity limit, analyze the first step, and compare it in experiments with the Willshaw model. Applying the very efficient CB memory model either in information retrieval systems or as a functional model for reciprocal cortico-cortical pathways requires more than robustness against random noise in the input: Our experiments show also the segmentation ability of CB-retrieval with addresses containing the superposition of pattens, provided even at high memory load. 1 INTRODUCTION From a technical point of view neural associative memories (N AM) provide data storage and retrieval. Neural models naturally imply parallel implementation of storage and retrieval algorithms by the correspondence to synaptic modification and neural activation. With distributed coding of the data the recall in N AM models is fault tolerant: It is robust against noise or superposition in the addresses and against local damage in the synaptic weight matrix. As biological models N AM 676 F. T. Sommer and G. Palm have been proposed as general working schemes of networks of pyramidal cells in many places of the cortex. An important property of a NAM model is its information capacity, measuring how efficient the synaptic weights are used. In the early sixties Steinbuch realized under the name "Lernmatrix" a memory model with binary synapses which is now known as Wills haw model [Ste6I, WBLH69]. The great variety of NAM models proposed since then, many triggered by Hopfield's work [Hop82], do not reach the high asymptotic information capacity of the Willshaw model. For finite network size, the Willshaw model does not optimally retrieve the stored information, since the inner product between matrix colum and input pattern determines the activity for each output neuron independently. For autoassociative pattern completion iterative retrieval can reduce cross talk noise [GM76, GR92, PS92, SSP96]. A simple bidirectional iteration - as in bidirectional associative memory (BAM) [Kos87] - can, however, not improve heteroassociative pattern mapping. For this task we propose CB-retrieval where each retrieval step forms the resulting activity pattern in an autoassociative process that uses the connectivity matrix twice before thresholding, thereby exploiting the stored information more efficiently. 2 WILLSHAW MODEL AND CB EXTENSION Here pattern mapping tasks XV -+ yV are considered for a set of memory patterns: {(XV,yV): XV E {O,I}n,yv E {o,I}m,v = I, ... ,M}. The number of I-components in a pattern is called pattern activity. The Willshaw model works efficiently, if the memories are sparse, i.e., if the memory patterns have the same activities: Ixvi = 2:~=I xi = a,lyvl = 2::1 Yi = b V v with a « nand b «m. During learning the set of memory patterns is transformed to the weight matrix by Cij = min(I, L xiv}) = supxiy'j· V V For a given initial pattern XJ1. the retrieval yields the output pattern yJ1. by forming in each neuron the dendritic sum [CxJ1.]j = 2:i Ci/if and by calculating the activity value by threshold comparison yj = H([CxJ1.jj - 9) Vj, (1) with the global threshold value 9 and H(x) denoting the Heaviside function. For finite sizes and with high memory load, Le., 0« PI := Prob[Cij = 1] « 0.5), the Willshaw model provides no tolerance with respect to errors in the address, see Fig. 1 and 2. A bidirectional iteration of standard simple retrieval (1), as proposed in BAM models [Kos87], can therefore be ruled out for further retrieval error reduction [SP97j. In the energy function of the Willshaw BAM E(x,y) = - LCijXiYj + 8' LXi + 8 LYj ij i j we now indroduce a factor accounting for the magnitudes of dendritic potentials at acti vated neurons (2) Bidirectional Retrieval from Associative Memory Differentiating the energy function (2) yields the gradient descent equations yrW = H( [CxU + L 'LCijCikXi Yk - 8 ) k i "'-v---' =:Wjk X~ew = H( [CT y); + L "LPiiCljYi Xl - 8' ) I i --------=:wfr 677 (3) (4) As new terms in (3) and (4) sums over pattern components weighted with the quantities wjk and wft occur. wjk is the overlap between the matrix columns j and k conditioned by the pattern X, which we call a conditioned link between y-units. Restriction on the conditioned link terms yields a new iterative retrieval scheme which we denote as crosswise bidirectional (eB) retrieval y(r+ I)i = H( 'L Cij[CT y(r-I))i - 8) iEx(r) H( L Cij[Cx(r-I))j - 8') iEy(r) (5) (6) For r = 0 pattern y(r:-I) has to be replaced by H([Cx(O)] - 0), for r > 2 Boolean ANDing with results from timestep r - 1 can be applied which has been shown to improve iterative retrieval in the Willshaw model for autoassociation [SSP96]. 3 MODEL EVALUATION Two possible retrieval error types can be distinguished: a "miss" error converts a I-entry in Y~ to '0' and a "add" error does the opposite. 35 ]. 30 " 25 2. 20 " 15 ,. 10 5 0 2. ]. 40 5 simple r. add error ..... C8-r. add error CB-r. miss error ..... 10 15 20 25 30 Figure 1: Mean retrieval error rates for n = 2000, M = 15000, a = b = 10 corresponding to a memory load of H = 0.3. The x-axes display the address activity: lilLl = 10 corresponds to a errorfree learning pattern, lower activities are due to miss errors, higher activities due to add errors. Left: Theory - Add errors for simple retrieval, eq. (7) (upper curve) and lower bound for the first step of CB-retrieval, eq. (9). Right: Simulations - Errors for simple and CB retrieval. The analysis of simple retrieval from the address i~ yields with optimal threshold setting 0 = k the add error rate, i.e, the expectation of spurious ones: & = (m - b)Prob [r ~ k] , (7) 678 F. T. Sommer and G. Palm with the binomial random variable Prob[r=l] = B(Lit'I,Pt}I, where B(n,p), := (7)pl(1 - p)n-l. a denotes the add error rate and k = lit'l - a the number of correct 1-s in the address. For the first step of CB-retrieval a lower bound of the add error rate a(l) can be derived by the analysis of CB-retrieval with fixed address x(O) = iIJ. and the perfect learning pattern ylJ. as starting patterns in the y-Iayer. In this case the add error rate is: (8) where the random variables rl and r2 have the distributions: Prob [rl = lib] = B(k, PI), and Prob [r2 = 1] = B(ab, (PI )2) l" Thus, k a(l) ~ (m - b) L B(k, PdsBS [ab, (PI )2, (k - s)b) , (9) 8=0 where BS [n,p, t] := L:~t B(n,p), is the binomial sum. In Fig. 1 the analytic results for the first step (7) and (9) can be compared with simulations (left versus right diagram). In the experiments simple retrieval is performed with threshold () = k. CB-retrieval is iterated in the y-Iayer (with fixed address x) starting with three randomly chosen 1-s from the simple retrieval result yt'. The iteration is stopped, if a stable pattern at threshold e = bk is reached. The memory capacity can be calculated per pattern component under the assumption that in the memory patterns each component is independent, i.e., the probabilities for a 1 are p = a/n or q = b/m respectively, and the probabilities of an add and a miss error are simply the renormalized rates denoted by a', {3' and a', {3' for x-patterns and by,', 6' for y-patterns. The information about the stored pattern contained in noisy initial or retrieved patterns is then given by the transinformation t(p,a',{3') := i(p) -i(p,a',{3'), where i(p) is the Shannon information, and i (p, a', {3') the conditional information. The heteroassociative mapping is evaluated by the output capacity: A(a', {3') := Mm t(q, ,', 6')/mn (in units bit/synapse). It depends on the initial noise since the performance drops with growing initial errors and assumes the maximum, if no fault tolerance is provided, that is, with noiseless initial patterns, see Fig. 2. Autoassociative completion of a distorted x-pattern is evaluated by the completion capacity: C(a', {3') := Mn(t(p, a', {3')-t(p, a', {3'))/mn. A BAM maps and completes at the same time and should be therefore evaluated by the search capacity S := C + A. The asymptotic capacity of the Willshaw model is strikingly high: The completion capacity (for autoassociation) is C+ = In[2] /4, the mapping capacity (for heteroassociation with input noise) is A+ = In[2] /2 bit/syn [Pal91]' leading to a value for the search capacity of (3 In[2])/4 = 0.52 bit/syn. To estimate S for general retrieval procedures one can consider a recognition process of stored patterns in the whole space of sparse initial patterns; an initial pattern is "recognized", if it is invariant under a bidirectional retrieval cycle. The so-called recognition capacity of this process is an upper bound of the completion capacity and it had been determined as In [2J/2, see [PS92]. This is achieved again with parameters M, p, q providing A = In[2] /2 yielding In[2] bit/syn as upper bound of the asymptotic search capacity. In summary, we know about the asymptotic search capacity of the CB-model: 0.52 ::; S+ ::; 0.69 bit/syn. For experimental results, see Fig. 4. Bidirectional Retrieval from Associative Memory 679 4 EXPERIMENTAL RESULTS The CB model has been tested in simulations and compared with the Willshaw model (simple retrieval) for addresses with random noise (Fig. 2) and for addresses composed by two learning patterns (Fig. 3). In Fig. 2 the widely enlarged range of high qualtity retrieval in the CB-model is demonstrated for different system sizes. output miss errors 6 10 7 7 5 simple r. .. ". : 8 6 6 4 CB·r. .. :" 5 5 6 4 4 3 ....... 4 .' , .. ~ '" 3 3 2 .' , .. " ... 2 2 1 2 ::' 1 0 0 0 0 5 10 15 20 25 30 5 10 15 20 25 30 101214161820 101214161820 output add errors 6 r--..--.--,--.----.---, 14 rcr-..---,----.~----, 5 simple r. " ." 12 CB·r.4 10 3 /" 8 2 6 ',' 4 1 2 10 10 .,.,.; , • .-:: .... os·,· 8 8 6 6 .:'1 ...... · 4 4 /", 2 2 o ~'--'--'--'--1----'-""'-'-----' 0 L-.C..:....I..!.-...I.-_~=:::::::..J 0 0 L..:....L-I.. ........ "--I 5 10 15 20 25 30 5 1015202530 101214161820 101214161820 transinformation in output pattern (bit) 50 ;c;::r;:::::r:::=r::=7---"J 1 00 1 00 r-r--,-,--,--,1 00 r-r--.-.-T'"""1 45 L ~ ~ ~ ~ 35 30 60 25 ...... . 60 40 20 ~ ~ 15 " . 1 0 simple r. ..". 20 20 5 ·CB-r. o '---''---'----1---'----'---' 5 10 15 20 25 30 5 10 15 20 25 30 Fig. 2: Retrieval from addresses with random noise. The x-axis labeling is as in Fig. 1. Small system with n = 100, M = 35 (left), system size as in Fig. 1, two trials (right). Output activities adjusted near Iyl = k by threshold setting. o 0 L-.I.---L.--1-.I..--J 101214161820 101214161820 Fig. 3: Retrieval from addresses composed by two learning patterns. Parameters as in right column of Fig. 2, explanation of left and right column, see text. In Fig. 3 the address contains one learning pattern and I-components of a second learning pattern successively added with increasing abscissae. On the right end of each diagram both patterns are completely superimposed. Diagrams in the left column show errors and transinformation, if retrieval results are compared with the learning pattern which is for li~ I < 20 dominantly addressed. Simple retrieval errors behave similiar as for random noise in the address (Fig. 2) while the error level of CB-retrieval raises faster if more than 7 adds from the second pattern are present. Diagrams in the right column show the same quantities, if the retrieval result is compared with the closest of the two learning patterns. It can be observed i) that a learning pattern is retrieved even if the address is a complete superposition and ii) if the second pattern is almost complete in the address the retrieved pattern corresponds in some cases to the second pattern. However, in all cases CBretrieval yields one of the learning pattern pairs and it could be used to generate a good address for further retrieval of the other by deletion of the corresponding I-components in the original address. 680 0.48 0.46 0.44 0.42 0.4 0.38 . output c. ..... . .... searchc. 8 10 12 14 16 18 F. T. Sommer and G. Palm Fig. 4: Output and search capacity of CB retrieval in bit/syn with x-axis labeling as in Fig. 2 for n = m = 2000, a = b = 10 M = 20000. The difference between both curves is the contribution due to x-pattern completion, the completion capacity C. It is zero for Ix(O}1 = 10, if the initial pattern is errorfree. The search capacity of the CB model in Fig. 4 is close to the theoretical expectations from Sect. 3, increasing with input noise due to the address completion. 5 SPARSE CODING To apply the proposed N AM model, for instance, in information retrieval, a coding of the data to be accessed into sparse binary patterns is required. A useful extraction of sparse features should take account of statistical data properties and the way the user is acting on them. There is evidence from cognitive psychology that such a coding is typically quite easy to find. The feature encoding, where a person is extracting feature sets to characterize complex situations by a few present features, is one of the three basic classes of cognitive processes defined by Sternberg [Ste77]. Similarities in the data are represented by feature patterns having a large number of present features in common, that is a high overlap: o(x, x'} := L:i XiX'i' For text retrieval word fragments used in existing indexing techniques can be directly taken as sparse binary features [Geb87]. For image processing sparse coding strategies [Zet90], and neural models for sparse feature extraction by anti-Hebbian learning [F6l90] have been proposed. Sparse patterns extracted from different data channels in heterogeneous data can simply be concatenated and processed simultaneously in N AM. If parts of the original data should be held in a conventional memory, also these addresses have to be represented by distributed and sparse patterns in order to exploit the high performance of the proposed NAM. 6 CONCLUSION A new bidirectional retrieval method (CB-retrieval) has been presented for the Willshaw neural associative memory model. Our analysis of the first CB-retrieval step indicates a high potential for error reduction and increased input fault tolerance. The asymptotic capacity for bidirectional retrieval in the binary Willshaw matrix has been determined between 0.52 and 0.69 bit/syn. In experiments CB-retrieval showed significantly increased input fault tolerance with respect to the standard model leading to a practical information capacity in the order of the theoretical expectations (0.5 bit/syn). Also the segmentation ability of CB-retrieval with ambiguous addresses has been shown. Even at high memory load such input patterns can be decomposed and corresponding memory entries returned individually. The model improvement does not require sophisticated individual threshold setting [GW95], strategies proposed for BAM like more complex learning procedures, or "dummy augmentation" in the pattern coding [WCM90, LCL95]. The demonstrated performance of the CB-model encourages applications as massively parallel search strategies in Information Retrieval. The sparse coding requirement has been briefly discussed regarding technical strategies and psychological plausibility. Biologically plausible variants of CB-retrieval contribute to more Bidirectional Retrieval from Associative Memory 681 refined cell assembly theories, see [SWP98]. Acknowledgement: One of the authors (F.T.S.) was supported by grant S0352/3-1 of the Deutsche Forschungsgemeinschaft. References [F6190] [Geb87] [GM76] [GR92] [GW95] [Hop82] [Kos87] P. F6ldiak. Forming sparse representations by local anti-hebbian learning. Biol. Cybern., 64:165-170, 1990. F. Gebhardt. Text signatures by superimposed coding of letter triplets and quadruplets. Information Systems, 12(2):151-156, 1987. A.R. Gardner-Medwin. The recall of events through the learning of associations between their parts. Proceedings of the Royal Society of London B, 194:375-402, 1976. W.G. Gibson and J. Robinson. Statistical analysis of the dynamics of a sparse associative memory. Neural Networks, 5:645-662, 1992. B. Graham and D. Willshaw. Improving recall from an associative memory. Biological Cybernetics, 72:337-346, 1995. J.J. Hopfield. Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences, USA, 79, 1982. B. Kosko. Adaptive bidirectional associative memories. Applied Optics, 26(23):4947-4971, 1987. [LCL95] C.-S. Leung, L.-W. Chan, and E. Lai. Stability, capacity and statistical dynamics of second-order bidirectional associative memory. IEEE Trans. Syst, Man Cybern., 25(10):1414-1424, 1995. [Pal91] G. Palm. Memory Capacities of Local Rules for Synaptic Modification. Concepts in Neuroscience, 2:97-128, 1991. [PS92] G. Palm and F. T. Sommer. Information capacity in recurrent McCulloch-Pitts networks with sparsely coded memory states. Network, 3:1-10, 1992. [SP97] F. T. Sommer and G. Palm. Improved bidirectional retrieval of sparse patterns stored by Hebbian learning. Submitted to Neural Networks, 1997. [SSP96] F. Schwenker, F. T. Sommer, and G. Palm. Iterative retrieval of sparsely coded associative memory patterns. Neural Networks, 9(3):445 - 455, 1996. [Ste61] K. Steinbuch. Die Lernmatrix. Kybernetik, 1:36-45, 1961. [Ste77] R. J. Sternberg. Intelligence, information processing and analogical reasoning. Hillsdale, NJ, 1977. [SWP98] F. T. Sommer, T. Wennekers, and G. Palm. Bidirectional completion of Cell Assemblies in the cortex. In Computational Neuroscience: Trends in Research. Plenum Press, 1998. [WBLH69] D. J. Willshaw, O. P. Buneman, and H. C. Longuet-Higgins. Nonholographic associative memory. Nature, 222:960-962, 1969. [WCM90] Y. F. Wang, J. B. Cruz, and J. H. Mulligan. Two coding stragegies for bidirectional associative memory. IEEE Trans. Neural Networks, 1(1):81-92, 1990. [Zet90] C. Zetsche. Sparse coding: the link between low level vision and associative memory. In R. Eckmiller, G. Hartmann, and G. Hauske, editors, Parallel Processing in Neural Systems a.nd Computers. Elsevier Science Publishers B. V. (North Holland), 1990.
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Using Helmholtz Machines to analyze multi-channel neuronal recordings Virginia R. de Sa desa@phy.ucsf.edu R. Christopher deC harms decharms@phy.ucsf.edu Michael M. Merzenich merz@phy.ucsf.edu Sloan Center for Theoretical Neurobiology and W. M. Keck Center for Integrative Neuroscience University of California, San Francisco CA 94143 Abstract One of the current challenges to understanding neural information processing in biological systems is to decipher the "code" carried by large populations of neurons acting in parallel. We present an algorithm for automated discovery of stochastic firing patterns in large ensembles of neurons. The algorithm, from the "Helmholtz Machine" family, attempts to predict the observed spike patterns in the data. The model consists of an observable layer which is directly activated by the input spike patterns, and hidden units that are activated through ascending connections from the input layer. The hidden unit activity can be propagated down to the observable layer to create a prediction of the data pattern that produced it. Hidden units are added incrementally and their weights are adjusted to improve the fit between the predictions and data, that is, to increase a bound on the probability of the data given the model. This greedy strategy is not globally optimal but is computationally tractable for large populations of neurons. We show benchmark data on artificially constructed spike trains and promising early results on neurophysiological data collected from our chronic multi-electrode cortical implant. 1 Introduction Understanding neural processing will ultimately require observing the response patterns and interactions of large populations of neurons. While many studies have demonstrated that neurons can show significant pairwise interactions, and that these pairwise interactions can code stimulus information [Gray et aI., 1989, Meister et aI., 1995, deCharms and Merzenich, 1996, Vaadia et al., 1995], there is currently little understanding of how large ensembles of neurons might function together to represent stimuli. This situation has arisen partly out of the historical 132 V R. d. Sa, R. C. deCharms and M. M. Merzenich difficulty of recording from large numbers of neurons simultaneously. Now that this is becoming technically feasible, the remaining analytical challenge is to understand how to decipher the information carried in distributed neuronal responses. Extracting information from the firing patterns in large neuronal populations is difficult largely due to the combinatorial complexity of the problem, and the uncertainty about how information may be encoded. There have been several attempts to look for higher order correlations [Martignon et al., 1997] or decipher the activity from multiple neurons, but existing methods are limited in the type of patterns they can extract assuming absolute reliability of spikes within temporal patterns of small numbers of neurons [Abeles, 1982, Abeles and Gerstein, 1988, Abeles et al., 1993, Schnitzer and Meister,] or considering only rate codes [Gat and Tishby, 1993, Abeles et al., 1995]. Given the large numbers of neurons involved in coding sensory events and the high variability of cortical action potentials, we suspect that meaningful ensemble coding events may be statistically similar from instance to instance while not being identical. Searching for these type of stochastic patterns is a more challenging task. One way to extract the structure in a pattern dataset is to construct a generative model that produces representative data from hidden stochastic variables. Helmholtz machines [Hinton et al., 1995, Dayan et al., 1995] efficiently [Frey et al., 1996] produce generative models of datasets by maximizing a lower bound on the log likelihood of the data. Cascaded Redundancy Reduction [de Sa and Hinton, 1998] is a particularly simple form of Helmholtz machine in which hidden units are incrementally added. As each unit is added, it greedily attempts to best model the data using all the previous units. In this paper we describe how to apply the Cascaded Redundancy Reduction algorithm to the problem of finding patterns in neuronal ensemble data, test the performance of this method on artificial data, and apply the method to example neuronal spike trains. 1.1 Cascaded Redundancy Reduction The simplest form of generative model is to model each observed (or input) unit as a stochastic binary random variable with generative bias bi. This generative input is passed through a transfer function to give a probability of firing. 1 p. = a(b·) = (1) ~ ~ 1 + e- bi While this can model the individual firing rates of binary units, it cannot account for correlations in firing between units. Correlations can be modeled by introducing hidden units with generative weights to the correlated observed units. By cascading hidden units as in Figure 1, we can represent higher order correlations. Lower units sum up their total generative input from higher units and their generative bias. Xi = bi + L Sj9j,i j>i (2) Finding the optimal generative weights (9j,i, bi) for a given dataset involves an intractable search through an exponential number of possible states of the hidden units. Helmholtz machines approximate this problem by using forward recognition connections to compute an approximate distribution over hidden states for each data pattern. Cascaded Redundancy Reduction takes this approximation one step further by approximating the distribution by a single state. This makes the search for recognition and generative weights much simpler. Given a data vector, d, considering the state produced by the recognition connections as Sd gives a lower bound on the log Using Helmholtz Machines to Analyze Multi-channel Neuronal Recordings -:.> generative connections ~ recognition connections T i,k TO,k 133 Figure 1: The Cascaded Redundancy Reduction Network. Hidden units are added incrementally to help better model the data. likelihood of the data. Units are added incrementally with the goal of maximizing this lower bound, C, C = 'L[(s% log a(bk ) + (l-s%) log(l-a(bk))+ L sf log a(xf)+ (I-sf) log(l-a(xf))] d i (3) Before a unit is added it is considered as a temporary addition. Once its weights have been learned, it is added to the permanent network only if adding it reduces the cost on an independent validation set from the same data distribution. This is to prevent overtraining. While a unit is considered for addition, all weights other than those to and from the new unit and the generative bias weights are fixed. The learning of the weights to and from the new unit is then a fairly simple optimization problem involving treating the unit as stochastic, and performing gradient descent on the resulting modified lower bound. 2 Method This generic pattern finding algorithm can be applied to multi-unit spike trains by treating time as another spatial dimension as is often done for time series data. The spikes are binned on the order of a few to tens of milliseconds and the algorithm looks for patterns in finite time length windows by sliding a window centered on each spike from a chosen trigger channel. An example extracted window using channel 4 as the trigger channel is shown in Figure 2. Because the number of spikes can be larger than one, the observed units (bins) are modeled as discrete Poisson random variables rather than binary random variables (the hidden units are still kept as binary units). To reflect the constraint that the expected number of spikes cannot be negative but may be larger than one, the transfer function for these observed bins was chosen to be exponential. Thus if Xi is the total summed generative input, Ai, the expected mean number of spikes in bin i, is calculated as eX; and the probability of finding s spikes in that bin is given by s! (4) 134 V.R. d. Sa, R. C. deChanns and M. M. Merzenich Figure 2: The input patterns for the algorithm are windows from the full spatiatemporal firing patterns. The full dataset is windows centered about every spike in the trigger channel. The terms in the lower bound objective function due to the observed bins are modified accordingly. 3 Experimental Results Before applying the algorithm to real neural spike trains we have characterized its properties under controlled conditions. We constructed sample data containing two random patterns across 10 units spanning 100 msec. The patterns were stochastic such that each neuron had a probability of firing in each time bin of the pattern. Sample patterns were drawn from the stochastic pattern templates and embedded in other "noise" spikes. The sample pattern templates are shown in the first column of Figure 3. 300 seconds of independent training, validation and test data were generated. All results are reported on the test data . After training the network, performance was assessed by stepping through the test data and observing the pattern of activation across the hidden units obtained from propagating activity through the forward (recognition) connections and their corresponding generative pattern {Ad obtained from the generative connections from the binary hidden unit pattern. Typically, many of the theoretically possible 2n hidden unit patterns do not occur. Of the ones that do, several may code for the noise background. A crucial issue for interpreting patterns in real neural data is to discover which of the hidden unit activity patterns correspond to actual meaningful patterns. We use a measure that calculates the quality of the match of the observed pattern and the generative pattern it invokes. As the algorithm was not trained on the test data, close matches between the generative pattern and the observed pattern imply real structure that is common to the training and test dataset. With real neural data, this question can also be addressed by correlating the occurrence of patterns to stimuli or behavioural states of the animal. One match measure we have used to pick out temporally modulated structure is the cost of coding the observed units using the hidden unit pattern compared to the cost of using the optimal rate code for that pattern (derived by calculating the firing rate for each channel in the window excluding the trigger bin). Match values were calculated for each hidden unit pattern by averaging the results across all its contributing observed patterns. Typical generative patterns of the added template patterns (in noise) are shown in the second column of Figure 3. The third column in the figure shows example matches from the test set, (Le. patterns that activated the hidden unit pattern corresponding to the generative pattern in column 2). Note that the instances of the patterns are missing some spikes present in the template, and are surrounded by many additional spikes. Using Helmholtz Machines to Analyze Multi-channel Neuronal Recordings 135 Template Pattern 1 Pattern 2 Generative Pattern Test set Example Figure 3: Pattern templates, resulting generative patterns after training (showing the expected number of spikes the algorithm predicts for each bin), and example test set occurrences. The size and shade of the squares represents the probability of activation of that bin (or 0/1 for the actual occurrences), the colorbars go from 0 to 1. We varied both the frequency of the pattern occurrences and that of the added background spikes. Performance as a function of the frequency of the background spikes is shown on the left in Figure 4 for a pattern frequency of .4 Hz. Performance as a function of the pattern frequency for a noise spike frequency of 15Hz is shown on the right of the Figure. False alarm rates were extremely low ranging from 0-4% across all the tested conditions. Also, importantly, when we ran three trials with no added patterns"no patterns were detected by the algorithm. ~! .... , .118, .~1 0(.88, .95, .97) i x(.73, .94, .94) e0.8 "i ~ s:: '" ~ 0.6 .. 0(0, .73, .82) E CD 0(0, .45, .9) .. co 0.4 Q. '0 .~ I, pattem 1 ~ I " pattem 2 ~ 0.2 shifted Pattem 1 x(O, 0, .51) Q. 0 14 15 16 17 18 19 20 21 average firing 'ale 01 back ground spikas (Hz) i e"i N "2 '" ~ .. E CD .. co Q. '0 .~ " ~ Q. x(.73, .94, .94) 0.8 0.6 .(0, .83, .90) 0.4 0(0, 0, .87) 0.2 x(O, 0, .32) OL--L __ ~ __ ~-L __ ~ __ ~~ __ ~ 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 frequency 01 pattem occurrence (Hz) Figure 4: Graphs showing the effect of adding more background spikes (left) and decreasing the number of pattern occurrences in the dataset (right) on the percentage of patterns correctly detected. The detection of shifted pattern is due to the presence of a second spike in channel 4 in the pattern (hits for this case are only calculated for the times when this spike was present - the others would all be missed). In fact in some cases the presence of the only slightly probable 3rd bin in channel 4 was enough to detect another shifted pattern 1. Means over 3 trials are plotted with the individual trial values given in braces The algorithm was then applied to recordings made from a chronic array of extracellular microelectrodes placed in the primary auditory cortex of one adult marmoset monkey and one adult owl monkey [deC harms and Merzenich, 1998]. On some elec136 V.R. d. Sa, R. C. deCharms and M. M. Merzenich Figure 5: Data examples (all but top left) from neural recordings in an awake marmoset monkey that invoke the same generative pattern (top left). The instances are patterns from the test data that activated the same hidden unit activity pattern resulting in the generative pattern in the top left. The data windows were centered around all the spikes in channel 4. The brightest bins in the generative pattern represent an expected number of spikes of 1. 7. In the actual patterns, The darkest and smallest bins represent a bin with 1 spike; each discrete grayscale/size jump represents an additional spike. Each subfigure is indiv!dually normalized to the bin with the most spikes. trodes spikes were isolated from individual neurons; others were derived from small clusters of nearby neurons. Figure 5 shows an example generative pattern (accounting for 2.8% of the test data) that had a high match value together with example occurrences in the test data. The data were responses recorded to vocalizations played to the marmoset monkey, channel 4 was used as the trigger channel and 7 hidden units were added. 4 Discussion We have introduced a procedure for searching for structure in multineuron spike trains, and particularly for searching for statistically reproducible stochastic temporal events among ensembles of neurons. We believe this method has great promise for exploring the important question of ensemble coding in many neuronal systems, a crucial part of the problem of understanding neural information coding. The strengths of this method include the ability to deal with stochastic patterns, the search for any type of reproducible structure including the extraction of patterns of unsuspected nature, and its efficient, greedy, search mechanism that allows it to be applied to large numbers of neurons. Acknowledgements We would like to acknowledge Geoff Hinton for useful suggestions in the early stages of this work, David MacKay for helpful comments on an earlier version of the manuscript, and the Sloan Foundation for financial support. Using Helmholtz Machines to Analyze Multi-channel Neuronal Recordings 137 References [Abeles, 1982] Abeles, M. (1982). Local Cortical Circuits An Electrophysiological Study, volume 6 of Studies of Brain Function. Springer-Verlag. [Abeles et al., 1995] Abeles, M., Bergman, H., Gat, I., Meilijson, I., Seidemann, E., Tishby, N., and Vaadia, E. (1995). Cortical activity flips among quasi-stationary states. Proceedings of the National Academy of Science, 92:8616- 8620. [Abeles et al., 1993] Abeles, M., Bergman, H., Margalit, E., and Vaadia, E. (1993). Spatiotemporal firing patterns in the frontal cortex of behaving monkeys. Journal of Neurophysiology, 70(4):1629-1638. [Abeles and Gerstein, 1988] Abeles, M. and Gerstein, G. L. (1988). Detecting spatiotemporal firing patterns among simultaneously recorded single neurons. Journal of Neurophysiology, 60(3). [Dayan et al., 1995] Dayan, P., Hinton, G. E., Neal, R. M., and Zemel, R. S. (1995). The helmholtz machine. Neural Computation, 7:889-904. [de Sa and Hinton, 1998] de Sa, V. R. and Hinton, G. E. (1998). Cascaded redundancy reduction. to appear in Network{February). [deC harms and Merzenich, 1996] deCharms, R. C. and Merzenich, M. M. (1996). Primary cortical representation of sounds by the coordination of action-potential timing. Nature, 381:610-613. [deCharms and Merzenich, 1998] deCharms, R. C. and Merzenich, M. M. (1998). Characterizing neurons in the primary auditory cortex of the awake primate using reverse correlation. this volume. [Frey et al., 1996] Frey, B. J., Hinton, G. E., and Dayan, P. (1996). Does the wakesleep algorithm produce good density estimators? In Touretzky, D., Mozer, M., and Hasselmo, M., editors, Advances in Neural Information Processing Systems 8, pages 661-667. MIT Press. [Gat and Tishby, 1993] Gat, I. and Tishby, N. (1993). Statistical modeling of cellassemblies activities in associative cortex of behaving monkeys. In Hanson, S., Cowan, J., and Giles, C., editors, Advances in Neural Information Processing Systems 5, pages 945-952. Morgan Kaufmann. [Grayet al., 1989] Gray, C. M., Konig, P., Engel, A. K., and Singer, W. (1989). Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature, 338:334-337. [Hinton et al., 1995] Hinton, G. E., Dayan, P., Frey, B. J., and Neal, R. M. (1995). The wake-sleep algorithm for unsupervised neural networks. Science, 268:1158116l. [Martignon et al., 1997] Martignon, L., Laskey, K., Deco, G., and Vaadia, E. (1997). Learning exact patterns of quasi-synchronization among spiking neurons from data on multi-unit recordings. In Mozer, M., Jordan, M., and Petsche, T., editors, Advances in Neural Information Processing Systems 9, pages 76-82. MIT Press. [Meister et al., 1995] Meister, M., Lagnado, L., and Baylor, D. (1995). Concerted signaling by retinal ganglion cells. Science, 270:95-106. [Schnitzer and Meister,] Schnitzer, M. J. and Meister, M. Information theoretic identification of neural firing patterns from multi-electrode recordings. in preparation. [Vaadia et al., 1995] Vaadia, E., Haalman, I., Abeles, M., Bergman, H., Prut, Y., Slovin, H., and Aertsen, A. (1995). Dynamics of neuronal interactions in monkey cortex in relation to behavioural events. Nature, 373:515-518.
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Silicon Retina with Adaptive Filtering Properties Shih-Chii Liu Computation and Neural Systems 136-93 California Institute of Technology Pasadena, CA 91125 shih@pcmp.caltech.edu Abstract This paper describes a small, compact circuit that captures the temporal and adaptation properties both of the photoreceptor and of the laminar layers of the fly. This circuit uses only six transistors and two capacitors. It is operated in the subthreshold domain. The circuit maintains a high transient gain by using adaptation to the background intensity as a form of gain control. The adaptation time constant of the circuit can be controlled via an external bias. Its temporal filtering properties change with the background intensity or signal-to-noise conditions. The frequency response of the circuit shows that in the frequency range of 1 to 100 Hz, the circuit response goes from highpass filtering under high light levels to lowpass filtering under low light levels (Le., when the signal-tonoise ratio is low). A chip with 20x20 pixels has been fabricated in 1.2J.Lm ORBIT CMOS nwell technology. 1 BACKGROUND The first two layers in the fly visual system are the retina layer and the laminar layer. The photo receptors in the retina synapse onto the monopolar cells in the laminar layer. The photoreceptors adapt to the background intensity, and use this adaptation as a form of gain control in maintaining a high response to transient signals. The laminar layer performs bandpass filtering under high background intensities, and reverts to low pass filtering in the case of low background intensities where the signal-to-noise (SIN) ratio is low. This adaptive filtering response in the temporal domain is analogous to the spatial center-surround response of the bipolar cells in the vertebrate retina. Silicon Retina with Acklptive Filtering Properties 713 Figure 1: Circuit diagram of retino-Iaminar circuit. The feedback consists of a resistor implemented by a pFET transistor, Q1. The conductance of the resistor is controlled by the external bias, Vm . ga Vr VI iin CdI CI ~ Figure 2: Small signal model of the circuit shown in Figure 1. Cr is the parasitic capacitance at the node, Vr . The Delbriick silicon receptor circuit (Delbriick, 1994) modeled closely the step responses and the adaptation responses of the biological receptors. By including two additional transistors, the retino-Iaminar (RL) circuit described here captures the properties of both the photoreceptor layer (Le., the adaptation properties and phototransduction) and the cells in the laminar layer (Le., the adaptive filtering). The time constant of the circuit is controllable via an external bias, and the adaptation behavior of the circuit over different background intensities is more symmetrical than that of Delbriick's photoreceptor circuit. 2 CIRCUIT DESCRIPTION The RL circuit which has the basic form of Delbriick's receptor circuit is shown in Figure 1. I have replaced the adaptive element in his receptor circuit by a nonlinear resistor consisting of a pFET transistor, Q1. The implementation of a floating, voltage-controlled resistor has been described earlier by (Banu and Tsividis, 1982). The bias for Q1, Vb, is generated by Q3 and Q4. The conductance of Q1 is determined by the output voltage, Vi, and an external bias, Vm . We give a brief description of the circuit operation here; details are described in (Delbriick, 1994). The receptor node, Vr , is clamped to the voltage needed to sink the current sourced 714 s-c. Liu Frequency (Hz) Figure 3: Frequency plot of the RL circuit over five decades of background intensity. The number next to each curve corresponds to the log intensity of the mean value; a log corresponds to the intensity of a red LED. The plot shows that, in the range of 1 to 100 Hz, the circuit is a bandpass filter at high light levels, and reduces to a lowpass filter at low light levels. by Q6, which is biased by an external voltage, Vu. Changes in the photo current are amplified by the transistors, Q2 and Q6, resulting in a change in Vi, This change in Vi is capacitively coupled through the capacitive divider, consisting of Cl and Cd, into Vp1 , so that Qs supplies the extra increase in photocurrent. The feedback transistor, Qs, is operated in subthreshold so that v;. and Vi is logarithmic in the photocurrent. A large change in the photocurrent resulting from a change in the background intensity, leads to a large change in the circuit output, Vi. Significant current then flows through Ql, thus charging or discharging Vp1 . 3 PROPERTIES OF RL CIRCUIT The temporal responses and adaptation properties of this circuit are expounded in the following sections. In Section 3.1, we solve for the transfer function of the circuit and in Section 3.2, we describe the dependence of the conductance of Ql on the background intensity. In Sections 3.3 and 3.4, we describe the temporal responses of this circuit, and compare the adaptation response of RL circuit with that of Delbriick's circuit. 3.1 TRANSFER FUNCTION We can solve for the transfer function of the RL circuit in Figure 1 by writing the KCL equations of the small-signal model shown in Figure 2. The transfer function, ~, is given by: '.n (1) -=-lin grnS Silicon Retina with Aooptive Filtering Properties 17 .. I 2 0~--::1l'::2 ----::0':.4 ----::0:';.6 ----::0:'::.8 ---! Time (sec) ~ .~ B .: .. . ~ j 715 Figure 4: Temporal responses of the circuit over five decades of background intensity. The input stimulus is a square-wave-modulated red LED of contrast 0.15. The circuit acts as a high pass filter (that is, a differentiator) at high intensities, and as a lowpass filter as the intensity drops. where Aamp = 9~2, gm is the transconductance, and gd is the output conductance of a transistor. We define the time constants, TI, TT' and Tid, as follows: Cl Cr Cd TI = -;Tr = --;Tld = --, gm2 gm5 gm2 where ga is the output conductance of Ql and Cr is the parasitic capacitance at the node, Yr' The frequency response curves in Figure 3 are measured from the fabricated circuit over five decades of background intensity. We obtain the curves by using a sinewave-modulated red LED source. The number next to each curve is the log intensity of the mean value; 0 log is the intensity of a red LED. We obtain the remaining curves by interposing neutral density filters between the LED source and the chip. Figure 3 shows that, in the range of 1 to 100 Hz, the circuit is a bandpass filter at high light levels, and reduces to a lowpass filter at low light levels. For each frequency curve, the gain is flat in the middle, and is given by Ad = CI~~d. The cutoff frequencies change with the background intensity; this change is analyzed in Section 3.2. 3.2 DEPENDENCE OF CIRCUIT'S TIME CONSTANT ON BACKGROUND INTENSITY The cutoff frequencies of the circuit depend on the conductance, ga, of Ql . Here, we analyze the dependence of ga on the background intensity. Since Ql is operated in subthreshold, the conductance depends on the current flowing through Ql' The I-V relationship for Ql can be written as I = 2IoP(fh r p e(l-Itnltp)y e-Itnltp t!.V/2 sinh (.6Yj2) on = lal~~-Itnltp )/ltn e(1-2ltn lt p)t!. V/2 sinh(~ Vj2) (2) (3) where V "l~VPI, ~V = Vi Vp1 , lph is the photo current , and fa = 2Iop (-f!: t p ((: ) (l-Itnltp)/ltn . The exponential relationship for Equations 2 and 3 is for a FET transistor operating in subthreshold, where lop is the quiescent leakage current of the transistor, and K. is the effectiveness of the gate in controlling 716 1.8 ~ "0 ~ :; 16 9-5 ti ... c . ~ b 1.2 C . ." ~ s-c. Liu Time (sec) Figure 5: Plots of adaptation responses of the RL circuit and of Delbriick's circuit. The input stimulus is a red LED driven by a square wave of contrast 0.18. The bottom curve corresponding to Delbriick's receptor has been shifted down so that we can compare the two curves. The adaptation response of the RL circuit is more symmetrical than that of Delbriick's circuit when the circuit goes from dark to light conditions and back. the surface potential of the channel of the transistor. Equation 3 shows that ga is proportional to the photocurrent, I ph , hence, the background intensity. A more intuitive way of understanding how ga changes with Iph is that the change in Vb with a fixed change in the output, Vi, depends on the output level of Vi. The change in Vb is larger for a higher DC output, Vi, because of the increased body effect at Q4 due to its higher source voltage. The larger change in Vb leads to an increase in the conductance, gao As Iph increases, ga increases, so the cutoff frequencies shift to the right, as seen in Figure 3. If we compare both the "0" curve and the "-1" curve, we can see that the cutoff frequencies are approximately different by a factor of 10. Thus, the exponent of Iph , (1- /'i,n/'i,p)//'i,n ::::: 1. Since the /'i, values change with the current through the transistor, the exponent also changes. The different values of the exponent with Iph can be seen from the different amounts of shifts in the cutoff frequencies of the curves. 3.3 TEMPORAL RESPONSES The adaptive temporal filtering of the circuit over five decades of background intensity can also be observed from the step response of the RL circuit to a squarewave-modulated LED of contrast 0.15, as shown in Figure 4. The data in Figure 4 show that the time constant of the circuit increases as the light level decreases. The temporal responses observed in these circuits are comparable to the contrast responses recorded from the LMCs by Juusola and colleagues (Juusola et aI., 1995). 3.4 ADAPTATION PROPERTIES The RL circuit also differs from Delbriick's circuit in that the adaptation time constant can be set by an external bias. In the Delbriick circuit, the adaptation time constant is predetermined at the design phase and by process parameters. In Figure 5, we compare the adaptation properties of the RL circuit with those of Delbriick's Silicon Retina with Atklptive Filtering Properties 717 V m..o.7l. 0.71. 0.8, 0.14. 0 .• 7. 0.9 '" 1.45 0 01 0.2 0 3 04 0.5 0.6 Time <sec) Figure 6: Step response of the RL circuit for different values of Vm . The input stimulus is a square-wave-modulated red LED source. The value of Vm was varied from 0.73 to 0.9 V. The curve with the longest time constant of decay corresponds to the lowest value of Vm . circuit. The input stimulus consists of a square-wave-modulated LED source with a contrast of about 0.18. We take the circuit from dark to light conditions, and back again, by using neutral density filters. The top curve corresponds to the response from the RL circuit, and the bottom curve corresponds to the response from the Delbriick circuit. The RL circuit adapts symmetrically, when it goes from light to dark conditions and back. In contrast, Delbriick's circuit shows an asymmetrical adaptative behavior; it adapts more slowly when it goes from dark to light conditions. The adaptation time constant of the RL circuit depends on the conductance, ga, and the capacitors, CI and Cd. From Equation 3, we see that ga is dependent on h which is set by the bias, Vm . Hence, we can change the adaptation time constant by varying Vm . The dependence of the time constant on Vm is further demonstrated by recording the step response of the circuit to a LED source of contrast 0.15 for various values of Vm . The output data is shown in Figure 6 for five different values of Vm . The time constant of the circuit decreases as Vm increases. A chip consisting of 20x20 pixels was fabricated in 1.2l-tm ORBIT CMOS nwell technology. An input stimulus consisting of a rotating flywheel, with black strips on a white background, was initially presented to the imager. The flywheel was then stopped, and the response of the chip was recorded one sec after the motion was ceased. I repeated the experiment for two adaptation time constants by changing the value of Vm . Figure 7a shows the output of the chip with the longer adaptation time constant. We see that the image is still present, whereas the image in Figure 7b has almost faded away; that is, the chip has adapted away the stationary image. 4 CONCLUSIONS I have described a small circuit captures the temporal and adaptation properties of both the photoreceptor and the laminar layers in the fly retina. By adapting to the background intensity, the circuit maintains a high transient gain. The temporal behavior of the circuit also changes with the background intensity, such that, at high SIN ratios, the circuit acts as a highpass filter and, at low SIN ratios, the circuit acts as a lowpass filter to average out the noise. The circuit uses only six transistors and two capacitors and is compact. The adaptation time constant of the 718 soc. Liu (a) (b) Figure 7: Adaptation results from a two-dimensional array of 20 x 20 pixels. The output of the array was recorded one sec after cessation of the pattern motion. The experiment was repeated for two different adaptation time constants. Figure (a) corresponds to the longer adaptation time constant. The image is still present, whereas the image in Figure (b) has almost faded away. circuit can be controlled via an external bias. Acknowledgments I thank Bradley A. Minch for discussions of this work, Carver Mead for supporting this work, and the MOSIS foundation for fabricating this circuit. I also thank Lyn Dupre for editing this document. This work was supported in part by the Office of Naval Research, by DARPA, and by the Beckman Foundation. References T. Delbriick, "Analog VLSI phototransduction by continous-time, adaptive, logarithmic photoreceptor circuits," eNS Memo No.SO, California Institute of Technology, Pasadena, CA, 1994. M. Banu and Y. Tsividis, "Floating voltage-controlled resistors in CMOS technology," Electronics Letters, 18:15, pp. 678-679, 1982. M. Juusola, R.O. Uusitola, and M. Weckstrom, " Transfer of graded potentials at the photoreceptor-interneuron synapse," J. of General Physiology, 105, pp. 115148,1995.
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Statistical Models of Conditioning Peter Dayan* Brain & Cognitive Sciences E25-2IDMIT Cambridge, MA 02139 Abstract Theresa Long 123 Hunting Cove Williamsburg, VA 23185 Conditioning experiments probe the ways that animals make predictions about rewards and punishments and use those predictions to control their behavior. One standard model of conditioning paradigms which involve many conditioned stimuli suggests that individual predictions should be added together. Various key results show that this model fails in some circumstances, and motivate an alternative model, in which there is attentional selection between different available stimuli. The new model is a form of mixture of experts, has a close relationship with some other existing psychological suggestions, and is statistically well-founded. 1 Introduction Classical and instrumental conditioning experiments study the way that animals learn about the causal texture of the world (Dickinson, 1980) and use this information to their advantage. Although it reached a high level of behavioral sophistication, conditioning has long since gone out of fashion as a paradigm for studying learning in animals, partly because of the philosophical stance of many practitioners, that the neurobiological implementation of learning is essentially irrelevant. However, more recently it has become possible to study how conditioning phenomena are affected by particular lesions or pharmacological treatments to the brain (eg Gallagher & Holland, 1994), and how particular systems, during simple learning tasks, report information that is consistent with models of conditioning (Gluck & Thompson, 1987; Gabriel & Moore, 1989). In particular, we have studied the involvement of the dopamine (DA) system in the ventral tegmental area of vertebrates in reward based learning (Montague et aI, 1996; Schultz et aI, 1997). The activity of these cells is consistent with a model in which they report a temporal difference (TO) based prediction error for reward "This work was funded by the Surdna Foundation. 118 P. Dayan and T. Long (Sutton & Barto, 1981; 1989). This prediction error signal can be used to learn correct predictions and also to learn appropriate actions (Barto, Sutton & Anderson, 1983). The DA system is important since it is crucially involved in normal reward learning, and also in the effects of drugs of addiction, self stimulation, and various neural diseases. The TO model is consistent with a whole body of experiments, and has even correctly anticipated new experimental findings. However, like the Rescorla-Wagner (RW; 1972) or delta rule, it embodies a particular additive model for the net prediction made when there are multiple stimuli. Various sophisticated conditioning experiments have challenged this model and found it wanting. The results support competitive rather than additive models. Although ad hoc suggestions have been made to repair the model, none has a sound basis in appropriate prediction. There is a well established statistical theory for competitive models, and it is this that we adopt. In this paper we review existing evidence and theories, show what constraints a new theory must satisfy, and suggest and demonstrate a credible candidate. Although it is based on behavioral data, it also has direct implications for our neural theory. 2 Data and Existing Models Table 1 describes some of the key paradigms in conditioning (Dickinson, 1980; Mackintosh, 1983). Although the collection of experiments may seem rather arcane (the standard notation is even more so), in fact it shows exactly the basis behind the key capacity of animals in the world to predict events of consequence. We will extract further biological constraints implied by these and other experiments in the discussion. In the table,l (light) and s (tone) are potential predictors (called conditioned stimuli or CSs), of a consequence, r, such a~ the delivery of a reward (called an unconditioned stimulus or US). Even though we use TO rules in practice, we discuss some of the abstract learning rules without much reference to the detailed time course of trials. The same considerations apply to TD. In Pavlovian conditioning, the light acquires a positive association with the reward in a way that can be reasonably well modeled by: ,6.Wl(t) = al(t)(r(t) - wl(t»l(t), (1) where let) E {O, I} represents the presence of the light in trial t (s(t) will similarly represent the presence of a tone), Wl(t) (we will often drop the index t) represents the strength of the expectation about the delivery of reward ret) in trial t if the light is also delivered, and al(t) is the learning rate. This is just the delta rule. It also captures well the probabilistic contingent nature of conditioning - for binary ret) E {O, I}, animals seem to assess il = P[r(t)ll(t) = 1J - P[r(t)ll(t) = OJ, and then only expect reward following the light (in the model, have WI > 0) if il > O. Pavlovian conditioning is easy to explain under a whole wealth of rules. The trouble comes in extending equation 1 to the case of multiple predictors (in this paper we consider just two). The other paradigms in table 1 probe different aspects of this. The one that is most puzzling is (perversely) called downwards unblocking (Holland, 1988). In a first set of trials, an association is established between the light and two presentations of reward separated by a few (u) seconds. In a second set, a tone is included with the light, but the second reward is dropped. The animal amasses less reward in conjunction with the tone. However, when presented with the tone Statistical Models of Conditioning 119 Name Set 1 Set 2 Test 1 Pavlovian I -t r l~ r 2 Overshadowing l+s-tr { I ~ r~ } 1 s~ r'i 3 Inhibitory { l-tr } Z+s-t· s~f 4 Blocking l-tr l+s-tr s~· 5 Upwards unblocking l-tr 1+ s -t rflur s~r .6 Downwards unblockinK I -t rflur l+s-tr s~ ±~ Table 1: Paradigms. Sets 1 and 2 are separate sets of learning trials, which are continued until convergence. Symbols land s indicate presentation of lights and tones as potential predictors. The 't+ in the test set indicates that the associations of the predictors are tested, prodUCing the listed results. In overshadowing, association with the reward can be divided between the light and the sound, indicated by r!. In some cases overshadowing favours one stimulus at the complete expense of the other; and at the end of very prolonged training, all effects of overshadowing can disappear. In blocking, the tone makes no prediction of r. In set 2 of inhibitory conditioning, the two types of trials are interleaved and the outcome is that the tone predicts the absence of reward. In upwards and downwards unblocking, the 6" indicates that the delivery of two rewards is separated by time u. For downwards unblocking, if u is small, then s is associated with the absence of r; if u is large, then s is associated with the presence of r. alone, the animal expects the presence rather than the absence of reward. On the face of it, this seems an insurmountable challenge to prediction-based theories. First we describe the existing theories, then we formalise some potential replacements. One theory (called a US-processing theory) is due to Rescorla & Wagner (RW; 1972), and, as pointed out by Sutton & Barto (1981), is just the delta rule. For RW, the animal constructs a net prediction: V(t) = wi(t)l(t) + ws{t)s{t) (2) for r(t), and then changes flWi(t) = Cti(t)(r(t) - V(t»l(t) (and similarly for ws(t» using the prediction error r(t) - V(t). Its foundation in the delta rule makes it computationally appropriate (Marr, 1982) as a method of making predictions. TD uses the same additive model in equation 2, but uses r(t) + V(t + 1) - V(t) as the prediction error. RW explains overshadowing, inhibitory conditioning, blocking, and upwards unblocking, but not downwards unblocking. In overshadowing, the terminal association between I and r is weaker if I and s are simultaneously trained - this is expected under RW since learning stops when V(t) = r{t), and W, and Ws will share the prediction. In inhibitory conditioning, the sound comes to predict the absence of r. The explanation of inhibitory conditioning is actually quite complicated (Konorski, 1967; Mackintosh, 1983); however RW provides the simple account that WI = r for the I -t r trials, forcing Ws = -r for the 1+ s -t . trials. In blocking, the prior association between I and r means that Wi = r in the second set of trials, leading to no learning for the tone (since V(t) - r(t) = 0). In upwards unblocking, Wi = r at the start of set 2. Therefore, r(t) - WI = r > 0, allowing Ws to share in the prediction. As described above, downwards unblocking is the key thorn in the side of RW. Since the TD rule combines the predictions from different stimuli in a similar way, 120 P. Dayan and T. Long it also fails to account properly for downwards unblocking. This is one reason why it is incorrect as a model of reward learning. The class of theories (called CS-processing theories) that is alternative to RW does not construct a net prediction V(t), but instead uses equation 1 for all the stimuli, only changing the learning rates O!l(t) and O!s(t) as a function of the conditioning history of the stimuli (eg Mackintosh, 1975; Pearce & Hall, 1980; Grossberg, 1982). A standard notion is that there is a competition between different stimuli for a limited capacity learning processor (Broadbent, 1958; Mackintosh, 1975; Pearce & Hall, 1980), translating into competition between the learning rates. In blocking, nothing unexpected happens in the second set of trials and equally, the tone does not predict anything novel. In either case as is set to '" 0 and so no learning happens. In these models, downwards unblocking now makes qualitative sense: the surprising consequences in set 2 can be enough to set as »0, but then learning according to equation 1 can make Ws > O. Whereas Mackintosh's (1975) and Pearce and Hall's (1980) models only consider competition between the stimuli for learning, Grossberg's (1982) model incorporates competition during representation, so the net prediction on a trial is affected by competitive interactions between the stimuli. In essence, our model provides a statistical formalisation of this insight. 3 New Models From the previous section, it would seem that we have to abandon the computational basis of the RW and TD models in terms of making collective predictions about the reward. The CS-processing models do not construct a net prediction of the reward, or say anything about how possibly conflicting information based on different stimuli should be integrated. This is a key flaw - doing anything other than well-founded prediction is likely to be maladaptive. Even quite successful pre-synaptic models, such as Grossberg (1982), do not justify their predictions. We now show that we can take a different, but still statistically-minded approach to combination in which we specify a parameterised probability distribution P[r(t)ls(t), l(t)] and perform a form of maximum likelihood (ML) inference, updating the parameters to maximise this probability over the samples. Consider three natural models of P[r(t)/s(t), l(t)]: Pa[r(t)ls(t),l(t)] N[w1l(t) + wss(t), (72] (3) P M[r(t)/s(t), l(t)] P J[r(t)/s(t), l(t)] 7l"1 (t)N[Wl' (72] + 7l"s(t)N[ws, (72] + 1i'(t).,v[w, r2] (4) N[WI7l"I(t)l(t) + wsnAt)s(t), (72] (5) where N[J.L, (72] is a normal distribution, with mean J.L and variance (72. In the latter two cases, 0 ::; 7l"1 (t) + 7l" s (t) ::; I, implementing a form of competition between the stimuli, and 7l".(t) = 0 if stimulus * is not presented. In equation 4, N[w, r2] captures the background expectation if neither the light nor the tone wins, and 1i'(t) = 1 - 7l"1(t) - n"s{t). We will show that the data argue against the first two and support the third of these models. Rescorla-Wagner: Pa[r(t)/s(t), l(t)] The RW rule is derived as ML inference based on equation 3. The only difference is the presence of the variance, (72. This is useful for capturing the partial reinforcement effect (see Mackintosh, 1983), in which if r(t) is corrupted by substantial noise (ie (72 »0), then learning to r is demonstrably slower. As we discussed above, Statistical Models of Conditioning 121 downwards unblocking suggests that animals are not using P G [r( t) Is( t), I (t)] as the basis for their predictions. Competitive mixture of experts: P M[r(t)ls(t), l(t)] PM[r(t)ls(t),l(t)] is recognisable as the generative distribution in a mixture of Gaussians model (Nowlan, 1991; Jacobs et ai, 1991b). Key in this model are the mixing proportions 7r1(t) and 7rs(t). Online variants of the E phase of the EM algorithm (Dempster et ai, 1977) compute posterior responsibilities as ql(t) + qs(t) + q(t) = 1, where ql(t) <X 7r1(t)e-(r(t)-w1I(t)),2/2(T2 (and similarly for the others), and then perform a partial M phase as L\wl(t) <X (r(t) - WI (t»ql(t) L\ws(t) <X (r(t) - ws(t»qs(t) (6) which has just the same character as the presynaptic rules (depending on how 7r1 (t) is calculated). As in the mixture of experts model, each expert (each stimulus here) that seeks to predict r(t) (ie each stimulus * for which q. (t) f; 0) has to predict the whole of r(t) by itself. This means that the model can capture downwards unblocking in the following way. The absence of the second r in the second set of trials forces 7r s (t) > 0, and, through equation 6, this in turn means that the tone will come to predict the presence of the first r. The time u between the rewards can be important because of temporal discounting. This means that there are sufficiently large values of u for which the inhibitory effect of the absence of the second reward will be dominated. Note also that the expected reward based on l(t) and s(t) is the sum (7) Although the net prediction given in equation 7 is indeed based on all the stimuli, it does not directly affect the course of learning. This means that the model has difficulty with inhibitory conditioning. The trouble with inhibitory conditioning is that the model cannot use Ws < 0 to counterbalance WI > 0 - it can at best set Ws = 0, which is experimentally inaccurate. Note, however, this form of competition bears some interesting similarities with comparator models of conditioning (see Miller & Matzel, 1989). It also has some problems in explaining overshadowing, for similar reasons. Cooperative mixture of experts: P J[r(t)ls(t), l(t)] The final model P J[r(t)ls(t), l(t)] is just like the mixture model that Jacobs et al (1991a) suggested (see also Bordley, 1982). One statistical formulation of this model considers that, independently, where Pl(t) and Ps(t) are inverse variances. This makes (72 = (Pl(t) + Ps(t»-l 7r1(t) = PI(t)(72 7rs(t) = Ps(t)(72. Normative learning rules should emerge from a statistical model of uncertainty in the world. Short of such a model, we used: 7r1 (t) L\WI = o:w-(-) 6(t) PI t where 6(t) = r(t) - 7r1(t)Wl (t) - 7rs (t)ws (t) is the prediction error; the 1/ Pl(t) term in changing WI makes learning slower if WI is more certainly related to r (ie if PI (t) is greater); the 0.1 substitutes for background noise; if 62 (t) is too large, then PI + Ps 122 I:SIOCKlng ana unDloCklng ~~L-~1~ 0 --~2~ 0 --~M~~~ Tim. to 2nd I8warn t'reolctrVe vanances; I:SloCklng 8 r-u"gj;\'--'--' 6 4 00 200 400 600 800 1000 Trial P. Dayan and T. Long t'reolctove vanances: UnDlocKlng ......... ... , ;:_1:1 i /~~ .... -"r . "" 00 200 400 600 800 1000 Trial Figure 1: Blocking and downwards unblocking with 5 steps to the first reward; and a variable number to the second. Here, the discount factor "y = 0.9, and O:w = 0.5, O:p = 0.02, f.L = 0.75. For blocking, the second reward remains; for unblocking it is removed after 500 trials. a) The terminal weight for the sound after learning - for blocking it is always small and positive; for downwards unblocking, it changes from negative at small ~u to positive at large ~ u. b,c) Predictive variances Pl(t) and P .. (t). In blocking, although there is a small change when the sound is introduced because of additivity of the variances, learning to the sound is substantially prevented. In downwards unblocking, the surprise omission of the second reward makes the sound associable and unblocks learning to it. is shared out in proportion of pr to capture the insight that there can be dramatic changes to variabilities; and the variabilities are bottom-limited. Figure 1 shows the end point and course of learning in blocking and downwards unblocking. Figure 1a confirms that the model captures downwards unblocking, making the terminal value of Ws negative for short separations between the rewards and positive for long separations. By comparison, in the blocking condition, for which both rewards are always presented, W s is always small and positive. Figures 1b,c show the basis behind this behaviour in terms of Pl(t) and Ps{t). In particular, the heightened associability of the sound in unblocking following the prediction error when the second reward is removed accounts for the behavior. As for the mixture of experts model (and also for comparator models), the presence of 11'j(t) and nAt) makes the explanation of inhibitory conditioning and overshadowing a little complicated. For instance, if the sound is associable (Ps(t) » 0), then it can seem to act as a conditioned inhibitor even if Ws = O. Nevertheless, unlike the mixture of experts model, the fact that learning is based on the joint prediction makes true inhibitory conditioning possible. 4 Discussion Downwards unblocking may seem like an extremely abstruse paradigm with which to refute an otherwise successful and computationally sound model. However, it is just the tip of a conditioning iceberg that would otherwise sink TD. Even in other reinforcement learning applications of TO, there is no a priori reason why predictions should be made according to equation 2 - the other statistical models in equations 4 and 5 could also be used. Indeed, it is easy to generate circumstances in which these more competitive models will perform better. For the neurobiology, experiments on the behavior of the DA system in these conditioning tasks will help specify the models further. The model is incomplete in various important ways. First, it makes no distinction between preparatory and consumatory conditioning (Konorski, 1967). There is evidence that the predictions a CS makes about the affective value of USs fall in a different class from the predictions it makes about the actual USs that appear. Statistical Models of Conditioning 123 For instance, an inhibitory stimulus reporting the absence of expected delivery of food can block learning to the delivery of shock, implying that aversive events form a single class. The affective value forms the preparatory aspect, is likely what is reported by the DA cells, and perhaps controls orienting behavior, the characteristic reaction of animals to the conditioned stimuli that may provide an experimental handle on the attention they are paid. Second, the model does not use opponency (Konorski, 1967; Solomon & Corbit, 1974; Grossberg, 1982) to handle inhibitory conditioning. This is particularly important, since the dynamics of the interaction between the opponent systems may well be responsible for the importance of the delay u in downwards unblocking. Serotonin is an obvious candidate as an opponent system to DA (Montague et a11996). We also have not specified a substrate for the associabilities or the attentional competition - the DA system itself may well be involved. Finally, we have not specified an overall model of how the animal might expect the contingency of the world to change over time - which is key to the statistical justification of appropriate learning rules. References [1] Barto, AG, Sutton, RS & Anderson, CW (1983). IEEE Transactions on Systems, Man, and Cybernetics, 13, pp 834-846. [2] Bordley, RF (1982). Journal of the Operational Research Society, 33, 171-174. [3] Broadbent, DE (1958). Perception and Communication. London: Pergamon. [4] Buhusi, CV & Schmajuk, NA. Hippocampus, 6, 621-642. [5] Dempster, AP, Laird, NM & Rubin, DB (1977). Proceedings of the Royal Statistical Society, B-39,1-38. [6] Dickinson, A (1980). Contemporary Animal Learning Theory. Cambridge, England: Cambridge University Press. [7] Gabriel, M & Moore, J, editors (1989). Learning and Computational Neuroscience. Cambridge, MA: MIT Press. [8] Gallagher, M & Holland, PC (1994). PNAS, 91, 11771-6. [9] Gluck, MA & Thompson, RF (1987). Psychological Reviews, 94, 176-191. [10] Grossberg, S (1982). Psychological Review, 89,529-572. [11] Holland, PC (1988). Journal of Experimental Psychology: Animal Behavior Processes, 14, 261-279. [12] Jacobs, RA, Jordan, MI & Barto, AG (1991). Cognitive Science, 15, 219-250. [13] Jacobs, RA, Jordan, MI, Nowlan, SJ & Hinton, GE (1991). Neural Computation, 3, 79-87. [14] Konorski, J (1967). Integrative Activity of the Brain. Chicago, 11: Chicago University Press. [15] Mackintosh, NJ (1975). Psychological Review, 82,276-298. [16] Mackintosh, NJ (1983). Conditioning and Associative Learning. Oxford, UK: Oxford University Press. [17] Marr, 0 (1982). Vision. New York, NY: Freeman. [18] Miller, RR & Matzel, LD (1989). In SB Klein & RR Mowrer, editors, Contemporary Learning Theories: Pavlovian Conditioning and the Status of Traditional Theory. Hillsdale, NJ: Lawrence Erlbaum. [19] Montague, PR, Dayan, P & Sejnowski, TK (1996). Journal of Neuroscience, 16, 1936-1947. [20] Nowlan, SJ (1991). Soft Competitive Adaptation: Neural Network Learning Algorithms Based on Fitting Statistical Mixtures. PhD Thesis, Department of Computer Science, CamegieMellon University. [21] Pearce, JM & Hall, G (1980). Psychological Review, 87,532-552. [22] Rescorla, RA & Wagner, AR (1972). In AH Black & WF Prokasy, editors, Classical Conditioning II: Current Research and Theory, pp 64-69. New York, NY: Appleton-CenturyCrofts. [23] Schultz, W, Dayan, P & Montague, PR (1997). Science, 275, 1593-1599. [24] Solomon, RL & Corbit, JD (1974). Psychological Review, 81, 119-145. [25] Sutton, RS & Barto, AG (1981). Psychological Review, 882, pp 135-170. [26] Sutton, RS & Barto, AG (1989). In Gabriel & Moore (1989).
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Incorporating Contextual Information in White Blood Cell Identification Xubo Song* Department of Electrical Engineering California Institute of Technology Pasadena, CA 91125 xubosong@fire.work.caltech.edu Yaser Abu-Mostafa Dept. of Electrical Engineering and Dept. of Computer Science California Institute of Technology Pasadena, CA 91125 Yaser@over. work.caltech.edu Joseph Sill Computation and Neural Systems Program California Institute of Technology Pasadena, CA 91125 joe@busy.work.caltech.edu Harvey Kasdan International Remote Imaging Systems 9162 Eton Ave., Chatsworth, CA 91311 Abstract In this paper we propose a technique to incorporate contextual information into object classification. In the real world there are cases where the identity of an object is ambiguous due to the noise in the measurements based on which the classification should be made. It is helpful to reduce the ambiguity by utilizing extra information referred to as context, which in our case is the identities of the accompanying objects. This technique is applied to white blood cell classification. Comparisons are made against "no context" approach, which demonstrates the superior classification performance achieved by using context. In our particular application, it significantly reduces false alarm rate and thus greatly reduces the cost due to expensive clinical tests. • Author for correspondence. Incorporating Contextual Information in White Blood Cell Identification 951 1 INTRODUCTION One of the most common assumptions made in the study of machine learning is that the examples are drawn independently from some joint input-output distribution. There are cases, however, where this assumption is not valid. One application where the independence assumption does not hold is the identification of white blood cell images. Abnormal cells are much more likely to appear in bunches than in isolation. Specifically, in a sample of several hundred cells, it is more likely to find either no abnormal cells or many abnormal cells than it is to find just a few. In this paper, we present a framework for pattern classification in situations where the independence assumption is not satisfied. In our case, the identity of an object is dependent of the identities of the accompanying objects, which provides the contextual information. Our method takes into consideration the joint distribution of all the classes, and uses it to adjust the object-by-object classification. In section 2, the framework for incorporating contextual information is presented, and an efficient algorithm is developed. In section 3 we discuss the application area of white blood cell classification, and address the importance of using context for this application. Empirical testing results are shown in Section 4, followed by conclusions in Section 5. 2 INCORPORATING CONTEXTUAL INFORMATION INTO CLASSIFICATION 2.1 THE FRAMEWORK Let Xi be the feature vector of an object, and Ci = C(Xi} be the classification for Xi, i = I, ... N, where N is the total number of objects. Ci E {I, ... , D}, where D is the number of total classes. According to Bayes rule, ( I ) - p(xlc}p(c} pC X p(x} It follows that the "with context" a posteriori probability of the class labels of all the objects assuming values Cl, C2, "', C N, given all the feature vectors, is It is reasonable to assume that the feature distribution given a class is independent of the feature distributions of other classes, i.e., P(Xb X2, ... , xNlcl, C2, ... , CN} = p(xllcd···p(XNlcN} Then Equation (1) can be rewritten as p(cllxd",p(CNlxN }P(Xl}···P(XN )P(Cl' C2, ... , CN) P(Cl}· .. P(CN }P(Xl ' X2, ... , XN} 952 X Song, Y. Abu-Mostafa, 1. Sill and H. Kasdan where p( cilxi) is the "no context" object-by-object Bayesian a posteriori probability, and p( Ci) is the a priori probability of the classes, p( Xi) is the marginal probability of the features, and P(Xl' X2, ... , XN) is the joint distribution of all the feature vectors. Since the features (Xl, X2, ... , XN) are given, p(Xb X2, ... , XN) and p(xd are constant, where (3) The quantity p( Cl, C2, •.. , C N ), which we call context ratio and through which the context plays its role, captures the dependence among the objects. In the case where all the objects are independent, p( Cl, C2, ..• , CN) equals one - there will be no context. In the dependent case, p( Cl, C2, ..• , CN) will not equal one, and the context has an effect on the classifications. We deal with the application of object classification where it is the count in each class, rather than the particular ordering or numbering of the objects, that matters. As a result, p ( Cl , C2, .•. , C N) is only a function of the count in each class. Let N d be the count in class d, and Vd = !ft, d = 1..., D, where Pd is the prior distribution of class d, for d = 1, ... D. 2:f=l Nd L:f=l Vd = l. The decision rule is to choose class labels Cl, C2, .. . , CN such that (4) Nand (Cl' C2, •.. , CN) = argmax P(Cll C2, .•. , cNIXl, X2, ... , XN) (5) (Cl ,C2 , ... ,CN) When implementing the decision rule, we need to compute and compare DN cases for Equation 5. In the case of white blood cell recognition, D = 14 and N is typically around 600, which makes it virtually impossible to implement. In many cases, additional constraints can be used to reduce computation, as is the case in white blood cell identification, which will be demonstrated in the following section. 3 WmTE BLOOD CELL RECOGNITION Leukocyte analysis is one of the major routine laboratory examinations. The utility of leukocyte classification in clinical diagnosis relates to the fact that in various physiological and pathological conditions the relative percentage composition of the blood leukocytes Incorporating Contextual Infonnation in White Blood Cell Identification 953 changes. An estimate of the percentage of each class present in a blood sample conveys information which is pertinent to the hematological diagnosis. Typical commercial differential WBC counting systems are designed to identify five major mature cell types. But blood samples may also contain other types of cells, i.e. immature cells. These cells occur infrequently in normal specimen, and most commercial systems will simply indicate the presence of these cells because they can't be individually identified by the systems. But it is precisely these cell types that relate to the production rate and maturation of new cells and thus are important indicators of hematological disorders. Our system is designed to differentiate fourteen WBC types which includes the five major mature types: segmented neutrophils, lymphocytes, monocytes, eosinophils, and basophils; and the immature types: bands (unsegmented neutrophils), metamyelocytes, myelocytes, promyelocytes, blasts, and variant lymphocytes; as well as nucleated red blood cells and artifacts. Differential counts are made based on the cell classifications, which further leads to diagnosis or prognosis. The data was provided by IRIS, Inc. Blood specimens are collected at Harbor UCLA Medical Center from local patients, then dyed with Basic Orange 21 metachromatic dye supravital stain. The specimen is then passed through a flow microscopic imaging and image processing instrument, where the blood cell images are captured. Each image contains a single cell with full color. There are typically 600 images from each specimen. The task of the cell recognition system is to categorize the cells based on the images. 3.1 PREPROCESSING AND FEATURE EXTRACTION The size of cell images are automatically tailored according to the size of the cell in the images. Images containing larger cells have bigger sizes than those with small cells. The range varies from 20x20 to 40x40 pixels. The average size is around 25x25. See Figure 3.1. At the preprocessing stage, the images are segmented to set the cell interior apart from the background. Features based on the interior of the cells are extracted from the images. The features include size, shape, color 1 and texture. See Table 1 for the list of features. 2 Figure 1: Example of some of the cell images. 3.2 CELL·BY·CELL CLASSIFICATION The features are fed into a nonlinear feed-forward neural network with 20 inputs, 15 hidden units with sigmoid transfer functions, and 14 sigmoid output units. A cross-entropy error 1 A color image is decomposed into three intensity images - red, green and blue respectively 2The red-blue distribution is the pixel-by-pixel log(red)- log(blue) distribution for pixels in cell interior. The red distribution is the distribution of the red intensity in cell interior. 954 X Song, Y. Abu-Mostafa, 1. Sill and H. Kasdan feature number feature description 1 cell area 2 number of pixels on cell edge 3 the 4th quantile of red-blue distribution 4 the 4th quantile of green-red distribution 5 the median of red-blue distribution 6 the median of green-red distribution 7 the median of blue-green distribution 8 the standard deviation of red-blue distribution 9 the standard deviation of green-red distribution 10 the standard deviation of blue-green distribution 11 the 4th quantile of red distribution 12 the 4th quantile of green distribution 13 the 4th quantile of blue distribution 14 the median of red distribution 15 the median of green distribution 16 the median of blue distribution 17 the standard deviation of red distribution 18 the standard deviation of green distribution 19 the standard deviation of blue distribution 20 the standard deviation of the distance from the edge to the mass center function is used in order to give the output a probability interpretation. Denote the input feature vector as x, the network outputs a D dimensional vector ( D = 14 in our case) p = {p(dlx)}, d = 1, ... , D, where p(dlx) is p(dlx) = Prob( a cell belongs to class dl feature x) The decision made at this stage is d(x) = argmax p(dlx) d 3.3 COMBINING CONTEXTUAL INFORMATION The "no-context" cell-by-cell decision is only based on the features presented by a cell, without looking at any other cells. When human experts make decisions, they always look at the whole specimen, taking into consideration the identities of other cells and adjusting the cell-by-cell decision on a single cell according to the company it keeps. On top of the visual perception of the cell patterns, such as shape, color, size, texture, etc., comparisons and associations, either mental or visual, with other cells in the same specimen are made to infer the final decision. A cell is assigned a certain identity if the company it keeps supports that identity. For instance, the difference between lymphocyte and blast can be very subtle sometimes, especially when the cell is large. A large unusual mononuclear cell with the characteristics of both blast and lymphocyte is more likely to be a blast if surrounded by or accompanied by other abnormal cells or abnormal distribution of the cells. Incorporating Contextual Infonnation in White Blood Cell Identification 955 This scenario fits in the framework we described in section 2. The Combining Contextual Information algorithm was used as the post-precessing of the cell-by-cell decisions. 3.4 OBSERVATIONS AND SIMPLIFICATIONS Direct implementation of the proposed algorithm is difficult due to the computational complexity. In the application of WBC identification, simplification is possible. We observed the following: First, we are primarily concerned with one class blast, the presence of which has clinical significance. Secondly, we only confuse blast with another class lymphocyte. In other words, for a potential blast, p(blastlx) » 0, p(lymphocytelx) » 0, p(any other classlx) ~ O. Finally, we are fairly certain about the classification of all other classes, i.e. p(a certain c1asslx) ~ 1, p(any other c1asslx) ~ O. Based on the above observations, we can simplify the algorithm, instead of doing an exhaustive search. Let pf = P(Ci = dlxi), i = 1, ... , N. More specifically, let pf = p(blastlxd, pf = p(lymphocytelxi) and pi = p(c1ass * IXi) where * is neither a blast nor a lymphocyte. Suppose there are K potential blasts. Order the pf, pf, ... , pf 's in a descending manner over i, such that B B B Pl ~ P2 ~ ... ~ PK then the probability that there are k blasts is PB(k) = PP ···pfpr+l···pj( PK+1"'piv p(VB = t, VL = v~ + K ;/, V3, "" VD) where v~ is the proportion of unambiguous lymphocytes and V3, "" VD are the proportions of the other cell types, We can compute the PB(k)'s recursively, for k:::: 1, "" K-l, and This way we only need to compute K terms to get PB(k)'s . Pick the optimal number of blasts k* that maximizes PB (k), k = 1, "., K, An important step is to calculate p(Vl, "" VD) which can be estimated from the database, 3.5 THE ALGORITHM Step 1 Estimate P(Vl' ,." VD) from the database, for d = 1"", D, Step 2 Compute the object-by-object "no context" a posteriori probability p(cilxi), i 1, "', N, and Ci E {I, ... , D}, Step 3 Compute PB (k) and find k* for k = 1, .'" K, and relabel the cells accordingly, 956 X Song, Y. Abu-Mostafa, 1. Sill and H. Kasdan 4 EMPIRICAL TESTING The algorithm has been intensively tested at IRIS, Inc. on the specimens obtained at Harbor UCLA medical center. We compared the performances with or without using contextual information on blood samples from 220 specimens (consisting of 13,200 cells). In about 50% of the cases, a false alarm would have occurred had context not been used. Most cells are correctly classified, but a few are incorrectly labelled as immature cells, which raises a flag for the doctors. Change of the classification of the specimen to abnormal requires expert intervention before the false alarm is eliminated, and it may cause unnecessary worry. When context is applied, the false alarms for most of the specimens were eliminated, and no false negative was introduced. methods cell normality false false classification identification positive negative no context 88% I"V 50% 1"V50% 0% with context 89% I"V 90% ,....10% 0% Table 2: Comparison of with and without using contextual information 5 CONCLUSIONS In this paper we presented a novel framework for incorporating contextual information into object identification, developed an algorithm to implement it efficiently, and applied it to white blood cell recognition. Empirical tests showed that the "with context" approach is significantly superior than the "no context" approach. The technique described could be generalized to a number of domains where contextual information plays an essential role, such a speech recognition, character recognition and other medical diagnosis regimes. Acknowledgments The authors would like to thank the members of Learning Systems Group at Caltech for helpful suggestions and advice: Dr. Amir Atiya, Zehra Cataltepe, Malik Magdon-Ismail, and Alexander Nicholson. References Richard, M.D., & Lippmann, R.P., (1991) Neural network classifiers estimate Bayesian a posteriori probabilities. Neural Computation 3. pp.461-483. Cambridge, MA: MIT Press. Kasdan, H.K., Pelmulder, J.P., Spolter, L., Levitt, G.B., Lincir, M.R., Coward, G.N., Haiby, S. 1., Lives, J., Sun, N.C.J., & Deindoerfer, F.H., (1994) The WhiteIRISTM Leukocyte differential analyzer for rapid high-precision differentials based on images of cytoprobereacted cells. Clinical Chemistry. Vol. 40, No.9, pp.1850-1861. Haralick, R.M., & Shapiro, L.G.,(l992),Computer and Robot Vision, Vol.1 , AddisonWelsley. Aus, H. A., Harms, H., ter Meulen, v., & Gunzer, U. (1987) Statistical evaluation of computer extracted blood cell features for screening population to detect leukemias. In Pierre A. Devijver and Josef Kittler (eds.) Pattern Recognition Theory and Applications, pp. 509518. Springer-Verlag. Kittler, J., (1987) Relaxation labelling. In Pierre A. Devijver and Josef Kittler (eds.) Pattern Recognition Theory and Applications, pp. 99-108. Springer-Verlag.
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The Efficiency and The Robustness of Natural Gradient Descent Learning Rule Howard Hua Yang Department of Computer Science Oregon Graduate Institute PO Box 91000, Portland, OR 97291, USA hyang@cse.ogi.edu Shun-ichi Amari Lab. for Information Synthesis RlKEN Brain Science Institute Wako-shi, Saitama 351-01, JAPAN amari@zoo.brain.riken.go.jp Abstract The inverse of the Fisher information matrix is used in the natural gradient descent algorithm to train single-layer and multi-layer perceptrons. We have discovered a new scheme to represent the Fisher information matrix of a stochastic multi-layer perceptron. Based on this scheme, we have designed an algorithm to compute the natural gradient. When the input dimension n is much larger than the number of hidden neurons, the complexity of this algorithm is of order O(n). It is confirmed by simulations that the natural gradient descent learning rule is not only efficient but also robust. 1 INTRODUCTION The inverse of the Fisher information matrix is required to find the Cramer-Rae lower bound to analyze the performance of an unbiased estimator. It is also needed in the natural gradient learning framework (Amari, 1997) to design statistically efficient algorithms for estimating parameters in general and for training neural networks in particular. In this paper, we assume a stochastic model for multilayer perceptrons. Considering a Riemannian parameter space in which the Fisher information matrix is a metric tensor, we apply the natural gradient learning rule to train single-layer and multi-layer perceptrons. The main difficulty encountered is to compute the inverse of the Fisher information matrix of large dimensions when the input dimension is high. By exploring the structure of the Fisher information matrix and its inverse, we design a fast algorithm with lower complexity to implement the natural gradient learning algorithm. 386 H H Yang and S. Amari 2 A STOCHASTIC MULTI-LAYER PERCEPTRON Assume the following model of a stochastic multi-layer perceptron: m z = L ail{J(wT x + bi) + ~ (1) i=l where OT denotes the transpose, ~ ,..., N(O, (72) is a Gaussian random variable, and l{J(x) is a differentiable output function for hidden neurons. Assume the multi-layer network has a n-dimensional input, m hidden neurons, a one dimensional output, and m S n. Denote a = (ai, ... ,am)T the weight vector of the output neuron, Wi = (Wli,···, Wni)T the weight vector of the i-th hidden neuron, and b = (bl ,···, bm)T the vector of thresholds for the hidden neurons. Let W = [WI,···, W m ] be a matrix formed by column weight vectors Wi, then (1) can be rewritten as z = aT I{J(WT x + b) +~. Here, the scalar function I{J operates on each component of the vector WT x + b. The joint probability density function (pdf) of the input and the output is p(x,z;W,a,b) = p(zlx; W,a,b)p(x). Define a loss function: L(x, z; 0) = -logp(x, Z; 0) = l(zlx; 0) -logp(x) where 0 = (wI,···, W~, aT, bTV includes all the parameters to be estimated and 1 l(zlx; 0) = -logp(zlx; 0) = 2(72 (z - aT I{J(WT x + b»2. Since ~ = -!b, the Fisher information matrix is defined by G(O) = E[8L(8L)T] = E[~(~)T] 80 80 80 80 (2) The inverse of G(O) is often used in the Cramer-Rao inequality: E[II6 - 0*112 I 0*] ~ Tr(G-I(O*» where 6 is an unbiased estimator of a true parameter 0*. For the on-line estimator Ot based on the independent examples {(xs, zs), s = 1,···, t} drawn from the probability law p(x, Z; 0*), the Cramer-Rao inequality for the on-line estimator is E[II6t - 0*112 I 0*] ~ ~Tr(G-I(O*» 3 NATURAL GRADIENT LEARNING (3) Consider a parameter space e = {O} in which the divergence between two points 01 and O2 is given by the Kullback-Leibler divergence D(OI, O2 ) = KL(P(x, z; OI)IIp(x, z; O2 )]. When the two points are infinitesimally close, we have the quadratic form D(O,O + dO) = ~dOTG(O)dO. (4) The Efficiency and the Robustness of Natural Gradient Descent Learning Rule 387 This is regarded as the square of the length of dO. Since G(8) depends on 8, the parameter space is regarded as a Riemannian space in which the local distance is defined by (4). Here, the Fisher information matrix G(8) plays the role of the Riemannian metric tensor. It is shown by Amari(1997) that the steepest descent direction of a loss function C(8) in the Riemannian space (3 is -VC(8) = -G- 1(8)\7C(8). The natural gradient descent method is to decrease the loss function by updating the parameter vector along this direction. By multiplying G- 1 (8), the covariant gradient \7C(8) is converted into its contravariant form G-1 (8)\7C(8) which is consistent with the contravariant differential form dC(8). Instead of using l(zlx; 8) we use the following loss function: 1 lr(zlx; 8) = "2(z - aT tp(WT x + b»2. We have proved in [5] that G(8) = ~A(8) where A(8) does not depend on the unknown u. So G-1(8)-lb = A -1(8)~. The on-line learning algorithms based on the gradient ~ and the natural gradient A -1(8)~ are, respectively, fl. Oil 8tH = 8t - t {)8 (ztlxt; 8t ), (5) fl.' -1 Oil 8t+1 = 8t - fA (8t ) {)8 (ztlXt; 8t) (6) where fl. and fl.' are learning rates. When the negative log-likelihood function is chosen as the loss function, the natural gradient descent algorithm (6) gives a Fisher efficient on-line estimator (Amari, 1997), i.e., the asymptotic variance of 8t driven by (6) satisfies (7) which gives the mean square error (8) The main difficulty in implementing the natural gradient descent algorithm (6) is to compute the natural gradient on-line. To overcome this difficulty, we studied the structure of the matrix A(8) in [5] and proposed an efficient scheme to represent this matrix. Here, we briefly describe this scheme. Let A(8) = [Aijlcm+2)x(m+2) be a partition of A(8) corresponding to the partition of 8 = (wf,.··,w?;.,aT,bT)T. Denote Ui = Wi/I/Will,i = 1, · · · ,m, U 1 = [U1,· .. ,um ] and [VI,·· . ,Vm ] = U 1 (UiU 1)-1. It has been proved in [5] that those blocks in A(8) are divided into three classes: C1 = {Aij,i,j = 1,· · · ,m}, C2 = {Ai,mH, A!:H,i' A i,m+2, A!:+2,i' i = 1,···, m} and C3 = {Am+i,m+j, i,j = 1,2}. Each block in C1 is a linear combination of matrices UkVf, k, 1 = 1,· · ·, m, and no = I E~1 ukvf. Each block in C2 is a matrix whose column is a linear combination of {Vk' k = 1,· .. ,m.}. The coefficients in these combinations are integrals with respect to the multivariate Gaussian distribution N(O, R 1 ) where 388 H. H. Yang and S. Amari HI = ufu 1 is m x m. Each block in C3 is an m x m matrix whose entries are also integrals with respect to N(O, HI)' Detail expressions for these integrals are given in [5]. When rp(x) = erf(.i2), using the techniques in (Saa.d and Solla, 1995), we can find the analytic expressions for most of these integrals. The dimension of A(9) is (nm + 2m) x (nm + 2m). When the input dimension n is much larger than the number of hidden neurons, by using the above scheme, the space for storing this large matrix is reduced from O(n2) to O(n). We also gave a fast algorithm in [5] to compute A-1(9) and the natural gradient with the time complexity O(n2) and O(n) respectively. The trick is to make use of the structure of the matrix A -1(9). 4 SIMULATION In this section, we give some simulation results to demonstrate that the natural gradient descent algorithm is efficient and robust . 4.1 Single-layer perceptron Assume 7-dimensional inputs Xt '" N(O, J) and rp(u) = ~+:=:. For the single-layer perceptron, Z = rp(wTx), the on-line gradient descent (GD) and the natural GD algorithms are respectively Wt+l = Wt + J.to(t)(Zt - rp(w[ Xt))rp'(w[ Xt)Xt and (9) Wt+l = Wt + J.tdt) A-I (Wt)(Zt - rp(W[Xt»rp'(wiXt)Xt (10) where (11) (12) (13) and j.to(t) and j.tl (t) are two learning rate schedules defined by J.ti(t) = J.t(1]i,Ci,Tijt),i = 0,1. Here, C t c t t2 J.t(1],C,Tjt) = 1](1 + --)/(1 + -- + -). (14) 1]T 1]T T is the search-then-converge schedule proposed by (Darken and Moody, 1992) . Note that t < T is a "search phase" and t > T is a "converge phase". When Ti = 1, the learning rate function J.ti(t) has no search phase but a weaker converge phase when 1]i is small. When t is large, J.ti (t) decreases as ¥. Randomly choose a 7-dimensional vector as w· for the teacher network: w· = [-1.1043,0.4302,1.1978,1.5317, -2.2946, -0.7866,0.4428f. Choose 1]0 = 1.25, 1]1 = 0.05, Co = 8.75, CI = 1, and TO = Tl = 1. These parameters are selected by trial and error to optimize the performance of the GD and the natural GD methods at the noise level u = 0.2. The training examples {(Xt, Zt)} are generated by Zt = rp(w·TXt) +~t where ~t '" N(0,u2) and u2 is unknown to the algorithms. The Efficiency and the Robustness o/Natural Gradient Descent Learning Rule 389 Let Wt and Wt be the weight vectors driven by the equations (9) and (to) respectively. Ilwt - w"'l1 and IIWt - w'"l1 are error functions for the GD and the natural GD. Denote w'" = IIw'"lI. From the equation (11), we obtain the Cramer-Rao Lower Bound (CRLB) for the deviation at the true weight vector w"': u CRLB(t) = Vi n -1 1 d1 (w"') + d2(w"'r (15) Figure 1: Performance of the GD and the natural GD at different noise levels u = 0.2,0.4,1. -- natural GO ----- GO CRLB --- .. _--- ----------"'- ..... _----- .... , -, , .... _-... _--- .... rtt\.vt~~ ---------10-2'--:'-:--~-_=_-=___:=____::::::__~:__:_:_:-=__:: o 50 100 150 200 250 300 350 400 450 500 Iteration It is shown in Figure 1 that the natural GD algorithin reaches CRLB at different noise levels while the GD algorithm reaches the CRLB only at the noise level u = 0.2. The robustness of the natural gradient descent against the additive noise in 390 H. H. Yang and S. Amari Figure 2: Performance of the GD and the natural GD when "10 1.25, 1. 75,2.25,2.75, "11 = 0.05,0.2,0.4425,0.443, and CO = 8.75 and Cl = 1 are fixed. the training examples is clearly shown by Figure 1. When the teacher signal is non-stationary, our simulations show that the natural GD algorithm also reaches the CRLB. Figure 2 shows that the natural GD algorithm is more robust than the GD algorithm against the change of the learning rate schedule. The performance of the GD algorithm deteriorates when the constant "10 in the learning rate schedule (..to(t) is different from that optimal one. On the contrary, the natural GD algorithm performs almost the same for all "11 within a interval [0.05,0.4425]. Figure 2 also shows that the natural GD algorithm breaks down when "11 is larger than the critical number 0.443. This means that the weak converge phase in the learning rate schedule is necessary. 4.2 Multi-layer perceptron Let us consider the simple multi-layer perceptron with 2-dimensional input and 2hidden neurons. The problem is to train the committee machine y = <p(w[ x) + <p(wf x) based on the examples {eXt, Zt), t = 1,· ·· , T} generated by the stochastic committee machine Zt = <p(wiTXt) + <P(W2TXt) + ~t. Assume IIwili = 1. We can reparameterize the weight vector to decrease the dimension of the parameter space from 4 to 2: [ COS(Oi) ] * _ [ COS(O;)] . _ 1 2 Wi = sin(oi) , Wi sin(oi) , '/, , . 10',-----r-----r-----r-----r-----r----. - - - GO -- natural GO CRLB 10~~--~~--~~--~~--~~----~--~ o 100 200 300 400 500 600 Heratlon Figure 3: The GD vs. the natural GD The parameter space is {8 = (01, (2)}. Assume that the true parameters are oi = 0 and 02 = 3;. Due to the symmetry, both 8r = (0, 3;) and 8; = e; ,0) are true parameters. Let 8t and 8~ be computed by the GD algorithm and the natural GD The Efficiency and the Robustness of Natural Gradient Descent Learning Rule 391 algorithm respectively. The errors are measured by ct = min{119t - 9rll, 119t 8~1I}, and c~ = min{lI~ - 9rll, 118~ - 9;11}. In this simulation, using 80 = (0.1,0.2) as an initial estimate, we first start the GD algorithm and run it for 80 iterations. Then, we use the estimate obtained from the GD algorithm at the 80-th iteration as an initial estimate for the natural GD algorithm and run the latter algorithm for 420 iterations. The noise level is (1 = 0.05. N independent runs are conducted to obtain the errors ctU) and c~U), i = 1,···, N. Define root mean square errors and c l '" tN ~ I:(cHi)2. j=1 Based on N = 10 independent runs, the errors Et and c't are computed and compared with the CRLB in Figure 3. The search-then-converge learning schedule (14) is used in the GD algorithm while the learning rate for the natural GD algorithm is simply the annealing rate t. 5 CONCLUSIONS The natural gradient descent learning rule is statistically efficient. It can be used to train any adaptive system. But the complexity of this learning rule depends on the architecture of the learning machine. The main difficulty in implementing this learning rule is to compute the inverse of the Fisher information matrix of large dimensions. For a multi-layer perceptron, we have shown an efficient scheme to represent the Fisher information matrix based on which the space for storing this large matrix is reduced from O(n2 ) to O(n). We have also shown an algorithm to compute the natural gradient. Taking advantage of the structure of the inverse of the Fisher information matrix, we found that the complexity of computing the natural gradient is O(n) when the input dimension n is much larger than the number of hidden neurons. The simulation results have confirmed the fast convergence and statistical efficiency of the natural gradient descent learning rule. They have also verified that this learning rule is robust against the changes of the noise levels in the training examples and the parameters in the learning rate schedules. References [1] S. Amari. Natural gradient works efficiently in learning. Accepted by Neural Computation, 1997. [2] S. Amari. Neural learning in structured parameter spaces - natural Riemannian gradient. In Advances in Neural Information Processing Systems, 9, ed. M. C. Mozer, M. 1. Jordan and T. Petsche, The MIT Press: Cambridge, MA., pages 127-133, 1997. [3] C. Darken and J. Moody. Towards faster stochastic gradient search. In Advances in Neural Information Processing Systems, 4, eds. Moody, Hanson, and Lippmann, Morgan Kaufmann, San Mateo, pages 1009-1016, 1992. [4] D. Saad and S. A. Solla. On-line learning in soft committee machines. Physical Review E, 52:4225-4243, 1995. [5] H. H. Yang and S. Amari. Natural gradient descent for training multi-layer perceptrons. Submitted to IEEE Tr. on Neural Networks, 1997.
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Function Approximat.ion with the Sweeping Hinge Algorithm Don R. Hush, Fernando Lozano Dept. of Elec. and Compo Engg. University of New Mexico Albuquerque, NM 87131 Abstract Bill Horne MakeWaves, Inc. 832 Valley Road Watchung, NJ 07060 We present a computationally efficient algorithm for function approximation with piecewise linear sigmoidal nodes. A one hidden layer network is constructed one node at a time using the method of fitting the residual. The task of fitting individual nodes is accomplished using a new algorithm that searchs for the best fit by solving a sequence of Quadratic Programming problems. This approach offers significant advantages over derivative-based search algorithms (e.g. backpropagation and its extensions). Unique characteristics of this algorithm include: finite step convergence, a simple stopping criterion, a deterministic methodology for seeking "good" local minima, good scaling properties and a robust numerical implementation. 1 Introduction The learning algorithm developed in this paper is quite different from the traditional family of derivative-based descent methods used to train Multilayer Perceptrons (MLPs) for function approximation. First, a constructive approach is used, which builds the network one node at a time. Second, and more importantly, we use piecewise linear sigmoidal nodes instead of the more popular (continuously differentiable) logistic nodes. These two differences change the nature of the learning problem entirely. It becomes a combinatorial problem in the sense that the number of feasible solutions that must be considered in the search is finite. We show that this number is exponential in the input dimension, and that the problem of finding the global optimum admits no polynomial-time solution. We then proceed to develop a heuristic algorithm that produces good approximations with reasonable efficiency. This algorithm has a simple stopping criterion, and very few user specified parameters. In addition, it produces solutions that are comparable to (and sometimes better than) those produced by local descent methods, and it does so 536 D. R. Hush, R Lozano and B. Horne using a deterministic methodology, so that the results are independent of initial conditions. 2 Background and Motivation We wish to approximate an unknown continuous function f(x) over a compact set with a one-hidden layer network described by n f~(x) = ao + L aiU(x, Wi) (1) i=l where n is the number of hidden layer nodes (basis functions), x E ~d is the input vector, and {u(x, w)} are sigmoidal functions parameterized by a weight vector w. A set of example data, S = {Xi, Yi}, with a total of N samples is available for training and test. The models in (1) have been shown to be universal approximators. More importantly, (Barron, 1993) has shown that for a special class of continuous functions, r c, the generalization error satisfies E[lIf - fn,NII2] ~ IIf - fnll 2 + E[lIfn - fn,NII2] = 0 (*) + 0 ( nd ~g N ) where 11·11 is the appropriate two-norm, f n is the the best n-node approximation to f, and fn,N is the approximation that best fits the samples in S. In this equation IIf - fnll 2 and E[lIfn - fn,NII2] correspond to the approximation and estimation error respectively. Of particular interest is the O(l/n) bound on approximation error, which for fixed basis functions is of the form O(1/n2/ d ) (Barron, 1993). Barron's result tells us that the (tunable) sigmoidal bases are able to avoid the curse of dimensionality (for functions in rc). Further, it has been shown that the O(l/n) bound can be achieved constructively (Jones, 1992), that is by designing the basis functions (nodes) one at a time. The proof of this result is itself constructive, and thus provides a framework for the development of an algorithm which can (in principle) achieve this bound. One manifestation of this algorithm is shown in Figure 1. We call this the iterative approximation algorithm (I1A) because it builds the approximation by iterating on the residual (Le. the unexplained portion of the function) at each step. This is the same algorithmic strategy used to form bases in numerous other settings, e.g. Grahm-Schmidt, Conjugate Gradient, and Projection Pursuit. The difficult part of the I1A algorithm is in the determination of the best fitting basis function Un in step 2. This is the focus of the remainder of this paper. 3 Algorithmic Development We begin by defining the hinging sigmoid (HS) node on which our algorithms are based. An HS node performs the function { -T- > w+, w, X _ w+ - T- TUh(X, w) = w, X, w_ ~ w, X ~ w+ - T- < w_, w, x _ w_ (2) where w T = [WI w+ w_] and x is an augmented input vector with a 1 in the first component. An example of the surface formed by an HS node on a two-dimensional input is shown in Figure 2. It is comprised of three hyperplanes joined pairwise Function Approximation with the Sweeping Hinge Algorithm Initialization: fo(x) = 0 for n = 1 to nma:c do 1. Compute Residual: en(x) = f(x) - fn-l (x) 2. Fit Residual: un(x) = argminO"EE lIen(x) - u(x)11 3. Update Estimate: fn(x) = o:fn-l (x) + f3un(x) where 0: and f3 are chosen to minimize IIf(x) - fn(x)1I endloop Figure 1: Iterative Approximation Algorithm (rIA). Figure 2: A Sigmoid Hinge function in two dimensions. 537 continuously at two hinge locations. The upper and middle hyperplanes are joined at "Hinge I" and the lower and middle hyperplanes are joined at "Hinge 2". These hinges induce linear partitions on the input space that divide the space into three regions, and the samples in 5 into three subsets, 5+ = {(Xi,Yi): Wr-Xi ~ w+} 5, = {(Xi,Yi): w_ ~ WTXi ~ w+} 5_ = {(Xi,Yi): WTXi ~ w_} (3) These subsets, and the corresponding regions of the input space, are referred to as the PLUS, LINEAR and MINUS subsets/regions respectively. We refer to this type of partition as a sigmoidal partition. A sigmoidal partition of 5 will be denoted P = {5+, 5" 5_}, and the set of all such partitions will be denoted II = {Pd. Input samples that fall on the boundary between two regions can be assigned to the set on either side. These points are referred to as hinge samples and playa crucial role in subsequent development. Note that once a weight vector w is specified, the partition P is completely determined, but the reverse is not necessarily true. That is, there are generally an infinite number of weight vectors that induce the same partition. We begin our quest for a learning algorithm with the development of an expression for the empirical risk. The empirical risk (squared error over the sample set) is defined (4) 538 D. R Hush, F. Lozano and B. Horne This expression can be expanded into three terms, one for each set in the partition, Ep(w) = ~ :E(Yi - W_)2 + ~ :E(Yi - W+)2 + ~ 2)Yi - WTXi)2 ~ ~ ~ After further expansion and rearrangement of terms we obtain 1 Ep(w) = 2wTRw - w T r + s; where R, = "L:s, XiX; r, = "L:s, XiYi s; = ! "L:s Y; st = "L:s+ Yi Sy = "L:s_ Yi ( R, R= ~ r=un (5) (6) (7) (8) and N+ , N, and N_ are the number of samples in S+ , S, and S_ respectively. The subscript P is used to emphasize that this criterion is dependent on the partition (i.e. P is required to form Rand r). In fact, the nature of the partition plays a critical role in determining the properties of the solution. When R is positive definite (i.e. full rank), P is referred to as a stable partition, and when R has reduced rank P is referred to as an unstable partition. A stable partition requires that R, > O. For purposes of algorithm development we will assume that R, > 0 when ISti > Nmin, where Nmin is a suitably chosen value greater than or equal to d + 1. With this, a necessary condition for a stable partition is that there be at least one sample in S+ and S_ and N, ~ Nmin. When seeking a minimizing solution for Ep(w) we restrict ourselves to stable partitions because of the potential nonuniqueness associated with solutions to unstable partitions. Determining a weight vector that simultaneously minimizes E p (w) and preserves the current partition can be posed as a constrained optimization problem. This problem takes on the form min !wTRw - w T r 2 subject to Aw ~ 0 (9) where the inequality constraints are designed to maintain the current partition defined by (3). This is a Quadratic Programming problem with inequality constraints, and because R > 0 it has a unique global minimum. The general Quadratic Programming problem is N P-hard and also hard to approximate (Bellare and Rogaway, 1993). However, the convex case which we restrict ourselves to here (i.e. R > 0) admits a polynomial time solution. In this paper we use the active set algorithm (Luenberger, 1984) to solve (9). With the proper implementation, this algorithm runs in O(k(~ + Nd)) time, where k is typically on the order of d or less. The solution to the quadratic programming problem in (9) is only as good as the current partition allows. The more challenging aspect of minimizing Ep(w) is in the search for a good partition. Unfortunately there is no ordering or arrangement of partitions that is convex in Ep(w), so the search for the optimal partition will be a computationally challenging problem. An exhaustive search is usually out of the question because of the prohibitively large number of partitions, as given by the following lemma. Lemma 1: Let S contain a total of N samples in Rd that lie in general position. Then the number of sigmoidal partitions defined in (3) is 8(Nd+l). Function Approximation with the Sweeping Hinge Algorithm 539 Proof: A detailed proof is beyond the scope of this paper, but an intuitive proof follows. It is well-known that the number of linear dichotomies of N points in d dimensions is 8(Nd) (Edelsbrunner, 1987). Each sigmoidal partition is comprised of two linear dichotomies, one formed by Hinge 1 and the other by Hinge 2, and these dichotomies are constrained to be simple translations of one another. Thus, to enumerate all sigmoidal partitions we allow one of the hinges, say Hinge 1, can take on 8(Nd) different positions. For each of these the other hinge can occupy only'" N unique positions. The total is therefore 8 (Nd+l ). The search algorithm developed here employs a Quadratic Programming (QP) algorithm at each new partition to determine the optimal weight vector for that partition (Le. the optimal orientation for the separating hyperplanes). Transitions are made from one partition to the next by allowing hinge samples to flip from one side of the hinge boundary to the next. The search is terminated when a minimum value of Ep(w) is found (Le. it can no longer be reduced by flipping hinge samples). Such an algorithm is shown in Figure 3. We call this the HingeDescent algorithm because it allows the hinges to "walk across" the data in a manner that descends the Ep(w) criterion. Note that provisions are made within the algorithm to avoid unstable partitions. Note also that it is easy to modify this algorithm to descend only one hinge at a time, simply by omitting one of the blocks of code that flips samples across the corresponding hinge boundary. {This routine is invoked with a stable feasible solution W = {w, R, r, A, S+, SI, S_ }.} procedure HingeDescent (W) { Allow hinges to walk across the data until a minimizing partition is found. } E_1wTRw-wTr 2 do Emin = E {Flip Hinge 1 Samples.} for each «Xi, Yi) on Hinge 1) do if «Xi, Yi) E S+ and N+ > 1) then Move (Xi,Yi) from S+ to S" and update R, r, and A elseif «Xi, Yi) E S, and N, > N min ) then Move (Xi, Yi) from S, to S+, and update R, r, and A endif endloop {Flip Hinge 2 Samples.} for each «Xi, Yi) on Hinge 2) do if «Xi,Yi) E S- and N_ > 1) then Move (Xi,Yi) from S- to S" and update R, r, and A elseif «Xi, Yi) E S, and N, > Nmin) then Move (Xi,Yi) from S, to S-, and update R, r, and A endif endloop {Compute optimal solution for new partition.} W = QPSolve(W}; E= ~wTRw-wTr while (E < Emin) j return(W)j end; {HingeDescent} Figure 3: The HingeDescent Algorithm. Lemma 2: When started at a stable partition, the HingeDescent algorithm will 540 D. R Hush, R Lozano and B. Horne converge to a stable partition of Ep(w) in a finite number of steps. Proof: First note that when R> 0, a QP solution can always be found in a finite number of steps. The proof of this result is beyond the scope of this paper, but can easily be found in the literature (Luenberger, 1984). Now, by design, HingeDescent always moves from one stable partition to the next, maintaining the R > 0 property at each step so that all QP solutions can be produced in a finite number of steps. In addition, Ep(w) is reduced at each step (except the last one) so no partitions are revisited, and since there are a finite number of partitions (see Lemma 1) this algorithm must terminate in a finite number of steps. QED. Assume that QPSol ve runs in O(k( cP + N d)) time as previously stated. Then the run time of HingeDescent is given by O(Np((k+Nh)cP+kNd)), where Nh is the number of samples flipped at each step and Np is the total number of partitions explored. Typical values for k and Nh are on the order of d, simplifying this expression to O(Np(d3 + NcP)). Np can vary widely, but is often substantially less than N. HingeDescent seeks a local minimum over II, and may produce a poor solution, depending on the starting partition. One way to remedy this is to start from several different initial partitions, and then retain the best solution overall. We take a different approach here, that always starts with the same initial condition, visits several local minima along the way, and always ends up with the same final solution each time. The SweepingHinge algorithm works as follows. It starts by placing one of the hinges, say Hinge 1, at the outer boundary of the data. It then sweeps this hinge across the data, M samples at a time (e.g. M = 1), allowing the other hinge (Hinge 2) to descend to an optimal position at each step. The initial hinge locations are determined as follows. A linear fit is formed to the entire data set and the hinges are positioned at opposite ends of the data so that the PLUS and MINUS regions meet the LINEAR region at the two data samples on either end. After the initial linear fit, the hinges are allowed to descend to a local minimum using HingeDescent. Then Hinge 1 is swept across the data M samples at a time. Mechanically this is achieved by moving M additional samples from S, to S+ at each step. Hinge 2 is allowed to descend to an optimal position at each of these steps using the Hinge2Descent algorithm. This algorithm is identical to HingeDescent except that the code that flips samples across Hinge 1 is omitted. The best overall solution from the sweep is retained and "fine-tuned" with one final pass through the HingeDescent algorithm to produce the final solution. The run time of SweepingHinge is no worse than N j M times that of HingeDescent. Given this, an upper bound on the (typical) run time for this algorithm (with M = 1) is O(NNp(d3 + NcP)). Consequently, SweepingHinge scales reasonably well in both Nand d, considering the nature of the problem it is designed to solve. 4 Empirical Results The following experiment was adapted from (Breiman, 1993). The function lex) = e-lIxll 2 is sampled at 100d points {xd such that IIxll ~ 3 and IIxll is uniform on [0,3]. The dimension d is varied from 4 to 10 (in steps of 2) and models of size 1 to 20 nodes are trained using the I1AjSweepingHinge algorithm. The number of samples traversed at each step of the sweep in SweepingHinge was set to M = 10. Nmin was set equal to 3d throughout. A refitting pass was employed after each new node was added in the I1A. The refitting algorithm used HingeDescent to "fine-tune" each node each node before adding the next node. The average sum of squared Function Approximation with the Sweeping Hinge Algorithm 3000 2500 2000 1500 1000 500 d=4 d=6 d=8 d=10 d=4 d=6 d=8 d=10 6 8 10 12 14 16 18 20 Number or Nodes Figure 4: Upper (lower) curves are for training (test) data. 541 error, e-2 , was computed for both the training data and an independent set of test data of size 200d. Plots of 1/e-2 versus the number of nodes are shown in Figure 4. The curves for the training data are clearly bounded below by a linear function of n (as suggested by inverting the O(l/n) result of Barron's). More importantly however, they show no significant dependence on the dimension d. The curves for the test data show the effect of the estimation error as they start to "bend over" around n = 10 nodes. Again however, they show no dependence on dimension. Acknowledgements This work was inspired by the theoretical results of (Barron, 1993) for Sigmoidal networks as well as the "Hinging Hyperplanes" work of (Breiman, 1993) , and the "Ramps" work of (Friedman and Breiman, 1994). This work was supported in part by ONR grant number N00014-95-1-1315. References Barron, A.R. (1993) Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transactions on Information Theory 39(3):930-945. Bellare, M. & Rogaway, P. (1993) The complexity of approximating a nonlinear program. In P.M. Pardalos (ed.), Complexity in numerical optimization, pp. 16-32, World Scientific Pub. Co. Breiman, L. (1993) Hinging hyperplanes for regression, classification and function approximation. IEEE Transactions on Information Theory 39(3):999-1013. Breiman, L. & Friedman, J.H. (1994) Function approximation using RAMPS. Snowbird Workshop on Machines that Learn. Edelsbrunner, H. (1987) In EATCS Monographs on Theoretical Computer Science V. 10, Algorithms in Combinatorial Geometry. Springer-Verlag. Jones, L.K. (1992) A simple lemma on greedy approximation in Hilbert space and convergence rates for projection pursuit regression and neural network training. The Annals of Statistics, 20:608-613. Luenberger, D.G. (1984) Introduction to Linear and Nonlinear Programming. Addison-Wesley.
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Correlates of Attention in a Model of Dynamic Visual Recognition* . Rajesh P. N. Rao Department of Computer Science University of Rochester Rochester, NY 14627 rao@cs.rochester.edu Abstract Given a set of objects in the visual field, how does the the visual system learn to attend to a particular object of interest while ignoring the rest? How are occlusions and background clutter so effortlessly discounted for when recognizing a familiar object? In this paper, we attempt to answer these questions in the context of a Kalman filter-based model of visual recognition that has previously proved useful in explaining certain neurophysiological phenomena such as endstopping and related extra-classical receptive field effects in the visual cortex. By using results from the field of robust statistics, we describe an extension of the Kalman filter model that can handle multiple objects in the visual field. The resulting robust Kalman filter model demonstrates how certain forms of attention can be viewed as an emergent property of the interaction between top-down expectations and bottom-up signals. The model also suggests functional interpretations of certain attentionrelated effects that have been observed in visual cortical neurons. Experimental results are provided to help demonstrate the ability of the model to perform robust segmentation and recognition of objects and image sequences in the presence of varying degrees of occlusions and clutter. 1 INTRODUCTION The human visual system possesses the remarkable ability to recognize objects despite the presence of distractors and occluders in the field of view. A popular suggestion is that an "attentional spotlight" mediates this ability to preferentially process a relevant object in a given scene (see [5, 9] for reviews). Numerous models have been proposed to simulate the control of this .. focus of attention" [10, 11, 15]. Unfortunately, there is inconclusive evidence for the existence of an explicit neural mechanism for implementing an attentional spotlight in the visual *This research was supported by NIH/PHS research grant 1-P41-RR09283. I am grateful to Dana Ballard for many useful discussions and suggestions. Author's current address: The Salk Institute, CNL, 10010 N. Torrey Pines Road, La Jolla, CA 92037. E-mail: rao@salk. edu . . ·· .. \.' .. ' Correlates of Attention in a Model of Dynamic Visual Recognition 81 cortex. Thus, an important question is whether there are alternate neural mechanisms which don't explicitly use a spotlight but whose effects can nevertheless be interpreted as attention. In other words, can attention be viewed as an emergent property of a distributed network of neurons whose primary goal is visual recognition? In this paper, we extend a previously proposed Kalman filter-based model of visual recognition [13, 12] to handle the case of multiple objects, occlusions, and clutter in the visual field. We provide simulation results suggesting that certain forms of attention can be viewed as an emergent property of the interaction between bottom-up signals and top-down expectations during visual recognition. The simulation results demonstrate how "attention" can be switched between different objects in a visual scene without using an explicit spotlight of attention. 2 A KALMAN FILTER MODEL OF VISUAL RECOGNITION We have previously introduced a hierarchical Kalman filter-based model of visual recognition and have shown how this model can be used to explain neurophysiological effects such as endstopping and neural response suppression during free-viewing of natural images [ 12, 13 ]. The Kalman filter [7] is essentially a linear dynamical system that attempts to mimic the behavior of an observed natural process. At any time instant t, the filter assumes that the internal state of the given natural process can be represented as a k x 1 vector r( t). Although not directly accessible, this internal state vector is assumed to generate ann x 1 measurable and observable output vector I( t) (for example, an image) according to: I(t) = Ur(t) + n(t) (1) where U is ann x k generative (or measurement) matrix, and n(t) is a Gaussian stochastic noise process with mean zero and a covariance matrix given by E = E[nilT] (E denotes the expectation operator and T denotes transpose). In order to specify how the internal state r changes with time, the Kalman filter assumes that the process of interest can be modeled as a Gauss-Markov random process [1]. Thus, given the state r(t- 1) at time instant t- 1, the next state r(t) is given by: r(t) = Vr(t- 1) + m(t- 1) (2) where Vis the state transition (or prediction) matrix and m is white Gaussian noise with mean m = E[m] and covariance II= E[(m- m)(m- m)T]. Given the generative model in Equation 1 and the dynamics in Equation 2, the goal is to optimally estimate the current internal state r( t) using only the measurable inputs I( t). An optimization function whose minimization yields an estimate of r is the weighted least-squares criterion: J = (I- Ur)TE-1 (I- Ur) + (r- rf M-1(r- r) (3) where r( t) is the mean of the state vector before measurement of the input data I( t) and M = E[(r - r)(r - r)T] is the corresponding covariance matrix. It is easy to show [1] that J is simply the sum of the negative log-likelihood of generating the data I given the stater, and the negative log of the prior probability of the stater. Thus, minimizing J is equivalent to maximizing the posterior probability p(rji) of the stater given the input data. The optimization function J can be minimized by setting ~J = 0 and solving for the minimum valuer of the stater (note thatr equals the mean ofr afte~ measurement of I). The resultant Kalman .filter equation is given by: · r(t) = r(t) + N(t)UTE(t)-1 (I(t)- Ur(t)) (4) r(t) = W(t- 1) + m(t- 1) (5) where N(t) = (UTE(t)-1U + M(t)-1 )-1 is a "normalization" matrix that maintains the covariance of the stater after measurement ofl. The matrix M, which is the covariance before 82 R. P. N. Rao measurement of I, is updated as M ( t) = V N ( t - 1) VT +II ( t - 1). Thus, the Kalman filter predicts one step into the future using Equation 5, obtains the next sensory input I( t), and then co~ects its Jj.rediction r(9 u~ing the sensory resi~ual e~or (I(t) --: Ur(t)) a~d ~e Kalman gam N(t)U ~(t)- 1 • This yields the corrected est1mate r(t) (Equation 4), wh1ch IS then used to make the next state predictionf(t + 1). The measurement (or generative) matrix U and the state transition (or prediction) matrix V used by the Kalman filter together encode an internal nwdel of the observed dynamic process. As suggested in [13], it is possible to learn an internal model of the input dynamics from observed data. Let u and v denote the vectorized forms of the matrices U and V respectively. For exam,fle, the n ~ k generative matrix U can be collapsed into an nk x 1 vector u = [U1U2 ••. un] where U' denotes the ith row of U. Note that (I- Ur) = (I- Ru) where R is then x nk matrix given by: = [ r; r~ ~ l R . . . . . . 0 0 rT (6) By minimizing an optimization function similar to J [13], one can derive a Kalman filter-like .. learning rule" for the generative matrix U: u(t) = ii(t) + Nu(t)R(t)T~(t)- 1 (I(t)- R(t)ii(t))- aNu(t)ii(t) (7) where ii(t) = u(t- 1), Nu(t) = (Nu(t- 1)-1 + R(t)T~(t)- 1 R(t) + ai)-1 , and I is the nk x nk identity matrix. The constant a determines the decay rate of ii. As in the case of U, an estimate of the prediction matrix V can be obtained via the following learning rule for v [13]: v(t) = v(t) + Nv(t)R(tf M(t)-1 [r(t + 1)- r(t + 1)]- f3Nv(t)v(t) (8) wherev(t) = v(t-1), Nv(t) = (Nv(t-1)- 1 +R(t)T M(t)-1R(t)+f3I)-1 andRisak X k2 matrix analogous to R (Equation 6) but with rT = rr. The constant /3 determines the decay rate for v while I denotes the k2 x k2 identity matrix. Note that in this case, the estimate ofV is corrected using the prediction residual error ( r( t + 1) - r( t + 1)), which denotes the difference between the actual state and the predicted state. One unresolved issue is the specification of values for r(t) (comprising R(t)) in Equation 7 and r(t + 1) in Equation 8. The ExpectationMaximization (EM) algorithm [4] suggests that in the case of static stimuli (f(t) = r(t 1)), one may use r(t) = r which is the converged optimal state estimate for the given static input. In the case of dynamic stimuli, the EM algorithm prescribes r( t) = r( tl N), which is the optimal temporally snwothed state estimate [1] for timet (5 N), given input data for each of the time instants 1, ... , N. Unfortunately, the smoothed estimate requires knowledge of future inputs and is computationally quite expensive. For the experimental results, we used the on-line estimates r(t) when updating the matrices u and v during training. 3 ROBUST KALMAN FILTERING The standard derivation of the Kalman filter minimizes Equation 3 but unfortunately does not specify how the covariance ~ is to be obtained. A common choice is to use a constant matrix or even a constant scalar. Making ~ constant however reduces the Kalman filter estimates to standard least-squares estimates,· which are highly susceptible to outliers or gross errors i.e. data points that lie far away from the bulk of the observed or predicted data [ 6]. For example, in the case where I represents an input image, occlusions and clutter will cause many pixels in I to deviate significantly from corresponding pixels in the predicted image Ur. The problem Correlates of AJtention in a Model of Dynamic Visual Recognition 83 Gating Feedforward Matrix Matrix Sensory G uT Robust Residual Kalman Filter I- ltd Estimate Nonnalization " Prediction Inhibition r0 r Matrix ~ Input I N v Itd=Ur Feedback Matrix Top-Down Prediction u Predicted State r of Ex ted n ut pee I p Figure 1: Recurrent Network Implementation of the Robust Kalman Filter. The gating matrix G is a non-linear function of the current residual error between the input I and its top-down prediction ur. G effectively filters out any high residuals, thereby preventing outliers in input data I from influencing the robust Kalman filter estimate r. Note that the entire filter can be implemented in a recurrent neural network, with U, UT, and V represented by the synaptic weights of neurons with linear activation functions and G being implemented by a set of threshold non-linear neurons with binary outputs. of outliers can be tackled using robust estimation procedures [6] such as M-estimation, which involves minimizing a function of the form: n (9) i=1 where Ii and Ui are the ith pixel and ith row of I and U respectively, and p is a function that increases less rapidly than the square. This reduces the influence oflarge residual errors (which correspond to outliers) on the optimization of J', thereby "rejecting" the outliers. A special case of the above function is the following weighted least squares criterion: J' =(I-:- Urf S(I- Ur) (10) where Sis a diagonal matrix whose diagonal entries Si,i determine the weight accorded to the corresponding pixel error (Ii - Uir). A simple but attractive choice for these weights is the non-linear function given by Si,i =min {1, c/(Ii- Uir)2}, wherecisa threshold parameter. To understand the behavior of this function, note that S effectively clips the ith summand in J' (Equation 10 above) to a constant value c whenever the ith squared residual (Ii - Uir? exceeds the threshold c; otherwise, the summand is set equal to the squared residual. By substituting E-1 = Sin the optimization function J (Equation 3), we can rederive the following robust Kalman filter equation: r(t) = r(t) + N(t)UTG(t)(I- Ur(t)) (11) where r(t) = W(t- 1)) + iii(t- 1), N(t) = (UTG(t)U + M(t)-1)- 1 , M(t) = V N(t1)VT + II(t -1), and G(t) is ann x n diagonal matrix whose diagonal entries at time instant t are given by: Gi,i _ { 0 if (Ii(t) - Uir(t))2 > c(t) 1 otherwise G can be regarded as the sensory residual gain or "gating" matrix, which determines the (binary) gain on the various components of the incoming sensory residual error vector. By effectively filtering out any high residuals, G allows the Kalman filter to ignore the corresponding outliers in the input I, thereby enabling it to robustly estimate the stater. Figure 1 depicts an implementation of the robust Kalman filter in the form of a recurrent network of linear and threshold non-linear neurons. In particular, the feedforward, feedback and prediction neurons possess linear activation functions while the gating neurons implementing G compute binary outputs based on a threshold non-linearity. 84 Training Objects (a) Input Image Robust Estimate I Input Image Outliers (c) I ' Robust Estimate (b) Robust Estimate 2 I. R. P.N.Rao Outliers Least Squares Estimate Figure 2: Correlates of Attention during Static Recognition. (a) Images of size 105 x 65 used to train a robust Kalman filter network. The generative matrix U was 6825 x 5. (b) Occlusions and background clutter are treated as outliers (white regions in the third image, depicting the diagonal of the gating matrix G). This allows the network to "attend to" and recognize the training object, as indicated by the accurate reconstruction (middle image) of the training image based on the final robust state estimate. (c) In the more interesting case of the training objects occluding each other, the network converges to one of the objects (the "dominant" one in the image -in this case, the object in the foreground). Having recognized one object, the second object is attended to and recognized by taking the complement of the outliers ( diagonal of G) and repeating the robust filtering process (third and fourth images). The fifth image is the image reconstruction obtained from the standard (least squares derived) Kalman filter estimate, showing an inability to resolve or recognize either of the two objects. 4 VISUAL ATIENTION IN A SIMULATED NETWORK The gating matrix G allows the Kalman filter network to "selectively attend .. to an object while treating the remaining components of the sensory input as outliers. We demonstrate this capability of the network using three different examples. In the first example, a network was trained on static grayscale images of a pair of 3D objects (Figure 2 (a)}. For learning static inputs, the prediction matrix Vis unnecessary since we may use r(t) = r(t -1) and M(t) = N(t -1). After training, the network was tested on images containing the training objects with varying degrees of occlusion and clutter (Figure 2 (b) and (c)). The outlier threshold c was initialized to the sum of the mean plus k standard deviations of the current distribution of squared residual errors (Ii - Uir )2 • The value of k was gradually decreased during each iteration in order to allow the network to refine its robust estimate by gradually pruning away the outliers as it converges to a single object estimate. After convergence, the diagonal of the matrix G contains zeros in the image locations containing the outliers and ones in the remaining locations. As shown in Figure 2 (b), the network was successful in recognizing the training object despite occlusion and background clutter. More interestingly, the outliers (white) produce a crude segmentation of the occluder and background clutter, which can subsequently .be used to focus "attention" on these previously ignored objects and recover their identity. In particular, an outlier mask m can be defined by taking the complement of the diagonal of G (i.e. m i = 1-Gi,i). By replacing the diagonal of G with m in Equation tt 1 and repeating the estimation process, the network can "attend to" 1 Although not implemented here, this "shifting of attentional focus" can be automated using a model of neuronal fatigue and active decay (see, for example, [3)). Correlates of AUention in a Model of Dynamic Visual Recognition 85 laput ~ Oulllen ..... I ~~~~~ -~ ~/' ~ . -------·--~- ·.. J~r5' ~ ~ (a) ••Ill I••• Predictions Outliers (c) (b) Inputs Predictions Outliers (d) Figure 3: Correlates of Attention during Dynamic Recognition. (a) A network was trained on a cyclic image sequence of gestures (top), each image of size 75 x 75, with U and V of size 5625 x 15 and 15 x 15 respectively. The panels below show how the network can ignore various fonns of occlusion and clutter (outliers}, "attending to" the sequence of gestures that it has been trained on. The outlier threshold c was computed as the mean plus 0.3 standard deviati9ns of the current distribution of squared residual errors. Results shown are those obtained after 5 cycles of exposure to the occluded images. (b) Three image sequences used to train a network. (c) and (d) show the response of the network to ambiguous stimuli comprised of images containing both a horizontal and a vertical bar. Note that the network was trained on a horizontal bar moving downwards and·a vertical bar moving rightwards (see (b)) but not both simultaneously. Given ambiguous stimuli containing both these stimuli, the network interprets the input differently depending on the initial "priming" input. When the initial input is a vertical bar as in (c), the network interprets the sequence as a vertical bar moving rightwards (with some minor artifacts due to the other training sequences). On the other hand, when the initial input is a horizontal bar as in (d), the sequence is interpreted as a horizontal bar moving downwards, not paying "attention" to the extraneous vertical bars, which are now treated as outliers. the image region(s) that were previously ignored as outliers. Such a two-step serial recognition process is depicted in Figure 2 (c). The network first recognizes the "doniinant" object, which was generally observed to be the object occupying a larger area of the input image or possessing regions with higher contrast.· The outlier mask m is subsequently used for "switching attention" and extracting the identity of the second object (lower arrow). Figure 3 shows examples of attention during recognition of dynamic stimuli. In particular, Figure 3 (c) and (d) show how the same image sequence can be interpreted in two different ways depending on which part of the stimulus is "attended to," which in tum depends on the initial priming input. 5 CONCLUSIONS The simulation results indicate that certain· experimental observations that have previously been interpreted using the metaphor of an attentional spotlight can also arise as a result of competition and cooperation during visual recognition within networks of linear and non-linear 86 R. P. N. Rao neurons. Although not explicitly designed to simulate attention, the robust Kalman filter networks nevertheless display some of the essential characteristics of visual attention, such as the preferential processing of a subset of the input signals and the consequent "switching" of attention to previously ignored stimuli. Given multiple objects or conflicting stimuli in their receptive fields (Figures 2 and 3), the responses of the feedforward, .feedback, and prediction neurons in the simulated network were modulated according to the current object being "attended to." The modulation in responses was mediated by the non-linear gating neurons G, taking into account both bottom-up signals as well top-down feedback signals. This suggests a network-level interpretation of similar forms of attentional response modulation in the primate visual cortex [2, 8, 14], with the consequent prediction that the genesis of attentional modulation in such cases may not necessarily lie within the recorded neurons themselves but within the distributed circuitry that these neurons are an integral part of. References [1] A.B. Bryson and Y.-C. Ho. Applied Optimtzl Control. New York: John Wiley, 1975. [2] L. Chelazzi, E.K. Miller, J. Duncan, and R. Desimone. A neural basis for visual search in inferior temporal cortex. Nature, 363:345-347, 1993. [3] P. Dayan. An hierarchical model of visual rivalry. In M. Mozer, M. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems 9, pages 48-54. Cambridge, MA: MIT Press, 1997. [4] A.P. Dempster, N.M. Laird, andD.B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. J. Royal Statistical Society Series B, 39:1-38, 1977. [5] R. Desimone and J. Duncan. Neural mechanisms of selective visual attention. Annual Review of Neuroscience, 18:193-222,1995. [6] P.J. Huber. Robust Statistics. New York: John Wiley, 1981. [7] R.E. Kalman. A new approach to linear filtering and prediction theory. Trans. ASME J. Basic Eng., 82:35-45, 1960. [8] J. Moran and R. Desimone. Selective attention gates visual processing in the extrastriate cortex. Science, 229:782-784, 1985. [9] W.T. Newsome. Spotlights, highlights and visual awareness. Current Biology, 6( 4 ):357360, 1996. [10] E. Niebur and C. Koch. Control of selective visual attention: Modeling the "where" pathway. In D. Touretzky, M. Mozer, and M. Hasselmo, editors, Advances in Neural lnformtztion Processing Systems 8, pages 802-808. Cambridge, MA: MIT Press, 1996. [11] B.A. Olshausen, D.C. Van Essen, and C.H. Anderson. A neurobiological model of visual attention and invariant pattern recognition based on dynamic routing of information. Journal of Neuroscience, 13:4700-4719, 1993. [12] R.P.N. Rao and D.H. Ballard. The visual cortex as a hierarchical predictor. Technical Report 96.4, National Resource Laboratory for the Study of Brain and Behavior, Department of Computer Science, University of Rochester, September 1996. [13] R.P.N. Rao and D.H. Ballard. Dynamic model of visual recognition predicts neural response properties in the visual cortex. Neural Computation, 9(4):721-763, 1997. [14] S. Treue and J.H.R. Maunsell. Attentional modulation of visual motion processing in cortical areas MT and MST. Nature, 382:539-541, 1996. [15] J .K. Tsotsos, S.M. Culhane, W. Y.K. Wai, Y. Lai, N. Davis, and F. Nuflo. Modeling visual attention via selective tuning. Artificial Intelligence, 78:507~545, 1995.
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Mapping a manifold of perceptual observations Joshua B. Tenenbaum Department of Brain and Cognitive Sciences Massachusetts Institute of Technology, Cambridge, MA 02139 jbt@psyche.mit.edu Abstract Nonlinear dimensionality reduction is formulated here as the problem of trying to find a Euclidean feature-space embedding of a set of observations that preserves as closely as possible their intrinsic metric structure - the distances between points on the observation manifold as measured along geodesic paths. Our isometric feature mapping procedure, or isomap, is able to reliably recover low-dimensional nonlinear structure in realistic perceptual data sets, such as a manifold of face images, where conventional global mapping methods find only local minima. The recovered map provides a canonical set of globally meaningful features, which allows perceptual transformations such as interpolation, extrapolation, and analogy - highly nonlinear transformations in the original observation space - to be computed with simple linear operations in feature space. 1 Introduction In psychological or computational research on perceptual categorization, it is generally taken for granted that the perceiver has a priori access to a representation of stimuli in terms of some perceptually meaningful features that can support the relevant classification. However, these features will be related to the raw sensory input (e.g. values of retinal activity or image pixels) only through a very complex transformation, which must somehow be acquired through a combination of evolution, development, and learning. Fig. 1 illustrates the featurediscovery problem with an example from visual perception. The set of views of a face from all possible viewpoints is an extremely high-dimensional data set when represented as image arrays in a computer or on a retina; for example, 32 x 32 pixel grey-scale images can be thought of as points in a 1 ,024-dimensional observation space. The perceptually meaningful structure of these images, however, is of much lower dimensionality; all of the images in Fig. 1 lie on a two-dimensional manifold parameterized by viewing angle. A perceptual system that discovers this manifold structure has learned a model of the appearance of this face that will support a wide range of recognition, classification, and imagery tasks (some demonstrated in Fig. 1), despite the absence of any prior physical knowledge about three-dimensional object geometry, surface texture, or illumination conditions. Learning a manifold of perceptual observations is difficult because these observations c: o ~ > <1) (i) Mapping a Manifold of Perceptual ObseIVations 683 : : " .. . . • •. Wi 11 L..... .... ------ .. --. --...; .... "-.... ...... --.. ~. ----. --... ... --i:::: ......... ~. ~--------------------------------------- ----~. l.. ~.-p--.. _--,.---' : ~ • C! azimuth A [11---.---.---.--- ---B. B fll---fii-{ii iI • • • cll-. W· Figure 1: Isomap recovers a global topographic map of face images varying in two viewing angle parameters, azimuth and elevation. Image interpolation (A), extrapolation (B), and analogy (C) can then be carried out by linear operations in this feature space. usually exhibit significant nonlinear structure. Fig. 2A provides a simplified version of this problem. A flat two-dimensional manifold has been nonlinearly embedded in a threedimensional observation space, 1 and must be "unfolded" by the learner_ For linearly embedded manifolds. principal component analysis (PCA) is guaranteed to discover the dimensionality of the manifold and produce a compact representation in the form of an orthonormal basis_ However, PCA is completely insensitive to the higher-order. nonlinear structure that characterizes the points in Fig. 2A or the images in Fig. 1. Nonlinear dimensionality reduction - the search for intrinsically low-dimensional structures embedded nonlinearly in high-dimensional observations - has long been a goal of computational learning research. The most familiar nonlinear techniques. such as the self-organizing map (SOM; Kohonen, 1988), the generative topographic mapping (GTM; Bishon, Svensen, & Williams, 1998), or autoencoder neural networks (DeMers & Cottrell, 1993), try to generalize PCA by discovering a single global low-dimensional nonlinear model of the observations. In contrast, local methods (Bregler & Omohundro. 1995; Hinton, Revow, & Dayan, 1995) seek a set of low-dimensional models, usually linear and hence valid only for a limited range of data. When appropriate, a single global model is IGiven by XI = ZI COS(ZI), X2 = ZI sin(zJ), X3 = Z2, for Zl E [311"/2,911"/2], Z2 E [0,15]. 684 J. B. Tenenbaum A B c 10 10 10 o o o o 10 o 10 10 10 Figure 2: A nonlinearly embedded manifold may create severe local minima for "top-down" mapping algorithms. (A) Raw data. (B) Best SOM fit. (C) Best GTM fit. more revealing and useful than a set of local models. However, local linear methods are in general far more computationally efficient and reliable than global methods. For example, despite the visually obvious structure in Fig. 2A, this manifold was not successfuly modeled by either of two popular global mapping algorithms, SOM (Fig. 2B) and GTM (Fig. 2C), under a wide range of parameter settings. Both of these algorithms try to fit a grid of predefined (usually two-dimensional) topology to the data, using greedy optimization techniques that first fit the large-scale (linear) structure of the data, before making small-scale (nonlinear) refinements. The coarse structure of such "folded" data sets as Fig. 2A hides their nonlinear structure from greedy optimizers, virtually ensuring that top-down mapping algorithms will become trapped in highly suboptimal solutions. Rather than trying to force a predefined map onto the data manifold, this paper shows how a perceptual system may map a set of observations in a "bottom-up" fashion, by first learning the topological structure of the manifold (as in Fig. 3A) and only then learning a metric map of the data (as in Fig. 3C) that respects this topology. The next section describes the goals and steps of the mapping procedure, and subsequent sections demonstrate applications to two challenging learning tasks: recovering a five-dimensional manifold embedded nonlinearly in 50 dimensions, and recovering the manifold of face images depicted in Fig. I. 2 Isometric feature mapping We assume our data lie on an unknown manifold M embedded in a high-dimensional observation space X. Let xci) denote the coordinates of the ith observation. We seek a mapping I : X Y from the observation space X to a low-dimensional Euclidean feature space Y that preserves as well as possible the intrinsic metric structure of the observations, i.e. the distances between observations as measured along geodesic (locally shortest) paths of M . The isometric feature mapping, or isomap, procedure presented below generates an implicit description of the mapping I, in terms of the corresponding feature points y(i) = I( xci)) for sufficiently many observations x(i). Explicit parametric descriptions of I or I-I can be found with standard techniques of function approximation (Poggio & Girosi, 1990) that interpolate smoothly between the known corresponding pairs {x( i) , y( i)}. A Euclidean map of the data's intrinsic geometry has several important properties. First, intrinsically similar observations should map to nearby points in feature space, supporting efficient similarity-based classification and informative visualization. Moreover, the geodesic paths of the manifold, which are highly nonlinear in the original observation space, should map onto straight lines in feature space. Then perceptually natural transfonnations along these paths, such as the interpolation, extrapolation and analogy demonstrated in Figs. IA-C, may be computed by trivial linear operations in feature space. Mapping a Manifold of Perr:eptual Observations 685 A B C ~bU~ 1~9 ~ 4 ~ ~ (lJ?t '0 c!Je~r!~ Q ~ ~ Qligo)", 1!@> 2 1t:°·1·\~ 0 "'. rt;,.. tje. 'fJ o . . . . . . ~ 10 Manifold Distance 5 10 15 Figure 3: The results of the three-step isomap procedure. (A) Discrete representation of manifold in Fig. 2A. (B) Correlation between measured graph distances and true manifold distances. (C) Correspondence of recovered two-dimensional feature points {Yl, Y2} (circles) with original generating vectors {ZI' Z2} (line ends). The isomap procedure consists of three main steps, each of which might be carried out by more or less sophisticated techniques. The crux of isomap is finding an efficient way to compute the true geodesic distance between observations, gi ven only their Euclidean distances in the high-dimensional observation space. Isomap assumes that distance between points in observation space is an accurate measure of manifold distance only locally and must be integrated over paths on the manifold to obtain global distances. As preparation for computing manifold distances, we first construct a discrete representation of the manifold in the form of a topology-preserving network (Fig. 3A). Given this network representation, we then compute the shortest-path distance between any two points in the network using dynamic programming. This polynomial-time computation provides a good approximation to the actual manifold distances (Fig. 3B) without having to search over all possible paths in the network (let alone the infinitely many paths on the unknown manifold!). Finally, from these manifold distances, we construct a global geometry-preserving map of the observations in a low-dimensional Euclidean space, using multidimensional scaling (Fig. 3C). The implementation of this procedure is detailed below. Step 1: Discrete representation of manifold (Fig. 3A). From the input data of n observations {x(1) , . • . , xC n)}, we randomly select a subset of T points to serve as the nodes {g(1) , .. . , gC r)} of the topology-preserving network. We then construct a graph G over these nodes by connecting gCi) and g(;) if and only if there exists at least one xCk) whose two closest nodes (in observation space) are gC i) and gCi) (Martinetz & Schulten, 1994). The resulting graph for the data in Fig. 2A is shown in Fig. 3A (with n = 104, T = )03). This graph clearly respects the topology of the manifold far better than the best fits with SOM (Fig. 2B) or GTM (Fig. 2C). In the limit of infinite data, the graph thus produced converges to the Delaunay triangulation of the nodes, restricted to the data manifold (Martinetz & Schulten, 1994). In practice, n = 104 data points have proven sufficient for all examples we have tried. This number may be reduced significantly if we know the dimensionality d of the manifold, but here we assume no a priori information about dimensionality. The choice of T, the number of nodes in G, is the only free parameter in isomap. If T is too small, the shortest-path distances between nodes in G will give a poor approximation to their true manifold distance. If T is too big (relative to n), G will be missing many appropriate links (because each data point XCi) contributes at most one link). In practice, choosing a satisfactory T is not difficult - all three examples presented in this paper use T = n /10, the first value tried. I am currently exploring criteria for selecting the optimal value T based on statistical arguments and dimensionality considerations. Step 2: Manifold distance measure (Fig. 3B). We first assign a weight to each link w,) in the graph G, equal to d1 = I\xCi) - xC)I\, the Euclidean distance between nodes i and j in the observation space X . The length of a path in G is defined to be the sum of link weights along that path. We then compute the geodesic distance d~ (i.e. shortest path length) between all pairs of nodes i and j in G, using Floyd's O( T 3 ) algorithm (Foster, 1995). Initialize d& = d1 if nodes i and j are connected 686 J. B. Tenenbaum A B Figure 4: Given iii 1 iii 1 lI! a ::> Oisomap ::> .PCA 5-dimensional manifold :2 "0 )1, m 'iii xMDS embedded nonlinearly in Q) Q) x '. a: a: a 50-dimensional space, -g 0,5 -g 0,5 ' '. 1 N X " isomap identifies the 'a ., x , '. intrinsic dimensionality E x'x','., 0 0 'X" (A), while PCA and z 0 z 0 0 5 10 0 5 10 MDS alone do not (B). Dimension Dimension and 00 otherwise. Then for each node k, set each d~ = min(d~, d~ + d"d). Fig. 3B plots the distances d~ computed between nodes i and j in the graph of Fig. 3A versus their actual manifold distances d~. Note that the correlation is almost perfect (R > .99), but d~ tends to overestimate d~ by a constant factor due to the discretization introduced by the graph. As the density of observations increases, so does the possible graph resolution. Thus, in the limit of infinite data, the graph-based approximation to manifold distance may be made arbitrarily accurate. Step 3: Isometric Euclidean embedding (Fig. 3C). We use ordinal multidimensional scaling (MOS; Cox & Cox, 1994; code provided by Brian Ripley), also called "non metric " MOS, to find a kdimensional Euclidean embedding that preserves as closely as possible the graph distances d~. In contrast to classical "metric" MOS, which explicitly tries to preserve distances, ordinal MOS tries to preserve only the rank ordering of distances. MOS finds a configuration of k-dimensional feature vectors {y(1) •...• y( r)}, corresponding to the high-dimensional observations {x(I), ... • x(r)}, that minimizes the stress function, S = min d;1 (1) Here d~ = II y( i) yU) II, the Euclidean distance between feature vectors i and j, and the d~ are some monotonic transformation of the graph distances d~. We use ordinal MOS because it is less senstitive to noisy estimates of manifold distance. Moreover, when the number of points scaled is large enough (as it is in all our examples), ordinal constraints alone are sufficient to reconstruct a precise metric map. Fig. 3C shows the projections of 100 random points on the manifold in Fig, 2A onto a two-dimensional feature space computed by MOS from the graph distances output by step 2 above. These points are in close correspondence (after rescaling) with the original two-dimensional vectors used to generate the manifold (see note 1), indicating that isomap has successfully unfolded the manifold onto a 2-dimensional Euclidean plane. 3 Example 1: Five-dimensional manifold This section demonstrates isomap's ability to discover and model a noisy five-dimensional manifold embedded within a 50-dimensional space. As the dimension of the manifold increases beyond two, SOM, GTM, and other constrained clustering approaches become impractical due to the exponential proliferation of cluster centers. Isomap, however, is quite practical for manifolds of moderate dimensionality, because the estimates of manifold distance for a fixed graph size degrade gracefully as dimensionality increases. Moreover, isomap is able to automatically discover the intrinsic dimensionality of the data, while conventional methods must be initialized with a fixed dimensionality. We consider a 5-dimensional manifold parameterized by {Z\, . . " zs} E [0,4]5. The first 10 of 50 observation dimensions were determined by nonlinear functions of these parameters. 2 2XI = cos( 1rzt}, X2 = sine 1rzI), X3 = cose; zI), X4 = sine; zI), Xs = cos( fzI), X6 = sin(fzl), X7 = z2cos\j~'zl)+z3sin2(lizl)' X8 = z2sin2(lizI)+Z3COS2(~zt}, X9 = Z4 cos2(lizt} + Zs sin2(~zl)' XIO = Z4 sin\j~'zl) + Zs COS2(~ZI)' Mapping a Manifold of Perceptual Observations 687 Low-amplitude gaussian noise (4-5% of variance) was added to each of these dimensions, and the remaining 40 dimensions were set to pure noise of similar variance. The isomap procedure applied to this data (n = 104 , r = 103) correctly recognized its intrinsic fivedimensionality, as indicated by the sharp decrease of stress (see Eq. 1) for embedding dimensions up to 5 and only gradual decrease thereafter (Fig. 4A). In contrast, both PCA and raw MDS (using distances in observation space rather than manifold distances) identify the lO-dimensional linear subspace containing the data, but show no sensitivity to the underlying five-dimensional manifold (Fig. 4B). 4 Example 2: 1Wo-dimensional manifold of face images This section illustrates the performance of isomap on the two-dimensional manifold of face images shown in Fig. 1. To generate this map, 32 x 32-pixel images of a face were first rendered in MATLAB in many different poses (azimuth E [-90°,90°], elevation E [-10°, 10°]), using a 3-D range image of an actual head and a combination oflambertian and specular reflectance models. To save computation, the data (n = 104 images) were first reduced to 60 principal components and then submitted to isomap (r = 103). The plot of stress S vs. dimension indicated a dimensionality of two (even more clearly than Fig. 4A). Fig. 1 shows the two-dimensional feature space that results from applying MDS to the computed graph distances, with 25 face images placed at their corresponding points in feature space. Note the clear topographic representation of similar views at nearby feature points. The principal axes of the feature space can be identified as the underlying viewing angle parameters used to generate the data. The correlations of the two isomap dimensions with the two pose angles are R = .99 and R = .95 respectively. No other global mapping procedure tried (PCA, MDS, SOM, GTM) produced interpretable results for these data. The human visual system's implicit knowledge of an object's appearance is not limited to a representation of view similarity, and neither is isomap's. As mentioned in Section 2, an isometric feature map also supports analysis and manipulation of data, as a consequence of mapping geodesics of the observation manifold to straight lines in feature space. Having found a number of corresponding pairs {x( i) , y( i)} of images x( i) and feature vectors y( i) , it is easy to learn an explicit inverse mapping 1-1 : y -+ X from low-dimensional feature space to high-dimensional observation space, using generic smooth interpolation techniques such as generalized radial basis function (GRBF) networks (Poggio & Girosi, 1990). All images in Fig. 1 have been synthesized from such a mapping. 3 Figs. lA-C show how learning this inverse mapping allows interpolation, extrapolation, and analogy to be carried out using only linear operations. We can interpolate between two images x(l) and x(2) by synthesizing a sequence of images along their connecting line (y(2) _ yO) in feature space (Fig. lA). We can extrapolate the transformation from one image to another and far beyond, by following the line to the edge of the manifold (Fig. IB). We can map the transformation between two images xCI) and x(2) onto an analogous transformation of another image x(3), by adding the transformation vector (y(2) y(1» to y(3) and synthesizing a new image at the resulting feature coordinates (Fig. 1 C). A number of authors (Bregler & Omohundro, 1995; Saul & Jordan, 1997; Beymer & Poggio, 1995) have previously shown how learning from examples allows sophisticated 3The map from feature vectors to images was learned by fitting a GRBF net to 1000 corresponding points in both spaces. Each point corresponds to a node in the graph G used to measure manifold distance, so the feature-space distances required to fit the GRBF net are given (approximately) by the graph distances d~ computed in step 2 of isomap. A subset C of m = 300 points were randomly chosen as RBF centers, and the standard deviation of the RBFs was set equal to max;,jEC d~rJ2m (as prescribed by Haykin, 1994). 688 J. B. Tenenbaum image manipulations to be carried out efficiently. However, these approaches do not support as broad a range of transformations as isomap does, because of their use of only locally valid models and/or the need to compute special-purpose image features such as optical flow. See Tenenbaum (1997) for further discussion, as well as examples of isomap applied to more complex manifolds of visual observations. 5 Conclusions The essence of the isomap approach to nonlinear dimensionality reduction lies in the novel problem formulation: to seek a low-dimensional Euclidean embedding of a set of observations that captures their intrinsic similarities, as measured along geodesic paths of the observation manifold. Here I have presented an efficient algorithm for solving this problem and shown that it can discover meaningful feature-space models of manifolds for which conventional "top-down" approaches fail. As a direct consequence of mapping geodesics to straight lines in feature space, isomap learns a representation of perceptual observations in which it is easy to perform interpolation and other complex transformations. A negative consequence of this strong problem formulation is that isomap will not be applicable to every data manifold. However, as with the classic technique of peA, we can state clearly the general class of data for which isomap is appropriate - manifolds with no "holes" and no intrinsic curvature - with a guarantee that isomap will succeed on data sets from this class, given enough samples from the manifold. Future work will focus on generalizing this domain of applicability to allow for manifolds with more complex topologies and significant curvature, as would be necessary to model certain perceptual manifolds such as the complete view space of an object. Acknowledgements Thanks to M. Bernstein, W. Freeman, S. Gilbert, W. Richards, and Y. Weiss for helpful discussions. The author is a Howard Hughes Medical Institute Predoctoral Fellow. References Beymer, D. & Poggio, T. (1995). Representations for visual learning, Science 272,1905. Bishop, c., Svensen, M., & Williams, C. (1998). GTM: The generative topographic mapping. Neural Computation 10(1). Bregler, C. & Omohundro, S. (1995). Nonlinear image interpolation using manifold learning. NIPS 7. MIT Press. Cox, T. & Cox, M. (1994). Multidimensional scaling. Chapman & Hall. DeMers, D. & Cottrell, G. (1993). Nonlinear dimensionality reduction. NIPS 5. Morgan Kauffman. Foster, I. (1995). Designing and building parallel programs. Addison-Wesley. Hayldn, S. (1994). Neural Networks: A Comprehensive Foundation. Macmillan. Hinton. G., Revow. M .• & Dayan, P. (1995). Recognizing handwritten digits using mixtures of linear models. NIPS 7. MIT Press. Kohonen. T. (1988). Self-Organization and Associative Memory. Berlin: Springer. Martinetz. T. & Schulten, K. (1994). Topology representing networks. Neural Networks 7, 507. Poggio, T. & Girosi. F. (1990). Networks for approximation and learning. Proc. IEEE 78, 1481 . Saul, L. & Jordan. M. (1997). A variational principle for model-based morphing. NIPS 9. MIT Press. Tenenbaum, J. (1997). Unsupervised learning of appearance manifolds. Manuscript submitted.
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The Rectified Gaussian Distribution N. D. Socci, D. D. Lee and H. S. Seung Bell Laboratories, Lucent Technologies Murray Hill, NJ 07974 {ndslddleelseung}~bell-labs.com Abstract A simple but powerful modification of the standard Gaussian distribution is studied. The variables of the rectified Gaussian are constrained to be nonnegative, enabling the use of nonconvex energy functions. Two multimodal examples, the competitive and cooperative distributions, illustrate the representational power of the rectified Gaussian. Since the cooperative distribution can represent the translations of a pattern, it demonstrates the potential of the rectified Gaussian for modeling pattern manifolds. 1 INTRODUCTION The rectified Gaussian distribution is a modification of the standard Gaussian in which the variables are constrained to be nonnegative. This simple modification brings increased representational power, as illustrated by two multimodal examples of the rectified Gaussian, the competitive and the cooperative distributions. The modes of the competitive distribution are well-separated by regions of low probability. The modes of the cooperative distribution are closely spaced along a nonlinear continuous manifold. Neither distribution can be accurately approximated by a single standard Gaussian. In short, the rectified Gaussian is able to represent both discrete and continuous variability in a way that a standard Gaussian cannot. This increased representational power comes at the price of increased complexity. While finding the mode of a standard Gaussian involves solution of linear equations, finding the modes of a rectified Gaussian is a quadratic programming problem. Sampling from a standard Gaussian can be done by generating one dimensional normal deviates, followed by a linear transformation. Sampling from a rectified Gaussian requires Monte Carlo methods. Mode-finding and sampling algorithms are basic tools that are important in probabilistic modeling. Like the Boltzmann machine[l], the rectified Gaussian is an undirected graphical model. The rectified Gaussian is a better representation for probabilistic modeling The Rectified Gaussian Distribution 351 (a) (c) Figure 1: Three types of quadratic energy functions. (a) Bowl (b) Trough (c) Saddle of continuous-valued data. It is unclear whether learning will be more tractable for the rectified Gaussian than it is for the Boltzmann machine. A different version of the rectified Gaussian was recently introduced by Hinton and Ghahramani[2, 3]. Their version is for a single variable, and has a singularity at the origin designed to produce sparse activity in directed graphical models. Our version lacks this singularity, and is only interesting in the case of more than one variable, for it relies on undirected interactions between variables to produce the multimodal behavior that is of interest here. The present work is inspired by biological neural network models that use continuous dynamical attractors[4]. In particular, the energy function of the cooperative distribution was previously studied in models of the visual cortex[5], motor cortex[6], and head direction system[7]. 2 ENERGY FUNCTIONS: BOWL, TROUGH, AND SADDLE The standard Gaussian distribution P(x) is defined as P(x) E(x) = Z -l -{3E(;r:) e , 1 _xT Ax - bTx 2 . (1) (2) The symmetric matrix A and vector b define the quadratic energy function E(x). The parameter (3 = lIT is an inverse temperature. Lowering the temperature concentrates the distribution at the minimum of the energy function. The prefactor Z normalizes the integral of P(x) to unity. Depending on the matrix A, the quadratic energy function E(x) can have different types of curvature. The energy function shown in Figure l(a) is convex. The minimum of the energy corresponds to the peak of the distribution. Such a distribution is often used in pattern recognition applications, when patterns are well-modeled as a single prototype corrupted by random noise. The energy function shown in Figure 1 (b) is flattened in one direction. Patterns generated by such a distribution come with roughly equal1ikelihood from anywhere along the trough. So the direction of the trough corresponds to the invariances of the pattern. Principal component analysis can be thought of as a procedure for learning distributions of this form. The energy function shown in Figure 1 (c) is saddle-shaped. It cannot be used in a Gaussian distribution, because the energy decreases without limit down the 352 N. D. Socci, D. D. Lee and H. S. Seung sides of the saddle, leading to a non-normalizable distribution. However, certain saddle-shaped energy functions can be used in the rectified Gaussian distribution, which is defined over vectors x whose components are all nonnegative. The class of energy functions that can be used are those where the matrix A has the property xT Ax > 0 for all x > 0, a condition known as copositivity. Note that this set of matrices is larger than the set of positive definite matrices that can be used with a standard Gaussian. The nonnegativity constraints block the directions in which the energy diverges to negative infinity. Some concrete examples will be discussed shortly. The energy functions for these examples will have multiple minima, and the corresponding distribution will be multimodal, which is not possible with a standard Gaussian. 3 MODE-FINDING Before defining some example distributions, we must introduce some tools for analyzing them. The modes of a rectified Gaussian are the minima of the energy function (2), subject to nonnegativity constraints. At low temperatures, the modes of the distribution characterize much of its behavior. Finding the modes of a rectified Gaussian is a problem in quadratic programming. Algorithms for quadratic programming are particularly simple for the case of nonnegativity constraints. Perhaps the simplest algorithm is the projected gradient method, a discrete time dynamics consisting of a gradient step followed by a rectification (3) The rectification [x]+ = max(x, 0) keeps x within the nonnegative orthant (x ~ 0). If the step size 7J is chosen correctly, this algorithm can provably be shown to converge to a stationary point of the energy function[8]. In practice, this stationary point is generally a local minimum. Neural networks can also solve quadratic programming problems. We define the synaptic weight matrix W = I - A, and a continuous time dynamics x+x = [b+ Wx]+ (4) For any initial condition in the nonnegative orthant, the dynamics remains in the nonnegative orthant, and the quadratic function (2) is a Lyapunov function of the dynamics. Both of these methods converge to a stationary point of the energy. The gradient of the energy is given by 9 = Ax - b. According to the Kiihn-Tucker conditions, a stationary point must satisfy the conditions that for all i, either gi = 0 and Xi > 0, or gi > 0 and Xi = O. The intuitive explanation is that in the interior of the constraint region, the gradient must vanish, while at the boundary, the gradient must point toward the interior. For a stationary point to be a local minimum, the Kiihn-Tucker conditions must be augmented by the condition that the Hessian of the nonzero variables be positive definite. Both methods are guaranteed to find a global minimum only in the case where A is positive definite, so that the energy function (2) is convex. This is because a convex energy function has a unique minimum. Convex quadratic programming is solvable in polynomial time. In contrast, for a nonconvex energy function (indefinite A), it is not generally possible to find the global minimum in polynomial time, because of the possible presence of local minima. In many practical situations, however, it is not too difficult to find a reasonable solution. The Rectified Gaussian Distribution 353 (a) (b) Figure 2: The competitive distribution for two variables. (a) A non-convex energy function with two constrained minima on the x and y axes. Shown are contours of constant energy, and arrows that represent the negative gradient of the energy. (b) The rectified Gaussian distribution has two peaks. The rectified Gaussian happens to be most interesting in the nonconvex case, precisely because of the possibility of multiple minima. The consequence of multiple minima is a multimodal distribution, which cannot be well-approximated by a standard Gaussian. We now consider two examples of a multimodal rectified Gaussian. 4 COMPETITIVE DISTRIBUTION The competitive distribution is defined by Aij -dij + 2 (5) bi = 1; (6) We first consider the simple case N = 2. Then the energy function given by X2 +y2 E(x,y)=2 +(x+y)2_(x+y) (7) has two constrained minima at (1,0) and (0,1) and is shown in figure 2(a). It does not lead to a normalizable distribution unless the nonnegativity constraints are imposed. The two constrained minima of this nonconvex energy function correspond to two peaks in the distribution (fig 2(b)). While such a bimodal distribution could be approximated by a mixture of two standard Gaussians, a single Gaussian distribution cannot approximate such a distribution. In particular, the reduced probability density between the two peaks would not be representable at all with a single Gaussian. The competitive distribution gets its name because its energy function is similar to the ones that govern winner-take-all networks[9]. When N becomes large, the N global minima of the energy function are singleton vectors (fig 3), with one component equal to unity, and the rest zero. This is due to a competitive interaction between the components. The mean of the zero temperature distribution is given by (8) The eigenvalues of the covariance 1 1 (XiXj) - (Xi)(Xj) = N dij - N2 (9) 354 N. D. Socci, D. D. Lee and H. S. Seung .:(a) : (c) r.: (b) ., r· · n n n 0 III I · u , 2 J a I • ., • • to '2 S ••• ., •• 'I . . , . . . , . . .. Figure 3: The competitive distribution for N = 10 variables. (a) One mode (zero temperature state) of the distribution. The strong competition between the variables results in only one variable on. There are N modes of this form, each with a different winner variable. (b) A sample at finite temperature (13 ~ 110) using Monte Carlo sampling. There is still a clear winner variable. (c) Sample from a standard Gaussian with matched mean and covariance. Even if we cut off the negative values this sample still bears little resemblance to the states shown in (a) and (b), since there is no clear winner variable. all equal to 1/ N, except for a single zero mode. The zero mode is 1, the vector of all ones, and the other eigenvectors span the N - 1 dimensional space perpendicular to 1. Figure 3 shows two samples: one (b) drawn at finite temperature from the competitive distribution, and the other (c) drawn from a standard Gaussian distribution with the same mean and covariance. Even if the sample from the standard Gaussian is cut so negative values are set to zero the sample does not look at all like the original distribution. Most importantly a standard Gaussian will never be able to capture the strongly competitive character of this distribution. 5 COOPERATIVE DISTRIBUTION To define the cooperative distribution on N variables, an angle fh = 27ri/N is associated with each variable Xi, so that the variables can be regarded as sitting on a ring. The energy function is defined by 1 4 Aij 6ij + N - N COS(Oi - OJ) (10) bi = 1; (11) The coupling Aij between Xi and X j depends only on the separation Oi - 03. between them on the ring. The minima, or ground states, of the energy function can be found numerically by the methods described earlier. An analytic calculation of the ground states in the large N limit is also possible[5]. As shown in Figure 4(a), each ground state is a lump of activity centered at some angle on the ring. This delocalized pattern of activity is different from the singleton modes of the competitive distribution, and arises from the cooperative interactions between neurons on the ring. Because the distribution is invariant to rotations of the ring (cyclic permutations of the variables xd, there are N ground states, each with the lump at a different angle. The mean and the covariance of the cooperative distribution are given by (Xi) = const (XiXj) - (Xi}(Xj) = C(Oi - OJ) (12) (13) A given sample of x, shown in Figure 4(a), does not look anything like the mean, which is completely uniform. Samples generated from a Gaussian distribution with The Rectified Gaussian Distribution 355 '(a) , (b) (c) r Figure 4: The cooperative distribution for N = 25 variables. (a) Zero temperature state. A cooperative interaction between the variables leads to a delocalized pattern of activity that can sit at different locations on the ring. (b) A finite temperature (/3 = 50) sample. (c) A sample from a standard Gaussian with matched mean and covariance. the same mean and covariance look completely different from the ground states of the cooperative distribution (fig 4(c)). These deviations from standard Gaussian behavior reflect fundamental differences in the underlying energy function. Here the energy function has N discrete minima arranged along a ring. In the limit of large N the barriers between these minima become quite small. A reasonable approximation is to regard the energy function as having a continuous line of minima with a ring geometry[5]. In other words, the energy surface looks like a curved trough, similar to the bottom of a wine bottle. The mean is the centroid of the ring and is not close to any minimum. The cooperative distribution is able to model the set of all translations of the lump pattern of activity. This suggests that the rectified Gaussian may be useful in invariant object recognition, in cases where a continuous manifold of instantiations of an object must be modeled. One such case is visual object recognition, where the images of an object from different viewpoints form a continuous manifold. 6 SAMPLING Figures 3 and 4 depict samples drawn from the competitive and cooperative distribution. These samples were generated using the Metropolis Monte Carlo algorithm. Since full descriptions of this algorithm can be found elsewhere, we give only a brief description of the particular features used here. The basic procedure is to generate a new configuration of the system and calculate the change in energy (given by eq. 2). If the energy decreases, one accepts the new configuration unconditionally. If it increases then the new configuration is accepted with probability e-{3AE. In our sampling algorithm one variable is updated at a time (analogous to single spin flips). The acceptance ratio is much higher this way than if we update all the spins simultaneously. However, for some distributions the energy function may have approximately marginal directions; directions in which there is little or no barrier. The cooperative distribution has this property. We can expect critical slowing down due to this and consequently some sort of collective update (analogous to multi-spin updates or cluster updates) might make sampling more efficient. However, the type of update will depend on the specifics of the energy function and is not easy to determine. 356 N D. Socci, D. D. Lee and H. S. Seung 7 DISCUSSION The competitive and cooperative distributions are examples of rectified Gaussians for which no good approximation by a standard Gaussian is possible. However, both distributions can be approximated by mixtures of standard Gaussians. The competitive distribution can be approximated by a mixture of N Gaussians, one for each singleton state. The cooperative distribution can also be approximated by a mixture of N Gaussians, one for each location of the lump on the ring. A more economical approximation would reduce the number of Gaussians in the mixture, but .make each one anisotropic[IO]. Whether the rectified Gaussian is superior to these mixture models is an empirical question that should be investigated empirically with specific real-world probabilistic modeling tasks. Our intuition is that the rectified Gaussian will turn out to be a good representation for nonlinear pattern manifolds, and the aim of this paper has been to make this intuition concrete. To make the rectified Gaussian useful in practical applications, it is critical to find tractable learning algorithms. It is not yet clear whether learning will be more tractable for the rectified Gaussian than it was for the Boltzmann machine. Perhaps the continuous variables of the rectified Gaussian may be easier to work with than the binary variables of the Boltzmann machine. Acknowledgments We would like to thank P. Mitra, L. Saul, B. Shraiman and H. Sompolinsky for helpful discussions. Work on this project was supported by Bell Laboratories, Lucent Technologies. 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] G. E. Hinton and Z. Ghahramani. Generative models for discovering sparse distributed representations. Phil. Trans. Roy. Soc., B352:1177-90, 1997. [3] Z. Ghahramani and G. E. Hinton. Hierarchical non-linear factor analysis and topographic maps. Adv. Neural Info. Proc. Syst., 11, 1998. [4] H. S. Seung. How the brain keeps the eyes still. Proc. Natl. Acad. Sci. USA, 93:13339-13344, 1996. [5] R. Ben-Yishai, R. L. Bar-Or, and H. Sompolinsky. Theory of orientation tuning in visual cortex. Proc. Nat. Acad. Sci. USA, 92:3844-3848, 1995. [6] A. P. Georgopoulos, M. Taira, and A. Lukashin. Cognitive neurophysiology of the motor cortex. Science, 260:47-52, 1993. [7] K. Zhang. Representation of spatial orientation by the intrinsic dynamics of the head-direction cell ensemble: a theory. J. Neurosci., 16:2112-2126, 1996. [8] D. P. Bertsekas. Nonlinear programming. Athena Scientific, Belmont, MA, 1995. [9] S. Amari and M. A. Arbib. Competition and cooperation in neural nets. In J. Metzler, editor, Systems Neuroscience, pages 119-165. Academic Press, New York, 1977. [10] G. E. Hinton, P. Dayan, and M. Revow. Modeling the manifolds of images of handwritten digits. IEEE Trans. Neural Networks, 8:65-74, 1997.
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A Revolution: Belief Propagation Graphs With Cycles Brendan J. Frey· http://wvw.cs.utoronto.ca/-frey Department of Computer Science University of Toronto David J. C. MacKay http://vol.ra.phy.cam.ac.uk/mackay Department of Physics, Cavendish Laboratory Cambridge University Abstract • In Until recently, artificial intelligence researchers have frowned upon the application of probability propagation in Bayesian belief networks that have cycles. The probability propagation algorithm is only exact in networks that are cycle-free. However, it has recently been discovered that the two best error-correcting decoding algorithms are actually performing probability propagation in belief networks with cycles. 1 Communicating over a noisy channel Our increasingly wired world demands efficient methods for communicating bits of information over physical channels that introduce errors. Examples of real-world channels include twisted-pair telephone wires, shielded cable-TV wire, fiber-optic cable, deep-space radio, terrestrial radio, and indoor radio. Engineers attempt to correct the errors introduced by the noise in these channels through the use of channel coding which adds protection to the information source, so that some channel errors can be corrected. A popular model of a physical channel is shown in Fig. 1. A vector of K information bits u = (Ut, ... ,UK), Uk E {O, I} is encoded, and a vector of N codeword bits x = (Xl! ... ,XN) is transmitted into the channel. Independent Gaussian noise with variance (12 is then added to each codeword bit, .. Brendan Frey is currently a Beckman Fellow at the Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign. 480 Gaussian noise with variance (J2 B. J Frey and D. J. C. MacKay --U--'~~I~ __ E __ nc_o_d_e_r __ ~1 x ~ y .I~ ___ D_e_c_od_e_r __ ~--U~.~ Figure 1: A communication system with a channel that adds Gaussian noise to the transmitted discrete-time sequence. producing the real-valued channel output vector y = (Y!, ... ,YN). The decoder must then use this received vector to make a guess U at the original information vector. The probability P" (e) of bit error is minimized by choosing the Uk that maximizes P(ukly) for k = 1, ... , K. The rate K/N of a code is the number of information bits communicated per codeword bit. We will consider rate ~ 1/2 systems in this paper, where N == 2K. The simplest rate 1/2 encoder duplicates each information hit: X2k-l = X2k = Uk, k = 1, ... , K. The optimal decoder for this repetition code simply averages together pairs of noisy channel outputs and then applies a threshold: Uk = 1 if (Y2k-l + Y2k)/2 > 0.5, 0 otherwise. (1) Clearly, this procedure has the effect of reducing the noise variance by a factor of 1/2. The resulting probability p,,(e) that an information bit will be erroneously decoded is given by the area under the tail of the noise Gaussian: ( -0.5) p,,(e) = 4> (J2/2 ' (2) where 4>0 is the cumulative standard normal distribution. A plot of p,,(e) versus (J for this repetition code is shown in Fig. 2, along with a thumbnail picture that shows the distribution of noisy received signals at the noise level where the repetition code gives p,,(e) == 10-5 • More sophisticated channel encoders and decoders can be used to increase the tolerable noise level without increasing the probability of a bit error. This approach can in principle improve performance up to a bound determined by Shannon (1948). For a given probability of bit error P,,(e), this limit gives the maximum noise level that can be tolerated, no matter what channel code is used. Shannon's proof was nonconstructive, meaning that he showed that there exist channel codes that achieve his limit, but did not present practical encoders and decoders. The curve for Shannon's limit is also shown in Fig. 2. The two curves described above define the region of interest for practical channel coding systems. For a given P,,(e), if a system requires a lower noise level than the repetition code, then it is not very interesting. At the other extreme, it is impossible for a system to tolerate a higher noise level than Shannon's limit. 2 Decoding Hamming codes by probability propagation One way to detect errors in a string of bits is to add a parity-check bit that is chosen so that the sum modulo 2 of all the bits is O. If the channel flips one bit, the receiver will find that the sum modulo 2 is 1, and can detect than an error occurred. In a simple Hamming code, the codeword x consists of the original vector A Revolution: Belief Propagation in Graphs with Cycles le-l Concatenated Code Shannon limit 481 le-5 '-----I~_----'c..........L..L_ ____ ....L._ ___ ..L____L_......u._~LI__...J.........J 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Standard deviation of Gaussian noise, U Figure 2: Probability of bit error Ph (e) versus noise level u for several codes with rates near 1/2, using 0/1 signalling. It is impossible to obtain a Ph(e) below Shannon's limit (shown on the far right for rate 1/2). "H-PP" = Hamming code (rate 4/7) decoded by probability propagation (5 iterations); "H-Exact" = Hamming code decoded exactly; "LDPCC-PP" = low-density parity-check coded decoded by probability propagation; "TC-PP" = turbo code decoded by probability propagation. The thumbnail pictures show the distribution of noisy received signals at the noise levels where the repetition code and the Shannon limit give Ph (e) = 10-5 . u in addition to several parity-check bits, each of which depends on a different subset of the information bits. In this way, the Hamming code can not only detect errors but also correct them. The code can be cast in the form of the conditional probabilities that specify a Bayesian network. The Bayesian network for a K = 4, N = 7 rate 4/7 Hamming code is shown in Fig. 3a. Assuming the information bits are uniformly random, we have P(Uk) = 0.5, Uk E {0,1}, k = 1,2,3,4. Codeword bits 1 to 4 are direct copies of the information bits: P(xkluk) = 6(Xk,Uk), k = 1,2,3,4, where 6(a, b) = 1 if a = b and 0 otherwise. Codeword bits 5 to 7 are parity-check bits: P(XSIUI,U2,U3) = 6(X5,Ul EB U2 EB U3), P(XaIU.,U2,U4) = 6(Xa,Ul EB U2 EB U4), P(x7Iu2,U3,U4) = 6(X7,u2EBU3EBu4), where EB indicates addition modulo 2 (XOR). Finally, the conditional channel probability densities are (3) for n = 1, ... , 7. The probabilities P(ukly) can be computed exactly in this belief network, using Lauritzen and Spiegelhalter's algorithm (1988) or just brute force computation. However, for the more powerful codes discussed below, exact computations are intractable. Instead, one way the decoder can approximate the probabilities P( Uk Iy) is by applying the probability propagation algorithm (Pearl 1988) to the Bayesian network. Probability propagation is only approximate in this case because the 482 B. 1. Frey and D. J. C. MacKay (a) (b) (d) (c) Figure 3: (a) The Bayesian network for a K = 4, N = 7 Hamming code. (b) The Bayesian network for a K = 4, N = 8 low-density parity-check code. (c) A block diagram for the turbocode linear feedback shift register. (d) The Bayesian network for a K = 6, N = 12 turbocode. network contains cycles (ignoring edge directions), e.g., UI-XS-U2-X6-UI. Once a channel output vector y is observed, propagation begins by sending a message from Yn to Xn for n = 1, ... ,7. Then, a message is sent from Xk to Uk for k = 1,2,3,4. An iteration now begins by sending messages from the information variables 'Ill, U2, U3, U4 to the parity-check variables Xs, X6, X7 in parallel. The iteration finishes by sending messages from the parity-check variables back to the information variables in parallel. Each time an iteration is completed, new estimates of P( Uk Iy) for k = 1,2,3,4 are obtained. The Pb (e) (j curve for optimal decoding and the curve for the probability propagation decoder (5 iterations) are shown in Fig. 2. Quite surprisingly, the performance of the iterative decoder is quite close to that of the optimal decoder. Our expectation was that short cycles would confound the probability propagation decoder. However, it seems that good performance can be obtained even when there are short cycles in the code network. For this simple Hamming code, the complexities of the probability propagation decoder and the exact decoder are comparable. However, the similarity in performance between these two decoders prompts the question: "Can probability propagation decoders give performances comparable to exact decoding in cases where exact decoding is computationally intractable?" A Revolution: Belief Propagation in Graphs with Cycles 483 3 A leap towards the limit: Low-density parity-check codes Recently, there has been an explosion of interest in the channel coding community in two new coding systems that have brought us a leap closer to Shannon's limit. Both of these codes can be described by Bayesian networks with cycles, and it turns out that the corresponding iterative decoders are performing probability propagation in these networks. Fig. 3b shows the Bayesian network for a simple low-density parity-check code (Gallager 1963). In this network, the information bits are not represented explicitly. Instead, the network defines a set of allowed configurations for the codewords. Each parity-check vertex qi requires that the codeword bits {Xn}nEQ; to which qi is connected have even parity: P(qil{xn}nEQ;) = 8(qi' EB xn), (4) nEQi where q is clamped to 0 to ensure even parity. Here, Qi is the set of indices of the codeword bits to which parity-check vertex qi is connected. The conditional probability densities for the channel outputs are the same as in Eq. 3. One way to view the above code is as N binary codeword variables along with a set of linear (modulo 2) equations. If in the end we want there to be K degrees of freedom, then the number of linearly independent parity-check equations should be N - K. In the above example, there are N = 8 codeword bits and 4 paritychecks, leaving K = 8 - 4 = 4 degrees of freedom. It is these degrees of freedom that we use to represent the information vector u. Because the code is linear, a K -dimensional vector u can be mapped to a valid x simply by multiplying by an N x K matrix (using modulo 2 addition). This is how an encoder can produce a low-density parity-check codeword for an input vector. Once a channel output vector y is observed, the iterative probability propagation decoder begins by sending messages from y to x. An iteration now begins by sending messages from the codeword variables x to the parity-check constraint variables q. The iteration finishes by sending messages from the parity-check constraint variables back to the codeword variables. Each time an iteration is completed, new estimates of P(xnly) for n = 1, . . . , N are obtained. After a valid (but not necessarily correct) codeword has been found, or a prespecified limit on the number of iterations has been reached, decoding stops. The estimate of the codeword is then mapped back to an estimate ii of the information vector. Fig. 2 shows the performance of a K = 32,621, N = 65,389 low-density paritycheck code when decoded as described above. (See MacKay and Neal (1996) for details.) It is impressively close to Shannon's limit significantly closer than the "Concatenated Code" (described in Lin and Costello (1983» which was considered the best practical code until recently. 4 Another leap: Turbocodes The codeword for a turbocode (Berrou et al. 1996) consists of the original information vector, plus two sets of bits used to protect the information. Each of these two sets is produced by feeding the information bits into a linear feedback shift register (LFSR), which is a type of finite state machine. The two sets differ in that one set is produced by a permuted set of information bits; i.e., the order of the bits is scrambled in a fixed way before the bits are fed into the LFSR. Fig. 3c shows a block diagram (not a Bayesian network) for the LFSR that was used in our experiments. 484 B. 1. Frey and D. J. C. MacKay Each box represents a delay (memory) element, and each circle performs addition modulo 2. When the kth information bit arrives, the machine has a state Sk which can be written as a binary string of state bits b4b3b2blbo as shown in the figure. bo of the state Sk is determined by the current input Uk and the previous state Sk-l' Bits b1 to b4 are just shifted versions of the bits in the previous state. Fig. 3d shows the Bayesian network for a simple turbocode. Notice that each state variable in the two constituent chains depends on the previous state and an information bit. In each chain, every second LFSR output is not transmitted. In this way, the overall rate of the code is 1/2, since there are K = 6 information bits and N = 6 + 3 + 3 = 12 codeword bits. The conditional probabilities for the states of the non permuted chain are P(sllsl-I' Uk) = 1 if state sl follows Sk-l for input Uk, 0 otherwise. (5) The conditional probabilities for the states in the other chain are similar, except that the inputs are permuted. The probabilities for the information bits are uniform, and the conditional probability densities for the channel outputs are the same as in Eq.3. Decoding proceeds with messages being passed from the channel output variables to the constituent chains and the information bits. Next, messages are passed from the information variables to the first constituent chain, SI. Messages are passed forward and then backward along this chain, in the manner of the forwardbackward algorithm (Smyth et al. 1997). After messages are passed from the first chain to the second chain s2, the second chain is processed using the forwardbackward algorithm. To complete the iteration, messages are passed from S2 to the information bits. Fig. 2 shows the performance of a K = 65,536, N = 131,072 turbocode when decoded as described above, using a fixed number (18) of iterations. (See Frey (1998) for details.) Its performance is significantly closer to Shannon's limit than the performances of both the low-density parity-check code and the textbook standard "Concatenated Code" . 5 Open questions We are certainly not claiming that the NP-hard problem (Cooper 1990) of probabilistic inference in general Bayesian networks can be solved in polynomial time by probability propagation. However, the results presented in this paper do show that there are practical problems which can be solved using approximate inference in graphs with cycles. Iterative decoding algorithms are using probability propagation in graphs with cycles, and it is still not well understood why these decoders work so well. Compared to other approximate inference techniques such as variational methods, probability propagation in graphs with cycles is unprincipled. How well do more principled decoders work? In (MacKay and Neal 1995), a variational decoder that maximized a lower bound on n~=1 P(ukly) was presented for low-density parity-check codes. However, it was found that the performance of the variational decoder was not as good as the performance of the probability propagation decoder. It is not difficult to design small Bayesian networks with cycles for which probability propagation is unstable. Is there a way to easily distinguish between those graphs for which propagation will work and those graphs for which propagation is unstable? A belief that is not uncommon in the graphical models community is that short cycles are particularly apt to lead probability propagation astray. Although it is possible to design networks where this is so, there seems to be a variety of interesting networks A Revolution: Belief Propagation in Graphs with Cycles 485 (such as the Hamming code network described above) for which propagation works well, despite short cycles. The probability distributions that we deal with in decoding are very special distributions: the true posterior probability mass is actually concentrated in one microstate in a space of size 2M where M is large (e.g., 10,000). The decoding problem is to find this most probable microstate, and it may be that iterative probability propagation decoders work because the true probability distribution is concentrated in this microstate. We believe that there are many interesting and contentious issues in this area that remain to be resolved. Acknowledgements We thank Frank Kschischang, Bob McEliece, and Radford Neal for discussions related to this work, and Zoubin Ghahramani for comments on a draft of this paper. This research was supported in part by grants from the Gatsby foundation, the Information Technology Research Council, and the Natural Sciences and Engineering Research Council. References C. Berrou and A. Glavieux 1996. Near optimum error correcting coding and decoding: Turbo-codes. IEEE 7hmsactions on Communications 44, 1261-1271. G. F. Cooper 1990. The computational complexity of probabilistic inference using Bayesian belief networks. Artificial Intelligence 42, 393-405. B. J. Frey 1998. Graphical Models for Machine Learning and Digital Communication, MIT Press, Cambridge, MA. See http://vwv . cs. utoronto. cal -frey. R. G. Gallager 1963. Low-Density Parity-Check Codes, MIT Press, Cambridge, MA. S. Lin and D. J. Costello, Jr. 1983. Error Control Coding: Fundamentals and Applications, Prentice-Hall Inc., Englewood Cliffs, NJ. S. L. Lauritzen and D. J. Spiegelhalter 1988. Local computations with probabilities on graphical structures and their application to expert systems. Journal of the Royal Statistical Society B 50, 157-224. D. J. C. MacKay and R. M. Neal 1995. Good codes based on very sparse matrices. In Cryptography and Coding. 5th IMA Conference, number 1025 in Lecture Notes in Computer Science, 100-111, Springer, Berlin Germany. D. J. C. MacKay and R. M. Neal 1996. Near Shannon limit performance of low density parity check codes. Electronics Letters 32, 1645-1646. Due to editing errors, reprinted in Electronics Letters 33, 457-458. J. Pearl 1988. Probabilistic Reasoning in Intelligent Systems, Morgan Kaufmann, San Mateo, CA. C. E. Shannon 1948. A mathematical theory of communication. Bell System Technical Journal 27, 379-423, 623-656. P. Smyth, D. Heckerman, and M. I. Jordan 1997. Probabilistic independence networks for hidden Markov probability models. Neural Computation 9, 227-270.
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RCC Cannot Compute Certain FSA, Even with Arbitrary Transfer Functions Mark Ring RWCP Theoretical Foundation GMD Laboratory GMD German National Research Center for Information Technology Schloss Birlinghoven D-53 754 Sankt Augustin, Germany email: Mark.Ring@GMD.de Abstract Existing proofs demonstrating the computational limitations of Recurrent Cascade Correlation and similar networks (Fahlman, 1991; Bachrach, 1988; Mozer, 1988) explicitly limit their results to units having sigmoidal or hard-threshold transfer functions (Giles et aI., 1995; and Kremer, 1996). The proof given here shows that for any finite, discrete transfer function used by the units of an RCC network, there are finite-state automata (FSA) that the network cannot model, no matter how many units are used. The proof also applies to continuous transfer functions with a finite number of fixed-points, such as sigmoid and radial-basis functions. 1 Introduction The Recurrent Cascade Correlation (RCC) network was proposed by Fahlman (1991) to offer a fast and efficient alternative to fully connected recurrent networks. The network is arranged such that each unit has only a single recurrent connection: the connection that goes from itself to itself. Networks with the same structure have been proposed by Mozer (Mozer, 1988) and Bachrach (Bachrach, 1988). This structure is intended to allow simplified training of recurrent networks in the hopes of making them computationally feasible. However, this increase in efficiency comes at the cost of computational power: the networks' computational capabilities are limited regardless of the power of their activation functions. The remaining input to each unit consists of the input to the network as a whole together with the outputs from all units lower in the RCC network. Since it is the structure of the network and not the learning algorithm that is of interest here, only the structure will be described in detail. 620 M. Ring Figure 1: This finite-state automaton was shown by Giles et al. (1995) to be unrepresentable by an Ree network whose units have hard-threshold or sigmoidal transfer functions. The arcs are labeled with transition labels of the FSA which are given as input to the Ree network. The nodes are labeled with the output values that the network is required to generate. The node with an inner circle is an accepting or halting state. Figure 2: This finite-state automaton is one of those shown by Kremer (1996) not to be representable by an Ree network whose units have a hard-threshold or sigmoidal transfer function. This FSA computes the parity of the inputs seen so far. The functionality of a network of N Ree units, Uo, .. . , UN-l can be described in the following way: /o([(t), Vo(t - 1» /x(i(t), Vx(t - 1), Vx-1(t), Vx-2(t), ... , Vo(t», (1) (2) where Vx(t) is the output value of Ux at time step t, and l(t) is the input to the network at time step t. The value of each unit is determined from: (1) the network input at the current time step, (2) its own value at the previous time step, and (3) the output values of the units lower in the network at the current time step. Since learning is not being considered here, the weights are assumed to be constant. 2 Existing Proofs The proof of Giles, et al (1995) showed that an Ree network whose units had a hard-threshold or sigmoidal transfer function cannot produce outputs that oscillate with a period greater than two when the network input is constant. (An oscillation has a period of x if it repeats itself every x steps.) Thus, the FSA shown in Figure 1 cannot be modeled by such an Ree network, since its output (shown as node labels) oscillates at a period greater than two given constant input. Kremer (1996) refined the class of FSA representable by an Ree network showing that, if the input to the net oscillates with period p, then the output can only oscillate with a period of w, where w is one of p's factors (or of 2p's factors if p is odd). An unrepresentable example, therefore, is the parity FSA shown in Figure 2, whose output has a period of four given the following input (of period two): 0,1,0,1, .... Both proofs, that by Giles et al. and that by Kremer, are explicitly designed with RCC Cannot Compute Certain FSA, Even with Arbitrary Transfer Functions 621 *0,1 Figure 3: This finite-state automaton cannot be modeled with any RCC network whose units are capable of representing only k discrete outputs. The values within the circles are the state names and the output expected from the network. The arcs describe transitions from state to state, and their values represent the input given to the network when the transition is made. The dashed lines indicate an arbitrary number of further states between state 3 and state k which are connected in the same manner as states 1,2, and 3. (All states are halting states.) hard-threshold and sigmoidal transfer functions in mind, and can say nothing about other transfer functions. In other words, these proofs do not demonstrate the limitations of the RCC-type network structure, but about the use of threshold units within this structure. The following proof is the first that actually demonstrates the limitations of the single-recurrent-link network structure. 3 Details of the Proof This section proves that RCC networks are incapable even in principle of modeling certain kinds of FSA, regardless of the sophistication of each unit's transfer function, provided only that the transfer function be discrete and finite, meaning only that the units of the RCC network are capable of generating a fixed number, k, of distinct output values. (Since all functions implemented on a discrete computer fall into this category, this assumption is minor. Furthermore, as will be discussed in Section 4, the outputs of most interesting continuous transfer functions reduce to only a small number of distinct values.) This generalized RCC network is proven here to be incapable of modeling the finite-state automaton shown in Figure 3. 622 MRing For ease of exposition, let us call any FSA of the form shown in Figure 3 an RFk+l for Ring FSA with k + 1 states. I Further, call a unit whose output can be any of k distinct values and whose input includes its own previous output, a DRUk for Discrete Recurrent Unit. These units are a generalization ofthe units used by RCC networks in that the specific transfer function is left unspecified. By proving the network is limited when its units are DRUbs proves the limitations of the network's structure regardless of the transfer function used. Clearly, a DRUk+1 with a sufficiently sophisticated transfer function could by itself model an RFk+1 by simply allocating one of its k + 1 output values for each of the k + 1 states. At each step it would receive as input the last state of the FSA and the next transition and could therefore compute the next state. By restricting the units in the least conceivable manner, i.e., by reducing the number of distinct output values to k, the RCC network becomes incapable of modeling any RFk+1 regardless of how many DRUk's the network contains. This will now be proven. The proof is inductive and begins with the first unit in the network, which, after being given certain sequences of inputs, becomes incapable of distinguishing among any states of the FSA. The second step, the inductive step, proves that no finite number of such units can 'assist a unit hi~her in the ReC network in making a distinction between any states of the RFk+ . Lemma 1 No DR Uk whose input is the current transition of an RFk+1 can reliably distinguish among any states of the RP+I. More specifically, at least one of the DR Uk,s k output values can be generated in all of the RP+I 's k + 1 states. Proof: Let us name the DRUbs k distinct output values VO, VI, ... , Vk-I. The mapping function implemented by the DRUk can be expressed as follows: ( V X , i) =} VY, which indicates that when the unit's last output was V X and its current input is i, then its next output is VY. Since an RFk is cyclical, the arithmetic in the following will also be cyclical (i.e., modular): xtfJy = { x+y if x + y < k x+y-k if x + y ~ k x8y { x-y if x 2: y x+k-y if x < y where 0 ~ x < k and 0 ~ y < k. Since it is impossible for the DRUk to represent each of the RFk+I,s k + 1 states with a distinct output value, at least two of these states must be represented ambiguously by the same value. That is, there are two RFk+l states a and b and one DRUk value V a/ b such that V a/ b can be generated by the unit both when the FSA is in state a and when it is in state b. Furthermore, this value will be generated by the unit given an appropriate sequence of inputs. (Otherwise the value is unreachable, serves no purpose, and can be discarded, reducing the unit to a DRUk- I.) Once the DRUk has generated V a/ b , it cannot in the next step distinguish whether the FSA's current state is a or b. Since the FSA could be in either state a or b, the next state after a b transition could be either a or b tfJ 1. That is: (va/b, b) =} Va/bEl'll, (3) IThanks to Mike Mozer for suggesting this catchy name. RCC Cannot Compute Certain FSA, Even with Arbitrary Transfer Functions 623 where a e b ~ be a and k > 1. This new output value Va/b$l can therefore be generated when the FSA is in either state a or state b EB 1. By repeatedly replacing b with b EB 1 in Equation 3, all states from b to a e 1 can be shown to share output values with state a, i.e., V a/ b, Va/b$l, V a/ b$2, ... , va/ae2, v a/ ae1 all exist. Repeatedly substituting a eland a for a and b respectively in the last paragraph produces values vx/y Vx, YEO, 1, ... , k + 1. There is, therefore, at least one value that can be generated by the unit in both states of every possible pair of states. Since there are (k! 1) distinct pairs but only k distinct output values, and since when k > 1, then not all of these pairs can be represented by unique V values. At least two of these pairs must share the same output value, and this implies that some v a / b/ e exists that can be output by the unit in any of the three FSA states a, b, and c. Starting with (Va/ b/ e, c) ::::} va/b/e$l, and following the same argument given above for V a/ b, there must be a vx/y/z for all triples of states x, Y, and z. Since there are (k ~ 1) distinct triples but only k distinct output values, and since fi+ll > 1, where k > 3, some va/ b/ e/ d must also exist. This argument can be followed repeatedly since: rer)l >1, for all m < k + 1, including when m = k. Therefore, there is at least one VO/l/2f..fk/k+l that can be output by the unit in all k + 1 states of the RFk+l. Call this value and any other that can be generated in all FSA states ~,k. All Vk>s are reachable (else they could be discarded and the above proof applied for DRUI, / < k). When a Vk is output by a DRUk , it does not distinguish any states of the RFH 1 . Lemma 2 Once a DRUk outputs a V k , all future outputs will a/so be Vk's. Proof: The proof is simply by inspection, and is shown in the following table: Actual State Transition Next State x x xEB1 xEB1 x x xEB2 x xEB2 xEB3 x xEB3 x82 x x82 x81 x x81 624 M. Ring If the unit's last output value was a Vk, then the FSA might be in any of its k + 1 possible states. As can be seen, if at this point any of the possible transitions is given as input, the next state can also be any of the k + 1 possible states. Therefore, no future inp'ut can ever serve to lessen the unit's ambiguity. Theorem 1 An RGG network composed of any finite number of DR Uk 's cannot model an Rpk+l. Proof: Let us describe the transitions of an RCC network of N units by using the following notation: ((VN-I , VN-2, ... , VI, Va), i) ~ (VN-I , VN-2, ... , V{, V~), where Vrn is the output value of the m'th unit (i.e., Urn) before the given input, i, is seen by the network, and V~ is Urn's value after i has been processed by the network. The first unit, Uo, receives only i and Va as input. Every other unit Ux receives as input i and Vx as well as v~, y < x. Lemma 1 shows that the first unit, Uo, will eventually generate a value vl, which can be generated in any of the RFk+1 states. From Lemma 2, the unit will continue to produce vl values after this point. Given any finite number N of DRUk,s, Urn-I, ... , Uo that are producing their Vk values, V~ -1' .. . , Vt, the next higher unit, UN, will be incapable of disambiguating all states by itself, i.e., at least two FSA states, a and b, will have overlapping output values, V;,p. Since none of the units UN-I, ... , Uo can distinguish between any states (including a and b), ( ( a / b k k k) b (a / b (JJ 1 k Vk k ) VN ,VN-I,·· ·, VI'VO ' )~ VN 'VN- I '·· ·, I'VO ' assuming that be a ~ ae b and k > 1. The remainder of the prooffollows identically along the lines developed for Lemmas 1 and 2. The result of this development is that UN also has a set of reachable output values V~ that can be produced in any state of the FSA. Once one such value is produced, no less-ambiguous value is ever generated. Since no RCC network containing any number of DRUk 's can over time distinguish among any states of an RFHI, no such RCC network can model such an FSA. 4 Continuous Transfer Functions Sigmoid functions can generate a theoretically infinite number of output values; if represented with 32 bits, they can generate 232 outputs. This hardly means, however, that all such values are of use. In fact, as was shown by Giles et al. (1995), if the input remains constant for a long enough period of time (as it can in all RFHI'S) , the output of sigmoid units will converge to a constant value (a fixed point) or oscillate between two values. This means that a unit with a sigmoid transfer function is in principle a DRU 2 . Most useful continuous transfer functions (radial-basis functions, for example), exhibit the same property, reducing to only a small number of distinct output values when given the same input repeatedly. The results shown here are therefore not merely theoretical, but are of real practical significance and apply to any network whose recurrent links are restricted to self connections. 5 Concl usion No RCC network can model any FSA containing an RFk+1 (such as that shown in Figure 3), given units limited to generating k possible output values, regardless RCC Cannot Compute Certain FSA, Even with Arbitrary Transfer Functions 625 of the sophistication of the transfer function that generates these values. This places an upper bound on the computational capabilities of an RCC network. Less sophisticated transfer functions, such as the sigmoid units investigated by Giles et al. and Kremer may have even greater limitations. Figure 2, for example, could be modeled by a single sufficiently sophisticated DRU 2, but cannot be modeled by an RCe network composed of hard-threshold or sigmoidal units (Giles et al., 1995; Kremer, 1996) because these units cannot exploit all mappings from inputs to outputs. By not assuming arbitrary transfer functions, previous proofs could not isolate the network's structure as the source of RCC's limitations. References Bachrach, J. R. (1988). Learning to represent state. Master's thesis, Department of Computer and Information Sciences, University of Massachusetts, Amherst, MA 01003. Fahlman, S. E. (1991) . The recurrent cascade-correlation architecture. In Lippmann, R. P., Moody, J. E., and Touretzky, D. S., editors, Advances in Neural Information Processing Systems 3, pages 190-196, San Mateo, California. Morgan Kaufmann Publishers. Giles, C., Chen, D., Sun, G., Chen, H., Lee, Y., and Goudreau, M. (1995). Constructive learning of recurrent neural networks: Problems with recurrent cascade correlation and a simple solution. IEEE Transactions on Neural Networks, 6(4):829. Kremer, S. C. (1996). Finite state automata that recurrent cascade-correlation cannot represent. In Touretzky, D. S., Mozer, M. C., and Hasselno, M. E., editors, Advances in Neural Information Processing Systems 8, pages 679-686. MIT Press. In Press. Mozer, M. C. (1988). A focused back-propagation algorithm for temporal pattern recognition. Technical Report CRG-TR-88-3, Department of Psychology, University of Toronto.
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Learning Generativ e Mo dels with the UpPropagation Algorithm JongHo on Oh and H Sebastian Seung Bell Labs Lucen t T ec hnologies Murra y Hill NJ fjhohseunggbell l abs c om Abstract Uppropagation is an algorithm for in v erting and learning neural net w ork generativ e mo dels Sensory input is pro cessed b y in v erting a mo del that generates patterns from hidden v ariables using topdo wn connections The in v ersion pro cess is iterativ e utilizing a negativ e feedbac k lo op that dep ends on an error signal propagated b y b ottomup connections The error signal is also used to learn the generativ e mo del from examples The algorithm is b enc hmark ed against principal comp onen t analysis in exp erimen ts on images of handwritten digits In his do ctrine of unconscious inference Helmholtz argued that p erceptions are formed b y the in teraction of b ottomup sensory data with topdo wn exp ectations According to one in terpretation of this do ctrine p erception is a pro cedure of sequen tial h yp othesis testing W e prop ose a new algorithm called uppropagation that realizes this in terpretation in la y ered neural net w orks It uses topdo wn connections to generate h yp otheses and b ottomup connections to revise them It is imp ortan t to understand the di erence b et w een uppropagation and its an cestor the bac kpropagation algorithm Bac kpropagation is a learning algorithm for r e c o gnition mo dels As sho wn in Figure a b ottomup connections recognize patterns while topdo wn connections propagate an error signal that is used to learn the recognition mo del In con trast uppropagation is an algorithm for in v erting and learning gener ative mo dels as sho wn in Figure b T opdo wn connections generate patterns from a set of hidden v ariables Sensory input is pro cessed b y in v erting the generativ e mo del reco v ering hidden v ariables that could ha v e generated the sensory data This op eration is called either pattern recognition or pattern analysis dep ending on the meaning of the hidden v ariables In v ersion of the generativ e mo del is done iterativ ely through a negativ e feedbac k lo op driv en b y an error signal from the b ottomup connections The error signal is also used for learning the connections error recognition (a) generation error (b) Figure
Bottomup and topdo wn pro cessing in neural net w orks a Bac kprop net w ork b Upprop net w ork in the generativ e mo del Uppropagation can b e regarded as a generalization of principal comp onen t analysis PCA and its v arian ts lik e Conic to nonlinear m ultila y er generativ e mo dels Our exp erimen ts with images of handwritten digits demonstrate that uppropagation learns a global nonlinear mo del of a pattern manifold With its global parametriza tion this mo del is distinct from lo cally linear mo dels of pattern manifolds INVER TING THE GENERA TIVE MODEL The generativ e mo del is a net w ork of L la y ers of neurons with la y er at the b ottom and la y er L at the top The v ectors x t t L are the activ ations of the la y ers The pattern x is generated from the hidden v ariables x L b y a topdo wn pass through the net w ork x t f W t x t t L The nonlinear function f acts on v ectors comp onen t b y comp onen t The matrix W t con tains the synaptic connections from the neurons in la y er t to the neurons in la y er t A bias term b t can b e added to the argumen t of f but is omitted here It is con v enien t to dene auxiliary v ariables x t b y x t f x t In terms of these auxiliary v ariables the topdo wn pass is written as x t W t f x t Giv en a sensory input d the topdo wn generativ e mo del can b e in v erted b y nding hidden v ariables x L that generate a pattern x matc hing d If some of the hid den v ariables represen t the iden tit y of the pattern the in v ersion op eration is called r e c o gnition Alternativ ely the hidden v ariables ma y just b e a more compact repre sen tation of the pattern in whic h case the op eration is called analysis or enc o ding The in v ersion is done iterativ ely as describ ed b elo w In the follo wing the op erator denotes elemen t wise m ultiplication of t w o v ectors so that z x y means z i x i y i for all i The b ottomup pass starts with the mismatc h b et w een the sensory data d and the generated pattern x f x d x whic h is propagated up w ards b y t f x t W T t t When the error signal reac hes the top of the net w ork it is used to up date the hidden v ariables x L x L W T L L This up date closes the negativ e feedbac k lo op Then a new pattern x is generated b y a topdo wn pass and the pro cess starts o v er again This iterativ e in v ersion pro cess p erforms gradien t descen t on the cost function jd x j sub ject to the constrain ts This can b e pro v ed using the c hain rule as in the traditional deriv ation of the bac kprop algorithm Another metho d of pro of is to add the equations as constrain ts using Lagrange m ultipliers jd f x j L X t T t x t W t f x t This deriv ation has the adv an tage that the b ottomup activ ations t ha v e an in ter pretation as Lagrange m ultipliers In v erting the generativ e mo del b y negativ e feedbac k can b e in terpreted as a pro cess of sequen tial h yp othesis testing The topdo wn connections generate a h yp othesis ab out the sensory data The b ottomup connections propagate an error signal that is the disagreemen t b et w een the h yp othesis and data When the error signal reac hes the top it is used to generate a revised h yp othesis and the generatetest revise cycle starts all o v er again P erception is the con v ergence of this feedbac k lo op to the h yp othesis that is most consisten t with the data LEARNING THE GENERA TIVE MODEL The synaptic w eigh ts W t determine the t yp es of patterns that the net w ork is able to generate T o learn from examples the w eigh ts are adjusted to impro v e the net w orks generation abilit y A suitable cost function for learning is the reconstruction error jd x j a v eraged o v er an ensem ble of examples Online gradien t descen t with resp ect to the synaptic w eigh ts is p erformed b y a learning rule of the form W t t x T t The same error signal that w as used to in v ert the generativ e mo del is also used to learn it The batc h form of the optimization is compactly written using matrix notation T o do this w e dene the matrices D X X L whose columns are the v ectors d x x L corresp onding to examples in the training set Then computation and learning are the minimization of min X L W t jD X j sub ject to the constrain t that X t f W t X t t L In other w ords upprop is a dual minimization with resp ect to hidden v ariables and synaptic connections Computation minimizes with resp ect to the hidden v ariables X L and learning minimizes with resp ect to the synaptic w eigh t matrices W t F rom the geometric viewp oin t uppropagation is an algorithm for learning pattern manifolds The topdo wn pass maps an n L dimensional v ector x L to an n dimensional v ector x Th us the generativ e mo del parametrizes a con tin uous n L dimensional manifold em b edded in n dimensional space In v erting the generativ e mo del is equiv alen t to nding the p oin t on the manifold that is closest to the sensory data Learning the generativ e mo del is equiv alen t to deforming the manifold to t a database of examples principal components W Figure
Onestep generation of handwritten digits W eigh ts of the upprop net w ork left v ersus the top principal comp onen ts righ t target image x0 t=0 0 5 10 0 2 4 x1 t=1 0 5 10 0 2 4 t=10 0 5 10 0 2 4 t=100 0 5 10 0 2 4 t=1000 0 5 10 0 2 4 Figure
Iterativ e in v ersion of a generativ e mo del as sequen tial h yp othesis testing A fully trained net w ork is in v erted to generate an appro ximation to a target image that w as not previously seen during training The stepsize of the dynamics w as xed to to sho w time ev olution of the system P attern manifolds are relev an t when patterns v ary con tin uously F or example the v ariations in the image of a threedimensional ob ject pro duced b y c hanges of view p oin t are clearly con tin uous and can b e describ ed b y the action of a transformation group on a protot yp e pattern Other t yp es of v ariation suc h as deformations in the shap e of the ob ject are also con tin uous ev en though they ma y not b e readily describable in terms of transformation groups Con tin uous v ariabilit y is clearly not conned to visual images but is presen t in man y other domains Man y existing tec hniques for mo deling pattern manifolds suc h as PCA or PCA mixtures de p end on linear or lo cally linear appro ximations to the manifold Upprop constructs a globally parametrized nonlinear manifold ONESTEP GENERA TION The simplest generativ e mo del of the form has just one step L Up propagation minimizes the cost function min X W jD f W X j F or a linear f this reduces to PCA as the cost function is minimized when the v ec tors in the w eigh t matrix W span the same space as the top principal comp onen ts of the data D Uppropagation with a onestep generativ e mo del w as applied to the USPS database whic h consists of example images of handwritten digits Eac h of the training and testing images w as normalized to a grid with pixel in tensities in the range A separate mo del w as trained for eac h digit class The nonlinearit y f w as the logistic function Batc h optimization of w as done b y 5 10 15 20 25 30 35 40 0 0.005 0.01 0.015 0.02 0.025 Error number of vectors Reconstruction Error PCA, training Up−prop, training PCA, test Up−prop, test Figure
Reconstruction error for n net w orks as a function of n The error of PCA with n principal comp onen ts is sho wn for comparison The upprop net w ork p erforms b etter on b oth the training set and test set gradien t descen t with adaptiv e stepsize con trol b y the Armijo rule In most cases the stepsize v aried b et w een and and the optimization usually con v erged within ep o c hs Figure sho ws the w eigh ts of a net w ork that w as trained on di eren t images of the digit t w o Eac h of the subimages is the w eigh t v ector of a toplev el neuron The top principal comp onen ts are also sho wn for comparison Figure sho ws the time ev olution of a fully trained net w ork during iterativ e in v ersion The error signal from the b ottom la y er x quic kly activ ates the top la y er x A t early times all the top la y er neurons ha v e similar activ ation lev els Ho w ev er the neurons with w eigh t v ectors more relev an t to the target image b ecome dominan t so on and the other neurons are deactiv ated The reconstruction error of the upprop net w ork w as m uc h b etter than that of PCA W e trained di eren t upprop net w orks one for eac h digit and these w ere compared with corresp onding PCA mo dels Figure sho ws the a v erage squared error p er pixel that resulted A upprop net w ork p erformed as w ell as PCA with principal comp onen ts TW OSTEP GENERA TION Tw ostep generation is a ric her mo del and is learned using the cost function min X W W jD f W f W X j Note that a nonlinear f is necessary for t w ostep generation to ha v e more represen tational p o w er than onestep generation When this t w ostep generativ e mo del w as trained on the USPS database the w eigh t v ectors in W learned features resem bling principal comp onen ts The activities of the X neurons tended to b e close to their saturated v alues of one or zero The reconstruction error of the t w ostep generativ e net w ork w as compared to that of the onestep generativ e net w ork with the same n um b er of neurons in the top la y er Our net w ork outp erformed our net w ork on the test set though b oth net w orks used nine hidden v ariables to enco de the sensory data Ho w ev er the learning time w as m uc h longer and iterativ e in v ersion w as also slo w While upprop for onestep generation con v erged within sev eral h undred ep o c hs upprop for t w ostep generation often needed sev eral thousand ep o c hs or more to con v erge W e often found long plateaus in the learning curv es whic h ma y b e due to the p erm utation symmetry of the net w ork arc hitecture DISCUSSION T o summarize the exp erimen ts discussed ab o v e w e constructed separate generativ e mo dels one for eac h digit class Relativ e to PCA eac h generativ e mo del w as sup erior at enco ding digits from its corresp onding class This enhanced generativ e abilit y w as due to the use of nonlinearit y W e also tried to use these generativ e mo dels for recognition A test digit w as classied b y in v erting all the generativ e mo dels and then c ho osing the one b est able to generate the digit Our tests of this recognition metho d w ere not encouraging The nonlinearit y of uppropagation tended to impro v e the generation abilit y of mo dels corresp onding to all classes not just the mo del corresp onding to the correct classication of the digit Therefore the impro v ed enco ding p erformance did not immediately transfer to impro v ed recognition W e ha v e not tried the exp erimen t of training one generativ e mo del on all the digits with some of the hidden v ariables represen ting the digit class In this case pattern recognition could b e done b y in v erting a single generativ e mo del It remains to b e seen whether this metho d will w ork Iterativ e in v ersion w as surprisingly fast as sho wn in Figure and ga v e solutions of surprisingly go o d qualit y in spite of p oten tial problems with lo cal minima as sho wn in Figure In spite of these virtues iterativ e in v ersion is still a problematic metho d W e do not kno w whether it will p erform w ell if a single generativ e mo del is trained on m ultiple pattern classes F urthermore it seems a rather indirect w a y of doing pattern recognition The upprop generativ e mo del is deterministic whic h handicaps its mo deling of pattern v ariabilit y The mo del can b e dressed up in probabilistic language b y den ing a prior distribution P x L for the hidden v ariables and adding Gaussian noise to x to generate the sensory data Ho w ev er this probabilistic app earance is only skin deep as the sequence of transformations from x L to x is still completely de terministic In a truly probabilistic mo del lik e a b elief net w ork ev ery la y er of the generation pro cess adds v ariabilit y In conclusion w e brie y compare uppropagation to other algorithms and arc hitec tures In bac kpropagation only the recognition mo del is explicit Iterativ e gra dien t descen t metho ds can b e used to in v ert the recognition mo del though this implicit generativ e mo del generally app ears to b e inaccurate Uppropagation has an explicit generativ e mo del and recognition is done b y in v erting the generativ e mo del The accuracy of this implicit recognition mo del has not y et b een tested empirically Iterativ e in v ersion of generativ e mo dels has also b een prop osed for linear net w orks and probabilistic b elief net w orks In the auto enco der and the Helmholtz mac hine there are separate mo dels of recognition and generation b oth explicit Recognition uses only b ottomup connections and generation uses only topdo wn connections Neither pro cess is iterativ e Both pro cesses can op erate completely inde p enden tly they only in teract during learning In attractor neural net w orks and the Boltzmann mac hine recog nition and generation are p erformed b y the same recurren t net w ork Eac h pro cess is iterativ e and eac h utilizes b oth b ottomup and topdo wn connec tions Computation in these net w orks is c hie y based on p ositiv e rather than negativ e feedbac k Bac kprop and upprop su er from a lac k of balance in their treatmen t of b ottomup and topdo wn pro cessing The auto enco der and the Helmholtz mac hine su er from inabilit y to use iterativ e dynamics for computation A ttractor neural net w orks lac k these deciencies so there is incen tiv e to solv e the problem of learning attractors This w ork w as supp orted b y Bell Lab oratories JHO w as partly supp orted b y the Researc h Professorship of the LGY onam F oundation W e are grateful to Dan Lee for helpful discussions References D E Rumelhart G E Hin ton and R J Williams Learning in ternal represen tations b y bac kpropagating errors Natur e D D Lee and H S Seung Unsup ervised learning b y con v ex and conic co ding A dv Neur al Info Pr o c Syst G E Hin ton P Da y an and M Rev o w Mo deling the manifolds of images of hand written digits IEEE T r ans Neur al Networks Y LeCun et al Learning algorithms for classication a comparison on handwritten digit recognition In JH Oh C Kw on and S Cho editors Neur al networks the statistic al me chanics p ersp e ctive pages Singap ore W orld Scien tic D P Bertsek as Nonline ar pr o gr amming A thena Scien tic Belmon t MA K Kang JH Oh C Kw on and Y P ark Generalization in a t w ola y er neural net w ork Phys R ev E J Kindermann and A Linden In v ersion of neural net w orks b y gradien t descen t Par al lel Computing Y Lee Handwritten digit recognition using K nearestneigh b or radialbasis function and bac kpropagation neural net w orks Neur al Comput R P N Rao and D H Ballard Dynamic mo del of visual recognition predicts neural resp onse prop erties in the visual cortex Neur al Comput L K Saul T Jaakk ola and M I Jordan Mean eld theory for sigmoid b elief net w orks J A rtif Intel l R es G W Cottrell P Munro and D Zipser Image compression b y bac k propagation an example of extensional programming In N E Shark ey editor Mo dels of c o gnition a r eview of c o gnitive scienc e Ablex Norw o o d NJ G E Hin ton P Da y an B J F rey and R M Neal The w ak esleep algorithm for unsup ervised neural net w orks Scienc e H S Seung P attern analysis and syn thesis in attractor neural net w orks In KY M W ong I King and DY Y eung editors The or etic al Asp e cts of Neur al Computation A Multidisciplinary Persp e ctive Singap ore SpringerV erlag H S Seung Learning con tin uous attractors in recurren t net w orks A dv Neur al Info Pr o c Syst D H Ac kley G E Hin ton and T J Sejno wski A learning algorithm for Boltzmann mac hines Co gnitive Scienc e
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Local Dimensionality Reduction Stefan Schaal 1,2,4 sschaal@usc.edu http://www-slab.usc.edulsschaal Sethu Vijayakumar 3, I sethu@cs.titech.ac.jp http://ogawawww.cs.titech.ac.jp/-sethu Christopher G. Atkeson 4 cga@cc.gatech.edu http://www.cc.gatech.edul fac/Chris.Atkeson IERATO Kawato Dynamic Brain Project (IST), 2-2 Hikaridai, Seika-cho, Soraku-gun, 619-02 Kyoto 2Dept. of Comp. Science & Neuroscience, Univ. of South. California HNB-I 03, Los Angeles CA 90089-2520 3Department of Computer Science, Tokyo Institute of Technology, Meguro-ku, Tokyo-I 52 4College of Computing, Georgia Institute of Technology, 801 Atlantic Drive, Atlanta, GA 30332-0280 Abstract If globally high dimensional data has locally only low dimensional distributions, it is advantageous to perform a local dimensionality reduction before further processing the data. In this paper we examine several techniques for local dimensionality reduction in the context of locally weighted linear regression. As possible candidates, we derive local versions of factor analysis regression, principle component regression, principle component regression on joint distributions, and partial least squares regression. After outlining the statistical bases of these methods, we perform Monte Carlo simulations to evaluate their robustness with respect to violations of their statistical assumptions. One surprising outcome is that locally weighted partial least squares regression offers the best average results, thus outperforming even factor analysis, the theoretically most appealing of our candidate techniques. 1 INTRODUCTION Regression tasks involve mapping a n-dimensional continuous input vector x E ~n onto a m-dimensional output vector y E ~m • They form a ubiquitous class of problems found in fields including process control, sensorimotor control, coordinate transformations, and various stages of information processing in biological nervous systems. This paper will focus on spatially localized learning techniques, for example, kernel regression with Gaussian weighting functions. Local learning offer advantages for real-time incremental learning problems due to fast convergence, considerable robustness towards problems of negative interference, and large tolerance in model selection (Atkeson, Moore, & Schaal, 1997; Schaal & Atkeson, in press). Local learning is usually based on interpolating data from a local neighborhood around the query point. For high dimensional learning problems, however, it suffers from a bias/variance dilemma, caused by the nonintuitive fact that " ... [in high dimensions] if neighborhoods are local, then they are almost surely empty, whereas if a neighborhood is not empty, then it is not local." (Scott, 1992, p.198). Global learning methods, such as sigmoidal feedforward networks, do not face this 634 S. School, S. Vijayakumar and C. G. Atkeson problem as they do not employ neighborhood relations, although they require strong prior knowledge about the problem at hand in order to be successful. Assuming that local learning in high dimensions is a hopeless, however, is not necessarily warranted: being globally high dimensional does not imply that data remains high dimensional if viewed locally. For example, in the control of robot anns and biological anns we have shown that for estimating the inverse dynamics of an ann, a globally 21dimensional space reduces on average to 4-6 dimensions locally (Vijayakumar & Schaal, 1997). A local learning system that can robustly exploit such locally low dimensional distributions should be able to avoid the curse of dimensionality. In pursuit of the question of what, in the context of local regression, is the "right" method to perfonn local dimensionality reduction, this paper will derive and compare several candidate techniques under i) perfectly fulfilled statistical prerequisites (e.g., Gaussian noise, Gaussian input distributions, perfectly linear data), and ii) less perfect conditions (e.g., non-Gaussian distributions, slightly quadratic data, incorrect guess of the dimensionality of the true data distribution). We will focus on nonlinear function approximation with locally weighted linear regression (L WR), as it allows us to adapt a variety of global linear dimensionality reduction techniques, and as L WR has found widespread application in several local learning systems (Atkeson, Moore, & Schaal, 1997; Jordan & Jacobs, 1994; Xu, Jordan, & Hinton, 1996). In particular, we will derive and investigate locally weighted principal component regression (L WPCR), locally weighted joint data principal component analysis (L WPCA), locally weighted factor analysis (L WF A), and locally weighted partial least squares (L WPLS). Section 2 will briefly outline these methods and their theoretical foundations, while Section 3 will empirically evaluate the robustness of these methods using synthetic data sets that increasingly violate some of the statistical assumptions of the techniques. 2 METHODS OF DIMENSIONALITY REDUCTION We assume that our regression data originate from a generating process with two sets of observables, the "inputs" i and the "outputs" y. The characteristics of the process ensure a functional relation y = f(i). Both i and yare obtained through some measurement device that adds independent mean zero noise of different magnitude in each observable, such that x == i + Ex and y = y + Ey • For the sake of simplicity, we will only focus on one-dimensional output data (m=l) and functions / that are either linear or slightly quadratic, as these cases are the most common in nonlinear function approximation with locally linear models. Locality of the regression is ensured by weighting the error of each data point with a weight from a Gaussian kernel: Wi = exp(-O.5(Xi - Xqf D(Xi - Xq)) (1) Xtt denotes the query point, and D a positive semi-definite distance metric which determmes the size and shape of the neighborhood contributing to the regression (Atkeson et aI., 1997). The parameters Xq and D can be determined in the framework of nonparametric statistics (Schaal & Atkeson, in press) or parametric maximum likelihood estimations (Xu et aI, 1995}- for the present study they are determined manually since their origin is secondary to the results of this paper. Without loss of generality, all our data sets will set !,q to the zero vector, compute the weights, and then translate the input data such that the locally weighted mean, i = L WI Xi / L Wi , is zero. The output data is equally translated to be mean zero. Mean zero data is necessary for most of techniques considered below. The (translated) input data is summarized in the rows of the matrix X, the corresponding (translated) outputs are the elements of the vector y, and the corresponding weights are in the diagonal matrix W. In some cases, we need the joint input and output data, denoted as Z=[X y). Local Dimensionality Reduction 635 2.1 FACTORANALYSIS(LWFA) Factor analysis (Everitt, 1984) is a technique of dimensionality reduction which is the most appropriate given the generating process of our regression data. It assumes the observed data z was produced. by a mean zero independently distributed k -dimensional vector of factors v, transformed by the matrix U, and contaminated by mean zero independent noise f: with diagonal covariance matrix Q: z=Uv+f:, where z=[xT,yt and f:=[f:~,t:yr (2) If both v and f: are normally distributed, the parameters Q and U can be obtained iteratively by the Expectation-Maximization algorithm (EM) (Rubin & Thayer, 1982). For a linear regression problem, one assumes that z was generated with U=[I, f3 Y and v = i, where f3 denotes the vector of regression coefficients of the linear model y = f31 x, and I the identity matrix. After calculating Q and U by EM in joint data space as formulated in (2), an estimate of f3 can be derived from the conditional probability p(y I x). As all distributions are assumed to be normal, the expected value ofy is the mean of this conditional distribution. The locally weighted version (L WF A) of f3 can be obtained together with an estimate of the factors v from the joint weighted covariance matrix 'I' of z and v: E{[: ] + [ ~ } ~ ~,,~,;'x, where ~ ~ [ZT, VT~~Jft: w; ~ (3) [Q+UUT U] ['I'II(=n x n) 'I'12(=nX(m+k»)] = UT I = '¥21(= (m + k) x n) '1'22(= (m + k) x (m + k») where E { .} denotes the expectation operator and B a matrix of coefficients involved in estimating the factors v. Note that unless the noise f: is zero, the estimated f3 is different from the true f3 as it tries to average out the noise in the data. 2.2 JOINT-SPACE PRINCIPAL COMPONENT ANALYSIS (LWPCA) An alternative way of determining the parameters f3 in a reduced space employs locally weighted principal component analysis (LWPCA) in the joint data space. By defining the . largest k+ 1 principal components of the weighted covariance matrix ofZ as U: U = [eigenvectors(I Wi (Zi - ZXZi - Z)T II Wi)] (4) max(l:k+1l and noting that the eigenvectors in U are unit length, the matrix inversion theorem (Hom & Johnson, 1994) provides a means to derive an efficient estimate of f3 ( T T( T )-1 T\ [Ux(=nXk)] f3=U x Uy -Uy UyUy -I UyUyt where U= Uy(=mxk) (5) In our one dimensional output case, U y is just a (1 x k) -dimensional row vector and the evaluation of (5) does not require a matrix inversion anymore but rather a division. If one assumes normal distributions in all variables as in L WF A, L WPCA is the special case of L WF A where the noise covariance Q is spherical, i.e., the same magnitude of noise in all observables. Under these circumstances, the subspaces spanned by U in both methods will be the same. However, the regression coefficients of L WPCA will be different from those of L WF A unless the noise level is zero, as L WF A optimizes the coefficients according to the noise in the data (Equation (3» . Thus, for normal distributions and a correct guess of k, L WPCA is always expected to perform worse than L WF A. 636 S. Schaal, S. Vijayakumar and C. G. Atkeson 2.3 PARTIAL LEAST SQUARES (LWPLS, LWPLS_I) Partial least squares (Wold, 1975; Frank & Friedman, 1993) recursively computes orthogonal projections of the input data and performs single variable regressions along these projections on the residuals of the previous iteration step. A locally weighted version of partial least squares (LWPLS) proceeds as shown in Equation (6) below. As all single variable regressions are ordinary univariate least-squares minim izations, L WPLS makes the same statistical assumption as ordinary linear regressions, i.e., that only output variables have additive noise, but input variables are noiseless. The choice of the projections u, however, introduces an element in L WPLS that remains statistically still debated (Frank & Friedman, 1993), although, interestingly, there exists a strong similarity with the way projections are chosen in Cascade Correlation (Fahlman & Lebiere, 1990). A peculiarity of L WPLS is that it also regresses the inputs of the previous step against the projected inputs s in order to ensure the orthogonality of all the projections u. Since L WPLS chooses projections in a very powerful way, it can accomplish optimal function fits with only one single projections (i.e., For Training: Initialize: Do = X, eo = y For i = 1 to k: For Lookup: Initialize: do = x, y= ° For i = 1 to k: s. = dT.u. I 1I (6) k= 1) for certain input distributions. We will address this issue in our empirical evaluations by comparing k-step L WPLS with I-step L WPLS, abbreviated L WPLS_I. 2.4 PRINCIPAL COMPONENT REGRESSION (L WPCR) Although not optimal, a computationally efficient techniques of dimensionality reduction for linear regression is principal component regression (LWPCR) (Massy, 1965). The inputs are projected onto the largest k principal components of the weighted covariance matrix of the input data by the matrix U: U = [eigenvectors(2: Wi (Xi - xX Xi - xt /2: Wi )] (7) max(l:k) The regression coefficients f3 are thus calculated as: f3 = (UTXTwxUtUTXTWy (8) Equation (8) is inexpensive to evaluate since after projecting X with U, UTXTWXU becomes a diagonal matrix that is easy to invert. L WPCR assumes that the inputs have additive spherical noise, which includes the zero noise case. As during dimensionality reduction L WPCR does not take into account the output data, it is endangered by clipping input dimensions with low variance which nevertheless have important contribution to the regression output. However, from a statistical point of view, it is less likely that low variance inputs have significant contribution in a linear regression, as the confidence bands of the regression coefficients increase inversely proportionally with the variance of the associated input. If the input data has non-spherical noise, L WPCR is prone to focus the regression on irrelevant projections. 3 MONTE CARLO EVALUATIONS In order to evaluate the candidate methods, data sets with 5 inputs and 1 output were randomly generated. Each data set consisted of 2,000 training points and 10,000 test points, distributed either uniformly or nonuniformly in the unit hypercube. The outputs were Local Dimensionality Reduction 637 generated by either a linear or quadratic function. Afterwards, the 5-dimensional input space was projected into a to-dimensional space by a randomly chosen distance preserving linear transformation. Finally, Gaussian noise of various magnitudes was added to both the 10-dimensional inputs and one dimensional output. For the test sets, the additive noise in the outputs was omitted. Each regression technique was localized by a Gaussian kernel (Equation (1)) with a to-dimensional distance metric D=IO*I (D was manually chosen to ensure that the Gaussian kernel had sufficiently many data points and no "data holes" in the fringe areas of the kernel) . The precise experimental conditions followed closely those suggested by Frank and Friedman (1993): • 2 kinds of linear functions y = {g.I for: i) 131 .. = [I, I, I, I, If , ii) I3Ii. = [1,2,3,4,sf • 2 kinds of quadratic functions y = f3J.I + f3::.aAxt ,xi ,xi ,X;,X;]T for: i) 1311. = [I, I, I, I, Wand f3q.ad = 0.1 [I, I, I, I, If, and ii) 131 .. = [1,2,3,4, sf and f3quad = 0.1 [I, 4, 9, 16, 2sf • 3 kinds of noise conditions, each with 2 sub-conditions: i) only output noise: a) low noise: local signal/noise ratio Isnr=20, and b) high noise: Isnr=2, ii) equal noise in inputs and outputs: a) low noise Ex •• = Sy = N(O,O.Ot2), n e[I,2, ... ,10], and b) high noise Ex •• =sy=N(0,0.12),ne[I,2, ... ,10], iii) unequal noise in inputs and outputs: a) low noise: Ex .• = N(0,(0.0In)2), n e[I,2, ... ,1O] and Isnr=20, and b) high noise: Ex .• = N(0,(0.0In)2), n e[I,2, ... ,1O] and Isnr=2, • 2 kinds of input distributions: i) uniform in unit hyper cube, ii) uniform in unit hyper cube excluding data points which activate a Gaussian weighting function (I) at c = [O.S,O,o,o,of with D=IO*I more than w=0.2 (this forms a "hyper kidney" shaped distribution) Every algorithm was run * 30 times on each of the 48 combinations of the conditions. Additionally, the complete test was repeated for three further conditions varying the dimensionality--called factors in accordance with L WF A-that the algorithms assumed to be the true dimensionality of the to-dimensional data from k=4 to 6, i.e., too few, correct, and too many factors. The average results are summarized in Figure I. Figure I a,b,c show the summary results of the three factor conditions. Besides averaging over the 30 trials per condition, each mean of these charts also averages over the two input distribution conditions and the linear and quadratic function condition, as these four cases are frequently observed violations of the statistical assumptions in nonlinear function approximation with locally linear models. In Figure I b the number of factors equals the underlying dimensionality of the problem, and all algorithms are essentially performing equally well. For perfectly Gaussian distributions in all random variables (not shown separately), LWFA's assumptions are perfectly fulfilled and it achieves the best results, however, almost indistinguishable closely followed by L WPLS. For the ''unequal noise condition", the two PCA based techniques, L WPCA and L WPCR, perform the worst since--as expected-they choose suboptimal projections. However, when violating the statistical assumptions, L WF A loses parts of its advantages, such that the summary results become fairly balanced in Figure lb. The quality of function fitting changes significantly when violating the correct number of factors, as illustrated in Figure I a,c. For too few factors (Figure la), L WPCR performs worst because it randomly omits one of the principle components in the input data, without respect to how important it is for the regression. The second worse is L WF A: according to its assumptions it believes that the signal it cannot model must be noise, leading to a degraded estimate of the data's subspace and, consequently, degraded regression results. L WPLS has a clear lead in this test, closely followed by L WPCA and L WPLS_I. * Except for LWFA, all methods can evaluate a data set in non-iterative calculations. LWFA was trained with EM for maximally 1000 iterations or until the log-likelihood increased less than I.e-lOin one iteration. 638 S. Schaal, S. Vljayakumar and C. G. Atkeson For too many factors than necessary (Figure Ie), it is now LWPCA which degrades. This effect is due to its extracting one very noise contaminated projection which strongly influences the recovery of the regression parameters in Equation (4). All other algorithms perform almost equally well, with L WF A and L WPLS taking a small lead. c o 0.1 ~ 0.01 ::::;; c II> C> ~ 0.001 ~ c:: o 0.0001 0.1 W 0.01 ~ c:: II> C) ~ 0.001 ~ ~ ~ 8 0.0001 0.1 W 0.01 ~ c:: g, ~ 0.001 ~ jj il f-a 0.0001 0.1 ~ 0.01 ::::;; c II> C) ~ 0.001 ~ 0.0001 OnlyOutpul Noise Equal NoIse In ell In puIS end OutpUIS Unequel NoIse In ell Inputs end OutpulS fl- I. E>O ~I. £ >>(I ~ J. &>O ~J , E » O ~ J .E>O fl- I.&» O ~I . & >O ~ I . & >>O ~I.& >O ~ I .£>>o p,. 1. s>O tJ-J .£>>O e) RegressIon Results with 4 Factors • LWFA • LWPCA • LWPCR 0 LWPLS • LWPLS_1 c) RegressIon Results with 6 Feclors d) Summery Results Figure I: Average summary results of Monte Carlo experiments. Each chart is primarily divided into the three major noise conditions, cf. headers in chart (a). In each noise condition, there are four further subdivision: i) coefficients of linear or quadratic model are equal with low added noise; ii) like i) with high added noise; iii) coefficients oflinear or quadratic model are different with low noise added; iv) like iii) with high added noise. Refer to text and descriptions of Monte Carlo studies for further explanations. Local Dimensionality Reduction 639 4 SUMMARY AND CONCLUSIONS Figure 1 d summarizes all the Monte Carlo experiments in a final average plot. Except for L WPLS, every other technique showed at least one clear weakness in one of our "robustness" tests. It was particularly an incorrect number of factors which made these weaknesses apparent. For high-dimensional regression problems, the local dimensionality, i.e., the number of factors, is not a clearly defined number but rather a varying quantity, depending on the way the generating process operates. Usually, this process does not need to generate locally low dimensional distributions, however, it often "chooses" to do so, for instance, as human ann movements follow stereotypic patterns despite they could generate arbitrary ones. Thus, local dimensionality reduction needs to find autonomously the appropriate number of local factor. Locally weighted partial least squares turned out to be a surprisingly robust technique for this purpose, even outperforming the statistically appealing probabilistic factor analysis. As in principal component analysis, LWPLS's number of factors can easily be controlled just based on a variance-cutoff threshold in input space (Frank & Friedman, 1993), while factor analysis usually requires expensive cross-validation techniques. Simple, variance-based control over the number of factors can actually improve the results of L WPCA and L WPCR in practice, since, as shown in Figure I a, L WPCR is more robust towards overestimating the number of factors, while L WPCA is more robust towards an underestimation. If one is interested in dynamically growing the number of factors while obtaining already good regression results with too few factors, L WPCA and, especially, L WPLS seem to be appropriate-it should be noted how well one factor L WPLS (L WPLS_l) already performed in Figure I! In conclusion, since locally weighted partial least squares was equally robust as local weighted factor analysis towards additive noise in. both input and output data, and, moreover, superior when mis-guessing the number of factors, it seems to be a most favorable technique for local dimensionality reduction for high dimensional regressions. Acknowledgments The authors are grateful to Geoffrey Hinton for reminding them of partial least squares. This work was supported by the ATR Human Information Processing Research Laboratories. S. Schaal's support includes the German Research Association, the Alexander von Humboldt Foundation, and the German Scholarship Foundation. S. Vijayakumar was supported by the Japanese Ministry of Education, Science, and Culture (Monbusho). C. G. Atkeson acknowledges the Air Force Office of Scientific Research grant F49-6209410362 and a National Science Foundation Presidential Young Investigators Award. References tures of experts and the EM algorithm." Neural Computation, 6, 2, pp.181-214. Atkeson, C. G., Moore, A. W., & Schaal, S, (1997a). Massy, W. F, (1965). "Principle component regression "Locally weighted learning." ArtifiCial Intelligence Re- in exploratory statistical research." Journal of the view, 11, 1-5, pp.II-73. American Statistical Association, 60, pp.234-246. Atkeson, C. G., Moore, A. W., & Schaal, S, (1997c). Rubin, D. B., & Thayer, D. T, (l982). "EM algorithms "Locally weighted learning for control." ArtifiCial Infor ML factor analysis." Psychometrika, 47, I, 69-76. telligence Review, 11, 1-5, pp.75-113. Schaal, S., & Atkeson, C. G, (in press). "Constructive Belsley, D. A., Kuh, E., & Welsch, R. E, (1980). Reincremental learning from only local information." gression diagnostics: Identifying influential data and Neural Computation. sources of collinearity. New York: Wiley. Scott, D. W, (1992). Multivariate Density Estimation. Everitt, B. S, (1984). An introduction to latent variable New York: Wiley. models. London: Chapman and Hall. Vijayakumar, S., & Schaal, S, (1997). "Local dimenFahlman, S. E. ,Lebiere, C, (1990). "The cascadesionality reduction for locally weighted learning." In: correlation learning architecture." In: Touretzky, D. S. International Conference on Computational Intelli(Ed.), Advances in Neural Information Processing gence in Robotics and Automation, pp.220-225, MonSystems II, pp.524-532. Morgan Kaufmann. teray, CA, July 10-11, 1997. Frank, I. E., & Friedman, 1. H, (1993). "A statistical Wold, H. (1975). "Soft modeling by latent variables: view of some chemometric regression tools." Techthe nonlinear iterative partial least squares approach." nometrics, 35, 2, pp.l09-135. In: Gani, J. (Ed.), Perspectives in Probability and StaGeman, S., Bienenstock, E., & Doursat, R. (1992). tistics, Papers in Honour ofM S. Bartlett. Aca<j. Press. "Neural networks and the bias/variance dilemma." Xu, L., Jordan, M.l., & Hinton, G. E, (1995). "An alNeural Computation, 4, pp.I-58. ternative model for mixture of experts." In: Tesauro, Hom, R. A., & Johnson, C. R, (1994). Matrix analySis. G., Touretzky, D. S., & Leen, T. K. (Eds.), Advances in Press Syndicate of the University of Cambridge. Neural Information Processing Systems 7, pp.633-640. Jordan, M.I., & Jacobs, R, (1994). "Hierarchical mixCambridge, MA: MIT Press. Serial Order in Reading Aloud: Connectionist Models and Neighborhood Structure Jeanne C. Milostan Computer Science & Engineering 0114 University of California San Diego La Jolla, CA 92093-0114 Garrison W. Cottrell Computer Science & Engineering 0114 University of California San Diego La Jolla, CA 92093-0114 Abstract Dual-Route and Connectionist Single-Route models ofreading have been at odds over claims as to the correct explanation of the reading process. Recent Dual-Route models predict that subjects should show an increased naming latency for irregular words when the irregularity is earlier in the word (e.g. chef is slower than glow) - a prediction that has been confirmed in human experiments. Since this would appear to be an effect of the left-to-right reading process, Coltheart & Rastle (1994) claim that Single-Route parallel connectionist models cannot account for it. A refutation of this claim is presented here, consisting of network models which do show the interaction, along with orthographic neighborhood statistics that explain the effect. 1 Introduction A major component of the task of learning to read is the development of a mapping from orthography to phonology. In a complete model of reading, message understanding must playa role, but many psycholinguistic phenomena can be explained in the context of this simple mapping task. A difficulty in learning this mapping is that in a language such as English, the mapping is quasiregular (Plaut et al., 1996); there are a wide range of exceptions to the general rules. As with nearly all psychological phenomena, more frequent stimuli are processed faster, leading to shorter naming latencies. The regularity of mapping interacts with this variable, a robust finding that is well-explained by connectionist accounts (Seidenberg and M.cClelland, 1989; Taraban and McClelland, 1987). In this paper we consider a recent effect that seems difficult to account for in terms of the standard parallel network models. Coltheart & Rastle (1994) have shown 60 1. C. Milostan and G. W. Cottrell Position of Irregular Phoneme Filler 1 2 3 4 5 Nonword Irregular 554 542 530 529 537 Regular Control 502 516 518 523 525 Difference 52 26 12 6 12 Exception Irregular 545 524 528 526 528 Regular Control 500 503 503 515 524 Difference 45 21 25 11 4 Avg. Difl'. 48.5 23.5 18.5 8.5 8 Table 1: Naming Latency vs. Irregularity Position that the amount of delay experienced in naming an exception word is related to the phonemic position of the irregularity in pronunciation. Specifically, the earlier the exception occurs in the word, the longer the latency to the onset of pronouncing the word. Table 1, adapted from (Coltheart and Rastle, 1994) shows the response latencies to two-syllable words by normal subjects. There is a clear left-to-right ranking of the latencies compared to controls in the last row of the Table. Coltheart et al. claim this delay ranking cannot be achieved by standard connectionist models. This paper shows this claim to be false, and shows that the origin of the effect lies in a statistical regularity of English, related to the number of "friends" and "enemies" of the pronunciation within the word's neighborhood 1. 2 Background Computational modeling of the reading task has been approached from a number of different perspectives. Advocates of a dual-route model of oral reading claim that two separate routes, one lexical (a lexicon, often hypothesized to be an associative network) and one rule-based, are required to account for certain phenomena in reaction times and nonword pronunciation seen in human subjects (Coltheart et al., 1993). Connectionist modelers claim that the same phenomena can be captured in a single-route model which learns simply by exposure to a representative dataset (Seidenberg and McClelland, 1989). In the Dual-Route Cascade model (DRC) (Coltheart et al., 1993), the lexical route is implemented as an Interactive Activation (McClelland and Rumelhart, 1981) system, while the non-lexical route is implemented by a set of grapheme-phoneme correspondence (GPC) rules learned from a dataset. Input at the letter identification layer is activated in a left-to-right sequential fashion to simulate the reading direction of English, and fed simultaneously to the two pathways in the model. Activation from both the GPC route and the lexicon route then begins to interact at the output (phoneme) level, starting with the phonemes at the beginning of the word. If the GPC and the lexicon agree on pronunciation, the correct phonemes will be activated quickly. For words with irregular pronunciation, the lexicon and GPC routes will activate different phonemes: the GPC route will try to activate the regular pronunciation while the lexical route will activate the irregular (correct) 1 Friends are words with the same pronunciations for the ambiguous letter-ta-sound correspondence; enemies are words with different pronunciations. Serial Ortier in Reading Aloud 61 pronunciation. Inhibitory links between alternate phoneme pronunciations will slow down the rise in activation, causing words with inconsistencies to be pronounced more slowly than regular words. This slowing will not occur, however, when an irregularity appears late in a word. This is because in the model the lexical node spreads activation to all of a word's phonemes as soon as it becomes active. If an irregularity is late in a word, the correct pronunciation will begin to be activated before the GPC route is able to vote against it. Hence late irregularities will not be as affected by conflicting information. This result is validated by simulations with the one-syllable DRC model (Coltheart and Rastle, 1994). Several connectionist systems have been developed to model the orthography to phonology process (Seidenberg and McClelland, 1989; Plaut et al., 1996). These connectionist models provide evidence that the task, with accompanying phenomena, can be learned through a single mechanism. In particular, Plaut et al. (henceforth PMSP) develop a recurrent network which duplicates the naming latencies appropriate to their data set, consisting of approximately 3000 one-syllable English words (monosyllabic words with frequency greater than 1 in the Kucera & Francis corpus (Kucera and Francis, 1967». Naming latencies are computed based on time-t~settle for the recurrent network, and based on MSE for a feed-forward model used in some simulations. In addition to duplicating frequency and regularity interactions displayed in previous human studies, this model also performs appr~ priately in providing pronunciation of pronounceable nonwords. This provides an improvement over, and a validation of, previous work with a strictly feed-forward network (Seidenberg and McClelland, 1989). However, to date, no one has shown that Coltheart's naming latency by irregularity of position interaction can be accounted for by such a model. Indeed, it is difficult to see how such a model could account for such a phenomenon, as its explanation (at least in the DRC model) seems to require the serial, left-t~right nature of processing in the model, whereas networks such as PMSP present the word orthography all at once. In the following, we fill this gap in the literature, and explain why a parallel, feed-forward model can account for this result. 3 Experiments & Results 3.1 The Data Pronunciations for approximately 100,000 English words were obtained through an electronic dictionary developed by CMU 2 . The provided format was not amenable to an automated method for distinguishing the number of syllables in the word. To obtain syllable counts, English tw~syllable words were gathered from the Medical Research Council (MRC) Psycholinguistic Database (Coltheart and Rastle, 1994), which is conveniently annotated with syllable counts and frequency (only those with Kucera-Francis written frequency of one or greater were selected). Intersecting the two databases resulted in 5,924 tw~syllable words. There is some noise in the data; ZONED and AERIAL, for example, are in this database of purported tw~syllable words. Due to the size of the database and time limitations, we did not prune the data of these errors, nor did we eliminate proper nouns or foreign words. Singlesyllable words with the same frequency criterion were also selected for comparison with previous work. 3,284 unique single-syllable words were obtained, in contrast to 2,998 words used by PMSP. Similar noisy data as in the tw~syllable set exists in this database. Each word was represented using the orthography and phonology representation scheme outlined by PMSP. 2 Available via ftp://ftp.cs.cmu.edu/project/fgdata/dict/ 62 1.0 S 0.8 I 0.6 "" it. J 0.4 >::' ... Ii r!I 02 1. C. Milostan and G. W Cottrell Figure 1: I-syllable network latency differences & neighborhood statistics 3.2 Methods For the single syllable words, we used an identical network to the feed-forward network used by PMSP, i.e., a 105-100-61 network, and for the two syllable words, we simply used the same architecture with the each layer size doubled. We trained each network for 300 epochs, using batch training with a cross entropy objective function, an initial learning rate of 0.001, momentum of 0.9 after the first 10 epochs, weight decay of 0.0001, and delta-bar-delta learning rate adjustment. Training exemplars were weighted by the log of frequency as found in the Kucera-Francis corpus. After this training, the single syllable feed-forward networks averaged 98.6% correct outputs, using the same evaluation technique outlined in PMSP. Two syllable networks were trained for 1700 epochs using online training, a learning rate of 0.05, momentum of 0.9 after the first 10 epochs, and raw frequency weighting. The two syllable network achieved 85% correct. Naming latency was equated with network output MSE; for successful results, the error difference between the irregular words and associated control words should decrease with irregularity position. 3.3 Results Single Syllable Words First, Coltheart's challenge that a single-route model cannot produce the latency effects was explored. The single-syllable network described above was tested on the collection of single-syllable words identified as irregular by (Taraban and McClelland, 1987). In (Coltheart and Rastle, 1994), control words are selected based on equal number of letters, same beginning phoneme, and Kucera-Francis frequency between 1 and 20 (controls were not frequency matched). For single syllable words used here, the control condition was modified to allow frequency from 1 to 70, which is the range of the "low frequency" exception words in the Taraban & McClelland set. Controls were chosen by drawing randomly from the words meeting the control criteria. Each test and control word input vector was presented to the network, and the MSE at the output layer (compared to the expected correct target) was calculated. From these values, the differences in MSE for target and matched control words were calculated and are shown in Figure 1. Note that words with an irregularity in the first phoneme position have the largest difference from their control words, with this (exception - regular control) difference decreasing as phoneme position increases. Contrary to the claims of the Dual-Route model, this network does show the desired rank-ordering of MSE/latency. Serial Order in Reading Aloud 02 I' 0.1 d I 1;l 0.0 ::I! o 4 I'boaeme 1 ........ larit)' PooIIioa 1.0 O.O+--~--r----.---~--' 6 o 2 " 6 Pbonane lnegularlty PosItioIl Figure 2: 2-syllable network latency differences & neighborhood statistics 63 Two Syllable Words Testing of the two-syllable network is identical to that of the one-syllable network. The difference in MSE for each test word and its corresponding control is calculated, averaging across all test pairs in the position set. Both test words and their controls are those found in (Coltheart and Rastle, 1994). The 2-syllable network appears to produce approximately the correct linear trend in the naming MSE/latency (Figure 2), although the results displayed are not monotonically decreasing with position. Note, however, that the results presented by Coltheart, when taken separately, also fail to exhibit this trend (Table 1). For correct analysis, several "subject" networks should be trained, with formal linear trend analysis then performed with the resulting data. These further simulations are currently being undertaken. 4 Why the network works: Neighborhood effects A possible explanation for these results relies on the fact that connectionist networks tend to extract statistical regularities in the data, and are affected by regularity by frequency interactions. In this case, we decided to explore the hypothesis that the results could be explained by a neighborhood effect: Perhaps the number of "friends" and "enemies" in the neighborhood (in a sense to be defined below) of the exception word varies in English in a position-dependent way. If there are more enemies (different pronunciations) than friends (identical pronunciations) when the exception occurs at the beginning of a word than at the end, then one would expect a network to reflect this statistical regularity in its output errors. In particular, one would expect higher errors (and therefore longer latencies in naming) if the word has a higher proportion of enemies in the neighborhood. To test this hypothesis, we created some data search engines to collect word neighborhoods based on various criteria. There is no consensus on the exact definition of the "neighborhood" of a word. There are some common measures, however, so we explored several of these. Taraban & McClelland (1987) neighborhoods (T&M) are defined as words containing the same vowel grouping and final consonant cluster. These neighborhoods therefore tend to consist of words that rhyme (MUST, DUST, TRUST). There is independent evidence that these word-body neighbors are psychologically relevant for word naming tasks (i.e., pronunciation) (Treiman and Chafetz, 1987). The neighborhood measure given by Coltheart (Coltheart and Rastle, 1994), N, counts same-length words which differ by only one letter, taking string position into account. Finally, edit-distance-1 (ED1) neighborhoods are those words which can be generated from the target word by making one change 64 J. C. Milostan and G. W Cottrell (Peereman, 1995): either a letter substitution, insertion or deletion. This differs from the Coltheart N definition in that "TRUST" is in the EDI neighborhood (but not the N neighborhood) of "RUST" , and provides a neighborhood measure which considers both pronunciation and spelling similarity. However, the N and the ED-l measure have not been shown to be psychologically real in terms of affecting naming latency (Treiman and Chafetz, 1987). We therefore extended T&M neighborhoods to multi-syllable words. Each vowel group is considered within the context of its rime, with each syllable considered separately. Consonant neighborhoods consist of orthographic clusters which correspond to the same location in the word. This results in 4 consonant cluster locations: first syllable onset, first syllable coda, second syllable onset, and second syllable coda. Consonant cluster neighborhoods include the preceeding vowel for coda consonants, and the following vowel for onset consonants. The notion of exception words is also not universally agreed upon. Precisely which words are exceptions is a function of the working definition of pronunciation and regularity for the experiment at hand. Given a definition of neighborhood, then, exception words can be defined as those words which do not agree with the phonological mapping favored by the majority of items in that particular neighborhood. Alternatively, in cases assuming a set of rules for grapheme-phoneme correspondence, exception words are those which violate the rules which define thp majority of pronunciations. For this investigation, single syllable exception words are those defined as exception by the T&M neighborhood definition. For instance, PINT would be considered an exception word compared to its neighbors MINT, TINT, HINT, etc. Coltheart, on the other hand, defines exception words to be those for which his G PC rules produce incorrect pronunciation. Since we are concerned with addressing Coltheart's claims, these 2-syllable exception words will also be used here. 4.1 Results Single syllable words For each phoneme position, we compare each word with irregularity at that position with its neighbors, counting the number of enemies (words with alternate pronunciation at the supposed irregularity) and friends (words with pronunciation in agreement) that it has. The T &M neighborhood numbers (words containing the same vowel grouping and final consonant cluster) used in Figure 1 are found in (Taraban and McClelland, 1987). For each word, we calculate its (enemy) / (friend+enemy) ratio; these ratios are then averaged over all the words in the position set. The results using neighborhoods as defined in Taraban & McClelland clearly show the desired rank ordering of effect. First-position-irregularity words have more "enemies" and fewer "friends" than third-position-irregularity words, with the second-position words falling in the middle as desired. We suggest that this statistical regularity in the data is what the above networks capture. However convincing these results may be, they do not fully address Coltheart's data, which is for two syllable words of five phonemes or phoneme clusters, with irregularities at each of five possible positions. Also, due to the size of the T&M data set, there are only 2 members in the position I set, and the single-syllable data only goes up to phoneme position 3. The neighborhoods for the two-syllable data set were thus examined. Two syllable results Recall that the two-syllable test words are those used in the (Coltheart and Rastle, 1994) subject study, for which naming latency differences are shown in Table 1. CoItheart's I-letter-different neighborhood definition Serial Order in Reading Aloud 65 is not very informative in this case, since by this criterion most of the target words provided in (Coltheart and Rastle, 1994) are loners (i.e., have no neighbors at all). However, using a neighborhood based on T&M-2 recreates the desired ranking (Figure 2) as indicated by the ratio of hindering pronunciations to the total of the helping and hindering pronunciations. As with the single syllable words, each test word is compared with its neighbor words and the (enemy)/(friend+enemy) ratio is calculated. Averaging over the words in each position set, we again see that words with early irregularities are at a support disadvantage compared to words with late irregularities. 5 Summary Dual-Route models claim the irregularity position effect can only be accounted for by two-route models with left-to-right activation of phonemes, and interaction between GPC rules and the lexicon. The work presented in this paper refutes this claim by presenting results from feed-forward connectionist networks which show the same rank ordering of latency. Further, an analysis of orthographic neighborhoods shows why the networks can do this: the effect is based on a statistical interaction between friend/enemy support and position. Words with irregular orthographicphonemic correspondence at word beginning have less support from their neighbors than words with later irregularities; it is this difference which explains the latency results. The resulting statistical regularity is then easily captured by connectionist networks exposed to representative data sets. References Coltheart, M., Curitis, B., Atkins, P., and Haller, M. (1993). Models of reading aloud: Dual-route and parallel-distributed-processing approaches. Psychological Review, 100(4):589-608. Coltheart, M. and Rastle, K. (1994). Serial processing in reading aloud: Evidence for dual route models of reading. Journal of Experimental Psychology: Human Perception and Performance, 20(6):1197-1211. Kucera, H. and Francis, W. (1967). Computational Analysis of Present-Day American English. Brown University Press, Providence, RI. McClelland, J. and Rumelhart, D. (1981). An interactive activation model of context effects in letter perception: Part 1. an account of basic findings. Psychological Review, 88:375-407. Peereman, R. (1995). Naming regular and exception words: Further examination of the effect of phonological dissension among lexical neighbours. European Journal of Cognitive Psychology, 7(3):307-330. Plaut, D., McClelland, J., Seidenberg, M., and Patterson, K. (1996). Understanding normal and impaired word reading: Computational principles in quasi-regular domains. Psychological Review, 103(1):56-115. Seidenberg, M. and McClelland, J. (1989). A distributed, developmental model of word recognition and naming. Psychological Review, 96:523-568. Taraban, R. and McClelland, J. (1987). Conspiracy effects in word pronunciation. Journal of Memory and Language, 26:608-631. Treiman, R. and Chafetz, J. (1987). Are there onset- and rime-like units in printed words? In Coltheart, M., editor, Attention and Performance XII: The Psychology of Reading. Erlbaum, Hillsdale, NJ.
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Adaptation in Speech Motor Control John F. Houde* UCSF Keck Center Box 0732 San Francisco, CA 94143 houde~phy.ucsf.edu Michael I. Jordan MIT Dept. of Brain and Cognitive Sci. EI0-034D Cambridge, MA 02139 jordan~psyche.mit.edu Abstract Human subjects are known to adapt their motor behavior to a shift of the visual field brought about by wearing prism glasses over their eyes. We have studied the analog of this effect in speech. U sing a device that can feed back transformed speech signals in real time, we exposed subjects to alterations of their own speech feedback. We found that speakers learn to adjust their production of a vowel to compensate for feedback alterations that change the vowel's perceived phonetic identity; moreover, the effect generalizes across consonant contexts and to different vowels. 1 INTRODUCTION For more than a century, it has been know that humans will adapt their reaches to altered visual feedback [8]. One of the most studied examples of this adaptation is prism adaptation, which is seen when a subject reaches to targets while wearing image-shifting prism glasses [2]. Initially, the subject misses the targets, but he soon learns to compensate and reach accurately. This compensation is retained beyond the time that the glasses are worn: when the glasses are removed, the subject's reaches now overshoot targets in the direction that he compensated. This retained compensation is called adaptation, and its generation from exposure to altered sensory feedback is called sensorimotor adaptation (SA). In the study reported here, we investigated whether SA could be observed in a motor task that is quite different from reaching - speech production. Specifically, we examined whether the control of phonetically relevant speech features would respond adaptively to altered auditory feedback. By itself, this is an important theoretical question because various aspects of speech production have already been shown to be sensitive to auditory feedback [5, 1, 4]. Moreover, we were particularly *To whom correspondence should be addressed. Adaptation in Speech Motor Control 39 interested in whether speech SA would also exhibit generalization. If so, speech SA could be used to examine the organization of speech motor control. For example, suppose we observed adaptation of [c) in "get". We could then examine whether we also see adaptation of [c) in "peg". IT so, then producing [c) in the two different words must access a common, adapted representation - evidence for a hierarchical speech production system in which word productions are composed from smaller units such as phonemes. We could also examine whether adapting [c) in "get" causes adaptation of [re) in "gat". IT so, then the production representations of [c) and [re] could not be independent, supporting the idea that vowels are produced by controlling a common set of features. Such theories about the organization of the speech production system have been postulated in phonology and phonetics, but the empirical evidence supporting these theories has generally been observational and hence not entirely conclusive [7,6]. 2 METHODS To study speech SA, we focused on vowel production because the phonetically relevant features of vowel sounds are formant frequencies, which are feasible to alter in real time.1 To alter the formants of a subject's speech feedback, we built the apparatus shown in Figure 1. The subject wears earphones and a microphone and sits in front of a PC video monitor that presents words to be spoken aloud. The signal from the microphone is sent to a Digital Signal Processing board, which collects a 64ms time interval from which a magnitude spectrum is calculated. From this spectrum, formant frequencies and amplitudes are estimated. To alter the speech, the first three formant frequencies are shifted, and the shifted formants drive a formant synthesizer that creates the output speech sent to the subject's earphones. This analysis-synthesis process was accomplished with only 16ms of feedback delay. To minimize how much the subject directly heard of his own voice via bone conduction, the subject produced only whispered speech, masked with mild noise. altered feedback Altered Fonnants earphones. ;-------... .... I I •. / Fl.F2.F3 Frequency Alteration. pep PC video monitor DSP board in PC I '. • Fonnants .'" I Fonnant Estimation I microphone .' V\ /\ • L....-_~ ..... L.....---' .. ~. Magnitude Spectrum Figure 1: The apparatus used in the study. For each subject in our experiment, we shifted formants along the path defined by the (F1,F2,F3) frequencies of a subject's productions of the vowels [i), [t.]' [c], [re], 1 See [3] for detailed discussion of the methods used in this study. 40 J. F. Houde and M. I Jordan and [a].2 Figure 2 shows examples of this shifting process in (Fl,F2) space for the feedback transformations that were used in the study. TOI shift formants along the subject's [i]-[a] path, we extend the path at both ends and we number the endpoints and vowels to make a path position measure that normalizes the distances between vowels. The formants of each speech sound F produced by the subject were then re-represented in terms of path projection - the path posiUon of nearest path point P, and path deviation - the distance D to this point P. Feedback transformations were constructed to alter path projections while preserving path deviations. Two different transformations were used. The +2.0 transformation added 2.0 to path projections: under this transform, if the subject produced speech sound F (a sound near [cD, he heard instead sound F+ (a sound near [aD. The subject could compensate for this transform and hear sound F only by shifting his production of F to F- (a sound near [iD. The -2.0 transformation subtracted 2.0 from path projections: under this transformation, if the subject produced F, he heard F-. Thus, in this case, the subject could compensate by shifting production to F+. FF+ F+ [ah)O 5 end /)6 end /)6 Fl Fl (a) +2.0 Transformation (b) -2.0 Transformation Figure 2: Feedback transformations used in the study. These feedback transformations were used in an experiment in which a subject was visually prompted to whisper words with a 300ms target duration_ Word promptings occurred in groups of ten called epochs. Within each epoch, the first six words came from a set of training words and the last four came from a set of testing words. The subject heard feedback of his first five word productions in each epoch, while masking noise blocked his hearing for his remaining five word productions in the epoch. Thus, the subject only heard feedback of his production of the first five training words and never heard his productions of the testing words. 2Where possible, we use standard phonetic symbols for vowel sounds: [i] as in "seat", [L] as in "hit", [c] as in "get", [re] as in "hat", and [a] as in "pop". Where font limitations prevent us from using these symbols, we use the alternate notation of [i], [ib], [eh] , rae], and lab], respectively, for the same vowel sounds. Adaptation in Speech Motor Control 41 The experiment lasted 2 hours and consisted of 422 epochs divided over five phases: 1. A 10 minute warmup phase used to acclimate the subject to the experimental setup. 2. A 17 minute baseline phase used to measure formants of the subject's normal vowel productions. 3. A 20 minute ramp phase in which the subject's feedback was increasingly altered up to a maximum value. 4. A 1 hour training phase in which the subject produced words while the feedback was maximally altered. 5. A 17 minute test phase used to measure formants of the subject's postexposure vowel productions while his feedback was maximally altered. By the end of the ramp phase, feedback alteration reached its maximum strength, which was +2.0 for half the subjects and -2.0 for the other subjects. In addition, all subjects were run in a control experiment in which feedback was never altered. The two word sets from which prompted words were selected were both sets of eve words. Training words (in which adaptation was induced) were all bilabials with [c] as the vowel ("pep", "peb", "bep", and "beb"). Testing words (in which generalization of the training word adaptation was measured) were divided into two subsets, each designed to measure a different type of generalization: (1) context generalization words, which had the same vowel [c] as the training words but varied the consonant context ("peg", "gep", and "teg"); (2) vowel target generalization words, which had the same consonant context as the training words but varied the vowel ("pip,", "peep,", "pap" , and "pop"). Eight male MIT students participated in the study. All were native speakers of North American English and all were naive to the purpose of the study. 3 RESULTS To illustrate how we measured compensation and adaptation in the experiments, we first show the results for an individual subject. Figure 3 shows (F1,F2) plots of response of subject OB in both the adaptation experiment (in which he was exposed to the -2.0 feedback transformation) and the control experiment. In each figure, the dotted line is OB's [i]-[a] path. Figure 3(a) shows OB's compensation responses, which were measured from his productions of the training words made when he heard feedback of his whispering. The solid arrow labeled "-2.0 xform" shows how much his mean vowel formants changed (testing phase - baseline phase) after being exposed to the -2.0 feedback transformation. It shows he shifted his production of [c] to something a bit past [re], which corresponds to a p;;tth projection change of slightly more than one vowel interval towards [a]. Thus, since the path projection shift of the transform was -2.0 (2.0 vowel intervals towards liD, the figure shows that OB compensates for over half the action of the transformation. The hollow arrow in Figure 3(a) shows how OB heard his compensation. It shows he heard his actual production shift from [c] towards [a] as a shift from [i] back towards [c]. Figure 3(b) shows how much of OB's compensation was retained when he whispered the training words with feedback blocked by noise .. This retained compensation is called adaptation, and it was measured from path projection changes by the same method used to measure compensation. In the figure, we see OB's adaptation 42 J. F. Houde and M. I Jordan response (the solid "-2.0 xform" arrow) is a path projection shift of slightly less than one vowel interval, so his adaptation is slightly less than half. Thus, the figure shows that OB retains an appreciable amount of his compensation in the absence of feedback. Finally, in both plots of Figure 3, the almost non-existent "control" arrows show that OB exhibited almost no formant change in the control experiment - as we would expect since feedback was never altered in this experiment. 2600 2400 2200 N 2000 X C'I 1800 ~ 1600 1400 1200 [i) ~ .. :~ [eb;···~ ~ \. ·2.0 dorm control "': rae) \ '. [ab) • 300 SOO 700 900 II 00 Fl (Hz) (a) compensation 2600 2400 2200 N 2000 X '-' C'I 1800 ~ 1600 1400 1200 [i) •. '\\ .\\. \~ [ib)", " ' .. [eb) '-. ~ . control ,\. 2.0 dorm [ae) \ \ [ab)· 300 SOO 700 900 1100 Fl (Hz) (b) adaptation Figure 3: Subject OB compensation and adaptation. The plots in Figure 4 show that there was significant compensation and adaptation across all subjects. In these plots, the vertical scale indicates how much the changes in mean vowel formants (testing phase - baseline phase) in each subject's productions of the training words compensated for the action of the feedback transformation he was exposed to. The filled circles linked by the solid line show compensation (Figure 4(a» and retained compensation, or adaptation (Figure 4(b)) across subjects in the adaptation experiment in which feedback was altered; the open circles linked by the dotted line show the same measures from the control experiment in which feedback was not altered. (The solid and dotted lines facilitate comparison of results across subjects but do not signify any relationship between subjects.) In the control experiment, for each subject, compensation and adaptation were measured with respect to the feedback transformation used in the adaptation experiment. The plots show that there are large variations in compensation and adaptation across subjects, but overall there was significantly more compensation (p < 0.006) and adaptation (p < 0.023) in the adaptation experiments that in the control experiments. Figure 5 shows plots of how much of the adaptation observed in the training words carried over the the testing words. For each testing word shown, a measure of this carryover called mean generalization is plotted, which was calculated as a ratio of adaptations: the adaptation seen in the testing word divided by the adaptation seen in the training words (adaptation values observed in the control experiment were Adaptation in Speech Motor Control 43 1.2 1.2 1.0 1.0 0.80 0.80 0 .60 0 .60 0.40 0.20 0.40 0.20 ro £ ; \ /rs \" 0.00 ah 0.00 i \~_ .J: . / .~ vs -0.20 sr ab -0.20 ro ty vs sr cw -0.40 -0 .40 (a) mean compensation (b) mean adaptatioll Figure 4: Mean compensation and adaptation across all subjects. subtracted out to remove any effects not arising from exposure to altered feedback). Figure 5(a) shows mean generalization for the context generalization words except for "pep" (since "pep" was also a training word). The plot shows large variance in mean generalization for each of the three words, but overall there was significant (p < 0.040) mean generalization. Thus, there was significant carryover of the adaptation of [c] in the training words to different consonant contexts. Figure 5(b) shows mean generalization for the vowel target generalization words. Not all of these words are shown: unfortunately, we weren't able to accurately estimate the formants of [i] and [a], so "peep" and "pop" were dropped from our generalization analysis. For the remaining two vowel target generalization words, the plot shows large variance in mean generalization for each of the words, but overall there was significant (p < 0.013) mean generalization. Thus, there was significant generalization of the adaptation of [c] to the vowels [t] and [eel. 4 DISCUSSION Several conclusions can be drawn from the experiment described above. First, comparison of the adaptation and control experiment results seen in Figure 4 shows a clear effect of exposure to the altered feedback: this exposure caused compensation responses in most subjects. Furthermore, the adaptation results show that this compensation was retained in absence of acoustic feedback. Next, the context generalization results seen in Figure 5(a) show that some adapted representation of [c] is shared across the training and testing words. These results provide evidence for a hierarchical speech production system in which words are composed from smaller phoneme-like units. Finally, the vowel target generalization results seen in Figure 5(b) show that the production representations of [t], [cl, and [ee] are not independent, suggesting that these vowels are produced by controlling a common set of features. 44 1. F. Houde and M. I. Jordan 2.0 2.0 1.5 1.5 peg .,: 1.0 .,: pap 1.0 Q) teg 0 gep c: <'S Q) 0.50 ~ 0.00 Q) ~ 0 c: pip <'S Q) 0.50 ~ A 0.00 -0.50 ·0.50 (a) context gen. words (b) vowel target gen. words Figure 5: Mean generalization for the testing words, averaged across subjects. Thus, in summary, our study has shown (1) that speech production, like reaching, can be made to exhibit sensorimotor adaptation, and (2) that this adaptation effect exhibits generalization that can be used to make inferences about the structure of the speech production system. Acknowledgments We thank J. Perkell, K. Stevens, R. Held and P. Sabes for helpful discussions. References [1] V. L. Gracco et al., (1994) J. Acoust. Soc. Am. 95:2821 [2] H. V. Helmholtz, (1867) Treatise on physiological optics, Vol. 3 (Eng. Trans. by Optical Soc. of America, Rochester, NY, 1925) [3] J. F. Houde (1997), Sensorimotor Adaptation in Speech Production, Doctoral Dissertation, M. I. T., Cambridge, MA. [4) H. Kawahara (1993) J. Acoust. Soc. Am. 94:1883. [5] B. S. Lee (1950) J. Acoust. Soc. Am. 22:639. [6] W. J. M. Levelt (1989), Speaking: from intention to articulation, MIT Press, Cambridge, MA. [7] A. S. Meyer (1992), Cognition 42:18l. [8) R. B. Welch (1986), Handbook of Perception and Human Performance, K. R. Boff, L. Kaufman, J. P. Thomas Eds., John Wiley and Sons, New York.
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An Incremental Nearest Neighbor Algorithm with Queries Joel Ratsaby· N.A.P. Inc. Hollis, New York Abstract We consider the general problem of learning multi-category classification from labeled examples. We present experimental results for a nearest neighbor algorithm which actively selects samples from different pattern classes according to a querying rule instead of the a priori class probabilities. The amount of improvement of this query-based approach over the passive batch approach depends on the complexity of the Bayes rule. The principle on which this algorithm is based is general enough to be used in any learning algorithm which permits a model-selection criterion and for which the error rate of the classifier is calculable in terms of the complexity of the model. 1 INTRODUCTION We consider the general problem of learning multi-category classification from labeled examples. In many practical learning settings the time or sample size available for training are limited. This may have adverse effects on the accuracy of the resulting classifier. For instance, in learning to recognize handwritten characters typical time limitation confines the training sample size to be of the order of a few hundred examples. It is important to make learning more efficient by obtaining only training data which contains significant information about the separability of the pattern classes thereby letting the learning algorithm participate actively in the sampling process. Querying for the class labels of specificly selected examples in the input space may lead to significant improvements in the generalization error (cf. Cohn, Atlas & Ladner, 1994, Cohn, 1996). However in learning pattern recognition this is not always useful or possible. In the handwritten recognition problem, the computer could ask the user for labels of selected patterns generated by the computer -The author's coordinates are: Address: Hamered St. #2, Ra'anana, ISRAEL. Eaail: jer@ee. technion. ac.il An Incremental Nearest Neighbor Algorithm with Queries 613 however labeling such patterns are not necessarily representative of his handwriting style but rather of his reading recognition ability. On the other hand it is possible to let the computer (learner) select particular pattern classes, not necessarily according to their a priori probabilities, and then obtain randomly drawn patterns according to the underlying unknown class-conditional probability distribution. We refer to such selective sampling as sample querying. Recent theory (cf. Ratsaby, 1997) indicates that such freedom to select different classes at any time during the training stage is beneficial to the accuracy of the classifier learnt. In the current paper we report on experimental results for an incremental algorithm which utilizes this sample-querying procedure. 2 THEORETICAL BACKGROUND We use the following setting: Given M distinct pattern classes each with a class conditional probability density fi (x), 1 ::; i ::; M, x E IR d, and a priori probabilities Pi, 1 ::; i ::; M. The functions fi (x), 1 ::; i ::; M, are assumed to be unknown while the Pi are assumed to be known or easily estimable as is the case oflearning character recognition. For a sample-size vector m = [m1' ... , mM] where L~1 mj = m denote by (m = {( x j , Yj ) } T= 1 a sample of labeled examples consisting of mi example from pattern class i where Yj, 1 ::; j ::; m, are chosen not necessarily at random from {I, 2 .... , M}, and the corresponding Xj are drawn at random i.i.d. according to the class conditional probability density fy) (x). The expected misclassification error of a classifier c is referred to as the loss of c and is denoted by L( c ). It is defined as the probability of miselassification of a randomly drawn x with respect to the underlying mixture probability density function f(x) = L~l pdi(X). The loss is commonly represented as L(c) = El{x :c(x);iy(x)}, where l{xEA} is the indicator function of a set A, expectation is taken with respect to the joint probability distribution fy (x )p(y) where p(y) is a discrete probability distribution taking values Pi over 1 ::; i ::; M, while y denotes the label of the class whose distribution fy(x) was used to draw x. The loss L(e) may also be written as L(e) = Lf!l PiEi1{c(x);ii} where Ei denotes expectation with respect to fi(X) . The pattern recognition problem is to learn based on em the optimal classifier, also known as the Bayes classifier, which by definition has minimum loss whkh we denote by L * . A multi-category classifier c is represented as a vector e(x) = [el(x), ... , CM(X)] of boolean classifiers, where Ci (x) = 1 if e( x) = i, and Ci (x) = 0 otherwise, 1 ::; i ::; M. The loss L(e) of a multi-category classifier c may then be expressed as the average of the losses of its component classifiers, i.e., L(e) = L~l PiL(ei) where for a boolean classifier ei the loss is defined as L(ed = Ed{c.(x);il}' As an estimate of L(e) we define the empirical loss Lm(c} = L~l p;Lm.(e) where Lm,(c) = ~, Lj :Y1=i l{c(x);ii} which may also can be expressed as Lm,(ci) = ~, Lj:YJ=i l{c.(x);il}' The family of all classifiers is assumed to be decomposed into a multi-structure 5 = 51 X 52 X .. , X 5M , where 5 j is a nested structure (cf. Vapnik, 1982) of boolean families Bk). ' ji = 1,2, ... , for 1 ::; i ::; M, i.e., 51 = Bkl , Bk 2 , •• • ,Bk)1 ' ... , 52 = BkllBk2,···,Bk12'oo" up to 5M = Bk1 ,Bk2 ,· .. ,Bk)M'·'" where ki, E71+ denotes the VC-dimension of BkJ , and Bk1 , ~ Bk1,+I' 1 ::; i ::; M . For any fixed positive integer vector j E 7l ~ consider the class of vector classifiers 1ik(j) = Bk x Bk x· .. X Bk == 1ik where we take the liberty in dropping the multi11.l2 1M index j and write k instead of k(j) . Define by (h the subfamily of 1ik consisting 614 J. Ratsaby of classifiers C that are well-defined, i.e., ones whose components Ci, 1 S; i S; M satisfy Ut!I{X: Ci(X) = I} = IRd and {x : Ci(X) = l}n{x: Cj(x) = I} = 0, for 1 S; i =1= j S; M . From the Vapnik-Chervonenkis theory (cf. Vapnik, 1982, Devroye, Gyorfi & Lugosi, 1996) it follows that the loss of any boolean classifier Ci E Bk ], is, with high confidence, related to its empirical loss as L( Ci ) S Lm. (Ci) + f( m i, kj ,) where f(mi' kj,) = const Jkj,ln mi!mi , 1 S; i S; M, where henceforth we denote by const any constant which does not depend on the relevant variables in the expression. Let the vectors m = [ml, "" mM] and k == k(j) = [kil , . ··, kjM] in 'lh~. Define f(m, k) = 2:f'!1 Pif(mi,kj,). It follows that the deviation between the empirical loss and the loss is bounded uniformly over all multi-category classifiers in a class (}k by f(m, k) . We henceforth denote by ck the optimal classifier in (}k , i.e., ck = argmincE~h L( c) and Ck = argmincEQk Lm (c) is the empirical loss minimizer over the class (}k. The above implies that the classifier Ck has a loss which is no more than L( c~) + f(m, k) . Denote by k* the minimal complexity of a class (}k which contains the Bayes classifier. We refer to it as the Bayes complexity and henceforth assume k: < 00, 1 S; i S; M. If k* was known then based on a sample of size m with a sample size vector m = [ml , "" mM] a classifier Ck o whose loss is bounded from above by L * + f( m, k*) may be determined where L * = L( c~o) is the Bayes loss. This bound is minimal with respect to k by definition of k* and we refer to it as the minimal criterion. It can be further minimized by selecting a sample of size vector m* = argmin{ 'WM ."\,,,M __ }f(m,k*). This basically says that more examples mEaJ+ '6,=1 m,_m should be queried from pattern classes which require more complex discriminating rules within the Bayes classifier. Thus sample-querying via minimization of the minimal criterion makes learning more efficient through tuning the subsample sizes to the complexity of the Bayes classifier. However the Bayes classifier depends on the underlying probability distributions which in most interesting scenarios are unknown thus k* should be assumed unknown. In (Ratsaby, 1997) an incremental learning algorithm, based on Vapnik's structural risk minimization, generates a random complexity sequence ken), corresponding to a sequence of empirical loss minimizers ck(n) over (}k(n), which converges to k* with increasing time n for learning problems with a zero Bayes loss. Based on this, a sample-query rule which achieves the same minimization is defined without the need to know k*. We briefly describe the main ideas next. At any time n, the criterion function is c(-, ken)) and is defined over the m-domain 'lhtt· A gradient descent step of a fixed size is taken to minimize the current criterion. After a step is taken, a new sample-size vector men + 1) is obtained and the difference m( n + 1) - m( n) dictates the sample-query at time n, namely, the increment in subsample size for each of the M pattern classes. With increasing n the vector sequence men) gets closer to an optimal path defined as the set which is comprised of the solutions to the minimization of f( m, k*) under all different constraints of 2:~1 mi = m, where m runs over the positive integers. Thus for all large n the sample-size vector m( n) is optimal in that it minimizes the minimal cri terion f(', k*) for the current total sample size m( n). This consti tutes the samplequerying procedure of the learning algorithm. The remaining part does empirical loss minimization over the current class (}k(n) and outputs ck(n)" By assumption, since the Bayes classifier is contained in (}k o , it follows that for all large n, the loss L(ck(n» S; L* + min{mE~ :2::1 m,=m'(n)} f(m, k*), which is basically the minimal criterion mentioned above. Thus the algorithm produces a classifier ck(n) with a An Incremental Nearest Neighbor Algorithm with Queries 615 minimal loss even when the Bayes complexity k* is unknown. In the next section we consider specific modf'l classes consisting of nearest-nf'ighbor classifiers on which we implement this incremental learning approach. 3 INCREMENTAL NEAREST-NEIGHBOR ALGORITHM Fix and Hodges, cf. Silverman & Jones (1989). introduced the simple but powerful nearest-neighbor classifier which based on a labeled training sample {(Xj,yd}i!:I' Xi E m,d, Yi E {I, 2, ... , M}, when given a pattern x, it outputs the label Yj corresponding to the example whose x j is closest to x. Every example in the training sample is used for this decision (we denote such an example as a prototype) thus the empirical loss is zero. The condensed nearest-neighbor algorithm (Hart, 1968) and the reduced nearest neighbor algorithm (G ates, 1972) are procedures which aim at reducing the number of prototypes while maintaining a zero empirical loss. Thus given a training sample of size m, after running either of these procedures, a nearest neighbor classifier having a zero empirical loss is generated based on s ~ m prototypes. Learning in this manner may be viewed as a form of empirical loss minimization with a complexity regularization component which puts a penalty proportional to the number of prototypes. A cell boundary ej,j of the voronoi diagram (cf. Preparata & Shamos, 1985) corresponding to a multi-category nearest-neighbor classifier c is defined as the (d - 1 )-dimensional perpendicular-bisector hyperplane <?f the line connecting the x-component of two prototypes Xi and Xj. For a fixed I E {1, ... ,M}, the collection of voronoi cell-boundaries based on pairs of prototypes of the form (xi,/), (Xj,q) where q =1= I, forms the boundary which separates the decision region labeled I from its complement and represents the boolean nearest-neighbor classifier CI. Denote by kl the number of suc.h cell-boundaries and denote by SI the number of prototypes from a total of ml examples from pattern class t. The value of kl may be calculated directly from the knowledge of the SI prototypes, 1 ~ I ~ M, using various algorithms. The boolean classifier Cl is an element of an infinite class of boolean classifiers based on partitions of m,d by arrangements of kl hyperplanes of dimensionality d - 1 where each of the cells of a partition is labeled either 0 or 1. It follows, cf. Devroye et. al. (1996), that the loss of a multi-category nearestneighbor classifier C which consists of 81 prototypes out of ml examples, 1 ~ I ~ M, is bounded as L(c) ~ Lm(c) + f(m, k), where the a priori probabilities are taken as known, m = [mI, ... ,mM)' k = [kI , ... ,kM] and f(m,k) = E~lPlf(ml,kl)' where f( ml, kz) = const J « d + 1 )kl In ml + (ekd d)d) / mi . Letting k* denote the Bayes complexity then f(-, k*) represents the minimal criterion. The next algorithm uses the Condense and Reduce procedures in order to generate a sequence of classifiers ck(n) with a complexity vector k( n) which tends t.o k* as n --+ 00. A sample-querying procedure referred to as Greedy Query (GQ) chooses at any time n to increment the single subsample of pattern class j*(n) where mjO(n) is the direction of maximum descent of the criterion f(', k( n)) at the current sample-size vector m( n). For the part of the algorithm which utilizes a Delaunay-Triangulation procedure we use the fast Fortune's algorithm (cf. 0 'Rourke) which can be used only for dimensionality d = 2. Since all we are interested is in counting Voronoi borders between all adjacent Voronoi cells then an efficient computation is possible also for dimensions d > 2 by resorting to linear programming for computing the adjacencies of facets of a polyhedron, d. Fukuda (1997). 616 1. Ratsaby Incremental Nearest Neighbor (INN) Algorithm Initialization: (Time n = 0) Let increment-size t::. be a fixed small positive integer. Start with m(O) = [e, ... , e], where e is a small posit.ive integer. Draw (m(o) = {(m](o)}§";l where (m)(O ) consists of mJ(O) randomly drawn i.i.d. examples from pattern class j. While (number of available examples 2: t::.) Do: 1. Call Procedure CR: chIn ) = CR«(m(n» . 2. Call Procedure GQ: m(n + 1) = GQ(n). 3. n:= n + 1. End While //Used up all examples . Output: NN-cIassifier ck(n). Procedure Condense-Reduce (CR) Input: Sample (m(n) stored in an array A[] of size m(n). Initialize: Make only the first example A[I] be a prototype. //Condense Do: ChangeOccl.lred := FALSE. For i= 1, . .. , m(n): • Classify A[i] based on available prototypes using the NN-Rule. • If not correct then Let A[i] be a prototype. ChangeOcel.lred:= TRUE. • End If End For While ( ChangeOecl.lred). //Reduce Do: ChangeOccl.lred := FALSE. For i = 1, ... . m(n): • If A[ i] is a prototype then classify it using the remaining prototypes by the NN-Rule. • If correct then Make A[i] be not a prototype. ChangeOccl.lred := TRUE. • End If End For While ( ChangeOec'Ured). Run Delaunay-Triangulation Let k(n) = [k~, ... , kM ], k. denotes the number of Voronoi-cell boundaries associated with the s, prototypes. Return (NN-classifier with complexity vector k(n». Procedure Greedy-Query (GQ) Input: Time n. j*(n) := argmaxl~J~M I a!) f(m, k(n»1 Im(n) Draw: t::. new i.i.d. examples from class j·(n). Denote them by ( . Update Sample: (m]·(n)(n+l) := Cn)·(n)Cn) U (, while (m,Cn+l) := (m,Cn), for 1 :S i:;i: j·(n) :S M . Return: (m(n)+ t::. eJ.Cn», where eJ is an all zero vector except 1 at jth element. An Incremental Nearest Neighbor Algorithm with Queries 617 3.1 EXPERIMENTAL RESULTS We ran algorithm INN on several two-dimensional (d = 2) multi-category classification problems and compared its generalization error versus total sample size m with that of batch learning, the latter uses Procedure CR (but not Procedure GQ) with uniform subsample proportions, i.e., mi = ~, 1 ~ i ~ M. We ran three classification problems consisting of 4 equiprobable pattern classes with a zero Bayes loss. The generalization curves represent the average of 15 independent learning runs of the empirical error on a fixed size test set. Each run (both for INN and Batch learning) consists of 80 independent experiments where each differs by 10 in the sample size used for training where the maximum sample size is 800. We call an experiment a success if INN results in a lower generalization error than Batch. Let p be the probability of INN beating Batch. We wish to reject the hypothesis H that p = ~ which says that INN and Batch are approximately equal in performance. The results are displayed in Figure 1 as a series of pairs, the first picture showing the pattern classes of the specific problem while the second shows the learning curves for the two learning algorithms. Algorithm INN outperformed the simple Batch approach with a reject level of less than 1 %, the latter ignoring the inherent Bayes complexity and using an equal subsample size for each of the pattern classes. In contrast, the INN algorithm learns, incrementally over time, which of the classes are harder to separate and queries more from these pattern classes. References Cohn D., Atlas L., Ladner R. (1994), Improving Generalization with Active Learning. Machine Learning, Vol 15, p.201-221. Devroye L., Gyorfi L. Lugosi G. (1996). "A Probabilistic Theory of Pattern Recognition", Springer Verlag. Fukuda K. (1997). Frequently Asked Questions in Geometric Computation. Technical report, Swiss Federal Institute of technology, Lausanne. Available at ftp://ftp.ifor.ethz.ch/pub/fukuda/reports. Gates, G. W. (1972) The Reduced Nearest Neighbor Rule. IEEE Trans. Info. Theo., p.431-433. Hart P. E. (1968) The Condensed Nearest Neighbor Rule. IEEE Trans. on Info. Thea., Vol. IT-14, No.3. O'rourke J . (1994). "Computational Geometry in C". Cambridge University Press. Ratsaby, J. (1997) Learning Classification with Sample Queries. Electrical Engineering Dept., Technion, CC PUB #196. Available at URL http://www.ee.technion.ac.il/jer/iandc.ps. Rivest R. L., Eisenberg B. (1990), On the sample complexity of pac-learning using random and chosen examples. Proceedings of the 1990 Workshop on Computational Learning Theory, p. 154-162, Morgan Kaufmann, San Maeto, CA. B. W. Silverman and M. C. Jones. E. Fix and J. 1. Hodges (1951): An important contribution to nonparametric discriminant analysis and density estimationcommentary on Fix and Hodges (1951). International statistical review, 57(3), p.233-247, 1989. Vapnik V.N., (1982), "Estimation of Dependences Based on Empirical Data", Springer-Verlag. Berlin. 1 618 ~" __ ---I'----""I----r----' PoltcmCI3.~S I '..00 PaltcmCla. ... 2 PoltcmCl3.'i.'i.l I(ICl( l'a"cmCJa.~s4 I I 100 1:-0 :!OO n.2 n.1 . ~ 0.1 .~ ~. I ~ 8o.0K 0.06 0.04 0.0 0.25 O. ~ 0.17 ..,; .~ 0.15 .~ O. IJ ;; Ii 0.1 c J. Ratsaby ~ , il ' \ \ , \. ',. ~ ":-. !t~ . ..... "<.., . fb'""'" ", .. : ' . "'-, , ' . 0 100 200 ~OO 400 500 600 700 ROO Tolal number of """"",I .... Balch ....... INN C 0.011 -~""''' .................... -.....:""," . ~.>...-,...". 0 l'aU<:m('I_1 (.<>,) PaucmCl_2 l'aUc:mCIass.~ Cit( l'aUcmCl-.l CoX .·~1~< : ., I( t.f. '~ 'J ~::·.t"<i 'S.:: - (II, IC( .. , ';;'{ ~"'x ~ ,(. xi/(o( I( .Itc>fC" _-.:-c 0 ~ PanemCI ..... 1 ",,,,. PanemClass2 PananCI .. ....u 1(( PanemCl ..... 4 w'Ic ~, ><'V-' ~t. i.8cl"« (# ( \·qllc ., . , .... I(~ « I ~PC( icc ~: .... I( • 100 I~ 200 0.05 0.03 O.J 0.2 0.26 ~ 0.24 ~ . ~ 0.22 ."§ 0.2 "ii J 0.1 0.16 0.14 0.12 0.1 0 100 Balch -(r INN 0 Batch -- INN 2110 JOn 400 SIlO TnUl numhcr of """"",lea 400 T oI.tl numher of cxamplCli Figure 1. Three different Pattern Classification Problems and Learning Curves of the INN-Algorithm compared to Batch Learning. ~ (1110 70n 1100
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Competitive On-Line Linear Regression V. Vovk pepartment of Computer Science Royal Holloway, University of London Egham, Surrey TW20 OEX, UK vovkGdcs.rhbnc.ac.uk Abstract We apply a general algorithm for merging prediction strategies (the Aggregating Algorithm) to the problem of linear regression with the square loss; our main assumption is that the response variable is bounded. It turns out that for this particular problem the Aggregating Algorithm resembles, but is slightly different from, the wellknown ridge estimation procedure. From general results about the Aggregating Algorithm we deduce a guaranteed bound on the difference between our algorithm's performance and the best, in some sense, linear regression function's performance. We show that the AA attains the optimal constant in our bound, whereas the constant attained by the ridge regression procedure in general can be 4 times worse. 1 INTRODUCTION The usual approach to regression problems is to assume that the data are generated by some stochastic mechanism and make some, typically very restrictive, assumptions about that stochastic mechanism. In recent years, however, a different approach to this kind of problems was developed (see, e.g., DeSantis et al. [2], Littlestone and Warmuth [7]): in our context, that approach sets the goal of finding an on-line algorithm that performs not much worse than the best regression function found off-line; in other words, it replaces the usual statistical analyses by the competitive analysis of on-line algorithms. DeSantis et al. [2] performed a competitive analysis of the Bayesian merging scheme for the log-loss prediction game; later Littlestone and Warmuth [7] and Vovk [10] introduced an on-line algorithm (called the Weighted Majority Algorithm by the Competitive On-line Linear Regression 365 former authors) for the simple binary prediction game. These two algorithms (the Bayesian merging scheme and the Weighted Majority Algorithm) are special cases of the Aggregating Algorithm (AA) proposed in [9, 11]. The AA is a member of a wide family of algorithms called "multiplicative weight" or "exponential weight" algorithms. Closer to the topic of this paper, Cesa-Bianchi et al. [1) performed a competitive analysis, under the square loss, of the standard Gradient Descent Algorithm and Kivinen and Warmuth [6] complemented it by a competitive analysis of a modification of the Gradient Descent, which they call the Exponentiated Gradient Algorithm. The bounds obtained in [1, 6] are of the following type: at every trial T, (1) where LT is the loss (over the first T trials) of the on-line algorithm, LT is the loss of the best (by trial T) linear regression function, and c is a constant, c > 1; specifically, c = 2 for the Gradient Descent and c = 3 for the Exponentiated Gradient. These bounds hold under the following assumptions: for the Gradient Descent, it is assumed that the L2 norm of the weights and of all data items are bounded by constant 1; for the Exponentiated Gradient, that the Ll norm of the weights and the Loo norm of all data items are bounded by 1. In many interesting cases bound (1) is weak. For example, suppose that our comparison class contains a "true" regression function, but its values are corrupted by an Li.d. noise. Then, under reasonable assumptions about the noise, LT will grow linearly in T, and inequality (1) will only bound the difference LT - LT by a linear function of T. (Though in other situations bound (1) can be better than our bound (2), see below. For example, in the case of the Exponentiated Gradient, the 0(1) in (1) depends on the number of parameters n logarithmically whereas our bound depends on n linearly.) In this paper we will apply the AA to the problem of linear regression. The AA has been proven to be optimal in some simple cases [5, 11], so we can also expect good performance in the problem of linear regression. The following is a typical result that can be obtained using the AA: Learner has a strategy which ensures that always LT ~ LT + nIn(T + 1) + 1 (2) (n is the number of predictor variables). It is interesting that the assumptions for the last inequality are weaker than those for both the Gradient Descent and Exponentiated Gradient: we only assume that the L2 norm of the weights and the Loo norm of all data items are bounded by constant 1 (these assumptions will be further relaxed later on). The norms L2 and Loo are not dual, which casts doubt on the accepted intuition that the weights and data items should be measured by dual norms (such as Ll-Loo or L2-L2). Notice that the logarithmic term nln(T + 1) of (2) is similar to the term ~ In T occurring in the analysis of the log-loss game and its generalizations, in particular in Wallace's theory of minimum message length, Rissanen's theory of stochastic complexity, minimax regret analysis. In the case n = 1 and Xt = 1, Vt, inequality (2) differs from Freund's [4] Theorem 4 only in the additive constant. In this paper we will see another manifestation of a phenomenon noticed by Freund [4]: for some important problems, the adversarial bounds of on-line competitive learning theory 366 V. Vovk are only a tiny amount worse than the average-case bounds for some stochastic strategies for Nature. A weaker variant of inequality (2) can be deduced from Foster's [3] Theorem 1 (if we additionally assume that the response variable take only two values, -1 or 1): Foster's result implies LT ~ LT + 8n In(2n(T + 1)) + 8 (a multiple of 4 arises from replacing Foster's set {O, 1} of possible values of the response variable by our {-1, 1}j we also replaced Foster's d by 2n: to span our set of possible weights we need 2n Foster's predictors). Inequality (2) is also similar to Yamanishi's [12] resultj in that paper, he considers a more general framework than ours but does not attempt to find optimal constants. 2 ALGORITHM We consider the following protocol of interaction between Learner and Nature: FOR t = 1,2, ... Nature chooses Xt € m.n Learner chooses prediction Pt E m. Nature chooses Yt E [-Y, Y] END FOR. This is a "perfect-information" protocol: either player can see the other player's moves. The parameters of our protocol are: a fixed positive number n (the dimensionality of our regression problem) and an upper bound Y > 0 on the value Yt returned by Nature. It is important, however, that our algorithm for playing this game (on the part of Learner) does not need to know Y. We will only give a description of our regression algorithmj its derivation from the general AA will be given in the future full version of this paper. (It is usually a nontrivial task to represent the AA in a computationally efficient form, and the case of on-line linear regression is not an exception.) Fix n and a > O. The algorithm is as follows: A :=alj b:=O FOR TRlAL t = 1,2, ... : read new Xt E m.n A:= A +XtX~ output prediction Pt := b' A -1 Xt read new Yt E m. b:= b+YtXt END FOR. In this description, A is an n x n matrix (which is always symmetrical and positive definite), bE mn , I is the unit n x n matrix, and 0 is the all-O vector. The naive implementation of this algorithm would require O(n3) arithmetic operations at every trial, but the standard recursive technique allows us to spend only O(n2 ) arithmetic operations per trial. This is still not as good as for the Gradient Descent Algorithm and Exponentiated Gradient Algorithm (they require only O(n) Competitive On-line Linear Regression 367 operations per trial); we seem to have a trade-off between the quality of bounds on predictive performance and computational efficiency. In the rest of the paper "AA" will mean the algorithm. described in the previous paragraph (which is the Aggregating Algorithm applied to a particular uncountable pool of experts with a particular Gaussian prior). 3 BOUNDS In this section we state, without proof, results describing the predictive performance of our algorithm. Our comparison class consists of the linear functions Yt = W· Xt, where W E m.n • We will call the possible weights w "experts" (imagine that we have continuously many experts indexed by W E m.n ; Expert w always recommends prediction w . Xt to Learner). At every trial t Expert w and Learner suffer loss (Yt - w . Xt)2 and (Yt - Pt)2, respectively. Our notation for the total loss suffered by Expert w and Learner over the first T trials will be and respectively. T LT(W) := L(Yt - W· Xt)2 t=1 T LT(Learner) := L(Yt - Pt)2, t=1 For compact pools of experts (which, in our setting, corresponds to the set of possible weights w being bounded and closed) it is usually possible to derive bounds (such as (2» where the learner's loss is compared to the best expert's loss. In our case of non-compact pool, however, we need to give the learner a start on remote experts. Specifically, instead of comparing Learner's performance to infw LT(W), we compare it to infw (LT(W) + allwlI2) (thus giving ~arner a start of allwII2 on Expert w), where a > 0 is a constant reflecting our prior expectations about the "complexity" IIwll := -IE:=1 w; of successful experts. This idea of giving a start to experts allows us to prove stronger results; e.g., the following elaboration of (2) holds: (3) (this inequality still assumes that IIXtiloo ~ 1 for all t but w is unbounded). Our notation for the transpose of matrix A will be A'; as usual, vectors are identified with one-column matrices. Theorem 1 For any fi:ted n, Leamer has a strategy which ensures that always 368 V. Vovk II, in addition, IIxt II 00 $ X, \It, (4) The last inequality of this theorem implies inequality (3): it suffices to put X = Y=a=1. The term lndet (1 +; t,x.x:) in Theorem 1 might be difficult to interpret. Notice that it can be rewritten as nlnT + lndet (~I + ~COV(Xl' ... ,Xn)) , where cov(Xl, ... , Xn) is the empirical covariance matrix of the predictor variables (in other words, cov(Xl, ... ,Xn) is the covariance matrix of the random vector which takes the values Xl, ... ,XT with equal probability ~). We can see that this term is typically close to n In T. Using standard transformations, it is easy to deduce from Theorem 1, e.g., the following results (for simplicity we assume n = 1 and Xt,Yt E [-1,1], 'It): • if the pool of experts consists of all polynomials of degree d, Learner has a strategy guaranteeing • if the pool of experts consists of all splines of degree d with k nodes (chosen a priori), Learner has a strategy guaranteeing LT(Learner) S inf (LT(W) + Ilw1l 2 ) + (d + k + 1) In(T + 1). w The following theorem shows that the constant n in inequality (4) cannot be improved. Theorem 2 Fix n (the number 01 attributes) and Y (the upper bound on IlItl). For any f > 0 there exist a constant C and a stochastic strategy lor Nature such that IIxtiloo = 1 and Illtl = Y, lor all t, and, lor any stochastic strategy lor Learner, E (LT(Learner) inf LT(W)) ~ (n - f)y21nT - C, 'IT. w:llwll:SY 4 COMPARISONS It is easy to see that the ridge regression procedure sometimes gives results that are not sensible in our framework where lit E [-Y, Y] and the goal is to compete Competitive On-line Linear Regression 369 against the best linear regression function. For example, suppose n = 1, Y = 1, and Nature generates outcomes (Xt, Yt), t = 1,2, ... , where _ { 1, if todd, a « Xl « X2 « ... , Yt -1, if t even. At trial t = 2,3, ... the ridge regression procedure (more accurately, its natural modification which truncates its predictions to [-1, 1]) will give prediction Pt = Yt-l equal to the previous response, and so will suffer a loss of about 4T over T trials. On the other hand, the AA's prediction will be close to 0, and so the cumulative loss of the AA over the first T trials will be about T, which is close to the best expert's loss. We can see that the ridge regression procedure in this situation is forced to suffer a loss 4 times as big as the AA's loss. The lower bound stated in Theorem 2 does not imply that our regression algorithm is better than the ridge regression procedure in our adversarial framework. (Moreover, the idea of our proof of Theorem 2 is to lower bound the performance of the ridge regression procedure in the situation where the expected loss of the ridge regression procedure is optimal.) Theorem 1 asserts that LT(Leamer) :S ~ (LT( w) + allwl12) + y2 t. In ( 1 + ~ t. X~.i) (5) when Learner follows the AA. The next theorem shows that the ridge regression procedure sometimes violates this inequality. Theorem 3 Let n = 1 (the number 0/ attributes) and Y = 1 (the upper bound on IYtl); fix a > O. Nature has a strategy such that, when Learner plays the ridge regression strategy, LT(Learner) = 4T + 0(1), inf (LT(w) + allwll2) = T + 0(1), UI In (1+ ~ t.x~) = TIn2+ 0(1) as T -4 00 (and, there/ore, (5) is violated). 5 CONCLUSION (6) (7) (8) A distinctive feature of our approach to linear regression is that our only assumption about the data is that IYt I ~ Y, 'tit; we do not make any assumptions about stochastic properties of the data-generating mechanism. In some situations (if the data were generated by a partially known stochastic mechanism) this feature is a disadvantage, but often it will be an advantage. This paper was greatly influenced by Vapnik's [8] idea of transductive inference. The algorithm analyzed in this paper is "transductive", in the sense that it outputs some prediction Pt for Yt after being given Xt, rather than to output a general rule for mapping Xt into Ptj in particular, Pt may depend non-linearly on Xt. (It is easy, however, to extract such a rule from the description of the algorithm once it is found.) 370 V. Vovk Acknowledgments Kostas Skouras and Philip Dawid noticed that our regression algorithm is different from the ridge regression and that in some situations it behaves very differently. Manfred Warmuth's advice about relevant literature is also gratefully appreciated. References [11 N. Cesa-Bianchi, P. M. Long, and M. K. Warmuth (1996), Worst-case quadratic loss bounds for on-line prediction of linear functions by gradient descent, IEEE 7rons. Neural Networks 7:604-619. [21 A. DeSantis, G. Markowsky, and M. N. Wegman (1988), Learning probabilistic prediction functions, in "Proceedings, 29th Annual IEEE Symposium on Foundations of Computer Science," pp. 110-119, Los Alamitos, CA: IEEE Comput. Soc. [31 D. P. Foster (1991), Prediction in the worst case, Ann. Statist. 19:1084-1090. [4] Y. Freund (1996), Predicting a binary sequence almost as well as the optimal biased coin, in "Proceedings, 9th Annual ACM Conference on Computational Learning Theory" , pp. 89-98, New York: Assoc. Comput. Mach. [5] D. Haussler, J. Kivinen, and M. K. Warmuth (1994), Tight worst-case loss bounds for predicting with expert advice, University of California at Santa Cruz, Technical Report UCSC-CRL-94-36, revised December. Short version in "Computational Learning Theory" (P. Vitanyi, Ed.), Lecture Notes in Computer Science, Vol. 904, pp. 69-83, Berlin: Springer, 1995. [6] J. Kivinen and M. K. Warmuth (1997), Exponential Gradient versus Gradient Descent for linear predictors, Inform. Computation 132:1-63. [7] N. Littlestone and M. K. Warmuth (1994), The Weighted Majority Algorithm, Inform. Computation 108:212-261. [8] V. N. Vapnik (1995), The Nature of Statistical Learning Theory, New York: Springer. [9] V. Vovk (1990), Aggregating strategies, in "Proceedings, 3rd Annual Workshop on Computational Learning Theory" (M. Fulk and J. Case, Eds.), pp. 371-383, San Mateo, CA: Morgan Kaufmann. [10] V. Vovk (1992), Universal forecasting algorithms, In/orm. Computation 96:245-277. [11] V. Vovk (1997), A game of prediction with expert advice, to appear in J. Comput. In/orm. Syst. Short version in "Proceedings, 8th Annual ACM Conference on Computational Learning Theory," pp. 51-60, New York: Assoc. Comput. Mach., 1995. [12] K. Yamanishi (1997), A decision-theoretic extension of stochastic complexity and its applications to learning, submitted to IEEE 7rons. In/orm. Theory.
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Nonlinear Markov Networks for Continuous Variables Reimar Hofmann and Volker Tresp* Siemens AG, Corporate Technology Information and Communications 81730 Munchen, Germany Abstract We address the problem oflearning structure in nonlinear Markov networks with continuous variables. This can be viewed as non-Gaussian multidimensional density estimation exploiting certain conditional independencies in the variables. Markov networks are a graphical way of describing conditional independencies well suited to model relationships which do not exhibit a natural causal ordering. We use neural network structures to model the quantitative relationships between variables. The main focus in this paper will be on learning the structure for the purpose of gaining insight into the underlying process. Using two data sets we show that interesting structures can be found using our approach. Inference will be briefly addressed. 1 Introduction Knowledge about independence or conditional independence between variables is most helpful in ''understanding'' a domain. An intuitive representation of independencies is achieved by graphical models in which independency statements can be extracted from the structure of the graph. The two most popular types of graphical stochastical models are Bayesian networks which use a directed graph, and Markov networks which use an undirected graph. Whereas Bayesian networks are well suited to represent causal relationships, Markov networks are mostly used in cases where the user wants to express statistical correlation between variables. This is the case in image processing where the variables typically represent the grey levels of pixels and the graph encourages smootheness in the values of neighboring pixels (Markov random fields, Geman and Geman, 1984). We believe that Markov networks might be a useful representation in many domains where the concept of cause and effect is somewhat artificial. The learned structure of a Markov network also seems to be more easily communicated to non-experts; in a Bayesian network not all arc directions can be uniquely identified based on training data alone which makes a meaningful interpretation for the non-expert rather difficult. As in Bayesian networks, direct dependencies between variables in Markov networks are represented by an arc between those variables and missing edges represent independencies (in Section 2 we will be more precise about the independencies represented in Markov networks). Whereas the graphical structure in Markov networks might be known a priori in some cases, f{eimar.Hofinann@mchp.siemens.de Volker. Tresp@mchp.siemens.de 522 R. Hofmann and V. Tresp the focus of this work is the case that structure is unknown and must be inferred from data. For both discrete variables and linear relationships between continuous variables algorithms for structure learning exist (Whittaker, 1990). Here we address the problem of learning structure for Markov networks of continuous variables where the relationships between variables are nonlinear. In particular we use neural networks for approximating the dependency between a variable and its Markov boundary. We demonstrate that structural learning can be achieved without a direct reference to a likelihood function and show how inference in such networks can be perfonned using Gibbs sampling. From a technical point of view, these Marlwv boundary networks perfonn multi-dimensional density estimation for a very general class of non-Gaussian densities. In the next section we give a mathematical description of Markov networks and a formulation of the joint probability density as a product of compatibility functions. In Section 3.1 we discuss strucurallearning in Markov networks based on a maximum likelihood approach and show that this approach is in general unfeasible. We then introduce our approach which is based on learning the Markov boundary of each variable. We also show how belief update can be performed using Gibbs sampling. In Section 4 we demonstrate that useful structures can be extraced from two data sets (Boston housing data., financial market) using our approach. 2 Markov Networks The following brief introduction to Markov networks is adapted from Pearl (1988). Consider a strictly positive I joint probability density p(x) over a set of variables X := {XI, ... , XN }. For each variable Xi, let the Marlwv boundary of Xi, Bi ~ X - {Xi}, be the smallest set of variables that renders Xi and X - ({ xd U Bd independent under p( x) (the Markov boundary is unique for strictly positive distributions). Let the Marlwv network 9 be the undirected graph with nodes Xl, ••• , xN and edges between Xi and Xj if and only if Xi E Bj (which also implies X j E Bi). In other words, a Markov network is generated by connecting each node to the nodes in its Markov boundary. Then for any set Z ~ (X - {Xi, Xj}), Xi is independent of X j given Z if and only if every path from Xi to X j goes through at least one node in Z. In other words, two variables are independent if any path between those variables is "blocked" by a known variable. In particular a variable is independent of the remaining variables if the variables in its Markov boundary are known. A clique in G is a maximal fully connected sub graph. Given a Markov Network G for p( x) it can be shown that p can be factorized as a product of positive functions on the cliques of G, i.e. (1) where the product is over all cliques in the graph. Xclique, is the projection of X to the variables of the i-th clique and the gi are the compatibility functions w.r.t. cliquej. K = J fli gi(Xclique.)dx is the normalization constant. Note, that a state whose clique functions have large values has high probability. The theorem of Hammersley and Clifford states that the nonnalized product in equation 1 embodies all the conditional independencies portrayed by the graph (Pearl, 1988? for any choice of the gi . If the graph is sparse, i.e. if many conditional independencies exist then the cliques might 1 To simplify the discussion we will assume strict positivity for the rest of this paper. For some of the statements weaker conditions may also be sufficient. Note that strict positivity implies that functional constraints (for example, a = b) are excluded. 2 In terms of graphical models: The graph G is an I-map of p. Nonlinear Markov Networks for Continuous Variables 523 be small and the product will be over low dimensional functions. Similar to Bayesian networks where the complexity of describing a joint probability density is greatly reduced by decomposing the joint density in a product of ideally low-dimensional conditional densities, equation 1 describes the decomposition of a joint probability density function into a product of ideally low-dimensional compatibility functions. It should be noted that Bayesian networks and Markov networks differ in which specific independencies they can represent (Pearl, 1988). 3 Learning the Markov Network 3.1 Likelihood Function Based Learning Learning graphical stochastical models is usually decomposed into the problems of learning structure (that is the edges in the graph) and of learning the parameters of the joint density function under the constraint that it obeys the independence statements made by the graph. The idea is to generate candidate structures according to some search strategy, learn the parameters for this structure and then judge the structure on the basis of the (penalized) likelihood of the model or, in a fully Bayesian approach, using a Bayesian scoring metric. Assume that the compatibility functions in equation 1 are approximated using a function approximator such as a neural network gi 0 ~ 9 i (x). Let {xP}:= 1 be a training set. With likelihood L = I1;=1 pM (xP) (where the M in pM indicates a probability density model in contrast to the true distribution), the gradient of the log-likelihood with respect to weight Wi in gi (.) becomes ~~I M( P)-~~l ~(P )_NI(i!v;loggi(Xclique,))I1jgj(XcliqueJ)dX a L-0gp x -L-a oggl Xclique, II1 W( )d Wi p=l p=l Wi j gj Xclique) X (2) where the sums are over N training patterns. The gradient decomposes into two terms. Note, that only in the first term the training patterns appear explicitly and that, conveniently, the first term is only dependent on the clique i which contains parameter Wi. The second term emerges from the normalization constant K in equation I. The difficulty is that the integrals in the second term can not be solved in closed form for universal types of compatibility functions gi and have to be approximated numerically, typically using a form of Monte Carlo integration. This is exactly what is done in the Boltzmann machine, which is a special case of a Markov network with discrete variables.3 Currently, we consider maximum likelihood learning based on the compatibility functions unsuitable, considering the complexity and slowness of Monte Carlo integration (Le. stochastic sampling). Note, that for structural learning the maximum likelihood learning is in the inner loop and would have to be executed repeatedly for a large number of structures. 3.2 Markov Boundary Learning The difficulties in using maximum likelihood learning for finding optimal structures motivated the approach pursued in this paper. If the underlying true probability density is known the structure in a Markov network can be found using either the edge deletion method or the 3 A fully connected Boltzmann machine does not display any independencies and we only have one clique consisting of all variables. The compatibility function is gO = exp (- L: WijSiSj). The Boltzmann machine typically contains hidden variables, such that not only the second tenn (corresponding to the unclamped phase) in equation 2 has to be approximated using stochastic sampling but also the first tenn. (In this paper we only consider the case that data are complete). 524 R. Hofmann and V Tresp Markov boundary method (Pearl, 1988). The edge deletion method uses the fact that variables a and b are not connected by an edge if and only if a and b are independent given all other variables. Evaluating this test for each pair of variables reveals the structure of the network. The Markov boundary method consists of determining - for each variable a - its Markov boundary and connecting a to each variable in its Markov boundary. Both approaches are simple if we have a reliable test for true conditional independence. Both methods cannot be applied directly for learning structure from data since here tests for conditional independence cannot be based on the true underlying probability distribution (which is unknown) but has to be inferred from a finite data set. The hope is that dependencies which are strong enough to be supported by the data can still be reliably identified. It is, however not difficult to construct cases where simply using an (unreliable) statistical test for conditional independence with the edge deletion method does not work wel1.4 We now describe our approach, which is motivated by the Markov boundary method. First, we start with a fully connected graph. We train a model ptt to approximate the conditional density of each variable i, given the current candidate variables for its Markov boundary Bi which initially are all other variables. For this we can use a wide variety of neural networks. We use conditional Parzen windows (3) where {XP};'=l is the training set and G(x; J-l, 1:) is our notation for a multidimensional Gaussian centered at J-l with covariance matrix 1: evaluated at x. The Gaussians in the nominator are centered at X~i}U8: which is the location of the p-th sample in the jointinput!output( {x;} UBi) space and the Gaussians in the denominator are centered at x~: which is the location of the p-th sample in the input space (Bi). There is one covariance matrix 1:i for each conditional density model which is shared between all the Gaussians in that model. 1:i is restricted to a diagonal matrix where the diagonal elements in all dimensions except the output dimension i, are the same. So there are only two free parameters in the matrix: The variance in the output dimension and the variance in all input dimensions. Ei 8' is equal to 1:i except that the row and column corresponding to the output dimension ha~e been deleted. For each conditional model ptt, 1:i was optimized on the basis of the leave-one-out cross validation log-likelihood. Our approach is based on tentatively removing edges from the model. Removing an edge decreases the size of the Markov boundary candidates of both affected variables and thus decreases the number of inputs in the corresponding two conditional density models. With the inputs removed, we retrain the two models (in our case, we simply find the optimal Ei for the two conditional Parzen windows). If the removal of the edge was correct, the leaveone-out cross validation log-likelihood (model-score) of the two models should improve since an unnecessary input is removed. (Removing an unnecessary input typically decreases model variance.) We therefore remove an edge if the model-scores of both models improve. Let's define as edge-removal-score the smaller ofthe two improvements in model-score. Here is the algorithm in pseudo code: • Start with a fully connected network 4The problem is that in the edge deletion method the decision is made independently for each edge whether or not it should be present There are however cases where it is obvious that at least one of two edges must be present although the edge deletion method which tests each edge individually removes both. . Nonlinear Markov Networksfor Continuous Variables • Until no edge-removal-score is positive: - for all edges edgeij in the network 525 * calculate the model-scores of the reduced models ptt (Xi IBi - {j}) and ptt (Xj IB; - {i}) * compare with the model-scores of the current models pM (xiIB~) and Mi t I Pi (XjIBj) * set the edge-removal-score to the smaller of both model-score improvements - remove the edge for which the edge-removal-score is in maximum. • end 3.3 Inference Note that we have learned the structure of the Markov network without an explicit representation of the probability density. Although the conditional densities p(.r i IBi) provide sufficient information to calculate the joint probability density the latter can not be easily computed. More precisely, the conditional densities overdetermine the joint density which might lead to problems if the conditional densities are estimated from data. For inference, we are typically interested in the expected value of an unknown variable, given an arbitrary set of known variables, which can be calculated using Gibbs sampling. Note, that the conditional densities pM (Xi IBi) which are required for Gibbs sampling are explicitly modeled in our approach by the conditional Parzen windows. Also note, that sampling from the conditional Parzen model (as well as many other neural networks, such as mixture of experts models) is easy.5 In Hofmann (1997) we show that Gibbs sampling from the conditional Parzen models gives significantly better results than running inference using either a kernel estimator or a Gaussian mixture model of the joint density. 4 Experiments In our first experiment we used the Boston housing data set, which contains 506 samples. Each sample consists of the housing price and 13 other variables which supposedly influence the housing price in a Boston neighborhood. Maximizing the cross validation log-likelihood as score as described in the previous chapters results in a Markov network with 68 edges. While cross validation gives an unbiased estimate of whether a direct dependency exists between two variables the estimate can have a large variance depending on the size of the given data set. If the goal of the experiment is to interpret the resulting structure one would prefer to see only those edges corresponding to direct dependencies which can be clearly identified from the given data set. In other words, if the relationship between two variables observed on the given data set is so weak that we can not be sure that it is not just an effect of the finite data set size, then we do not want to display the corresponding edge. This can be achieved by adding a penalty per edge to the score of the conditional density models. (figure 1). Figure 2 shows the resulting Markov network for a penalty per edge of 0.2. The goal of the original experiment for which the Boston housing data were collected was to examine whether the air quality (5) has direct influence on the housing price (14). Our algorithm did not find such an influence - in accordance with the original study. It found that the percentage of low status population (13) and the average number of rooms (6) are in direct relationship with the housing price. The pairwise relationships between these three variables are displayed in figure 3. 5 Readers not familiar with Gibbs sampling, please consult Geman and Geman (1984). 526 R. Hofmann and V. Tresp ° 0 0011 01 016 02 on 0 3 0» oc o.~ os ......... Figure I: Number of edges in the Markov network for the Boston housing data as a function of the penalty per edge. I crime rate 2 percent land zoned for lots 3 percent nooretail bu.. in"", 4 located on Charies river'! 5 nitrogen oxide concentration 6 average number of room., 7 percent bui I t before 1940 8 weighted distance to employment center 9 acces. to radial highways 10 tax rate II pupi IIteacher ratio 12 percent black 13 percent lower-status population 14 median value ofbomes Figure 2: Final structure of a run on the full Boston housing data set (penalty = 0.2). The scatter plots visualize the relationship between variables 13 and 14, 6 and 14 and between 6 and 13 (from left to right). The left and the middle correspond to edges in the Markov network whereas for the right diagram the corresponding edge (6-13) is missing even though both variables are clearly dependent. The reason is, that the dependency between 6 and 13 can be explained as indirect relationship via variable 14. The Markov network tells us that 13 and 6 are independent given 14, but dependent if 14 is unknown. In a second experiment we used a financial dataset. Each pattern corresponds to one business day. The variables in our model are relative changes in certain economic variables from the last business day to the present day which were expected to possibly influence the development of the German stock index DAX and the composite DAX, which contains a larger selection of stocks than the DAX. We used 500 training patterns consisting of 12 variables (figure 4). In comparison to the Boston housing data set most relationships are very weak. Using a penalty per edge of 0.2 leads to a very sparse model with only three edges (2-12, 12-1,5-11) (not shown). A penalty of 0.025 results in the model shown in figure 4. Note, that the composite o-- - ----~-- ...J o 10 ~ ~ ~ Pc Low Status Population 50~- - - .... -- ~---.. • - -' . .. ,I : :. 10 O'--~ . _ _ __________ -..l 3456 7 89 A. Number of Rooms .0--- ----- -- ---Figure 3: Pairwise relationship between the variables 6, 13 and 14. Displayed are all data points in the Boston housing data set. Nonlinear Markov Networks/or Continuous Variables 527 DAX composite DAX 3 month interest rates Gennany 4 rerum Gennany 5 Morgan Stanley tndex Germany 6 I)(JW' Jones mdustrial index 7 DM-USD exchange rate 8 US treasury bonds 9 gold price in DM 10 N.kkei index Japan II Morgan Stanley index Europe 12 price earning ratio (DAX stocks) Figure 4: Final structure of a run on the financial data set with a penalty of 0.025. The small numbers next to the edges indicate the strength of the connection, i.e. the decrease in score (excluding the penalty) when the edge is removed. All variables are relative changes - not absolute values. DAX is connected to the DAX mainly through the price earning ratio. While the DAX has direct connections to the Nikkei index and to the DM-USD exchange rate the composite DAX has a direct connection to the Morgan Stanley index for Germany. Recall, that composite DAX contains the stocks of many smaller companies in addition to the DAX stocks. The graph structure might be interpreted (with all caution) in the way that the composite DAX (including small companies) has a stronger dependency on national business whereas the DAX (only including the stock of major companies) reacts more to international indicators. 5 Conclusions We have demonstrated, to our knowledge for the first time, how nonlinear Markov networks can be learned for continuous variables and we have shown that the resulting structures can give interesting insights into the underlying process. We used a representation based on models of the conditional probability density of each variable given its Markov boundary. These models can be trained locally. We showed how searching in the space of all possible structures can be done using this representation. We suggest to use the conditional densities of each variable given its Markov boundary also for inference by Gibbs sampling. Since the required conditional densities are modeled explicitly by our approach and sampling from these is easy, Gibbs sampling is easier and faster to realize than with a direct representation of the joint density. A topic of further research is the variance in resulting structures, i.e. the fact that different structures can lead to almost equally good models. It would for example be desirable to indicate to the user in a principled way the certainty of the existence or nonexistence of edges. References Geman, S., and Geman, D. (1984). Stochastic relaxations, Gibbs distributions and the Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence PAMI-6 (no. 6):721-42 Hofinann, R. (1997). Inference in Markov Blanket Models. Technical report, in preparation. Monti, S., and Cooper, G. (1997). Learning Bayesian belief networks with neural network estimators. In Neural Information Processing Systems 9., MIT Press. Pearl, J. (1988). Probabilistic reasoning in intelligent systems. San Mateo: Morgan Kaufinann. Whittaker, J. (1990). Graphical models in applied multivariate statistics. Chichester, UK: John Wiley and Sons.
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Online learning from finite training sets in nonlinear networks Peter Sollich* Department of Physics University of Edinburgh Edinburgh ERg 3JZ, U.K. P.Sollich~ed.ac.uk David Barbert Department of Applied Mathematics Aston University Birmingham B4 7ET, U.K. D.Barber~aston . ac.uk Abstract Online learning is one of the most common forms of neural network training. We present an analysis of online learning from finite training sets for non-linear networks (namely, soft-committee machines), advancing the theory to more realistic learning scenarios. Dynamical equations are derived for an appropriate set of order parameters; these are exact in the limiting case of either linear networks or infinite training sets. Preliminary comparisons with simulations suggest that the theory captures some effects of finite training sets, but may not yet account correctly for the presence of local minima. 1 INTRODUCTION The analysis of online gradient descent learning, as one of the most common forms of supervised learning, has recently stimulated a great deal of interest [1, 5, 7, 3]. In online learning, the weights of a network ('student') are updated immediately after presentation of each training example (input-output pair) in order to reduce the error that the network makes on that example. One of the primary goals of online learning analysis is to track the resulting evolution of the generalization error - the error that the student network makes on a novel test example, after a given number of example presentations. In order to specify the learning problem, the training outputs are assumed to be generated by a teacher network of known architecture. Previous studies of online learning have often imposed somewhat restrictive and • Royal Society Dorothy Hodgkin Research Fellow tSupported by EPSRC grant GR/J75425: Novel Developments in Learning Theory for Neural Networks 358 P. SolIich and D. Barber unrealistic assumptions about the learning framework. These restrictions are, either that the size of the training set is infinite, or that the learning rate is small[l, 5, 4]. Finite training sets present a significant analytical difficulty as successive weight updates are correlated, giving rise to highly non-trivial generalization dynamics. For linear networks, the difficulties encountered with finite training sets and noninfinitesimal learning rates can be overcome by extending the standard set of descriptive ('order') parameters to include the effects of weight update correlations[7]. In the present work, we extend our analysis to nonlinear networks. The particular model we choose to study is the soft-committee machine which is capable of representing a rich variety of input-output mappings. Its online learning dynamics has been studied comprehensively for infinite training sets[l, 5]. In order to carry out our analysis, we adapt tools originally developed in the statistical mechanics literature which have found application, for example, in the study of Hopfield network dynamics[2]. 2 MODEL AND OUTLINE OF CALCULATION For an N-dimensional input vector x, the output of the soft committee machine is given by (I) where the nonlinear activation function g(hl ) = erf(hz/V2) acts on the activations hi = wtxl.JFi (the factor 1/.JFi is for convenience only). This is a neural network with L hidden units, input to hidden weight vectors WI, 1 = I..L, and all hidden to output weights set to 1. In online learning the student weights are adapted on a sequence of presented examples to better approximate the teacher mapping. The training examples are drawn, with replacement, from a finite set, {(X/",yl-') ,j.t = I..p}. This set remains fixed during training. Its size relative to the input dimension is denoted by a = piN. We take the input vectors xl-' as samples from an N dimensional Gaussian distribution with zero mean and unit variance. The training outputs y'" are assumed to be generated by a teacher soft committee machine with hidden weight vectors w~, m = I..M, with additive Gaussian noise corrupting its activations and output. The discrepancy between the teacher and student on a particular training example (x, y), drawn from the training set, is given by the squared difference of their corresponding outputs, E= H~9(hl) -yr = H~9(hl) - ~g(km +em) -eor where the student and teacher activations are, respectively h, = {J;wtx km = {J;(w:n?x, (2) and em, m = I..M and eo are noise variables corrupting the teacher activations and output respectively. Given a training example (x, y), the student weights are updated by a gradient descent step with learning rate "I, w; - W, = -"I\1wIE = - JNx8h l E (3) On-line Learning from Finite Training Sets in Nonlinear Networks 359 The generalization error is defined to be the average error that the student makes on a test example selected at random (and uncorrelated with the training set), which we write as €g = (E). Although one could, in principle, model the student weight dynamics directly, this will typically involve too many parameters, and we seek a more compact representation for the evolution of the generalization error. It is straightforward to show that the generalization error depends, not on a detailed description of all the network weights, but only on the overlap parameters Qll' = ~ W r WI' and Rim = ~ W r w':n [1, 5, 7]. In the case of infinite 0, it is possible to obtain a closed set of equations governing the overlap parameters Q, R [5]. For finite training sets, however, this is no longer possible, due to the correlations between successive weight updates[7]. In order to overcome this difficulty, we use a technique developed originally to study statistical physics systems [2] . Initially, consider the dynamics of a general vector of order parameters, denoted by 0, which are functions of the network weights w. If the weight updates are described by a transition probability T(w -+ w'), then an approximate update equation for 0 is 0' - 0 = IfdW' (O(w') - O(w)) T(w -+ W')) (4) \ P(w)oc6(O(w)-O) Intuitively, the integral in the above equation expresses the average changel of 0 caused by a weight update w -+ w', starting from (given) initial weights w. Since our aim is to develop a closed set of equations for the order parameter dynamics, we need to remove the dependency on the initial weights w. The only information we have regarding w is contained in the chosen order parameters 0, and we therefore average the result over the 'subshell' of all w which correspond to these values of the order parameters. This is expressed as the 8-function constraint in equation(4). It is clear that if the integral in (4) depends on w only through O(w), then the average is unnecessary and the resulting dynamical equations are exact. This is in fact the case for 0 -+ 00 and 0 = {Q, R}, the standard order parameters mentioned above[5]. If this cannot be achieved, one should choose a set of order parameters to obtain approximate equations which are as close as possible to the exact solution. The motivation for our choice of order parameters is based on the linear perceptron case where, in addition to the standard parameters Q and R, the overlaps projected onto eigenspaces of the training input correlation matrix A = ~ E:=l xl' (xl') T are required2 . We therefore split the eigenvalues of A into r equal blocks ('Y = 1 ... r) containing N' = N Ir eigenvalues each, ordering the eigenvalues such that they increase with 'Y. We then define projectors p'Y onto the corresponding eigenspaces and take as order parameters: R'Y _ 1 Tp'Y .. 1m N'w, wm U'Y - ~ Tp'Yb I. Nt W, II (5) where the b B are linear combinations of the noise variables and training inputs, (6) 1 Here we assume that the system size N is large enough that the mean values of the parameters alone describe the dynamics sufficiently well (i. e., self-averaging holds). 2The order parameters actually used in our calculation for the linear perceptron[7] are Laplace transforms of these projected order parameters. 360 P. Sollich and D. Barber As r -+ 00, these order parameters become functionals of a continuous variable3 . The updates for the order parameters (5) due to the weight updates (3) can be found by taking the scalar products of (3) with either projected student or teacher weights, as appropriate. This then introduces the following activation 'components', k'Y = ff(w* )Tp'"Yx m VNi m so that the student and teacher activations are h, = ~ E'"Y hi and km = ~ E'"Y k~, respectively. For the linear perceptron, the chosen order parameters form a complete set - the dynamical equations close, without need for the average in (4). For the nonlinear case, we now sketch the calculation of the order parameter update equations (4). Taken together, the integral over Wi (a sum of p discrete terms in our case, one for each training example) and the subshell average in (4), define an average over the activations (2), their components (7), and the noise variables ~m, ~o. These variables turn out to be Gaussian distributed with zero mean, and therefore only their covariances need to be worked out. One finds that these are in fact given by the naive training set averages. For example, = (8) where we have used p'"Y A = a'"YP'"Y with a'"Y 'the' eigenvalue of A in the ,-th eigenspace; this is well defined for r -+ 00 (see [6] for details of the eigenvalue spectrum). The correlations of the activations and noise variables explicitly appearing in the error in (3) are calculated similarly to give, (h,h,,) = ~ L:; Q~, '"Y (h,km) = ~ L :; Rim (9) '"Y (h,~s) = ~ L ~U,~ '"Y where the final equation defines the noise variances. The T~m' are projected overlaps between teacher weight vectors, T~m' = ~ (w~)Tp'"Yw:n,. We will assume that the teacher weights and training inputs are uncorrelated, so that T~m' is independent of ,. The required covariances of the 'component' activations are (kinh,) a'"YR'"Y a 'm (k~km') = a'"YT'"Y a mm' (k~~s) 0 (c] h,) a'"YU'"Y (c]km, ) 0 (C]~8' ) a'"Y 2 a ls -(7s588, a (hih" ) a'"YQ'"Y a II' (hJkm,) a'"YR'"Y a 'm (hJ~s) = .!.U'"Y a 's (10) 3Note that the limit r -+ 00 is taken after the thermodynamic limit, i.e., r ~ N. This ensures that the number of order parameters is always negligible compared to N (otherwise self-averaging would break down). On-line Learning from Finite Training Sets in Nonlinear Networks 0.03 rfII:-----........ -------, 0.025 I 0.02 o 0 0 000000000000000000000000 0.01 '------~-----~ o 50 t 100 (a) 0.25 0.2 0.15 OOOOOOOOC 000000000 0000 L.. ... o~ooo ~ aaaoaaaaaaaaaaaaaaaac I 'NNNoaaoa \ ,,-----------o 50 t 100 361 (b) Figure 1: fg vs t for student and teacher with one hidden unit (L = M = 1); a = 2, 3, 4 from above, learning rate "I = 1. Noise of equal variance was added to both activations and output (a) O'~ = 0'5 = 0.01, (b) O'~ = 0'5= 0.1. Simulations for N = 100 are shown by circles; standard errors are of the order of the symbol size. The bottom dashed lines show the infinite training set result for comparison. r = 10 was used for calculating the theoretical predictions; the curved marked "+" in (b), with r = 20 (and a = 2), shows that this is large enough to be effectively in the r -+ 00 limit. Using equation (3) and the definitions (7), we can now write down the dynamical equations, replacing the number of updates n by the continuous variable t = n/ N in the limit N -+ 00: OtRim OtU?s OtQIz, -"I (k-:nOh,E) -"I (c~oh,E) -"I (h7 Oh" E) - "I (h~ Oh, E) + "12 a-y (Oh,Eoh" E) a (11) where the averages are over zero mean Gaussian variables, with covariances (9,10). Using the explicit form of the error E, we have oh,E = g'(h,) [L9(hl') - Lg(km + em) - eo] (12) I' m which, together with the equations (11) completes the description of the dynamics. The Gaussian averages in (11) can be straightforwardly evaluated in a manner similar to the infinite training set case[5], and we omit the rather cumbersome explicit form of the resulting equations. We note that, in contrast to the infinite training set case, the student activations hI and the noise variables Cs and es are now correlated through equation (10). Intuitively, this is reasonable as the weights become correlated, during training, with the examples in the training set. In calculating the generalization error, on the other hand, such correlations are absent, and one has the same result as for infinite training sets. The dynamical equations (11), together with (9,10) constitute our main result. They are exact for the limits of either a linear network (R, Q, T -+ 0, so that g(x) ex: x) or a -+ 00, and can be integrated numerically in a straightforward way. In principle, the limit r -+ 00 should be taken but, as shown below, relatively small values of r can be taken in practice. 3 RESULTS AND DISCUSSION We now discuss the main consequences of our result (11), comparing the resulting predictions for the generalization dynamics, fg(t), to the infinite training set theory 362 0.25k 02 100000000000000000000000 . 1 ______________ ~ 0.15 , 0.1 0.05 \ ... ----O~--~------~----~~~ o 10 20 30 40 t 50 (a) P. Sollich and D. Barber 0.4 ..----------~--~----, ,'-0.3 0.2 0.1 ~ooooooooooooooooooo OL---~----------~----~ o W 100 1W t 200 (b) Figure 2: €g VS t for two hidden units (L = M = 2). Left: a = 0.5, with a = 00 shown by dashed line for comparison; no noise. Right: a = 4, no noise (bottom) and noise on teacher activations and outputs of variance 0.1 (top). Simulations for N = 100 are shown by small circles; standard errors are less than the symbol size. Learning rate fJ = 2 throughout. and to simulations. Throughout, the teacher overlap matrix is set to Tij = c5ij (orthogonal teacher weight vectors of length V'ii). In figure(l), we study the accuracy of our method as a function of the training set size for a nonlinear network with one hidden unit at two different noise levels. The learning rate was set to fJ = 1 for both (a) and (b). For small activation and output noise (0'2 = 0.01), figure(la) , there is good agreement with the simulations for a down to a = 3, below which the theory begins to underestimate the generalization error, compared to simulations. Our finite a theory, however, is still considerably more accurate than the infinite a predictions. For larger noise (0'2 = 0.1, figure(lb», our theory provides a reasonable quantitative estimate of the generalization dynamics for a > 3. Below this value there is significant disagreement, although the qualitative behaviour of the dynamics is predicted quite well, including the overfitting phenomenon beyond t ~ 10. The infinite a theory in this case is qualitatively incorrect. In the two hidden unit case, figure(2), our theory captures the initial evolution of €g(t) very well, but diverges significantly from the simulations at larger t; nevertheless, it provides a considerable improvement on the infinite a theory. One reason for the discrepancy at large t is that the theory predicts that different student hidden units will always specialize to individual teacher hidden units for t --+ 00, whatever the value of a. This leads to a decay of €g from a plateau value at intermediate times t. In the simulations, on the other hand, this specialization (or symmetry breaking) appears to be inhibited or at least delayed until very large t. This can happen even for zero noise and a 2:: L, where the training data should should contain enough information to force student and teacher weights to be equal asymptotically. The reason for this is not clear to us, and deserves further study. Our initial investigations, however, suggest that symmetry breaking may be strongly delayed due to the presence of saddle points in the training error surface with very 'shallow' unstable directions. When our theory fails, which of its assumptions are violated? It is conceivable that multiple local minima in the training error surface could cause self-averaging to break down; however, we have found no evidence for this, see figure(3a). On the other hand, the simulation results in figure(3b) clearly show that the implicit assumption of Gaussian student activations - as discussed before eq. (8) - can be violated. On-line Learning from Finite Training Sets in Nonlinear Networks 363 (a) (b) / Variance over training histories 10"'" '---------------' 102 N Figure 3: (a) Variance of fg(t = 20) vs input dimension N for student and teacher with two hidden units (L = M = 2), a = 0.5, 'fJ = 2, and zero noise. The bottom curve shows the variance due to different random choices of training examples from a fixed training set ('training history'); the top curve also includes the variance due to different training sets. Both are compatible with the liN decay expected if selfaveraging holds (dotted line). (b) Distribution (over training set) of the activation hI of the first hidden unit of the student. Histogram from simulations for N = 1000, all other parameter values as in (a). In summary, the main theoretical contribution of this paper is the extension of online learning analysis for finite training sets to nonlinear networks. Our approximate theory does not require the use of replicas and yields ordinary first order differential equations for the time evolution of a set of order parameters. Its central implicit assumption (and its Achilles' heel) is that the student activations are Gaussian distributed. In comparison with simulations, we have found that it is more accurate than the infinite training set analysis at predicting the generalization dynamics for finite training sets, both qualitatively and also quantitatively for small learning times t. Future work will have to show whether the theory can be extended to cope with non-Gaussian student activations without incurring the technical difficulties of dynamical replica theory [2], and whether this will help to capture the effects of local minima and, more generally, 'rough' training error surfaces. Acknowledgments: We would like to thank Ansgar West for helpful discussions. References [1] M. Biehl and H. Schwarze. Journal of Physics A, 28:643-656, 1995. [2] A. C. C. Coolen, S. N. Laughton, and D. Sherrington. In NIPS 8, pp. 253-259, MIT Press, 1996; S.N. Laughton, A.C.C. Coolen, and D. Sherrington. Journal of Physics A, 29:763-786, 1996. [3] See for example: The dynamics of online learning. Workshop at NIPS'95. [4] T. Heskes and B. Kappen. Physical Review A, 44:2718-2762, 1994. [5] D. Saad and S. A. Solla Physical Review E, 52:4225, 1995. [6] P. Sollich. Journal of Physics A, 27:7771-7784, 1994. [7] P. Sollich and D. Barber. In NIPS 9, pp.274-280, MIT Press, 1997; Europhysics Letters, 38:477-482, 1997.
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Automated Aircraft Recovery via Reinforcement Learning: Initial Experiments Jeffrey F. Monaco Barron Associates, Inc. Jordan Building 1160 Pepsi Place, Suite 300 Charlottesville VA 22901 monaco@bainet.com David G. Ward Barron Associates, Inc. Jordan Building 1160 Pepsi Place, Suite 300 Charlottesville VA 22901 ward@bainet.com Andrew G. Barto Department of Computer Science University of Massachusetts Amherst MA 01003 barto@cs.umass.edu Abstract Initial experiments described here were directed toward using reinforcement learning (RL) to develop an automated recovery system (ARS) for high-agility aircraft. An ARS is an outer-loop flight-control system designed to bring an aircraft from a range of out-of-control states to straightand-level flight in minimum time while satisfying physical and physiological constraints. Here we report on results for a simple version of the problem involving only single-axis (pitch) simulated recoveries. Through simulated control experience using a medium-fidelity aircraft simulation, the RL system approximates an optimal policy for pitch-stick inputs to produce minimum-time transitions to straight-and-Ievel flight in unconstrained cases while avoiding ground-strike. The RL system was also able to adhere to a pilot-station acceleration constraint while executing simulated recoveries. Automated Aircraft Recovery via Reinforcement Learning 1023 1 INTRODUCTION An emerging use of reinforcement learning (RL) is to approximate optimal policies for large-scale control problems through extensive simulated control experience. Described here are initial experiments directed toward the development of an automated recovery system (ARS) for high-agility aircraft. An ARS is an outer-loop flight control system designed to bring the aircraft from a range of initial states to straight, level, and non-inverted flight in minimum time while satisfying constraints such as maintaining altitude and accelerations within acceptable limits. Here we describe the problem and present initial results involving only single-axis (pitch) recoveries. Through extensive simulated control experience using a medium-fidelity simulation of an F-16, the RL system approximated an optimal policy for longitudinal-stick inputs to produce near-minimum-time transitions to straight and level flight in unconstrained cases, as well as while meeting a pilot-station acceleration constraint. 2 AIRCRAFT MODEL The aircraft was modeled as a dynamical system with state vector x = {q, 0, p, r, {3, Vi}, where q = body-axes pitch rate, 0 = angle of attack, p = body-axes roll rate, r = body-axes yaw rate, {3 = angle of sideslip, Vi = total airspeed, and control vector fl = {flse , flae, fla/' Orud} of effector and pseudo-effector displacements. The controls are defined as: flse = symmetric elevon, oae = asymmetric elevon, oal = asymmetric flap, and Orud = rudder. (A pseudo-effector is a mathematically convenient combination of real effectors that, e.g., contributes to motion in a limited number of axes.) The following additional descriptive variables were used in the RL problem formulation: h = altitude, h = vertical component of velocity, e = pitch attitude, N z = pilot-station normal acceleration. For the initial pitch-axis experiment described here, five discrete actions were available to the learning agent in each state; these were longitudinal-stick commands selected from {-6, -3,0, +3, +6} lbf. The command chosen by the learning agent was converted into a desired normal-acceleration command through the standard F-16 longitudinal-stick command gradient with software breakout. This gradient maps pounds-of-force inputs into desired acceleration responses. We then produce an approximate relationship between normal acceleration and body-axes pitch rate to yield a pitch-rate flying-qualities model. Given this model, an inner-loop linear-quadratic (LQ) tracking control algorithm determined the actuator commands to result in optimal model-following of the desired pitch-rate response. The aircraft model consisted of complete translational and rotational dynamics, including nonlinear terms owing to inertial cross-coupling and orientation-dependent gravitational effects. These were obtained from a modified linear F-16 model with dynamics of the form j; = Ax + Bfl + b + N where A and B were the F-16 aero-inertial parameters (stability derivatives) and effector sensitivities (control derivatives). These stability and control derivatives and the bias vector, b, were obtained from linearizations of a high-fidelity nonlinear, six-degree-of-freedom model. Nonlinearities owing to inertial cross-coupling and orientation-dependent gravitational effects were accounted for through the term N, which depended nonlinearly on the state. Nonlinear actuator dynamics were modeled via the incorporation ofF-16 effector-rate and effector-position limits. See Ward et al. (1996) for additional details. 3 PROBLEM FORMULATION The RL problem was to approximate a minimum-time control policy capable of bringing the aircraft from a range of initial states to straight, level, and non-inverted flight, while satisfying given constraints, e.g., maintaining the normal acceleration at the pilot station within 1024 1. F. Monaco, D. G. Ward and A G. Barto an acceptable range. For the single-axis (pitch-axis) flight control problem considered here, recovered flight was defined by: q = q = it = h = i't = o. (1) Successful recovery was achieved when all conditions in Eq. 1 were satisfied simultaneously within pre-specified tolerances. Because we wished to distinguish between recovery supplied by the LQ tracker and that learned by the RL system, special attention was given to formulating a meaningful test to avoid falsely attributing successes to the RL system. For example, if initial conditions were specified as off-trim perturbations in body-axes pitch rate, pitch acceleration, and true airspeed, the RL system may not have been required because the LQ controller would provide all the necessary recovery, i.e., zero longitudinal-stick input would result in a commanded body-axes pitch rate of zero deg./ sec. Because this controller is designed to be highly responsive, its tracking and integral-error penalties usually ensure that the aircraft responses attain the desired state in a relatively short time. The problem was therefore formulated to demand recovery from aircraft orientations where the RL system was primarily responsible for recovery, and the goal state was not readily achieved via the stabilizing action of the LQ control law. A pitch-axis recovery problem of interest is one in which initial pitch attitude, e, is selected to equal etrim +U(80Tn,n' 8 0Tna:l:)' where etrim == atrim (by definition), U is a uniformly distributed random number, and eOTnin and eoTnaz define the boundaries of the training region, and other variables are set so that when the aircraft is parallel to the earth (80 = 0), it is "pancaking" toward the ground (with positive trim angle of attack). Other initial conditions correspond to purely-translational climb or descent of the aircraft. For initial conditions where eo < atrim, the flight vehicle will descend, and in the absence of any corrective longitudinal-stick force, strike the ground or water. Because it imposes no constraints on altitude or pitch-angle variations, the stabilizing response of the LQ controller is inadequate for providing the necessary recovery. 4 REINFORCEMENT LEARNING ALGORITHM Several candidate RL algorithms were evaluated for the ARS. Initial efforts focused primarily on (1) Q-Learning, (2) alternative means for approximating the action-value function (Q function), and (3) use of discrete versus continuous-action controls. During subsequent investigations, an extension of Q-Learning called Residual Advantage Learning (Baird, 1995; Harmon & Baird, 1996) was implemented and successfully applied to the pitch-axis ARS problem. As with action-values in Q-Learning, the advantage function, A(x, u), may be represented by a function approximation system of the form A(x,u) = ¢(x,ufO, (2) where ¢( x, u) is a vector of relevant features and 0 are the corresponding weights. Here, the advantage function is linear in the weights, 0, and these weights are the modifiable, learned parameters. For advantage functions of the form in Eq. 2, the update rule is: Ok+l Ok - a ((r + "Y~t A(y, b*)) K~t + (1 - K~t) A(x, a*) - A(x, a)) • ( ~"Y~t¢(y, b*) K~t + ~ (1 - K~t) ¢(x, a*) - ¢(x, a)) , where a* = argminaA(x, a) and b* = argminbA(y, b), !l.t is the system rate (0.02 sec. in the ARS), "Y~t is the discount factor, and K is an fixed scale factor. In the above notation, Automated Aircraft Recovery via Reinforcement Learning 1025 y is the resultant state, i.e., the execution of action a results in a transition from state x to its successor y. The Residual Advantage Learning update collapses to the Q-Learning update for the case ~ = 0, K = L. The parameter ~ is a scalar that controls the trade-off between residualgradient descent when ~ = 1, and a faster, direct algorithm when ~ = O. Harmon & Baird (1996) address the choice of ~, suggesting the following computation of ~ at each time step: ;F,. l:o WdWrg 'J!'= +J-L l:o(Wd - wrg)wrg where Wd and Wrg are traces (one for each (J of the function approximation system) associated with the direct and residual gradient algorithms, respectively, and J-L is a small, positive constant that dictates how rapidly the system forgets. The traces are updated during each cycle as follows Wd f(1-J-L)Wd-J-L[(r+'Y~tA(y,b*)) K~t+(1- K~t)A(X,a*)] • [- :(JA(x, a*)] wrg f(1-J-L)Wrg-J-L[(r+'Y~tA(y,b*»K~t+(1- K~t)A(x,a*)-A(X,a)] • ['Y~t ;(JA(y,b*) K~t + (1- K~t) ;(JA(x,a*) - ;(JA(X, a)] . Advantage Learning updates of the weights, including the calculation of an adaptive ~ as discussed above, were implemented and interfaced with the aircraft simulation. The Advantage Learning algorithm consistently outperformed its Q-Learning counterpart. For this reason, most of our efforts have focused on the application of Advantage Learning to the solution of the ARS. The feature vector 4>(x, u) consisted of normalized (dimensionless) states and controls, and functions ofthese variables. Use ofthese nondimensionalized variables (obtained via the Buckingham 7r-theorem; e.g., Langharr, 1951) was found to enhance greatly the stability and robustness of the learning process. Furthermore, the RL system appeared to be less sensitive to changes in parameters such as the learning rate when these techniques were employed. 5 TRAINING Training the RL system for arbitrary orientations was accomplished by choosing random initial conditions on e as outlined above. With the exception of h, all other initial conditions corresponded to trim values for a Mach 0.6, 5 kIt. flight condition. Rewards were -1 per-time-step until the goal state was reached. In preliminary experiments, the training region was restricted to ± 0.174 rad.(l0 deg.) from the trim pitch angle. For this range of initial conditions, the system was able to learn an appropriate policy given only a handful of features (approximately 30). The policy was significantly mature after 24 hours oflearning on an HP-730 workstation and appeared to be able to achieve the goal for arbitrary initial conditions in the aforementioned domain. We then expanded the training region and considered initial e values within ± 0.785 rad. (45 deg.) of trim. The policy previously learned for the more restricted training domain performed well here too, and learning to recover for these more drastic off-trim conditions was trivial. No boundary restrictions were imposed on the system, but a report of whether the aircraft would have struck the ground was maintained. It was noted 1026 1. R Monaco, D. G. Ward and A. G. Barto that recovery from all possible initial conditions could not be achieved without hitting the ground. Episodes in which the ground would have been encountered were a result of inadequate control authority and not an inadequate RL policy. For example, when the initial pitch angle was at its maximum negative value, maximum-allowable positive stick (6 lbf.) was not sufficient to pull up the aircraft nose in time. To remedy this in subsequent experiments, the number of admissible actions was increased to include larger-magnitude commands: {-12, -9, -6, -3,0, +3, +6, +9, +12} lbf. Early attempts at solving the pitch-axis recovery problem with the expanded initial conditions in conjunction with this augmented action set proved challenging. The policy that worked well in the two previous experiments was no longer able to attain the goal state; it was only able to come close and oscillate indefinitely about the goal region. The agent learned to pitch up and down appropriately, e.g., when h was negative it applied a corrective positive action, and vice versa. However, because of system and actuator dynamics modeled in the simulation, the transient response caused the aircraft to pass through the goal state. Once beyond the goal region, the agent applied an opposite action, causing it to approach the goal state again, repeating the process indefinitely (until the system was reset and a new trial was started). Thus, the availability of large-amplitude commands and the presence of actuator dynamics made it difficult for the agent to fonnulate a consistent policy that afforded all goal state criteria being satisfied simultaneously. One might remedy the problem by removing the actuator dynamics; however, we did not wish to compromise simulation fidelity, and chose to use an expanded feature set to improve RL perfonnance. Using a larger collection offeatures with approximately 180 inputs, the RL agent was able to formulate a consistent recovery policy. The learning process required approximately 72 hours on an HP-730 workstation. (On this platform, the combined aircraft simulation and RL software execution rate was approximately twice that of real-time.) At this point performance was evaluated. The simulation was run in evaluation mode, i.e., learning rate was set to zero and random exploration was disabled. Performance is summarized below. 6 RESULTS 6.1 UNCONSTRAINED PITCH-AXIS RECOVERY Fig. 1 shows the transition times from off-trim orientations to the goal state as a function of initial pitch (inclination) angle. Recovery times were approximately 11-12 sec. for the worst-case scenarios. i.e .• 1801 = 45 deg. off-trim. and decrease (almost) monotonically for points closer to the unperturbed initial conditions. The occasional "blips" in the figure suggest that additional learning would have improved the global RL performance slightly. For 180 1 = 45 deg. off-trim, maximum altitude loss and gain were each approximately 1667 ft. (0.33 x 5000 f t. ). These excursions may seem substantial. but when one looks atthe time histories for these maneuvers, it is apparent that the RL-derived policy was perfonning well. The policy effectively minimizes any altitude variation; the magnitude of these changes are principally governed by available control authority and the severity of the flight condition from which the policy must recover. Fig. 2 shows time histories of relevant variables for one of the limiting cases. The first column shows body-axes pitch rate (Qb) and commanded body-axes pitch rate (Qbmodel) in (deg./sec.), pilot station nonnal acceleration (Nz) in (g), angle of attack (Alpha) in (deg.). and pitch attitude (Theta) in (deg.), respectively. The second column shows the longitudinal stick action executed by the RL system (lbf.), the left and right elevator deflections (deg.). total airspeed (ft./ sec.), and altitude (ft.). The majority ofthe 1600+ ft. altitude loss occurs between zero and five sec.; during this time, the RL system is applying maximum (allowable) positive stick. Thus, this altitude excursion is principally attributed to limited control authority as well as significant off-trim initial orientations. Automated Aircraft Recovery via Reinforcement Learning 20 18 16 14 12 Recovery Time (sec.) 10 8 6 4 2 O~~rn~~~~~~~~~~~~~~~~ ·50 ·40 ·30 ·20 ·10 0 10 20 30 40 50 Figure 1: Simulated Aircraft Recovery Times for Unconstrained Pitch-Axis ARS 00. _. 00.- ",,-L ~ , • 2 • • " 12 11 j cC. • • " ,'z -IU - ........... -_ ....... jV ~ , he • 2 • • " 12 o 2 • I 10 --:t: ~ , • 2 • • 1. 12 -.-~ , , t== • 2 • • • " " • 2 • "0 1027 Figure 2: TIme Histories During Unconstrained Pitch-Axis Recovery for 8 0 = 8 trim 45 deg. 1028 I. R Monaco, D. G. Ward and A G. Barto 6.2 CONSTRAINED PITCH-AXIS RECOVERY The requirement to execute aircraft recoveries while adhering to pilot-safety constraints was a deciding factor in using RL to demonstrate the automated recovery system concept. The need to recover an aircraft while minimizing injury and, where possible, discomfort to the flight crew, requires that the controller incorporate constraints that can be difficult or impossible to express in forms suitable for linear and nonlinear programming methods. In subsequent ARS investigations, allowable pilot-station normal acceleration was restricted to the range -1.5 9 ~ N z ~ 3.5 g. These values were selected because the unconstrained ARS was observed to exceed these limits. Several additional features (for a total of 189) were chosen, and the learning process was continued. Initial weights for the original 180 inputs corresponded to those from the previously learned policy; the new features were chosen to have zero weights initially. Here, the RL system learned to avoid the normal acceleration limits and consistently reach the goal state for initial pitch angles in the region [-45 + 8 trim , 35 + 8 trim] deg. Additional learning should result in improved recovery policies in this bounded acceleration domain for all initial conditions. Nonetheless, the results showed how an RL system can learn to satisfy these kinds of constraints. 7 CONCLUSION In addition to the results reported here, we conducted extensive analysis of the degree to which the learned policy successfully generalized to a range of initial conditions not experienced in training. In all cases, aircraft responses to novel recovery scenarios were stable and qualitatively similar to those previously executed in the training region. We are also conducting experiments with a multi-axes ARS, in which longitudinal-stick and lateral-stick sequences must be coordinated to recover the aircraft. Initial results are promising, but substantially longer training times are required. In summary, we believe that the results presented here demonstrate the feasibility of using RL algorithms to develop robust recovery strategies for high-agility aircraft, although substantial further research is needed. Acknowledgments This work was supported by the Naval Air Warfare Center Aircraft Division (NAWCAD), Flight Controls/Aeromechanics Division under Contract N62269-96-C-0080. The authors thank Marc Steinberg, the Program Manager and Chief Technical Monitor. The authors also express appreciation to Rich Sutton and Mance Harmon for their valuable help, and to Lockheed Martin Tactical Aircraft Systems for authorization to use their ATLAS software, from which F-16 parameters were extracted. References Baird, L. C. (1995) Residual algorithms: reinforcement learning with function approximation. In A. Prieditis and S. Russell (eds.), Machine Learning: Proceedings of the Twelfth International Conference, pp. 30-37. San Francisco, CA: Morgan Kaufmann. Harmon, M. E. & Baird, L. C. (1996) Multi-agent residual advantage learning with general function approximation. Wright Laboratory Technical Report, WPAFB, OH. Langharr, H. L. (1951) Dimensional Analysis and Theory of Models. New York: Wiley and Sons. Ward, D. G., Monaco, J. E, Barron, R. L., Bird, R.A., Virnig, J.C., & Landers, T.E (1996) Self-designing controller. Final Tech. Rep. for Directorate of Mathematics and Computer Sciences, AFOSR, Contract F49620-94-C-0087. Barron Associates, Inc.
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Adaptive choice of grid and time reinforcement learning Stephan Pareigis stp@numerik.uni-kiel.de Lehrstuhl Praktische Mathematik Christian-Albrechts-U ni versitiit Kiel Kiel, Germany Abstract • In We propose local error estimates together with algorithms for adaptive a-posteriori grid and time refinement in reinforcement learning. We consider a deterministic system with continuous state and time with infinite horizon discounted cost functional. For grid refinement we follow the procedure of numerical methods for the Bellman-equation. For time refinement we propose a new criterion, based on consistency estimates of discrete solutions of the Bellmanequation. We demonstrate, that an optimal ratio of time to space discretization is crucial for optimal learning rates and accuracy of the approximate optimal value function. 1 Introduction Reinforcement learning can be performed for fully continuous problems by discretizing state space and time, and then performing a discrete algorithm like Q-Iearning or RTDP (e.g. [5]). Consistency problems arise if the discretization needs to be refined, e.g. for more accuracy, application of multi-grid iteration or better starting values for the iteration of the approximate optimal value function. In [7] it was shown, that for diffusion dominated problems, a state to time discretization ratio k/ h of Ch'r, I > 0 has to hold, to achieve consistency (i.e. k = o(h)). It can be shown, that for deterministic problems, this ratio must only be k / h = C, C a constant, to get consistent approximations of the optimal value function. The choice of the constant C is crucial for fast learning rates, optimal use of computer memory resources and accuracy of the approximation. We suggest a procedure involving local a-posteriori error estimation for grid refinement, similar to the one used in numerical schemes for the Bellman-equation (see [4]). For the adaptive time discretization we use a combination from step size conAdaptive Choice of Grid and Time in Reinforcement Learning 1037 trol for ordinary differential equations and calculations for the rates of convergence of fully discrete solutions of the Bellman-equation (see [3]). We explain how both methods can be combined and applied to Q-Iearning. A simple numerical example shows the effects of suboptimal state space to time discretization ratio, and provides an insight in the problems of coupling both schemes. 2 Error estimation for adaptive choice of grid We want to approximate the optimal value function V : n -+ IR in a state space n C IRd of the following problem: Minimize J(x, u(.)) := 1 00 e- pr g(Yx,u(.)(r), u(r))dr, u(.): IR+ -+ A measurable, (1) where 9 : n X A -+ IR+ is the cost function, and Yx,u( .)(.) is the solution of the differential equation y(t) = f(y(t), u(t)), y(O) = x. (2) As a trial space for the approximation of the optimal value function (or Q-function) we use locally linear elements on simplizes Si, i = 1, ... , N s which form a triangulation of the state space, N s the number of simplizes. The vertices shall be called Xi, i = 1, . .. , N, N the dimension of the trial space1 . This approach has been used in numerical schemes for the Bellman-equation ([2], [4]). We will first assume, that the grid is fixed and has a discretization parameter k = maxdiam{Si}. i Other than in the numerical case, where the updates are performed in the vertices of the triangulation, in reinforcement learning only observed information is available. We will assume, that in one time step of size h > 0, we obtain the following information: • the current state Yn E n, • an action an E A, • the subsequent state Yn+1 := YYn,an (h) • the local cost rn = r(Yn, an) = Joh e-PTg(YYn,an(r),an(r))dr. The state Yn, in which an update is to be made, may be any state in n. A shall be finite, and an locally constant. The new value of the fully discrete Q-function Qi (Yn, an) should be set to shall be -phTTk ( ) rn + e v h Yn+l , where V; (Yn+d = minaQi(Yn+l,a). We call the right side the update function (3) We will update Qi in the vertices {Xd~l of the triangulation in one ofthe following two ways: 1 When an adaptive grid is used, then N s and N depend on the refinement. Kaczmarz-update. Let >.7 (AI, .. . , AN) be the vector of barycentric coordinates, such that Then update N Yn = 2: Aixi, O:SAi:Sl, foralli=I, ... ,N. i=1 (4) Kronecker-update. Let 53 Yn and x be the vertex of 5, closest to Yn (if there is a draw, then the update can be performed in all winners). Then update Q~ only in x according to (5) Each method has some assets and drawbacks. In our computer simulations the Kaczmarz-update seemed to be more stable over the Kronecker-update (see [6]) . However, examples may be constructed where a (Holder-) continuous bounded optimal value function V is to be approximated, and the Kaczmarz-update produces an approximation with arbitrarily high "."sup-norm (place a vertex x of the triangulation in a point where d: V is infinity, and use as update states the vertex x in turn with an arbitrarily close state x) . Kronecker-update will provide a bounded approximation if V is bounded. Let Vhk be the fully-discrete optimal value function Vhk (xd = min{r(xi' a) + e-PhVhk (Yxi,a(h)), i = 1, . . . , N. a Then it can be shown, that an approximation yerformed by Kronecker-update will eventually be caught in an c-neighborhood of VI: (with respect to the "."sup-norm), if the data points Yo, Yl, Y2, . . . are sufficiently dense. Under regularity conditions on V, c may be bounded by2 (6) As a criterion for grid refinement we choose a form of a local a posteriori error estimate as defined in [4] . Let vI: (x) = mina Q~ (x, a) be the current iterate of the optimal value function. Let ax E U be the minimizing control ax = argmina Q~ (x, a). Then we define (7) If Vhk is in the c-neighborhood of vI:, then it can be shown, that (for every x E n and simplex Sx with x E Sx, ax as above) O:S e(x) :S sup P(z, az , Vhk) inf P(z, az , Vhk). z ESz: z ES:t If Vhk is Lipschitz-continuous, then an estimate using only Gronwall's inequality bounds the right side and therefore e(x) by C p\' where C depends on the Lipschitzconstants of vl' and the cost g. 2With respect to the results in [3] we assume, that also E: ~ C(h + 7;:) can be shown. Adaptive Choice of Grid and Time in Reinforcement Learning 1039 The value ej := maXxesj eh(x) defines a function, which is locally constant on every simplex. We use ej, j = 1, ... , N as an indicator function for grid refinement. The (global) tolerance value tolk for ej shall be set to Ns tolk = C * (L; edlNs, i=l where we have chosen 1 :::; C :::; 2. We approximate the function e on the simplizes in the following way, starting in some Yn E Sj: 1. apply a control a E U constantly on [T, T + h] 2. receive value rn and subsequent state Yn+l 3. calculate the update value Ph(x, a, Vf) 4. if (IPh(x,a, vt) - Vt(x)l ~ ej) then ej := IPh(x,a, Vhk) - Vt(x)1 It is advisable to make grid refinements in one sweep. We also store (different to the described algorithm) several past values of ej in every simplex, to be able to distinguish between large e j due to few visits in that simplex and the large e j due to space discretization error. For grid refinement we use a method described in ([1]). 3 A local criterion for time refinement Why not take the smallest possible sampling rate? There are two arguments for adaptive time discretization. First, a bigger time step h naturally improves (decreases) the contraction rate of the iteration, which is e- ph . The new information is conveyed from a point further away (in the future) for big h, without the need to store intermediate states along the trajectory. It is therefore reasonable to start with a big h and refine where needed. The second argument is, that the grid and time discretization k and h stand in a certain relation. In [3] the estimate k lV(x) - vt(x)1 :::; C(h + ..Jh)' for all x En, C a constant is proven (or similar estimates, depending on the regularity of V). For obvious reasons, it is desirable to start with a coarse grid (storage, speed), i.e. k large. Having a too small h in this case will make the approximation error large. Also here, it is reasonable to start with a big h and refine where needed. What can serve as a refinement criterion for the time step h? In numerical schemes for ordinary differential equations, adaptive step size control is performed by estimating the local truncation error of the Taylor series by inserting intermediate points. In reinforcement learning, however, suppose the system has a large truncation error (i.e. it is difficult to control) in a certain region using large h and locally constant control functions. If the optimal value function is nearly constant in this region, we will not have to refine h. The criterion must be, that at an intermediate point, e.g. at time h12, the optimal value function assumes a value considerably smaller (better) than at time h. However, if this better value is due to error in the state discretization, then do not refine the time step. We define a function H on the simplices of the triangulation. H(S) > ° holds the time-step which will be used when in simplex S. Starting at a state Yn E n, Yn E Sn at time T > 0, with the current iterate of the Q-function Q~ (Vhk respectively) the following is performed: 1040 s. Pareigis 1. apply a control a E U constantly on [T, T + h] 2. take a sample at the intermediate state z = YYn,a(h/2) 3. if (H(Sn) < C*vdiam{Sn}) then end. else: 4. compute Vl(z) = millb Q~(z, b) 5. compute Ph/2(Yn, a, Vt) = rh/2(Yn, a) + e- ph/2Vt(z) 6. compute Ph(Yn, a, Vt) = rh(Yn, a) +e-phVl(Yn+d 7. if (Ph/2(Yn, a, Vhk) S Ph(Yn , a, Vhk)-tol) update H(Sn) = H(Sn)/2 The value C is currently set to 2 C = C(Yn, a) = -lrh/2(Yn, a) - rh(Yn, a)/, p whereby a local value of MI~gh2 is approximated, MJ (x) = maxa If(x, a)l, Lg an approximation of l\7g(x, a)1 (if 9 is sufficiently regular). tol depends on the local value of Vhk and is set to tOl(x) = 0.1 * vt(x). How can a Q-function Q:~:~(x, a), with state dependent time and space discretisation be approximated and stored? We have stored the time discretisation function H locally constant on every simplex. This implies (if H is not constant on 0), that there will be vertices Xj, such that adjacent triangles hold different values of H . The Q-function, which is stored in the vertices, then has different choices of H(xj). We solved this problem, by updating a function Q'H(Xj, a) with Kaczmarz-update and the update value PH(Yn) (Yn , a, Vt), Yn in an to Xj adjacent simplex, regardless of the different H-values in Xj. Q'H(Xj, a) therefore has an ambiguous semantic: it is the value if a is applied for 'some time', and optimal from there on. 'some time'depends here on the value of H in the current simplex. It can be shown, that IQ~(Xj)/2(xj,a) - Q'H(Xj)(xj,a)1 is less than the space discretization error. 4 A simple numerical example We demonstrate the effects of suboptimal values for space and time discretisation with the following problem. Let the system equation be iJ = f(y, u) := (~1 ~) (y - v), ( .375 ) v = .375 ' yEO = [0,1] x [0,1] (8) The stationary point of the uncontrolled system is v. The eigenvalues of the system are {u + i, U - i}, u E [-c, cJ. The system is reflected at the boundary. The goal of the optimal control shall be steer the solution along a given trajectory in state space (see figure 1), minimizing the integral over the distance from the current state to the given trajectory. The reinforcement or cost function is therefore chosen to be g(y) = dist(L, y)t, (9) where L denotes the set of points in the given trajectory. The cost functional takes the form ( 10) Adaptive Choice of Grid and Time in Reinforcement Learning 1041 0.5 IL ~ o~ ______ ~ ______ ~ o 0.5 Figure 1: The left picture depicts the L-form of the given trajectory. The stationary point of the system is at (.375, .375) (depicted as a big dot). The optimal value function computed by numerical schemes on a fine fixed grid is depicted with too large time discretization (middle) and small time discretization (right) (rotated by about 100 degrees for better viewing). The waves in the middle picture show the effect of too large time steps in regions where 9 varies considerably. In the learning problem, the adaptive grid mechanism tries to resolve the waves (figure 1, middle picture) which come from the large time discretization. This is depicted in figure 2. We used only three different time step sizes (h = 0.1, 0.05 and 0.025) and started globally with the coarsest step size 0.1. Figure 2: The adaptive grid mechanism refines correctly. However, in the left picture, unnecessary effort is spended in resolving regions, in which the time step should be refined urgently. The right picture shows the result, if adaptive time is also used. Regions outside the L-form are refined in the early stages of learning while h was still large. An additional coarsening should be considered in future work. We used a high rate of random jumps in the process and locally a certainty equivalence controller to produce these pictures. 1042 S. Pareigis 5 Discussion of the methods and conclusions We described a time and space adaptive method for reinforcement learning with discounted cost functional. The ultimate goal would be, to find a self tuning algorithm which locally adjusted the time and space discretization automatically to the optimal ratio. The methods worked fine in the problems we investigated, e.g. nonlinearities in the system showed no problems. Nevertheless, the results depended on the choice of the tolerance values C, tol and tolk' We used only three time discretization steps to prevent adjacent triangles holding time discretization values too far apart. The smallest state space resolution in the example is therefore too fine for the finest time resolution. A solution can be, to eventually use controls that are of higher order (in terms of approximation of control functions) than constant (e.g. linear, polynomial, or locally constant on subintervals of the finest time interval). This corresponds to locally open loop controls. The optimality of the discretization ratio time/space could not be proven. Some discontinuous value functions 9 gave problems, and we had problems handling stiff systems, too. The learning period was considerably shorter (about factor 100 depending on the requested accuracy and initial data) in the adaptive cases as opposed to fixed grid and time with the same accuracy. From our experience, it is difficult in numerical analysis to combine adaptive time and space discretization methods. To our knowledge this concept has not yet been applied to the Bellman-equation. Theoretical work is still to be done. We are aware, that triangulation of the state space yields difficulties in implementation in high dimensions. In future work we will be using rectangular grids. We will also make some comparisons with other algorithms like Parti-game ([5]). To us, a challenge is seen in handling discontinuous systems and cost functions as they appear in models with dry friction for example, as well as algebro-differential systems as they appear in robotics. References [1] E. Bansch. Local mesh refinement in 2 and 3 dimensions. IMPACT Comput. Sci. Engrg. 3, Vol. 3:181-191, 1991. [2] M. Falcone. A numerical approach to the infinite horizon problem of deterministic control theory. Appl Math Optim 15:1-13, 1987. [3] R. Gonzalez and M. Tidball. On the rates of convergence of fully discrete solutions of Hamilton-Jacobi equations. INRIA, Rapports de Recherche, No 1376, Programme 5, 1991. [4] L. Griine. An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation. Numerische Mathematik, Vol. 75, No. 3:319-337, 1997. [5] A. W. Moore and C. G. Atkeson. The parti-game algorithm for variable resolution reinforcement learning in multidimensional state-spaces. Machine Learning, Volume 21, 1995. [6] S. Pareigis. Lernen der Losung der Bellman-Gleichung durch Beobachtung von kontinuierlichen Prozepen. PhD thesis, Universitat Kiel, 1996. [7] S. Pareigis. Multi-grid methods for reinforcement learning in controlled diffusion processes. In D. S. Touretzky, M. C. Mozer, and M. E. Hasselmo, editors, Advances in Neural Information Processing Systems, volume 9. The MIT Press, Cambridge, 1997.
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Dynamic Stochastic Synapses as Computational Units Wolfgang Maass Institute for Theoretical Computer Science Technische Universitat Graz, A-B01O Graz, Austria. email: maass@igi.tu-graz.ac.at Abstract Anthony M. Zador The Salk Institute La Jolla, CA 92037, USA email: zador@salk.edu In most neural network models, synapses are treated as static weights that change only on the slow time scales of learning. In fact, however, synapses are highly dynamic, and show use-dependent plasticity over a wide range of time scales. Moreover, synaptic transmission is an inherently stochastic process: a spike arriving at a presynaptic terminal triggers release of a vesicle of neurotransmitter from a release site with a probability that can be much less than one. Changes in release probability represent one of the main mechanisms by which synaptic efficacy is modulated in neural circuits. We propose and investigate a simple model for dynamic stochastic synapses that can easily be integrated into common models for neural computation. We show through computer simulations and rigorous theoretical analysis that this model for a dynamic stochastic synapse increases computational power in a nontrivial way. Our results may have implications for the processing of time-varying signals by both biological and artificial neural networks. A synapse 8 carries out computations on spike trains, more precisely on trains of spikes from the presynaptic neuron. Each spike from the presynaptic neuron mayor may not trigger the release of a neurotransmitter-filled vesicle at the synapse. The probability of a vesicle release ranges from about 0.01 to almost 1. Furthermore this release probability is known to be strongly "history dependent" [Dobrunz and Stevens, 1997]. A spike causes an excitatory or inhibitory potential (EPSP or IPSP, respectively) in the postsynaptic neuron only when a vesicle is released. A spike train is represented as a sequence 1 of firing times, i.e. as increasing sequences of numbers tl < t2 < ... from R+ := {z E R: z ~ O} . For each spike train 1 the output of synapse 8 consists of the sequence 8W of those ti E 10n which vesicles are "released" by 8 , i.e. of those t, E 1 which cause an excitatory or inhibitory postsynaptic potential (EPSP or IPSP, respectively). The map 1 -+ 8(1) may be viewed as a stochastic function that is computed by synapse S. Alternatively one can characterize the output SW of a synapse 8 through its release pattern q = qlq2 ... E {R, F}· , where R stands for release and F for failure of release. For each t, E 1 one sets q, = R if ti E 8(1) , and qi = F if ti ¢ 8W . Dynamic Stochastic Synapses as Computational Units 195 1 Basic model The central equation in our dynamic synapse model gives the probability PS(ti) that the ith spike in a presynaptic spike train t = (tl,"" tk) triggers the release of a vesicle at time ti at synapse S, (1) The release probability is assumed to be nonzero only for t E t, so that releases occur only when a spike invades the presynaptic terminal (i.e. the spontaneous release probability is assumed to be zero). The functions C(t) ~ 0 and V(t) ~ 0 describe, respectively, the states of facilitation and depletion at the synapse at time t . The dynamics of facilitation are given by C(t) = Co + L c(t - ti) , (2) t. <t where Co is some parameter ~ 0 that can for example be related to the resting concentration of calcium in the synapse. The exponential response function c( s) models the response of C(t) to a presynaptic spike that had reached the synapse at time t - s: c(s) = a' e- a/ TC , where the positive parameters Te and a give the decay constant and magnitude, respectively, of the response. The function C models in an abstract way internal synaptic processes underlying presynaptic facilitation, such as the concentration of calcium in the presynaptic terminal. The particular exponential form used for c( s) could arise for example if presynaptic calcium dynamics were governed by a simple first order process. The dynamics of depletion are given by V(t) = max( 0, Vo (3) t.: t.<t and t.ES(!) for some parameter Vo > O. V(t) depends on the subset of those ti E t with ti < t on which vesicles were actually released by the synapse, i.e. ti E SW. The function v(s) models the response of V (t) to a preceding release of the same synapse at time t - s ~ t . Analogously as for c(s) one may choose for v(s) a function with exponential decay v(s) = e-a/ TV , where Tv > 0 is the decay constant. The function V models in an abstract way internal synaptic processes that support presynaptic depression, such as depletion of the pool of readily releasable vesicles. In a more specific synapse model one could interpret Vo as the maximal number of vesicles that can be stored in the readily releasable pool, and V(t) as the expected number of vesicles in the readily releasable pool at time t. In summary, the model of synaptic dynamics presented here is described by five parameters: Co, Vo, Te, Tv and a. The dynamics of a synaptic computation and its internal variables C(t) and V(t) are indicated in Fig. 1. For low release probabilities, Eq. 1 can be expanded to first order around r(t) := C(t) . V(t) = 0 to give (4) Similar expressions have been widely used to describe synaptic dynamiCS for mUltiple synapses [Magie by, 1987, Markram and Tsodyks, 1996, Varela et al., 1997]. In our synapse model, we have assumed a standard exponential form for the decay of facilitation and depression (see e.g. [Magleby, 1987, Markram and Tsodyks, 1996, Varela et al., 1997, Dobrunz and Stevens, 1997]}. We have further assumed a multiplicative interaction between facilitation and depletion. While this form has not been validated 196 presynaptic spike train function C(t) (facilitation) function V(t) (depression) " " W. Maass and A. M. Zador function p(t,) '[ (release 0'-------------probabilities) F FR R FRF F R release pattern --------------II I I I I I I )' t, Figure 1: Synaptic computation on a spike train i, together with the temporal dynamics of the internal variables C and V of our model. Note that V(t) changes its value only when a presynaptic spike causes release. at single synapses, in the limit of low release probability (see Eq. 4), it agrees with the multiplicative term employed in [Varela et al., 19971 to describe the dynamics of mUltiple synapses. The assumption that release at individual release sites of a synapse is binary, i. e. that each release site releases 0 or I-but not more than I-vesicle when invaded by a spike, leads to the exponential form of Eq. 1 [Dobrunz and Stevens, 19971. We emphasize the formal distinction between release site and synapse. A synapse might consist of several release sites in parallel, each of which has a dynamics similar to that of the stochastic "synapse model" we consider. 2 Results 2.1 Different "Weights" for the First and Second Spike in a Train We start by investigating the range of different release probabilities ps(td,PS(t2) that a synapse S can assume for the first two spikes in a given spike train. These release probabilities depend on t2 - tt as well as on the values of the internal parameters Co, Va,re,'TV,O of the synapse S. Here we analyze the potential freedom of a synapse to choose values for ps(tt} and PS(t2)' We show in Theorem 2.1 that the range of values for the release probabilities for the first two spikes is quite large, and that the entire attainable range can be reached through through suitable choices of Co and Vo . Theorem 2.1 Let (tt, t2) be some arbitrary spike train consisting of two spikes, and let PI ,P2 E (0,1) be some arbitrary given numbers with P2 > Pl' (1 - pd. Furthermore assume that arbitrary positive values are given for the parameters 0, re, 'TV of a synapse S. Then one can always find values for the two parameters Co and Va of the synapse S so that ps(tt) = PI and PS(t2) = P2. Furthermore the condition P2 > Pt . (1 - Pt) is necessary in a strong sense. If P2 ~ Pt · (1 - pt) then no synapse S can achieve ps(td = Pt and PS(t2) = P2 for any spike train (tl' t2) and for any values of its parameters Co, Vo, re, 'TV, 0. If one associates the current sum of release probabilities of multiple synapses or release sites between two neurons u and v with the current value of the "connection strength" wu,v between two neurons in a formal neural network model, then the preceding result points Dynamic Stochastic Synapses as Computational Units 197 Figure 2: The dotted area indicates the range of pairs (Pl,P2) of release probabilities for the first and second spike through which a synapse can move (for any given interspike interval) by varying its parameters Co and Vo . to a significant difference between the dynamics of computations in biological circuits and formal neural network models. Whereas in formal neural network models it is commonly assumed that the value of a synaptic weight stays fixed during a computation, the release probabilities of synapses in biological neural circuits may change on a fast time scale within a single computation. 2.2 Release Patterns for the First Three Spikes In this section we examine the variety of release patterns that a synapse can produce for spike trains tl, t2, t3, ' " with at least three spikes. We show not only that a synapse can make use of different parameter settings to produce 'different release patterns, but also that a synapse with a fixed parameter setting can respond quite differently to spike trains with different interspike intervals. Hence a synapse can serve as pattern detector for temporal patterns in spike trains. It turns out that the structure of the triples of release probabilities (PS(tl),PS(t2),PS(t3)) that a synapse can assume is substantially more complicated than for the first two spikes considered in the previous section. Therefore we focus here on the dependence of the most likely release pattern q E {R, FP on the internal synaptic parameters and on the interspike intervals II := t2 - fi and 12 := t3 - t2. This dependence is in fact quite complex, as indicated in Fig. 3. RRR RFR FRF FFF RRF interspike interval IJ interspike interval IJ Figure 3: (A, left) Most likely release pattern of a synapse in dependence of the interspike intervals It and 12. The synaptic parameters are Co = 1.5, Vo = 0.5, rc = 5, 'TV = 9, a = 0.7. (B, right) Release patterns for a synapse with other values of its parameters (Co = 0.1, Vo = 1.8, rc = 15, 'TV = 30, a = 1). 198 W. Maass and A. M Zador Fig. 3A shows the most likely release pattern for each given pair of interspike intervals (11,12 ), given a particular fixed set of synaptic parameters. One can see that a synapse with fixed parameter values is likely to respond quite differently to spike trains with different interspike intervals. For example even if one just considers spike trains with 11 = 12 one moves in Fig. 3A through 3 different release patterns that take their turn in becoming the most likely release pattern when II varies. Similarly, if one only considers spike trains with a fixed time interval t3 - t1 = II + 12 = ~, but with different positions of the second spike within this time interval of length ~, one sees that the most likely release pattern is quite sensitive to the position of the second spike within this time interval~. Fig. 3B shows that a different set of synaptic parameters gives rise to a completely different assignment of release patterns. We show in the next Theorem that the boundaries between the zones in these figures are "plastic": by changing the values of Co, Vo, Ct the synapse can move the zone for most of the release patterns q to any given point (11,12 )' This result provides another example for a new type of synaptic plasticity that can no longer be described in terms of a decrease or increase of the synaptic "weight". Theorem 2.2 Assume that an arbitrary number p E (0,1) and an arbitrary pattern (11,12 ) of interspike intervals is given. Furthermore assume that arbitrary fixed pOlJitive val;.;.p.s are given for the parameters rc and TV of a synapse S. Then for any pattern q E {R, FP except RRF, FFR one can assign values to the other parameters Ct, Co, Vo of this-synapse S so that the probability of release pattern q for a spike train with interspike intervals 11,12 becomes larger than p. It is shown in the full version oftbis paper [Maass and Zador, 19971 that it is not possible to make the release patterns RRF and FFR arbitrarily likely for any given spike train with interspike intervals (11,12 ) • 2.3 Computing with Firing Rates So far we have considered the effect of short trains of two or three presynaptic spikes on synaptic release probability. Our next result (cf. Fig.5) shows that also two longer Poisson spike trains that represent the same firing rate can produce quite different numers of synaptic releases, depending on the synaptic parameters. To emphasize that this is due to the pattern of interspike intervals, and not simply to the number of spikes, we compared the outputs in response to two Poisson spike trains A and B with the same number (lO)·of spikes. These examples indicate that even in the context of rate coding, synaptic efficacy may not be well described in terms of a single scalar parameter w. 2.4 Burst Detection Here we show that the computational power of a spiking (e.g. integrate-and-fire) neuron with stochastic dynamic synapses is strictly larger than that of a spiking neuron with traditional "static" synapses (cf Lisman, 1997). Let T be a some given time window, and consider the computational task of detecting whether at least one of n presynaptic neurons a1, . .. ,an fire at least twice during T ("burst detection"). To make this task computationally feasible we assume that none of the neurons al, ... ,an fires outside of this time window. Theorem 2.3 A spiking neuron v with dynamic stochastic synapses can solve this burst detection task (with arbitrarily high reliability). On the other hand no spiking neuron with static synapses can solve this task (for any assignment of "weights" to its synapses). 1 lWe assume here that neuronal transmission delays differ by less than (n - 1) . T), where by transmission delay we refer to the temporal delay between the firing of the presynaptic neuron and its effect on the postsynaptic target. Dynamic Stochastic Synapses as Computational Units 199 , .. •• r ....... ~ ... 11I II! r. III , M » • • • • n u , r ... u Il 1 • • .. '10 • 10 .. .. , ...... ,....". , , i f r , I i I , I .4 r I I l111 I ! I , I I i • . M » • d • • .1 • • Figure 4: Release probabilities of two synapses for two Poisson spike trains A and B with 10 spikes each. The release probabilities for the first synapse are shown on the left hand side, and for the second synapse on the right hand side. For both synapses the release probabilities for spike train A are shown at the top, and for spike train B at the bottom. The first synapse has for spike train A a 22 % higher average release probability, whereas the second synapse has for spike train B a 16 % higher average release probability. Note that the fourth spike in spike train B has for the first synapse a release probability of nearly zero and so is not visible. 2.5 Translating Interval Coding into Population Coding Assume that information is encoded in the length I of the interspike interval between the times tl and t2 when a certain neuron v fires, and that different motor responses need to be initiated depending on whether I < a or I > a, where a is some given parameter (c.f. [Hopfield, 1995]). For that purpose it would be useful to translate the information encoded in the interspike interval I into the firing activity of populations of neurons ("population coding"). Fig. 5 illustrates a simple mechanism for that task based on dynamic synapses. The synaptic parameters are chosen so that facilitation dominates (i.e., Co should be small and a large) at synapses between neuron v and the postsynaptic population of neurons. The release probability for the first spike is then close to 0, whereas the release probability for the second spike is fairly large if I < a and significantly smaller if I is substantially larger than a. If the resulting firing activity of the postsynaptic neurons is positively correlated with the total number of releases of these synapses, then their population response is also positively correlated with the length of the interspike interval I. 1 presynaptic spikes { FR • if 1 < a FF • if 1> a synaptic response { I. if 1 < a o • if I> a resulting activation of postsynaptic neurons Figure 5: A mechanism for translating temporal coding into population coding. 200 W. Maass and A. M Zador 3 Discussion We have explored computational implications of a dynamic stochastic synapse model. Our model incorporates several features of biological synapses usually omitted in the connections or weights conventionally used in artificial neural network models. Our main result is that a neural circuit in which connections are dynamic has fundamentally greater power than one in which connections are static. We refer to [Maass and Zador, 1997] for details. Our results may have implications for computation in both biological and artificial neural networks, and particularly for the processing of signals with interesting temporal structure. Several groups have recently proposed a computational role for one form of usedependent short term synaptic plasticity [Abbott et al., 1997, Tsodyks and Markram, 1997]. They showed that, under the experimental conditions tested, synaptic depression (of a form analogous to Vet) in our Eq. (3) can implement a form of gain control in which the steadystate synaptic output is independent of the input firing rate over a wide range of firing rates. We have adopted a more general approach in which, rather than focussing on a particular role for short term plasticity, we allow the dynamic synapse parameters to vary. This approach is analogous to that adopted in the study of artificial neural networks, in which few if any constraints are placed on the connections between units. In our more general framework, standard neural network tasks such as supervised and unsupervised learning can be formulated (see also [Liaw and Berger, 1996]). Indeed, a backpropagation-like gradient descent algorithm can be used to adjust the parameters of a network connected by dynamic synapses (Zador and Maass, in preparation). The advantages of dynamic synapses may become most apparent in the processing of time-varying Signals. References [Abbott et al., 1997] Abbott, L., Varela, J., Sen, K., and S.B., N. (1997). Synaptic depression and cortical gain control. Science, 275:220-4. [Dobrunz and Stevens, 1997] Dobrunz, L. and Stevens, C. (1997). Heterogeneity of release probability, facilitation and depletion at central synapses. Neuron, 18:995-1008. [Hopfield, 1995] Hopfield, J. (1995). Pattern recognition computation using action potential timing for stimulus representation. Nature, 376:33-36. [Liaw and Berger, 1996] Liaw, J.-S. and Berger, T. (1996). Dynamic synapse: A new concept of neural representation and computation. Hippocampus, 6:591-600. [Lisman, 1997] Lisman, J. (1997). Bursts as a unit of neural information: making unreliable synapses reliable. TINS, 20:38-43. [Maass and Zador, 1997] Maass, W. and Zador, A. (1997). Dynamic stochastic synapses as computational units. http://www.sloan.salk.edu/-zador/publications.html . [MagIe by, 1987] Magleby, K. (1987). Short term synaptic plasticity. In Edelman, G. M., Gall, W. E., and Cowan, W. M., editors, Synaptic function. Wiley, New York. [Markram and Tsodyks, 1996] Markram, H. and Tsodyks, M. (1996). Redistribution of synaptic efficacy between neocortical pyramidal neurons. Nature, 382:807-10. [Stevens and Wang, 1995] Stevens, C. and Wang, Y. (1995). Facilitation and depression at . single central synapses. Neuron, 14:795-802. [Tsodyks and Markram, 1997] Tsodyks, M. and Markram, H. (1997). The neural code between neocortical pyramidal neurons depends on neurotransmitter release probability. Proc. Natl. Acad. Sci., 94:719-23. [Varela et al., 1997] Varela, J. A., Sen, K., Gibson, J., Fost, J., Abbott, L. F., and Nelson, S. B. (1997). A quantitative description of short-term plasticity at excitatory synapses in layer 2/3 of rat primary visual cortex. J. Neurosci, 17:7926-7940.
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Graph Matching with Hierarchical Discrete Relaxation Richard C. Wilson and Edwin R. Hancock Department of Computer Science, University of York York, YOl 5DD, UK. Abstract Our aim in this paper is to develop a Bayesian framework for matching hierarchical relational models. The goal is to make discrete label assignments so as to optimise a global cost function that draws information concerning the consistency of match from different levels of the hierarchy. Our Bayesian development naturally distinguishes between intra-level and inter-level constraints. This allows the impact of reassigning a match to be assessed not only at its own (or peer) level ofrepresentation, but also upon its parents and children in the hierarchy. 1 Introd uction Hierarchical graphical structures are of critical importance in the interpretation of sensory or perceptual data. For instance, following the influential work of Marr [6] there has been sustained efforts at effectively organising and processing hierarchical information in vision systems. There are a plethora of concrete examples which include pyramidal hierarchies [3] that are concerned with multi-resolution information processing and conceptual hierarchies [4] which are concerned with processing at different levels of abstraction. Key to the development of techniques for hierarchical information processing is the desire to exploit not only the intra-level constraints applying at the individual levels of representation but also inter-level constraints operating between different levels of the hierarchy. If used effectively these interlevel constraints can be brought to bear on the interpretation of uncertain image entities in such a way as to improve the fidelity of interpretation achieved by single level means. Viewed as an additional information source, inter-level constraints can be used to resolve ambiguities that would persist if single-level constraints alone were used. 690 R. C. Wilson and E. R. Hancock In the connectionist literature graphical structures have been widely used to represent probabilistic causation in hierarchical systems [5, 9]. Although this literature has provided a powerful battery of techniques, they have proved to be of limited use in practical sensory processing systems. The main reason for this is that the underpinning independence assumptions and the resulting restrictions on graph topology are rarely realised in practice. In particular there are severe technical problems in dealing with structures that contain loops or are not tree-like. One way to overcome this difficulty is to edit intractable structures to produce tractable ones [8]. Our aim in this paper is to extend this discrete relaxation framework to hierarchical graphical structures. We develop a label-error process to model the violation of both inter-level and intra-level constraints. These two sets of constraints have distinct probability distributions. Because we are concerned with directly comparing the topology graphical structures rather than propagating causation, the resulting framework is not restricted by the topology of the hierarchy. In ,particular, we illustrate the effectiveness of the method on amoral graphs used to represent scene-structure in an image interpretation problem. This is a heterogeneous structure [2, 4] in which different label types and different classes of constraint operate at different levels of abstraction. This is to be contrasted with the more familiar pyramidal hierarchy which is effectively homogeneous [1, 3]. Since we are dealing with discrete entities inter-level information communication is via a symbolic interpretation of the objects under consideration. 2 Hierarchical Consistency The hierarchy consists of a number of levels, each containing objects which are fully described by their children at the level below. Formally each level is described by an attributed relational graph GI = (Vi, EI, Xl), Vi E L, with L being the index-set of levels in the hierarchy; the indices t and b are used to denote the top and bottom levels of the hierarchy respectively. According to our notation for level i of the hierarchy, Vi is the set of nodes, EI is the set of intra-level edges and Xl = {~~, Vu E Vi} is a set of unary attributes residing on the nodes. The children or descendents which form the representation of an element j at a lower level are denoted by V j . In other words, if U l - I is in Vj then there is a link in the hierarchy between element j at level i and element u at level i-I. According to our assumptions, the elements of Vj are drawn exclusively from Vi-I. The goal of performing relaxation operations is to find the match between scene graph G1 and model graph G2 • At each individual level of the hierarchy this match is represented by a mapping function p, Vi E L, where II: Vi -t Vi. The development of a hierarchical consistency measure proceeds along a similar line to the Single-level work of Wilson and Hancock [10]. The quantity of interest is the MAP estimate for the mapping function I given the available unary attributes, i.e. I = argmaxj P(jt, Vi E LIXI , Vi E L). We factorize the measurement information over the set of nodes by application of Bayes rule under the assumption of measurement independence on the nodes. As a result P(/, Vi E L/Xl , Vi E L) = (Xl ~i E L) {II II p(X~ll(u))}P(fI, Vi E L) (1) p, IELuEVI The critical modelling ingredient in developing a discrete relaxation scheme from the above MAP criterion is the joint prior for the mapping function, i.e. p(fl, Vi E L) Graph Matching with Hierarchical Discrete Relaxation Parents Children Possible mappings or children: 1,2,3 -A,B,C C,B,A A B c Figure 1: Example constrained children mappings 691 which represents the influence of structural information on the matching process. The joint measurement density, p(XI, 'VI E L), on the other hand is a fixed property of the hierarchy and can be eliminated from further consideration. Raw perceptual information resides on the lowest level of the hierarchy. Our task is to propagate this information upwards through the hierarchy. To commence our development, we assume that individual levels are conditionally dependent only on the immediately adjacent descendant and ancestor levels. This assumption allows the factorisation of the joint probability in a manner analogous to a Markov chain [3]. Since we wish to draw information from the bottom upwards, the factorisation commences from the highest level of labelling. The expression for the joint probability of the hierarchical labelling is p(fl, 'VI E L) = p(fb) II P(fl+Ill) (2) IEL,I#t We can now focus our attention on the conditional probabilities P(fI+1lfl). These quantities express the probability of a labelling at the level I + 1 given the previously defined labelling at the descendant level l. We develop tractable expressions for these probabilities by decomposing the hierarchical graph into convenient structural units. Here we build on the idea of decomposing Single-level graphs into supercliques that was successfully exploited in our previous work [10]. Super-cliques are the sets of nodes connected to a centre-object by intra-level edges. However, in the hierarchical case the relational units are more complex since we must also consider the graph-structure conveyed by inter-level edges. We follow the philosophy adopted in the single-level case [10] by averaging the superclique probabilities to estimate the conditional matching probabilities P(fI+1lfl). If r~ C fl denotes the current match of the super-clique centred on the object j E ~l then we write P(f'lfl-I) = ~l L p(r~lfl-l) I I jEV' (3) In order to model this probability, we require a dictionary of constraint relations for the corresponding graph sub-units (super-cliques) from the model graph G2 • The allowed mappings between the model graph and the data graph which preserve the topology of the graph structure at a particular level of representation are referred 692 R. C. Wilson and E R. Hancock to as "structure preserving mappings" or SPM's. It is important to note that we need only explore those mappings which are topologically identical to the superclique centred on object j and therefore the possible mappings of the child nodes are heavily constrained by the mappings of their parents (Figure 1). We denote the set of SPM's by P. Since the set P is effectively the state-space of legal matching, we can apply the Bayes theorem to compute the conditional super-clique probability in the following manner p(r~I/I-l) = 2: p(r~15,/I-l)P(5Ii-l) (4) SEP According to this expression, there are two distinct components to our model. The first involves the comparison between our mapped realisation of the super-clique from graph G1 , i.e. q, with the selected unit from graph G2 and the mapping from level 1 - 1. Here we take the view that once we have hypothesised a particular mapping 5 from P, the mapping P-l provides us with no further information, i.e. p(r~ 15, /1-1) = p(r~ 15). The matched super-clique r~ is conditionally independent given a mapping from the set of SPM's and we may write the first term as p(r~15). In other words, this first conditional probability models only intra-level constraints. The second term is the significant one in evaluating the impact inter-level constraints on the labelling at the previous level. In this term the probability of the hypothesised mapping 5 is conditioned according to the match of the child levell. All that remains now is to evaluate the conditional probabilities. Under the assumption of memoryless matching errors, the first term may be factorised over the marginal probabilities for the assigned matches lIon the individual nodes of the matched super-clique q given their counterparts Si belonging to the structure preserving mapping 5. In other words, p(r;15) = II P('~lsi) 1'! Ef~ (5) In order to proceed we need to specify a probability distribution for the different matching possibilities. There are three cases. Firstly, the match Ii may be to a dummy-node d inserted into q to raise it to the same size as 5 so as to facilitate comparison. This process effectively models structural errors in the data-graph. The second and third cases, relate to whether the match is correct or in error. Assuming that dummy node insertions may be made with probability Ps and that matching errors occur with probability Pe , then we can write down the following distribution rule if Ii = d or Si = d 'f 1 1 Ii = Si otherwise (6) The second term in Equation (5) is more subtle; it represents the conditional probability of the SPM 5 given a previously determined labelling at the level below. However, the mapping contains labels only from the current levell, not labels from level I - 1. We can reconcile this difference by noting that selection of a particular mapping at level I limits the number of consistent mappings allowed topologically at the level below. In other words if one node is mapped to another at level I, Graph Matching with Hierarchical Discrete Relaxation 693 the consistent interpretation is that the children of the nodes must match to each other. Provided that a set of mappings is available for the child-nodes, then this constraint can be used to model P(SljI-1). The required child-node mappings are referred to as "Hierarchy Preserving Mappings" or HPM's. It is these hierarchical mappings that lift the requirements for moralization in our matching scheme, since they effectively encode potentially incestuous vertical relations. We will denote the set of HPM's for the descendants of the SPM S as Qs and a member of this set as Q = {qi, 'Vi E Vj}. Using this model the conditional probability P(SIfI-l) is given by p(SIfI-1) = L P(SIQ,/I-1)P(Qll- 1 ) (7) QEQs Following our modelling of the intra-level probabilities, in this inter-level case assume that S is conditionally independent of 11- 1 given Q, i.e. P(SIQ, / 1- 1) = P(SIQ)· Traditionally, dictionary based hierarchical schemes have operated by using a labelling determined at a preceding level to prune the dictionary set by elimination of vertically inconsistent items [4]. This approach can easily be incorporated into our scheme by setting P(QI/I-l) equal to unity for consistent items and to zero for those which are inconsistent. However we propose a different approach; by adopting the same kind of label distribution used in Equation 6 we can grade the SPM's according to their consistency with the match at level 1 - 1, i.e. jI-l. The model is developed by factorising over the child nodes qi E Q in the following manner P(Qll- 1 ) = II P(qih,!-1) (8) qiEQ The conditional probabilities are assigned by a re-application of the distribution rule given in Equation (6), i.e. if dummy node match 'f 1-1 1 qi = Ii (9) otherwise For the conditional probability of the SPM given the HPM Q, we adopt a simple uniform model under the assumption that all legitimate mappings are equivalent, i.e. P(SIQ) = P(S) = I~I' The various simplifications can be assembled along the lines outlined in [10] to develop a discrete update rule for matching the two hierarchical structures. The MAP update decision depends only on the label configurations residing on levels 1 - 1, 1 and 1 + 1 together with the measurements residing on levell. Specifically, the level 1 matching configuration satisfies the condition II = argm!F{ II p(~~ljt(j)) }P(fI-llil )P(PI11+1 ) f jEV! (10) Here consistency of match between levels land 1 - 1 of the hierarchy is gauged by 694 the quantity PUl-llfl) = :1 L L K(rD exp 1 iEVI SEP Qs 1 L K(r!-l) exp QEQs R. C. Wilson and E. R. Hancock [-(keH(rL S) + ks~(rL S))] [ - (keH (r~-l ,Q) + ks ~(r~-l, Q)) }11) In the above expression H (r j, S) is the "Hamming distance" which counts the number of label conflicts between the assigned match rj and the structure preserving mapping S. This quantity measures the consistency of the matched labels. The number of dummy nodes inserted into r j by the mapping S is denoted by ~ (r j, S). This second quantity measures the structural compatibility of the two hierarchical graphs. The exponential constants ke = In (l-PeMI-P,) and ks = In 1Ft, are related to the probabilities of structural errors and mis-assignment errors. Finally, K(rj) = (1- Pe){1- Pe)lrjl is a normalisation constant. Finally, it is worth pointing out that the discrete relaxation scheme of Equation (10) can be applied at any level in the hierarchy. In other words the process can be operated in top-down or bottom-up modes if required. 3 Matching SAR Data In our experimental evaluation of the discrete relaxation scheme we will focus on the matching of perceptual groupings of line-segments in radar images. Here the model graph is elicited from a digital map for the same area as the radar image. The line tokens extracted from the radar data correspond to hedges in the landscape. These are mapped as quadrilateral field boundaries in the cartographic model. To support this application, we develop a hierarchical matching scheme based on linesegments and corner groupings. The method used to extract these features from the radar images is explained in detail in [10]. Straight line segments extracted from intensity ridges are organised into corner groupings. The intra-level graph is a constrained Delaunay triangulation of the line-segments. Inter-level relations represent the subsumption of the bottom-level line segments into corners. The raw image data used in this study is shown in Figure 2a. The extracted linesegments are shown in Figure 2c. The map used for matching is shown in Figure 2b. The experimental matching study is based on 95 linear segments in the SAR data and 30 segments contained in the map. However only 23 of the SAR segments have feasible matches within the map representation. Figure 2c shows the matches obtained by non-hierarchical means. The lines are coded as follows; the black lines are correct matches while the grey lines are matching errors. With the same coding scheme Figure 2d. shows the result obtained using the hierarchical method outlined in this paper. Comparing Figures 2c and 2d it is clear that the hierarchical method has been effective at grouping significant line structure and excluding clutter. To give some idea of relative performance merit, in the case of the non-hierarchical method, 20 of the 23 matchable segments are correctly identified with 75 incorrect matches. Application of the hierarchical method gives 19 correct matches, only 17 residual clutter segments with 59 nodes correctly labelled as clutter. 4 Concl usions We have developed graph matching technique which is tailored to hierarchical relational descriptions. The key element is this development is to quantify the matchGraph Matching with Hierarchical Discrete Relaxation 695 a) b) , \ \ " "\ ' ~\ '\ /\{-\'\(" " c) d) Figure 2: Graph editing: a) Original image, b) Digital map, c) Non hierarchical match, d) Hierarchical match. ing consistency using the concept of hierarchy preserving mappings between two graphs. Central to the development of this novel technique is the idea of computing the probability of a particular node match by drawing on the topologically allowed mappings of the child nodes in the hierarchy. Results on image data with lines and corners as graph nodes reveal that the technique is capable of matching perceptual groupings under moderate levels of corruption. References [1] F. Cohen and D. Cooper. Simple Parallel Hierarchical and Relaxation Algorithms for Segmenting Non-Causal Markovian Random Fields. IEEE PAMI, 9, 1987, pp.195219. [2] L. Davis and T. Henderson. Hierarchical Constraint Processes for Shape Analysis. IEEE PAMI, 3, 1981, pp.265-277. [3] B. Gidas. A Renormalization Group Approach to Image Processing Problems. IEEE PAMI, 11, 1989, pp.164-180. [4] T. Henderson. Discrete Relaxation Techniques. Oxford University Press, 1990. [5] D.J. Spiegelhalter and S.L. Lauritzen, Sequential updating of conditional probabilities on directed Graphical structures, Networks, 1990, 20, pp.579-605. [6] D. Marr, Vision. W.H. Freeman and Co., San Francisco. [7] J. Pearl, Probabilistic Reasoning in Intelligent Systems, Morgan Kaufmann, 1988. [8] M. Meila and M. Jordan, Optimal triangulation with continuous cost functions, Advances in Neural Information Processing Systems 9, to appear 1997. [9] P.Smyth, D. Heckerman, M.1. Jordan, Probabilistic independence networks for hidden Markov probability models, Neural Computation, 9, 1997, pp. 227-269. [10] R.C. Wilson and E. R. Hancock, Structural Matching by Discrete Relaxation. IEEE PAMI, 19, 1997, pp.634- 648. IEEE PAMI, June 1997.
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Minimax and Hamiltonian Dynamics of Excitatory-Inhibitory Networks H. S. Seung, T. J. Richardson Bell Labs, Lucent Technologies Murray Hill, NJ 07974 {seungltjr}~bell-labs.com J. C. Lagarias AT&T Labs-Research 180 Park Ave. D-130 Florham Park, NJ 07932 jcl~research.att.com J. J. Hopfield Dept. of Molecular Biology Princeton University Princeton, N J 08544 jhopfield~vatson.princeton.edu Abstract A Lyapunov function for excitatory-inhibitory networks is constructed. The construction assumes symmetric interactions within excitatory and inhibitory populations of neurons, and antisymmetric interactions between populations. The Lyapunov function yields sufficient conditions for the global asymptotic stability of fixed points. If these conditions are violated, limit cycles may be stable. The relations of the Lyapunov function to optimization theory and classical mechanics are revealed by minimax and dissipative Hamiltonian forms of the network dynamics. The dynamics of a neural network with symmetric interactions provably converges to fixed points under very general assumptions[l, 2]. This mathematical result helped to establish the paradigm of neural computation with fixed point attractors[3]. But in reality, interactions between neurons in the brain are asymmetric. Furthermore, the dynamical behaviors seen in the brain are not confined to fixed point attractors, but also include oscillations and complex nonperiodic behavior. These other types of dynamics can be realized by asymmetric networks, and may be useful for neural computation. For these reasons, it is important to understand the global behavior of asymmetric neural networks. The interaction between an excitatory neuron and an inhibitory neuron is clearly asymmetric. Here we consider a class of networks that incorporates this fundamental asymmetry of the brain's microcircuitry. Networks of this class have distinct populations of excitatory and inhibitory neurons, with antisymmetric interactions 330 H. S. Seung, T. 1. Richardson, J. C. Lagarias and 1. 1. Hopfield between populations and symmetric interactions within each population. Such networks display a rich repertoire of dynamical behaviors including fixed points, limit cycles[4, 5] and traveling waves[6]. After defining the class of excitatory-inhibitory networks, we introduce a Lyapunov function that establishes sufficient conditions for the global asymptotic stability of fixed points. The generality of these conditions contrasts with the restricted nature of previous convergence results, which applied only to linear networks[5]' or to nonlinear networks with infinitely fast inhibition[7]. The use of the Lyapunov function is illustrated with a competitive or winner-take-all network, which consists of an excitatory population of neurons with recurrent inhibition from a single neuron[8]. For this network, the sufficient conditions for global stability of fixed points also happen to be necessary conditions. In other words, we have proved global stability over the largest possible parameter regime in which it holds, demonstrating the power of the Lyapunov function. There exists another parameter regime in which numerical simulations display limit cycle oscillations[7]. Similar convergence proofs for other excitatory-inhibitory networks may be obtained by tedious but straightforward calculations. All the necessary tools are given in the first half of the paper. But the rest of the paper explains what makes the Lyapunov function especially interesting, beyond the convergence results it yields: its role in a conceptual framework that relates excitatory-inhibitory networks to optimization theory and classical mechanics. The connection between neural networks and optimization[3] was established by proofs that symmetric networks could find minima of objective functions[l, 2]. Later it was discovered that excitatory-inhibitory networks could perform the minimax computation of finding saddle points[9, 10, 11], though no general proof of this was given at the time. Our Lyapunov function finally supplies such a proof, and one of its components is the objective function of the network's minimax computation. Our Lyapunov function can also be obtained by writing the dynamics of excitatoryinhibitory networks in Hamiltonian form, with extra velocity-dependent terms. If these extra terms are dissipative, then the energy of the system is nonincreasing, and is a Lyapunov function. If the extra terms are not purely dissipative, limit cycles are possible. Previous Hamiltonian formalisms for neural networks made the more restrictive assumption of purely antisymmetric interactions, and did not include the effect of dissipation[12]. This paper establishes sufficient conditions for global asymptotic stability of fixed points. The problem of finding sufficient conditions for oscillatory and chaotic behavior remains open. The perspectives of minimax and Hamiltonian dynamics may help in this task. 1 EXCITATORY-INHIBITORY NETWORKS The dynamics of an excitatory-inhibitory network is defined by TxX+X f(u+Ax-By) , TyY+y = g(v+BTx-Cy). (1) (2) The state variables are contained in two vectors x E Rm and y E Rn, which represent the activities of the excitatory and inhibitory neurons, respectively. The symbol f is used in both scalar and vector contexts. The scalar function f : R ~ R is monotonic nondecreasing. The vector function f : Rm ~ Rm is Minimax and Hamiltonian Dynamics of Excitatory-Inhibitory Networks 331 defined by applying the scalar function 1 to each component of a vector argument, i.e., l(x) = (J(xt) , ... ,1(xm)). The symbol 9 is used similarly. The symmetry of interaction within each population is imposed by the constraints A = AT and C = CT. The antisymmetry of interaction between populations is manifest in the occurrence of - B and BT in the equations. The terms "excitatory" and "inhibitory" are appropriate with the additional constraint that the entries of matrices A, B, and C are nonnegative. Though this assumption makes sense in a neurobiological context the mathematics does not depends on it. The constant vectors u and v represent tonic input from external sources, or alternatively bias intrinsic to the neurons. The time constants Tz and Ty set the speed of excitatory and inhibitory synapses, respectively. In the limit of infinitely fast inhibition, Ty = 0, the convergence theorems for symmetric networks are applicable[l, 2], though some effort is required in applying them to the case C =/; 0. If the dynamics converges for Ty = 0, then there exists some neighborhood of zero in which it still converges[7]. Our Lyapunov function goes further, as it is valid for more general T y • The potential for oscillatory behavior in excitatory-inhibitory networks like (1) has long been known[4, 7]. The origin of oscillations can be understood from a simple two neuron model. Suppose that neuron 1 excites neuron 2, and receives inhibition back from neuron 2. Then the effect is that neuron 1 suppresses its own activity with an effective delay that depends on the time constant of inhibition. If this delay is long enough, oscillations result. However, these oscillations will die down to a fixed point, as the inhibition tends to dampen activity in the circuit. Only if neuron 1 also excites itself can the oscillations become sustained. Therefore, whether oscillations are damped or sustained depends on the choice of parameters. In this paper we establish sufficient conditions for the global stability of fixed points in (1). The violation of these sufficient conditions indicates parameter regimes in which there may be other types of asymptotic behavior, such as limit cycles. 2 LYAPUNOV FUNCTION We will assume that 1 and 9 are smooth and that their inverses 1-1 and g-1 exist. If the function 1 is bounded above and/or below, then its inverse 1-1 is defined on the appropriate subinterval of R. Note that the set of (x, y) lying in the range of (J,g) is a positive invariant set under (1) and that its closure is a global attractor for the system. The scalar function F is defined as the antiderivative of 1, and P as the Legendre transform P(x) maxp{px - F(p)}. The derivatives of these conjugate convex functions are, F'(x) = l(x) , (3) The vector versions of these functions are defined componentwise, as in the definition of the vector version of 1. The conjugate convex pair G, (; is defined similarly. The Lyapunov function requires generalizations of the standard kinetic energies Tzx2/2 and Tyy2/2. These are constructed using the functions ~ : Rm x Rm ~ R and r : Rn x Rn ~ R, defined by ~(p,x) r(q,y) = ITF(p) -xTp+lTP(x) , ITG(q) _yTq+ IT(;(y) . (4) (5) 332 H. S. Seung, T. 1. Richardson, J. C. Lagarias and J. J. Hopfield The components of the vector 1 are all ones; its dimensionality should be clear from context. The function ~(p, x) is lower bounded by zero, and vanishes on the manifold I(p) = x, by the definition of the Legendre transform. Setting p = U + Ax - By, we obtain the generalized kinetic energy T;l~(u + Ax - By, x), which vanishes when x = 0 and is positive otherwise. It reduces to T;xx 2/2 in the special case where I is the identity function. To construct the Lyapunov function, a multiple of the saddle function S = _uT x - !xT Ax + vT Y - !yTCy + ITP(x) + yTBT x - ITG(y) (6) 2 2 is added to the kinetic energy. The reason for the name "saddle function" will be explained later. Then L = T;l~(U + Ax - By,x) + T;lr(v + BT x - Cy, y) + rS (7) is a Lyapunov function provided that it is lower bounded, nonincreasing, and t only vanishes at fixed points of the dynamics. Roughly speaking, this is enough to prove the global asymptotic stability of fixed points, although some additional technical details may be involved. In the next section, the Lyapunov function will be applied to an example network, yielding sufficient conditions for the global asymptotic stability of fixed points. In this particular network, the sufficient conditions also happen to be necessary conditions. Therefore the Lyapunov function succeeds in delineating the largest possible parameter regime in which point attractors are globally stable. Of course, there is no guarantee of this in general, but the power of the Lyapunov function is manifest in this instance. Before proceeding to the example network, we pause to state some general conditions for L to be nonincreasing. A lengthy but straightforward calculation shows that the time derivative of L is given by t = xT Ax - iJTCiJ (8) _(T;l + r)j;T(J-l (T;xX + x) - I-I (x)J _(T;l - r)iJT[g-l(TyiJ + y) - g-l(y)J . Therefore, L is nonincreasing provided that (a-b)TA(a-b) max ( T [ < 1 + rT z , a,b a - b) I-l(a) - I-l(b)] (9) . (a - b)TC(a - b) mm T[ > 1 - rTy . a,b (a - b) g-l(a) - g-l(b)] (10) The quotients in these inequalities are generalizations of the Rayleigh-Ritz ratios of A and C. If I and 9 were linear, the left hand sides of these inequalities would be equal to the maximum eigenvalue of A and the minimum eigenvalue of C. 3 AN EXAMPLE: COMPETITIVE NETWORK The competitive or winner-take-all network is a classic example of an excitatoryinhibitory network[8, 7J. Its population of excitatory neurons Xi receives selffeedback of strength a and recurrent feedback from a single inhibitory neuron y, Tzii + Xi I(Ui + aXi - y) , (11) T.Y + y = 9 ( ~>i) . (12) Minimax and Hamiltonian Dynamics of Excitatory-Inhibitory Networks 333 This is a special case of (1), with A = aI, B = 1, and C = o. The global inhibitory neuron mediates a competitive interaction between the excitatory neurons. If the competition is very strong, a single excitatory neuron "wins," shutting off all the rest. If the competition is weak, more than one excitatory neuron can win, usually those corresponding to the larger Ui. Depending on the choice of f and g, self-feedback a, and time scales Tx and Ty, this network exhibits a variety of dynamical behaviors, including a single point attractor, multiple point attractors, and limit cycles[5, 7]. We will consider the specific case where f and 9 are the rectification nonlinearity [x]+ == max{ x, o}. The behavior ofthis network will be described in detail elsewhere; only a brief summary is given here. With either of two convenient choices for r, r = T;1 or r = a - T;1, it can be shown that the resulting L is bounded below for a < 2 and nonincreasing for a < T;1 + T;1. These are sufficient conditions for the global stability of fixed points. They also turn out to be necessary conditions, as it can be verified that the fixed points are locally unstable if the conditions are violated. The behaviors in the parameter regime defined by these conditions can be divided into two rough categories. For a < 1, there is a unique point attractor, at which more than one excitatory neuron can be active, in a soft form of winnertake-all. For a > 1, more than one point attractor may exist. Only one excitatory neuron is active at each of these fixed points, a hard form of winner-take-all. 4 MINIMAX DYNAMICS In the field of optimization, gradient descent-ascent is a standard method for finding saddle points of an objective function. This section of the paper explains the close relationship between gradient descent-ascent and excitatory-inhibitory networks[9, 10]. Furthermore, it reviews existing results on the convergence of gradient descentascent to saddle points[13, 10], which are the precedents of the convergence proofs of this paper. The similarity of excitatory-inhibitory networks to gradient descent-ascent can be seen by comparing the partial derivatives of the saddle function (6) to the velocities x and ii, as - ax as ay (13) (14) The notation a '" b means that the vectors a and b have the same signs, component by component. Because f and 9 are monotonic nondecreasing functions, x has the same signs as -as/ax, while iJ has the same signs as as/ay. In other words, the dynamics of the excitatory neurons tends to minimize S, while that of the inhibitory neurons tends to maximize S. If the sign relation", is replaced by equality in (13), we obtain a true gradient descent-ascent dynamics, . as . as ( 5) TxX = - ax ' Tyy = ay . 1 Sufficient conditions for convergence of gradient descent-ascent to saddle points are known[13, 10]. The conditions can be derived using a Lyapunov function constructed from the kinetic energy and the saddle function, L = ~Txlxl2 + ~Tylill2 + rS . (16) 334 H. S. Seung, T. 1. Richardson, 1. C. Lagarias and 1. 1. Hopfield The time derivative of L is given by L· 'T82S. 'T82S . ·2 · 2 (17) = -x 8x2 X + y 8y2 Y - rTxx + rTyy . Weak sufficient conditions can be derived with the choice r = 0, so that L includes only kinetic energy terms. Then L is obviously lower bounded by zero. Furthermore, L is nonincreasing if 82 S /8x2 is positive definite for all y and 82 S / 8y2 is negative definite for all x. In this case, the existence of a unique saddle point is guaranteed, as S is convex in x for all y , and concave in y for all x[13, 10]. If there is more than one saddle point, the kinetic energy by itself is generally not a Lyapunov function. This is because the dynamics may pass through the vicinity of more than one saddle point before it finally converges, so that the kinetic energy behaves nonmonotonically as a function of time. In this situation, some appropriate nonzero r must be found. The Lyapunov function (7) for excitatory-inhibitory networks is a generalization of the Lyapunov function (16) for gradient descent-ascent. This is analogous to the way in which the Lyapunov function for symmetric networks generalizes the potential function of gradient descent. It should be noted that gradient descent-ascent is an unreliable way of finding a saddle point. It is easy to construct situations in which it leads to a limit cycle. The unreliability of gradient descent-ascent contrasts with the reliability of gradient descent at finding local minimum of a potential function. Similarly, symmetric networks converge to fixed points, but excitatory-inhibitory networks can converge to limit cycles as well. 5 HAMILTONIAN DYNAMICS The dynamics of an excitatory-inhibitory network can be written in a dissipative Hamiltonian form. To do this, we define a phase space that is double the dimension ofthe state space, adding momenta (Px,Py) that are canonically conjugate to (x, y). The phase space dynamics TxX + X f(Px) , (18) TyY + y = g(py) , (19) (r+ :t) (u+Ax-By-px) = o , (20) (r+ !) (v+BTx-Cy-py) o , (21) reduces to the state space dynamics (1) on the affine space A = {(Px, PY' x, y) : Px = u + Ax - By,py = v + BTx - Cy}. Provided that r > 0, the affine space A is an attractive invariant manifold. Defining the Hamiltonian H(px, X'PY' y) = T;l~(Px, x) + T;lr(py, y) + rS(x, y) , the phase space dynamics (18) can be written as 8H 8px ' 8H 8py , (22) (23) (24) Minimax and Hamiltonian Dynamics of Excitatory-Inhibitory Networks Py = - ~~ + Ax - By - (r;l + r)[pX - i-leX)] , _ BH + BT x _ Gy _ (r- l _ r)r~ _ g-l(y)] By y lJ'y +2r(v+BT x-Gy-py) . 335 (25) (26) (27) On the invariant manifold A, the Hamiltonian is identical to the Lyapunov function (7) defined previously. The rate of change of the energy is given by H xT Ax - (r;l + r)xT[px - i-lex)] -yTGy _ (r;l _ r)yT[py _ g-l(y)] +2ryT(v + BT x - Gy - Py) . (28) The last term vanishes on the invariant manifold, leaving a result identical to (8). Therefore, if the noncanonical terms in the phase space dynamics (18) dissipate energy, then the Hamiltonian is nonincreasing. It is also possible that the velocitydependent terms may pump energy into the system, rather than dissipate it, in which case oscillations or chaotic behavior may arise. Acknowledgments This work was supported by Bell Laboratories. We would like to thank Eric Mjolsness for useful discussions. References [1] M. A. Cohen and S. Grossberg. Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE, 13:815-826, 1983. [2] J. J. Hopfield. Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. USA, 81:3088-3092, 1984. [3] J. J. Hopfield and D. W. Tank. Computing with neural circuits: a model. Science, 233:625-633, 1986. [4] H. R. Wilson and J. D. Cowan. A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik, 13:55-80, 1973. [5] Z. Li and J. J. Hopfield. Modeling the olfactory bulb and its neural oscillatory processings. Bioi. Cybern., 61:379-392, 1989. [6] S. Amari. Dynamics of pattern formation in lateral-inhibition type neural fields. Bioi. Cybern., 27:77-87, 1977. [7] B. Ermentrout. Complex dynamics in winner-take-all neural nets with slow inhibition. Neural Networks, 5:415-431, 1992. [8} S. Amari and M. A. Arbib. Competition and cooperation in neural nets. In J. Metzler, editor, Systems Neuroscience, pages 119-165. Academic Press, New York, 1977. [9} E. Mjolsness and C. Garrett. Algebraic transformations of objective functions. Neural Networks, 3:651-669, 1990. [10} J. C. Platt and A. H. Barr. Constrained differential optimization. In D. Z. Anderson, editor, Neural Information Processing Systems, page 55, New York, 1987. American Iristitute of Physics. [11] 1. M. Elfadel. Convex potentials and their conjugates in analog mean-field optimization. Neural Computation, 7(5):1079-1104, 1995. [12] J. D. Cowan. A statistical mechanics of nervous activity. In Some mathematical questions in biology, volume III. AMS, 1972. [13] K. J. Arrow, L. Hurwicz, and H. Uzawa. Studies in linear and non-linear programming. Stanford University, Stanford, 1958.
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Hybrid NNIHMM-Based Speech Recognition with a Discriminant Neural Feature Extraction Daniel Willett, Gerhard RigoU Department of Comfuter Science Faculty of Electrica Engineering Gerhard-Mercator-University Duisburg, Germany { willett,rigoll}@tb9-ti.uni-duisburg.de Abstract In this paper, we present a novel hybrid architecture for continuous speech recognition systems. It consists of a continuous HMM system extended by an arbitrary neural network that is used as a preprocessor that takes several frames of the feature vector as input to produce more discriminative feature vectors with respect to the underlying HMM system. This hybrid system is an extension of a state-of-the-art continuous HMM system, and in fact, it is the first hybrid system that really is capable of outperforming these standard systems with respect to the recognition accuracy. Experimental results show an relative error reduction of about 10% that we achieved on a remarkably good recognition system based on continuous HMMs for the Resource Management 1 OOO-word continuous speech recognition task. 1 INTRODUCTION Standard state-of-the-art speech recognition systems utilize Hidden Markov Models (HMMs) to model the acoustic behavior of basic speech units like phones or words. Most commonly the probabilistic distribution functions are modeled as mIXtures of Gaussian distributions. These mixture distributions can be regarded as output nodes of a Radial-BasisFunction (RBF) network that is embedded in the HMM system [1]. Contrary to neural training procedures the parameters of the HMM system, including the RBF network, are usually estimated to maximize the training observations' likelihood. In order to combine the time-warping abilities of HMMs and the more discriminative power of neural networks, several hybrid approaches arose during the past five years, that combine HMM systems and neural networks. The best known approach is the one proposed by Bourlard [2]. It replaces the HMMs' RBF -net with a Multi-Layer-Perceptron (MLP) which is trained to output each HMM state's posterior probability. At last year's NIPS our group presented a novel hybrid speech recognition approach that combines a discrete HMM speech recognition system and a neural quantizer (3). By maximizing the mutual information between the VQ-Iabels and the assigned phoneme-classes, this apl'roach outperforms standard discrete recognition systems. We showed that this approach IS capable of building up very accurate systems with an extremely fast likelihood computation, that only consists of a quantization and a table lookup. This resulted in a hybrid system with recognition performance equivalent to the best 764 x (t-P) x(t) x(t+F) Neural Network (linear transfonnation, MLP or recurrent MLP) feature extraction x'(t) p(x(t)IW[) D. Willett and G. Rigoll HMM-System (RBF-network) p(x(t)l~ Figure I: Architecture of the hybrid NN/HMM system continuous systems, but with a much faster decoding. Nevertheless, it has turned out that this hybrid approach is not really capable of substantially outperforming very good continuous systems with respect to the recognition accuracy. This observation is similar to experiences with Bourlard's MLP approach. For the decoding procedure, this architecture offers a very efficient pruning technique (phone deactivation pruning [4)) that is much more efficient than pruning on likelihoods, but until today this approach did not outperform standard continuous HMM systems in recognition performance. 2 HYBRID CONTINUOUS HMMlMLP APPROACH Therefore, we followed a different approach, namely the extension of a state-of-the-art continuous system that achieves extremely good recognition rates with a neural net that is trained with MMI-methods related to those in [5]. The major difference in this approach is the fact that the acoustic processor is not replaced by a neural network, but that the Gaussian probability density component is retained and combined with a neural component in an appropriate manner. A similar approach was presented in [6] to improve a speech recognitIOn system for the TIMIT database. We propose to regard the additional neural component as being part of the feature extraction, and to reuse it in recognition systems of higher complexity where discriminative training is extremely expensive. 2.1 ARCHITECTURE The basic architecture of this hybrid system is illustrated in Figure 1. The neural net functions as a feature transformation that takes several additional past and future feature vectors into account to produce an improved more discriminant feature vector that is fed into the HMM system. This architecture allows (at least) three ways of interpretation; 1. as a hybrid system that combines neural nets and continuous HMMs, 2. as an LDA-like transformation that incorporates the HMM parameters into the calculation of the transformation matrix and 3. as feature extraction method, that allows the extraction offeatures according to the underlying HMM system. The considered types of neural networks are linear transformations, MLPs and recurrent MLPs. A detailed 3escription of the possible topologies is given in Section 3. With this architecture, additional past and future feature vectors can be taken into account in the probability estimation process without increasing the dimensionality of the Gaussian mixture components. Instead of increasing the HMM system's number of parameters the neural net is trained to produce more discriminant feature vectors with respect to the trained HMM system. Of course, adding some kind of neural net increases the number of parameters too, but the increase is much more moderate than it would be when increasing each Gaussian's dimensionality. Speech Recognition with a Discriminant Neural Feature Extraction 765 2.2 TRAINING OBJECTIVE The original purpose of this approach was the intention to transfer the hybrid approach presented in [3], based on MMI neural network, to (semi-) continuous systems. ThIS way, we hoped to be able to achieve the same remarkable improvements that we obtained on discrete systems now on continuous systems, which are the much better and more flexible baseline systems. The most natural way to do this would be the re-estimation of the codebook of Gaussian mean vectors of a semi-continuous system using the neural MMI training algorithm presented in [31. Unfortunately though, this won't work, as this codebook of a semi-continuous system does not determine a separation of the feature space, but is used as means of Gaussian densities. The MMI-principle can be retained, however, by leaving the original HMM system unmodified and instead extending it with a neural component, trained according to a frame-based MMI approach, related to the one in [3]. The MMI criterion is usually formulated in the following way: )..MMI = argmaxi).(X, W) = argmax(H>.(X) - H>.(XIW)) = argmax P>.(~l~) >. >. >. P>. (1) This means that following the MMI criterion the system's free parameters ,\ have to be estimated to maximize the quotient of the observation's likelihood p>.(XIW) for the known transcription Wand its overall likelihood P>. (X). With X = (x(l), x(2), ... x(T)) denoting the training observations and W = (w(l), w(2) , ... w(T)) denoting the HMM statesassigned to the observation vectors in a Viterbi-alignment - the frame-based MMI criterion becomes T )..MMI ~ arg~ax L i).(x(i), w(i)) i=1 IT T p>.(x(i)lw(i)) ITT p>.(x(i)lw(i)) =arg~ax . p>.(x(i)) :::::::arg~ax . s. (2) 1:1 %=1 2: P>.(X(Z)lwk)p(Wk) k=1 where S is the total number ofHMM states, (W1 , '" ws) denotes the HMM states and p( Wk) denotes each states' prior-probability that is estimated on the alignment of the training data or by an analysis of the lan~age model. Eq. 2 can be used to re-estImate the Gaussians of a continuous HMM system directly. In [7] we reported the slight improvements in recognition accuracy that we achieved with this parameter estimation. However, it turned out, that only the incorporation of additional features in the probability calculation pipeline can provide more discriminative emission probabilities and a major advance in recognition accuracy. Thus, we experienced it to be more convenient to train an additional neural net in order to maximize Eq. 2. Besides, this approach offers the possibility of improving a recognition system by applying a trained feature extraction network taken from a different system. Section 5 will report our positive experiences with this procedure. At first, for matter of simplicity, we will consider a linear network that takes P past feature vectors and F future feature vectors as additional input. With the linear net denoted as a (P + F + 1) x N matrix NET, each component x' (t)[c] of the network output x'(t) computes to P+F N x'(t)[c] = L L x(t - P + i)[jJ . N ET[i * N + j][c] Vc E {L.N} (3) i=O j=1 so that the derivative with respect to a component of NET easily computes to 8x'(t)[cJ _ 6 -x(t - P i)['J (4) 8NET[i*N+j][c] C,c + J In a continuous HMM system with diagonal covariance matrices the pdf of each HMM state w is modeled by a mixture of Gaussian components like N 2 C 1 ~ (m;[I)-x[l]) 1U 1 -"2 L, ajll) p>.(xlw) = Ldwj e 1=1 (5) j=1 y'(2rr)nIUjl 766 D. Willett and G. Rigoll A pdfs derivative with respect to a component x'[e] of the net's output becomes N C 1 ~ (mj[ll-",/[I))2 8p,\.(x'lw) _ ~ d . (:e[e] - mj[e)) 1 e- 2 ~ aj[!} (6) 8x'[e) - ~ WJ oAe) y'(21l')nIO'jl With x(t) in Eq. 2 now replaced by the net output x'(t) the partial derivative ofEq. 2 with respect to a probabilistic distribution function p( x' (i) IWk) computes to 8h(x'(i), w(i)) _ <5W(i) ,Wk 8p,\.(x'(i)lwk) - p,\.(x(i)lwk) s 2: p,\.(x(i)lwI)p(wd 1=1 (7) Thus, using the chain rule the derivative of the net's parameters with respect to the framebased MMI criterion can be computed as displayed in Eq. 8 8h(X, W) = t(t(8h(:e(i)IW(i))) 8p,\.(x'(i)lwk) ox' (i)[c] ) (8) 8N ET[lHe) i=l k=l op,\.(x'(i)lwk) ox' (i)[e] oN ET[lHe) and a gradient descent procedure can be used to determine the optimal parameter estimates. 2.3 ADVANTAGES OF THE PROPOSED APPROACH When using a linear network, the proposed approach strongly resembles the well known Linear Discriminant Analysis (LDA) [8] in architecture and training objective. The main difference is the way the transformation is set up. In the proposed approach the transformation is computed by taking directly the HMM parameters into account whereas the LDA only tries to separate the features according to some class assignment. With the incorporation of a trained continuous HMM system the net's parameters are estimated to produce feature vectors that not only have a good separability in general, but also have a distribution that can be modeled with mixtures ofGaussians very well. Our experiments given at the end ofthis paper prove this advantage. Furthermore, contrary to LDA, that produces feature vectors that don't have much in common with the original vectors, the proposed approach only slightly modifies the input vectors. Thus, a well trained continuous system can be extended by the MMI-net approach, in order to improve its recognition performance without the need for completely rebuilding it. In addition to that, the approach offers a fairly easy extension to nonlinear networks (MLP) and recurrent networks (recurrent MLP). This will be outlined in the following Section. And, maybe as the major advantage, the approach allows keeping up the division of the input features into streams of features that are strongly uncorrelated and which are modeled with separate pdfs. The case of multiple streams is discussed in detail in Section 4. Besides) the MMI approach offers the possibility of a unified training of the HMM system and the reature extraction network or an iterative procedure of training each part alternately. 3 NETWORK TOPOLOGIES Section 2 explained how to train a linear transformation with respect to the frame-based MMI criterion. However, to exploit all the advantages of the proposed hybrid approach the network should be able to perform a nonlinear mapping, in order to produce features whose distribution is (closer to) a mixture of Gaussians although the original distribution is not. 3.1 MLP When using a fully connected MLP as displayed in Figure 2 with one hidden layer of H nodes, that perform the nonlinear function /, the activation of one of the output nodes x'(t)[e] becomes H P+F N x'(tHe) = t; L2[hJ[e)· f( BIAS. + ~ ~ x(t - P + i)li)· L1[i * N + jJ[h)) (9) Speech Recognition with a Discriminant Neural Feature Extraction 767 original features (multiple frames) x(t-l) x(t) x(t+1) 1 x'(t) RBFnetwork ~ ~ ~ ~ : :~--:EA6 ::: ~~~~ I I ' "''"~' ~... \NlIl,~l/\f)(JI\L.-i.M/I/ I: I \,,~~ : ': Ll "'~~~'~~~~~11111~ ! : : '~~:\ ... ::: ~ I I o : : : I I : : : : : : I I I : : ~ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ transformed features 1- '--_-_-.:_-_-_-_-_-_-_-_-_-_-_-_-_ - __ p(x(t)lw ~ p(x(t)lwi p(x(t)lwj §~~ context-dependent Hidden Markov Model Figure 2: Hybrid system with a nonlinear feature transfonnation which is easily differentiable with respect to the nonlinear network's parameters. In our experiments we chose f to be defined as the hyperbolic tangents f(x) := tanh(x) = (2(1+ e -X) -1 1) so that the partial derivative with respect to i.e. a weight L 1 [i . N + J] [h] of the first layer computes to ax' (t)[e] x(t - p + i)[J] . L2[h][c] aL1[i . N + 3][h] (10) .COSh( BIAS. + % ~ x(t - P + i)lil· L1[i * N + j][h1r' and the gradient can be set up according to Eq. 8. 3.2 RECURRENT MLP With the incorporation of several additional past feature vectors as explained in Section 2, more discriminant feature vectors can be generated. However, this method is not capable of modeling longer term relations, as it can be achieved by extending the network with some recurrent connections. For the sake of simplicity, in our experiments we simply extended the MLP as indicated with the dashed lines in Figure 2 by propagating the output x (t) back to the input of the network (with a delay of one discrete time step). This type of recurrent neural net is often referred to as a 'Jordan' -network. Certainly, the extension of the network with additional hidden nodes in order to model the recurrence more independently would be possible as well. 4 MULTI STREAM SYSTEMS In HMM-based recognition systems the extracted features are often divided into streams that are modeled independently. This is useful the less correlated the divided features are. In this case the overall likelihood of an observation computes to M p>.(xlw) = IT p$>.(xlw)w, (11) where each of the stream pdfs p$>.(xlw) only uses a subset of the features in x. The stream weights W$ are usually set to unity. 768 D. Willett and G. Rigoll Table 1: Word error rates achieved in the experiments A multi stream system can be improved by a neural extraction for each stream and an independent training of these neural networks. However, it has to be considered that the subdivided features usually are not totally independent and by considering multiple input frames as illustrated in Figure 1 this dependence often increases. It is a common practice, for instance, to model the features' first and second order delta coefficients in independent streams. So, for sure the streams lose independence when considering multiple frames, as these coefficients are calculated using the additional frames. Nevertheless, we found it to give best results to maintain this subdivision into streams, but to consider the stronger correlation by training each stream's net dependent on the other nets' outputs. A training criterion follows straight from Eq. 11 inserted in Eq. 2. \ _ rrT p).(x(i)lw(i)) _ rrT 11M (P3).(X(i)IW(i)))W' (12) /\MMI - argmax ( (.)) - argmax ( (.)) ). i=l P). X t .). i=13=1 Ps). X z The derivative of this equation with respect to the pdf P.;). (xlw) ofa specific stream s depends on the other streams' pdfs. With the Ws set to unity it is 8h(x'(;): w(i)) = (rr ps).(X(i)I~(i))) ( 6w (i):Wk _ s p(Wk) ) 8ps).(x (Z)IWk) ~. P3).(X(Z)) ps).(X(Z)IWk) 'I\' ((')1) ( ) 3 r 3 W P.;). x Z WI P WI 1=1 (13) Neglecting the correlation among the streams the training of each stream's net can be done independently. However, the more the incorporation of additional features increases the streams' correlation, the more important it gets to train the nets in a unified training procedure according to Eq. 13. 5 EXPERIMENTS AND RESULTS We applied the proposed approach to improve a context-independent (monophones) and a context-dependent (triphones) continuous speech reco~tion system for the 1000-wordResource Management (RM) task. The systems used lmear HMMs of three emitting states each. The tying of Gaussian mixture components was perfonned with an adaptive procedure according to [9]. The HMM states of the word-internal triphone system were clustered in a tree-based phonetic clustering procedure. Decoding was perfonned with a Viterbidecoder and the standard wordpair-grammar of perplexity 60. Training of the MLP was perfonned with the RPROP algorithm. For training the weights of the recurrent connections we chose real-time recurrent learning. The average error rates were computed using the test-sets Feb89, Oct89, Feb91 and Sep92. The table above shows the recognition results with single stream systems in its first section. These systems simply use a 12-value Cepstrum feature vector without the incorporation of delta coefficients. The systems with an input transfonnation use one additional past and one additional future feature vector as input. The proposed approach achieves the same perfonnance as the LDA, but it is not capable of outperfonning It. The second section of the table lists the recognition results with four stream systems that use the first and second order delta coefficients in additional streams plus log energy and this values' delta coefficients in a forth stream. The MLP system trained according to Eq. Speech Recognition with a Discriminant Neural Feature Extraction 769 II slightly outperforms the other approaches. The incorporation of recurrent network connections does not improve the system's performance. The third section of the table lists the recognition results with four stream systems with a context-dependent acoustic modeling (triphones). The applied LDA and the MMI -networks were taken from the monophone four stream system. On the one hand, this was done to avoid the computational complexity that the MMI training objective causes on contextdependent systems. On the other hand, this demonstrates that the feature vectors produced by the trained networks have a good discrimination for continuous systems in general. Again, the MLP system outperforms the other· approaches and achieves a very remarkable word error rate. It should be pointed out here, that the structure of the continuous system as reported in (9) is already highly optimized and it is almost impossible to further reduce the error rate by means of any acoustic modeling method. This is reflected in the fact that even a standard LDA cannot improve this system. Only the new neural approach leads to a 10% reduction in error rate which is a large improvement considering the fact that the error rate of the baseline system is among the best ever reported for the RM database, 6 CONCLUSION The paper has presented a novel approach to discriminant feature extraction. A MLP network has successfully been used to compute a feature transformation that outputs extremely suitable features for continuous HMM systems. The experimental results have proven that the proposed approach is an appropriate method for including several feature frames in the probability estimation process without increasing the dimensionality of the Gaussian mixture components in the HMM system. Furthermore did the results on the triphone speech recognition system prove that the approach provides discriminant features, not only for the system that the mapping is computed on, but for HMM systems with a continuous modeling in general: The application of recurrent networks did not improve the recognition accuracy. The longer range relations seem to be very weak and they seem to be covered well by using the neighboring feature vectors and first and second order delta coefficients. The proposed unified training procedure for multiple nets in multi-stream systems allows keeping up the subdivision of features of weak correlations, and gave us best profits in recognition accuracy. References [1) H. Ney, "Speech Recognition in a Neural Network Framework: Discriminative Training of Gaussian Models and Mixture Densities as Radial Basis Functions", Proc. IEEEICASSp, 1991, pp. 573-576. [2) H Bouriard, N. Morgan, "Connectionist Speech Recognition - A Hybrid Approach", Kluwer Academic Press, 1994. [3) G. Rigoll, C. Neukirchen, "A new approach to hybrid HMMIANN speech recognition using mutual information neural networks", Advances in Neural Information Processing Systems (NIPS-96), Denver, Dec. 1996, pp. 772-778. [4] M. M. Hochberg, G. D. Cook, S. J. Renals, A. J. Robinson, A. S. Schechtman, "The 1994 ABBOT Hybrid Connectionist-HMM Large-Vocabulary Recognition System", Proc. ARPA Spoken Language Systems Technology Workshop, 1995. [5] G. Rigoll, "Maximum Mutual Information Neural Networks for Hybrid ConnectionistHMM Speech Recognition", IEEE-Trans. Speech Audio Processing, Vol. 2, No.1, Jan. 1994,pp.175-184. [6] Y. Bengio et aI., "Global Optimization of a Neural Network - Hidden Markov Model Hybrid" IEEE-Transcations on NN, Vol. 3, No. 2, 1992, pp. 252-259. [7] D. Willett, C. Neukirchen, R. Rottland, "Dictionary-Based Discriminative HMM Parameter Estimation for Continuous Speech Recognition Systems", Proc. IEEE-ICASSp, 1997,pp.1515-1518. [8] X. Aubert, R. Haeb-Umbach, H. Ney, "Continuous mixture densities and linear discriminant analysis for improved context-dependent acoustic models", Proc. IEEE-ICASSp, 1993, pp. II 648-651. [9) D. Willett, G. Rigoll, "A New Approach to Generalized Mixture Tying for Continuous HMM-Based Speech Recognition",Proc. EUROSPEECH, Rhodes, 1997.
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2D Observers for Human 3D Object Recognition? Zili Liu NEC Research Institute . Abstract Daniel Kersten University of Minnesota Converging evidence has shown that human object recognition depends on familiarity with the images of an object. Further, the greater the similarity between objects, the stronger is the dependence on object appearance, and the more important twodimensional (2D) image information becomes. These findings, however, do not rule out the use of 3D structural information in recognition, and the degree to which 3D information is used in visual memory is an important issue. Liu, Knill, & Kersten (1995) showed that any model that is restricted to rotations in the image plane of independent 2D templates could not account for human performance in discriminating novel object views. We now present results from models of generalized radial basis functions (GRBF), 2D nearest neighbor matching that allows 2D affine transformations, and a Bayesian statistical estimator that integrates over all possible 2D affine transformations. The performance of the human observers relative to each of the models is better for the novel views than for the familiar template views, suggesting that humans generalize better to novel views from template views. The Bayesian estimator yields the optimal performance with 2D affine transformations and independent 2D templates. Therefore, models of 2D affine matching operations with independent 2D templates are unlikely to account for human recognition performance. 1 Introduction Object recognition is one of the most important functions in human vision. To understand human object recognition, it is essential to understand how objects are represented in human visual memory. A central component in object recognition is the matching of the stored object representation with that derived from the image input. But the nature of the object representation has to be inferred from recognition performance, by taking into account the contribution from the image information. When evaluating human performance, how can one separate the con830 Z Liu and D. Kersten tributions to performance of the image information from the representation? Ideal observer analysis provides a precise computational tool to answer this question. An ideal observer's recognition performance is restricted only by the available image information and is otherwise optimal, in the sense of statistical decision theory, irrespective of how the model is implemented. A comparison of human to ideal performance (often in terms of efficiency) serves to normalize performance with respect to the image information for the task. We consider the problem of viewpoint dependence in human recognition. A recent debate in human object recognition has focused on the dependence of recognition performance on viewpoint [1, 6]. Depending on the experimental conditions, an observer's ability to recognize a familiar object from novel viewpoints is impaired to varying degrees. A central assumption in the debate is the equivalence in viewpoint dependence and recognition performance. In other words, the assumption is that viewpoint dependent performance implies a viewpoint dependent representation, and that viewpoint independent performance implies a viewpoint independent representation. However, given that any recognition performance depends on the input image information, which is necessarily viewpoint dependent, the viewpoint dependence of the performance is neither necessary nor sufficient for the viewpoint dependence of the representation. Image information has to be factored out first, and the ideal observer provides the means to do this. The second aspect of an ideal observer is that it is implementation free. Consider the GRBF model [5], as compared with human object recognition (see below). The model stores a number of 2D templates {Ti} of a 3D object 0, and reco~nizes or rejects a stimulus image S by the following similarity measure ~iCi exp UITi - SI1 2 j2(2), where Ci and a are constants. The model's performance as a function of viewpoint parallels that of human observers. This observation has led to the conclusion that the human visual system may indeed, as does the model, use 2D stored views with GRBF interpolation to recognize 3D objects [2]. Such a conclusion, however, overlooks implementational constraints in the model, because the model's performance also depends on its implementations. Conceivably, a model with some 3D information of the objects can also mimic human performance, so long as it is appropriately implemented. There are typically too many possible models that can produce the same pattern of results. In contrast, an ideal observer computes the optimal performance that is only limited by the stimulus information and the task. We can define constrained ideals that are also limited by explicitly specified assumptions (e.g., a class of matching operations). Such a model observer therefore yields the best possible performance among the class of models with the same stimulus input and assumptions. In this paper, we are particularly interested in constrained ideal observers that are restricted in functionally Significant aspects (e.g., a 2D ideal observer that stores independent 2D templates and has access only to 2D affine transformations). The key idea is that a constrained ideal observer is the best in its class. So if humans outperform this ideal observer, they must have used more than what is available to the ideal. The conclusion that follows is strong: not only does the constrained ideal fail to account for human performance, but the whole class of its implementations are also falsified. A crucial question in object recognition is the extent to which human observers model the geometric variation in images due to the projection of a 3D object onto a 2D image. At one extreme, we have shown that any model that compares the image to independent views (even if we allow for 2D rigid transformations of the input image) is insufficient to account for human performance. At the other extreme, it is unlikely that variation is modeled in terms of rigid transformation of a 3D object 2D Observers/or Hwnan 3D Object Recognition? 831 template in memory. A possible intermediate solution is to match the input image to stored views, subject to 2D affine deformations. This is reasonable because 2D affine transformations approximate 3D variation over a limited range of viewpoint change. In this study, we test whether any model limited to the independent comparison of 2D views, but with 2D affine flexibility, is sufficient to account for viewpoint dependence in human recognition. In the following section, we first define our experimental task, in which the computational models yield the provably best possible performance under their specified conditions. We then review the 2D ideal observer and GRBF model derived in [4], and the 2D affine nearest neighbor model in [8]. Our principal theoretical result is a closed-form solution of a Bayesian 2D affine ideal observer. We then compare human performance with the 2D affine ideal model, as well as the other three models. In particular, if humans can classify novel views of an object better than the 2D affine ideal, then our human observers must have used more information than that embodied by that ideal. 2 The observers Let us first define the task. An observer looks at the 2D images of a 3D wire frame object from a number of viewpoints. These images will be called templates {Td. Then two distorted copies of the original 3D object are displayed. They are obtained by adding 3D Gaussian positional noise (i.i.d.) to the vertices of the original object. One distorted object is called the target, whose Gaussian noise has a constant variance. The other is the distract or , whose noise has a larger variance that can be adjusted to achieve a criterion level of performance. The two objects are displayed from the same viewpoint in parallel projection, which is either from one of the template views, or a novel view due to 3D rotation. The task is to choose the one that is more similar to the original object. The observer's performance is measured by the variance (threshold) that gives rise to 75% correct performance. The optimal strategy is to choose the stimulus S with a larger probability p (OIS). From Bayes' rule, this is to choose the larger of p (SIO). Assume that the models are restricted to 2D transformations of the image, and cannot reconstruct the 3D structure of the object from its independent templates {Ti}. Assume also that the prior probability p(Td is constant. Let us represent S and Ti by their (x, y) vertex coordinates: (X Y )T, where X = (Xl, x2, ... , xn), y = (yl, y2 , ... , yn). We assume that the correspondence between S and T i is solved up to a reflection ambiguity, which is equivalent to an additional template: Ti = (xr yr )T, where Xr = (xn, ... ,x2,xl ), yr = (yn, ... ,y2,yl). We still denote the template set as {Td. Therefore, (1) In what follows, we will compute p(SITi)p(Ti ), with the assumption that S = F (Ti) + N (0, crI2n), where N is the Gaussian distribution, 12n the 2n x 2n identity matrix, and :F a 2D transformation. For the 2D ideal observer, :F is a rigid 2D rotation. For the GRBF model, F assigns a linear coefficient to each template T i , in addition to a 2D rotation. For the 2D affine nearest neighbor model, :F represents the 2D affine transformation that minimizes liS - Ti11 2 , after Sand Ti are normalized in size. For the 2D affine ideal observer, :F represents all possible 2D affine transformations applicable to T i. 832 Z Liu and D. Kersten 2.1 The 2D ideal observer The templates are the original 2D images, their mirror reflections, and 2D rotations (in angle ¢) in the image plane. Assume that the stimulus S is generated by adding Gaussian noise to a template, the probability p(SIO) is an integration over all templates and their reflections and rotations. The detailed derivation for the 2D ideal and the GRBF model can be found in [4]. Ep(SITi)p(Ti) ex: E J d¢exp (-liS - Ti(¢)112 /2(2 ) • (2) 2.2 The GRBF model The model has the same template set as the 2D ideal observer does. Its training requires that EiJ;7r d¢Ci(¢)N(IITj - Ti(¢)II,a) = 1, j = 1,2, ... , with which {cd can be obtained optimally using singular value decomposition. When a pair of new stimuli is} are presented, the optimal decision is to choose the one that is closer to the learned prototype, in other words, the one with a smaller value of 111- E 127r d¢ci(¢)exp (_liS -2:~(¢)1I2) II. (3) 2.3 The 2D affine nearest neighbor model It has been proved in [8] that the smallest Euclidean distance D(S, T) between S and T is, when T is allowed a 2D affine transformation, S ~ S/IISII, T ~ T/IITII, D2(S, T) = 1 - tr(S+S . TTT)/IITII2, (4) where tr strands for trace, and S+ = ST(SST)-l. The optimal strategy, therefore, is to choose the S that gives rise to the larger of E exp (_D2(S, Ti)/2a2) , or the smaller of ED2(S, Ti). (Since no probability is defined in this model, both measures will be used and the results from the better one will be reported.) 2.4 The 2D affine ideal observer We now calculate the Bayesian probability by assuming that the prior probability distribution of the 2D affine transformation, which is applied to the template T i, AT + Tr = (~ ~) Ti + (~: ::: ~:), obeys a Gaussian distribution N(Xo,,,,/16), where Xo is the identity transformation xl' = (a,b,c,d,tx,ty) = (1,0,0,1,0,0). We have Ep(SITi ) = E i: dX exp (-IIATi + Tr - SII2/2(2) (5) = EC(n, a, ",/)deC 1 (QD exp (tr (KfQi(QD-1QiKi) /2(12), (6) where C(n, a, ",/) is a function of n, a, "'/; Q' = Q + ",/-212, and Q _ ( XT . XT XT · Y T ) QK _ ( XT· Xs Y T . Xs) -21 YT ·XT YT ·YT ' XT ·Ys YT .Ys +"'/ 2· (7) The free parameters are "'/ and the number of 2D rotated copies for each T i (since a 2D affine transformation implicitly includes 2D rotations, and since a specific prior probability distribution N(Xo, ",/1) is assumed, both free parameters should be explored together to search for the optimal results). 2D Observers for Hwnan 3D Object Recognition? 833 • • • • • • Figure 1: Stimulus classes with increasing structural regularity: Balls, Irregular, Symmetric, and V-Shaped. There were three objects in each class in the experiment. 2.5 The human observers Three naive subjects were tested with four classes of objects: Balls, Irregular, Symmetric, and V-Shaped (Fig. 1). There were three objects in each class. For each object, 11 template views were learned by rotating the object 60° /step, around the X- and Y-axis, respectively. The 2D images were generated by orthographic projection, and viewed monocularly. The viewing distance was 1.5 m. During the test, the standard deviation of the Gaussian noise added to the target object was (J"t = 0.254 cm. No feedback was provided. Because the image information available to the humans was more than what was available to the models (shading and occlusion in addition to the (x, y) positions of the vertices), both learned and novel views were tested in a randomly interleaved fashion. Therefore, the strategy that humans used in the task for the learned and novel views should be the same. The number of self-occlusions, which in principle provided relative depth information, was counted and was about equal in both learned and novel view conditions. The shading information was also likely to be equal for the learned and novel views. Therefore, this additional information was about equal for the learned and novel views, and should not affect the comparison of the performance (humans relative to a model) between learned and novel views. We predict that if the humans used a 2D affine strategy, then their performance relative to the 2D affine ideal observer should not be higher for the novel views than for the learned views. One reason to use the four classes of objects with increasing structural regularity is that structural regularity is a 3D property (e.g., 3D Symmetric vs. Irregular), which the 2D models cannot capture. The exception is the planar V-Shaped objects, for which the 2D affine models completely capture 3D rotations, and are therefore the "correct" models. The V-Shaped objects were used in the 2D affine case as a benchmark. If human performance increases with increasing structural regularity of the objects, this would lend support to the hypothesis that humans have used 3D information in the task. 2.6 Measuring performance A stair-case procedure [7] was used to track the observers' performance at 75% correct level for the learned and novel views, respectively. There were 120 trials for the humans, and 2000 trials for each of the models. For the GRBF model, the standard deviation of the Gaussian function was also sampled to search for the best result for the novel views for each of the 12 objects, and the result for the learned views was obtained accordingly. This resulted in a conservative test of the hypothesis of a GRBF model for human vision for the following reasons: (1) Since no feedback was provided in the human experiment and the learned and novel views were randomly intermixed, it is not straightforward for the model to find the best standard deviation for the novel views, particularly because the best standard deviation for the novel views was not the same as that for the learned 834 Z Liu and D. Kersten ones. The performance for the novel views is therefore the upper limit of the model's performance. (2) The subjects' performance relative to the model will be defined as statistical efficiency (see below). The above method will yield the lowest possible efficiency for the novel views, and a higher efficiency for the learned views, since the best standard deviation for the novel views is different from that for the learned views. Because our hypothesis depends on a higher statistical efficiency for the novel views than for the learned views, this method will make such a putative difference even smaller. Likewise, for the 2D affine ideal, the number of 2D rotated copies of each template Ti and the value I were both extensively sampled, and the best performance for the novel views was selected accordingly. The result for the learned views corresponding to the same parameters was selected. This choice also makes it a conservative hypothesis test. 3 Results Learned Views 25 • Human IJ 20 Ideal eO GRBF O 20 Affine Nearest NtMghbor .£. rn 20 Affine kIoai :g 0 ~ 1.5 81 l! l0.5 Object Type e.£. :!2 0 ~ 1.5 ~ ~ INovel Views • Human EJ 20 Ideal o GRBF o 20 Affine Nearesl N.tghbor ~ 2DAfllna~ Object Type Figure 2: The threshold standard deviation of the Gaussian noise, added to the distractor in the test pair, that keeps an observer's performance at the 75% correct level, for the learned and novel views, respectively. The dotted line is the standard deviation of the Gaussian noise added to the target in the test pair. Fig. 2 shows the threshold performance. We use statistical efficiency E to compare human to model performance. E is defined as the information used by humans relative to the ideal observer [3] : E = (d~uman/d~deal)2, where d' is the discrimination index. We have shown in [4] that, in our task, E = ((a~1!f;actor)2 - (CTtarget)2) / ((CT~~~~~tor)2 - (CTtarget)2) , where CT is the threshold. Fig. 3 shows the statistical efficiency of the human observers relative to each of the four models. We note in Fig. 3 that the efficiency for the novel views is higher than those for the learned views (several of them even exceeded 100%), except for the planar V-Shaped objects. We are particularly interested in the Irregular and Symmetric objects in the 2D affine ideal case, in which the pairwise comparison between the learned and novel views across the six objects and three observers yielded a significant difference (binomial, p < 0.05). This suggests that the 2D affine ideal observer cannot account for the human performance, because if the humans used a 2D affine template matching strategy, their relative performance for the novel views cannot be better than for the learned views. We suggest therefore that 3D information was used by the human observers (e.g., 3D symmetry). This is supported in addition by the increasing efficiencies as the structural regularity increased from the Balls, Irregular, to Symmetric objects (except for the V-Shaped objects with 2D affine models). 2D Observers for Hwnan 3D Object Recognition? 835 300 "" l 300 >300 20 Ideal GRBF Modol j 20 Aftlne Nearest Ighbor l 20 Affine Ideal l 250 250 250 o Learned I 0 l&arnedl ~ o Learned ,.. 250 l " o Learned .. • Novel .Noval • Novel j • Novel " 200 '" ~ 200 200 .. f " ..! '50 t ~ " "" 150 '50 $: ~ j w i "- .'" ---------------II! " '" I Q ! N ~ 0 ObJect Type Q Object Type ObjOGtType ObjoctTypo N Figure 3: Statistical efficiencies of human observers relative to the 2D ideal observer, the GRBF model, the 2D affine nearest neighbor model, and the 2D affine ideal observer_ 4 Conclusions Computational models of visual cognition are subject to information theoretic as well as implementational constraints. When a model's performance mimics that of human observers, it is difficult to interpret which aspects of the model characterize the human visual system. For example, human object recognition could be simulated by both a GRBF model and a model with partial 3D information of the object. The approach we advocate here is that, instead of trying to mimic human performance by a computational model, one designs an implementation-free model for a specific recognition task that yields the best possible performance under explicitly specified computational constraints. This model provides a well-defined benchmark for performance, and if human observers outperform it, we can conclude firmly that the humans must have used better computational strategies than the model. We showed that models of independent 2D templates with 2D linear operations cannot account for human performance. This suggests that our human observers may have used the templates to reconstruct a representation of the object with some (possibly crude) 3D structural information. References [1] Biederman I and Gerhardstein P C. Viewpoint dependent mechanisms in visual object recognition: a critical analysis. J. Exp. Psych.: HPP, 21: 1506-1514, 1995. [2] Biilthoff H H and Edelman S. Psychophysical support for a 2D view interpolation theory of object recognition. Proc. Natl. Acad. Sci., 89:60-64, 1992. [3] Fisher R A. Statistical Methods for Research Workers. Oliver and Boyd, Edinburgh, 1925. [4] Liu Z, Knill D C, and Kersten D. Object classification for human and ideal observers. Vision Research, 35:549-568, 1995. [5] Poggio T and Edelman S. A network that learns to recognize three-dimensional objects. Nature, 343:263-266, 1990. [6] Tarr M J and Biilthoff H H. Is human object recognition better described by geon-structural-descriptions or by multiple-views? J. Exp. Psych.: HPP, 21:1494-1505,1995. [7] Watson A B and Pelli D G. QUEST: A Bayesian adaptive psychometric method. Perception and Psychophysics, 33:113-120, 1983. [8] Werman M and Weinshall D. Similarity and affine invariant distances between 2D point sets. IEEE PAMI, 17:810-814,1995.
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A Neural Net w ork Based Head T rac king System D. D. Lee and H. S. Seung Bell Lab oratories, Lucen t T ec hnologies 00 Moun tain Av e. Murra y Hill, NJ 0 fddlee|seungg@b el l-l ab s.c om Abstract W e ha v e constructed an inexp ensiv e, video-based, motorized trac king system that learns to trac k a head. It uses real time graphical user inputs or an auxiliary infrared detector as sup ervisory signals to train a con v olutional neural net w ork. The inputs to the neural net w ork consist of normalized luminance and c hrominance images and motion information from frame di erences. Subsampled images are also used to pro vide scale in v ariance. During the online training phase, the neural net w ork rapidly adjusts the input w eigh ts dep ending up on the reliabilit y of the di eren t c hannels in the surrounding en vironmen t. This quic k adaptation allo ws the system to robustly trac k a head ev en when other ob jects are mo ving within a cluttered bac kground. In tro duction With the proliferation of inexp ensiv e m ultimedia computers and p eripheral equipmen t, video conferencing nally app ears ready to en ter the mainstream. But p ersonal video conferencing systems t ypically use a stationary camera, t ying the user to a xed lo cation m uc h as a corded telephone tethers one to the telephone jac k. A simple solution to this problem is to use a motorized video camera that can trac k a sp eci c p erson as he or she mo v es ab out. Ho w ev er, this presen ts the dicult y of ha ving to con tin ually con trol the mo v emen ts of the camera while one is comm unicating. In this pap er, w e presen t a protot yp e, neural net w ork based system that learns the c haracteristics of a p erson's head in real time and automatically trac ks it around the ro om, th us alleviating the user of m uc h of this burden. The camera mo v emen ts in this video conferencing system closely resem ble the mo v emen ts of h uman ey es. The task of the biological o culomotor system is to direct PC Color CCD Camera (Eye) Directional Microphones (Ears) Servo Motors (Oculomotor Muscles) IR Detector Frame Serial Sound Grabber Port Card Reinforcement Signals GUI Mouse Figure : Sc hematic hardw are diagram of Marvin, our head trac king system. \in teresting" parts of the visual w orld on to the small, high resolution areas of the retinas. F or this task, complex neural circuits ha v e ev olv ed in order to con trol the ey e mo v emen ts. Some examples include the saccadic and smo oth pursuit systems that allo w the ey es to rapidly acquire and trac k mo ving ob jects [, ]. Similarly , an activ e video conferencing system also needs to determine the appropriate face or feature to follo w in the video stream. Then the camera m ust trac k that p erson's mo v emen ts o v er time and transmit the image to the other part y . In the past few y ears, the problem of face detection in images and video has attracted considerable atten tion [ , , ]. Rule-based metho ds ha v e concen trated on lo oking for generic c haracteristics of faces suc h as o v al shap es or skin h ue. Since these t yp es of algorithms are fairly simple to implemen t, they are commonly found in real-time systems [ , ]. But b ecause other ob jects ha v e similar shap es and colors as faces, these systems can also b e easily fo oled. A p oten tially more robust approac h is to use a con v olutional neural net w ork to learn the appropriate features of a face [ , ]. Because most suc h implemen tations learn in batc h mo de, they are b eset b y the dicult y of constructing a large enough training set of lab elled images with and without faces. In this pap er, w e presen t a video based system that uses online sup ervisory signals to train a con v olutional neural net w ork. F ast online adaptation of the net w ork's w eigh ts allo ws the neural net w ork to learn ho w to discriminate an individual head at the b eginning of a session. This enables the system to robustly trac k the head ev en in the presence of other mo ving ob jects. Hardw are Implemen tation Figure sho ws a sc hematic of the trac king system w e ha v e constructed and ha v e named \Marvin" b ecause of an early v ersion's similarit y to a carto on c haracter. Marvin's ey e consists of a small CCD camera with a eld of view that is attac hed to a motorized platform. Tw o R C serv o motors giv e Marvin the abilit y to rapidly pan and tilt o v er a wide range of viewing angles, with a t ypical maxim um v elo cit y of 00 deg/sec. The system also includes t w o microphones or ears that giv e Marvin the abilit y to lo cate auditory cues. In tegrating auditory information with visual inputs allo ws the system to nd salien t ob jects b etter than with either sound or video alone. But these pro ceedings will fo cus exclusiv ely on ho w a visual represen tation is learned. RGB Images Y U V D Figure : Prepro cessing of the video stream. Luminance, c hromatic and motion information are separately represen ted in the Y, U, V, D c hannels at m ultiple resolutions. Marvin is able to learn to trac k a visual target using t w o di eren t sources of sup ervisory signals. One metho d of training uses a small KHz mo dulated infrared ligh t emitter ( 00 nm ) attac hed to the ob ject that needs to b e trac k ed. A heat lter renders the infrared ligh t in visible to Marvin's video camera so that the system do es not merely learn to follo w this signal. But moun ted next to the CCD camera and mo ving with it is a small infrared detector with a collimating lens that signals when the ob ject is lo cated within a narro w angular cone in the direction that the camera is p oin ting. This reinforcemen t signal can then b e used to train the w eigh ts of the neural net w ork. Another more natural w a y for the system to learn o ccurs in an actual video conferencing scenario. In this situation, a user who is activ ely w atc hing the video stream has man ual o v erride con trol of the camera using graphical user in terface inputs. Whenev er the user rep ositions the camera to a new lo cation, the neural net w ork w ould then adjust its w eigh ts to trac k whatev er is in the cen ter p ortion of the image. Since Marvin w as built from readily a v ailable commercial comp onen ts, the cost of the system not including the PC w as under $00. The input devices and motors are all con trolled b y the computer using custom-written Matlab driv ers that are a v ailable for b oth Microsoft Windo ws and the Lin ux op erating system. The image pro cessing computations as w ell as the graphical user in terface are then easily implemen ted as simple Matlab op erations and function calls. The follo wing section describ es the head trac king neural net w ork in more detail. Neural Net w ork Arc hitecture Marvin uses a con v olutional neural net w ork arc hitecture to detect a head within its eld of view. The video stream from the CCD camera is rst digitized with a video capture b oard in to a series of ra w 0 0 R GB images as sho wn in Figure . Eac h R GB color image is then con v erted in to its YUV represen tation, and a di erence (D) Y U V D Hidden Units Saliency Map WU WY WV WD Winner Take All Figure : Neural net w ork uses a con v olutional arc hitecture to in tegrate the di eren t sources of information and determine the maximally salien t ob ject. image is also computed as the absolute v alue of the di erence from the preceding frame. Of the four resulting images, the Y comp onen t represen ts the luminance or gra yscale information while the U and V c hannels con tain the c hromatic or color information. Motion information in the video stream is captured b y the D image where mo ving ob jects app ear highligh ted. The four YUVD c hannels are then subsampled successiv ely to yield represen tations at lo w er and lo w er resolutions. The resulting \image p yramids" allo w the net w ork to ac hiev e recognition in v ariance across man y di eren t scales without ha ving to train separate neural net w orks for eac h resolution. Instead, a single neural net w ork with the same set w eigh ts is run with the di eren t resolutions as inputs, and the maximally activ e resolution and p osition is selected. Marvin uses the con v olutional neural net w ork arc hitecture sho wn in Figure to lo cate salien t ob jects at the di eren t resolutions. The YUVD input images are ltered with separate k ernels, denoted b y W Y , W U , W V , and W D resp ectiv ely . This results in the ltered images Y s , U s , V s , D s : A s (i; j ) = W A A s = X i 0 ;j 0 W A (i 0 ; j 0 ) A s (i + i 0 ; j + j 0 ) () where s denotes the scale resolution of the inputs, and A is an y of the Y , U , V , or D c hannels. These ltered images represen t a single la y er of hidden units in the neural net w ork. These hidden units are then com bined to form the saliency map X s in the follo wing manner: X s (i; j ) = c Y g [ Y s (i; j )] + c U g [ U s (i; j )] + c V g [ V s (i; j )] + c D g [ D s (i; j )] + c 0 : () Since g (x) = tanh(x) is sigmoidal, the saliency X s is computed as a nonlinear, pixel-b y-pixel com bination of the hidden units. The scalar v ariables c Y , c U , c V , and c D represen t the relativ e imp ortance of the di eren t luminance, c hromatic, and motion c hannels in the o v erall saliency of an ob ject. With the bias term c 0 , the function g [X s (i; j )] ma y then b e though t of as the relativ e probabilit y that a head exists at lo cation (i; j ) at input resolution s. The nal output of the neural net w ork is then determined in a comp etitiv e manner b y nding the lo cation (i m ; j m ) and scale s m of the b est p ossible matc h: g [X m ] = g [X s m (i m ; j m )] = max i;j;s g [X s (i; j )]: () After pro cessing the visual inputs in this manner, saccadic camera mo v emen ts are generated in order to k eep the maximally salien t ob ject lo cated near the cen ter of the eld of view. T raining and Results Either GUI user inputs or the infrared detector ma y b e used as a sup ervisory signal to train the k ernels W A and scalar w eigh ts c A of the neural net w ork. The neural net w ork is up dated when the maximally salien t lo cation of the neural net w ork (i m ; j m ) do es not corresp ond to the desired ob ject's true p osition (i n ; j n ) as identi ed b y the external sup ervisory signal. A cost function prop ortional to the sum squared error terms at the maximal lo cation and new desired lo cation is used for training: e m = jg m g [X s m (i m ; j m )j ; () e n = min s jg n g [X s (i n ; j n )j : () In the follo wing examples, the constan ts g m = 0 and g n = are used. The gradien ts to Eqs. { are then bac kpropagated through the con v olutional net w ork [ , 0 ], resulting in the follo wing up date rules: c A = e m g 0 (X m )g [ A(i m ; j m )] + e n g 0 (X n )g [ A (i n ; j n )]; () W A = e m g 0 (X m )g 0 ( A m )c A A m + e n g 0 (X n )g 0 ( A n )c A A n : () In t ypical batc h learning applications of neural net w orks, the learning rate is set to b e some small p ositiv e n um b er. Ho w ev er in this case, it is desirable for Marvin to learn to trac k a head in a new en vironmen t as quic kly as p ossible. Th us, rapid adaptation of the w eigh ts during ev en a single training example is needed. A natural w a y of doing this is to use a fairly large learning rate ( = 0:), and to rep eatedly apply the up date rules in Eqs. { un til the calculated maximally salien t lo cation is v ery close to the actual desired p osition. An example of ho w quic kly Marvin is able to learn to trac k one of the authors as he mo v ed around his oce is giv en b y the learning curv e in Figure . The w eigh ts w ere rst initialized to small random v alues, and Marvin w as corrected in an online fashion using mouse inputs to lo ok at the author's head. After only a few seconds of training with a pro cessing time lo op of around 00 ms, the system w as able to lo cate the head to within four pixels of accuracy , as determined b y hand lab elling the video data afterw ards. As saccadic ey e mo v emen ts w ere initiated at 0 10 20 30 40 50 0 2 4 6 8 10 12 14 16 18 20 Frame Number Pixel Error Figure : F ast online adaptation of the neural net w ork. The head lo cation error in pixels in a 0 0 image is plotted as a function of frame n um b er ( frames/sec). the times indicated b y the arro ws in Fig. , new en vironmen ts of the oce w ere sampled and an o ccasional large error is seen. Ho w ev er, o v er time as these errors are corrected, the neural net w ork learns to robustly discriminate the head from the oce surroundings. Discussion Figure sho ws the inputs and w eigh ts of the net w ork after a min ute of training as the author w alk ed around his oce. The k ernels necessarily app ear a little smeared b ecause they are in v arian t to sligh t c hanges in head p osition, rotation, and scale. But they clearly depict the dark hair, facial features, and skin color of the head. The relativ e w eigh ting (c Y ; c U ; c V > c D ) of the di eren t input c hannels sho ws that the luminance and color information are the most reliable for trac king the head. This is probably b ecause it is relativ ely dicult to distinguish in the frame di erence images the head from other mo ving b o dy parts. W e are curren tly considering more complicated neural net w ork arc hitectures for com bining the di eren t input streams to giv e b etter trac king p erformance. Ho wev er, this example sho ws ho w a simple con v olutional arc hitecture can b e used to automatically in tegrate di eren t visual cues to robustly trac k a head. Moreo v er, b y using fast online adaptation of the neural net w ork w eigh ts, the system is able to learn without needing large hand-lab elled training sets and is also able to rapidly accomo date c hanging en vironmen ts. F uture impro v emen ts in hardw are and neural net w ork arc hitectures and algorithms are still necessary , ho w ev er, in order to approac h h uman sp eeds and p erformance in this t yp e of sensory pro cessing and recognition task. W e ac kno wledge the supp ort of Bell Lab oratories, Lucen t T ec hnologies. W e also thank M. F ee, A. Jacquin, S. Levinson, E. P eta jan, G. Pingali, and E. Rietman for helpful discussions. Y U V D c =0.15 Y c =0.12 U c =0.11 V c =0.08 D Figure : Example sho wing the inputs and w eigh ts used in trac king a head. The head p osition as calculated b y the neural net w ork is mark ed with a b o x. References [] Horiuc hi, TK, Bishofb erger, B & Ko c h, C ( ). An analog VLSI saccadic ey e mo v emen t system. A dvanc es in Neur al Information Pr o c essing Systems , { . [] Rao, RPN, Zelinsky , GJ, Ha yho e, MM & Ballard, DH ( ). Mo deling saccadic targeting in visual searc h. A dvanc es in Neur al Information Pr o c essing Systems , 0{. [] Sung, KK & P oggio, T ( ). Example-based learning for view-based h uman face detection. Pr o c. r d Image Understanding Workshop, {0. [] Eleftheriadis, A & Jacquin, A ( ). Automatic face lo cation detection and trac king for mo del-assisted co ding of video teleconferencing sequences at lo w bit-rates. Signal Pr o c essing: Image Communic ation , . [] P eta jan, E & Graf, HP ( ). Robust face feature analysis for automatic sp eec hreading and c haracter animation. Pr o c. nd Int. Conf. A utomatic F ac e and Gestur e R e c o gnition, -. [] Darrell, T, Maes, P , Blum b erg, B, & P en tland, AP ( ). A no v el en vironmen t for situated vision and b eha vior. Pr o c. IEEE Workshop for Visual Behaviors, {. [] Y ang, J & W aib el, A ( ). A real-time face trac k er. Pr o c. r d IEEE Workshop on Applic ation of Computer Vision, {. [] No wlan, SJ & Platt, JC ( ). A con v olutional neural net w ork hand trac k er. A dvanc es in Neur al Information Pr o c essing Systems , 0{ 0. [ ] Ro wley , HA, Baluja, S & Kanade, T ( ). Human face detection in visual scenes. A dvanc es in Neur al Information Pr o c essing Systems , {. [0] Le Cun, Y, et al. ( 0). Handwritten digit recognition with a bac k propagation net w ork. A dvanc es in Neur al Information Pr o c essing Systems , {0.
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Perturbative M-Sequences for Auditory Systems Identification Mark Kvale and Christoph E. Schreiner· Sloan Center for Theoretical Neurobiology, Box 0444 University of California, San Francisco 513 Parnassus Ave, San Francisco, CA 94143 Abstract In this paper we present a new method for studying auditory systems based on m-sequences. The method allows us to perturbatively study the linear response of the system in the presence of various other stimuli, such as speech or sinusoidal modulations. This allows one to construct linear kernels (receptive fields) at the same time that other stimuli are being presented. Using the method we calculate the modulation transfer function of single units in the inferior colli cui us of the cat at different operating points and discuss nonlinearities in the response. 1 Introduction A popular approach to systems identification, i.e., identifying an accurate analytical model for the system behavior, is to use Volterra or Wiener expansions to model behavior via functional Taylor or orthogonal polynomial series, respectively [Marmarelis and Marmarelis1978]. Both approaches model the response r(t) as a linear combination of small powers of the stimulus s(t). Although effective for mild nonlinearities, deriving the linear combinations becomes numerically unstable for highly nonlinear systems. A more serious problem is that many biological systems are adaptive, i.e., the system behavior is dependent on the stimulus ensemble. For instance, [Rieke et al.1995] found that in the auditory nerve of the bullfrog linearity and information rates depended sensitively on whether a white noise or naturalistic ensemble is used. One approach to handling these difficulties is to forgo the full expansion, and simply compute the linear response to small (perturbative) stimuli in the presence of various different ensembles, or operating points. By collecting linear responses • Email: kvale@phy.ucsf.edu and chris@phy.ucsf.edu Perturbative M-Sequences for Auditory Systems Identification 181 from different operating points, one may fit nonlinear responses as one fits a nonlinear function with a piecewise linear approximation. For adaptive systems the same procedure would be applied, with different operating points corresponding to different points along the time axis. Perturbative stimuli have wide application in condensed-matter physics, where they are used to characterize linear responses such as resistance, elasticity and viscosity, and in engineering, perturbative analyses are used in circuit analysis (small signal models) and structural diagnostics (vibration analysis). In neurophysiology, however, perturbative stimuli are unknown. An effective stimulus for calculating the perturbative linear response of a system is the m-sequence. M-sequences have a long history of use in engineering and the physical sciences, with applications ranging from systems identification to cryptography and cellular communication. In physiology, m-sequences have been used primarily to compute system kernels [Marmarelis and Marmarelisl978], especially in the visual system [Pinter and Nabet1987]. In this work, we use perturbative msequences to study the linear response of single units in the inferior colli cui us of a cat to amplitude-modulated (AM) stimuli. We add a small m-sequence signal to an AM carrier, which allows us to study the linear behavior of the system near a particular operating point in a non-destructive manner, i.e., without changing the operating point. Perturbative m-sequences allow one to calculate linear responses near the particular stimuli under study with only a little extra effort, and allow us to characterize the system over a wide range of stimuli, such as sinusoidal AM and naturalistic stimuli. The auditory system we selected to study was the response of single units in the central nucleus of the inferior colliculus (IC) of an anaesthetised cat. Single unit responses were recorded extraceUularly. Action potentials were amplified and stored on DAT tape, and were discriminated offline using a commercial computer-based spike sorter (Brainwave). 20 units were recorded, of which 10 yielded sufficiently stable responses to be analyzed. 2 M-Sequences and Linear Systems A binary m-sequence is a two-level pseudo-random sequence of +1's and -1's. The sequence length is L = 2n - 1, where n is the order of the sequence. Typically, a binary m-sequence can be generated by a shift register with n bits and feedback connections derived from an irreducible polynomial over the multiplicative group Z2 [Golomb1982]. For linear systems identification, m-sequences have two important properties. The first is that m-sequences have nearly zero mean: ~~':OI m[t] = -l. The second is that the autocorrelation function takes on the impulse-like form L-l {L if T = 0 Smm(T) = ~ m[t]m[t + T] = -1 otherwise (1) Impulse stimuli also have a 8-function autocorrelation function. In the context of perturbative stimuli, the advantage of an m-sequence stimulus over an impulse stimulus is that for a given signal to noise ratio, an m-sequence perturbation stays much closer to the original signal (in the least squares sense) than an impulse perturbation. Thus the perturbed signal does not stray as far from the operating point and measurement of linear response about that operating point is more accurate. We model the IC response with a system F through which a scalar stimulus s(t) is passed to give a response r(t): r(t) = F[s(t)]. (2) 182 M. Kvale and C. E. Schreiner For the purposes of this section, the functional F is taken to be a linear functional plus a DC component. In real experiments, the input and output signal are sampled into discrete sequences with t becoming an integer indexing the sequence. Then the system can be written as the discrete convolution L-1 r[t] = ho + L h[ttls[t - t1] (3) with kernels ho and h[tt] to be determined. We assume that the system has a finite memory of M time steps (with perhaps a delay) so that at most M of the h[t] coefficients are nonzero. To determine the kernels perturbatively, we add a small amount of m-sequence to a base stimulus so: s[t] = so[t] + am[t]. (4) Cross-correlating the response with the original m-sequence yields L-l L-l L-l L- l L m[t]r[t + r] = L m[t]ho + L L h[ttlm[t]so[t + r - ttl t=o t=o t=o tl =0 L-1 L-l + L L ah[tdm[t]m[t + r - tl]' (5) t=o tl=O Using the sum formula for am -sequence above, the first sum in Eq. (5) can be simplified to -ho. Using the autocorrelation Eq. (1), the third sum in Eq. (5) simplifies, and we find L-l L-l L-l Rrm(r) = a(L + l)h[r] - ho - a L h[tl] + L L h[tt]m[t]so[t + r - ttl (6) tl =0 t=o tl =0 Although the values for the kernels h(t) are set implicitly by this equation, the terms on the right hand side of Eq. (6) are widely different in size for large Land the equation can be simplified. As is customary in auditory systems, we assume the DC response ho is small. To estimate the size of the other terms, we compute statistical estimates of their sizes and look at their scaling with the parameters. The term a L:~-==~ h[tt] is a sum of M kernel elements; they may be correlated or uncorrelated, so a conservative estimate of their size is on the order of O( aM). The last term in (6) is more subtle. We rewrite it as L-l L-l L L h[tdm[t]so[t + r - ttl = h=O t=o L-l L h[tt]p[r, ttl tl=O L-l L m[t]so[t + r - ttl t=o (7) The time series of the ambient stimulus so[t] and m-sequence m[t] are assumed to be uncorrelated. By the central limit theorem, the sum p[r, tl] will then have an average of zero with a standard deviation of 0(L1/ 2 ). If in turn, the terms p[r, ttl are un correlated with the kernels h[tl], we have that L-l L-l L L h[tt]m[t]so[t + r - ttl '" 0(MI/2 Ll/2) (8) tl=O t=o Perturbative M-Sequences for Auditory Systems Identification 183 If N cycles of the m-sequence are performed, in which sort] is different for each cycle, all the terms in Eq. (6) scale with N as O(N), except for the double sum. By the same central limits arguments above, the double sum scales as O(Nl/2). Putting all these results together into Eq. (6) and solving for the kernels yields h(r) a(L 1+ 1) Rrm(r) - 0 ( ~) + 0 (aN~~~1/2 ) . 1 M Ml/2 a(L + 1) Rrm(r) - C1 L + C2 aNI/2£1/2' (9) with the constants C1 , C2 '" O(h[r]) depending neural firing rate, statistics, etc., determined from experiment. If we take the kernel element h(r) to be the first term in Eq. 9, then the last two terms in Eq. (9) contribute errors in determining the kernel and can be thought of as noise. Both error terms vanish as L -+ 00 and the procedure is asymptotically exact for arbitrary uncorrelated stimuli sort]. In order for the cross-correlation Ram (r) to yield a good estimate, the inequalities (10) must hold. In practice, the kernel memory is much smaller than the sequence length, and the second inequality is the stricter bound. The second inequality represents a tradeoff among sequence length, number of trials and the size of the perturbation for a given level of systematic noise in the kernel estimate. For instance, if L = 215 - 1, N = 10, M = 30, and noise floor at 10%, the perturbation should be larger than a = 0.095. If no signal sort] is present, then the O(Ml/2a-1(NL)-1/2) term drops out and the usual m-sequence cross-correlation result is recovered. 3 M-Sequences for Modulation Response Previous work, e.g., [M011er and Rees1986, Langner and Schreinerl988] has shown that many of the cells in the inferior colliculus are tuned not only to a characteristic frequency, but are also tuned to a best frequency of modulation of the carrier. A highly simplified model of the IC unit response to sound stimuli is the Ll- N - L2 cascade filter, with L1 a linear tank circuit with a transfer function matching that of the frequency tuning curve, N a nonlinear rectifying unit, and L2 a linear circuit with a transfer function matching that of the modulation transfer function. Detecting this modulation is an inherently nonlinear operation and N is not well approximated by a linear kernel. Thus IC modulation responses will not be well characterized by ordinary m-sequence stimuli using the methods described in Section 2. A better approach is to bypass the Ll - N demodulation step entirely and concentrate on measuring L2. This can be accomplished by creating a modulation m-sequence: s[t] = a (so[t] + bm[t]) sin[wet], (11) where Iso[t]1 :::; 1 is the ambient signal, i.e., the operating point, m[t] E [-1,1] is an m-sequence added with amplitude b, and We is the carrier frequency. Demodulation gives the effective input stimulus sm[t] = a (so[t] + bm[t]) . (12) Note that there is little physiological evidence for a purely linear rectifier N. In fact, both the work of [M011er and Rees1986, Rees and M011er1987] and ours below show that there is a nonlinear modulation response. Taking a modulation transfer 184 M. Kvale and C. E. Schreiner function seriously, however, implies that one assumes that modulation response is linear, which implies that the static nonlinearity used is something like a halfwave rectifier. Linearity is used here as a convenient assumption for organizing the stimulus and asking whether nonlinearities exist. For full m-sequence modulation (so[t] = 1 and b = 1) the stimulus Sm and the neural response can be used to compute, via the Lee--Schetzen cross-correlation, the modulation transfer function for the L2 system. Alternatively, for b « 1, the m-sequence is a perturbation on the underlying modulation envelope sort]. The derivation above shows that the linear modulation kernel can also be calculated using a Lee--Schetzen cross-correlation. M-sequences at full modulation depth were first used by [M0ller and Rees1986, Rees and M011erI987] to calculate white-noise kernels. Here, we are using m-sequence in a different way-we are calculating the small-signal properties around the stimulus sort]. The m-sequences used in this experiment were of length 215 -1 = 32,767. For each unit, 10 cycles of the m-sequence were presented back-to-back. After determining the characteristic frequency of a unit, stimuli were presented which never differed from the characteristic frequency by more than 500 Hz. Figure 1 depicts the sinusoidal and m-sequence components and their combined result. The stimuli were presented in random order so as to mitigate adaptation effects. Figure 1: A depiction of stimuli used in the experiment. The top graph shows a pure sine wave modulation at modulation depth 0.8. The middle graph shows an m-sequence modulation at depth 1.0. The bottom graph shows a perturbative m-sequence modulation at depth 0.2 added to a sinusoidal modulation at depth 0.8. 4 Results Figure 2 shows the spike rates for both the pure sinusoid and the combined sinusoid and m-sequence stimuli. Note that the rates are nearly the same, indicating that the perturbation did not have a large effect on the average response of the unit. The unit shows an adaptation in firing rate over the 10 trials, but we did not find Perturbative M-Sequences for Auditory Systems Identification 185 a statistically significant change in the kernels of different trials in any of the units. .-... o Q) 80.0 ~ C/) Q) ~ '5. ~ 60.0 Q) «i "40.0 G----e sinusoid ~ sinusoid + m-sequence 100.0 200.0 300.0 400.0 500.0 Time (sec) Figure 2: A plot of the unit firing rates for both the pure sinusoid and the sinusoid + m-sequence stimuli. The carrier frequency is 9 kHz and is close to the characteristic frequency of the neuron. The sinusoidal modulation has a frequency of 20 Hz and the m-sequence modulation has a frequency of 800 sec-I . Figure 3 shows modulation response kernels at several different values of the modulation depth. Note that if the system was a linear, superposition would cause all the kernels to be equivalent; in fact it is seen that the nonlinearities are of the same magnitude as the linear response. In this particular unit, the triphasic behavior at small modulation depths gives way to monophasic behavior at high modulation depths and an FFT of the kernel shows that the bandwidth of the modulation transfer function also broadens with increasing depth. 5 Discussion In this paper, we have introduced a new type of stimulus, perturbative m-sequences, for the study of auditory systems and derived their properties. We then applied perturbative m-sequences to the analysis of the modulation response of units in the Ie, and found the linear response at a few different operation point. We demonstrated that the nonlinear response in the presence of sinusoidal modulations are nearly as large as the linear response and thus a description of unit response with only an MTF is incomplete. We believe that perturbative stimuli can be an effective tool for the analysis of many systems whose units phase lock to a stimulus. The main limiting factor is the systematic noise discussed in section 2, but it is possible to trade off duration of measurement and size of the perturbation to achieve good results. The m-sequence stimuli also make it possible to derive higher order information [Sutter1987] and with a suitable noise floor, it may be possible to derive second-order kernels as well. This work was supported by The Sloan foundation and ONR grant number N0001494-1-0547. 186 M. Kvale and C. E. Schreiner Response vs. modulation depth sine wave @40Hz + pert. m-sequence 90.0 -0.2 -0.4 70.0 -0.6 -0.8 -1.0 50.0 CD "C 30.0 :::> "" a. E ns 10.0 -10.0 -30.0 -50.0 0.0 5.0 10.0 15.0 20.0 time from spike (milliseconds) Figure 3: A plot of the temporal kernels derived from perturbative m-sequence stimuli in conjunction with sinusoidal modulations at various modulation depth. The y-axis units are amplitude per spike and the x-axis is in milliseconds before the spike. References [Golomb1982] S. W. Golomb. Shift Register Sequences. Aegean Park Press, Laguna Hills, CA, 1982. [Langner and Schreiner1988] G. Langner and C. E. Schreiner. Periodicity coding in the inferior colliculus of the cat: 1. neuronal mechanisms. Journal of Neurophysiology, 60: 1799-1822, 1988. [Marmarelis and Marmarelis1978] Panos Z. Marmarelis and Vasilis Z. Marmarelis. Analysis of Physiological Systems. Plenum Press, New York, NY, 10011, 1978. [M011er and Rees1986] Aage R. M011er and Adrian Rees. Dynamic properties of single neurons in the inferior colliculus of the rat. Hearing Research, 24:203-215, 1986. [Pinter and Nabet1987] Robert B. Pinter and Bahram Nabet. Nonlinear Vision. CRC Press, Boca Raton, FL, 1987. [Rees and M011er1987] Adrian Rees and Aage R. M01ler. Stimulus properties influencing the responses of inferior colliculus neurons to amplitude-modulated sounds. Hearing Research, 27:129-143, 1987. [Rieke et al.1995] F. Rieke, D. A. Bodnar, and W. Bialek. Naturalistic stimuli increase the rate and efficiency of information transmission by primary auditory afi"erents. Proceedings of the Royal Society of London. Series B, 262:259-265, 1995. [Sutter1987] E. E. Sutter. A practical non-stochastic approach to nonlinear timedomain analysis. In Vasilis Z. Marmarelis, editor, Advanced Methods of Physiological Modeling, Vol. 1, pages 303-315. Biomedical Simulations Resource, University of Southern California, Los Angeles, CA 90089-1451, 1987.
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Bach in a Box - Real-Time Harmony Randall R. Spangler and Rodney M. Goodman* Computation and Neural Systems California Institute of Technology, 136-93 Pasadena, CA 91125 Jim Hawkinst 88B Milton Grove Stoke Newington, London N16 8QY, UK Abstract We describe a system for learning J. S. Bach's rules of musical harmony. These rules are learned from examples and are expressed as rule-based neural networks. The rules are then applied in realtime to generate new accompanying harmony for a live performer. Real-time functionality imposes constraints on the learning and harmonizing processes, including limitations on the types of information the system can use as input and the amount of processing the system can perform. We demonstrate algorithms for generating and refining musical rules from examples which meet these constraints. We describe a method for including a priori knowledge into the rules which yields significant performance gains. We then describe techniques for applying these rules to generate new music in real-time. We conclude the paper with an analysis of experimental results. 1 Introduction The goal of this research is the development of a system to learn musical rules from examples of J.S. Bach's music, and then to apply those rules in real-time to generate new music in a similar style. These algorithms would take as input a melody such *rspangle@micro.caltech.edu, rogo@micro.caltech.edu tjhawkins@cix.compulink.co.uk 958 R. R. Spangler; R. M. Goodman and J Hawkins I~II- JIJ Figure 1: Melody for Chorale #1 "Aus meines Herzens Grunde" Figure 2: J. S. Bach's Harmony For Chorale #1 as Figure 1 and produce a complete harmony such as Figure 2. Performance of this harmonization in real-time is a challenging problem. It also provides insight into the nature of composing music. We briefly review the representation of input data and the process of rule base generation. Then we focus on methods of increasing the performance of rule-based systems. Finally we present our data on learning the style of Bach. 1.1 Constraints Imposed by Real-Time Functionality A program which is to provide real-time harmony to accompany musicians at live performances faces two major constraints. First, the algorithms must be fast enough to generate accompaniment without detectable delay between the musician playing the melody and the algorithm generating the corresponding harmony. For musical instrument sounds with sharp attacks (plucked and percussive instruments, such as the harp or piano), delays of even a few tens of milliseconds between the start of the melody note and the start of the harmony notes are noticeable and distracting. This limits the complexity of the algorithm and the amount of information it can process for each timestep. Second, the algorithms must base their output only on information from previous timesteps. This differentiates our system from HARMONET (Hild, Feulnzer and Menzel, 1992) which required knowledge of the next note in the future before generating harmony for the current note. 1.2 Advantages of a Rule-Based Algorithm A rule-based neural network algorithm was chosen over a recurrent network or a non-linear feed-forward network. Neural networks have been previously used for harmonizing music with some success (Mozer, 1991)(Todd, 1989). However, rulebased algorithms have several advantages when dealing with music. Almost all music has some sort of rhythm and is tonal, meaning both pitch and duration of individual notes are quantized. This presents problems in the use of continuous networks, which must be overtrained to reasonably approximate discrete behavior. Bach in a Box-Real-Time Harmony 959 Rule-based systems are inherently discrete, and do not have this problem. Furthermore it is very difficult to determine why a non-linear multi-layer network makes a given decision or to extract the knowledge contained in such a network. However, it is straightforward to determine why a rule-based network produced a given result by examining the rules which fired. This aids development of the algorithm, since it is easier to determine where mistakes are being made. It allows comparison of the results to existing knowledge of music theory as shown below, and may provide insight into the theory of musical composition beyond that currently available. Rule-based neural networks can also be modified via segmentation to take advantage of additional a priori knowledge. 2 Background 2.1 Representation of Input Data The choice of input representation greatly affects the ability of a learning algorithm to generate meaningful rules. The learning and inferencing algorithms presented here speak an extended form of the classical figured bass representation common in Bach's time. Paired with a melody, figured bass provides a sufficient amount of information to reconstruct the harmonic content of a piece of music. Figured bass has several characteristics which make it well-disposed to learning rules. It is a symbolic format which uses a relatively small alphabet of symbols. It is also hierarchical - it specifies first the chord function that is to be played at the current note/timestep, then the scale step to be played by the bass voice, then additional information as needed to specify the alto and tenor scale steps. This allows our algorithm to fire sets of rules sequentially, to first determine the chord function which should be associated with a new melody note, and then to use that chord function as an input attribute to subsequent rulebases which determine the bass, alto, and tenor scale steps. In this way we can build up the final chord from simpler pieces, each governed by a specialized rulebase. 2.2 Generation of Rulebases Our algorithm was trained on a set of 100 harmonized Bach chorales. These were translated from MIDI format into our figured bass format by a preprocessing program which segmented them into chords at points where any voice changed pitch. Chord function was determined by simple table lookup in a table of 120 common Bach chords based on the scale steps played by each voice in the chord. The algorithm was given information on the current timestep (MelO-TeO), and the previous two timesteps (Mell-Func2). This produced a set of 7630 training examples, a subset of which are shown below: MelO FuncO 800 BaO AIO TeO Mell Funcl 801 Bal All Tel Me12 Func2 D V 82 Bl A2 TO E I 81 BO AO T2 C I E 17 81 B3 AO T2 D V 82 Bl A2 TO E I F IV 80 Bl A2 Tl E 17 81 B3 AO T2 D V G V 80 BO Al T2 F IV 80 Bl A2 Tl E 17 960 R. R. Spangler; R. M. Goodman and 1. Hawkins A rulebase is a collection of rules which predict the same right hand side (RHS) attribute (for example, FunctionO). All rules have the form IF Y=y... THEN X=x. A rule's order is the number of terms on its left hand side (LHS). Rules are generated from examples using a modified version of the ITRULE algorithm. (Goodman et al., 1992) All possible rules are considered and ranked by a measure of the information contained in each rule defined as J(X; Y = y) = p(y) [P(x1Y)log (p;~~~)) + (I - p(xly))log (11-!;~~~)) ] (1) This measure trades off the amount of information a rule contains against the probability of being able to use the rule. Rules are less valuable if they contains little information. Thus, the J-measure is low when p{xly) is not much higher than p(x). Rules are also less valuable if they fire only rarely (p(y) is small) since those rules are unlikely to be useful in generalizing to new data. A rulebase generated to predict the current chord's function might start with the following rules: 1. IF HelodyO 2. IF Function1 AND Helody1 AND HelodyO 3. IF Function1 AND HelodyO p(corr) J-meas E THEN FunctionO I 0.621 0.095 V THEN FunctionO V7 0.624 0.051 D D V THEN FunctionO V7 0.662 0.049 D 2.3 Inferencing Using Rulebases Rule based nets are a form of probabilistic graph model. When a rulebase is used to infer a value, each rule in the rule base is checked in order of decreasing rule J-measure. A rule can fire if it has not been inhibited and all the clauses on its LHS are true. When a rule fires, its weight is added to the weight of the value which it predicts, After all rules have had a chance to fire, the result is an array of weights for all predicted values. 2.4 Process of Harmonizing a Melody Input is received a note at a time as a musician plays a melody on a MIDI keyboard. The algorithm initially knows the current melody note and the data for the last two timesteps. The system first uses a rule base to determine the chord function which should be played for the current melody note. For example, given the melody note "e" , "it might playa chord function "IV", corresponding to an F -Major chord. The program then uses additional rulebases to specify how the chord will be voiced. In the example, the bass, alto, and tenor notes might be set to "BO", "AI", and "T2" , corresponding to the notes "F", "A", and "e". The harmony notes are then converted to MIDI data and sent to a synthesizer, which plays them in real-time to accompany the melody. Bach in a Box-Real-Time Harmony 961 3 Improvement of Rulebases The J-measure is a good measure for determining the information-theoretic worth of rules. However, it is unable to take into account any additional a priori knowledge about the nature of the problem - for example, that harmony rules which use the current melody note as input are more desirable because they avoid dissonance between the melody and harmony. 3.1 Segmentation A priori knowledge of this nature is incorporated by segmenting rulebases into moreand less-desirable rules based on the presence or absence of a desired LHS attribute such as the current melody note (MelodyO). Rules lacking the attribute are removed from the primary set of rules and placed in a second "fallback" set. Only in the event that no primary rules are able to fire is the secondary set allowed to fire. This gives greater impact to the primary rules (since they are used first) without the loss of domain size (since the less desirable rules are not actually deleted). Rulebase segmentation provides substantial improvements in the speed of the algorithm in addition to improving its inferencing ability. When an unsegmented rule base is fired, the algorithm has to compare the current input data with the LHS of every rule in the rulebase. However, processing for a segmented rulebase stops after the first segment which fires a rule on the input data. The algorithm does not need to spend time examining rules in lower-priority segments of that rulebase. This increase in efficiency allows segmented rule bases to contain more rules without impacting performance. The greater number of rules provides a richer and more robust knowledge base for generating harmony. 3.2 Realtime Dependency Pruning When rules are used to infer a value, the rules weights are summed to generate probabilities. This requires that all rules which are allowed to fire must be independent of one another. Otherwise, one good rule could be overwhelmed by the combined weight of twenty mediocre but virtually identical rules. To prevent this problem, each segment of a rulebase is analyzed to determine which rules are dependent with other rules in the same segment. Two rules are considered dependent if they fire together on more than half the training examples where either rule fires. For each rule, the algorithm maintains a list of lower rank rules which are dependent with the rule. This list is used in real-time dependency pruning. Whenever a rule fires on a given input, all rules dependent on it are inhibited for the duration of the input. This ensures that all rules which are able to fire for an input are independent. 3.3 Conflict Resolution When multiple rules fire and predict different values, an algorithm must be used to resolve the conflict. Simply picking the value with the highest weight, while most likely to be correct, leads to monotonous music since a given melody then always produces the same harmony. To provide a more varied harmony, our system exponentiates the accumulated rule 962 R. R. Spangler, R. M Goodman and J Hawkins Table 1: Rulebase Segments RHS REQUIRED LHS FOR SEGMENT RULES FunctionO MelodyO, Functionl, Function2 llO MelodyO,Functionl 380 MelodyO 346 SopranoO MelodyO, FunctionO 74 BassO FunctionO, SopranoO 125 (none) 182 AltoO SopranoO, BassO 267 (none) 533 TenorO SopranoO, BassO, AltoO. FunctionO 52 SopranoO, Bas80, AltoO 164 (none) 115 Table 2: Rulebase Performance RHS RULEBASE RULES AVG EVAL CORRECT FunctionO un8egmented 1825 1825 55% segmented 816 428 56% unsegmented # 2 428 428 50% SopranoO un8egmented 74 74 95% Bas80 unsegmented 307 307 70% 8egmented 307 162 70% unsegmented #2 162 162 65% AltoO un8egmented 800 800 63% segmented 800 275 63% unsegmented #2 275 275 59% TenorO un8egmented 331 331 73% segmented 331 180 74% unsegmented #2 180 180 67% weights for the possible outcomes to produce probabilities for each value, and the final outcome is chosen randomly based on those probabilities. It is because we use the accumulated rule weights to determine these probabilities that all rules which are allowed to fire must be independent of each other. If no rules at all fire, the system uses a first-order Bayes classifier to determine the RlIS value based on the current melody note. This ensures that the system will always return an outcome compatible with the melody. 4 Results Rulebases were generated for each attribute. Up to 2048 rules were kept in each rule base. Rules were retained if they were correct at least 30% of the time they fired, and had a J-measure greater than 0.001. The rulebases were then segmented. These rulebases were tested on 742 examples derived from 27 chorales not used in the training set. The number of examples correctly inferenced is shown for each rule base before and after segmentation. Also shown is the average number of rules evaluated per test example; the speed of inferencing is proportional to this number. To determine whether segmentation was in effect only removing lower J-measure rules, we removed low-order rules from the unsegmented rule bases until they had the same average number of rules evaluated as the segmented rule bases. In all cases, segmenting the rulebases reduced the average rules fired per example without lowering the accuracy of the rule bases (in some cases, segmentation even increased accuracy). Speed gains from segmentation ranged from 80% for TenorO up to 320% for FunctionO. In comparison, simply reducing the size of the unsegmented Bach in a Box-Real-Time Harmony 963 rulebase to match the speed of the segmented rulebase reduced the number of correctly inferred examples by 4% to 6%. The generated rules for harmony have a great deal of similarity to accepted harmonic transitions (Ottman, 1989). For example, high-priority rules specify common chord transitions such as V-V7-I (a classic way to end a piece of music). 5 Remarks The system described in this paper meets the basic objectives described in Section 1. It learns harmony rules from examples of the music of J.S. Bach. The system is then able to harmonize melodies in real-time. The generated harmonies are sometimes surprising (such as the diminished 7th chord near the end of "Happy Birthday"), yet are consistent with Bach harmony. 1\ I .. I I I Figure 3: Algorithm's Bach-Like Harmony for "Happy Birthday" Rulebase segmentation is an effective method for incorporating a priori knowledge into learned rulebases. It can provides significant speed increases over unsegmented rule bases with no loss of accuracy. Acknowledgements Randall R. Spangler is supported in part by an NSF fellowship. References J. Bach (Ed.: A. Riemenschneider) (1941) 371 Harmonized Chorales and 96 Chorale Melodies. Milwaukee, WI: G. Schirmer. H. Hild, J. Feulner & W. Menzel. (1992) HARMONET: A Neural Net for Harmonizing Chorales in the Style of J. S. Bach. In J. Moody (ed.), Advances in Neural Information Processing Systems 4,267-274. San Mateo, CA: Morgan Kaufmann. M. Mozer, T. Soukup. {1991} Connectionist Music Composition Based on Melodic and Stylistic Constraints. In R. Lippmann (ed.), Advances in Neural Information Processing Systems 3. San Mateo, CA: Morgan Kaufmann. P. Todd. (1989) A Connectionist Approach to Algorithmic Composition. Computer Music Joumal13(4}:27-43. R. Goodman, P. Smyth, C. Higgins, J. Miller. {1992} Rule-Based Neural Networks for Classification and Probability Estimation. Neural Computation 4(6}:781-804. R. Ottman. (1989) Elementary Harmony. Englewood Cliffs, NJ: Prentice Hall.
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Reinforcement Learning for Continuous Stochastic Control Problems Remi Munos CEMAGREF, LISC, Pare de Tourvoie, BP 121, 92185 Antony Cedex, FRANCE. Rerni.Munos@cemagref.fr Paul Bourgine Ecole Polyteclmique, CREA, 91128 Palaiseau Cedex, FRANCE. Bourgine@poly.polytechnique.fr Abstract This paper is concerned with the problem of Reinforcement Learning (RL) for continuous state space and time stocha.stic control problems. We state the Harnilton-Jacobi-Bellman equation satisfied by the value function and use a Finite-Difference method for designing a convergent approximation scheme. Then we propose a RL algorithm based on this scheme and prove its convergence to the optimal solution. 1 Introduction to RL in the continuous, stochastic case The objective of RL is to find -thanks to a reinforcement signal- an optimal strategy for solving a dynamical control problem. Here we sudy the continuous time, continuous state-space stochastic case, which covers a wide variety of control problems including target, viability, optimization problems (see [FS93], [KP95])}or which a formalism is the following. The evolution of the current state x(t) E 0 (the statespace, with 0 open subset of IRd), depends on the control u(t) E U (compact subset) by a stochastic differential equation, called the state dynamics: dx = f(x(t), u(t))dt + a(x(t), u(t))dw (1) where f is the local drift and a .dw (with w a brownian motion of dimension rand (j a d x r-matrix) the stochastic part (which appears for several reasons such as lake of precision, noisy influence, random fluctuations) of the diffusion process. For initial state x and control u(t), (1) leads to an infinity of possible traj~tories x(t). For some trajectory x(t) (see figure I)., let T be its exit time from 0 (with the convention that if x(t) always stays in 0, then T = 00). Then, we define the functional J of initial state x and control u(.) as the expectation for all trajectories of the discounted cumulative reinforcement : J(x; u(.)) = Ex,u(.) {loT '/r(x(t), u(t))dt +,,{ R(X(T))} 1030 R. Munos and P. Bourgine where rex, u) is the running reinforcement and R(x) the boundary reinforcement. 'Y is the discount factor (0 :S 'Y < 1). In the following, we assume that J, a are of class C2 , rand Rare Lipschitzian (with constants Lr and LR) and the boundary 80 is C2 . • · all · II • • xirJ • • • • • • • • Figure 1: The state space, the discretized ~6 (the square dots) and its frontier 8~6 (the round ones). A trajectory Xk(t) goes through the neighbourhood of state ~. RL uses the method of Dynamic Program~ing (DP) which generates an optimal (feed-back) control u*(x) by estimating the value function (VF), defined as the maximal value of the functional J as a function of initial state x : Vex) = sup J(x; u(.). u(.) (2) In the RL approach, the state dynamics is unknown from the system ; the only available information for learning the optimal control is the reinforcement obtained at the current state. Here we propose a model-based algorithm, i.e. that learns on-line a model of the dynamics and approximates the value function by successive iterations. Section 2 states the Hamilton-Jacobi-Bellman equation and use a Finite-Difference (FD) method derived from Kushner [Kus90] for generating a convergent approximation scheme. In section 3, we propose a RL algorithm based on this scheme and prove its convergence to the VF in appendix A. 2 A Finite Difference scheme Here, we state a second-order nonlinear differential equation (obtained from the DP principle, see [FS93J) satisfied by the value function, called the Hamilton-JacobiBellman equation. Let the d x d matrix a = a.a' (with' the transpose of the matrix). We consider the uniformly pambolic case, Le. we assume that there exists c > 0 such that V$ E 0, Vu E U, Vy E IRd ,2:t,j=l aij(x, U)YiYj 2: c1lY112. Then V is C2 (see [Kry80J). Let Vx be the gradient of V and VXiXj its second-order partial derivatives. Theorem 1 (Hamilton-Jacohi-Bellman) The following HJB equation holds : Vex) In 'Y + sup [rex, u) + Vx(x).J(x, u) + ! 2:~j=l aij VXiXj (x)] = 0 for x E 0 uEU Besides, V satisfies the following boundary condition: Vex) = R(x) for x E 80. Reinforcement Learningfor Continuous Stochastic Control Problems 1031 Remark 1 The challenge of learning the VF is motivated by the fact that from V, we can deduce the following optimal feed-back control policy: u*(x) E arg sup [r(x, u) + Vx(x).f(x, u) + ! L:7,j=l aij VXiXj (x)] uEU In the following, we assume that 0 is bounded. Let eI, ... , ed be a basis for JRd. Let the positive and negative parts of a function 4> be : 4>+ = ma.x(4),O) and 4>- = ma.x( -4>,0). For any discretization step 8, let us consider the lattices: 8Zd = {8. L:~=1 jiei} where j}, ... ,jd are any integers, and ~6 = 8Zd n O. Let 8~6, the frontier of ~6 denote the set of points {~ E 8Zd \ 0 such that at least one adjacent point ~ ± 8ei E ~6} (see figure 1). Let U6 cUbe a finite control set that approximates U in the sense: 8 ~ 8' => U6' C U6 and U6U6 = U. Besides, we assume that: Vi = l..d, (3) By replacing the gradient Vx(~) by the forward and backward first-order finitedifference quotients: ~;, V(~) = l [V(~ ± 8ei) V(~)l and VXiXj (~) by the secondorder finite-difference quotients: ~XiXi V(~) -b [V(~ + 8ei) + V(,' - 8ei) - 2V(O] ~;iXj V(~) = 2P[V(~ + 8ei ± 8ej) + V(~ - 8ei =F 8ej) -V(~ + 8ei) V(~ - 8ei) V(~ + 8ej) V(~ - 8ej) + 2V(~)] in the HJB equation, we obtain the following : for ~ E :£6, V6(~)In,+SUPUEUh {r(~,u) + L:~=1 [f:(~,u)'~~iV6(~) fi-(~,U)'~;iV6(~) + aii (~.u) ~ . . V(C) + " . . (at; (~,'U) ~ + . V(C) _ a~ (~,'U) ~ - . . V(C))] } = 0 2 X,X,'" wJ'l=~ 2 x,x.1'" 2 x,xJ '" Knowing that (~t In,) is an approximation of ( ,l:l.t -1) as ~t tends to 0, we deduce: V6(~) SUPuEUh [,"'(~'U)L(EEbP(~,U,()V6«()+T(~,u)r(~,u)] (4) with T(~, u) (5) which appears as a DP equation for some finite Markovian Decision Process (see [Ber87]) whose state space is ~6 and probabilities of transition: p(~,u,~ ± 8ei) p(~, u, ~ + 8ei ± 8ej) p(~,u,~ - 8ei ± 8ej) p(~,u,() "'~~r) [28Ift(~, u)1 + aii(~' u) - Lj=l=i laij(~, u)l] , "'~~r)a~(~,u)fori=f:j, (6) "'~~r)a~(~,u) for i =f: j, o otherwise. Thanks to a contraction property due to the discount factor" there exists a unique solution (the fixed-point) V to equation (4) for ~ E :£6 with the boundary condition V6(~) = R(~) for ~ E 8:£6. The following theorem (see [Kus90] or [FS93]) insures that V 6 is a convergent approximation scheme. 1032 R. Munos and P. Bourgine Theorem 2 (Convergence of the FD scheme) V D converges to V as 8 1 0 : lim /)10 VD(~) = Vex) un~formly on 0 ~-x Remark 2 Condition (3) insures that the p(~, u, () are positive. If this condition does not hold, several possibilities to overcome this are described in [Kus90j. 3 The reinforcement learning algorithm Here we assume that f is bounded from below. As the state dynami,:s (J and a) is unknown from the system, we approximate it by building a model f and a from samples of trajectories Xk(t) : we consider series of successive states Xk = Xk(tk) and Yk = Xk(tk + Tk) such that: - "It E [tk, tk + Tk], x(t) E N(~) neighbourhood of ~ whose diameter is inferior to kN.8 for some positive constant kN, - the control u is constant for t E [tk, tk + Tk], - T k satisfies for some positive kl and k2, (7) Then incrementally update the model : .1 ",n Yk - Xk n ~k=l Tk an(~,u) 1 n (Yk - Xk - Tk.fn(~, u)) (Yk - Xk - Tk·fn(~, u))' -;;; Lk=l Tk (8) and compute the approximated time T( x, u) ~d the approximated probabilities of transition p(~, u, () by replacing f and a by f and a in (5) and (6). We obtain the following updating rule of the V D -value of state ~ : V~+l (~) = sUPuEU/) [,~/:(x,u) L( p(~, u, ()V~(() + T(x, u)r(~, u)] (9) which can be used as an off-line (synchronous, Gauss-Seidel, asynchronous) or ontime (for example by updating V~(~) as soon as a trajectory exits from the neighbourood of ~) DP algorithm (see [BBS95]). Besides, when a trajectory hits the boundary [JO at some exit point Xk(T) then update the closest state ~ E [JED with: (10) Theorem 3 (Convergence of the algorithm) Suppose that the model as well as the V D -value of every state ~ E :ED and control u E UD are regularly updated (respectively with (8) and (9)) and that every state ~ E [JED are updated with (10) at least once. Then "Ie> 0, :3~ such that "18 ~ ~, :3N, "In 2: N, sUP~EE/) IV~(~) V(~)I ~ e with probability 1 Reinforcement Learningfor Continuous Stochastic Control Problems 1033 4 Conclusion This paper presents a model-based RL algorithm for continuous stochastic control problems. A model of the dynamics is approximated by the mean and the covariance of successive states. Then, a RL updating rule based on a convergent FD scheme is deduced and in the hypothesis of an adequate exploration, the convergence to the optimal solution is proved as the discretization step 8 tends to 0 and the number of iteration tends to infinity. This result is to be compared to the model-free RL algorithm for the deterministic case in [Mun97]. An interesting possible future work should be to consider model-free algorithms in the stochastic case for which a Q-Iearning rule (see [Wat89]) could be relevant. A Appendix: proof of the convergence Let M f ' Ma, M fr. and Ma .• be the upper bounds of j, a, f x and 0' x and m f the lower bound of f. Let EO = SUP€EI:h !V0(';) - V(';)I and E! = SUP€EI:b \V~(';) - VO(.;)\. A.I Estimation error of the model fn and an and the probabilities Pn Suppose that the trajectory Xk(t) occured for some occurence Wk(t) of the brownian motion: Xk(t) = Xk + f!k f(Xk(t),u)dt + f!" a(xk(t),U)dwk. Then we consider a trajectory Zk (t) starting from .; at tk and following the same brownian motion: Zk(t) ='; + fttk. f(Zk(t), u)dt + fttk a(zk(t), U)dWk' Let Zk = Zk(tk + Tk). Then (Yk - Xk) - (Zk -.;) = ftk [f(Xk(t), u) - f(Zk(t), u)] dt + ftt:.+Tk [a(xk(t), u) - a(zk(t), u)J dWk. Thus, from the C1 property of f and a, II(Yk - Xk) - (Zk - ';)11 ~ (Mf'" + M aJ.kN.Tk.8. (11) The diffusion processes has the following property ~ee for example the ItO-Taylor majoration in [KP95j) : Ex [ZkJ = ';+Tk.f(';, U)+O(Tk) which, from (7), is equivalent to: Ex [z~:g] = j(';,u) + 0(8). Thus from the law of large numbers and (11): li~-!~p Ilfn(';, u) - f(';, u)11 li;;:s~p II~ L~=l [Yk;kX& - ¥.] II + 0(8) (Mf:r: + M aJ·kN·8 + 0(8) = 0(8) w.p. 1 (12) Besides, diffusion processes have the following property (again see [KP95J): Ex [(Zk -.;) (Zk - .;)'] = a(';, U)Tk + f(';, u).f(';, U)'.T~ + 0(T2) which, from (7), is equivalent to: Ex [(Zk-€-Tkf(S'U)~(kZk-S-Tkf(S'U»/] = a(';, u) + 0(82). Let rk = Zk -.; - Tkf(';, u) and ik = Yk - Xk - Tkfn(';, u) which satisfy (from (11) and (12» : Ilrk - ikll = (Mf:r: + M aJ.Tk.kN.8 + Tk.o(8) (13) From the definition of Ci;;(';,u), we have: Ci;;(';,u) - a(';,u) = ~L~=l '\:1.' Ex [r~':k] + 0(82 ) and from the law of large numbers, (12) and (13), we have: li~~~p 11~(';,u) - a(';,u)11 = li~-!~p II~ L~=l rJ./Y - r~':k II + 0(82 ) Ilik -rkllli:!s!p~ fl (II~II + II~II) +0(82 ) = 0(82 ) 1034 R. Munos and P. Bourgine "In(';, u) - I(';, u)" ~ kf·8 w.p. 1 1Ill;;(';, u) - a(';, u)1I ~ ka .82 w.p. 1 (14) Besides, from (5) and (14), we have: 1 (c ) _ (c )1 < d.(k[.6 2+d.k,,62 ) J:2 < k J:2 T r.",U Tn r.",U _ (d.m,.6)2 U _ T'U (15) and from a property of exponential function, I,T(~.u) _ ,7' .. (€ .1£) I = kT.In ~ .82. (16) We can deduce from (14) that: . 1 ( ) -( )1 (2.6.Mt+d.Ma)(2.kt+d.k,,)62 k J: limsupp';,u,( -Pn';,u,( ~ 6mr(2.k,+d.ka)62 S; puw.p.l n-+oo (17) A.2 Estimation of IV~+l(';) - V6(.;) 1 Mter having updated V~(';) with rule (9), let A denote the difference IV~+l(';) - V6(.;) I. From (4), (9) and (8), A < ,T(€.U) L: [P(';, u, () - p(.;, u, ()] V6 (() + ( ,T(€.1£) ,7'(~'1£») L p(.;, u, () V 6 (() ( ( +,7' (€.u) . L:p(.;, u, () [V6(() V~(()] + L:p(.;, u, ().T(';, u) [r(';, u) - F(';, u)] ( ( + L:( p(.;, u, () [T(';, u) - T(';, u)] r(';, u) for all u E U6 As V is differentiable we have : Vee) = V(';) + VX ' (( -.;) + 0(1I( - ';11). Let us define a linear function V such that: Vex) = V(';) + VX ' (x - ';). Then we have: [P(';, u, () - p(.;, u, ()] V6(() = [P(';, u, () - p(.;, u, ()] . [V6(() - V(()] + [P(';,u,()-p(';,u,()]V((), thus: L:([p(';,u,()-p(';,u,()]V6(() = kp .E6.8 + L([P(';,U,()-p(.;,u,()] [V(() +0(8)] = [V(7J)-VUD] + kp .E6.8 + 0(8) = [V(7J) - V(1j)] + 0(8) with: 7J = L:( p(';, u, () (( -.;) and 1j = L:( p(.;, u, () (( - .;). Besides, from the convergence of the scheme (theorem 2), we have E6.8 = 0(8). From the linearity of V, IV(() - V(Z) I ~ II( -ZII·Mv", S; 2kp 82 . Thus IL( [P(';, u, () - p(.;, u, ()] V6 (() I = 0(8) and from (15), (16) and the Lipschitz property of r, A = 1'l'(€'U), L:( p(.;, u, () [V6(() - V~ (()] 1+ 0(8). As ,..,.7'(€.u) < 1 7'(€.U) In 1 < 1 _ T(€.u)-k.,.62 In 1 < 1 _ ( 6 _ !ix..82) In 1 I 2 'Y 2 'Y 2d(M[+d.M,,) 2 'Y ' we have: A = (1 k.8)E~ + 0(8) (18) with k = 2d(M[~d.M,,). Reinforcement Learning for Continuous Stochastic Control Problems 1035 A.3 A sufficient condition for sUP€EE~ IV~(~) V6(~)1 :S C2 Let us suppose that for all ~ E ~6, the following conditions hold for some a > 0 E~ > C2 =} IV~+I(O V6(~)1 :S E~ - a (19) E~ :S c2=}IV~+I(~)_V6(~)I:Sc2 (20) From the hypothesis that all states ~ E ~6 are regularly updated, there exists an integer m such that at stage n + m all the ~ E ~6 have been updated at least once since stage n. Besides, since all ~ E 8C6 are updated at least once with rule (10), V~ E 8C6, IV~(~) V6(~)1 = IR(Xk(T)) R(~)I :S 2.LR.8 :S C2 for any 8 :S ~3 = 2~lR' Thus, from (19) and (20) we have: E! > C2 =} E!+m :S E! - a E! :S C2 =} E!+m :S C2 Thus there exists N such that: Vn ~ N, E~ :S C2. A.4 Convergence of the algorithm Let us prove theorem 3. For any c > 0, let us consider Cl > 0 and C2 > 0 such that Cl +C2 = c. Assume E~ > £2, then from (18), A = E! - k.8'£2+0(8) :S E~ -k.8.~ for 8 :S ~3. Thus (19) holds for a = k.8.~. Suppose now that E~ :S £2. From (18), A :S (1 - k.8)£2 + 0(8) :S £2 for 8 :S ~3 and condition (20) is true. Thus for 8 :S min { ~1, ~2, ~3}, the sufficient conditions (19) and (20) are satisfied. So there exists N, for all n ~ N, E~ :S £2. Besides, from the convergence of the scheme (theorem 2), there exists ~o st. V8:S ~o, sUP€EE~ 1V6(~) V(~)I :S £1· Thus for 8 :S min{~o, ~1, ~2, ~3}, "3N, Vn ~ N, sup IV~(~) V(~)I :S sup IV~(~) V6(~)1 + sup 1V6(~) V(~)I :S £1 + c2 = £. €EE6 €EEh €EE6 References [BBS95j Andrew G. Barto, Steven J. Bradtke, and Satinder P. Singh. Learning to act using real-time dynamic programming. Artificial Intelligence, (72):81138, 1995. [Ber87j Dimitri P. Bertsekas. Dynamic Programming: Deterministic and Stochastic Models. Prentice Hall, 1987. [FS93j Wendell H. Fleming and H. Mete Soner. Controlled Markov Processes and Viscosity Solutions. Applications of Mathematics. Springer-Verlag, 1993. [KP95j Peter E. Kloeden and Eckhard Platen. Numerical Solutions of Stochastic Differential Equations. Springer-Verlag, 1995. [Kry80j N.V. Krylov. Controlled Diffusion Processes. Springer-Verlag, New York, 1980. [Kus90j Harold J. Kushner. Numerical methods for stochastic control problems in continuous time. SIAM J. Control and Optimization, 28:999-1048, 1990. [Mun97j [Wat89j Remi Munos. A convergent reinforcement learning algorithm in the continuous case based on a finite difference method. International Joint Conference on Art~ficial Intelligence, 1997. Christopher J.C.H. Watkins. Learning from delayed reward. PhD thesis, Cambridge University, 1989. Use of a Multi-Layer Percept ron to Predict Malignancy in Ovarian Tumors Herman Verrelst, Yves Moreau and Joos Vandewalle Dept. of Electrical Engineering Katholieke Universiteit Leuven Kard. Mercierlaan 94 B-3000 Leuven, Belgium Abstract Dirk Timmerman Dept. of Obst. and Gynaec. University Hospitals Leuven Herestraat 49 B-3000 Leuven, Belgium We discuss the development of a Multi-Layer Percept ron neural network classifier for use in preoperative differentiation between benign and malignant ovarian tumors. As the Mean Squared classification Error is not sufficient to make correct and objective assessments about the performance of the neural classifier, the concepts of sensitivity and specificity are introduced and combined in Receiver Operating Characteristic curves. Based on objective observations such as sonomorphologic criteria, color Doppler imaging and results from serum tumor markers, the neural network is able to make reliable predictions with a discriminating performance comparable to that of experienced gynecologists. 1 Introd uction A reliable test for preoperative discrimination between benign and malignant ovarian tumors would be of considerable help to clinicians. It would assist them to select patients for whom minimally invasive surgery or conservative management suffices versus those for whom urgent referral to a gynecologic oncologist is needed. We discuss the development of a neural network classifier/diagnostic tool. The neural network was trained by supervised learning, based on data from 191 thoroughly examined patients presenting with ovarian tumors of which 140 were benign and 51 malignant. As inputs to the network we chose indicators that in recent studies have proven their high predictive value [1, 2, 3]. Moreover, we gave preference to those indicators that can be obtained in an objective way by any gynecologist. Some of these indicators have already been used in attempts to make one single protocol or decision algorithm [3, 4]. Use of a MLP to Predict Malignancy in Ovarian Tumors 979 In order to make reliable assessments on the practical performance of the classifier, it is necessary to work with other concepts than Mean Squared classification Error (MSE), which is traditionally used as a measure of goodness in the training of a neural network. We will introduce notions as specificity and sensitivity and combine them into Receiver Operating Characteristic (ROC) curves. The use of ROC-curves is motivated by the fact that they are independent of the relative proportion of the various output classes in the sample population. This enables an objective validation of the performance of the classifier. We will also show how, in the training of the neural network, MSE optimization with gradient methods can be refined and/or replaced with the help of ROC-curves and simulated annealing techniques. The paper is organized as follows. In Section 2 we give a brief description of the selected input features. In Section 3 we state some drawbacks to the MSE criterion and introduce the concepts of sensitivity, specificity and ROC-curves. Section 4 then deals with the technicalities of training the neural network. In Section 5 we show the results and compare them to human performance. 2 Data acquisition and feature selection The data were derived from a study group of 191 consecutive patients who were referred to a single institution (University Hospitals Leuven, Belgium) from August 1994 to August 1996. Table 1 lists the different indicators which were considered, together with their mean value and standard deviations or together with the relative presence in cases of benign and malignant tumors. Table 1 Indicator Benign Malignant Demographic Age 49.3 ± 16.0 58.3 ± 14.3 Postmenopausal 40% 70.6% Serum marker CA 125 (log) 2.8±1.1 5.2 ± 1.9 CD! Blood flow present 72.9% 100% Morphologic Abdominal fluid 12.1% 52.9% Bilateral mass 11.4% 35.3% Unilocular cyst 42.1% 5.9% Multiloc/solid cyst 16.4% 49.0% Smooth wall 58.6% 2.0% Irregular wall 32.1% 76.5% Papillations 7.9% 74.5% Table 1: Demographic, serum marker, color Doppler imaging and morphologic indicators. For the continuous valued features the mean and standard deviation for each class are reported. For binary valued indicators, the last two columns give the presence of the feature in both classes e.g. only 2% of malignant tumors had smooth walls. First, all patients were scanned with ultrasonography to obtain detailed gray-scale images of the tumors. Every tumor was extensively examined for its morphologic characteristics. Table 1 lists the selected morphologic features: presence of abdominal fluid collection, papillary structures (> 3mm), smooth internal walls, wall irregularities, whether the cysts were unilocular, multilocular-solid and/or present on both pelvic sides. All outcomes are binary valued: every observation relates to the presence (1) or absence (0) of these characteristics. Secondly, all tumors were entirely surveyed by color Doppler imaging to detect presence or absence of blood flow within the septa, cyst walls, solid tumor areas or ovarian tissue. The outcome is also binary valued (1/0). 980 H. Verrelst, Y. Moreau, 1. Vandewalle and D. TImmennan Thirdly, in 173 out of the total of 191 patients, serum CA 125 levels were measured, using CA 125 II immunoradiometric assays (Centocor, Malvern, PA). The CA 125 antigen is a glycoprotein that is expressed by most epithelial ovarian cancers. The numerical value gives the concentration in U Iml. Because almost all values were situated in a small interval between 0 and 100, and because a small portion took values up to 30,000, this variable was rescaled by taking its logarithm. Since age and menopausal status of the patient are considered to be highly relevant, these are also included. The menopausal score is -1 for premenopausal, + 1 for postmenopausal. A third class of patients were assigned a 0 value. These patients had had an hysterectomy, so no menopausal status could be appointed to them. It is beyond the scope of this paper to give a complete account of the meaning of the different features that are used or the way in which the data were acquired. We will limit ourselves to this short description and refer the reader to [2, 3] and gynecological textbooks for a more detailed explanation. 3 Receiver Operating Characteristics 3.1 Drawbacks to Mean Squared classification Error Let us assume that we use a one-hidden-Iayer feed-forward NN with m inputs xl, nh hidden neurons with the tanh(.) as activation function, and one output i1k, nh m Yk(B) = L Wj tanh(L VijX~ + {3j)' (1) j=l i=l parameterized by the vector 0 consisting of the network's weights Wj and Vij and bias terms {3j. The cost function is often chosen to be the squared difference between the desired dk and the actual response Yk. averaged over all N samples [12], 1 N J(O) = N 2:)dk - Yk(9))2. (2) k=l This type of cost function is continuous and differentiable, so it can be used in gradient based optimization techniques such as steepest descent (back-propagation), quasi-Newton or Levenberg-Marquardt methods [8, 9, 11, 12]. However there are some drawbacks to the use of this type of cost function. First of all, the MSE is heavily dependent on the relative proportion of the different output classes in the training set. In our dichotomic case this can easily be demonstrated by writing the cost function, with superscripts b and m respectively meaning benign and malignant, as J(O) Nb 1 ""Nb (db )2 Nm 1 ""Nm (dm )2 Nb + Nm Nb wk=l k Yk + Nb + Nm N m wk=l k Yk (3) ~ ~ ,\ (1-,\) If the relative proportion in the sample population is not representative for reality, the .x parameter should be adjusted accordingly. In practice this real proportion is often not known accurately or one simply ignores the meaning of .x and uses it as a design parameter in order to bias the accuracy towards one of the output classes. A second drawback of the MSE cost function is that it is not very informative towards practical usage of the classification tool. A clinician is not interested in the averaged deviation from desired numbers, but thinks in terms of percentages found, missed or misclassified. In the next section we will introduce the concepts of sensitivity and specificity to express these more practical measures. Use of a MLP to Predict Malignancy in Ovarian Tumors 981 3.2 Sensitivity, specificity and ROC-curves If we take the desired response to be 0 for benign and 1 for malignant cases, the way to make clear cut (dichotomic) decisions is to compare the numerical outcome of the neural network to a certain threshold value T between 0 and 1. When the outcome is above the threshold T, the prediction is said to be positive. Otherwise the prediction is said to be negative. With this convention, we say that the prediction was True Positive (TP) True Negative (TN) False Positive (FP) False Negative (FN) if the prediction was positive when the sample was malignant. if the prediction was negative when the sample was benign. if the prediction was positive when the sample was benign. if the prediction was negative when the sample was malignant. To every of the just defined terms T P, TN, F P and F N, a certain subregion of the total sample space can be associated, as depicted in Figure 1. In the same sense, we can associate to them a certain number counting the samples in each subregion. We can then define sensitivity as Tl:FN' the proportion of malignant cases that Total opulation ¥~li~nant .... . " ... TP - ', \ , , " Figure 1: The concepts of true and false positive and negative illustrated. The dashed area indicates the malignant cases in the total sample population. The positive prediction of an imperfect classification (dotted area) does not fully coincide with this sub area. are predicted to be malignant and specificity as F::r N' the proportion of benign cases that are predicted to be benign. The false positive rate is I-specificity. When varying the threshold T, the values of T P, TN, F P, F N and therefore also sensitivity and specificity, will change. A low threshold will detect almost all malignant cases at the cost of many false positives. A high threshold will give less false positives, but will also detect less malignant cases. Receiver Operating Characteristic (ROC) curves are a way to visualize this relationship. The plot gives the sensitivity versus false positive rate for varying thresholds T (e.g. Figure 2). The ROC-curve is useful and widely used device for assessing and comparing the value of tests [5, 7]. The proportion of the whole area of the graph which lies below the ROC-curve is a one-value measure of the accuracy of a test [6]. The higher this proportion, the better the test. Figure 2 shows the ROC-curves for two simple classifiers that use only one single indicator. (Which means that we classify a tumor being malignant when the value of the indicator rises above a certain value.) It is seen that the CA 125 level has high predictive power as its ROC-curve spans 87.5% of the total area (left Figure 2). For the age parameter, the ROC-curve spans only 65.6% (right Figure 2). As indicated by the horizontal line in the plot, a CA 125 level classification will only misclassify 15% of all benign cases to reach a 80% sensitivity, whereas using only age, one would then misclassify up to 50% of them. 982 H. Verrelst, Y. Moreau, 1. Vandewalle and D. Timmennan ::I( .. .. ' f . , , r o ••• ,J " . , .. t f 1 t, '1 U U •• ,.I .. 0 ' •• " 1 Figure 2: The Receiver Operating Characteristic (ROC) curve is the plot of the sensitivity versus the false positive rate of a classifier for varying thresholds used. Only single indicators (left: CA 125, right: age) are used for these ROC-curlVes. The horizontal line marks the 80% specificity level. Since for every set of parameters of the neural network the area under the ROCcurve can be calculated numerically, this one-value measure can also be used for supervised training, as will be shown in the next Section. 4 Simulation results 4.1 Inputs and architecture The continuous inputs were standardized by subtracting their mean and dividing by their standard deviation (both calculated over the entire population). Binary valued inputs were left unchanged. The desired outputs were labeled 0 for benign examples, 1 for malignant cases. The data set was split up: 2/3 of both benign and malignant samples were randomly selected to form the training set. The remaining examples formed the test set. The ratio of benign to all examples is >. ~ j. Since the training set is not large, there is a risk of overtraining when too many parameters are used. We will limit the number of hidden neurons to nh = 3 or 5. As the CA 125 level measurement is more expensive and time consuming, we will investigate two different classifiers: one which does use the CA 125 level and one which does not. The one-hidden-Iayer MLP architectures that are used, are 11-3-1 and 10-5-1. A tanh(.) is taken for the activation function in the hidden layer. 4.2 Training A first way of training was MSE optimization using the cost function (3). By taking >. = ~ in this expression, the role of malignant examples is more heavily weighted. The parameter vector e was randomly initialized (zero mean Gaussian distribution, standard deviation a = 0.01). Training was done using a quasi-Newton method with BFGS-update of the Hessian (fminu in Matlab) [8, 9]. To prevent overtraining, the training was stopped before the MSE on the test set started to rise. Only few iterations (~ 100) were needed. A second way of training was through the use of the area spanned by the ROC-curve of the classifier and simulated annealing techniques [10]. The area-measure AROC was numerically calculated for every set of trial parameters: first the sensitivity and false positive rate were calculated for 1000 increasing values of the threshold T between 0 and 1, which gave the ROC-curve; secondly the area AROC under the curve was numerically calculated with the trapezoidal integration rule. Use of a MLP to Predict Malignancy in Ovarian Tumors 983 We used Boltzmann Simulated Annealing to maximize the ROC-area. At time k a trial parameter set of the neural network OHl is randomly generated in the neighborhood of the present set Ok (Gaussian distribution, a = O.OO~. The trial set 8H1 is always accepted if the area Af.2? 2: Afoc. If Af'?? < Ak OC, Ok+! is accepted if A r:g? - A r;oc ( ROC )/T. e Ak > Q with Q a uniformly distributed random variable E [0,1] and Te the temperature. As cooling schedule we took Te = 1/(100 + 10k), so that the annealing was low-temperature and fast-cooling. The optimization was stopped before the ROC-area calculated for the test set started to decrease. Only a few hundred annealing epochs were allowed. 4.3 Results Table 2 states the results for the different approaches. One can see that adding the CA 125 serum level clearly improves the classifier's performance. Without it, the ROC-curve spans about 96.5% of the total square area of the plot, whereas with the CA 125 indicator it spans almost 98%. Also, the two training methods are seen to give comparable results. Figure 3 shows the ROC-curve calculated for the total population for the 11-3-1 MLP case, trained with simulated annealing Table 2 Training set Test set Total population 10-5-1 MLP, MSE 96.7% 96.4% 96.5% 10-5-1 MLP, SA 96.6% 96.2% 96.4% 11-3-1 MLP, MSE 97.9% 97.4% 97.7% 11-3-1 MLP, SA 97.9% 97.5% 97.8% Table 2: For the two architectures (10-5-1 and 11-3-1) of the MLP and for the gradient (MSE) and the simulated annealing (SA) optimization techniques, this table gives the resulting areas under the ROC-curves . .. 07 °0 0.1 G.2 ":I 0 4 0.' 0, 07 01 0, , Figure 3: ROC-curves of 11-3-1 MLP (with CA 125 level indicator), trained with simulated annealing. The curve, calculated for the total population, spans 97.8% of the total region. All patients were examined by two gynecologists, who gave their subjective impressions and also classified the ovarian tumors into (probably) benign and malignant. Histopathological examinations of the tumors afterwards showed these gynecologists 984 H. Vendst, Y. Moreau, 1. Vandewalle and D. Timmerman to have a sensitivity up to 98% and a false positive rate of 13% and 12% respectively. As can be seen in Figure 3, the 11-3-1 MLP has a similar performance. For a sensitivity of 98%, its false positive rate is between 10% and 15%. 5 Conclusion In this paper we have discussed the development of a Multi-Layer Perceptron neural network classifier for use in preoperative differentiation between benign and malignant ovarian tumors. To assess the performance and for training the classifiers, the concepts of sensitivity and specificity were introduced and combined in Receiver Operating Characteristic curves. Based on objective observations available to every gynecologist, the neural network is able to make reliable predictions with a discriminating performance comparable to that of experienced gynecologists. Acknowledgments This research work was carried out at the ESAT laboratory and the Interdisciplinary Center of Neural Networks ICNN of the Katholieke Universiteit Leuven, in the following frameworks : the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture (IUAP P4-02 and IUAP P4-24), a Concerted Action Project MIPS (Model based Information Processing Systems) of the Flemish Community and the FWO (Fund for Scientific Research - Flanders) project G.0262.97 : Learning and Optimization: an Interdisciplinary Approach . The scientific responsibility rests with its authors. References [1] Bast R. C., Jr., Klug T.L. St. John E., et aI, "A radioimmunoassay using a monoclonal antibody to monitor the course of epithelial ovarian cancer," N. Engl. J. Med., Vol. 309, pp. 883-888, 1983 [2] Timmerman D., Bourne T ., Tailor A., Van Assche F.A., Vergote 1., "Preoperative differentiation between benign and malignant adnexal masses," submitted [3] Tailor A., Jurkovic D., Bourne T.H., Collins W.P., Campbell S., "Sonographic prediction of malignancy in adnexal masses using multivariate logistic regression analysis," Ultrasound Obstet. Gynaecol. in press, 1997 [4] Jacobs 1., Oram D., Fairbanks J., et aI., "A risk of malignancy index incorporating CA 125, ultrasound and menopausal status for the accurate preoperative diagnosis of ovarian cancer," Br. J. Obstet. Gynaecol., Vol. 97, pp. 922-929, 1990 [5] Hanley J.A., McNeil B., "A method of comparing the areas under the receiver operating characteristics curves derived from the same cases," Radiology, Vol. 148, pp. 839-843, 1983 [6J Swets J.A., "Measuring the accuracy of diagnostic systems," Science, Vol. 240, pp. 1285-1293, 1988 [7J Galen R.S., Gambino S., Beyond normality: the predictive value and efficiency of medical diagnosis, John Wiley, New York, 1975. [8J Gill P., Murray W., Wright M., Practical Optimization, Acad. Press, New York, 1981 [9] Fletcher R., Practical methods of optimization, 2nd ed., John Wiley, New York, 1987. [10J Kirkpatrick S., Gelatt C.D., Vecchi M., "Optimization by simulated annealing," Science, Vol. 220, pp. 621-680, 1983. [11] Rumelhart D.E., Hinton G.E., Williams R.J., "Learning representations by backpropagating errors," Nature, Vol. 323, pp. 533-536, 1986. [12] Bishop C., Artificial Neural Networks for Pattern Recognition, OUP, Oxford, 1996
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Learning nonlinear overcomplete representations for efficient coding Michael S. Lewicki lewicki~salk.edu Terrence J. Sejnowski terry~salk.edu Howard Hughes Medical Institute Computational Neurobiology Lab The Salk Institute 10010 N. Torrey Pines Rd. La Jolla, CA 92037 Abstract We derive a learning algorithm for inferring an overcomplete basis by viewing it as probabilistic model of the observed data. Overcomplete bases allow for better approximation of the underlying statistical density. Using a Laplacian prior on the basis coefficients removes redundancy and leads to representations that are sparse and are a nonlinear function of the data. This can be viewed as a generalization of the technique of independent component analysis and provides a method for blind source separation of fewer mixtures than sources. We demonstrate the utility of overcomplete representations on natural speech and show that compared to the traditional Fourier basis the inferred representations potentially have much greater coding efficiency. A traditional way to represent real-values signals is with Fourier or wavelet bases. A disadvantage of these bases, however, is that they are not specialized for any particular dataset. Principal component analysis (PCA) provides one means for finding an basis that is adapted for a dataset, but the basis vectors are restricted to be orthogonal. An extension of PCA called independent component analysis (Jutten and Herault, 1991; Comon et al., 1991; Bell and Sejnowski, 1995) allows the learning of non-orthogonal bases. All of these bases are complete in the sense that they span the input space, but they are limited in terms of how well they can approximate the dataset's statistical density. Representations that are overcomplete, i. e. more basis vectors than input variables, can provide a better representation, because the basis vectors can be specialized for Learning Nonlinear Overcomplete Representations for Efficient Coding 557 a larger variety of features present in the entire ensemble of data. A criticism of overcomplete representations is that they are redundant, i.e. a given data point may have many possible representations, but this redundancy is removed by the prior probability of the basis coefficients which specifies the probability of the alternative representations. Most of the overcomplete bases used in the literature are fixed in the sense that they are not adapted to the structure in the data. Recently Olshausen and Field (1996) presented an algorithm that allows an overcomplete basis to be learned. This algorithm relied on an approximation to the desired probabilistic objective that had several drawbacks, including tendency to breakdown in the case of low noise levels and when learning bases with higher degrees of overcompleteness. In this paper, we present an improved approximation to the desired probabilistic objective and show that this leads to a simple and robust algorithm for learning optimal overcomplete bases. 1 Inferring the representation The data, X 1 :L ' are modeled with an overcomplete linear basis plus additive noise: x = AS+i (1) where A is an L x M matrix, whose columns are the basis vectors, where M ~ L . We assume Gaussian additive noise so that 10gP(xIA, s) ()( -A(X - As)2/2, where A = 1/(12 defines the precision of the noise. The redundancy in the overcomplete representation is removed by defining a density for the basis coefficients, P(s), which specifies the probability of the alternative representations. The most probable representation, 5, is found by maximizing the posterior distribution s = maxP(sIA,x) = maxP(s)P(xIA,s) 8 8 (2) P(s) influences how the data are fit in the presence of noise and determines the uniqueness of the representation. In this model, the data is a linear function of s, but s is not, in general, a linear function of the data. IT the basis function is complete (A is invertible) then, assuming broad priors and low noise, the most probable internal state can be computed simply by inverting A. In the case of an overcomplete basis, however, A can not be inverted. Figure 1 shows how different priors induce different representations. Unlike the Gaussian prior, the optimal representation under the Laplacian prior cannot be obtained by a simple linear operation. One approach for optimizing sis to use the gradient of the log posterior in an optimization algorithm. An alternative method for finding the most probable internal state is to view the problem as the linear program: min 1 T s such that As = x. This can be generalized to handle both positive and negative s and solved efficiently and exactly with interior point linear programming methods (Chen et al., 1996). 558 a if L2 L1 b G) :::l ~ 0.1 8 \" M. S. Lewicki and T. 1. Sejnowski ~10'05~\ o 20 -~ - - ~ - - 80 100 120 Figure 1: Different priors induce different representations. (a) The 2D data distribution has three main axes which form an overcomplete representation. The graphs marked "L2" and "L1" show the optimal scaled basis vectors for the data point x under the Gaussian and Laplacian prior, respectively. Assuming zero noise, a Gaussian for P{s) is equivalent to finding the exact fitting s with minimum L2 norm, which is given by the pseudoinverse s = A+x. A Laplacian prior (P{Sm) ex: exp[-OlsmlJ) yields the exact fit with minimum L1 norm, which is a nonlinear operation which essentially selects a subset of the basis vectors to represent the data (Chen et al., 1996). The resulting representation is sparse. (b) A 64-sample segment of speech was fit to a 2x overcomplete Fourier representation (128 basis vectors). The plot shows rank order distribution of the coefficients of s under a Gaussian prior (dashed); and a Laplacian prior (solid). Far more significantly positive coefficients are required under the Gaussian prior than under the Laplacian prior. 2 Learning The learning objective is to adapt A to maximize the probability of the data which is computed by marginalizing over the internal states P(xIA) = J ds P(s)P(xIA, s) (3) general, this integral cannot be evaluated analytically but can be approximated with a Gaussian integral around s, yielding log P(xIA) ~ const. + log pes) - ~ (x - As)2 ~ log det H (4) where H is the Hessian of the log posterior at S, given by )'ATA - VVlogP(s). To avoid a singularity under the Laplacian prior, we use the approximation (logP(sm»)' ~ -8tanh(,8sm) which gives the Hessian full rank and positive determinant. For large ,8 this approximates the true Laplacian prior. A learning rule can be obtained by differentiating log P(xIA) with respect to A. In the following discussion, we will present the derivations of the three terms in (4) and simplifying assumptions that lead to the following simple form of the learning rule (5) Learning Nonlinear Ollercomplete Representations for Efficient Cod(ng 559 2.1 Deriving V log pes) This term specifies how to change A so as to make the probability of the representation s more probable. IT we assume a Laplacian prior, this component changes A to make the representation more sparse. We assume pes) = rIm P(Sm). In order to obtain 8sm/8aij, we need to describe s as a function of A. If the basis is complete (and we assume low noise), then we may simply invert A to obtain s = A -IX. When A is overcomplete, however, there is no simple expression, but we may still make an approximation. Under priors, the most probable solution, s, will yield at most L non-zero elements. In effect, this selects a complete basis from A. Let A represent this reduced basis under s. We then have s = A -1(X- €) where s is equal to s with M - L zero-valued elements removed. A-I obtained by removing the columns of A corresponding to the M - L zero-valued elements of s. This allows us to use results obtained for the case when A is invertible. Following MacKay (1996) we obtain (6) Rewriting in matrix notation we have 810gP(s) _ A~ -Tv~T 8A - zs (7) We can use this to obtain an expression in terms of the original variables. We simply invert the mapping s ~ s to obtain Z f- z and W T f- A -T (row-wise) with Zm = 0 and row m ofWT = 0 if 8m = O. We then have 8 log P(s) WT T 8A = zs (8) 2.2 Deriving Vex - As)2 The second term specifies how to change A so as to minimize the data misfit. Letting ek = [x - AS]k and using the results and notation from above we have: 8 A~ 2 ~ ~ 8s, -8 .. "2 L-ek = AeiSj + A L-ek L-ak'~ a" k k I alJ (9) = AeiSj + A Lek L -aklWliSj (10) k I = AeiSj - AeiSj = 0 (11) Thus no gradient component arises from the error term. 2.3 Deriving V log det H The third term in the learning rule specifies how to change the weights so as to minimize the width of the posterior distribution P(xIA) and thus increase the overall probability of the data. An element of H is defined by Hmn = Cmn + bmn 560 M. S. Lewicki and T. J. Sejnowski where Cmn = Ek Aakmakn and bmn = [-V'V' log P(s)]mn. This gives 8logdetH _ ""H-1 [8emn + 8bmn ] 8a·· - ~ nm 8a.. 8a·· ~ mn U ~ (12) First considering 8Cmn/8aij, we can obtain L H~~ ~~~ = L H~;.\aim + L Hj~.\aim + Hj/ 2Aaij (13) mn ~3 m:f.j m:f.j Using the fact that H~; = Hj~ due to the symmetry of the Hessian, we have (14) Next we derive 8bmn/8aij. We have that V'V'logP(s) is diagonal, because we assume pes) = nm P(sm). Letting 2Ym = H~!n8bmm/8sm and using the result under the reduced representation (6) we can obtain (15) 2.4 Stabilizing and simplifying the learning rule Putting the terms together yields a problematic expression due to the matrix inverses. This can be alleviated by multiplying the gradient by an appropriate positive definite matrix, which rescales the gradient components but preserves a direction valid for optimization. Noting that ATWT = I we have (16) H'\ is large (low noise) then the Hessian is dominated by AATA and we have (17) The vector y hides a computation involving the inverse Hessian. IT the basis vectors in A are randomly distributed, then as the dimensionality of A increases the basis vectors become approximately orthogonal and consequently the Hessian becomes approximately diagonal. It can be shown that if log pes) and its derivatives are smooth, Ym vanishes for large A. Combining the remaining terms yields equation (5). Note that this rule contains no matrix inverses and the vector z involves only the derivative of the log prior. In the case where A is square, this form of the rule is similar to the natural gradient independent component analysis (ICA) learning rule (Amari et al., 1996). The difference in the more general case where A is rectangular is that s must maximize the posterior distribution P(slx, A) which cannot be done simply with the filter matrix as in standard ICA algorithms. Learning Nonlinear Overcomplete Representations/or Efficient Coding 561 3 Examples More sources than inputs. In these 2D examples, the bases were initialized to random, normalized vectors. The coefficients were solved using BPMPD and publicly available interior point linear programming package (Meszaros, 1997) which gives the most probable solution under the Laplacian prior assuming zero noise. The algorithm was run for 30 iterations using equation (5) with a stepsize of 0.001 and a batchsize of 200. Convergence was rapid, typically requiring less than 20 iterations. In all cases, the direction of the learned vectors matched those of the true generating distribution; the magnitude was estimated less precisely, possibly due to the approximation oflogP(xIA). This can be viewed as a source separation problem, but true separation will be limited due to the projection of the sources down to a smaller subspace which necessarily loses information . . ' '. ~ Figure 2: Examples illustrating the fitting of 2D distributions with overcomplete bases. The first example is equivalent to 3 sources mixed into 2 channels; the second to 4 sources mixed into 2 channels. The data in both examples were generated from the true basis A using x = As with the elements of s distributed according to an exponential distribution with unit mean. Identical results were obtained by drawing s from a Laplacian prior (positive and negative coefficients). The overcomplete bases allow the model to capture the true underlying statistical structure in the 2D data space. Overcomplete representations of speech. Speech data were obtained from the TIMIT database, using a single speaker was speaking ten different example sentences with no preprocessing. The basis was initialized to an overcomplete Fourier basis. A conjugate gradient routine was used to obtain the most probable basis coefficients. The stepsize was gradually reduced over 10000 iterations. Figure 3 shows that the learned basis is quite different from the Fourier representation. The power spectrum for the learned basis vectors can be multimodal and/or broadband. The learned basis achieves greater coding efficiency: 2.19 ± 0.59 bits per sample compared to 3.86 ± 0.28 bits per sample for a 2x overcomplete Fourier basis. 4 Summary Learning overcomplete representations allows a basis to better approximate the underlying statistical density of the data and consequently the learned representations have better encoding and denoising properties than generic bases. Unlike the case for complete representations and the standard ICA algorithm, the transformation 562 M S. Lewicki and T. 1. Sejnowski ~ ~ ..... ~ ~ ~~J~~ JvV ~ ........... ~ ~ ~ ~ ~~~ + ~ fIN ~ ~ A- L -L J-.-L ~ ~ ~ ."... f\;vJ'v JL~~JL..... .. -JVV H ~ i~~L~ ... + ..... ..... ....... ~J~--L~ Figure 3: An example of fitting a 2x overcomplete representation to segments of from natural speech. Each segment consisted of 64 samples, sampled at a frequency of 8000 Hz (8 msecs). The plot shows a random sample of 30 of the 128 basis vectors (each scaled to full range). The right graph shows the corresponding power spectral densities (0 to 4000 Hz). from the data to the internal representation is non-linear. The probabilistic formulation of the basis inference problem offers the advantages that assumptions about the prior distribution on the basis coefficients are made explicit and that different models can be compared objectively using log P(xIA). References Amari, S., Cichocki, A., and Yang, H. H. (1996). A new learning algorithm for blind signal separation. In Advances in Neural and Information Processing Systems, volume 8, pages 757-763, San Mateo. Morgan Kaufmann. Bell, A. J. and Sejnowski, T. J. (1995). An information maximization approach to blind separation and blind deconvolution. Neural Computation, 7(6}:1129-1159. Chen, S., Donoho, D. L., and Saunders, M. A. (1996). Atomic decomposition by basis pursuit. Technical report, Dept. Stat., Stanford Univ., Stanford, CA. Comon, P., Jutten, C., and Herault, J. (1991). Blind separation of sources .2. problems statement. Signal Processing, 24(1}:11-20. Jutten, C. and Herault, J. (1991). Blind separation of sources .1. an adaptive algorithm based on neuromimetic architecture. Signal Processing, 24(1):1-10. MacKay, D. J. C. (1996). Maximum likelihood and covariant algorithms for independent component analysis. University of Cambridge, Cavendish Laboratory. Available at ftp: / /wol. ra. phy. cam. ac. uk/pub/mackay / ica. ps. gz. Meszaros, C. (1997). BPMPD: An interior point linear programming solver. Code available at ftp: / /ftp.netlib. org/ opt/bpmpd. tar. gz. Olshausen, B. A. and Field, D. J. (1996). Emergence of simple-cell receptive-field properties by learning a sparse code for natural images. Nature, 381:607-609.
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Receptive field formation in natural scene environments: comparison of single cell learning rules Brian S. Blais N.lntrator Brown University Physics Department Providence, Rl 02912 School of Mathematical Sciences Tel-Aviv University Ramat-Aviv, 69978 ISRAEL H. Shouval Institute for Brain and Neural Systems Brown University Providence, Rl 02912 Leon N Cooper Brown University Physics Department and Institute for Brain and Neural Systems Brown University Providence, Rl 02912 Abstract We study several statistically and biologically motivated learning rules using the same visual environment, one made up of natural scenes, and the same single cell neuronal architecture. This allows us to concentrate on the feature extraction and neuronal coding properties of these rules. Included in these rules are kurtosis and skewness maximization, the quadratic form of the BCM learning rule, and single cell ICA. Using a structure removal method, we demonstrate that receptive fields developed using these rules depend on a small portion of the distribution. We find that the quadratic form of the BCM rule behaves in a manner similar to a kurtosis maximization rule when the distribution contains kurtotic directions, although the BCM modification equations are computationally simpler. 424 B. S. Blais, N. Intrator, H. Shouval and L N. Cooper 1 INTRODUCTION Recently several learning rules that develop simple cell-like receptive fields in a natural image environment have been proposed (Law and Cooper, 1994; Olshausen and Field, 1996; Bell and Sejnowski, 1997). The details of these rules are different as well as their computational reasoning, however they all depend on statistics of order higher than two and they all produce sparse distributions. In what follows we investigate several specific modification functions that have the. general properties of BCM synaptic modification functions (Bienenstock et al., 1982), and study their feature extraction properties in a natural scene environment. Several of the rules we consider are derived from standard statistical measures (Kendall and Stuart, 1977), such as skewness and kurtosis, based on polynomial moments. We compare these with the quadratic form of BCM (Intrator and Cooper, 1992), though one should note that this is not the only form that could be used. By subjecting all of the learning rules to the same input statistics and retina/LGN preprocessing and by studying in detail the single neuron case, we eliminate possible network/lateral interaction effects and can examine the properties of the learning rules themselves. We compare the learning rules and the receptive fields they form, and introduce a procedure for directly measuring the sparsity of the representation a neuron learns. This gives us another way to compare the learning rules, and a more quantitative measure of the concept of sparse'representations. 2 MOTIVATION We use two methods for motivating the use of the particular rules. One comes from Projection Pursuit (Friedman, 1987) and the other is Independent Component Analysis (Comon, 1994). These methods are related, as we shall see, but they provide two different approaches for the current work. 2.1 EXPLORATORY PROJECTION PURSUIT Diaconis and Freedman (1984) show that for most high-dimensional clouds (of points), most low-dimensional projections are approximately Gaussian. This finding suggests that important information in the data is conveyed in those directions whose single dimensional projected distribution is far from Gaussian. Intrator (1990) has shown that a BCM neuron can find structure in the input distribution that exhibits deviation from Gaussian distribution in the form of multimodality in the projected distributions. This type of deviation is particUlarly useful for finding clusters in high dimensional data. In the natural scene environment, however, the structure does not seem to be contained in clusters. In this work we show that the BCM neuron can still find interesting structure in non-clustered data. The most common measures for deviation from Gaussian distribution are skewness and. kurtosis which are functions of the first three and four moments of the distribution respectively. Rules based on these statistical measures satisfy the BCM conditions proposed in Bienenstock et aI. (1982), including a threshold-based stabilization. The details of these rules and some of the qualitative features of the stabilization are different, however. In addition, there are some learning rules, such as the ICA rule of Bell and Sejnowski (1997) and the sparse coding algorithm of Olshausen and Field (1995), which have been used with natural scene inputs to produce oriented receptive fields. We do not include these in our comparison beRF Formation in Natural Scenes: Comparison of Single Cell Learning Rules 425 cause they are not single cell learning rules, and thus detract from our immediate goal of comparing rules with the same input structure and neuronal architecture. 2.2 INDEPENDENT COMPONENT ANALYSIS Recently it has been claimed that the independent components of natural scenes are the edges found in simple cells (Bell and Sejnowski, 1997). This was achieved through the maximization of the mutual entropy of a set of mixed signals. Others (Hyvarinen and Oja, 1996) have claimed that maximizing kurtosis can also lead to the separation of mixed signals into independent components. This alternate connection between kurtosis and receptive fields leads us into a discussion of ICA. Independent Component AnalYSis (ICA) is a statistical signal processing technique whose goal is to express a set of random variables as a linear mixture of statistically independent variables. The problem of ICA is then to find the transformation from the observed mixed signals to the "unmixed" independent sources. The search for independent components relies on the fact that a linear mixture of two nonGaussian distributions will become more Gaussian than either of them. Thus, by seeking projections which maximize deviations from Gaussian distribution, we recover the original (independent) signals. This explains the connection of ICA to the framework of exploratory projection pursuit. 3 SYNAPTIC MODIFICATION RULES In this section we outline the derivation for the learning rules in this study. Neural activity is assumed to be a positive quantity, so for biological plausibility we denote by c the rectified activity (T(d . m), where (T(.) is a smooth monotonic function with a positive output (a slight negative output is also allowed). (T' denotes the derivative of the sigmoidal. The rectification is required for all rules that depend on odd moments because these vanish in symmetric distributions such as natural scenes. We study the following measures(Kendall and Stuart, 1977, for review): Skewness 1 This measures the deviation from symmetry, and is of the form: 51 = E[c3]j E1.5[C2]. (1) A maximization of this measure via gradient ascent gives "V51 = \.5E [c (c - E[c3]jE[c2]) (TId] = \ .5E [c (c - E[c3]jeM) (TId] (2) eM eM where em is defined as E[c2]. Skewness 2 Another skewness measure is given by 52 = E[c3] - E1.5[C2]. (3) This measure requires a stabilization mechanism which we achieve by requiring that the vector of weights, denoted by m, has norm of 1. The gradient of 52 is "V52 = 3E [c2 - cJE[c2]] = 3E [c (c - JeM) (TId] ,II m 11= 1 (4) Kurtosis 1 Kurtosis measures deviation from Gaussian distribution mainly in the tails of the distribution. It has the form Kl = E[c4]jE2[C2] - 3. (5) This measure has a gradient of the form 1 1 "VKl = -2E [c (c2 - E[c4]jE[c2]) (TId] = -2E [c (c2 - E[c4]jeM) (TId]. (6) eM eM 426 B. S. Blais, N. Intrator. H. Shouval and L N. Cooper Kurtosis 2 As before, there is a similar form which requires some stabilization: K2 = E[c4] - 3E2[C2]. (7) This measure has a gradient of the form 'V K2 = 4E [c3 - cE[c2]] = 3E [c(c2 - eM )](1'd], II m 11= 1. (8) Kurtosis 2 and ICA It has been shown that kurtosis, defined as K2 = E [c4] - 3E2 [c2] can be used for ICA(Hyvarinen and Oja, 1996). Thus, finding the extrema of kurtosis of the projections enables the estimation of the independent components. They obtain the following expression m = ~ (E- l [ddT] E [d(m· d)3] - 3m). which leads to an iterative "fixed-point algorithm". (9) Quadratic BCM The Quadratic BCM (QBCM) measure as given in (Intrator and Cooper, 1992) is of the form QBCM = !E[c3] - !E2[C2]. (10) 3 4 Maximizing this form using gradient ascent gives the learning rule: (11) 4 METHODS We use 13x13 circular patches from 12 images of natural scenes, presented to the neuron each iteration of the learning. The natural scenes are preprocessed either with a Difference of Gaussians (DOG) filter(Law and Cooper, 1994) or a whitening filter (Oja, 1995; Bell and Sejnowski, 1995), which eliminates the second order correlations. The moments of the output, c, are calculated iteratively, and when it is needed (Le. K2 and 8 2) we also normalize the weights at each iteration. For Oja's fixed-point algorithm, the learning was done in batches of 1000 patterns over which the expectation values were performed. However, the covariance matrix was calculated over the entire set of input patterns. 5 RESULTS 5.1 RECEPTIVE FIELDS The resulting receptive fields (RFs) formed are shown in Figure 1 for both the DOGed and whitened images. Every learning rule developed oriented receptive fields, though some were more sensitive to the preprocessing than others. The additive versions of kurtosis and skewness, K2 and 82 respectively, developed RFs with a higher spatial frequency, and more orientations, in the whitened environment than in the DOGed environment. The multiplicative versions of kurtosis and skewness, Kl and 8 1 respectively, as well as QBCM, sampled from many orientations regardless of the preprocessing. 8 1 gives receptive fields with lower spatial frequencies than either QBCM or Kl. RF Formation in Natural Scenes: Comparison of Single Cell Learning Rules 427 This disappears with the whitened inputs, which implies that the spatial frequency of the RF is related to the dependence of the learning rule on the second moment. Example receptive fields using Oja's fixed-point ICA algorithm not surprisingly look qualitatively similar to those found using the stochastic maximization of K 2 • The output distributions for all of the rules appear to be double exponential. This distribution is one which we would consider sparse, but it would be difficult to compare the sparseness of the distributions merely on the appearance of the output distribution alone. In order to determine the sparseness of the code, we introduce a method for measuring it directly. Receptive Fields from Natural Scene Input DOGed Whitened Output Distribution § LI g fIJ ~~~I A I -20 0 20 Output Distribution § 11.11 ~~~I 1\ I -20 0 20 ~.11 i1 ~~~I A I ~.II a ~~~I A I -20 0 20 -20 0 20 ~t! Ii1Ii ~~~VSJ ~ ••• ~~:I 1\ I -20 0 20 -20 0 20 1\1 o 20 Figure 1: Receptive fields using DOGed (left) and whitened (right) image input obtained from learning rules maximizing (froni top to bottom) the Quadratic BCM objective function, Kurtosis (multiplicative), Kurtosis (additive), Skewness (multiplicative), and Skewness (additive). Shown are three examples (left to right) from each learning rule as well as the log of the normalized output distribution, before the application of the rectifying sigmoid. 5.2 STRUCTURE REMOVAL: SENSITIVITY TO OUTLIERS Learning rules which are dependent on large polynomial moments, such as Quadratic BCM and kurtosis, tend to be sensitive to the tails of the distribution. In the case of a sparse code the outliers, or the rare and interesting events, are what is important. Measuring the degree to which the neurons form a sparse code can be done in a straightforward and systematic fashion. The procedure involves simply eliminating from the environment those patterns for which the neuron responds strongly. These patterns tend to be the high contrast edges, and are thus the structure found in the image. The percentage of patterns that needs to be removed in order to cause a change in the receptive field gives a direct measure of the sparsity of the coding. The results of this structure removal 428 B. S. Blais, N. Intrator, H Shouval and L N. Cooper are shown in Figure 2. For Quadratic BCM and kurtosis, one need only eliminate less than one half of a percent of the input patterns to change the receptive field significantly. To make this more precise, we define a normalized difference between two mean zero vectors as V == H1 - cos a), where a is the angle between the two vectors. This measure has a value of zero for identical vectors, and a maximum value of one for orthogonal vectors. Also shown in Figure 2 is the normalized difference as a function of the percentage eliminated, for the different learning rules. RF differences can be seen with as little as a tenth of a percent, which suggests that the neuron is coding the information in a very sparse manner. Changes of around a half a percent and above are visible as significant orientation, phase, or spatial frequency changes. Although both skewness and Quadratic BCM depend primarily on the third moment, QBCM behaves more like kurtosis with regards to sparse coding. Structure Removal for BCM, Kurtosis, and Skew 0.3 BCM ~ BCM BCM • • • • • ~0.25 ~ - - 0 Kurtosis 1 C> ' 0 Skew 1 , " ". 2l 0.2 <= Kl II • I! 1'1 II e CI> ~0.15 i --S, N -II rI [I ~ 0.1 -SI rI) II E 0 , . " ' . 0 -;' ZO.05 Figure 2: Example receptive fields (left), and normalized difference measure (right), resulting from structure removal using QBCM, Kl, and 8 1 , The RFs show the successive deletion of top 1% of the distribution. On the right is the normalized difference between RFs as a function of the percentage deleted in structure removal. The maximum possible value of the difference is 1. 6 DISCUSSION This study attempts to compare several learning rules which have some statistical or biological motivation, or both. For a related study discussing projection pursuit and BCM see (Press and Lee, 1996). We have used natural scenes to gain some more insight about the statistics underlying natural images. There are several outcomes from this study: • All rules used, found kurtotic distributions. • The single cell lCA rule we considered, which used the subtractive form of kurtosis, achieved receptive fields qualitatively similar to other rules discussed. • The Quadratic BCM and the multiplicative version of kurtosis are less sensitive to the second moments of the distribution and produce oriented RFs even when the data is not whitened. The subtractive versions of kurtosis and skewness are sensitive and produces oriented RFs only after sphering the data (Friedman, 1987; Field, 1994). RF Fonnation in Natural Scenes: Comparison of Single Cell Learning Rules 429 • Both Quadratic BCM and kurtosis are sensitive to the elimination of the upper 1/2% portion of the distribution. The sensitivity to small portions of the distribution represents the other side of the coin of sparse coding. • The skew rules' sensitivity to the upper parts of the distribution is not so strong. • Quadratic BCM learning rule, which has been advocated as a projection index for finding multi-modality in high dimensional distribution, can find projections emphasizing high kurtosis when no cluster structure is present in the data. ACKNOWLEDGMENTS This work, was supported by the Office of Naval Research, the DANA Foundation and the National Science Foundation. References Bell, A. J. and Sejnowski, T. J. (1995). An information-maximisation approach to blind separation and blind deconvolution. Neural Computation, 7(6}:1129-1159. Bell, A. J. and Sejnowski, T. J. (1997). The independent components of natural scenes are edge filters. Vision Research. in press. Bienenstock, E. L., Cooper, L. N., and Munro, P. W. (1982). Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex. Journal of Neuroscience, 2:32-48. Comon, P. (1994). Independent component analysis, a n'ew concept? Signal Processing, 36:287-314. Field, D. J. (1994). What is the goal of sensory coding. Neural Computation, 6:559-601. Friedman, J. H. (1987). Exploratory projection pursuit. Journal of the American Statistical Association, 82:249-266. Hyvarinen, A. and Oja, E. (1996). A fast fixed-point algorithm for independent component analysis. Int. Journal of Neural Systems, 7(6):671-687. Intrator, N. (1990). A neural network for feature extraction. In Touretzky, D. S. and Lippmann, R. P., editors, Advances in Neural Information Processing Systems, volume 2, pages 719-726. Morgan Kaufmann, San Mateo, CA. Intrator, N. and Cooper, L. N. (1992) . Objective function formulation of the BCM theory of visual cortical plasticity: Statistical connections, stability conditions. Neural Networks, 5:3-17. Kendall, M. and Stuart, A. (1977). The Advanced Theory of Statistics, volume 1. MacMillan Publishing, New York. Law, C. and Cooper, L. (1994). Formation of receptive fields according to the BCM theory in realistic visual environments. Proceedings National Academy of Sciences, 91:7797-7801. Oja, E. (1995). The nonlinear pca learning rule and signal separation - mathematical analysis. Technical Report A26, Helsinki University, CS and Inf. Sci. Lab. Olshausen, B. A. and Field, D. J. (1996). Emergence of simple cell receptive field properties by learning a sparse code for natural images. Nature, 381:607-609. Press, W. and Lee, C. W. (1996). Searching for optimal visual codes: Projection pursuit analysis of the statistical structure in natural scenes. In The Neurobiology of Computation: Proceedings of the fifth annual Computation and Neural Systems conference. Plenum Publishing Corporation.
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Self-similarity properties of natural images ANTONIO TURIEL; GERMAN MATOt NESTOR PARGA t Departamento de Fisica Te6rica. Universidad AutOnoma de Madrid Cantoblanco, 28049 Madrid, Spain and JEAN-PIERRE N ADAL§ Laboratoire de Physique Statistique de I 'E. N.S. , Ecole Normale Superieure 24, rue Lhomond, F-75231 Paris Cedex OS, France Abstract Scale invariance is a fundamental property of ensembles of natural images [1]. Their non Gaussian properties [15, 16] are less well understood, but they indicate the existence of a rich statistical structure. In this work we present a detailed study of the marginal statistics of a variable related to the edges in the images. A numerical analysis shows that it exhibits extended self-similarity [3, 4, 5]. This is a scaling property stronger than self-similarity: all its moments can be expressed as a power of any given moment. More interesting, all the exponents can be predicted in terms of a multiplicative log-Poisson process. This is the very same model that was used very recently to predict the correct exponents of the structure functions of turbulent flows [6]. These results allow us to study the underlying multifractal singularities. In particular we find that the most singular structures are one-dimensional: the most singular manifold consists of sharp edges. Category: Visual Processing. 1 Introduction An important motivation for studying the statistics of natural images is its relevance for the modeling of the visual system. In particular, the epigenetic development • e-mail: amturiel@delta.ft.uam.es t e-mail: matog@cab.cnea.edu.ar +To whom correspondence should be addressed. e-mail: parga@delta.ft.uam.es §e-mail: nadal@lps.ens.fr 'Laboratoire associe au C.N.R.S. (U.R.A. 1306), a l'ENS, et aux Universites Paris VI et Paris VII. Self-similarity PropeT1ies of Natural Images 837 could lead to the adaptation of visual processing to the statistical regularities in the visual scenes [8, 9, 10, 11, 12, 13]. Most of these predictions on the development of receptive fields have been obtained using a gaussian description of the environment contrast statistics. However non Gaussian properties like the ones found by [15, 16] could be important. To gain further insight into non Gaussian aspects of natural scenes we investigate the self similarity properties of an edge type variable [14]. Scale invariance in natural images is a well-established property. In particular it appears as a power law behaviour of the power spectrum of luminosity contrast: S(f) ex: IfIL'! (the parameter 1] depends on the particular images that has been included in the dataset). A more detailed analysis of the scaling properties of the luminosity contrast was done by [15, 16]. These authors noted the possible analogy between the statistics of natural images and turbulent flows. There is however no model to explain the scaling behaviour that they observed. On the other hand, a large amount of effort has been put to understand the statistics of turbulent flows and to develop predictable models (see e.g. [17]). Qualitative and quantitative theories of fully developed turbulence elaborate on the original argument of Kolmogorov [2]. The cascade of energy from one scale to another is described in terms of local energy dissipation per unit mass within a box of linear size r. This quantity, fr, is given by: (1) where Vi(X) is the ith component of the velocity at point x. This variable has Sel/Similarity (SS) properties that is, there is a range of scales r (called the inertial range) where: (2) here < f~ > denotes the pth moment of the energy dissipation marginal distribution. A more general scaling relation, called Extended Self-Similarity (ESS) has been found to be valid in a much larger scale domain. This relation reads (3) where p(p, q) is the ESS exponent of the pth moment with respect to the qth moment. Let us notice that if SS holds then Tp = Tqp(p, q). In the following we will refer all the moments to < f; >. 2 The Local Edge Variance For images the basic field is the contrast c(x), that we define as the difference between the luminosity and its average. By analogy with the definition in eq. (1) we will consider a variable that accumulates the value of the variation of the contrast. We choose to study two variables, defined at position x and at scale r. The variable fh,r(X) takes contributions from edges transverse to a horizontal segment of size r: l1xl+r (ac(x/))2 fh,r(X) = -ady r Xl y X'={y,X2} (4) A vertical variable fv,r(X) is defined similarly integrating along the vertical direction. We will refer to the value of the derivative of the contrast along a given direction as an edge transverse to that direction. This is justified in the sense that in the presence of borders this derivative will take a great value, and it will almost vanish 838 A. Turiel, G. Mato, N. Parga and l-P. Nadal if evaluated inside an almost-uniformly illuminated surface. Sharp edges will be the maxima ofthis derivative. According to its definition, €/,r(x) ( 1 = h, v) is the local linear edge variance along the direction 1 at scale r. Let us remark that edges are well known to be important in characterizing images. A recent numerical analysis suggests that natural images are composed of statistically independent edges [18]. We have analyzed the scaling properties of the local linear edge variances in a set of 45 images taken into a forest, of 256 x 256 pixels each (the images have been provided to us by D. Ruderman; see [16] for technical details concerning them). An analysis of the image resolution and of finite size effects indicates the existence of upper and lower cut-offs. These are approximately r = 64 and r = 8, respectively. First we show that SS holds in a range of scales r with exponents Th,p and Tv,p. This is illustrated in Fig. (1) where the logarithm of two moments of horizontal and vertical local edge variances are plotted as a function of In r; we see that SS holds, but not in the whole range. ESS holds in the whole considered range; two representative graphs are shown in Fig. (2). The linear dependence of In < €f,r > vs In < €f,r > is observed in both the horizontal (l = h) and the vertical (l = v) directions. This is similar to what is found in turbulence, where this property has been used to obtain a more accurate estimation of the exponents of the structure functions (see e.g. [17] and references therein) . The exponents Ph(p,2) and Pv(p,2), estimated with a least squares regression, are shown in Fig. (3) as a function of p. The error bars refer to the statistical dispersion. From figs. (1-3) one sees that the horizontal and vertical directions have similar statistical properties. The SS exponents differ, as can be seen in Fig(I); but, surprisingly, ESS not only holds in both directions, but it does it with the same ESS exponents, i.e. Ph(P,2) '" Pv(p, 2). 3 ESS and multiplicative processes Let us now consider scaling models to predict the Jrdependence of the ESS exponents Pl(p, 2). (Since ESS holds, the SS exponents Tl,p can be obtained from the Pl(p, 2)' s by measuring 72,2). The simplest scaling hypothesis is that, for a random variable €r(x) observed at the scale r (such as €/,r(x)), its probability distribution Pr(€r(x) = €) can be obtained from any other scale L by Pr(€) = a(r~ L) PL (a(r~ L)) (5) From this one derives easily that a(r, L) = [~:~~P/p (for any p) and p(p, 2) ex: p; if SS holds, Tp ex: p: for turbulent flows this corresponds to the Kolmogorov prediction for the SS exponents [2]. Fig (3) shows that this naive scaling is violated. This discrepancy becomes more dramatic if eq. (5) is expressed in terms of a normalized variable. Taking €~ = limp-+oo < €~+l > / < €~ > ( that can be shown to be the maximum value of €r, which in fact is finite) the new variable is defined as ir = €r/€~ ; 0 < ir < 1. If Pr(J) is the distribution of ir, the scaling relation eq.(5) reads Pr(J) = PL(J); this identity does not hold as can be seen in Fig. (4). A way to generalize this scaling hypothesis is to say that a is no longer a constant as in eq. (5), but an stochastic variable. Thus, one has for Pr(J) : (6) This scaling relation has been first introduced in the context of turbulent flows [6, 19, 7]. Eq. (6) is an integral representation of ESS with general (not necessarily Self-similarity Properties of Natural Images 839 linear) exponents: once the kernel GrL is chosen, the p(p, 2)'s can be predicted. It can also be phrased in terms of multiplicative processes [20, 21] : now ir = aiL, where the factor a itself becomes a stochastic variable determined by the kernel GrL (1na). Since the scale L is arbitrary (scale r can be reached from any other scale L') the kernel must obey a composition law, GrLI ®G L' L = GrL. Consequently ir can be obtained through a cascade of infinitesimal processes G6 == Gr,r+6r' Specific choices of G6 define different models of ESS. The She-Leveque (SL) [6] model corresponds to a simple process such that a is 1 with probability 1 - sand is a constant f3 with probability s. One can see that s = ll,lF In( <~tl;» and that this stochastic process yields a log-Poisson distribution for a [22]. It also gives ESS with exponents p(p, q) that is expressed in terms of the parameter f3 as follows [6]: 1 f3P - (1 - (3)p p(p,q) = 1- f3 q - (1- (3)q (7) We can now test this models with the ESS exponents obtained with the image data set. The resulting fit for the SL model is shown in Fig. (3). Both the vertical and horizontal ESS exponents can be fitted with {3 = 0.50 ± 0.03. The integral representation of ESS can also be directly tested on the probability distributions evaluated from the data. In Fig. (4) we show the prediction for Pr (f) obtained from PL(f) using eq. (6), compared with the actual Pr(f). The parameter f3 allows us to predict all the ESS exponents p(p,2). To obtain the SS exponents 7p we need another parameter. This can be chosen e.g. as 72 or as the asymptotic exponent ~, given by f~ ex: r-t::., r » 1; we prefer~. As 7p = 72 p(P, 2), then from the definition of f~ one can see that ~ = -1":!13' A least square fit of 7p was used to determine ~, obtaining ~h = 0.4 ± 0.2 for the horizontal variable and ~v = 0.5 ± 0.2. for the vertical one. 4 Multifractal analysis Let us now partition the image in sets of pixels with the same singularity exponent h of the local edge variance: fr ex: rh. This defines a multifractal with dimensions D(h) given by the Legendre transform of 7p (see e.g. [17]): D(h) = inip{ph+d-7p}, where d = 2 is the dimension of the images. We are interested in the most singular of these manifolds; let us call Doo its dimension and hmin its singularity exponent. Since f~ is the maximum value of the variable fr, the most singular manifold is given by the set of points where fr = f~, so hmin = -~. Using again that 7p = -~ (1- {3) p(P, 2) with p(P, 2) given by the SL model, one has Doo = d- (1~13)' From our data we obtain Doo,h = 1.3 ± 0.3 and Doo,v = 1.1 ± 0.3. As a result we can say that Doo,h "" Doo,v "" 1: the most singular structures are almost onedimensional. This reflects the fact that the most singular manifold consists of sharp edges. 5 Conclusions We insist on the main result of this work, which is the existence of non trivial scaling properties for the local edge variances. This property appears very similar to the one observed in turbulence for the local energy dissipation. In fact, we have seen that the SL model predicts all the relevant exponents and that, in particular, it describes the scaling behaviour of the sharpest edges in the image ensemble. It would also be interesting to have a simple generative model of images which - apart 840 A. Turiel, G. Mato, N. Parga and J-P. Nadal from having the correct power spectrum as in [23] - would reproduce the self-similar properties found in this work. Acknowledgements We are grateful to Dan Ruderman for giving us his image data base. We warmly thank Bernard Castaing for very stimulating discussions and Zhen-Su She for a discussion on the link between the scaling exponents and the dimension of the most singular structure. We thank Roland Baddeley and Patrick Tabeling for fruitful discussions. We also acknowledge Nicolas BruneI for his collaboration during the early stages of this work. This work has been partly supported by the FrenchSpanish program "Picasso" and an E.V. grant CHRX-CT92-0063. References [1] Field D. J., 1. Opt. Soc. Am. 4 2379-2394 (1987). [2] Kolmogorov, Dokl. Akad. Nauk. SSSR 30, 301-305 (1941). [3] Benzi R., Ciliberto S., Baudet C., Ruiz Chavarria G. and Tripiccione C., Europhys. Lett. 24 275-279 (1993) [4] Benzi, Ciliberto, Tripiccione, Baudet, Massaioli, and Succi, Phys. Rev. E 48, R29 (1993) [5] Benzi, Ciliberto, Baudet and Chavarria Physica D 80 385-398 (1995) [6] She and Leveque, Phys. Rev. Lett. 72,336-339 (1994). [7] Castaing, 1. Physique II, France 6, 105-114 (1996) [8] Barlow H. B., in Sensory Communication (ed. Rosenblith W.) pp. 217. (M.I.T. Press, Cambridge MA, 1961). [9] Laughlin S. B., Z. Naturf. 36 910-912 (1981). [10] van Hateren J.H. 1. Compo Physiology A 171157-170,1992. [11] Atick J. J. Network 3 213-251, 1992. [12] Olshausen B.A. and Field D. J., Nature 381, 607-609 (1996). [13] Baddeley R., Cognitive Science, in press (1997). [14] Turiel A., Mato G., Parga N. and Nadal J.-P., to appear in Phys. Rev. Lett., 1998. [15] Ruderman D. and Bialek, Phys. Rev. Lett. 73,814 (1994) [16] Ruderman D., Network 5,517-548 (1994) [17] Frisch V., Turbulence, Cambridge Vniv. Press (1995). [18] Bell and Sejnowski, Vision Research 37 3327-3338 (1997). [19] Dubrulle B., Phys. Rev. Lett. 73 959-962 (1994) [20] Novikov, Phys. Rev. E 50, R3303 (1994) [21] Benzi, Biferale, Crisanti, Paladin, Vergassola and Vulpiani, Physica D 65, 352-358 (1993). [22] She and Waymire, Phys. Rev. Lett. 74, 262-265 (1995). [23] Ruderman D., Vision Research 37 3385-3398 (1997). Self-similarity Properties of Natural Images 841 In < €~ > a b .. . " " .. Inr Inr Figure 1: Test of SS. We plot In < €f r > vs. In r for p = 3 and 5; r from 8 to 64 pixels. a) horizontal direction, l = h. b) vertical direction, l = v. In < €~ > a b In < €~ > ~ , // ....... /...-' // / .-/ ' · "" ....• ' / .... / '" / ." .. , ,/ "./'" / / ././ · // . ~. -' ./ ... ",/ // .. , ....... , ..... · . " . ' .... ,,,,,,-.-/ .. ",/' .... . " , .,..,/ /./ ./ ./ // In < €~ > Figure 2: Test of ESS. We plot In < €f.r > vs. In < €~, r > for p=3, 5; r from 8 to r = 64 pixels. a) horizontal direction, l = h. b) vertical direction, l = v. 842 A. Turiel, G. Mato, N. Parga and J-P. Nadal p(p, 2) a b 12 1. p p Figure 3: ESS exponents p(p, 2), for the vertical and horizontal variables. a) horizontal direction, Ph (P, 2). b) vertical direction, pv (p, 2). The solid line represents the fit with the SL model. The best fit is obtained with (3v '" (3h '" 0.50. P 18 16 14 12 10 8 ++ + 6 + + + 4 + + + 2 + + ++++ 0 0 0.05 0.1 0.15 0.2 f Figure 4: Verification of the validity of the integral representation of ESS, eq.(6) with a log-Poisson kernel, for horizontal local edge variance. The largest scale is L = 64. Starting from the histogram Pdf) (denoted with crosses), and using a log-Poisson distribution with parameter (3 = 0.50 for the kernel GrL , eq.(6) gives a prediction for the distribution at the scale r = 16 (squares). This has to be compared with the direct evaluation of Pr (I) (diamonds). Similar results hold for other pairs of scales. Although not shown in the figure, the test for vertical case is as good as for horizontal variable.
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Hippocampal Model of Rat Spatial Abilities Using Temporal Difference Learning David J Foster* Centre for Neuroscience Edinburgh University Richard GM Morris Centre for Neuroscience Edinburgh University Abstract Peter Dayan E25-210, MIT Cambridge, MA 02139 We provide a model of the standard watermaze task, and of a more challenging task involving novel platform locations, in which rats exhibit one-trial learning after a few days of training. The model uses hippocampal place cells to support reinforcement learning, and also, in an integrated manner, to build and use allocentric coordinates. 1 INTRODUCTION Whilst it has long been known both that the hippocampus of the rat is needed for normal performance on spatial tasksl3, 11 and that certain cells in the hippocampus exhibit place-related firing,12 it has not been clear how place cells are actually used for navigation. One of the principal conceptual problems has been understanding how the hippocampus could specify or learn paths to goals when spatially tuned cells in the hippocampus respond only on the basis of the rat's current location. This work uses recent ideas from reinforcement learning to solve this problem in the context of two rodent spatial learning results. Reference memory in the watermazell (RMW) has been a key task demonstrating the importance of the hippocampus for spatial learning. On each trial, the rat is placed in a circular pool of cloudy water, the only escape from which is a platform which is hidden (below the water surface) but which remains in a constant position. A random choice of starting pOSition is used for each trial. Rats take asymptotically short paths after approximately 10 trials (see figure 1 a). Delayed match-to-place (DMP) learning is a refined version in which the platform'S location is changed on each day. Figure 1 b shows escape latencies for rats given four trials per day for nine days, with the platform in a novel position on each day. On early days, acquisition ·Crichton Street, Edinburgh EH8 9LE, United Kingdom. Funded by Edin. Univ. Holdsworth Scholarship, the McDonnell-Pew foundation and NSF grant IBN-9634339. Email: djf@cfn.ed.ac.uk 146 D. J Foster; R. G. M. Mo"is and P. Dayan 100 100 90 b 90 80 a _ 70 '" i oo '" 50 -' ~40 ~30 20 10 13 17 21 2S Figure 1: a) Latencies for rats on the reference memory in the watermaze (RMW) task (N=8). b) Latencies for rats on the Delayed Match-to-Place (DMP) task (N=62). is gradual but on later days, rats show one-trial learning, that is, near asymptotic performance on the second trial to a novel platform position. The RMW task has been extensively modelled. 6,4,5,20 By contrast, the DMP task is new and computationally more challenging. It is solved here by integrating a standard actor-critic reinforcement learning system2,7 which guarantees that the rat will be competent to perform well in arbitrary mazes, with a system that learns spatial coordinates in the maze. Temporal difference learning 1 7 (TO) is used for actor, critic and coordinate learning. TO learning is attractive because of its generality for arbitrary Markov decision problems and the fact that reward systems in vertebrates appear to instantiate it. 14 2 THEMODEL The model comprises two distinct networks (figure 2): the actor-critic network and a coordinate learning network. The contribution of the hippocampus, for both networks, is to provide a state-space representation in the form of place cell basis functions. Note that only the activities of place cells are required, by contrast with decoding schemes which require detailed information about each place cell.4 ACTOR-CRITIC SYSTEM COORDINATE SYSTEM r------------1 Remembered 1 Goal coordinates 1 1 VECTOR COMPUTA nONI ~ Coordinate Representation 1 1 ______ -------1 Figure 2: Model diagram showing the interaction between actor-critic and coordinate system components. Hippocampal Model of Rat Spatial Abilities Using TD Learning 147 2.1 Actor-Critic Learning Place cells are modelled as being tuned to location. At position p, place cell i has an output given by h(p) = exp{ -lip - sdI2/2(12}, where Si is the place field centre, and (1 = 0.1 for all place fields. The critic learns a value function V(p) = L:i wih(p) which comes to represent the distance of p from the goal, using the TO rule 6.w~ ex: 8t h(pt), where (1) is the TD error, pt is position at time t, and the reward r(pt, pt+I) is 1 for any move onto the platform, and 0 otherwise. In a slight alteration of the original rule, the value V (p) is set to zero when p is at the goal, thus ensuring that the total future rewards for moving onto the goal will be exactly 1. Such a modification improves stability in the case of TD learning with overlapping basis functions. The discount factor, I' was set to 0.99. Simultaneously the rat refines a policy, which is represented by eight action cells. Each action cell (aj in figure 2) receives a parameterised input at any position p: aj (p) = L:i qjdi (p). An action is chosen stochastically with probabilities given by P(aj) = exp{2aj}/ L:k exp{2ak}. Action weights are reinforced according to:2 (2) where 9j((Jt) is a gaussian function of the difference between the head direction (Jt at time t and the preferred direction of the jth action cell. Figure 3 shows the development of a policy over a few trials. V(p)l Triall V(p) 1 TrialS V(P)l Triall3 0.5 0.5 0.5 I 0. 01 0: 0.5 0.5 0.5 0.5 0.5 .---'---0.5 0 ------0 -0.5 -0.5 Figure 3: The RMW task: the value function gradually disseminates information about reward proximity to all regions of the environment. Policies and paths are also shown. There is no analytical guarantee for the convergence of TD learning with policy adaptation. However our simulations show that the algorithm always converges for the RMW task. In a simulated arena of diameter 1m and with swimming speeds of 20cm/s, the simulation matched the performance of the real rats very closely (see figure S). This demonstrates that TD-based reinforcement learning is adequately fast to account for the learning performance of real animals. 148 D. 1. Foster, R. G. M Morris and P. Dayan 2.2 Coordinate Learning Although the learning of a value function and policy is appropriate for finding a fixed platform, the actor-critic model does not allow the transfer of knowledge from the task defined by one goal position to that defined by any other; thus it could not generate the sort of one-trial learning that is shown by rats on the DMP task (see figure 1 b). This requires acquisition of some goal-independent know ledge about s~ace. A natural mechanism for this is the path integration or self-motion system. 0,10 However, path integration presents two problems. First, since the rat is put into the maze in a different position for each trial, how can it learn consistent coordinates across the whole maze? Second, how can a general, powerful, but slow, behavioral learning mechanism such as TO be integrated with a specific, limited, but fast learning mechanism involving spatial coordinates? Since TO critic learning is based on enforcing consistency in estimates of future reward, we can also use it to learn spatially consistent coordinates on the basis of samples of self-motion. It is assumed that the rat has an allocentric frame of reference.1s The model learns parameterised estimates of the x and y coordinates of all positions p: x(p) = Li w[ fi(P) and y(p) = Li wY h(p), Importantly, while place cells were again critical in supporting spatial representation, they do not embody a map of space. The coordinate functions, like the value function previously, have to be learned. As the simulated rat moves around, the coordinate weights {w[} are adjusted according to: t Llwi ()( (Llxt + X (pt+l ) - X(pt)) L At - k h(pk) (3) k=1 where Llxt is the self-motion estimate in the x direction. A similar update is applied to {wn. In this case, the full TO(A) algorithm was used (with A = 0.9); however TD(O) could also have been used, taking slightly longer. Figure 4a shows the x and y coordinates at early and late phases of learning. It is apparent that they rapidly become quite accurate - this is an extremely easy task in an open field maze. An important issue in the learning of coordinates is drift, since the coordinate system receives no direct information about the location of the origin. It turns out that the three controlling factors over the implicit origin are: the boundary of the arena, the prior setting of the coordinate weights (in this case all were zero) and the position and prior value of any absorbing area (in this case the platform). If the coordinate system as a whole were to drift once coordinates have been established, this would invalidate coordinates that have been remembered by the rat over long periods. However, since the expected value of the prediction error at time steps should be zero for any self-consistent coordinate mapping, such a mapping should remain stable. This is demonstrated for a single run: figure 4b shows the mean value of coordinates x evolving over trials, with little drift after the first few trials. We modeled the coordinate system as influencing the choice of swimming direction in the manner of an abstract action. I5 The (internally specified) coordinates of the most recent goal position are stored in short term memory and used, along with the current coordinates, to calculate a vector heading. This vector heading is thrown into the stochastic competition with the other possible actions, governed by a single weight which changes in a similar manner to the other action weights (as in equation 2, see also fig 4d), depending on the TO error, and on the angular proximity of the current head direction to the coordinate direction. Thus, whether the the coordinate-based direction is likely to be used depends upon its past performance. One simplification in the model is the treatment of extinction. In the DMP task, Hippocampal Model of Rat Spatial Abilities Using 1D Learning " TJUAL d i: ~Ol ~o !" ~o ,. 149 III .1 26 16 TRIAL Figure 4: The evolution of the coordinate system for a typical simulation run: a.) coordinate outputs at early and late phases of learning, b.) the extent of drift in the coordinates, as shown by the mean coordinate value for a single run, c.) a measure f d· A2 ~ ~ {X r (Pr.)-Xr -X(pr.)}2 o coor mate error for the same run (7E = r r. (Np-l)Nr ' where k indexes measurement points (max Np ) and r indexes runs (max Nr), Xr(Pk) is the model estimate of X at position Pk, X(Pk) is the ideal estimate for a coordinate system centred on zero, and Xr is the mean value over all the model coordinates, d.) the increase during training of the probability of choosing the abstract action. This demonstrates the integration of the coordinates into the control system. real rats extinguish to a platform that has moved fairly quickly whereas the actorcritic model extinguishes far more slowly. To get around this, when a simulated rat reaches a goal that has just been moved, the value and action weights are reinitialised, but the coordinate weights wf and wf, and the weights for the abstract action, are not. 3 RESULTS The main results of this paper are the replication by simulation of rat performance on the RMW and DMP tasks. Figures la and b show the course of learning for the rats; figures Sa and b for the model. For the DMP task, one-shot acquisition is apparent by the end of training. 4 DISCUSSION We have built a model for one-trial spatial learning in the watermaze which uses a single TD learning algorithm in two separate systems. One system is based on a reinforcement learning that can solve general Markovian decision problems, and the other is based on coordinate learning and is specialised for an open-field water maze. Place cells in the hippocampus offer an excellent substrate for learning the actor, the critic and the coordinates. The model is explicit about the relationship between the general and specific learning systems, and the learning behavior shows that they integrate seamlessly. As currently constituted, the coordinate system would fail if there were a barrier in the maze. We plan to extend the model to allow the coordinate system to specify abstract targets other than the most recent platform position - this could allow it fast navigation around a larger class of environments. It is also important to improve the model of learning 'set' behavior - the information about the nature of 150 D. 1. Foster; R. G. M. Mo"is and P. Dayan 14 a 12 b 12 10 §. z ~ S> .. j:10\ ~ . ~ ~ . '"~ ............................................ 0~D.~yl~~y~2~D~.y~3~D~.y~47~~yS~~~.~~~y~7~~~y~.7D.~y9~ Figure 5: a.) Performance of the actor-critic model on the RMW task, and b.) performance of the full model on the DMP task. The data for comparison is shown in figures la and b. the DMP task that the rats acquire over the course of the first few days of training. Interestingly, learning set is incomplete - on the first trial of each day, the rats still aim for the platform position on the previous day, even though this is never correct.16 The significant differences in the path lengths on the first trial of each day (evidence in figure Ib and figure 5b) come from the relative placements of the platforms. However, the model did not use the same positions as the empirical data, and, in any case, the model of exploration behavior is rather simplistic. The model demonstrates that reinforcement learning methods are perfectly fast enough to match empirical learning curves. This is fortunate, since, unlike most models specifically designed for open-field navigation,6,4,5,2o RL methods can provably cope with substantially more complicated tasks with arbitrary barriers, etc, since they solve the temporal credit assignment problem in its full generality. The model also addresses the problem that coordinates in different parts of the same environment need to be mutually consistent, even if the animal only experiences some parts on separate trials. An important property of the model is that there is no requirement for the animal to have any explicit knowledge of the relationship between different place cells or place field position, size or shape. Such a requirement is imposed in various models.9,4,6,2o Experiments that are suggested by this model (as well as by certain others) concern the relationship between hippocampally dependent and independent spatial learning. First, once the coordinate system has been acquired, we predict that merely placing the rat at a new location would be enough to let it find the platform in one shot, though it might be necessary to reinforce the placement e.g. by first placing the rat in a bucket of cold water. Second, we know that the establishment of place fields in an environment happens substantiallr faster than establishment of one-shot or even ordinary learning to a platform.2 We predict that blocking plasticity in the hippocampus following the establishment of place cells (possibly achieved without a platform) would not block learning of a platform. In fact, new experiments show that after extensive pre-training, rats can perform one-trial learning in the same environment to new platform positions on the DMP task without hippocampal synaptic plasticity. 16 This is in contrast to the effects of hippocampal lesion, which completely disrupts performance. According to the model, coordinates will have been learned during pre-training. The full prediction remains untested: that once place fields have been established, coordinates could be learned in the absence of hippocampal synaptic plasticity. A third prediction follows from evidence that rats with restricted hippocampal lesions can learn the fixed platform Hippocampal Model of Rat Spatial Abilities Using TD Learning 151 task, but much more slowly, based on a gradual "shaping" procedure.22 In our model, they may also be able to learn coordinates. However, a lengthy training procedure could be required, and testing might be complicated if expressing the knowledge required the use of hippocampus dependent short-term memory for the last platform location.I6 One way of expressing the contribution of the hippocampus in the model is to say that its function is to provide a behavioural state space for the solution of complex tasks. Hence the contribution of the hippocampus to navigation is to provide place cells whose firing properties remain consistent in a given environment. It follows that in different behavioural situations, hippocampal cells should provide a representation based on something other than locations and, indeed, there is evidence for this.8 With regard to the role of the hippocampus in spatial tasks, the model demonstrates that the hippocampus may be fundamentally necessary without embodying a map. References [1] Barto, AG & Sutton, RS (1981) BioI. Cyber., 43:1-8. [2] Barto, AG, Sutton, RS & Anderson, CW (1983) IEEE Trans. on Systems, Man and Cybernetics 13:834-846. [3] Barto, AG, Sutton, RS & Watkins, CJCH (1989) Tech Report 89-95, CAIS, Univ. Mass., Amherst, MA. [4] Blum, KI & Abbott, LF (1996) Neural Computation, 8:85-93. [5] Brown, MA & Sharp, PE (1995) Hippocampus 5:171-188. [6] Burgess, N, Reece, M & O'Keefe, J (1994) Neural Networks, 7:1065-1081. [7] Dayan, P (1991) NIPS 3, RP Lippmann et aI, eds., 464-470. [8] Eichenbaum, HB (1996) Curro Opin. Neurobiol., 6:187-195. [9] Gerstner, W & Abbott, LF (1996) J. Computational Neurosci. 4:79-94. [10] McNaughton, BL et a1 (1996) J. Exp. BioI., 199:173-185. [11] Morris, RGM et al (1982) Nature, 297:681-683. [12] O'Keefe, J & Dostrovsky, J (1971) Brain Res., 34(171). [13] Olton, OS & Samuelson, RJ (1976) J. Exp. Psych: A.B.P., 2:97-116. Rudy, JW & Sutherland, RW (1995) Hippocampus, 5:375-389. [14] SchUltz, W, Dayan, P & Montague, PR (1997) Science, 275, 1593-1599. [15] Singh, SP Reinforcement learning with a hierarchy of abstract models. [16] Steele, RJ & Morris, RGM in preparation. [17] Sutton, RS (1988) Machine Learning, 3:9-44. [18] Taube, JS (1995) J. Neurosci. 15(1):70-86. [19] Tsitsiklis, IN & Van Roy, B (1996) Tech Report LIDS-P-2322, M.LT. [20] Wan, HS, Touretzky, OS & Redish, AD (1993) Proc. 1993 Connectionist Models Summer School, Lawrence Erlbaum, 11-19. [21] Watkins, CJCH (1989) PhD Thesis, Cambridge. [22] Whishaw, IQ & Jarrard, LF (1996) Hippocampus [23] Wilson, MA & McNaughton, BL (1993) Science 261:1055-1058. Reinforcement Learning for Call Admission Control and Routing in Integrated Service Networks Peter Marbach" LIDS MIT Cambridge, MA, 02139 email: marbach@mi t . edu Miriam Schulte Zentrum Mathematik Technische UniversWit Miinchen D-80290 Munich Germany Oliver Mihatsch Siemens AG Corporate Technology, ZT IK 4 0-81730 Munich, Germany email:oliver.mihatsch@ mchp.siemens.de John N. Tsitsiklis LIDS MIT Cambridge, MA, 02139 email: jnt@mit. edu Abstract In integrated service communication networks, an important problem is to exercise call admission control and routing so as to optimally use the network resources. This problem is naturally formulated as a dynamic programming problem, which, however, is too complex to be solved exactly. We use methods of reinforcement learning (RL), together with a decomposition approach, to find call admission control and routing policies. The performance of our policy for a network with approximately 1045 different feature configurations is compared with a commonly used heuristic policy. 1 Introduction The call admission control and routing problem arises in the context where a telecommunication provider wants to sell its network resources to customers in order to maximize long term revenue. Customers are divided into different classes, called service types. Each service type is characterized by its bandwidth demand, its average call holding time and the immediate reward the network provider obtains, whenever a call of that service type is • Author to whom correspondence should be addressed. Reinforcement Learning for Call Admission Control and Routing 923 accepted. The control actions for maximizing the long term revenue are to accept or reject new calls (Call Admission Control) and, if a call is accepted, to route the call appropriately through the network (Routing). The problem is naturally formulated as a dynamic programming problem, which, however, is too complex to be solved exactly. We use the methodology of reinforcement learning (RL) to approximate the value function of dynamic programming. Furthermore, we pursue a decomposition approach, where the network is viewed as consisting of link processes, each having its own value function. This has the advantage, that it allows a decentralized implementation of the training methods of RL and a decentralized implementation of the call admission control and routing policies. Our method learns call admission control and routing policies which outperform the commonly used heuristic "Open-Shortest-Path-First" (OSPF) policy. In some earlier related work, we applied RL to the call admission problem for a single communication link in an integrated service environment. We found that in this case, RL methods performed as well, but no better than, well-designed heuristics. Compared with the single link problem, the addition of routing decisions makes the network problem more complex and good heuristics are not easy to derive. 2 Call Admission Control and Routing We are given a telecommunication network consisting of a set of nodes N = {I, ... , N} and a set of Iinks .c = {I, ... , L}, where link I has a a total capacity of B(l) units of bandwidth. We support a set M = {I, "', M} of different service types, where a service type m is characterized by its bandwidth demand b(m), its average call holding time I/v(m) (here we assume that the call holding times are exponentially distributed) and the immediate reward c( m) we obtain, whenever we accept a call of that service type. A link can carry simultaneously any combination of calls, as long as the bandwidth used by these calls does not exceed the total bandwidth of the link (Capacity Constraint). When a new call of service type m requests a connection between a node i and a node j, we can either reject or accept that request (Call Admission Control). If we accept the call, we choose a route out of a list of predefined routes (Routing). The call then uses b(m) units of bandwidth on each link along that route for the duration of the call. We can, therefore, only choose a route, which does not violate the capacity constraints of its links, if the call is accepted. Furthermore, if we accept the call, we obtain an immediate reward c( m). The objective is to exercise call admission control and routing in such a way that the long term revenue obtained by accepting calls is maximized. We can formulate the call admission control and routing problem using dynamic programming (e. g. Bertsekas, 1995). Events w which incur state transitions, are arrivals of new calls and call terminations. The state Xt at time t consists of a list for each route, indicating how many calls of each service type are currently using that route. The decision/control Ut applied at the time t of an arrival of a new call is to decide, whether to reject or accept the call, and, if the call is accepted, how to route it through the network. The objective is to learn a policy that assigns decisions to each state so as to where E{·} is the expectation operator, tk is the time when the kth event happens, g( Xtk' Wk, Ut,,) is the immediate reward associated with the kth event, and f3 is a discount factor that makes immediate rewards more valuable than future ones. 924 P Marbach, O. Mihatsch, M. Schulte and 1. N. Tsitsiklis 3 Reinforcement Learning Solution RL methods solve optimal control (or dynamic programming) problems by learning good approximations to the optimal value function r, given by the solution to the Bellman optimality equation which takes the following form for the caB admission control and routing problem J*(x) = Er {e- th } Ew { max [g(x,w, u) + J*(X I )]} ueU(x) where U ( x) is the set of control actions available in the current state x, T is the time when the first event w occurs and x' is the successor state. Note that x' is a deterministic function of the current state x, the control u and the event w. RL uses a compact representation j (', 0) to learn and store an estimate of J" (.). On each event, i(., 0) is both used to make decisions and to update the parameter vector e. In the caB admission control and routing problem, one has only to choose a control action when a new call requests a connection. In such a case, J (,,0) is used to choose a control action according to the formula u=arg max [g(x,w,u) + J(X', e)] ueU(x) (1) This can be expressed in words as follows. Decision Making: When a new call requests a connection, use J (', e) to evaluate, for each permissible route, the successor state x' we transit to, when we choose that route, and pick a route which maximizes that value. If the sum of the immediate reward and the value associated with this route is higher than the value of the current state, route the call over that route; otherwise reject the call. Usually, RL uses a global feature extractor f(x) to form an approximate compact representation of the state of the system, which forms the input to a function approximator i(., e). Sutton's temporal difference (TO()'» algorithms (Sutton, 1988) can then be used to train i(., 0) to learn an estimate of J*. Using ID(O), the update at the kth event takes the following form where dk e-/J(t/c-t/c-d (g(Xt/c, Wk, Ut/c) + J(!(Xt/c), ek-I)) -J(I(Xt/C_l)' Ok-I) and where 'Yk is a small step size parameter and Utk is the control action chosen according to the decision making rule described above. Here we pursue an approach where we view the network as being composed of link processes. Furthermore, we decompose immediate rewards g( Xtk' Wk, Ut/c) associated with the kth event, into link rewards g(l) (Xt/c, Wk, Ut/c) such that L g(Xtk' Wk. Ut/c) = L gil) (Xtl:' Wk, UtI:) 1=1 We then define, for each link I, a value function J(I) (I(l) (x), e( I»), which is interpreted as an estimate of the discounted long term revenue associated with that link. Here, f(l) defines a local feature, which forms the input to the value function associated with link I. To obtain Reinforcement Learning for Call Admission Control and Routing an approximation of J* (x), the functions ](1) (J(l) (x), 0(1)) are combined as follows L L ](1) (J(I) (x), (J(l)). 1=1 925 At each event, we update the parameter vector (J(l) of link 1, only if the event is associated with the link. Events associated with a link 1 are arrivals of new calls which are potentially routed over link 1 and termination of calls which were routed over the link I. The update rule of the parameter vector 0(1) is very similar to the TD(O) algorithm described above (J(l) (J(l) + "V(I)d(I)V ()](l) (/(1) (x ) (J(l) ) k k-l Ik k 9 I tk_1 , k-l (2) where e - i3(t~') -t~/~ I) (g(l) (x t(l), Wkl ) , Ut(l)) + ](l) (J(l) (xt(l»), (Jkl~ 1)) k k k (3) _](1) (J(l) (xt (/) ), (Jkl~l) k-I and where Ill) is a small step size parameter and tr) is the time when the kth event will associated with link 1 occurs. Whenever a new call of a service of type m is routed over a route r which contains the link i, the immediate reward g(l) associated with the link i is equal to c( m) / #r, where #r is the number of links along the route r. For all other events, the immediate reward associated with link 1 is equal to O. The advantage of this decomposition approach is that it allows decentralized training and decentralized decision making. Furthermore, we observed that this decomposition approach leads to much shorter training times for obtaining an approximation for J* than the approach without decomposition. All these features become very important if one considers applying methods of RL to large integrated service networks supporting a fair number of different service types. We use exploration to obtain the states at which we update the parameter vector O. At each state, with probability p == 0.5, we apply a random action, instead of the action recommended by the current value function, to generate the next state in our training trajectory. However, the action Ut(I), that is used in the update rule (3), is still the one chosen ack cording to the rule given in (1). Exploration during the training significantly improved the performance of the policy. Table I: Service Types. SERVICE TYPE m 1 2 3 BANDWIDTH DEMAND b( m) 1 3 5 AVERAGE HOLDING TIME l/v(m) 10 10 2 IMMEDIATE REWARD c( m) 1 2 50 4 Experimental Results In this section, we present experimental results obtained for the case of an integrated service network consisting of 4 nodes and 12 unidirectional links. There are two different classes of links with a total capacity of 60 and 120 units of bandwidth, respectively (indicated by thick and thin arrows in Figure 1). We assume a set M == {I, 2, 3} of three different service types. The corresponding bandwidth demands, average holding times and immediate 926 P. Marbach, O. Mihatsch, M Schulte and 1. N. Tsitsiklis Figure 1: Telecommunication Network Consisting of 4 Nodes and 12 Unidirectional Links. PERFORMANCE DURING LEAANING Figure 2: Average Reward per TIme Unit During the Whole Training Phase of 107 Steps (Solid) and During Shorter Time Windows of 105 Steps (Dashed). rewards are given in Table 1. Call arrivals are modeled as independent Poisson processes, with a separate mean for each pair of source and destination nodes and each service type. Furthermore, for each source and destination node pair, the list of possible routes consists of three entries: the direct path and the two alternative 2-hop-routes. We compare the policy obtained through RL with the commonly used heuristic OSPF (Open Shortest Path First). For every pair of source and destination nodes, OSPF orders the list of predefined routes. When a new call arrives, it is routed along the first route in the corresponding list, that does not violate the capacity constraint; if no such a route exists, the call is rejected. We use the average reward per unit time as performance measure to compare the two policies. For the RL approach, we use a quadratic approximator, which is linear with respect to the parameters ()(I), as a compact representation of ](1). Other approximation architectures were tried, but we found that the quadratic gave the best results with respect to both the speed of convergence and the final performance. As inputs to the compact representation Reinforcement Learning for Call Admission Control and Routing o o 5 AVERAGE REWARD potential reward reward obtained by RL reward obtained by OSPF 50 10 15 100 150 reward per time un~ COMPARISON OF REJECTION RATES 20 25 30 35 percentage of calls rejected 927 200 250 40 45 50 Figure 3: Comparison of the Average Rewards and Rejection Rates of the RL and OSPF Policies. o 10 o 10 ROUTING (OSPF) direct link ----~~-----.0:-= alternative route no. 1 alternative route no. 2 .. 20 30 ~ 50 ~ ro ~ percentage of calls routed on direct and alternative paths direct link alternative route no. 1 alternative route no. 2 ROUTING (RL) 20 30 40 50 60 70 80 percentage of calls routed on direct and alternative paths 90 90 100 100 Figure 4: Comparison of the Routing Behaviour of the RL and OSPF Policies. ](/), we use a set of local features, which we chose to be the number of ongoing calls of each service type on link l. For the 4-node network, there are approximately 1.6. 1045 different feature configurations. Note that the total number of possible states is even higher. The results of the case studies are given in in Figure 2 (Training Phase), Figure 3 (Performance) and Figure 4 (Routing Behaviour). We give here a summary of the results. Training Phase: Figure 2 shows the average reward of the RL policy as a function of the training steps. Although the average reward increases during the training, it does not exceed 141, the average reward of the heuristic OSPF. This is due to the high amount of exploration in the training phase. Performance Comparison: The policy obtained through RL gives an average reward of 212, which as about 50% higher than the one of 141 achieved by OSPF. Furthermore, the RL policy reduces the number of rejected calls for all service types. The most significant reduction is achieved for calls of service type 3, the service type, which has the highest 928 P. Marbach, O. Mihatsch, M Schulte and I. N. Tsitsiklis immediate reward. Figure 3 also shows that the average reward of the RL policy is close to the potential average reward of 242, which is the average reward we would obtain if all calls were accepted. This leaves us to believe that the RL policy is close to optimal. Figure 4 compares the routing behaviour of the RL control policy and OSPF. While OSPF routes about 15% - 20% of all calls along one of the alternative 2-hop-routes, the RL policy almost uses alternative routes for calls of type 3 (about 25%) and routes calls of the other two service types almost exclusively over the direct route. This indicates, that the RL policy uses a routing scheme, which avoids 2-hop-routes for calls of service type 1 and 2, and which allows us to use network resources more efficiently. 5 Conclusion The call admission control and routing problem for integrated service networks is naturally formulated as a dynamic programming problem, albeit one with a very large state space. Traditional dynamic programming methods are computationally infeasible for such large scale problems. We use reinforcement learning, based on Sutton's (1988) T D(O), combined with a decomposition approach, which views the network as consisting of link processes. This decomposition has the advantage that it allows decentralized decision making and decentralized training, which reduces significantly the time of the training phase. We presented a solution for an example network with about 1045 different feature configurations. Our RL policy clearly outperforms the commonly used heuristic OSPF. Besides the game of backgammon (Tesauro, 1992), the elevator scheduling (Crites & Barto, 1996), the jop-shop scheduling (Zhang & Dietterich, 1996) and the dynamic channel allocation (Singh & Bertsekas, 1997), this is another successful application of RL to a large-scale dynamic programming problem for which a good heuristic is hard to find. References Bertsekas, D. P; (1995) Dynamic Programming and Optimal Control. Athena Scientific, Belmont, MA. Crites, R. H., Barto, A. G. (1996) Improving elevator performance using reinforcement learning. In D. S. Touretzky, M. C. Mozer and M. E. Hasselmo (eds.), Advances in Neural Information Processing Systems 8, pp. 1017-1023. Cambridge, MA: MIT Press. Singh, S., Bertsekas, D. P. (1997) Reinforcement learning for dynamic channel allocation in cellular telephone systems. To appear in Advances in Neural Information Processing Systems 9, Cambridge, MA: MIT Press. Sutton, R. S. (1988) Learning to predict by the method of temporal differences. Machine Learning, 3:9-44. Tesauro, G. J. (1992) Practical issues in temporal difference learning. Machine Learning, 8(3/4):257-277. Zhang, W., Dietterich, T. G. (1996) High performance job-shop scheduling with a timedelay TD(>.) network. In D. S. Touretzky, M. C. Mozer and M. E. Hasselmo (eds.), Advances in Neural Information Processing Systems 8, pp. 1024-1030. Cambridge. MA: MIT Press.
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The Error Coding and Substitution PaCTs GARETH JAMES and TREVOR HASTIE Department of Statistics, Stanford University Abstract A new class of plug in classification techniques have recently been developed in the statistics and machine learning literature. A plug in classification technique (PaCT) is a method that takes a standard classifier (such as LDA or TREES) and plugs it into an algorithm to produce a new classifier. The standard classifier is known as the Plug in Classifier (PiC). These methods often produce large improvements over using a single classifier. In this paper we investigate one of these methods and give some motivation for its success. 1 Introduction Dietterich and Bakiri (1995) suggested the following method, motivated by Error Correcting Coding Theory, for solving k class classification problems using binary classifiers. • Produce a k by B (B large) binary coding matrix, ie a matrix of zeros and ones. We will denote this matrix by Z, its i, jth component by Zij, its ith row by Zi and its j th column by zj. • Use the first column of the coding matrix (Zl) to create two super groups by assigning all groups with a one in the corresponding element of Zl to super group one and all other groups to super group zero. • Train your plug in classifier (PiC) on the new two class problem. • Repeat the process for each of the B columns (Zl, Z2, ... ,ZB) to produce B trained classifiers. • For a new test point apply each of the B classifiers to it. Each classifier will produce a 'Pi which is the estimated probability the test point comes from the jth super group one. This will produce a vector of probability estimates, f> = (PI , ih., . .. ,P B) T . The Error Coding and Substitution PaCTs 543 • To classify the point calculate Li = 2::=1 IPi - Zij I for each of the k groups (ie for i from 1 to k). This is the LI distance between p and Zi (the ith row of Z). Classify to the group with lowest L 1 distance or equivalently argi min Li We call this the ECOC PaCT. Each row in the coding matrix corresponds to a unique (nonminimal) coding for the appropriate class. Dietterich's motivation was that this allowed errors in individual classifiers to be corrected so if a small number of classifiers gave a bad fit they did not unduly influence the final classification. Several PiC's have been tested. The best results were obtained by using tree's, so all the experiments in this paper are stated using a standard CART PiC. Note however, that the theorems are general to any Pic. In the past it has been assumed that the improvements shown by this method were attributable to the error coding structure and much effort has been devoted to choosing an optimal coding matrix. In this paper we develop results which suggest that a randomized coding matrix should match (or exceed) the performance of a designed matrix. 2 The Coding Matrix Empirical results (see Dietterich and Bakiri (1995» suggest that the ECOC PaCT can produce large improvements over a standard k class tree classifier. However, they do not shed any light on why this should be the case. To answer this question we need to explore its probability structure. The coding matrix, Z, is central to the PaCT. In the past the usual approach has been to choose one with as large a separation between rows (Zi) as possible (in terms of hamming distance) on the basis that this allows the largest number of errors to be corrected. In the next two sections we will examine the tradeoffs between a designed (deterministic) and a completely randomized matrix. Some of the results that follow will make use of the following assumption. k E[fji I Z,X] = LZiiqi = ZjT q j = 1, ... ,B (1) i=l where qi = P ( Gil X) is the posterior probability that the test observation is from group i given that our predictor variable is X. This is an unbiasedness assumption. It states that on average our classifier will estimate the probability of being in super group one correctly. The assumption is probably not too bad given that trees are considered to have low bias. 2.1 Deterministic Coding Matrix Let Di = 1 - 2Ld B for i = 1 ... k. Notice that ar& min Li = argi max Di so using Di to classify is identical to the ECOC PaCT. Theorem 3 in section 2.2 explains why this is an intuitive transformation to use. Obviously no PaCT can outperform the Bayes Classifier. However we would hope that it would achieve the Bayes Error Rate when we use the Bayes Classifier as our PiC for each 2 class problem. We have defined this property as Bayes Optimality. Bayes Optimality is essentially a consistency result It states, if our PiC converges to the Bayes Classifier, as the training sample size increases, then so will the PaCT. Definition 1 A PaCT is said to be Bayes Optimal if, for any test set, it always classifies to the bayes group when the Bayes Classifier is our PiC. For the ECOC PaCT this means that argi max qi = argi max Di , for all points in the predictor space, when we use the Bayes Classifier as our Pic. However it can be shown that in this case i = 1, ... ,k 544 G. James and T. Hastie It is not clear from this expression why there should be any guarantee that argi max Vi = argi max qi. In fact the following theorem tells us that only in very restricted circumstances will the ECOC PaCT be Bayes Optimal. Theorem 1 The Error Coding method is Bayes Optimal iff the Hamming distance between every pair of rows of the coding matrix is equal. The hamming distance between two binary vectors is the number of points where they differ. For general B and k there is no known way to generate a matrix with this property so the ECOC PaCT will not be Bayes Optimal. 2.2 Random Coding Matrix We have seen in the previous section that there are potential problems with using a deterministic matrix. Now suppose we randomly generate a coding matrix by choosing a zero or one with equal probability for every coordinate. Let Pi = E(I- 21ih - Zilll T) where T is the training set. Then Pi is the conditional expectation of D i and we can prove the following theorem. Theorem 2 For a random coding matrix, conditional on T, argi max Vi --+ argi max Pi a.s. as B --+ 00. Or in other words the classification from the ECOC PaCT approaches the classification from just using argi max Pi a.s. This leads to corollary 1 which indicates we have eliminated the main concern of a deterministic matrix. Corollary 1 When the coding matrix is randomly chosen the ECOC PaCT is asymptotically Bayes Optimal ie argi max Di --+ argi max qi a.s. as B --+ 00 This theorem is a consequence of the strong law. Theorems 2 and 3 provide motivation for the ECOC procedure. Theorem 3 Under assumption 1 for a randomly generated coding matrix E j) i = E Pi = qi i = 1 ... k This tells us that Vi is an unbiased estimate of the conditional probability so classifying to the maximum is in a sense an unbiased estimate of the Bayes classification. Now theorem 2 tells us that for large B the ECOC PaCT will be similar to classifying using argi max ILi only. However what we mean by large depends on the rate of convergence. Theorem 4 tells us that this rate is in fact exponential. Theorem 4 lfwe randomly choose Z then, conditional on T, for any fixed X Pr(argi max Vi i argi max ILi) ::; (k - 1) . e- mB for some constant m. Note that theorem 4 does not depend on assumption 1. This tells us that the error rate for the ECOC PaCT is equal to the error rate using argi max Pi plus a tenn which decreases exponentially in the limit. This result can be proved using Hoeffding's inequality (Hoeffding (1963». Of course this only gives an upper bound on the error rate and does not necessarily indicate the behavior for smaller values of B. Under certain conditions a Taylor expansion indicates that Pr(argi maxDi i argi maxPi) :::::: 0.5 - mVE for small values of mVE. So we The Error Coding and Substitution PaCTs o '" o o CO> o 50 1/sqrt(B) convergence lIB convergence 100 150 B Figure 1: Best fit curves for rates 1/ VB and 1/ B 545 200 might expect that for smaller values of B the error rate decreases as some power of B but that as B increases the change looks more and more exponential. To test this hypothesis we calculated the error rates for 6 different values of B (15,26,40,70,100,200) on the LEITER data set (available from the Irvine Repository of machine learning). Each value of B contains 5 points corresponding to 5 random matrices. Each point is the average over 20 random training sets. Figure 1 illustrates the results. Here we have two curves. The lower curve is the best fit of 1/ VB to the first four groups. It fits those groups well but under predicts errors for the last two groups. The upper curve is the best fit of 1/ B to the last four groups. It fits those groups well but over predicts errors for the first two groups. This supports our hypothesis that the error rate is moving through the powers of B towards an exponential fit. We can see from the figure that even for relatively low values of B the reduction in error rate has slowed substantially. This indicates that almost all the remaining errors are a result of the error rate of argi max J-li which we can not reduce by changing the coding matrix. The coding matrix can be viewed as a method for sampling from the distribution of 1- 21pj - Zij I. If we sample randomly we will estimate J-li (its mean). It is well known that the optimal way to estimate such a parameter is by random sampling so it is not possible to improve on this by designing the coding matrix. Of course it may be possible to improve on argi max J-li by using the training data to influence the sampling procedure and hence estimating a different quantity. However a designed coding matrix does not use the training data. It should not be possible to improve on random sampling by using such a procedure (as has been attempted in the past). 3 Why does the ECOC PaCT work? The easiest way to motivate why the ECOC PaCT works, in the case of tree classifiers, is to consider a very similar method which we call the Substitution PaCT. We will show that under certain conditions the ECOC PaC!' is very similar to the Substitution PaCT and then motivate the success of the later. 546 G. James and T. Hastie 3.1 Substitution PaCT The Substitution PaCT uses a coding matrix to fonn many different trees just as the ECOC PaCT does. However, instead of using the transformed training data to fonn a probability estimate for each two class problem, we now plug the original (ie k-class) training data back into the new tree. We use this training data to fonn probability estimates and classifications just as we would with a regular tree. The only difference is in how the tree is fonned. Therefore, unlike the ECOC PaCT, each tree will produce a probability estimate for each of the k classes. For each class we simply average the probability estimate for that class over our B trees. So if pf is the probability estimate for the Substitution PaCT, then 1 B pf = B LPij j=l (2) where Pij is the probability estimate for the ith group for the tree fonned from the jth column of the coding matrix. Theorem 5 shows that under certain conditions the ECOC PaCT can be thought of as an approximation to the Substitution PaCT. Theorem 5 Suppose that Pij is independentfrom the jth column of the coding matrix, for all i and j. Then as B approaches infinity the ECOC PaCT and Substitution PaCT will converge ie they will give identical classification rules. The theorem depends on an unrealistic assumption. However, empirically it is well known that trees are unstable and a small change in the data set can cause a large change in the structure of the tree so it may be reasonable to suppose that there is a low correlation. To test this empirically we ran the ECOC and Substitution PaCT's on a simulated data set. The data set was composed of 26 classes. Each class was distributed as a bivariate normal with identity covariance matrix and uniformly distributed means. The training data consisted of 10 observations from each group. Figure 2 shows a plot of the estimated probabilities for each of the 26 classes and 1040 test data points averaged over 10 training data sets. Only points where the true posterior probability is greater than 0.01 have been plotted since groups with insignificant probabilities are unlikely to affect the classification. If the two groups were producing identical estimates we would expect the data points to lie on the dotted 45 degree.line. Clearly this is not the case. The Substitution PaCT is systematically shrinking the probability estimates. However there is a very clear linear relationship (R2 :::::: 95%) and since we are only interested in the arg max for each test point we might expect similar classifications. In fact this is the. case with fewer than 4% of points correctly classified by one group but not the other. 3.2 Why does the Substitution PaCT work? The fact that pf is an average of probability estimates suggests that a reduction in variability may be an explanation for the success of the Substitution PaCT. Unfortunately it has been well shown (see for example Friedman (1996» that a reduction in variance of the probability estimates does not necessarily correspond to a reduction in the error rate. However theorem 6 provides simplifying assumptions under which a relationship between the two quantities exists. Theorem 6 Suppose that pT and pf (a[ > 0) (af > 0) (3) (4) The Error Coding and Substitution PaCTs 547 ~ II) d co ~ i co ~ d Q. C .g .... :0 ] d :0 CI) '" 0 0 d 0.0 0.2 0.4 0.6 08 1.0 Eeoc probabilitJea Figure 2: Probability estimates from both the ECOC and Substitution PaCT's where eS and eT have identical joint distributions with variance 1. pT is the probability estimate of the ith group for a k class tree method, ao and al are constants and qi is the true posterior probability. Let Var(p'!'jaT) 'V _ ~ 1 ,S V ar(pil / a 1 ) and p = corr(pil ,Pi2) (assumed constantfor all i). Then Pr(argmaxP7 = argmaxqi) ~ Pr(argmaxp; = argmaxqi) (5) if and I-p B>-'Y - P (6) (7) The theorem states that under fairly general conditions, the probability that the Substitution PaCT gives the same classification as the Bayes classifier is at least as great as that for the tree method provided that the standardized variability is low enough. It should be noted that only in the case of two groups is there a direct correspondence between the error rate and 5. The inequality in 5 is strict for most common distributions (e.g. normal, uniform, exponential and gamma) of e. Now there is reason to believe that in general p will be small. This is a result of the empirical variability of tree classifiers. A small change in the training set can cause a large change in the structure of the tree and also the final probability estimates. So by changing the super group coding we might expect a probability estimate that is fairly unrelated to previous estimates and hence a low correlation. To test the accuracy of this theory we examined the results from the simulation performed in section 3.1. We wished to estimate 'Y and p. The following table summarizes our estimates for the variance and standardizing (al) terms from the simulated data set. I Classifier Substitution PaCT Tree Method 548 '" o • o 5 Tree ECOC s..t>stilUtion ------------------------10 50 100 B (log scale) G. James and T. Hastie Figure 3: Error rates on the simulated data set for tree method, Substitution PaCf and ECOC PaCT plotted against B (on log scale) These quantities give us an estimate for, of l' = 0.227 We also derived an estimate for p of p = 0.125 We see that p is less than, so provided B ~ ~=~ ~ 8.6 we should see an improvement in the Substitution PaCT over a k class tree classifier. Figure 3 shows that the Substitution error rate drops below that of the tree classifier at almost exactly this point. 4 Conclusion The ECOC PaCT was originally envisioned as an adaption of error coding ideas to classification problems. Our results indicate that the error coding matrix is simply a method for randomly sampling from a fixed distribution. This idea is very similar to the Bootstrap where we randomly sample from the empirical distribution for a fixed data set. There you are trying to estimate the variability of some parameter. Your estimate will have two sources of error, randomness caused by sampling from the empirical distribution and the randomness from the data set itself. In our case we have the same two sources of error, error caused by sampling from 1 - 2!ftj - Zij! to estimate J-ti and error's caused by J-t itself. In both cases the first sort of error will reduce rapidly and it is the second type we are really interested in. It is possible to motivate the reduction in error rate of using argi max J-ti in terms of a decrease in variability, provided B is large enough and our correlation (p) is small enough. References Dietterich, T.G. and Bakiri G. (1995) Solving Multiclass Learning Problems via ErrorCorrecting Output Codes, Journal of Artificial Intelligence Research 2 (1995) 263-286 Diet~rich, T. G. and Kong, E. B. (1995) Error-Correcting Output Coding Corrects Bias and Variance, Proceedings of the 12th International Conference on Machine Learning pp. 313-321 Morgan Kaufmann Friedman, 1.H. (1996) On Bias, Variance, Oil-loss, and the Curse of Dimensionality, Dept of Statistics, Stanford University, Technical Report Hoeffding, W. (1963) Probability Inequalities for Sums of Bounded Random Variables. "Journal of the American Statistical Association", March, 1963
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Modeling Complex Cells in an A wake Macaque During Natural Image Viewing William E. Vinje vinjeCsocrates.berkeley.edu Department of Molecular and Cellular Biology, Neurobiology Division University of California, Berkeley Berkeley, CA, 94720 Jack L. Gallant gallantCsocrates.berkeley.edu Department of Psychology University of California, Berkeley Berkeley, CA, 94720 Abstract We model the responses of cells in visual area VI during natural vision. Our model consists of a classical energy mechanism whose output is divided by nonclassical gain control and texture contrast mechanisms. We apply this model to review movies, a stimulus sequence that replicates the stimulation a cell receives during free viewing of natural images. Data were collected from three cells using five different review movies, and the model was fit separately to the data from each movie. For the energy mechanism alone we find modest but significant correlations (rE = 0.41, 0.43, 0.59, 0.35) between model and data. These correlations are improved somewhat when we allow for suppressive surround effects (rE+G = 0.42, 0.56, 0.60, 0.37). In one case the inclusion of a delayed suppressive surround dramatically improves the fit to the data by modifying the time course of the model's response. 1 INTRODUCTION Complex cells in the primary visual cortex (area VI in primates) are tuned to localized visual patterns of a given spatial frequency, orientation, color, and drift direction (De Valois & De Valois, 1990). These cells have been modeled as linear spatio-temporal filters whose output is rectified by a static nonlinearity (Adelson & Bergen, 1985); more recent models have also included a divisive contrast gain control mechanism (Heeger, 1992; Wilson & Humanski, 1993; Geisler & Albrecht, 1997). We apply a modified form of these models to a stimulus that simulates natural vision. Our model uses relatively few parameters yet incorporates the cells' temporal response properties and suppressive influences from beyond the classical receptive field (C RF). Modeling Complex Cells during Natural Image Viewing 237 2 METHODS Data Collection: Data were collected from One awake behaving Macaque monkey, using single unit recording techniques described elsewhere (Connor et al., 1997).1 First, the cell's receptive field size and location were estimated manually, and tuning curves were objectively characterized using two-dimensional sinusoidal gratings. Next a static color image of a natural scene was presented to the animal and his eye position was recorded continuously as he freely scanned the image for 9 seconds (Gallant et al., 1998).2 Image patches centered on the position of the cell's C RF (and 2-4 times the CRF diameter) were then extracted using an automated procedure. The sequence of image patches formed a continuous 9 second review movie that simulated all of the stimulation that had occurred in and around the C RF during free viewing.3 Although the original image was static, the review movies contain the temporal dynamics of the saccadic eye movements made by the animal during free viewing. Finally, the review movies were played in and around the C RF while the animal performed a fixation task. During free viewing each eye position is unique, so each image patch is likely to enter the C RF only once. The review movies were therefore replayed several times and the cell's average response with respect to the movie timestream was computed from the peri-stimulus time histogram (PSTH). These review movies also form the model's stimulus input, while its output is relative spike probability versus time (the model cell's PSTH). Before applying the model each review movie was preprocessed by converting to gray scale (since the model does not consider color tuning), setting the average luminance level to zero (on a frame by frame basis) and prefiltering with the human contrast sensitivity function to more accurately reflect the information reaching cells in VI. Divisive Normalization Model: The model consists of a classical receptive field energy mechanism, ECRF, whose output is divided by two nonclassical suppressive mechanisms, a gain control field, G, and a texture contrast field, T. ECRF(t) PSTHmodel(t) ex 1 + Q G(t - d) + f3T(t - d) (1) We include a delay parameter for suppressive effects, consistent with the hypothesis that these effects may be mediated by local cortical interactions (Heeger, 1992; Wilson & Humanski, 1993). Any latency difference between the central energy mechanism and the suppressive surround will be reflected as a positive delay offset (15 > 0 in Equation 1). Classical Receptive Field Energy Mechanism: The energy mechanism, ECRF, is composed of four phase-dependent subunits, Uti>. Each subunit computes an inner product in space and a convolution in time between the model cell's space-time classical receptive field, CRFtI>(x, y, r), and the image, I(x, y, t). U<P(t) = J J J CRFtI>(x, y, r) . I(x, y, t - r) dx dydr (2) 1 Recorcling was performed under a university-approved protocol and conformed to all relevant NIH and USDA guidelines. 2 Images were taken from a Corel Corporation photo-CD library at 1280xl024 resolution. 3Eye position data were collected at 1 KHz, whereas the monitor clisplay rate was 72.5 Hz (14 ms per frame). Therefore each review movie frame was composed of the average stimulation occurring during the corresponcling 13.8 ms of free viewing. 238 W. E. Vinje and 1. L Gallant The model presented here incorporates the simplifying assumption of a space-time separable receptive field structure, CRF4>(x, y, r) = CRF4>(x, y) CRF(r). u4>(t) = L: CRF(r) (L: L: CRF4>(x, y) . I(x, y, t - r)) T X Y (3) Time is discretized into frames and space is discretized into pixels that match the review movie input. CRF4>(x, y) is modeled as a sinusoidal grating that is spatially weighted by a Gaussian envelope (i.e. a Gabor function). In this paper CRF(r) is approximated as a delta function following a constant latency. This minimizes model parameters and highlights the model's responses to the stimulus present at each fixation. The latency, orientation and spatial frequency of the grating, and the size of the C RF envelope, are all determined empirically by maximizing the fit between model and data. 4 A static non-linearity ensures that the model PSTH does not become negative. We have e~amined both half-wave rectification, fj4>(t) = max[U4>(t), O], and halfsquaring, U4>(t) = (max[U4>(t) , 0])2; here we present the results from half-wave rectification. Half-squaring produces small changes in the model PSTH but does not improve the fit to the data. The energy mechanism is made phase invariant by averaging over the rectified phasedependent subunits: (4) Gain Control Field: Cells in V 1 incorporate a contrast gain control mechanism that compensates for changes in local luminance. The gain control field, G, models this effect as the total image power in a region encompassing the C RF and surround. G(t-<5) = L:CRF(r) (L:L:VP(kx,ky,r) ) T k% ky P(kx, ky, r) = F FT[PG(x, y, r)] F FT*[JlG(x, y, r)] JlG(x, y, r) = vG(x, y) I(x, y, (t - <5) - r) (5) (6) (7) P(kx, ky, r) is the spatial Fourier power of JlG(x, y, r) and VG is a two dimensional Gaussian weighting function whose width sets the size of the gain control field. Heeger's (1992) divisive gain control term sums over many discrete energy mechanisms that tile space in and around the area of the C RF. Equation 5 approximates Heeger's approach in the limiting case of dense tiling. Texture Contrast Field: Cells in area VI can be affected by the image surrounding the region of the CRF (Knierim & Van Essen, 1992). The responses of many VI cells are highest when the optimal stimulus is presented alone within the CRF, and lowest when that stimulus is surrounded with a texture of similar orientation and frequency. The texture contrast field, T, models this effect as the image power 4 As a fit statistic we use the linear correlation coefficient (Pearson's r) between model and data. Fitting is done with a gradient ascent algorithm. Our choice of correlation as a statistic eliminates the need to explicitly consider model normalization as a variable, and is very sensitive to latency mismatches between model and data. However, linear correlation is more prone to noise contamination than is X2 • Modeling Complex Cells during Natural Image Viewing 239 in the spatial region surrounding the C RF that matches the C RF's orientation and spatial frequency. 1 90,180,270 [ ( )] T(t-J) = 4 1: 1: CRF(r) "£1: Jp4>(kx,ky,r) 4>=0 T k", ky P4>(kx, ky, r) = F FT[p~(x, y, r)] F FT·[p~(x, y, r)] J.t~(x, y, r) = ~*(x, y) (1 - lICRF(X, y)) I(x, y, (t - J) - r) (8) (9) (10) ~* is a Gabor function whose orientation and spatial frequency match those of the best' fit C RF4> (x, y). The envelope of ~* defines the size of the texture contrast field. lICRF is a two dimensional Gaussian weighting function whose width matches the C RF envelope, and which suppresses the image center. Thus the texture contrast term picks up oriented power from an annular region of the image surrounding the C RF envelope. T is made phase invariant by averaging over phase. 3 RESULTS Thus far our model has been evaluated on a small data set collected as part of a different study (Gallant et ai., 1998). Two cells, 87A and 98C, were examined with one review movie each, while cell 97 A was examined with three review movies. Using this data set we compare the model's response in two interesting situations: cell 97 A, which had high orientation-selectivity, versus cell 87 A, which had poor orientation-selectivity; and cell 98C, which was directionally-selective, versus cell 97 A, which was not directionally-selective. CRF Energy Mechanism: We separately fit the energy mechanism parameters to each of the three different cells. For cell 97 A the three review movies were fit independently to test for consistency of the best fit parameters. Table 1 shows the correlation between model and data using only the C RF energy mechanism (a = f3 = 0 in Equation 1). The significance of the correlations was assessed via a permutation test. The correlation values for cells 97 A and 98C, though modest, are significant (p < 0.01). For these cells the 95% confidence intervals on the best fit parameter values are consistent with estimates from the flashed grating tests. The best fit parameter values for cell 97 A are also consistent across the three independently fit review movies. The model best accounts for the data from cell 97 A. This cell was highly selective for vertical gratings and was not directionally-selective. Figure 1 compares the PSTH obtained from cell 97 A with movie B to the model PSTH. The model generally responds to the same features that drive the real cell, though the match is imperfect. Much of the discrepancy between the model and data arises from our approximation of CRF(t) as a delta function. The model's response is roughly constant during Cell 87A 97A 97A 97A 98C Movie A A B C A Oriented No Yes Yes Yes Yes Directional No No No No Yes rE NA 0.41 0.43 0.59 0.35 Table 1: Correlations between model and data PSTHs. Oriented cells showed orientation-selectivity in the flashed grating test while Directional cells showed directional-selectivity during manual characterization. rE is the correlation between ECRF and the data. No fit was obtained for cell 87 A. 240 W. E. Vinje and J L. Gallant 1~--~~--~----~----~----T-----~--~----~----~ cO.8 ..-. ~ ~ £ 0.6 ~ ~ 'a ~ 0.4 ~ .~ .-. ~ ~ 0.2 1 2 3 4 5 6 7 8 Time (seconds) Figure 1: CRF energy mechanism versus data (Cell 97A, Movie B). White indicates that the model response is greater than the data, while black indicates the data is greater than the model and gray indicates regions of overlap. A perfect match between model and data would result in the entire area under the curve being gray. Our approximation of CRF(t) leads to a relatively constant model PSTH during each fixation. In contrast the real cell generally gives a phasic response as each saccade brings a new stimulus into the CRF. In general the same movie features drive both model and cell. each fixation, which causes the model PSTH to appear stepped. In contrast the data PSTH shows a strong phasic response at the beginning of each fixation when a new stimulus patch enters the cell's CRF. The model is less successful at accounting for the responses of the directionallyselective cell, 98C. This is probably because the model's space-time separable receptive field misses motion energy cues that drive the cell. The model completely failed to fit the data from cell 87 A. This cell was not orientation-selective, so the fitting procedure was unable to find an appropriate orientation for the CRF¢(x, y) Gabor function. 5 CRF Energy Mechanism with Suppressive Surround: Table 2 lists the improvements in correlation obtained by adding the gain control term (a > 0, fJ = 0 in Equation 1). For cell 97 A (all three movies) the best correlations are obtained when the surround effects are delayed by 56 ms relative to the center. The best correlation for cell 98C is obtained when the surround is not delayed. In three out of four cases the correlation values are barely improved when the surround effects are included, suggesting that the cells were not strongly surroundinhibited by these review movies. However, the improvement is quite striking in the SFor cell 87 A the correlation values in the orientation and spatial frequency parameter subspace contained three roughly equivalent maxima. Contamination by multiple cells was unlikely due to this cell's excellent isolation. 9 Modeling Complex Cells during Natural Image Viewing 241 Cell 97A 97A 97A 98C Movie A B C A rE+G 0.42 0.56 0.60 0.37 ~r +0.01 +0.13 +0.01 +0.02 Table 2: Correlation improvements due to surround gain control mechanism. rE+G gives the correlation value between the best fit model and the data. ~r gives the improvement over rEo Including G in Equation 1 leads to a dramatic correlation increase for cell 97 A, movie B, but not for the other review movies. case of cell 97 A, movie B. Figure 2 compares the data with a model using both Ecr f and G in Equation 1. Here the delayed surround suppresses the sustained responses seen in Figure 1 and results in a more phasic model PSTH that closely matches the data. We consider G and T fields both independently and in combination. For each we independently fit for Q, {3, &, and the size of the suppressive fields. However, the oriented Fourier power correlates with the total Fourier power for our sample of natural images, so that G and T are highly correlated. Combined fitting of G and T terms leads to competition and dominance by G (i.e. (3 -r 0). In this paper we only report the effects of the gain control mechanism; the texture contrast mechanism results in similar (though slightly degraded) results. 1~--~----~----~----~----~----~---.----~----~ cO.8 ..... ...... ..... ~ ~ ct 0.6 ] ..... 0~ 0.4 > ..... ~ ...... (I) 0::: 0.2 o o 1 2 3 456 Time (seconds) 7 8 Figure 2: C RF energy mechanism with delayed surround gain control versus data (Cell 97A, Movie B). Color scheme as in Figure 1. The inclusion of the delayed G term results in a more phasic model response which greatly improves the match between model and data. 9 242 W. E. Vinje and 1. L. Gallant 4 DISCUSSION This preliminary study suggests that models of the form outlined here show great promise for describing the responses of area V1 cells during natural vision. For comparison consider the correlation values obtained from an earlier neural network model that attempted to reproduce V1 cells' responses to a variety of spatial patterns (Lehky et al. 1992). They report a median correlation value of 0.65 for complex stimuli, whereas the average correlation score from Table 2 is 0.49. This is remarkable considering that our model has only 7 free parameters, a very hmited data set for fitting, doesn't yet consider color tuning or directional-selectivity and considers response across time. Future implementations of the model will use a more sophisticated energy mechanism that allows for nonseparable space time receptive field structure and more realistic temporal response dynamics. We will also incorporate more detail into the surround mechanisms, such as asymmetric surround structure and a broadband texture contrast term. By abstracting physiological observation into approximate functional forms our model balances explanatory power against parametric complexity. A cascaded series of these models may form the foundation for future modeling of cells in extra-striate areas V2 and V4. Natural image stimuli may provide an appropriate stimulus set for development and validation of these extrastriate models. Acknowledgements We thank Joseph Rogers for assistance in this study, Maneesh Sahani for the extremely useful suggestion of fitting the CRF parameters, Charles Connor for help with data collection and David Van Essen for support of data collection. References Adelson, E. H. & Bergen, J. R. (1985) Spatiotemporal energy models for the perception of motion. Journal of the Optical Society of America, A, 2, 284-299. Connor, C. C., Preddie, D. C., Gallant, J. L. & Van Essen, D. C. (1997) Spatial attention effects in macaque area V4. Journal of Neuroscience, 77, 3201-3214. De Valois, R. L. & De Valois, K. K. (1990) Spatial Vision. New York: Oxford University Press. Gallant, J. L., Connor, C. E., & Van Essen, D. C. (1998) Neural Activity in Areas V1, V2 and V4 During Free Viewing of Natural Scenes Compared to Controlled Viewing. NeuroReport, 9. Geisler, W. S., Albrecht, D. G. (1997) Visual cortex neurons in monkeys and cats: Detection, discrimination, and identification. Visual Neuroscience, 14, 897-919. Heeger, D. J. (1992) Normalization of cell responses in cat striate cortex. Visual Neuroscience, 9, 181-198. Knierim, J . J . & Van Essen, D. C. (1992) Neuronal responses to static texture patterns in area V1 of the alert macaque monkey. Journal of Neurophysiology, 67, 961-980. Lehky, S. R., Sejnowski, T . J . & Desimone, R. (1992) Predicting Responses of Nonlinear Neurons in Monkey Striate Cortex to Complex Patterns. Journal of Neuroscience, 12, 3568-3581. Wilson, H. R. & Humanski, R. (1993) Spatial frequency adaptation and contrast gain control. Vision Research, 33, 1133-1149.
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Generalization in decision trees and DNF: Does size matter? Mostefa Golea\ Peter L. Bartletth , Wee Sun Lee2 and Llew Mason1 1 Department of Systems Engineering Research School of Information Sciences and Engineering Australian National University Canberra, ACT, 0200, Australia 2 School of Electrical Engineering University College UNSW Australian Defence Force Academy Canberra, ACT, 2600, Australia Abstract Recent theoretical results for pattern classification with thresholded real-valued functions (such as support vector machines, sigmoid networks, and boosting) give bounds on misclassification probability that do not depend on the size of the classifier, and hence can be considerably smaller than the bounds that follow from the VC theory. In this paper, we show that these techniques can be more widely applied, by representing other boolean functions as two-layer neural networks (thresholded convex combinations of boolean functions). For example, we show that with high probability any decision tree of depth no more than d that is consistent with m training examples has misclassification probability no more than o ( (~ (Neff VCdim(U) log2 m log d)) 1/2), where U is the class of node decision functions, and Neff ::; N can be thought of as the effective number of leaves (it becomes small as the distribution on the leaves induced by the training data gets far from uniform). This bound is qualitatively different from the VC bound and can be considerably smaller. We use the same technique to give similar results for DNF formulae. • Author to whom correspondence should be addressed 260 M. Golea, P Bartlett, W. S. Lee and L Mason 1 INTRODUCTION Decision trees are widely used for pattern classification [2, 7]. For these problems, results from the VC theory suggest that the amount of training data should grow at least linearly with the size of the tree[4, 3]. However, empirical results suggest that this is not necessary (see [6, 10]). For example, it has been observed that the error rate is not always a monotonically increasing function of the tree size[6]. To see why the size of a tree is not always a good measure of its complexity, consider two trees, A with N A leaves and B with N B leaves, where N B « N A . Although A is larger than B, if most of the classification in A is carried out by very few leaves and the classification in B is equally distributed over the leaves, intuition suggests that A is actually much simpler than B, since tree A can be approximated well by a small tree with few leaves. In this paper, we formalize this intuition. We give misclassification probability bounds for decision trees in terms of a new complexity measure that depends on the distribution on the leaves that is induced by the training data, and can be considerably smaller than the size of the tree. These results build on recent theoretical results that give misclassification probability bounds for thresholded real-valued functions, including support vector machines, sigmoid networks, and boosting (see [1, 8, 9]), that do not depend on the size of the classifier. We extend these results to decision trees by considering a decision tree as a thresholded convex combination of the leaf functions (the boolean functions that specify, for a given leaf, which patterns reach that leaf). We can then apply the misclassification probability bounds for such classifiers. In fact, we derive and use a refinement of the previous bounds for convex combinations of base hypotheses, in which the base hypotheses can come from several classes of different complexity, and the VC-dimension of the base hypothesis class is replaced by the average (under the convex coefficients) of the VC-dimension of these classes. For decision trees, the bounds we obtain depend on the effective number of leaves, a data dependent quantity that reflects how uniformly the training data covers the tree's leaves. This bound is qualitatively different from the VC bound, which depends on the total number of leaves in the tree. In the next section, we give some definitions and describe the techniques used. We present bounds on the misclassification probability of a thresholded convex combination of boolean functions from base hypothesis classes, in terms of a misclassification margin and the average VC-dimension of the base hypotheses. In Sections 3 and 4, we use this result to give error bounds for decision trees and disjunctive normal form (DNF) formulae. 2 GENERALIZATION ERROR IN TERMS OF MARGIN AND AVERAGE COMPLEXITY We begin with some definitions. For a class ti of { -1,1 }-valued functions defined on the input space X, the convex hull co(ti) ofti is the set of [-1, l]-valued functions of the form :Ei aihi, where ai ~ 0, :Ei ai = 1, and hi E ti. A function in co(ti) is used for classification by composing it with the threshold function, sgn : IR ~ {-I, I}, which satisfies sgn(a) = 1 iff a ~ O. So f E co(ti) makes a mistake on the pair (x,y) E X x {-1,1} iff sgn(f(x» =F y. We assume that labelled examples (x,y) are generated according to some probability distribution V on X x {-I, I}, and we let Pv [E] denote the probability under V of an event E. If S is a finite subset of Z, we let Ps [E] denote the empirical probability of E (that is, the proportion of points in S that lie in E). We use Ev [.] and Es [.] to denote expectation in a similar way. For a function class H of {-I, l}-valued functions defined on the input Generalization in Decision Trees and DNF: Does Size Matter? 261 space X, the growth function and VC dimension of H will be denoted by IIH (m) and VCdim(H) respectively. In [8], Schapire et al give the following bound on the misclassification probability of a thresholded convex combination of functions, in terms of the proportion of training data that is labelled to the correct side of the threshold by some margin. (Notice that Pv [sgn(f(x» # y] ~ Pv [yf(x) ~ 0].) Theorem 1 ([8]) Let V be a distribution on X x {-I, I}, 1£ a hypothesis class with VCdim(H) = d < 00, and 8> O. With probability at least 1- 8 over a training set S of m examples chosen according to V, every function f E co(1£) and every 8> 0 satisfy ( 1 (dl 2( /d) ) 1/2) Pv [yf(x) ~ 0] ~ Ps [yf(x) ~ 8] + 0..;m og 82m + log(1/8) . In Theorem 1, all of the base hypotheses in the convex combination f are elements of a single class 1£ with bounded VC-dimension. The following theorem generalizes this result to the case in which these base hypotheses may be chosen from any of k classes, 1£1, ... , 1£k, which can have different VC-dimensions. It also gives a related result that shows the error decreases to twice the error estimate at a faster rate. Theorem 2 Let V be a distribution on X x {-I, I}, 1£1, ... ,1£k hypothesis classes with VCdim(Hi) = di , and 8 > O. With probability at least 1 - 8 over a training set S of m examples chosen according to V, every function f E co (U~=1 1£i) and every 8 > 0 satisfy both Pv [yf(x) ~ 0] ~ Ps [yf(x) ~ 8] + ( 1 (1 )1/2) o ..;m 82 (dlogm + logk) log (m82/d) + log(1/8) , Pv [yf(x) ~ 0] ~ 2Ps [yf(x) ~ 8] + o (! (812 (dlogm + logk) log (m82/d) +IOg(1/8»)), where d = E · aidj; and the ai and ji are defined by f = Ei aihi and hi E 1£j; for jiE{l, ... ,k}. Proof sketch: We shall sketch only the proof of the first inequality of the theorem. The proof closely follows the proof of Theorem 1 (see [8]). We consider { N A A } a number of approximating sets of the form eN,1 = (l/N) Ei=1 hi : hi E 1£1; , where I = (h, ... , IN) E {I, ... , k}N and N E N. Define eN = Ul eN,I' For a given f = Ei aihi from co (U~=1 1£i ), we shall choose an approximation 9 E eN by choosing hI, .. . , hN independently from {hI, h2 , ... ,}, according to the distribution defined by the coefficients ai. Let Q denote this distribution on eN. As in [8], we can take the expectation under this random choice of 9 E eN to show that, for any 8 > 0, Pv [yf(x) ~ 0] ~ Eg_Q [PD [yg(x) ~ 8/2]] + exp(-N82/8). Now, for a given I E {I, .. . ,k}N, the probability that there is a 9 in eN,1 and a 8 > 0 for which Pv [yg(x) ~ 8/2] > Ps [yg(x) ~ 8/2] + fN,1 is at most 8(N + 1) rr~1 (2:/7) d l ; exp( -mf~,zl32). Applying the union bound 262 M. Golea, P. Bartlett, W S. Lee andL Mason (over the values of 1), taking expectation over 9 I'V Q, and setting EN,l = ( ~ In (8(N + 1) n~1 (2;;; )". kN / 6N ) ) 1'2 shows that, with probability at least 1 - 6N, every f and 8 > 0 satisfy Pv [yf(x) ~ 0] ~ Eg [Ps [yg(x) ~ 8/2]] + Eg [EN,d. As above, we can bound the probability inside the first expectation in terms of Ps [yf(x) ~ 81. Also, Jensen's inequality implies that Eg [ENtd ~ (~ (In(8(N + 1)/6N) + Nln k + N L..i aidj; In(2em))) 1/2. Setting 6N = 6/(N(N + 1)) and N = r /-I In ( mf) 1 gives the result. I Theorem 2 gives misclassification probability bounds only for thresholded convex combinations of boolean functions. The key technique we use in the remainder of the paper is to find representations in this form (that is, as two-layer neural networks) of more arbitrary boolean functions. We have some freedom in choosing the convex coefficients, and this choice affects both the error estimate Ps [yf(x) ~ 81 and the average VC-dimension d. We attempt to choose the coefficients and the margin 8 so as to optimize the resulting bound on misclassification probability. In the next two sections, we use this approach to find misclassification probability bounds for decision trees and DNF formulae. 3 DECISION TREES A two-class decision tree T is a tree whose internal decision nodes are labeled with boolean functions from some class U and whose leaves are labeled with class labels U from {-I, +1}. For a tree with N leaves, define the leaf functions, hi : X -+ {-I, I} by hi(X) = 1 iff x reaches leaf i, for i = 1, ... ,N. Note that hi is the conjunction of all tests on the path from the root to leaf i. For a sample S and a tree T, let Pi = Ps [hi(X) = 1]. Clearly, P = (PI, .. " PN) is a probability vector. Let Ui E {-I, + I} denote the class assigned to leaf i. Define the class of leaf functions for leaves up to depth j as 1lj = {h : h = UI /\ U2 /\ •.• /\ U r I r ~ j, Ui E U}. It is easy to show that VCdim(1lj) ~ 2jVCdim(U) In(2ej). Let di denote the depth of leaf i, so hi E 1ld;, and let d = maxi di. The boolean function implemented by a decision tree T can be written as a thresholded convex combination of the form T(x) = sgn(f(x», where f(x) = L..~I WWi «hi(x) + 1)/2) = L..~I WWihi(X)/2 + L..~l wwd2, with Wi > 0 and L..~I Wi = 1. (To be precise, we need to enlarge the classes 1lj slightly to be closed under negation. This does not affect the results by more than a constant.) We first assume that the tree is consistent with the training sample. We will show later how the results extend to the inconsistent case. The second inequality of Theorem 2 shows that, for fixed 6 > 0 there is a constant c such that, for any distribution V, with probability at least 1 - 6 over the sample S we have Pv [T(x) 'I y] ~ 2Ps [yf(x) ~ 8] + -b L~I widiB, where B = ~ VCdim(U) log2 m log d. Different choices of the WiS and the 8 will yield different estimates of the error rate of T. We can assume (wlog) that PI ~ ... ~ PN. A natural choice is Wi = Pi and Pj+I ::.; 8 < Pj for some j E {I, ... ,N} which gives N dB Pv [T(x) 'I y] ~ 2 L Pi + (i2' i=j+I (1) Generalization in Decision Trees and DNF: Does Size Matter? 263 where d = L:~1 Pidi . We can optimize this expression over the choices of j E {I ... ,N} and () to give a bound on the misclassification probability of the tree. Let pep, U) = L:~1 (Pi - IIN)2 be the quadratic distance between the prob-ability vector P = (PI, ... ,PN ) and the uniform probability vector U = (liN, liN, ... , liN). Define Neff == N (1 - pep, U». The parameter Neff is a measure of the effective number of leaves in the tree. Theorem 3 For a fixed d > 0, there is a constant c that satisfies the following. Let V be a distribution on X x { -1, I}. Consider the class of decision trees of depth 'Up to d, with decision functions in U. With probability at least 1 - d over the training set S (of size mY, every decision tree T that is consistent with S has • 2 ) 1/2 Pv [T(x) 1= y] ~ c ( Neff VCdlm(~ log m log d , where Neff is the effective number of leaves of T. Proof: Supposing that () ~ (aIN)I/2 we optimize (1) by choice of (). If the chosen () is actually smaller than ca/ N)I/2 then we show that the optimized bound still holds by a standard VC result. If () ~ (a/N)I/2 then L:~i+l Pi ~ (}2 Neff/d. So (1) implies that P v [T (x) 1= y] ~ 2(}2 Neff /d + dB / (}2. The optimal choice of () is then (~iB/Neff)I/4. So if (~iB/Neff)I/4 ~ (a/N)I/2, we have the result. Otherwise, the upper bound we need to prove satisfies 2(2NeffB)I/2 > 2NB, and this result is implied by standard VC results using a simple upper bound for the growth function of the class of decision trees with N leaves. I Thus the parameters that quantify the complexity of a tree are: a) the complexity of the test function class U, and b) the effective number of leaves Neff. The effective number of leaves can potentially be much smaller than the total number of leaves in the tree [5]. Since this parameter is data-dependent, the same tree can be simple for one set of PiS and complex for another set of PiS. For trees that are not consistent with the training data, the procedure to estimate the error rate is similar. By defining Qi = Ps [YO'i = -1 I hi(x) = 1] and PI = Pi (l- Qi)/ (1 - Ps [T(x) 1= V]) we obtain the following result. Theorem 4 For a fixed d > 0, there is a constant c that satisfies the following. Let V be a distribution on X x { -1, 1}. Consider the class of decision trees of depth up to d, with decision functions in U. With probability at least 1 - d over the training set S (of size mY, every decision tree T has ( . () 2 ) 1/3 Pv [T(x) 1= y] ~ Ps [T(x) 1= y] + c Neff VCdim ~ log mlogd , where c is a universal constant, and Neff = N(1- pep', U» is the effective number of leaves ofT. Notice that this definition of Neff generalizes the definition given before Theorem 3. 4 DNF AS THRESHOLDED CONVEX COMBINATIONS A DNF formula defined on {-1, I}n is a disjunction of terms, where each term is a conjunction of literals and a literal is either a variable or its negation. For a given DNF formula g, we use N to denote the number of terms in g, ti to represent the ith 264 M. Golea, P. Bartlett, W S. Lee and L Mason term in f, Li to represent the set of literals in ti, and Ni the size of Li . Each term ti can be thought of as a member of the class HNi' the set of monomials with Ni literals. Clearly, IHi I = et). The DNF 9 can be written as a thresholded convex combination of the form g(x) = -sgn( - f(x)) = -sgn ( - L:f:,l Wi «ti + 1)/2)) . (Recall that sgn(a) = 1 iff a ~ 0.) Further, each term ti can be written as a thresholded convex combination of the form ti(X) = sgn(Ji(x)) = sgn (L:lkELi Vik «lk(x) - 1)/2)) . Assume for simplicity that the DNF is consistent (the results extend easily to the inconsistent case). Let ')'+ (')'-) denote the fraction of positive (negative) examples under distribution V. Let Pv + [.] (Pv - [.]) denote probability with respect to the distribution over the positive (negative) examples, and let Ps+ [.] (Ps- [.]) be defined similarly, with respect to the sample S. Notice that Pv [g(x) :f:. y] = ')'+Pv+ [g(x) = -l]+,),-Pv - [(3i)ti(X) = 1], so the second inequality of Theorem 2 shows that, with probability at least 1- 8, for any 8 and any 8i s, ( dB) N ( B) Pv [g(x) :f:. y] :::; ')'+ 2Ps+ [I(x) :::; 8] + ¥ + ')'- ~ 2Ps- [- fi(x) ~ 8i ] + 8; where d = L:f:,l WiNi and B = c(lognlog2m+log(N/8)) /m. As in the case of decision trees, different choices of 8, the 8is, and the weights yield different estimates of the error. For an arbitrary order of the terms, let Pi be the fraction of positive examples covered by term ti but not by terms ti-l, ... ,tl' We order the terms such that for each i, with ti-l. ... ,tl fixed, Pi is maximized, so that PI 2:: ... ~ PN, and we choose Wi = Pi. Likewise, for a given term ti with literals 11,'" ,LN. in an arbitrary order, let p~i) be the fraction of negative examples uncovered by literal lk but not uncovered by lk-l, ... ,11' We order the literals of term ti in the same greedy way as above so that pi i) ~ ... 2:: P~:, and we choose Vik = P~ i). For PHI:::; 8 < Pi and pLiL ~ 8i < Pi~iL, where 1 :::; j :::; Nand 1 ~ ji :::; Ni, we get ( N dB) N (N' . B) P D [g(x) :f:. y] :::; ')'+ 2 i~l Pi + ¥ + ')'- ~ 2 kf+l p~,) + 8; Now, let P = (Pl,,,,,PN) and for each term i let p(i) = (pii), ... ,p~:). Define Neff = N(1 - pcP, U)) and N~~ = Ni(1 - p(p(i) , U)), where U is the relevant uniform distribution in each case. The parameter Neff is a measure of the effective number of terms in the DNF formula. It can be much smaller than N; this would be the case if few terms cover a large fraction of the positive examples. The parameter N~~ is a measure of the effective number of literals in term ti. Again, it can be much smaller than the actual number of literals in ti: this would be the case if few literals of the term uncover a large fraction of the negative examples. Optimizing over 8 and the 8i s as in the proof of Theorem 3 gives the following result. Theorem 5 For a fixed 8 > 0, there is a constant c that satisfies the following. Let V be a distribution on X x {-I, I}. Consider the class of DNF formuLae with up to N terms. With probabiLity at Least 1 - 8 over the training set S (of size mY, every DNF formulae 9 that is consistent with S has N PD [g(x):f:. y]:::; ,),+(NeffdB)1/2 + ')'-I~)N~~B)1/2 i=l where d = maxf:.l N i , ')'± = Pv [y = ±1] and B = c(lognlog2 m + log(N/8))/m. Generalization in Decision Trees and DNF: Does Size Matter? 265 5 CONCLUSIONS The results in this paper show that structural complexity measures (such as size) of decision trees and DNF formulae are not always the most appropriate in determining their generalization behaviour, and that measures of complexity that depend on the training data may give a more accurate descriptirm. Our analysis can be extended to multi-class classification problems. A similar analysis implies similar bounds on misclassification probability for decision lists, and it seems likely that these techniques will also be applicable to other pattern classification methods. The complexity parameter, Neff described here does not always give the best possible error bounds. For example, the effective number of leaves Neff in a decision tree can be thought of as a single number that summarizes the probability distribution over the leaves induced by the training data. It seems unlikely that such a number will give optimal bounds for all distributions. In those cases, better bounds could be obtained by using numerical techniques to optimize over the choice of (J and WiS. It would be interesting to see how the bounds we obtain and those given by numerical techniques reflect the generalization performance of classifiers used in practice. Acknowledgements Thanks to Yoav Freund and Rob Schapire for helpful comments. References [1] P. L. Bartlett. For valid generalization, the size of the weights is more important than the size of the network. In Neural Information Processing Systems 9, pages 134-140. Morgan Kaufmann, San Mateo, CA, 1997. [2] L. Breiman, J.H. Friedman, R.A. Olshen, and C.J. Stone. Classification and Regression Trees. Wadsworth, Belmont, 1984. [3] A. Ehrenfeucht and D. Haussler. Learning decision trees from random examples. Information and Computation, 82:231-246, 1989. [4] U .M. Fayyad and K.B. Irani. What should be . '1inimized in a decision tree? In AAAI-90, pages 249-754,1990. [5] R. C. Holte. Very simple rules perform well on most commonly used databases. Machine learning, 11:63-91, 1993. [6] P.M. Murphy and M.J. pazzani. Exploring the decision forest: An empirical investigation of Occam's razor in decision tree induction. Journal of Artificial Intelligence Research, 1:257-275, 1994. [7] J.R. Quinlan. C4.5: Programs for Machine Learning. Morgan Kaufmann, 1992. [8] R. E. Schapire, Y. Freund, P. L. Bartlett, and W. S. Lee. Boosting the margin: a new explanation for the effectiveness of voting methods. In Machine Learning: Proceedings of the Fourteenth International Conference, pages 322-330, 1997. [9] J. Shawe-Taylor, P. L. Bartlett, R. C. Williamson, and M. Anthony. A framework for structural risk minimisation. In Proc. 9th COLT, pages 68-76. ACM Press, New York, NY, 1996. [10] G.L. Webb. Further experimental evidence against the utility of Occam's razor. 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Local Dimensionality Reduction Stefan Schaal 1,2,4 sschaal@usc.edu http://www-slab.usc.edulsschaal Sethu Vijayakumar 3, I sethu@cs.titech.ac.jp http://ogawawww.cs.titech.ac.jp/-sethu Christopher G. Atkeson 4 cga@cc.gatech.edu http://www.cc.gatech.edul fac/Chris.Atkeson IERATO Kawato Dynamic Brain Project (IST), 2-2 Hikaridai, Seika-cho, Soraku-gun, 619-02 Kyoto 2Dept. of Comp. Science & Neuroscience, Univ. of South. California HNB-I 03, Los Angeles CA 90089-2520 3Department of Computer Science, Tokyo Institute of Technology, Meguro-ku, Tokyo-I 52 4College of Computing, Georgia Institute of Technology, 801 Atlantic Drive, Atlanta, GA 30332-0280 Abstract If globally high dimensional data has locally only low dimensional distributions, it is advantageous to perform a local dimensionality reduction before further processing the data. In this paper we examine several techniques for local dimensionality reduction in the context of locally weighted linear regression. As possible candidates, we derive local versions of factor analysis regression, principle component regression, principle component regression on joint distributions, and partial least squares regression. After outlining the statistical bases of these methods, we perform Monte Carlo simulations to evaluate their robustness with respect to violations of their statistical assumptions. One surprising outcome is that locally weighted partial least squares regression offers the best average results, thus outperforming even factor analysis, the theoretically most appealing of our candidate techniques. 1 INTRODUCTION Regression tasks involve mapping a n-dimensional continuous input vector x E ~n onto a m-dimensional output vector y E ~m • They form a ubiquitous class of problems found in fields including process control, sensorimotor control, coordinate transformations, and various stages of information processing in biological nervous systems. This paper will focus on spatially localized learning techniques, for example, kernel regression with Gaussian weighting functions. Local learning offer advantages for real-time incremental learning problems due to fast convergence, considerable robustness towards problems of negative interference, and large tolerance in model selection (Atkeson, Moore, & Schaal, 1997; Schaal & Atkeson, in press). Local learning is usually based on interpolating data from a local neighborhood around the query point. For high dimensional learning problems, however, it suffers from a bias/variance dilemma, caused by the nonintuitive fact that " ... [in high dimensions] if neighborhoods are local, then they are almost surely empty, whereas if a neighborhood is not empty, then it is not local." (Scott, 1992, p.198). Global learning methods, such as sigmoidal feedforward networks, do not face this 634 S. School, S. Vijayakumar and C. G. Atkeson problem as they do not employ neighborhood relations, although they require strong prior knowledge about the problem at hand in order to be successful. Assuming that local learning in high dimensions is a hopeless, however, is not necessarily warranted: being globally high dimensional does not imply that data remains high dimensional if viewed locally. For example, in the control of robot anns and biological anns we have shown that for estimating the inverse dynamics of an ann, a globally 21dimensional space reduces on average to 4-6 dimensions locally (Vijayakumar & Schaal, 1997). A local learning system that can robustly exploit such locally low dimensional distributions should be able to avoid the curse of dimensionality. In pursuit of the question of what, in the context of local regression, is the "right" method to perfonn local dimensionality reduction, this paper will derive and compare several candidate techniques under i) perfectly fulfilled statistical prerequisites (e.g., Gaussian noise, Gaussian input distributions, perfectly linear data), and ii) less perfect conditions (e.g., non-Gaussian distributions, slightly quadratic data, incorrect guess of the dimensionality of the true data distribution). We will focus on nonlinear function approximation with locally weighted linear regression (L WR), as it allows us to adapt a variety of global linear dimensionality reduction techniques, and as L WR has found widespread application in several local learning systems (Atkeson, Moore, & Schaal, 1997; Jordan & Jacobs, 1994; Xu, Jordan, & Hinton, 1996). In particular, we will derive and investigate locally weighted principal component regression (L WPCR), locally weighted joint data principal component analysis (L WPCA), locally weighted factor analysis (L WF A), and locally weighted partial least squares (L WPLS). Section 2 will briefly outline these methods and their theoretical foundations, while Section 3 will empirically evaluate the robustness of these methods using synthetic data sets that increasingly violate some of the statistical assumptions of the techniques. 2 METHODS OF DIMENSIONALITY REDUCTION We assume that our regression data originate from a generating process with two sets of observables, the "inputs" i and the "outputs" y. The characteristics of the process ensure a functional relation y = f(i). Both i and yare obtained through some measurement device that adds independent mean zero noise of different magnitude in each observable, such that x == i + Ex and y = y + Ey • For the sake of simplicity, we will only focus on one-dimensional output data (m=l) and functions / that are either linear or slightly quadratic, as these cases are the most common in nonlinear function approximation with locally linear models. Locality of the regression is ensured by weighting the error of each data point with a weight from a Gaussian kernel: Wi = exp(-O.5(Xi - Xqf D(Xi - Xq)) (1) Xtt denotes the query point, and D a positive semi-definite distance metric which determmes the size and shape of the neighborhood contributing to the regression (Atkeson et aI., 1997). The parameters Xq and D can be determined in the framework of nonparametric statistics (Schaal & Atkeson, in press) or parametric maximum likelihood estimations (Xu et aI, 1995}- for the present study they are determined manually since their origin is secondary to the results of this paper. Without loss of generality, all our data sets will set !,q to the zero vector, compute the weights, and then translate the input data such that the locally weighted mean, i = L WI Xi / L Wi , is zero. The output data is equally translated to be mean zero. Mean zero data is necessary for most of techniques considered below. The (translated) input data is summarized in the rows of the matrix X, the corresponding (translated) outputs are the elements of the vector y, and the corresponding weights are in the diagonal matrix W. In some cases, we need the joint input and output data, denoted as Z=[X y). Local Dimensionality Reduction 635 2.1 FACTORANALYSIS(LWFA) Factor analysis (Everitt, 1984) is a technique of dimensionality reduction which is the most appropriate given the generating process of our regression data. It assumes the observed data z was produced. by a mean zero independently distributed k -dimensional vector of factors v, transformed by the matrix U, and contaminated by mean zero independent noise f: with diagonal covariance matrix Q: z=Uv+f:, where z=[xT,yt and f:=[f:~,t:yr (2) If both v and f: are normally distributed, the parameters Q and U can be obtained iteratively by the Expectation-Maximization algorithm (EM) (Rubin & Thayer, 1982). For a linear regression problem, one assumes that z was generated with U=[I, f3 Y and v = i, where f3 denotes the vector of regression coefficients of the linear model y = f31 x, and I the identity matrix. After calculating Q and U by EM in joint data space as formulated in (2), an estimate of f3 can be derived from the conditional probability p(y I x). As all distributions are assumed to be normal, the expected value ofy is the mean of this conditional distribution. The locally weighted version (L WF A) of f3 can be obtained together with an estimate of the factors v from the joint weighted covariance matrix 'I' of z and v: E{[: ] + [ ~ } ~ ~,,~,;'x, where ~ ~ [ZT, VT~~Jft: w; ~ (3) [Q+UUT U] ['I'II(=n x n) 'I'12(=nX(m+k»)] = UT I = '¥21(= (m + k) x n) '1'22(= (m + k) x (m + k») where E { .} denotes the expectation operator and B a matrix of coefficients involved in estimating the factors v. Note that unless the noise f: is zero, the estimated f3 is different from the true f3 as it tries to average out the noise in the data. 2.2 JOINT-SPACE PRINCIPAL COMPONENT ANALYSIS (LWPCA) An alternative way of determining the parameters f3 in a reduced space employs locally weighted principal component analysis (LWPCA) in the joint data space. By defining the . largest k+ 1 principal components of the weighted covariance matrix ofZ as U: U = [eigenvectors(I Wi (Zi - ZXZi - Z)T II Wi)] (4) max(l:k+1l and noting that the eigenvectors in U are unit length, the matrix inversion theorem (Hom & Johnson, 1994) provides a means to derive an efficient estimate of f3 ( T T( T )-1 T\ [Ux(=nXk)] f3=U x Uy -Uy UyUy -I UyUyt where U= Uy(=mxk) (5) In our one dimensional output case, U y is just a (1 x k) -dimensional row vector and the evaluation of (5) does not require a matrix inversion anymore but rather a division. If one assumes normal distributions in all variables as in L WF A, L WPCA is the special case of L WF A where the noise covariance Q is spherical, i.e., the same magnitude of noise in all observables. Under these circumstances, the subspaces spanned by U in both methods will be the same. However, the regression coefficients of L WPCA will be different from those of L WF A unless the noise level is zero, as L WF A optimizes the coefficients according to the noise in the data (Equation (3» . Thus, for normal distributions and a correct guess of k, L WPCA is always expected to perform worse than L WF A. 636 S. Schaal, S. Vijayakumar and C. G. Atkeson 2.3 PARTIAL LEAST SQUARES (LWPLS, LWPLS_I) Partial least squares (Wold, 1975; Frank & Friedman, 1993) recursively computes orthogonal projections of the input data and performs single variable regressions along these projections on the residuals of the previous iteration step. A locally weighted version of partial least squares (LWPLS) proceeds as shown in Equation (6) below. As all single variable regressions are ordinary univariate least-squares minim izations, L WPLS makes the same statistical assumption as ordinary linear regressions, i.e., that only output variables have additive noise, but input variables are noiseless. The choice of the projections u, however, introduces an element in L WPLS that remains statistically still debated (Frank & Friedman, 1993), although, interestingly, there exists a strong similarity with the way projections are chosen in Cascade Correlation (Fahlman & Lebiere, 1990). A peculiarity of L WPLS is that it also regresses the inputs of the previous step against the projected inputs s in order to ensure the orthogonality of all the projections u. Since L WPLS chooses projections in a very powerful way, it can accomplish optimal function fits with only one single projections (i.e., For Training: Initialize: Do = X, eo = y For i = 1 to k: For Lookup: Initialize: do = x, y= ° For i = 1 to k: s. = dT.u. I 1I (6) k= 1) for certain input distributions. We will address this issue in our empirical evaluations by comparing k-step L WPLS with I-step L WPLS, abbreviated L WPLS_I. 2.4 PRINCIPAL COMPONENT REGRESSION (L WPCR) Although not optimal, a computationally efficient techniques of dimensionality reduction for linear regression is principal component regression (LWPCR) (Massy, 1965). The inputs are projected onto the largest k principal components of the weighted covariance matrix of the input data by the matrix U: U = [eigenvectors(2: Wi (Xi - xX Xi - xt /2: Wi )] (7) max(l:k) The regression coefficients f3 are thus calculated as: f3 = (UTXTwxUtUTXTWy (8) Equation (8) is inexpensive to evaluate since after projecting X with U, UTXTWXU becomes a diagonal matrix that is easy to invert. L WPCR assumes that the inputs have additive spherical noise, which includes the zero noise case. As during dimensionality reduction L WPCR does not take into account the output data, it is endangered by clipping input dimensions with low variance which nevertheless have important contribution to the regression output. However, from a statistical point of view, it is less likely that low variance inputs have significant contribution in a linear regression, as the confidence bands of the regression coefficients increase inversely proportionally with the variance of the associated input. If the input data has non-spherical noise, L WPCR is prone to focus the regression on irrelevant projections. 3 MONTE CARLO EVALUATIONS In order to evaluate the candidate methods, data sets with 5 inputs and 1 output were randomly generated. Each data set consisted of 2,000 training points and 10,000 test points, distributed either uniformly or nonuniformly in the unit hypercube. The outputs were Local Dimensionality Reduction 637 generated by either a linear or quadratic function. Afterwards, the 5-dimensional input space was projected into a to-dimensional space by a randomly chosen distance preserving linear transformation. Finally, Gaussian noise of various magnitudes was added to both the 10-dimensional inputs and one dimensional output. For the test sets, the additive noise in the outputs was omitted. Each regression technique was localized by a Gaussian kernel (Equation (1)) with a to-dimensional distance metric D=IO*I (D was manually chosen to ensure that the Gaussian kernel had sufficiently many data points and no "data holes" in the fringe areas of the kernel) . The precise experimental conditions followed closely those suggested by Frank and Friedman (1993): • 2 kinds of linear functions y = {g.I for: i) 131 .. = [I, I, I, I, If , ii) I3Ii. = [1,2,3,4,sf • 2 kinds of quadratic functions y = f3J.I + f3::.aAxt ,xi ,xi ,X;,X;]T for: i) 1311. = [I, I, I, I, Wand f3q.ad = 0.1 [I, I, I, I, If, and ii) 131 .. = [1,2,3,4, sf and f3quad = 0.1 [I, 4, 9, 16, 2sf • 3 kinds of noise conditions, each with 2 sub-conditions: i) only output noise: a) low noise: local signal/noise ratio Isnr=20, and b) high noise: Isnr=2, ii) equal noise in inputs and outputs: a) low noise Ex •• = Sy = N(O,O.Ot2), n e[I,2, ... ,10], and b) high noise Ex •• =sy=N(0,0.12),ne[I,2, ... ,10], iii) unequal noise in inputs and outputs: a) low noise: Ex .• = N(0,(0.0In)2), n e[I,2, ... ,1O] and Isnr=20, and b) high noise: Ex .• = N(0,(0.0In)2), n e[I,2, ... ,1O] and Isnr=2, • 2 kinds of input distributions: i) uniform in unit hyper cube, ii) uniform in unit hyper cube excluding data points which activate a Gaussian weighting function (I) at c = [O.S,O,o,o,of with D=IO*I more than w=0.2 (this forms a "hyper kidney" shaped distribution) Every algorithm was run * 30 times on each of the 48 combinations of the conditions. Additionally, the complete test was repeated for three further conditions varying the dimensionality--called factors in accordance with L WF A-that the algorithms assumed to be the true dimensionality of the to-dimensional data from k=4 to 6, i.e., too few, correct, and too many factors. The average results are summarized in Figure I. Figure I a,b,c show the summary results of the three factor conditions. Besides averaging over the 30 trials per condition, each mean of these charts also averages over the two input distribution conditions and the linear and quadratic function condition, as these four cases are frequently observed violations of the statistical assumptions in nonlinear function approximation with locally linear models. In Figure I b the number of factors equals the underlying dimensionality of the problem, and all algorithms are essentially performing equally well. For perfectly Gaussian distributions in all random variables (not shown separately), LWFA's assumptions are perfectly fulfilled and it achieves the best results, however, almost indistinguishable closely followed by L WPLS. For the ''unequal noise condition", the two PCA based techniques, L WPCA and L WPCR, perform the worst since--as expected-they choose suboptimal projections. However, when violating the statistical assumptions, L WF A loses parts of its advantages, such that the summary results become fairly balanced in Figure lb. The quality of function fitting changes significantly when violating the correct number of factors, as illustrated in Figure I a,c. For too few factors (Figure la), L WPCR performs worst because it randomly omits one of the principle components in the input data, without respect to how important it is for the regression. The second worse is L WF A: according to its assumptions it believes that the signal it cannot model must be noise, leading to a degraded estimate of the data's subspace and, consequently, degraded regression results. L WPLS has a clear lead in this test, closely followed by L WPCA and L WPLS_I. * Except for LWFA, all methods can evaluate a data set in non-iterative calculations. LWFA was trained with EM for maximally 1000 iterations or until the log-likelihood increased less than I.e-lOin one iteration. 638 S. Schaal, S. Vljayakumar and C. G. Atkeson For too many factors than necessary (Figure Ie), it is now LWPCA which degrades. This effect is due to its extracting one very noise contaminated projection which strongly influences the recovery of the regression parameters in Equation (4). All other algorithms perform almost equally well, with L WF A and L WPLS taking a small lead. c o 0.1 ~ 0.01 ::::;; c II> C> ~ 0.001 ~ c:: o 0.0001 0.1 W 0.01 ~ c:: II> C) ~ 0.001 ~ ~ ~ 8 0.0001 0.1 W 0.01 ~ c:: g, ~ 0.001 ~ jj il f-a 0.0001 0.1 ~ 0.01 ::::;; c II> C) ~ 0.001 ~ 0.0001 OnlyOutpul Noise Equal NoIse In ell In puIS end OutpUIS Unequel NoIse In ell Inputs end OutpulS fl- I. E>O ~I. £ >>(I ~ J. &>O ~J , E » O ~ J .E>O fl- I.&» O ~I . & >O ~ I . & >>O ~I.& >O ~ I .£>>o p,. 1. s>O tJ-J .£>>O e) RegressIon Results with 4 Factors • LWFA • LWPCA • LWPCR 0 LWPLS • LWPLS_1 c) RegressIon Results with 6 Feclors d) Summery Results Figure I: Average summary results of Monte Carlo experiments. Each chart is primarily divided into the three major noise conditions, cf. headers in chart (a). In each noise condition, there are four further subdivision: i) coefficients of linear or quadratic model are equal with low added noise; ii) like i) with high added noise; iii) coefficients oflinear or quadratic model are different with low noise added; iv) like iii) with high added noise. Refer to text and descriptions of Monte Carlo studies for further explanations. Local Dimensionality Reduction 639 4 SUMMARY AND CONCLUSIONS Figure 1 d summarizes all the Monte Carlo experiments in a final average plot. Except for L WPLS, every other technique showed at least one clear weakness in one of our "robustness" tests. It was particularly an incorrect number of factors which made these weaknesses apparent. For high-dimensional regression problems, the local dimensionality, i.e., the number of factors, is not a clearly defined number but rather a varying quantity, depending on the way the generating process operates. Usually, this process does not need to generate locally low dimensional distributions, however, it often "chooses" to do so, for instance, as human ann movements follow stereotypic patterns despite they could generate arbitrary ones. Thus, local dimensionality reduction needs to find autonomously the appropriate number of local factor. Locally weighted partial least squares turned out to be a surprisingly robust technique for this purpose, even outperforming the statistically appealing probabilistic factor analysis. As in principal component analysis, LWPLS's number of factors can easily be controlled just based on a variance-cutoff threshold in input space (Frank & Friedman, 1993), while factor analysis usually requires expensive cross-validation techniques. Simple, variance-based control over the number of factors can actually improve the results of L WPCA and L WPCR in practice, since, as shown in Figure I a, L WPCR is more robust towards overestimating the number of factors, while L WPCA is more robust towards an underestimation. If one is interested in dynamically growing the number of factors while obtaining already good regression results with too few factors, L WPCA and, especially, L WPLS seem to be appropriate-it should be noted how well one factor L WPLS (L WPLS_l) already performed in Figure I! In conclusion, since locally weighted partial least squares was equally robust as local weighted factor analysis towards additive noise in. both input and output data, and, moreover, superior when mis-guessing the number of factors, it seems to be a most favorable technique for local dimensionality reduction for high dimensional regressions. Acknowledgments The authors are grateful to Geoffrey Hinton for reminding them of partial least squares. This work was supported by the ATR Human Information Processing Research Laboratories. S. Schaal's support includes the German Research Association, the Alexander von Humboldt Foundation, and the German Scholarship Foundation. S. Vijayakumar was supported by the Japanese Ministry of Education, Science, and Culture (Monbusho). C. G. Atkeson acknowledges the Air Force Office of Scientific Research grant F49-6209410362 and a National Science Foundation Presidential Young Investigators Award. References tures of experts and the EM algorithm." Neural Computation, 6, 2, pp.181-214. Atkeson, C. G., Moore, A. W., & Schaal, S, (1997a). Massy, W. F, (1965). "Principle component regression "Locally weighted learning." ArtifiCial Intelligence Re- in exploratory statistical research." Journal of the view, 11, 1-5, pp.II-73. American Statistical Association, 60, pp.234-246. Atkeson, C. G., Moore, A. W., & Schaal, S, (1997c). Rubin, D. B., & Thayer, D. T, (l982). "EM algorithms "Locally weighted learning for control." ArtifiCial Infor ML factor analysis." Psychometrika, 47, I, 69-76. telligence Review, 11, 1-5, pp.75-113. Schaal, S., & Atkeson, C. G, (in press). "Constructive Belsley, D. A., Kuh, E., & Welsch, R. E, (1980). Reincremental learning from only local information." gression diagnostics: Identifying influential data and Neural Computation. sources of collinearity. New York: Wiley. Scott, D. W, (1992). Multivariate Density Estimation. Everitt, B. S, (1984). An introduction to latent variable New York: Wiley. models. London: Chapman and Hall. Vijayakumar, S., & Schaal, S, (1997). "Local dimenFahlman, S. E. ,Lebiere, C, (1990). "The cascadesionality reduction for locally weighted learning." In: correlation learning architecture." In: Touretzky, D. S. International Conference on Computational Intelli(Ed.), Advances in Neural Information Processing gence in Robotics and Automation, pp.220-225, MonSystems II, pp.524-532. Morgan Kaufmann. teray, CA, July 10-11, 1997. Frank, I. E., & Friedman, 1. H, (1993). "A statistical Wold, H. (1975). "Soft modeling by latent variables: view of some chemometric regression tools." Techthe nonlinear iterative partial least squares approach." nometrics, 35, 2, pp.l09-135. In: Gani, J. (Ed.), Perspectives in Probability and StaGeman, S., Bienenstock, E., & Doursat, R. (1992). tistics, Papers in Honour ofM S. Bartlett. Aca<j. Press. "Neural networks and the bias/variance dilemma." Xu, L., Jordan, M.l., & Hinton, G. E, (1995). "An alNeural Computation, 4, pp.I-58. ternative model for mixture of experts." In: Tesauro, Hom, R. A., & Johnson, C. R, (1994). Matrix analySis. G., Touretzky, D. S., & Leen, T. K. (Eds.), Advances in Press Syndicate of the University of Cambridge. Neural Information Processing Systems 7, pp.633-640. Jordan, M.I., & Jacobs, R, (1994). "Hierarchical mixCambridge, MA: MIT Press.
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A mathematical model of axon guidance by diffusible factors Geoffrey J. Goodhill Georgetown Institute for Cognitive and Computational Sciences Georgetown University Medical Center 3970 Reservoir Road Washington DC 20007 geoff@giccs.georgetown.edu Abstract In the developing nervous system, gradients of target-derived diffusible factors play an important role in guiding axons to appropriate targets. In this paper, the shape that such a gradient might have is calculated as a function of distance from the target and the time since the start of factor production. Using estimates of the relevant parameter values from the experimental literature, the spatiotemporal domain in which a growth cone could detect such a gradient is derived. For large times, a value for the maximum guidance range of about 1 mm is obtained. This value fits well with experimental data. For smaller times, the analysis predicts that guidance over longer ranges may be possible. This prediction remains to be tested. 1 Introduction In the developing nervous system, growing axons are guided to targets that may be some distance away. Several mechanisms contribute to this (reviewed in TessierLavigne & Goodman (1996». One such mechanism is the diffusion of a factor from the target through the extracellular space, creating a gradient of increasing concentration that axons can sense and follow. In the central nervous system, such a process seems to occur in at least three cases: the guidance ofaxons from the trigeminal ganglion to the maxillary process in the mouse (Lumsden & Davies, 1983, 1986), of commissural axons in the spinal cord to the floor plate (TessierLavigne et al., 1988), and ofaxons and axonal branches from the corticospinal tract to the basilar pons (Heffner et al., 1990). The evidence for this comes from both in vivo and in vitro experiments. For the latter, a piece of target tissue is embedded in a three dimensional collagen gel near to a piece of tissue containing the appropriate 160 G. J Goodhill population of neurons. Axon growth is then observed directed towards the target, implicating a target-derived diffusible signal. In vivo, for the systems described, the target is always less than 500 J..lm from the population ofaxons. In vitro, where the distance between axons and target can readily be varied, guidance is generally not seen for distances greater than 500 - 1000 J..lm. Can such a limit be explained in terms of the mathematics of diffusion? There are two related constraints that the distribution of a diffusible factor must satisfy to provide an effective guidance cue at a point. Firstly, the absolute concentration of factor must not be too small or too large. Secondly, the fractional change in concentration of factor across the width of the gradient-sensing apparatus, generally assumed to be the growth cone, must be sufficiently large. These constraints are related because in both cases the problem is to overcome statistical noise. At very low concentrations, noise exists due to thermal fluctuations in the number of molecules of factor in the vicinity of the growth cone (analyzed in Berg & Purcell (1977». At higher concentrations, the limiting source of noise is stochastic variation in the amount of binding of the factor to receptors distributed over the growth cone. At very high concentrations, all receptors will be saturated and no gradient will be apparent. The closer the concentration is to the upper or lower limits, the higher the gradient that is needed to ensure detection (Devreotes & Zigmond, 1988; Tessier-Lavigne & Placzek, 1991). The limitations these constraints impose on the guidance range of a diffusible factor are now investigated. For further discussion see Goodhill (1997; 1998). 2 Mathematical model Consider a source releasing factor with diffusion constant D cm2/sec, at rate q moles/sec, into an infinite, spatially uniform three-dimensional volume. Initially, zero decay of the factor is assumed. For radially symmetric Fickian diffusion in three dimensions, the concentration C(r, t) at distance r from the source at time t is given by q r C(r, t) = -4 D erfc r;-r:;-; 7r r v4Dt (1) (see e.g. Crank (1975», where erfc is the complementary error function. The percentage change in concentration p across a small distance D.r (the width of the growth cone) is given by D.r [ r e-r2/4Dt 1 p = --;:1 + J7rDt erfc(r/J4Dt) (2) This function has two perhaps surprising characteristics. Firstly, for fixed r, Ipi decreases with t. That is, the largest gradient at any distance occurs immediately after the source starts releaSing factor. For large t, Ipi asymptotes at D.r Jr. Secondly, for fixed t < 00, numerical results show that p is nonmonotonic with r. In particular it decreases with distance, reaches a minimum, then increases again. The position of this minimum moves to larger distances as t increases. The general characteristics of the above constraints can be summarized as follows. (1) At small times after the start of production the factor is very unevenly distributed. The concentration C falls quickly to almost zero moving away from the source, the gradient is steep, and the percentage change across the growth cone p is everywhere large. (2) As time proceeds the factor becomes more evenly distributed. C everywhere increases, but p everywhere decreases. (3) For large times, C tends to an inverse variation with the distance from the source r, while Ipi tends A Mathematical Model of Axon Guidance by Diffusible Factors 161 to 6.r/r independent of all other parameters. This means that, for large times, the maximum distance over which guidance by diffusible factors is possible scales linearly with growth cone diameter 6.r. 3 Parameter values Diffusion constant, D. Crick (1970) estimated the diffusion constant in cytoplasm for a molecule of mass 0.3 - 0.5 kDa to be about 10-6 cm2 /sec. Subsequently, a direct determination of the diffusion constant for a molecule of mass 0.17 kDa in the aqueous cytoplasm of mammalian cells yielded a value of about 3.3 x 10-6 cm2 / sec (Mastro et aL, 1984). By fitting a particular solution of the diffusion equation to their data on limb bud determination by gradients of a morphogenetically active retinoid, Eichele & Thaller (1987) calculated a value of 10-7 cm2 /sec for this molecule (mass 348.5 kDa) in embryonic limb tissue. One chemically identified diffusible factor known to be involved in axon guidance is the protein netrin-1, which has a molecular mass of about 75 kDa (Kennedy et al., 1994). D should scale roughly inversely with the radius of a molecule, Le. with the cube root of its mass. Taking the value of 3.3 x 10-6 cm2 /sec and scaling it by (170/75,000)1/3 yields 4.0 x 10-7 cm2 /sec. This paper therefore considers D = 10-6 cm2 /sec and D = 10-7 cm2 /sec. Rate of production of factor q. This is hard to estimate in vivo: some insight can be gained by considering in vitro experiments. Gundersen & Barrett (1979) found a turning response in chick spinal sensory axons towards a nearby pipette filled with a solution of NGF. They estimated the rate of outflow from their pipette to be 1 /LI/hour, and found an effect when the concentration in the pipette was as low as 0.1 nM NGF (Tessier-Lavigne & Placzek, 1991). This corresponds to a q of 3 x 1O-11 nM/sec. Lohof et al. (1992) studied growth cone turning induced by a gradient of cell-membrane permeant cAMP from a pipette containing a 20 mM solution and a release rate of the order of 0.5 pI/sec: q = 10-5 nM/sec. Below a further calculation for q is performed, which suggests an appropriate value may be q = 10-7 nM/sec. Growth cone diameter, 6.r. For the three systems mentioned above, the diameter of the main body of the growth cone is less than 10 /Lm. However, this ignores filopodia, which can increase the effective width for gradient sensing purposes. The values of 10 /Lm and 20 /Lm are considered below. Minimum concentration for gradient detection. Studies of leukocyte chemotaxis suggest that when gradient detection is limited by the dynamics of receptor binding rather than physical limits due to a lack of molecules of factor, optimal detection occurs when the concentration at the growth cone is equal to the dissociation constant for the receptor (Zigmond, 1981; Devreotes & Zigmond, 1988). Such studies also suggest that the low concentration limit is about 1 % of the dissociation constant (Zigmond, 1981). The transmembrane protein Deleted in Colorectal Cancer (DeC) has recently been shown to possess netrin-1 binding activity, with an order of magnitude estimate for the dissociation constant of 10 nM (Keino-Masu et aI, 1996). For comparison, the dissociation constant of the low-affinity NGF receptor P75 is about 1 nM (Meakin & Shooter, 1992). Therefore, low concentration limits of both 10-1 nM and 10-2 nM will be considered. Maximum concentration for gradient detection. Theoretical considerations suggest that, for leukocyte chemotaxis, sensitivity to a fixed gradient should fall off symmetrically in a plot against the log of background concentration, with the peak at the dissociation constant for the receptor (Zigmond, 1981). Raising the con162 G. 1. Goodhill centration to several hundred times the dissociation constant appears to prevent axon guidance (discussed in Tessier-Lavigne & Placzek (1991». At concentrations very much greater than the dissociation constant, the number of receptors may be downregulated, reducing sensitivity (Zigm0nd, 1981). Given the dissociation constants above, 100 nM thus constitutes a reasonable upper bound on concentration. Minimum percentage change detectable by a growth cone, p. By establishing gradients of a repellent, membrane-bound factor directly on a substrate and measuring the response of chick retinal axons, Baier & Bonhoeffer (1992) estimated p to be about 1 %. Studies of cell chemotaxis in various systems have suggested optimal values of 2%: for concentrations far from the dissociation constant for the receptor, p is expected to be larger (Devreotes & Zigmond, 1988). Both p = 1% and p = 2% are considered below. 4 Results In order to estimate bounds for the rate of production of factor q for biological tissue, the empirical observation is used that, for collagen gel assays lasting of the order of one day, guidance is generally seen over distances of at most 500 I'm (Lumsden & Davies, 1983,1986; Tessier-Lavigne et al., 1988). Assume first that this is constrained by the low concentration limit. Substituting the above parameters (with D = 10-7 cm2/sec) into equation 1 and specifying that C(500pm, 1 day) = 0.01 nM gives q :::::: 10-9 nM/ sec. On the other hand, assuming constraint by the high concentration limit, i.e. C(500pm, 1 day) = 100 nM, gives q :::::: 10-5 nM/sec. Thus it is reasonable to assume that, roughly, 10-9 nM/ sec < q < 10-5 nM/ sec. The results discussed below use a value in between, namely q = 10-7 nM/sec. The constraints arising from equations 1 and 2 are plotted in figure 1. The cases of D = 10-6 cm2/sec and D = 10-7 cm2/sec are shown in (A,C) and (B,D) respectively. In all four pictures the constraints C = 0.01 nM and C = 0.1 nM are plotted. In (A,B) the gradient constraint p = 1 % is shown, whereas in (C,D) p = 2% is shown. These are for a growth cone diameter of 10 I'm. The graph for a 2% change and a growth cone diameter of 20 I'm is identical to that for a 1% change and a diameter of 10 I'm. Each constraint is satisfied for regions to the left of the relevant line. The line C = 100 nM is approximately coincident with the vertical axis in all cases. For these parameters, the high concentration limit does not therefore prevent gradient detection until the axons are within a few microns of the source, and it is thus assumed that it is not an important constraint. As expected, for large t the gradient constraint asymptotes at D.r Ir = p, i.e. r = 1000 I'm for p = 1% and r = 500 I'm for p = 2% and a 10 I'm growth cone. That is, the gradient constraint is satisfied at all times when the distance from the source is less than 500 I'm for p = 2% and D.r = 10 I'm. The gradient constraint lines end to the right because at earlier times p exceeds the critical value over all distances (since the formula for p is non-monotonic with r, there is sometimes another branch of each p curve (not shown) off the graph to the right). As t increases from zero, guidance is initially limited only by the concentration constraint. The maximum distance over which guidance can occur increases smoothly with t, reaching for instance 1500 I'm (assuming a concentration limit of 0.01 nM) after about 2 hours for D = 10-6 cm2/sec and about 6 hours for D = 10-7 cm2/sec. However at a particular time, the gradient constraint starts to take effect and rapidly reduces the maximum range of guidance towards the asymptotic value as t increases. This time (for p = 2%) is about 2 hours for D = 10-6 cm2/sec, and about one day for D = 10-7 cm2 / sec. It is clear from these pictures that although the exact size of A Mathematical Model of Axon Guidance by Diffusible Factors 163 A B Ci) Ci) "'-J >4.0 >4.0 '" '" ~ ~ CD 3.5 C=O.OlnM CD 3.5 E ... E i= " . C=O.lnM i= 3.0 p=O.OI 3.0 ... C=O.OlnM ... C = O.lnM 2.5 2.5 p= 0.01 2.0 2.0 1.5 1.5 1.0 1.0 ", " 0.5 0.5 , .. " ....... " . 0.0 0.0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 Distance (microns) Distance (microns) C D Ci) ~ >4.0 4.0 '" '" ~ ~ ... C = O.OlnM CD 3.5 ... C = O.OlnM CD 3.5 E E ., . C=O.lnM i= ., . C = O.lnM i= P =0.02 . 3.0 p=0.02 3.0 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0 " " " 0.5 0.5 .' . ,' .. .. ' .. ' ." 0.0 0.0 !~ ......... 0 1500 2000 2500 0 500 1000 1500 2000 2500 Distance (microns) Distance (microns) Figure 1: Graphs showing how the gradient constraint (solid line) interacts with the minimum concentration constraint (dashed/dotted lines) to limit guidance range, and how these constraints evolve over time. The top row (A,B) is for p = 1%, the bottom row (C,D) for p = 2%. The left column (A,C) is for D = 10-6 cm2 / sec, the right column (B,D) for D = 10-7 cm2 /sec. Each constraint is satisfied to the left of the appropriate curve. It can be seen that for D = 1O-6cm2 / s~c the gradient limit quickly becomes the dominant constraint on maximum guidance range. In contrast for D = 10-7 cm2 / sec, the concentration limit is the dominant constraint at times up to several days. However after this the gradient constraint starts to take effect and rapidly reduces the maximum guidance range. 164 G. 1. Goodhill the diffusion constant does not affect the position of the asymptote for the gradient constraint, it does play an important role in the interplay of constraints while the gradient is evolving. The effect is however subtle: reducing D from 10-6 cm2 /sec to 10-7 cm2 /sec increases the time for the C = 0.01 nM limit to reach 2000 J..tm, but decreases the time for the C = 0.1 nM limit to reach 2000 f..lm. 5 Discussion Taking the gradient constraint to be a fractional change of at least 2% across a growth cone of width of 10 f..lm or 20 J..tm yields asymptotic values for the maximum distance over which guidance can occur once the gradient has stabilized of 500 f..lm and 1000 f..lm respectively. This fits well with both in vitro data, and the fact that for the systems mentioned in the introduction, the growing axons are always less than 500 f..lm from the target in vivo. The concentration limits seem to provide a weaker constraint than the gradient limit on the maximum distances possible. However, this is very dependent on the value of q, which has only been very roughly estimated: if q is significantly less than 10-7 nM/sec, the low concentration limits will provide more restrictive constraints (q may well have different values in different target tissues). The gradient constraint curves are independent of q. The gradient constraint therefore provides the most robust explanation for the observed guidance limit. The model makes the prediction that guidance over longer distances than have hitherto been observed may be possible before the gradient has stabilized. In the early stages following the start of factor production the concentration falls off more steeply, providing more effective guidance. The time at which guidance range is a maximum defends on the diffusion constant D. For a rapidly diffusing molecule (D >::::: 1O-6cm jsec) this occurs after only a few hours. For a more slowly diffusing molecule however (D >::::: 10-7 cm2 jsec) this occurs after a few days, which would be easier to investigate in vitro. In vivo, molecules such as netrin-1 may thus be large because, during times immediately following the start of production by the source, there could be a definite benefit (i.e. steep gradient) to a slowly-diffusing molecule. Also, it is conceivable that Nature has optimized the start of production of factor relative to the time that guidance is required in order to exploit an evolving gradient for extended range. This could be especially important in larger animals, where axons may need to be guided over longer distances in the developing embryo. Bibliography Baier, H. & Bonhoeffer, F. (1992). Axon guidance by gradients of a target-derived component. Science, 255,472-475. Berg, H.C. and Purcell, E.M. (1977). Physics of chemoreception. Biophysical Journal, 20,193-219. Crick, F. (1970). Diffusion in embryogenesis. Nature, 255,420-422. Crank, J. (1975). The mathematics of diffusion, Second edition. Oxford, Clarendon. Devreotes, P.N. & Zigmond, S.H. (1988). Chemotaxis in eukaryotic cells: a focus on leukocytes and Dictyostelium. Ann. Rev. Cell. Bioi., 4, 649-686. Eichele, G. & Thaller, C. (1987). Characterization of concentration gradients of a morphogenetically active retinoid in the chick limb bud. J. Cell. Bioi., 105, 19171923. A Mathematical Model of Axon Guidance by Diffusible Factors 165 Goodhill, G.J. (1997). Diffusion in axon guidance. Eur. J. Neurosci., 9,1414-1421. Goodhill, G.J. (1998). Mathematical guidance for axons. Trends. Neurosci., in press. Gundersen, R.W. & Barrett, J.N. (1979). Neuronal chemotaxis: chick dorsal-root axons tum toward high concentrations of nerve growth factor. Science, 206, 1079-1080. Heffner, C.D., Lumsden, AG.5. & O'Leary, D.D.M. (1990). Target control of collateral extension and directional growth in the mammalian brain. Science, 247, 217-220. Keino-Masu, K., Masu, M., Hinck, L., Leonardo, E.D., Chan, S.5.-Y., Culotti, J.G. & Tessier-Lavigne, M. (1996). Deleted in Colorectal Cancer (DCC) encodes a netrin receptor. Cell, 87, 175-185. Kennedy, T.E., Serafini, T., de al Torre, J.R. & Tessier-Lavigne, M. (1994). Netrins are diffusible chemotropic factors for commissural axons in the embryonic spinal cord. Cell, 78,425-435. Lohof, A.M., Quillan, M., Dan, Y, & Poo, M-m. (1992). Asymmetric modulation of cytosolic cAMP activity induces growth cone turning. J. Neurosci., 12, 12531261. Lumsden, AG.s. & Davies, A.M. (1983). Earliest sensory nerve fibres are guided to peripheral targets by attractants other than nerve growth factor. Nature, 306, 786-788. Lumsden, A.G.5. & Davies, AM. (1986). Chemotropic effect of specific target epithelium in the developing mammalian nervous system. Nature, 323, 538-539. Mastro, AM., Babich, M.A, Taylor, W.D. & Keith, AD. (1984). Diffusion of a small molecule in the cytoplasm of mammalian cells. Proc. Nat. Acad. Sci. USA, 81, 3414-3418. Meakin, S.O. & Shooter, E.M. (1992). The nerve growth family of receptors. Trends. Neurosci., 15, 323-331. Tessier-Lavigne, M. & Placzek, M. (1991). Target attraction: are developing axons guided by chemotropism? Trends Neurosci., 14,303-310. Tessier-Lavigne, M. & Goodman, C.S. (1996). The molecular biology of axon guidance. Science, 274, 1123-1133. Tessier-Lavigne, M., Placzek, M., Lumsden, A.G.5., Dodd, J. & Jessell, T.M. (1988). Ch~motropic guidance of developing axons in the mammalian central nervous system. Nature, 336, 775-778. Tranquillo, R.T. & Lauffenburger, D.A. (1987). Stochastic model of leukocyte chemosensory movement. J. Math. BioI., 25,229-262. Zigmond, S.H. (1981). Consequences of chemotactic peptide receptor modulation for leukocyte orientation. J. Cell. BioI., 88, 644-647.
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Asymptotic Theory for Regularization: One-Dimensional Linear Case Petri Koistinen Rolf Nevanlinna Institute, P.O. Box 4, FIN-00014 University of Helsinki, Finland. Email: PetrLKoistinen@rnLhelsinkLfi Abstract The generalization ability of a neural network can sometimes be improved dramatically by regularization. To analyze the improvement one needs more refined results than the asymptotic distribution of the weight vector. Here we study the simple case of one-dimensional linear regression under quadratic regularization, i.e., ridge regression. We study the random design, misspecified case, where we derive expansions for the optimal regularization parameter and the ensuing improvement. It is possible to construct examples where it is best to use no regularization. 1 INTRODUCTION Suppose that we have available training data (Xl, Yd, .. 0' (Xn' Yn) consisting of pairs of vectors, and we try to predict Yi on the basis of Xi with a neural network with weight vector w. One popular way of selecting w is by the criterion (1) 1 n - L £(Xi' Yi, w) + >..Q(w) = min!, n I where the loss £(x,y,w) is, e.g., the squared error Ily - g(x,w)11 2 , the function g(., w) is the input/output function of the neural network, the penalty Q(w) is a real function which takes on small values when the mapping g(o, w) is smooth and high values when it changes rapidly, and the regularization parameter >.. is a nonnegative scalar (which might depend on the training sample). We refer to the setup (1) as (training with) regularization, and to the same setup with the choice >.. = 0 as training without regularization. Regularization has been found to be very effective for improving the generalization ability of a neural network especially when the sample size n is of the same order of magnitude as the dimensionality of the parameter vector w, see, e.g., the textbooks (Bishop, 1995; Ripley, 1996). Asymptotic Theory for Regularization: One-Dimensional Linear Case 295 In this paper we deal with asymptotics in the case where the architecture of the network is fixed but the sample size grows. To fix ideas, let us assume that the training data is part of an Li.d. (independent, identically distributed) sequence (X,Y);(Xl'Yl),(X2'Y2)"" of pairs of random vectors, i.e., for each i the pair (Xi, Yi) has the same distribution as the pair (X, Y) and the collection of pairs is independent (X and Y can be dependent). Then we can define the (prediction) risk of a network with weights w as the expected value (2) r(w) := IE:f(X, Y, w). Let us denote the minimizer of (1) by Wn (.),) , and a minimizer of the risk r by w*. The quantity r(wn (>.)) is the average prediction error for data independent of the training sample. This quantity r(wn (>.)) is a random variable which describes the generalization performance of the network: it is bounded below by r( w*) and the more concentrated it is about r(w*), the better the performance. We will quantify this concentration by a single number, the expected value IE:r(wn(>.)) . We are interested in quantifying the gain (if any) in generalization for training with versus training without regularization defined by (3) When regularization helps, this is positive. However, relatively little can be said about the quantity (3) without specifying in detail how the regularization parameter is determined. We show in the next section that provided>' converges to zero sufficiently quickly (at the rate op(n- 1/ 2 )), then IE: r(wn(O)) and IE: r(wn(>.)) are equal to leading order. It turns out, that the optimal regularization parameter resides in this asymptotic regime. For this reason, delicate analysis is required in order to get an asymptotic approximation for (3). In this article we derive the needed asymptotic expansions only for the simplest possible case: one-dimensional linear regression where the regularization parameter is chosen independently of the training sample. 2 REGULARIZATION IN LINEAR REGRESSION We now specialize the setup (1) to the case of linear regression and a quadratic smoothness penalty, i.e., we take f(x,y,w) = [y-xTwJ2 and Q(w) = wTRw, where now y is scalar, x and w are vectors, and R is a symmetric, positive definite matrix. It is well known (and easy to show) that then the minimizer of (1) is (4) 1 n 1 n [ ] -1 wn (>') = ~ ~ XiX! + >'R ~ ~ XiYi. This is called the generalized ridge regression estimator, see, e.g., (Titterington, 1985); ridge regression corresponds to the choice R = I, see (Hoerl and Kennard, 1988) for a survey. Notice that (generalized) ridge regression is usually studied in the fixed design case, where Xi:s are nonrandom. Further, it is usually assumed that the model is correctly specified, i.e., that there exists a parameter such that Yi = Xr w* + €i , and such that the distribution of the noise term €i does not depend on Xi. In contrast, we study the random design, misspecified case. Assuming that IE: IIXI12 < 00 and that IE: [XXT] is invertible, the minimizer of the risk (2) and the risk itself can be written as (5) w* = A-lIE: [XY], with A:=IE:[XXT] (6) r(w) = r(w*) + (w - w*f A(w - w*). 296 P. Koistinen If Zn is a sequence of random variables, then the notation Zn = open-a) means that na Zn converges to zero in probability as n -+ 00. For this notation and the mathematical tools needed for the following proposition see, e.g., (Serfiing, 1980, Ch. 1) or (Brockwell and Davis, 1987, Ch. 6). Proposition 1 Suppose that IE: y4 < 00, IE: IIXII4 < 00 and that A = IE: [X XTj is invertible. If,\ = op(n-I/2), then both y'n(wn(O) -w*) and y'n(wn('\) - w*) converge in distribution to N (0, C), a normal distribution with mean zero and covariance matrix C. The previous proposition also generalizes to the nonlinear case (under more complicated conditions). Given this proposition, it follows (under certain additional conditions) by Taylor expansion that both IE:r(wn('\)) - r(w*) and IEr(wn(O)) - r(w*) admit the expansion f31 n -} + o( n -}) with the same constant f3I. Hence, in the regime ,\ = op(n-I/2) we need to consider higher order expansions in order to compare the performance of wn(,\) and wn(O). 3 ONE-DIMENSIONAL LINEAR REGRESSION We now specialize the setting of the previous section to the case where x is scalar. Also, from now on, we only consider the case where the regularization parameter for given sample size n is deterministic; especially ,\ is not allowed to depend on the training sample. This is necessary, since coefficients in the following type of asymptotic expansions depend on the details of how the regularization parameter is determined. The deterministic case is the easiest one to analyze. We develop asymptotic expansions for the criterion (7) where now the regularization parameter k is deterministic and nonnegative. The expansions we get turn out to be valid uniformly for k ~ O. We then develop asymptotic formulas for the minimizer of I n, and also for In(O) - inf I n. The last quantity can be interpreted as the average improvement in generalization performance gained by optimal level of regularization, when the regularization constant is allowed to depend on n but not on the training sample. From now on we take Q(w) = w2 and assume that A = IEX2 = 1 (which could be arranged by a linear change of variables). Referring back to formulas in the previous section, we see that (8) r(wn(k)) - r(w*) = ern - kw*)2/(Un + 1 + k)2 =: h(Un, Vn, k), whence In(k) = IE:h(Un, Vn , k), where we have introduced the function h (used heavily in what follows) as well as the arithmetic means Un and Vn (9) (10) _ 1 n Vn:= - L Vi, with n I Vi := XiYi - w* xl For convenience, also define U := X2 - 1 and V := Xy - w* X2 . Notice that U; UI, U2 , • .. are zero mean Li.d. random variables, and that V; Vi, V2 ,. " satisfy the same conditions. Hence Un and Vn converge to zero, and this leads to the idea of using the Taylor expansion of h(u, v, k) about the point (u, v) = (0,0) in order to get an expansion for In(k). Asymptotic Theory for Regularization: One-Dimensional Linear Case 297 To outline the ideas, let Tj(u,v,k) be the degree j Taylor polynomial of (u,v) f-7 h(u, v, k) about (0,0), i.e., Tj(u, v, k) is a polynomial in u and v whose coefficients are functions of k and whose degree with respect to u and v is j. Then IETj(Un,Vn,k) depends on n and moments of U and V. By deriving an upper bound for the quantity IE Ih(Un, Vn, k) - Tj(Un, Vn, k)1 we get an upper bound for the error committed in approximating In(k) by IE Tj(Un, Vn, k). It turns out that for odd degrees j the error is of the same order of magnitude in n as for degree j - 1. Therefore we only consider even degrees j. It also turns out that the error bounds are uniform in k ~ 0 whenever j ~ 2. To proceed, we need to introduce assumptions. Assumption 1 IE IXlr < 00 and IE IYls < 00 for high enough rand s. Assumption 2 Either (a) for some constant j3 > 0 almost surely IXI ;::: j3 or (b) X has a density which is bounded in some neighborhood of zero. Assumption 1 guarantees the existence of high enough moments; the values r = 20 and s = 8 are sufficient for the following proofs. E.g., if the pair (X, Y) has a normal distribution or a distribution with compact support, then moments of all orders exist and hence in this case assumption 1 would be satisfied. Without some condition such as assumption 2, In(O) might fail to be meaningful or finite. The following technical result is stated without proof. Proposition 2 Let p > 0 and let 0 < IE X 2 < 00. If assumption 2 holds, then where the expectation on the left is finite (a) for n ~ 1 (b) for n > 2p provided that assumption 2 (a), respectively 2 (b) holds. Proposition 3 Let assumptions 1 and 2 hold. Then there exist constants no and M such that In(k) = JET2(Un, Vn, k) + R(n, k) where _ _ (w*)2k2 -1 [IEV2 (w*)2k2JEU2 W*kIEUV] IET2(Un, Vn, k) = (1+k)2 +n (1+k)2 +3 (1+k)4 +4 (1+k)3 IR(n, k) I :s; Mn-3/2(k + 1)-1, "In;::: no, k ;::: o. PROOF SKETCH The formula for IE T2(Un , Vn. k) follows easily by integrating the degree two Taylor polynomial term by term. To get the upper bound for R(n, k), consider the residual where we have omitted four similar terms. Using the bound 298 P. Koistinen the Ll triangle inequality, and the Cauchy-Schwartz inequality, we get IR(n, k)1 = IJE [h(Un, Vn, k) - T2(Un, Vn, k)]1 ., (k+ W' {Ii: [(~ ~Xl)-'] r {2(k + 1)3[JE (lUnI2IVnI 4)]l/2 + 4(w*)2k2(k + 1)[18 IUnI6]l/2 ... } By proposition 2, here 18 [(~ 2:~ X[)-4] = 0(1). Next we use the following fact, cf. (Serfiing, 1980, Lemma B, p. 68). Fact 1 Let {Zd be i.i.d. with 18 [Zd = 0 and with 18 IZI/v < 00 for some v ~ 2. Then v Applying the Cauchy-Schwartz inequality and this fact, we get, e.g., that [18 (IUnI2 IVnI 4 )]l/2 ~ [(18 IUnI4)1/2(E IVnI8)1/2p/2 = 0(n- 3/ 2). Going through all the terms carefully, we see that the bound holds. Proposition 4 Let assumptions 1 and 2 hold, assume that w* :j; 0, and set al := (18 V2 - 2w*E [UVD/(w*)2. o If al > 0, then there exists a constant ni such that for all n ~ nl the function k ~ ET2(Un, Vn,k) has a unique minimum on [0,(0) at the point k~ admitting the expanszon k~ = aIn-1 + 0(n-2); further, In(O) - inf{Jn(k) : k ~ O} = In(O) - In(aln- 1 ) = ar(w*)2n- 2 + 0(n- 5/ 2). If a ~ 0, then PROOF SKETCH The proof is based on perturbation expansio!1 c~nsidering lin a small parameter. By the previous proposition, Sn(k) := ET2(Un , Vn , k) is the sum of (w*)2k2/(1 + k)2 and a term whose supremum over k ~ ko > -1 goes to zero as n ~ 00. Here the first term has a unique minimum on (-1,00) at k = O. Differentiating Sn we get S~(k) = [2(w*)2k(k + 1)2 + n-1p2(k)]/(k + 1)5, where P2(k) is a second degree polynomial in k. The numerator polynomial has three roots, one of which converges to zero as n ~ 00. A regular perturbation expansion for this root, k~ = aln-I + a2n-2 + ... , yields the stated formula for al. This point is a minimum for all sufficiently large n; further, it is greater than zero for all sufficiently large n if and only if al > O. The estimate for J n (0) - inf { J n (k) : k ~ O} in the case al > 0 follows by noticing that In(O) - In(k) = 18 [h(Un, Vn, 0) - h(Un, Vn, k)), where we now use a third degree Taylor expansion about (u, v, k) = (0,0,0) h(u,v,O) - h(u,v,k) = 2w* kv - (w*)2k2 - 4w*kuv + 2(w*?k2u + 2kv2 - 4w*k2v + 2(W*)2k3 + r(u, v, k). Asymptotic Theory for Regularization: One-Dimensional Linear Case 0.2 0.18 0.16 0.14 0.12 0.1 ~~ __ ~ __ ~ __ ~ __ ~ __ ~ __ ~ __ L-~ __ ~ o 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 299 Figure 1: Illustration of the asymptotic approximations in the situation of equation (11). Horizontal axis kj vertical axis .In(l£) and its asymptotic appr~ximations. Legend: markers In(k); solid line IE T2(Un, Vn, k)j dashed line IET4 (Un, Vn, k). Usin~ t~e techniques of the previous proposition, it can be shown that IE Ir(Un , Vn , k~)1 = O(n-S/ 2 ). Integrating the Taylor polynomial and using this estimate gives In(O) - In(aI/n) = af(w*)2n-2 + O(n- S/ 2 ). Finally, by the mean value theorem, In(O) -inf{ In(k) : k ~ O} = In(O) -In(aI/n) + ! (In(O) - In(k)]lk=8(k~ -aI/n) = In(O) - In(aI/n) + O(n-1)O(n-2) where () lies between k~ and aI/n, and where we have used the fact that the indicated derivative evaluated at () is of order O(n- 1), as can be shown with moderate effort. 0 Remark In the preceding we assumed that A = IEX 2 equals 1. If this is not the case, then the formula for a1 has to be divided by A; again, if a1 > 0, then k~ = a1n-1 + O(n-2 ) . If the model is correctly specified in the sense that Y = w* X + E, where E is independent of X and IE E = 0, then V = X E and IE [UV] = O. Hence we have a1 = IE [E2]j(w*)2, and this is strictly positive expect in the degenerate case where E = 0 with probability one. This means that here regularization helps provided the regularization parameter is chosen around the value aI/n and n is large enough. See Figure 1 for an illustration in the case (11) X "'" N(O, 1), Y = w* X + f , f "'" N(O, 1), w* = 1, where E and X are independent. In(k) is estimated on the basis of 1000 repetitions of the task for n = 8. In addition to IE T2(Un, Vn, k) the function IE T4(Un, lin, k) is also plotted. The latter can be shown to give In(k) correctly up to order O(n-s/2(k+ 1)-3). Notice that although IE T2(Un, Vn, k) does not give that good an approximation for In(k), its minimizer is near the minimizer of In(k), and both of these minimizers lie near the point al/n = 0.125 as predicted by the theory. In the situation (11) it can actually be shown by lengthy calculations that the minimizer of In(k) is exactly al/n for each sample size n ~ 1. It is possible to construct cases where a1 < O. For instance, take X "'" Uniform (a, b), Y = cjX + d+ Z, 1 1 a=- b=-(3Vs-l) 2' 4 c= -5,d= 8, 300 P. Koistinen and Z '" N (0, a 2) with Z and X independent and 0 :::; a < 1.1. In such a case regularization using a positive regularization parameter only makes matters worse; using a properly chosen negative regularization parameter would, however, help in this particular case. This would, however, amount to rewarding rapidly changing functions. In the case (11) regularization using a negative value for the regularization parameter would be catastrophic. 4 DISCUSSION We have obtained asymptotic approximations for the optimal regularization parameter in (1) and the amount of improvement (3) in the simple case of one-dimensional linear regression when the regularization parameter is chosen independently of the training sample. It turned out that the optimal regularization parameter is, to leading order, given by Qln-1 and the resulting improvement is of order O(n-2 ). We have also seen that if Ql < 0 then regularization only makes matters worse. Also (Larsen and Hansen, 1994) have obtained asymptotic results for the optimal regularization parameter in (1). They consider the case of a nonlinear network; however, they assume that the neural network model is correctly specified. The generalization of the present results to the nonlinear, misspecified case might be possible using, e.g., techniques from (Bhattacharya and Ghosh, 1978). Generalization to the case where the regularization parameter is chosen on the basis of the sample (say, by cross validation) would be desirable. Acknowledgements This paper was prepared while the author was visiting the Department for Statistics and Probability Theory at the Vienna University of Technology with financial support from the Academy of Finland. I thank F. Leisch for useful discussions. References Bhattacharya, R. N. and Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion. The Annals of Statistics, 6(2):434-45l. Bishop, C. M. (1995). Neural Networks for Pattern Recognition. Oxford University Press. Brockwell, P. J. and Davis, R. A. (1987). Time Series: Theory and Methods. Springer series in statistics. Springer-Verlag. Hoerl, A. E. and Kennard, R. W. (1988). Ridge regression. In Kotz, S., Johnson, N. L., and Read, C. B., editors, Encyclopedia of Statistical Sciences. John Wiley & Sons, Inc. Larsen, J. and Hansen, L. K. (1994). Generalization performance of regularized neural network models. In Vlontos, J., Whang, J.-N., and Wilson, E., editors, Proc. of the 4th IEEE Workshop on Neural Networks for Signal Processing, pages 42-51. IEEE Press. Ripley, B. D. (1996). Pattern Recognition and Neural Networks. Cambridge University Press. Serfiing, R. J. (1980). Approximation Theorems of Mathematical Statistics. John Wiley & Sons, Inc. Titterington, D. M. (1985). Common structure of smoothing techniques in statistics. International Statistical Review, 53:141-170.
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Instabilities in Eye Movement Control: A Model of Periodic Alternating Nystagmus ErnstR. Dow Center for Biophysics and Computational Biology, Beckman Institute University of Illinois at UrbanaChampaign,Urbana, IL 61801. edow@uiuc.edu Thomas J. Anastasio Department of Molecular and Integrative Physiology, Center for Biophysics and Computational Biology, Beckman Institute University of Illinois at UrbanaChampaign, Urbana, IL 61801. tstasio@uiuc.edu Abstract Nystagmus is a pattern of eye movement characterized by smooth rotations of the eye in one direction and rapid rotations in the opposite direction that reset eye position. Periodic alternating nystagmus (PAN) is a form of uncontrollable nystagmus that has been described as an unstable but amplitude-limited oscillation. PAN has been observed previously only in subjects with vestibulo-cerebellar damage. We describe results in which PAN can be produced in normal subjects by prolonged rotation in darkness. We propose a new model in which the neural circuits that control eye movement are inherently unstable, but this instability is kept in check under normal circumstances by the cerebellum. Circumstances which alter this cerebellar restraint, such as vestibulocerebellar damage or plasticity due to rotation in darkness, can lead to PAN. 1 INTRODUCTION Visual perception involves not only an operating visual sensory system, but also the ability to control eye movements. The oculomotor subsystems provide eye movement control. For example, the vestibulo-ocular reflex (VOR) maintains retinal image stability by making slow-phase eye rotations that counterbalance head rotations, making it possible to move and see at the same time (Wilson and Melvill Jones, 1979). The VOR makes slowphase eye rotations that are directed opposite to head rotations. When these ongoing slow-phase eye rotations are interrupted by fast-phase eye rotations that reset eye position, the resulting eye movement pattern is called nystagmus. Periodic alternating nysA Model of Periodic Alternating Nystagmus 139 tagmus (PAN) is a congenital or acquired eye movement disorder characterized by uncontrollable nystagmus that alternates direction roughly sinusoidally with a period of 200 s to 400 s (Baloh et al., 1976; Leigh et al., 1981; Furman et aI., 1990). Furman and colleagues (1990) have determined that PAN in humans is caused by lesions of parts of the vestibulo-cerebellum known as the nodulus and uvula (NU). Lesions to the NU cause PAN in the dark (Waespe et aI., 1985; Angelaki and Hess, 1995). NU lesions also prevent habituation (Singleton, 1967; Waespe et aI, 1985; Torte et al., 1994), which is a semi-permanent decrease in the gain (eye velocity / head velocity) of the VOR response that can be brought about by prolonged low-frequency rotational stimulation in the dark. Vestibulo-cerebellectomy in habituated goldfish causes VOR dishabituation (Dow and Anastasio, 1996). Temporary inactivation of the vestibulo-cerebellum in habituated goldfish causes temporary dishabituation and can result in a temporary PAN (Dow and Anastasio, in press). Stimulation of the NU temporarily abolish the VOR response (Fernandez and Fredrickson, 1964). Cerebellar influence on the VOR may be mediated by connections between the NU and vestibular nucleus neurons, which have been demonstrated in many species (Dow, 1936; 1938). We have previously shown that intact goldfish habituate to prolonged low-frequency (0.01 Hz) rotation (Dow and Anastasio, 1996) and that rotation at higher frequencies (0.05-0.1 Hz) causes PAN (Dow and Anastasio, 1997). We also proposed a limit-cycle model of PAN in which habituation or PAN result from an increase or decrease, respectively, of the inhibition of the vestibular nuclei by the NU. This model suggested that velocity storage, which functions to increase low-frequency VOR gain above the biophysical limits of the semicircular canals (Robinson, 1977;1981), is mediated by a potentially unstable low-frequency resonance. This instability is normally kept in check by constant suppression by the NU. 2 METHODS PAN was studied in intact, experimentally naive, comet goldfish (carassius auratus). Each goldfish was restrained horizontally underwater with the head at the center of a cylindrical tank. Eye movements were measured using the magnetic search coil technique (Robinson, 1963). For technical details see Dow and Anastasio (1996). The tank was centered on a horizontal rotating platform. Goldfish were rotated continuously for various durations (30 min to 2 h) in darkness at various single frequencies (0.03 - 0.17 Hz). Some data have been previously reported (Dow and Anastasio, 1997). All stimuli had peak rotational velocities of 60 degls. Eye position and rotator (i.e. head) velocity signals were digitiZed for analysis. Eye position data were digitally differentiated to compute eye velocity and fast-phases were removed. Data were analyzed and simulated using MATLAB and SIMULINK (The Mathworks, Inc.). 3 RESULTS Prolonged rotation in darkness at frequencies which produced some habituation in naive goldfish (0.03-0.17 Hz) could produce a lower-frequency oscillation in slow-phase eye velocity that was superimposed on the normal VOR response (fig 1). This lowerfrequency oscillation produced a periodic alternating nystagmus (PAN). When PAN occurred, it was roughly sinusoidal and varied in period, amplitude, and onset-time. Habituation could occur simultaneously with PAN (fig IB) or habituation could completely 140 60 ~ 40 ~ 20 ~ 'g 0 g! -20 Q) >Q) -40 E. R. Dow and T. J. Anastasio Initial response A ~-60~------~--------~~--~--~------~--------~ Q) ; -:8 ~JJJvAvAJ~\VV~AvA~¥~AvA~\AvAJv\Av¥v¥~AJfv\¥M~¥v¥JvA~ i .r; 60 ~ 40 ~ 20 ~ '8 0 Q) > -20 g!. Q) -40 tn-60 t Response after 1 h. rotation iJ ~ V ~ ~ B ,A A A ~j l~J v~ I¥ v y~ v~ ~ 50~A66A6GGGA66666AGGAAG6A666A6GA6AAAA666AGA666GA6A6 1 -58 V V vv rv V V IlVlJlJ V V V V V V U V V V rv V V VlJ u vrvvrvwvlv vvrVV V VVt '0 al 0 200 400 600 800 1 000 .r; time (s) Figure 1: Initial 1000 s 0.05 Hz rotation showing PAN (A). Slow-phase eye velocity shows that PAN starts almost immediately and there is a slight reduction in VOR gain after 1000 s. Following 1 h continuous rotation in the same goldfish (B), VOR gain has decreased. suppress PAN (fig 4). PAN observed at lower frequencies (0.03 and 0.05 Hz) typically decreased in amplitude as rotation continued. Previous work has shown that PAN was most likely to occur during prolonged rotations at frequencies between 0.05 and 0.1 Hz (Dow and Anastasio, 1997). At these frequencies, habituation also caused a slight decrease in VOR gain (1.3 to 1.8 times, initial gain I final gain) following 1 h of rotation. At higher frequencies, neither habituation nor PAN were observed. At lower frequencies (0.03 Hz) PAN could occur before habituation substantially reduced VOR gain (fig 4). PAN, was not observed in naIve goldfish rotated a lower frequency (0.01 Hz) where VOR gain fell by 22 times due to habituation (Dow and Anastasio, 1997). A Model of Periodic Alternating Nystagmus 141 4 MODEL Previously, a non-linear limit cycle model was constructed by Leigh, Robinson, and Zee (1981; see also Furman, 1989) to simulate PAN in humans. This model included a velocity storage loop with saturation, and a central adaptation loop. This second order system would spontaneously oscillate, producing PAN, if the gain of the velocity storage loop was greater than 1. We adjusted Robinson's model to simulate rotation inducible PAN and habituation in the goldfish. Input to and output from the model (fig 2) represent head and slow-phase eye velocity, respectively. The time constants of the canal (S'tc/(S'tc+ 1» and velocity-storage (g.l(S'ts+l» elements were set to the value of the canal time constant as determined experimentally in goldfish ('Cc = '1:s = 3 s) (Hartman and Klinke, 1980). The time constant of the central adaptation element (l/S'ta) was 10 times longer ('1:a = 30 s). The Laplace variable (s) is complex frequency (s = jro where / is -1 and ro is frequency in rad/s). The gain of the velocity-storage loop (gs) is 1.05 while that of the central adaptation loop (ga) is 1. The central adaptation loop represents in part a negative feedback loop onto vestibular nucleus neurons through inhibitory Purkinje cells of the NU. The vestibulo-cerebellum is known to modulate the gain of the VOR (Wilson and Melvill Jones, 1979). The static nonlinearity in the velocity storage loop consists of a threshold (± 0.0225) and a saturation (± 1.25). The threshold was added to model the decay in PAN following termination of rotation (Dow and Anastasio, 1997), which is not modeled here. Increases or decreases in the absolute value of ga will cause VOR habituation or PAN, respectively. However, it was more common for VOR habituation and PAN to occur simultaneously (fig tB). This behavior could not be reproduced with the lumped model (fig 2). It would be necessary on one hand to increase ga to decrease overall VOR gain while, on the other hand, decrease ga to produce PAN. A distributed system would address this problem, with multiple parallel pathways, each having velocity-storage and adaptive control through the NU. The idea can be illustrated using the simplest distributed system which has 2 lumped models in parallel (not shown), each having an independently adjustable gao The results from such a two parallel pathway model are shown in fig 3. In one pathway, ga(h) was increased to model habituation, and gio) was decreased to start oscillations. Paradoxically, although the ultimate effect of increasing ga(h) is to decrease VOR gain, the initial effect as ga(h) is increased is to increase gain. This is due to the resonant frequency of the system continuously shifting to higher frequencies and temporarily matching the frequency of rotation (see DISCUSSION). Conversely, when ga(o) is decreased, there is a temporary decrease in gain as the resonant frequency moves away from the frequency of rotation. The two results are combined after the gain is reduced by half (fig 3B). canal transfer head velocity function L...--'r---' central adaptation eye velocity Figure 2: Model used to simulation PAN (Dow and Anastasio 1997). Used with permission. 142 E. R Dow and T. 1. Anastasio The combined result shows a continual decrease in VOR gain with the oscillations superimposed. 5 DISCUSSION If the nonlinearities (i.e. threshold and saturation) in the model are ignored, linear analysis shows that the model will be unstable when [(1 - gs)/ts + gJ'tal is negative, and will oscillate with a period of [2m1(tstJga)]. With the initial parameters, the model is stable because the central adaptation loop can compensate for the unstable gain of the velocity storage loop. The natural frequency of the system, calculated from the above equation, is 0.017 Hz. This resonance, which peaks at the resonant frequency but is stiII pronounced at nearby frequencies, produces an enhancement of the VOR response. The hypothesis that low frequency VOR gain enhancement is produced by a potentially unstable resonance is a novel feature of our model. The natural frequency increases with increases in ga and can alter the frequency specific enhancement. Q) 8~ 2[--B il=--~ Q) -2L---------------~------~--------~---~ :::l . . •. > ..... < ..... c al 4k .... ~2 / ::: I:: 0 L _____ --=::. ____________ ........ __ o 1000 2000 time (sec) 3000 Figure 3: Model at 0.03 Hz. Two simulations with differing values of ga (A) are combined in (B) with the values of ga in (C). Decreases in ga, in addition to decreasing the natural frequency, also cause the model to become unstable. (If ga is reduced to zero, the model becomes first order and the equations are no longer valid). The ability to get either habituation or PAN by varying only one parameter suggests that habituation and PAN are a related phenomena Through the process of habituation, prolonged low frequency rotation (0.01 Hz) in goldfish severely decreased VOR gain, often abruptly and unilaterally (Dow and Anastasio, in press). The decrease in gain due to habituation can effectively eliminate PAN at the lowest frequency at which PAN was observed (0.03 Hz) as shown in fig 4. In this example the naive VOR responds symmetrically for the first cycle of rotation. It then becomes markedly asymmetrical, with a strong, unilateral response in one direction for -10 cycles followed by another in the opposite direction for -17 cycles. The VOR response abruptly habituates after that with no PAN. Complete habituation can be simulated by further increases in the value of ga' in the limit-cycle model (fig 2). Unilateral habituation has been simulated previously with a bilateral network model of the VOR in which the cerebellum inhibits the vestibular nuclei unilaterally (Dow and Anastasio, in press). A Model of Periodic Alternating Nystagmus 60 40 ~ 20 Q) "0 --~ ·0 0 o CD > ~-20 Q) -40 .l bJ A ~tLL ~ r~ I'~ "'1 o 200 400 600 time (5) 800 Figure 4: PAN superimposed on the VOR response to continuous rotation at 0.03 Hz. Upper trace, slow-phase eye velocity (fast phases removed); lower trace, head velocity (not to scale). 143 The cerebellum has several circuits which could provide an increase in firing rate of some Purkinje cells with a concurrent decrease in the firing rate of other Purkinje cells suggested by the model. There are many lateral inhibitory pathways including inhibition of Purkinje cells to neighboring Purkinje cells (Llimis and Walton, 1990). Therefore, if one Purkinje cell were to increase its firing rate, this circuitry suggests that neighboring Purkinje cells would decrease their firing rates. Also, experimental evidence shows that during habituation, not all vestibular nuclei gradually decrease their firing rate as might be expected. Kileny and colleagues (1980) recorded from vestibular nucleus neurons during habituation. They could divide the neurons into 3 roughly equal groups based on response over time: continual decrease, constant followed by a decrease, and increase followed by decrease. The cerebellar circuitry and the single-unit recording data support multiple, variable levels of inhibition from the NU. How this mechanism may work is being explored with a more biologically realistic distributed model. 6 CONCLUSION Our experimental results are consistent with a multi-parallel pathway model of the VOR. In each pathway an unstable, positive velocity-storage loop is stabilized by an inhibitory, central adaptation loop, and their interaction produces a low-frequency resonance that enhances the low-frequency response of the VOR. Prolonged rotation at specific frequencies could produce a decrease in central adaptation VOR gain in some pathways resulting in an unstable, low-frequency oscillation resembling PAN in these pathways. An 144 E. R. Dow and T. J. Anastasio increase in adaptation loop gain in the other pathways would result in a decrease in VOR gain resembling habituation. The sum over the VOR pathways would show PAN and habituation occurring together. We suggest that resonance enhancement and mUltiple parallel (i.e. distributed) pathways are necessary to model the interrelationship between PAN and habituation. Acknowledgments The work was supported by grant MH50577 from the National Institutes of Health. We thank M. Zelaya and X. Feng for experimental assistance. References Angelaki DE and Hess BJM. J Neurophysl73 1729-1751 (1995). Baloh RW, Honrubia V and Konrad HR. Brain 99 11-26 (1976). Dow ER and Anastasio TJ. NeuroReport 71305-1309 (1996). Dow ER and Anastasio TJ. NeuroReport 82755-2759 (1997). Dow ER and Anastasio TJ. J. Computat. Neuro. in press. Dow RS. J Comp Neurol63 527-548 (1936). Dow RS. J Comp Neurol68 297-305 (1938). Fernandez C and Fredrickson lM. Acta Otolaryngol Suppl192 52-62 (1964). Furman JMR, Wall C and Pang D. Brain 113 1425-1439 (1990). Furman lMR, Hain TC and Paige GO. Biol Cybern 61255-264 (1989). Hartmann R and Klinke R. Pflugers Archiv 388 111-121 (1980). Kileny P, Ryu JH, McCabe BF and Abbas PJ. Acta Otolaryngol90 175-183 (1980). Leigh RJ, Robinson DA and Zee OS. Ann NY A cad Sci 374619-635 (1981). LIinas RR and Walton KD. Cerebellum. In: Shepherd GM ed. The Synaptic Organization o/the Brain. Oxford: Oxford University Press, 1990: 214-245. Remmel RS. IEEE Trans Biomed Eng 31 388-390 (1984). Robinson DA. IEEE Trans Biomed Eng 10 137-145 (1963). Robinson DA. Exp Brain Res 30447-450 (1977). Robinson DA. Ann Rev Neurosci 4463-503 (1981). Singleton GT. Laryngoscope 77 1579-1620 (1967). Torte MP, Courjon JH, Flandrin JM, et al. Exp Brain Res 99 441-454 (1994). Waespe W, Cohen Band Raphan T. Science 228 199-202 (1985). Wilson V and Melvill Jones G. Mammalian Vestibular Physiology, New York: Plenum Press, 1979.
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Neural Basis of Object-Centered Representations Sophie Deneve and Alexandre Pouget Georgetown Institute for Computational and Cognitive Sciences Georgetown University Washington, DC 20007-2197 sophie, alex@giccs.georgetown.edu Abstract We present a neural model that can perform eye movements to a particular side of an object regardless of the position and orientation of the object in space, a generalization of a task which has been recently used by Olson and Gettner [4] to investigate the neural structure of object-centered representations. Our model uses an intermediate representation in which units have oculocentric receptive fields- just like collicular neurons- whose gain is modulated by the side of the object to which the movement is directed, as well as the orientation of the object. We show that these gain modulations are consistent with Olson and Gettner's single cell recordings in the supplementary eye field. This demonstrates that it is possible to perform an object-centered task without a representation involving an object-centered map, viz., without neurons whose receptive fields are defined in object-centered coordinates. We also show that the same approach can account for object-centered neglect, a situation in which patients with a right parietal lesion neglect the left side of objects regardless of the orientation of the objects. Several authors have argued that tasks such as object recognition [3] and manipulation [4] are easier to perform if the object is represented in object-centered coordinates, a representation in which the subparts of the object are encoded with respect to a frame of reference centered on the object. Compelling evidence for the existence of such representations in the cortex comes from experiments on hemineglect- a neurological syndrome resulting from unilateral lesions of the parietal cortex such that a right lesion, for example, leads patients to ignore stimuli located on the left side of their egocentric space. Recently, Driver et al. (1994) showed that the deficit can also be object-centered. Hence, hemineglect patients can detect a gap in the upper edge of a triangle when this gap is associated with the right side of the object Neural Basis of Object-Centered Representations 25 A. B. Obje~tnt.recl I Three possible locations: " , cu·"'S " " ~ I Spotiol . / e I ; 1! cueing I, .1 I • : 1 ;.~ 1 I e ' I e I e I ! • ! I 1'2 I ! i e , ! I [4 15/ !1 , 13 2 l . • C. Time Figure 1: A- Driver et al. (1994) experiment demonstrating object-centered neglect. Subjects were asked to detect a gap in the upper part of the middle triangle, while fixating at the cross, when the overall figure is tilted clockwise (top) or counterclockwise (bottom). Patients perform worse for the clockwise condition, when the gap is perceived to be on the left of the overall figure. B- Sequence of screens presented on each trial in Olson and Gettner experiment (1995). 1- Fixation, 2- apparition of a cue indicating where the saccade should go, either in object-centered coordinates (object-centered cueing), or in screen coordinates (spatial cueing), 3- delay period, 4- apparition of the bar in one of three possible locations (dotted lines), and 5- saccade to the cued location. C- Schematic response of an SEF neuron for 4 different conditions. Adapted from [4]. but not when it belongs to the left side (figure I-A). What could be the neural basis of these object-centered representations? The simplest scheme would involve neurons with receptIve fields defined in object-centered coordinates, i.e., the cells respond to a particular side of an object regardless of the position and orientation of the object. A recent experiment by Olson and Gettner (1995) supports this possibility. They recorded the activity of neurons in the supplementary eye field (SEF) while the monkey was performing object-directed saccades. The task consisted of making a saccade to the right or left side of a bar, independently of the position of the bar on the screen and according to the instruction provided by a visual cue. For instance, the cue corresponding to the instruction 'Go to the right side of the bar' was provided by highlighting the right side of a small bar briefly flashed at the beginning of the trial (step 2 in figure I-B). By changing the position of the object on the screen, it is possible to compare the activity of a neuron for movements involving different saccade directions but directed to the same side of the object, and vice-versa, for movements involving the same saccade direction but directed to opposite sides of the object. Olson and Gettner found that many neurons responded more prior to saccades directed to a particular side of the bar, independently of the direction of the saccades. For example, some neurons responded more for an upward right saccade directed to the left side of the bar but not at all for the same upward right saccade directed to the right side of the bar (column 1 and 3, figure I-C). This would suggest that these neurons have bar-centered receptive! fields, i.e., their receptive fields are centered lwe use the term receptive field in a general sense, meaning either receptive or motor 26 "Do E -;:" ... i~ ·c !: o~ ~ . ~ mop s. Deneve andA. Pouget Figure 2: Schematic structure of the network with activity patterns in response to the horizontal bar shown in the VI map and the command 'Go to the right'. Only one SEF map is active in this case, the one selective to the right edge of the bar (where right is defined in retinal coordinates), object orientation of 0° and the command 'Go to the right'. The letter a, b, c and d indicate which map would be active for the same command but for various orientations of the object, respectively, 0°, 90°, 180°, 270°. The dotted lines on the maps indicate the outline of the bar. Only a few representative connections are shown. on the bar and not on the retina. This would correspond to what we will call an explicit object-centered representation. We argue in this paper that these data are compatible with a different type of representation which is more suitable for the task performed by the monkey. We describe a neural network which can perform a saccade to the right, or left, boundary of an object, regardless of its orientation, position or size- a generalization of the task used by Olson and Gettner. This network uses units with receptive fields defined in oculocentric coordinates, i.e., they are selective for the direction and amplitude of saccades with respect to the fixation point, just like collicular neurons. These tuning curves, however, are also modulated by two types of signals, the orientation of the object, and the command indicating the side of the object to which the saccade should be directed. We show that these response properties are compatible with the Olson and Gettner data and provide predictions for future experiments. We also show that a simulated lesion leads to object-centered neglect as observed by Driver et al. (1994). 1 Network Architecture The network performs a mapping from the image of the bar and the command (indicating the side of the object to which the saccade must be directed) to the appropriate motor command in oculocentric coordinates (the kind of command observed in the frontal eye field, FEF). We use a bar whose left and right sides are defined with respect to a a triangle appearing on the top of the bar (see figure 2). The network is composed of four parts. The first two parts of the network models the field. Neural Basis of Object-Centered Representations 27 lower areas in visual cortex, where visual features are segmented within retinotopic maps. In the first layer, the image is projected on a very simple VI-like map (10 by 10 neurons with activity equal to one if a visual feature appears within their receptive field, and zero otherwise). The second part on the network contains 4 different V2 retinotopic maps, responding respectively to the right, left, top and bottom boundary of the bar. This model of V2 is intended to reproduce the response properties of a subset of cells recently discovered by Zhou et al. (1996). These cells respond to oriented edges, like VI cells, but when the edge belongs to a closed figure, they also show a selectivity for the side on which the figure appears with respect to the edge. For example, a cell might respond to a vertical edge only if this edge is on the right side of the figure but not on the left (where right and left are defined with respect to the viewer, not the object itself). This was observed for any orientation of the edge, but we limit ourselves in this model to horizontal and vertical ones. The third part of the network models the SEF and is divided into 4 groups of 4 maps, each group receiving connections from the corresponding map in V2 (figure 2). Within each group of maps, visual activity is modulated by signals related to the orientation of the object (assumed to be computed in temporal cortex) such that each of the 4 maps respond best for one particular orientation (respectively 0°, 90°, 180° and 270°). For example, a neuron in the second map of the top group responds maximally if: 1- there is an edge in its receptive field and the figure is below, and 2- the object has an orientation of 90° counterclockwise. Note that this situation arises only if the left side of the object appears in the cell's receptive field; it will never occur for the right side. However, the cell is only partially selective to the left side of the object, e.g., it does not respond when the left side is in the retinal receptive field and the orientation of the object is 270° counterclockwise. These collection of responses can be used to generate an object-centered saccade by selecting the maps which are partially selective for the side of the object specified by the command. This is implemented in our network by modulating the SEF maps by signals related to the command. For example, the unit encoding 'go to the right' send a positive weight to any map compatible with the right side of the object while inhibiting the other maps (figure 2). Therefore, the activity, Bfj' of a neuron at position ij on the map k in the SEF is the product of three functIOns: where Vij is the visual receptive field from the V2 map, h((J) is a gaussian function of orientation centered on the cell preferred orientation, (Jk, and gk(C) is the modulation by the command unit. Olson and Gettner also used a condition with spatial cueing, viz., the command was provided by a spatial cue indicating where the saccade should go, as opposed to an object-centered instruction (see figure I-B). We modeled this condition by simply multiplying the activity of neurons coding for this location in all the SEF maps by a fixed constant (10 in the simulations presented here). We also assume that there is no modulation by orientation of the object since this information is irrelevant in this experimental condition. Finally, the fourth part of the network consists of an oculocentric map similar to the one found in the frontal eye field (FEF) or superior colliculus (SC) in which the command for the saccade is generated in oculocentric coordinates. The activity in the output map, {Oij}, is obtained by simply summing the activities of all the 28 B. ..., ... , : ... I 06L ___ -G ___ ..t.. I I .. ; ..t.. .6J [!1] S. Deneve and A. Pouget Figure 3: A- Polar plots showing the selectivity for saccade direction of a representative SEF units. The first three plots are for various combinations of command (L: left, R: right, P: spatial cueing) and object orientation. The left plot corresponds to saccades to a single dot. B- Data for the same unit but for a subset of the conditions. The first four columns can be directly compared to the experimental data plotted in figure 1-C from Olson and Gettner. The 5th and 6th columns show responses when the object is inverted. The seventh columns corresponds to spatial cueing. SEF maps. The result is typically a broad two dimensional bell-shaped pattern of activity from which one can read out the horizontal and vertical components, Xs and Ys, of the intended saccade by applying a center-of-mass operator. (1) 2 Results This network is able to generate a saccade to the right or to left of a bar, whatever its position, size and orientation. This architecture is basically like a radial basis function network, i.e., a look-up table with broad tuning curves allowing for interpolation. Consequently, one or two of the SEF maps light up at the appropriate location for any combination of the command and, position and orientation of the bar. Neurons in the SEF maps have the following property: 1- they have an invariant tuning curve for the direction (and amplitude) of saccadic eye movements in oculacentric coordinates, just like neurons in the FEF or the SC, and 2- the gain of this tuning curve is modulated by the orientation of the object as well as the command. Figure 3-A shows how the tuning curve for saccade direction of one particular unit varies as a function of these two variables. In this particular case, the cell responds best to a right-upward saccade directed to the left side of the object when the object is horizontal. Therefore, the SEF units in our model do not have an invariant receptive field in object-centered coordinates but, nevertheless, the gain modulation is sufficient to perform the object-centered saccade. We predict that similar response properties should be found in the SEF, and perhaps parietal cortex, of a monkey trained on a task analogous to the one described here. Since Olson and Gettner (1995) tested only three positions of the bar and held its Neural Basis of Object-Centered Representations 29 orientation constant, we cannot determine from their data whether SEF neurons are gain modulated in the way we just described. However, they found that all the SEF neurons had oculocentric receptive fields when tested on saccades to a single dot (personal communication), an observation which is consistent with our hypothesis (see fourth plot in figure 3-A) while being difficult to reconcile with an explicit object-centered representation. Second, if we sample our data for the conditions used by Olson and Gettner, we find that our units behave like real SEF cells. The first four columns of figure 3-B shows a unit with the same response properties as the cell represented in figure I-C. Figure 3-B also shows the response of the same unit when the object is upside down and when we use a spatial cue. Note that this unit responds to the left side of the bar when the object is upright (1st column), but not when it is rotated by 1800 (5th column), unless we used a spatial cue (7th column). The absence of response for the inverted object is due to the selectivity of the cell to orientation. The cell nevertheless responds in the spatial cueing conditions because we have assumed that orientation does not modulate the activity of the units in this case, since it is irrelevant to the task. Therefore, the gain modulation observed in our units is consistent with available experimental data and makes predictions for future experiments. 3 Simulation of Neglect Our representational scheme can account for neglect if the parietal cortex contains gain modulated cells like the ones we have described and if each cortical hemisphere contains more units selective for the contralateral side of space. This is known to be the case for the retinal input; hence most cells in the left hemisphere have their receptive field on the right hemiretina. We propose that the left hemisphere also over-represents the right side of objects and vice-versa (where right is defined in object-centered coordinates). Recall that the SEF maps in our model are partially selective for the side of objects. A hemispheric preference for the contralateral side of objects could therefore be achieved by having all the maps responding to the left side of objects in the right hemisphere. Clearly, in this case, a right lesion would lead to left object-centered neglect; our network would no longer be able to perform a saccade to the left side of an object. IT we add the retinal gradient and make the previous gradient not quite as binary, then we predict that a left lesion leads to a syndrome in which the network has difficulty with sac cades to the left side of an object but more so if the object is shown in the left hemiretina. Preliminary data from Olson and Gettner (personal communication) are compatible with this prediction. The same model can also account for Driver et al. (1994) experiment depicted in figure I-A. If the hemispheric gradients are as we propose, a right parietal lesion would lead to a situation in which the overall activity associated with the gap, i.e., the summed activity of all the neurons responding to this retinal location, is greater when the object is rotated counterclockwise- the condition in which the gap is perceived as belonging to the right side of the object- than in the clockwise condition. This activity difference, which can be thought as being a difference in the saliency of the upper edge of the triangle, may be sufficient to account for patients' performance. Note that object-centered neglect should be observed only if the orientation of the object is taken into consideration by the SEF units. If the experimental conditions 30 S. Deneve and A. Pouget are such that the orientation of the object can be ignored by the subject -a situation similar to the spatial cueing condition modeled here- we do not expect to observe neglect. This may explain why several groups (such as Farah et al., 1990) have failed to find object-centered neglect even though they used a paradigm similar to Driver et al. (1994). 4 Discussion We have demonstrated how object-centered saccades can be performed using neurons with oculocentric receptive fields, gain modulated by the orientation of the object and the command. The same representational scheme can also account for object-centered neglect without invoking an explicit object-centered representation, i.e., representation in which neurons' receptive fields are defined in object-centered coordinates. The gain modulation by the command is consistent with the single cell data available [4], but the modulation by the orientation of the object is a prediction for future experiments. Whether explicit object-centered representations exist, remains an empirical issue. In some cases, such representations would be computationally inefficient. In the Olson and Gettner experiment, for instance, having a stage in which motor commands are specified in object-centered coordinates does not simplify the task. Encoding the motor command in object-centered coordinates in the intermediate stage of processing requires (i) recoding the sensory input into object-centered coordinates, (ii) decoding the object-centered command into an oculocentric command, which is ultimately what the oculomotor system needs to ,generate the appropriate saccade. Each of these steps are computationally as complex as performing the overall transformation directly as we have done in this paper. Therefore, gain modulation provides a simple algorithm for performing objectcentered saccades. Interestingly, the same basic mechanism underlies spatial representations in other frames of reference, such as head-centered and body-centered. We have shown previously that these responses can be formalized as being basis functions of their sensory and postures inputs, a set of function which is particularly useful for sensory-motor transformations [5]. The same result applies to the SEF neurons considered in this paper, suggesting that basis functions may provide a unified theory of spatial representations in any spatial frame of reference. References [1] J. Driver, G. Baylis, S. Goodrich, and R. Rafal. Axis-based neglect of visual shapes. Neuropsychologia, 32{11}:1353-1365, 1994. [2] M. Farah, J. Brunn, A. Wong, M. Wallace, and P. Carpenter. Frames of reference for allocating attention to space: evidence from the neglect syndrome. Neuropsychologia, 28(4):335-47, 1990. [3] G. Hinton. Mapping part-whole hierarchies into connectionist networks. Artificial Intelligence, 46(1}:47-76, 1990. [4] C. Olson and S. Gettner. Object-centered direction selectivity in the macaque supplementary eye. Science, 269:985-988, 1995. [5] A. Pouget and T. Sejnowski. Spatial transformations in the parietal cortex using basis functions. Journal of Cognitive Neuroscience, 9(2):222-237, 1997. [6] H. Zhou, H. Friedman, and R. von der Heydt. Edge selective cells code for figure-ground in area V2 of monkey visual cortex. In Society For Neuroscience Abstracts, volume 22, page 160.1, 1996.
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Modelling Seasonality and Trends in Daily Rainfall Data Peter M Williams School of Cognitive and Computing Sciences University of Sussex Falmer, Brighton BN1 9QH, UK. email: peterw@cogs.susx.ac.uk Abstract This paper presents a new approach to the problem of modelling daily rainfall using neural networks. We first model the conditional distributions of rainfall amounts, in such a way that the model itself determines the order of the process, and the time-dependent shape and scale of the conditional distributions. After integrating over particular weather patterns, we are able to extract seasonal variations and long-term trends. 1 Introduction Analysis of rainfall data is important for many agricultural, ecological and engineering activities. Design of irrigation and drainage systems, for instance, needs to take account not only of mean expected rainfall, but also of rainfall volatility. In agricultural planning, changes in the annual cycle, e.g. advances in the onset of winter rain, are significant in determining the optimum time for planting crops. Estimates of crop yields also depend on the distribution of rainfall during the growing season, as well as on the overall amount. Such problems require the extrapolation of longer term trends as well as the provision of short or medium term forecasts. 2 Occurrence and amount processes Models of daily precipitation commonly distinguish between the occurrence process, i.e. whether or not it rains, and the amount process, i.e. how much it rains, if it does. The occurrence process is often modelled as a two-state Markov chain of first or higher order. In discussion of [12], Katz traces this approach back to Quetelet in 1852. A first order chain has been considered adequate for some weather stations, but second or higher order models may be required for others, or at different times of year. Non-stationary Markov chains have been used by a number of investigators, and several approaches have been taken 986 P. M Williams to the problem of seasonal variation, e.g. using Fourier series to model daily variation of parameters [16, 12, 15]. The amount of rain X on a given day, assuming it rains, normally has a roughly exponential distribution. Smaller amounts of rain are generally more likely than larger amounts. Several models have been used for the amount process. Katz & Parlange [9], for example, assume that \IX has a normal distribution, where n is a positive integer empirically chosen to minimise the skewness of the resulting historical distribution. But use has more commonly been made of a gamma distribution [7,8, 12] or a mixture of two exponentials [16, 15]. 3 Stochastic model The present approach is to deal with the occurrence and amount processes jointly, by assuming that the distribution of the amount of rain on a given day is a mixture of a discrete and continuous component. The discrete component relates to rainfall occurrence and the continuous component relates to rainfall amount on rainy days. We use a gamma distribution for the continuous component. l This has density proportional to x v - 1 e-x to within an adjustable scaling of the x-axis. The shape parameter v > 0 controls the ratio of standard deviation to mean. It also determines the location of the mode, which is strictly positive if v > 1. For certain patterns of past precipitation, larger amounts may be more likely on the following day than smaller amounts. Specifically the distribution of the amount X of rain on a given day is modelled by the three parameter family where 0 ~ a ~ 1 and v,O > 0 and if x < 0 if x ~ 0 r(v,z) = r(V)-l 1 00 yv-l e- y dy (1) is the incomplete gamma function. For a < 1, there is a discontinuity at x = 0 corresponding to the discrete component. Putting x = 0, it is seen that a = P(X > 0) is the probability of rain on the day in question. The mean daily rainfall amount is avO and the variance is aV{l + v(l - a)}02. 4 Modelling time dependency The parameters a, v, 0 determining the conditional distribution for a given day, are understood to depend on the preceding pattern of precipitation, the time of year etc. To model this dependency we use a neural network with inputs corresponding to the conditioning events, and three outputs corresponding to the distributional parameters.2 Referring to the activations of the three output units as zO:, ZV and zO, we relate these to the distributional parameters by 1 v = expzv 0= expzo (2) a=---1 + expzO: in order to ensure an unconstrained parametrization with 0 < a < 1 and v,O > 0 for any real values of zO:, zV, zO. 1 It would be straightforward to use a mixture of gammas, or exponentials, with time-dependent mixture components. A single gamma was chosen for simplicity to illustrate the approach. 2 A similar approach to modelling conditional distributions, by having the network output distributional parameters, is used, for example, by Ghabramani & Jordan [6], Nix & Weigend [10], Bishop & Legleye [3], Williams [14], Baldi & Chauvin [2]. Modelling Seasonality and Trends in Daily Rainfall Data 987 On the input side, we first need to make additional assumptions about the statistical properties of the process. Specifically it is assumed that the present is stochastically independent of the distant past in the sense that (t > T) (3) for a sufficiently large number of days T. In fact the stronger assumption will be made that P(Xt>X!Xt-1,,,,,XO) = P(Xt>X!Rt-1, ,, ,,Rt-T) (t>T) (4) where Rt = (Xt > 0) is the event of rain on day t. This assumes that today's rainfall amount depends stochastically only on the occurrence or non-occurrence of rain in the recent past, and not on the actual amounts. Such a simplification is in line with previous approaches [8, 16, 12J. For the present study T was taken to be 10. To assist in modelling seasonal variations, cyclic variables sin T and cos T were also provided as inputs, where T = 21rt/ D and D = 365.2422 is the length of the tropical year. This corresponds to using Fourier series to model seasonality [16, 12J but with the number of harmonics adaptively determined by the model.3 To allow for non-periodic nonstationarity, the current value of t was also provided as input. 5 Model fitting Suppose we are given a sequence of daily rainfall data of length N. Equation (4) implies that the likelihood of the full data sequence (x N -1 I ••• I Xo) factorises as N-1 p(XN-1 , .. . I Xo; w) = p(XT-1I" . I Xo) II p(Xt ! Tt-1 I' .. I Tt-T; w) (5) t=T where the likelihood p(XT-1I'" IXO) of the initial sequence is not modelled and can be considered as a constant (compare [14]). Our interest is in the likelihood (5) of the actual sequence of observations, which is understood to depend on the variable weights w of the neural network. Note that p(Xt ! Tt-1 I' •• I Tt-T; w) is computed by means of the neural network outputs zf I zf I zf, using weights wand the inputs corresponding to time t. The log likelihood of the data can therefore be written, to within a constant, as N-1 logp(xN-1 I' .. IXO; w) = L logp(xt ! Tt-1,· .. I Tt-T; w) t=T or, more simply, N-1 L(w) = L Lt(w) t=T where from (1) L () {log(1 - at) t w = log at + (lit -1) logxt -lit logOt -logr(lIt) - xt/Ot where dependence of at, lit I Ot on w, and also on the data, is implicit. if Xt = 0 if Xt > 0 (6) (7) To fit the model, it is useful to know the gradient 'VL(w). This can be computed using backpropagation if we know the partial derivatives of L(w) with respect to network outputs. In view of (6) we can concentrate on a single observation and perform a summation. 3Note that both sin nr and cos nr can be expressed as non-linear functions of sin r and cos r. which can be approximated by the network. 988 P. M. Williams Omitting subscript references to t for simplicity, and recalling the links between network outputs and distributional parameters given by (2), we have 8L { -a if x = 0 8zQ 1-0 if x > 0 8L { 0 if x = 0 = x (8) 8zv v'I/J (v) - v log (j ifx> 0 8L { 0 if x = 0 8zo x if x > 0 v--() where d r'(v) 'I/J(v) = -logr(v) = -dv r(v) is the digamma function of v. Efficient algorithms for computing log r(v) in (7) and 'I/J(v) in (8) can be found in Press et al. [11] and Amos [1]. 6 Regularization Since neural networks are universal approximators, some form of regularization is needed. As in all statistical modelling, it is important to strike the right balance between jumping to conclusions (overfitting) and refusing to learn from experience (underfitting). For this purpose, each model was fitted using the techniques of [13] which automatically adapt the complexity of the model to the information content of the data, though other comparable techniques might be used. The natural interpretation of the regularizer is as a Bayesian prior. The Bayesian analysis is completed by integration over weight space. In the present case, this was achieved by fitting several models and taking a suitable mixture as the solution. On account of the large datasets used, however, the results are not particularly sensitive to this aspect of the modelling process. 7 Results for conditional distributions The process was applied to daily rainfall data from 5 stations in south east England and 5 stations in central Italy.4 The data covered approximately 40 years providing some 15,000 observations for each station. A simple fully connected network was used with a single layer of 13 input units, 20 hidden units and 3 output units corresponding to the 3 parameters of the conditional distribution shown in (2). As a consequence of the pruning features of the regularizer, the models described here used an average of roughly 65 of the 343 available parameters. To illustrate the general nature of the results, Figure 1 shows an example from the analysis of an early part of the Falmer series. It is worth observing the succession of 16 rainy days from day 39 to day 54. The lefthand figure shows that the conditional probability of rain increases rapidly at first, and then levels out after about 5-7 days.s Similar behaviour is observed for successive dry days, for example between days 13 and 23. This suggests that the choice of 10 time lags was sufficient. Previous studies have used mainly first or second order Markov chains [16, 12]. Figure 1 confirms that conditional dependence 4The English stations were at Cromptons, FaImer, Kemsing, Petworth, Rothertield; the Italian stations were at Monte Oliveto, Pisticci, Pomarico, Siena, Taverno d' Arbia. sIn view of the number of lags used as inputs, the conditional probability would necessarily be constant after 10 days apart from seasonal effects. In fact this is the last quarter of 1951 and the incidence of rain is increasing here at that time of year. Modelling Seasonality and Trends in Daily Rainfall Data 989 F ALMER: conditional probability FALMER: conditional mean 20 ii I II II 0.8 II II 15 0.6 I II i 10 I 0.4 0.2 5 I II 0 0 10 20 30 40 50 60 70 10 20 30 40 50 60 70 Figure 1: Results for the 10 weeks from 18 September to 27 November, 1951. The lefthand figure shows the conditional probability of rain for each day, with days on which rain occurred indicated by vertical lines. The righthand figure shows the conditional expected amount of rain in millimeters for the same period, together with the actual amount recorded. decays rapidly at this station, at this time of year, but also indicates that it can persist for up to at least 5 days (compare [5,4]). 8 Seasonality and trends Conditional probabilities and expectations displayed in Figure 1 show considerable noise since they are realisations of random variables depending on the rainfall pattern for the last 10 days. For the purpose of analysing seasonal effects and longer term trends, it is more indicative to integrate out the noise resulting from individual weather patterns as follows. Let Rt denote the event (Xt > 0) and let Rt denote the complementary event (Xt = 0). The expected value of Xt can then be expressed as E(Xt ) = L E(Xt I At - 1, . .. ,At-T) P(At- 1, . .. ,At-T) (9) where each event At stands for either Rt or R t , and summation is over the 2T possible combinations. Equation (9) takes the full modelled jOint distribution over the variables X N -1, .. . ,X 0 and extracts the marginal distribution for X t . This should be distinguished from an unconditional distribution which might be estimated by pooling the data over all 40 years. E(Xt ) relates to a specific day t. Note that (9) also holds if X t is replaced by any integrable function of X t , in particular by the indicator function of the event (Xt > 0) in which case (9) expresses the probability of rain on that day. Examining (9) we see that the conditional expectations in the first term on the right are known from the model, which supplies a conditional distribution not only for the sequence of events which actually occurred, but for any possible sequence over the previous T days. It therefore only remains to calculate the probabilities P( At- 1 , ... , At-T) of T -day sequences preceding a given day t. Note that these are again time-dependent marginal probabilities, which can be calculated recursively from P(At, .. . , At-T+t} = P(At I At- 1,· . . , At-T+1 Rt-T) P(At- 1 , . .• , At-T+1 Rt-T) + P(At I At-I,·· ., At-T+IRt-T) P(At- 1, . .. , At-T+IRt-T) provided we assume a prior distribution over the 2T initial sequences (AT-I, . .. , Ao) as a base for the recursion. The conditional probabilities on the right are given by the model, 990 POMARICO: probability of rain 0.35 .----r---..---.---~-___r-__, 0.3 HHHI"*-lHI-IHHHH1·-lH ·lI-H~-f+H 0.25 M-I++H++I-tI+i1~++HHI#H-fHHHHI-{HI-1HHHHI·H-II-fHli 0.2 I\I*H*I*+H1flHl-+l!#~iHHIl-Iffi~+++H,HIf~KH+IHIH·HH1ffHII 0.15 H+1-HrH-l1-f , I ! 0.1 L--_-'--_-'--_~ _ __'_ _ _'__~ 1955 1960 1965 1970 1975 1980 1985 P. M. Williams POMARICO: mean and standard deviation 2~~~~~~~~~~~~~-~ 1 OL----'----'---~---'---'---~ 1955 1960 1965 1970 1975 1980 1985 Figure 2: Integrated results for Pomarico from 1955-1985. The lefthand figure shows the daily probability of rain, indicating seasonal variation from a summer minimum to a winter maximum. The righthand figure shows the daily mean (above) and standard deviation (below) of rainfall amount in millimeters. as before, and the unconditional probabilities are given by the recursion. It turns out that results are insensitive to the choice of initial distribution after about 50 iterations, verifying that the occurrence process, as modelled here, is in fact ergodic. 9 Integrated results Results for the integrated distribution at one of the Italian stations are shown in Figure 2. By integrating out the random shocks we are left with a smooth representation of time dependency alone. The annual cycles are clear. Trends are also evident over the 30 year period. The mean rainfall amount is decreasing significantly, although the probability of rain on a given day of the year remains much the same. Rain is occurring no less frequently, but it is occurring in smaller amounts. Note also that the winter rainfall (the upper envelope of the mean) is decreasing more rapidly than the summer rainfall (the lower envelope of the mean) so that the difference between the two is narrowing. 10 Conclusions This paper provides a new example of time series modelling using neural networks. The use of a mixture of a discrete distribution and a gamma distribution emphasises the general principle that the "error function" for a neural network depends on the particular statistical model used for the target data. The use of cyclic variables sin T and cos T as inputs shows how the problem of selecting the number of harmonics required for a Fourier series analysis of seasonality can be solved adaptively. Long term trends can also be modelled by the use of a linear time variable, although both this and the last feature require the presence of a suitable regularizer to avoid overfitting. Lastly we have seen how a suitable form of integration can be used to extract the underlying cycles and trends from noisy data. These techniques can be adapted to the analysis of time series drawn from other domains. Modelling Seasonality and Trends in Daily Rainfall Data 991 Acknowledgement I am indebted to Professor Helen Rendell of the School of Chemistry, Physics and Environmental Sciences, University of Sussex, for kindly supplying the rainfall data and for valuable discussions. References [1] D. E. Amos. A portable fortran subroutine for derivatives of the psi function. ACM Transactions on Mathematical Software, 9:49~502, 1983. [2] P. Baldi and Y. Chauvin. Hybrid modeling, HMM/NN architectures, and protein applications. Neural Computation, 8:1541-1565, 1996. [3] C. M. Bishop and C. Legleye. Estimating conditional probability densities for periodic variables. In G. Tesauro, D. Touretzky, and T. Leen, editors, Advances in Neural Information Processing Systems 7, pages 641-648. The MIT Press, 1995. [4] E. H. Chin. Modelling daily precipitation occurrence process with Markov chain. Water Resources Research, 13:949-956,1977. [5] P. Gates and H. Tong. On Markov chain modelling to some weather data. Journal of AppliedMeteorology, 15:1145-1151, 1976. [6] Z. Ghahramani and M. 1. Jordan. Supervised learning from incomplete data via an EM approach. In Jack D. Cowan, Gerald Tesauro, and Joshua Alspector, editors, Advances in Neural Information Processing Systems 6, pages 120-127. Morgan Kaufmann, 1994. [7] N. T. Ison, A. M. Feyerherm, andL. D. Bark. Wet period precipitation and the gamma distribution. Journal of Applied Meteorology, 10:658-665, 1971. [8] R. W. Katz. Precipitation as a chain-dependent process. Journal of Applied Meteorology, 16:671-676,1977. [9] R. W. Katz and M. B. Parlange. Effects of an index of atmospheric circulation on stochastic properties of precipitation. Water Resources Research, 29:2335-2344, 1993. [10] D. A. Nix and A. S. Weigend. Learning local error bars for nonlinear regression. In Gerald Tesauro, David S. Touretzky, and Todd K. Leen, editors, Advances in Neural Information Processing Systems 7, pages 489-496. MIT Press, 1995. [11] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes in C. Cambridge University Press, 2nd edition, 1992. [12] R. D. Stern and R. Coe. A model fitting analysis of daily rainfall data, with discussion. Journal of the Royal Statistical Society A, 147(Part 1):1-34,1984. [13] P. M. Williams. Bayesian regularization and pruning using a Laplace prior. Neural Computation, 7:117-143,1995. [14] P. M. Williams. Using neural networks to model conditional multivariate densities. Neural Computation, 8:843-854, 1996. [15] D. A. Woolhiser. Modelling daily precipitation-progress and problems. In Andrew T. Walden and Peter Guttorp, editors, Statistics in the Environmental and Earth Sciences, chapter 5, pages 71-89. Edward Arnold, 1992. [16] D. A. Woolhiser and G. G. S. Pegram. Maximum likelihood estimation of Fourier coefficients to describe seasonal variation of parameters in stochastic daily precipitation models. Journal of Applied Meteorology, 18:34-42, 1979.
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Multiple Threshold Neural Logic Vasken Bohossian Jehoshua Bruck California Institute of Technology Mail Code 136-93 Pasadena, CA 91125 ~mail: {vincent, bruck}~paradise.caltech.edu Abstract We introduce a new Boolean computing element related to the Linear Threshold element, which is the Boolean version of the neuron. Instead of the sign function, it computes an arbitrary (with polynornialy many transitions) Boolean function of the weighted sum of its inputs. We call the new computing element an LT M element, which stands for Linear Threshold with Multiple transitions. The paper consists of the following main contributions related to our study of LTM circuits: (i) the creation of efficient designs of LTM circuits for the addition of a multiple number of integers and the product of two integers. In particular, we show how to compute the addition of m integers with a single layer of LT M elements. (ii) a proof that the area of the VLSI layout is reduced from O(n2 ) in LT circuits to O(n) in LTM circuits, for n inputs symmetric Boolean functions, and (iii) the characterization of the computing power of LT M relative to LT circuits. 1 Introduction Human brains are by far superior to computers in solving hard problems like combinatorial optimization and image and speech recognition, although their basic building blocks are several orders of magnitude slower. This observation has boosted interest in the field of artificial neural networks [Hopfield 82], [Rumelhart 82]. The latter are built by interconnecting artificial neurons whose behavior is inspired by that of biological neurons. In this paper we consider the Boolean version of an artificial neuron, namely, a Linear Threshold (LT) element, which computes a neural-like MUltiple Threshold Neural Logic 253 WI t3 0 WI t3 0 1 1 1 -Wo 0 t2 0 t2 0 Wn t1 1 Wn t1 1 LT gate SYM gate LTM gate Figure 1: Schematic representation of LT, SYM and LTM computing elements. Boolean function of n binary inputs [Muroga 71]. An LT element outputs the sign of a weighted sum of its Boolean inputs. The main issues in the study of networks (circuits) consisting of LT elements, called LT circuits, include the estimation of their computational capabilities and limitations and the comparison of their properties with those of traditional Boolean logic circuits based on AND, OR and NOT gates (called AON circuits). For example, there is a strong evidence that LT circuits are more efficient than AON circuits in implementing a number of important functions including the addition, product and division of integers [Siu 94], [Siu 93]. Motivated by our recent work on the VLSI implementation of LT elements [Bohossian 95b], we introduce in this paper a more powerful computing element, a multiple threshold neuron, which we call LTM, which stands for Linear Threshold with Multiple transitions, see [Haring 66] and [Olafsson 88]. Instead of the sign function in the LT element it computes an arbitrary (with polynomialy many transitions) Boolean function of the weighted sum of its inputs. The main issues in the study of LTM circuits (circuits consisting of LTM elements) include the estimation of their computational capabilities and limitations and the comparison of their properties to those of AON circuits. A natural approach in this study is first to understand the relation between LT circuits and LT M circuits. Our main contributions in this paper are: • We demonstrate the power of LTM by deriving efficient designs of LTM circuits for the addition of m integers and the product of two integers. • We show that LT M circuits are more amenable in implementation than LT circuits. In particular, the area of the VLSI layout is reduced from O(n2 ) in LT circuits to O(n) in LTM circuits, for n input symmetric Boolean functions. • We characterize the computing power of LT M relative to LT circuits. Next we describe the formal definitions of LT and LT M elements. 1.1 Definitions and Examples Definition 1 (Linear Threshold Gate - LT) A linear threshold gate computes a Boolean function of its binary inputs : n f(X) = sgn(wo + L WiXi) i=l 254 V. Bohossian and J. Bruck where the Wi are integers and sgn(.) outputs 1 if its argument is greater or equal to 0, and 0 otherwise. Figure 1 shows an-input LT element; if L~ WiXi ~ -Wo the element outputs 1, otherwise it outputs o. A single LT gate is unable to compute parity. The latter belongs to the general class of symmetric functions - SY M. Definition 2 (Symmetric Functions - SY M) A Boolean function f is symmetric if its value depends only on the number of ones in the input denoted by IX I. Figure 1 shows an example of a symmetric function; it has three transitions, it outputs 1 for IXI < tl and for t2 ~ IXI < t3, and 0 otherwise. AND, OR and parity are examples of symmetric functions. A single LT element can implement only a limited subset of symmetric functions. We define LT M as a generalization of SY M. That is, we allow the weights to be arbitrary as in the case of LT, rather than fixed to 1 (see Figure 1 ). Definition 3 (Linear Threshold Gate with Multiple Transitions - LT M) A function f is in LTM if there exists a set of weights Wi E Z, 1 ~ i ~ n and a function h : Z ---+ {O, 1} such that n f(X) = h(L wixd for all X E {O,l}n i=l The only constraint on h is that it undergoes polynomialy many transitions as its input scans [- L~=l IWi I, L~=l IWi I]· Notice that without the constraint on the number of transitions, an LTM gate is capable of computing any Boolean function. Indeed, given an arbitrary function f, let Wi = 2i - 1 and h(L~ 2i - 1xd = f(xt, .•• , x n ). Example 1 (XOR E LTM) XOR(X) outputs 1 if lXI, the number of l's in X, is odd. Otherwise it outputs O. To implement it choose Wi = 1 and h(k) = ~(1 - (_l)k) for 0 ~ k ~ n. Note that h(k) needs not be defined for k < 0 and k > n, and has polynomialy many transitions. Another useful function that LTM can compute is ADD (X, Y), the sum of two n-bit integers X and Y. Example 2 (ADD E LT M) To implement addition we set fl (X, Y) = h, (L~=l 2i (Xi + yd) where h, (k) = 1 for k E [2' ,2 x 2' - 1] U [3 X 2', +00). Defined thus, fl computes the m-th bit of X + Y. 1.2 Organization The paper is organized as follows. In Section 2, we study a number of applications as well as the VLSI implementations of LTM circuits. In particular, we show how to compute the addition of m integers with a single layer of LT!vI elements. In Section 3, we prove J..he characterization results of LT M - inclusion relations, in particular LTM ~ LT2. In addition, we indicate which inclusions are proper and exhibit functions to demonstrate the separations. Multiple Threshold Neural Logic 255 2 LT M Constructions The theoretical results about LT M can be applied to the VLSI implementation of Boolean functions. The idea of a gate with multiple thresholds came to us as we were looking for an efficient VLSI implementation of symmetric Boolean functions. Even though a single LT gate is not powerful enough to implement any symmetric function, a 2-layer LT circuit is. FUrthermore, it is well known that such a circuit performs much better than the traditional logic circuit based on AND, OR and NOT gates. The latter has exponential size (or unbounded depth) [Wegener 91]. Proposition 4 (LT2 versus LT M for symmetric function implementation) The LT2 layout of a symmetric function requires area of O(n2), while using LT M one needs only area of O( n). PROOF: Implementing a generalized symmetric function in LT2 requires up to n LT gates in the first layer. Those have the same weights Wi except for the threshold Woo Instead of laying out n times the same linear sum E~ WiXi we do it once and compare the result to n different thresholds. The resulting circuit corresponds to a single LT M gate. 0 The LT2 layout is redundant, it has n copies of each weight, requiring area of at least O(n2). On the other hand, LTM performs a single weighted sum, its area requirement is O(n). A single LT M gate can compute the addition of m n-bit integers M ADD. The only constraint is that m be polynomial in n. Theorem 5 (MADD E LTM) A single layer of LT M gates can compute the sum of m n-bit integers, provided that m is at most polynomial in n. PROOF: MAD D returns an integer of at most n + log m bits. We need one LT M gate per bit. The least significant bit is computed by a simple m-bit XOR. For all other bits we use h(X(l), .. ,x(m») = hl(E~=12i Ej=l x~j») to compute the l-th bit ofthe mm. 0 Corollary 6 (PRODUCT E PTM) A single layer of PTM (which is defined below) gates, can compute the product of m n-bit integers, provided that m is at most polynomial in n. PROOF: By analogy with PTb defined in [Bruck 90], in PT Ml (or simply PT M) we allow a polynomial rather than a linear sum: f(X) = h(WIXl + ... +wnxn +W(1,2)XIX2+ ... ) However we restrict the sum to have polynomialy many terms (else, any Boolean function could be realized with a single gate). The product of two n-bit integers X and Y can be written as PRODUCT(X, Y) = E~=l XiY. We use the construction of MADD in order to implement PRODUCT. PRODUCT(X, Y) = "n "I i MADD(x1Y,x2Y , ... ,xnY). fleX, Y) = hi (LJj=l LJi=12 XjYi) b outputs the l-th bit of the product. 0 256 V. Bohossian and J Bruck Figure 2: Relationship between Classes 3 Classification of LTM .- --We me a hat to indicate small (polynomialy growing) weights, e.g. LT, LT M [Bohossian 95a], [Siu 91], and a subscript to indicate the depth (number of layers) of the circuit of more than a single layer. All the circuits we consider in this paper are of polynomial size (number of elements) in n (number of inputs). For example, the class fr2 consists of those B0...2!ean functions that can be implemented by a depth-2 polynomial size circuit of LT elements. Figure 2 depicts the membership relations between five classes of Boolean functions, including, LT, ilr, LTM, LTM and ilr2, along with the functions used to establish the separations. In this section we will prove the relations illustrated by Figure 2 . Theorem 7 (Classification of LTM ) The inclusions and separations shown in Figure 12 hold. That is, .1. LT ~ LT ; LTM --12. LT ~ LTM ; LTM .9. LTM; LT2 4· XOR E CTM but XOR tJ. LT 5. CaMP E LT but CaMP tJ. LTM 6. ADD E LTM but ADD tJ. LTULTM 7. IPk E fr2 but 1Pk tJ. LTM PROOF: We show only the outline of the proof. The complete version can be found in [Bohossian 96]. Claims 1 and 2 follow from the definition. The first part of Claim 4 was shown in Example 1 and the second is well known. In Claim 5, CaMP stands for the Comparison functio!lt the proof mes the pigeonhole principle and is related to the proof of CaMP tJ. LT which can be found in [Siu 91]. In Claim 6 to show that ADD tJ. LTM we use the same idea as for CaMP. Claim 3 is proved using a result from [Goldman 93]: a single LT gate with arbitrary weights can be realized by an LT2 circuit. Claim 7 introduces the function IPk(X, Y) = 1 iff L:~ XiYi ~ k, Multiple Threshold Neural Logic 257 o otherwise. If IPk E LTM, using the result from [Goldman 93], we can construct .a LT2 circuit that computes IP2 (Inner Product mod 2) which is known to be false [Hajnal 94]. 0 What remains to be shown in order to complete the classification picture is fr = LT n LTM. We conjecture that this is true. 4 Conclusions Our original goal was to use theoretical results in order to efficiently layout a generalized symmetric function. During that process we came to the conclusion that the LT2 implementation is partially redundant, which lead to the definition of LTM, a new, more powerful computing element. We characterized the power of LTM relative to LT. We showed how it can be used to reduce the area of VLSI layouts from O(n2 ) to O(n) and derive efficient designs for multiple addition and product. Interesting directions for future investigation are (i) to prove the conjecture: fr = LT n LTM, (ii) to apply spectral techniques ([Bruck 90)) to the analysis of LT M, in particular show how PT M fits into the classification picture (Figure 2 ). Another direction for future research consists in introducing the ideas described above in the domain of VLSI. We have fabricated a programmable generalized symmetric function on a 2J,L, analog chip using the model described above. Floating gate technology is used to program the weights. We store a weight on a single transistor by injecting and tunneling electrons on the floating gate [Hasler 95]. Acknowledgments This work was supported in part by the NSF Young Investigator Award CCR9457811 and by the Sloan Research Fellowship. References [Bohossian 95a] V. Bohossian and J. Bruck. On Neural Networks with Minimal Weights. In Advances in Neural Information Processing Systems 8, MIT Press, Cambridge, MA, 1996, pp.246-252. [Bohossian 95b] V. Bohossian, P. Hasler and J. Bruck. Programmable Neural Logic. Proceedings of the second annual IEEE International Conference on Innovative Systems in Silicon, pp. 13-21, October 1997. [Bohossian 96] V. Bohossian and J. Bruck. Multiple Threshold N eural Logic. Technical Report, ETR010, June 1996. (available at http://paradise.caltech.edu/ETR.html) [Bruck 90] J. Bruck. Harmonic Analysis of Polynomial Threshold Functions. SIAM J. Disc. Math, Vol. 3(No. 2)pp. 168- 177, May 1990. [Goldman 93] M. Goldmann and M. Karpinski. Simulating threshold circuits by majority circuits. In Proc. 25th ACM STOC, pages pp. 551-560, 1993. 258 V. Bohossian and J Bruck [Hajnal 94] A. Hajnal, W. Maass, P. Pudlak, M. Szegedy, G. Turan. Threshold Circuits of Bounded Depth. Journal of Computer and System Sciences, Vol. 46(No. 2):pp. 129-154, April 1993. [Haring 66] D.R. Haring. Multi-Threshold Threshold Elements. IEEE Transactions on Electronic Computers, Vol. EC-15, No.1, February 1966. [Hasler 95] P. Hasler, C. Diorio, B.A. Minch and C.A. Mead. Single Transistor Learning Synapses. Advances in Neural Information Processing Systems 7, MIT Press, Cambridge, MA, 1995, pp.817-824. [Hastad 94] J. Hastad. On the size of weights for threshold gates. SIAM. J. Disc. Math., 7:484-492, 1994. [Hofmeister 96] T. Hofmeister. A Note on the Simulation of Exponential Threshold Weights. 1996 CONCOON conference. [Hopfield 82] J. Hopfield. Neural networks and physical systems with emergent collective computational abilities. Proc. of the USA National Academy of Sciences, 79:2554-2558, 1982. [Muroga 71] M. Muroga. Threshold Logic and its Applications. Wiley-Interscience, 1971. [Olafsson 88] S. Olafsson and Y.S. Abu-Mostafa. The Capacity of Multilevel Threshold Functions. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol.10, No.2, March 1988. [Rumelhart 82] D. Rumelhart and J. McClelland. Parallel distributed processing: Explorations in the microstructure of cognition. MIT Press, 1982. [Siu 91] K. Siu and J. Bruck. On the power of threshold circuits with small weights. SIAM J. Disc. Math., Vol. 4(No. 3):pp. 423-435, August 1991. [Siu 93] K. Siu, J. Bruck, T. Kailath, and T. Hofmeister. Depth Efficient Neural Networks for Division and Related Problems. IEEE Trans. on Information Theory, Vol. 39(No. 3), May 1993. [Siu 94] K. Siu and V.P. Roychowdhury. On Optimal Depth Threshold Circuits for Multiplication and Related Problems. SIAM J. Disc. Math., Vol. 7(No. 2):pp. 284-292, May 94. [Szegedy 89] M. Szegedy. Algebraic Methods in Lower Bounds for Computational Models with Limited Communication. PhD Thesis, Dep. Computer Science, Chicago Univ., December 1989. [Wegener 91] 1. Wegener. The complexity of the parity function in unbounded fanin unbounded depth circuits. In Theoretical Computer Science, Vol. 85, pp. 155-170, 1991.
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The Storage Capacity of a Fully-Connected Committee Machine Yuansheng Xiong Department of Physics, Pohang Institute of Science and Technology, Hyoja San 31, Pohang, Kyongbuk, Korea xiongOgalaxy.postech.ac.kr Chulan Kwon Department of Physics, Myong Ji University, Yongin, Kyonggi, Korea ckwonOwh.myongji.ac.kr Jong-Hoon Oh Lucent Technologies, Bell Laboratories, 600 Mountain Ave., Murray Hill, NJ07974, U. S. A. jhohOphysics.bell-labs.com Abstract We study the storage capacity of a fully-connected committee machine with a large number K of hidden nodes. The storage capacity is obtained by analyzing the geometrical structure of the weight space related to the internal representation. By examining the asymptotic behavior of order parameters in the limit of large K, the storage capacity Q c is found to be proportional to ]{ Jln ]{ up to the leading order. This result satisfies the mathematical bound given by Mitchison and Durbin, whereas the replica-symmetric solution in a conventional Gardner's approach violates this bound. 1 INTRODUCTION Since Gardner's pioneering work on the storage capacity of a single layer perceptron[1], there have been numerous efforts to use the statistical mechanics formulation to study feed-forward neural networks. The storage capacity of multilayer neural networks has been of particular interest, together with the generalization problem. Barkai, Hansel and Kanter[2] studied a parity machine with a The Storage Capacity of a Fully-Connected Committee Machine 379 non-overlapping receptive field of continuous weights within a one-step replica symmetry breaking (RSB) scheme, and their result agrees with a mathematical bound previously found by Mitchison and Durbin (MD)[3] . Subsequently Barkai, Hansel and Sompolinsky[4] and Engel et al.[5] have studied the committee machine, which is closer to the multi-layer perceptron architecture and is most frequently used in real-world applications. Though they have derived many interesting results, particularly for the case of a finite number of hidden units, it was found that their the replica-symmetric (RS) result violates the MD bound in the limit where the number of hidden units K is large. Recently, Monasson and O'Kane[6] proposed a new statistical mechanics formalism which can analyze the weight-space structure related to the internal representations of hidden units. It was applied to single layer perceptrons[7, 8, 9] as well as multilayer networks[10, 11, 12]. Monasson and Zecchina[lO] have successfully applied this formalism to the case of both committee and parity machines with non-overlapping receptive fields (NRF)[lO]. They suggested that analysis of the RS solution under this new statistical mechanics formalism can yield results just as good as the onestep RSB solution in the conventional Gardner's method. In this letter, we apply this formalism for a derivation of the storage capacity of a fully-connected committee machine, which is also called a committee machine with overlapping receptive field (ORF) and is believed to be a more relevant architecture. In particular, we obtain the value of the critical storage capacity in the limit of large K , which satisfies the MD bound. It also agrees with a recent one-step RSB calculation, using the conventional Gardner method, to within a small difference of a numerical prefactor[13]. Finally we will briefly discuss the fully-connected parity machine. 2 WEIGHT SPACE STRUCTURE OF THE COMMITTEE MACHINE We consider a fully-connected committee machine with N input units, K hidden units and one output unit, where weights between the hidden units and the output unit are set to 1. The network maps input vectors {xf}, where J1. = 1, ... , P , to output yPo as: (1) where Wji is the weight between the ith input node and the jth hidden unit. hj ~ sgn(E~l WjiXn is the jth component of the internal representation for input pattern {xn . We consider continuous weights with spherical constraint, EfWji=N . Given P = aN patterns, the learning process in a layered neural network can be interpreted as the selection of cells in the weight space corresponding to a set of suitable internal representations h = {hj}, each of which has a non-zero elementary 380 Y. Xiong, C. Kwon and J-H. Oh volume defined by: (2) where 8(x) is the Heaviside step function. The Gardner's volume VG, that is, the volume of the weight space which satisfies the given input-output relations, can be written as the sum of the cells over all internal representations: VG=LVh. (3) h The method developed by Monasson and his collaborators [6, 10] is based on analysis of the detailed internal structure, that is, how the Gardner's volume VG is decomposed into elementary volumes Vb associated with a possible internal representation. The distribution of the elementary volumes can be derived from the free energy, (4) where ((- . -») denotes the average over patterns. The entropy N[w(r)] of the volumes whose average sizes are equal to w (r) = -1/ N In (( Vb)), can be given by the Legendre relations respecti vely. og(r) N[w(r)] = - o(l/r)' w(r) = o[rg(r)] or (5) The entropies ND = N[w(r = 1)] and NR = N[w(r = 0)) are of most importance, and will be discussed below. In the thermodynamic limit, 1/ N ((In(VG))) = -g(r = 1) is dominated by elementary volumes of size w(r = 1), of which there are exp(N ND). Furthermore, the most numerous elementary volumes have the size w(r = 0) and number exp(NNR). The vanishing condition for the entropies is related to the zero volume condition for VG and thus gives the storage capacity. We focus on the entropy N D of elementary volumes dominating the weight space VG. 3 ORDER PARAMETERS AND PHASE TRANSITION For a fully-connected machine, the overlaps between different hidden units should be taken into account, which makes this problem much more difficult than the treelike (NRF) architecture studied in Ref. [10] . The replicated partition function for the fully-connected committee machine reads: (( (~ Vh)")) = (( Thh;·ThWj· IT e (It hr") ,n. e (hr" ~ Wj~.X;) )), (6) with a = 1,···, rand 0' = 1, · ··, n. Unlike Gardner's conventional approach, we need two sets of replica indices for the weights. We introduce the order parameters, (7) where the indices a, b originate from the integer power r of elementary volumes, and 0', {3 are the standard replica indices. The replica symmetry ansatz leads to five The Storage Capacity of a Fully-Connected Committee Machine order parameters as: I q'" QOI{3ab _ q 'k C J d'" d (j = k, 0: = (3, a =1= b), (j = k, 0: =1= (3), (j =1= k,o: = (3,a = b), (j =1= k,o:= (3,a =1= b), (j =1= k, 0: =1= (3), 381 (8) where q'" and q are, respectively, the overlaps between the weight vectors connected to the same hidden unit of the same (0: = (3) and different (0: =1= (3) replicas corresponding to the two different internal representations. The order parameters c, d'" and d describe the overlaps between weights that are connected to different hidden units, of which c and d'" are the overlaps within the same replica whereas d correlates different replicas. Using a standard replica trick, we obtain the explicit form of g( r). One may notice that the free energy evaluated at r = 1 is reduced to the RS results obtained by the conventional method on the committee machine[4, 5], which is independent of q'" and d"'. This means that the internal structure of the weight space is overlooked by conventional calculation of the Gardner's volume. When we take the limit r ~ 1, the free energy can be expanded as: ( '" d'" d) ( d) ( )og(r,q"',q,c,d"',d) I gr,q,q,c, , =gl,q,c, + r-l or r=l' (9) As noticed, g(r, q"', q, c, d"', d) is the same as the RS free energy in the Gardner's method. From the relation: N: _ og(r) I _ og(r) I D - o(l/r) r=l - -a;:- r=l ' (10) we obtain the explicit form of ND . In the case of the NRF committee machine, where each of the hidden units is connected to different input units, we do not have a phase transition. Instead, a single solution is applicable for the whole range of 0:. In contrast, the phase-space structure of the fully-connected committee machine is more complicated than that of the NRF committee machine. When a small number of input patterns are given, the system is in the permutation-symmetry (PS) phase[4, 5, 14), where the role of each hidden unit is not specialized. In the PS phase, the Gardner's volume is a single connected region. The order parameters associated with different hidden units are equal to the corresponding ones associated with the same hidden unit. When a critical number of patterns is given, the Gardner's volume is divided into many islands, each one of which can be transformed into other ones by permutation of hidden units. This phenomenon is called permutation symmetry breaking (PSB), and is usually accompanied by a first-order phase transition. In the PSB phase, the role of each hidden unit is specialized to store a larger number of patterns effectively. A similar breaking of symmetry has been observed in the study of generalization[14, 15], where the first-order phase transition induces discontinuity of the learning curve. It was pointed out that the critical storage capacity is attained in the PSB phase[4, 5), and our recent one-step replica symmetry breaking calculation confirmed this picture[13). Therefore, we will focus on the analysis of the PSB solution near the storage capacity, in which q*, q ~ 1, and c, d"', d are of order 11K. 382 Y. Xiong. C. Kwon and J-H. Oh 4 STORAGE CAPACITY When we analyze the results for free energy, the case with q( r = 1), c( r = 1) and d(r = 1) is reduced to the usual saddle-point solutions of the replica symmetric expression of the Gardner's volume g(r = 1)[4, 5). When K is large, the trace over all allowed internal representations can be evaluated similarly to Ref.[4]. The saddle-point equations for q* and d* are derived from the derivative of the free energy in the limit r -+ 1, as in Eq. (9). The details of the self-consistent equations are not shown for space consideration. In the following, we only summarize the asymptotic behavior of the order parameters for large a: 128 K2 1 - q + d - c '" (1r _ 2)2 0'2 ' " 32 ]{ 1 - q + (Ii - 1)( c - d) '" 1r _ 2 a 2 ' 1r-2 q + (K - l)d '" -a ' 1r2r 2 1 - q* + (I{ - l)(c - d*) '" 20'2 ' where r = -[..j1r J duH(u)lnH(u)]-l ~ 0.62. (11) ( 12) (13) (14) (15) It is found that all the overlaps between weights connecting different hidden units have scaling of -1/ K, whereas the typical overlaps between weights connecting the same hidden unit approach one. The order parameters c, d and d* are negative, showing antiferromagnetic correlations between different hidden units, which implies that each hidden unit attempts to store patterns different from those of the others[4, 5). Finally, the asymptotic behavior of the entropy N D in the large K limit can be derived using the scaling given above. Near the storage capacity, ND can be written, up to the leading order, as: N '" "I r (1r - 2)20'2 D - K n '\ 256K (16) Being the entropy of a discrete system, ND cannot be negative. Therefore, ND = a gives an indication of the upper bound of storage capacity, that is, a c '" 7r~2 K Vln K. The storage capacity per synapse, 7r~2 Vln K, satisfies the rigorous bound", In K derived by Mitchison and Durbin (MD)[3], whereas the conventional RS result[4, 5], which scales as .JR, violates the MD bound. 5 DISCUSSIONS Recently, we have studied this problem using a conventional Gardner approach in the one-step RSB scheme[13]. The result yields the same scaling with respect to K, but a coefficient smaller by a factor v'2. In the present paper, we are dealing with the fine structure of version space related to internal representations. On the other hand, the RSB calculation seems to handle this fine structure in association with symmetry breaking between replicas. Although the physics of the two approaches seems to be somehow related, it is not clear which of the two can yield a better The Storage Capacity of a Fully-Connected Committee Machine 383 estimate of the storage capacity. It is possible that the present RS calculation does not properly handle the RSB picture of the system. Monasson and his co-workers reported that the Almeida-Thouless instability of the RS solutions decreases with increasing K, in the NRF case[1O, 11]. A similar analysis for the fully-connected case certainly deserves further research. On the other hand, the one-step RSB scheme also introduces approximation, and possibly it cannot fully explain the weight-space structure associated with internal representations. It is interesting to compare our result with that of the NRF committee machine along the same lines[10]. Based on the conventional RS calculation, Angel et al. suggested that the same storage capacity per synapse for both fully-connected and NRF committee machines will be similar, as the overlap between the hidden nodes approaches zero.[5]. While the asymptotic scaling with respect to K is the same, the storage capacity in the fully-connected committee machine is larger than in the NRF one. It is also consistent with our result from one-step RSB calculation[13]. This implies that the small, but nonzero negative correlation between the weights associated with different hidden units, enhances the storage capacity. This may be good news for those people using a fully connected multi-layer perceptron in applications. From the fact that the storage capacity of the NRF parity machine is In Kj In 2[2, 10], which saturates the MD bound, one may guess that the storage capacity of a fully-connected parity machine is also proportional to Kin K. It will be interesting to check whether the storage capacity per synapse of the fully-connected parity machine is also enhanced compared to the NRF machine[16]. Acknowledgements This work was partially supported by the Basic Science Special Program of POSTECH and the Korea Ministry of Education through the POSTECH Basic Science Research Institute(Grant No. BSRI-96-2438). It was also supported by non-directed fund from Korea Research Foundation, 1995, and by KOSEF grant 971-0202-010-2. References [1] E. Gardner, Europhys, Lett. 4(4), 481 (1987); E. Gardner, J. Phys. A21, 257 (1988); E. Gardner and B. Derrida, J . Phys. A21, 271 (1988). [2] E. Barkai, D. Hansel and 1. Kanter, Phys. Rev. Lett. V 65, N18, 2312 (1990). [3] G. J. Mitchison and R. M. Durbin, Boil. Cybern. 60,345 (1989). [4] E. Barkai, D. Hansel and H. Sompolinsky, Phys. Rev. E45, 4146 (1992) . [5] A. Engel, H. M. Kohler, F. Tschepke, H. Vollmayr, and A. Zippeelius, Phys. Rev. E45, 7590 (1992). [6] R. Monasson and D. O'Kane, Europhys. Lett. 27,85(1994). [7] B. Derrida, R. B. Griffiths and A Prugel-Bennett, J. Phys. A 24,4907 (1991). [8] M. Biehl and M. Opper, Neural Networks: The Statistical Mechanics Perspective, Jong-Hoon Oh, Chulan Kwon, and Sungzoon Cho (eds.) (World Scientific, Singapore, 1995). [9] A. Engel and M. Weigt, Phys. Rev. E53, R2064 (1996). [10] R. Monasson and R. Zecchina, Phys. Rev. Lett. 75, 2432 (1995); 76, 2205 (1996). 384 Y. Xiong, C. Kwon and J-H. Oh [11] R. Monasson and R. Zecchina, Mod. Phys. B, Vol. 9 , 1887-1897 (1996) . [12] S. Cocco, R. Monasson and R. Zecchina, Phys. Rev. E54, 717 (1996) . [13] C. Kwon and J. H. Oh, J . Phys. A, in press. [14] K. Kang, J. H. Oh, C. Kwon and Y. Park, Phys. Rev. E48, 4805 (1993). [15] H. Schwarze and J. Hertz, Europhys. Lett. 21, 785 (1993) . [16] Y. Xiong, C. Kwon and J .-H. Oh, to be published (1997).
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Structural Risk Minimization for Nonparametric Time Series Prediction Ron Meir* Department of Electrical Engineering Technion, Haifa 32000, Israel rmeir@dumbo.technion.ac.il Abstract The problem of time series prediction is studied within the uniform convergence framework of Vapnik and Chervonenkis. The dependence inherent in the temporal structure is incorporated into the analysis, thereby generalizing the available theory for memoryless processes. Finite sample bounds are calculated in terms of covering numbers of the approximating class, and the tradeoff between approximation and estimation is discussed. A complexity regularization approach is outlined, based on Vapnik's method of Structural Risk Minimization, and shown to be applicable in the context of mixing stochastic processes. 1 Time Series Prediction and Mixing Processes A great deal of effort has been expended in recent years on the problem of deriving robust distribution-free error bounds for learning, mainly in the context of memory less processes (e.g. [9]). On the other hand, an extensive amount of work has been devoted by statisticians and econometricians to the study of parametric (often linear) models of time series, where the dependence inherent in the sample, precludes straightforward application of many of the standard results form the theory of memoryless processes. In this work we propose an extension of the framework pioneered by Vapnik and Chervonenkis to the problem of time series prediction. Some of the more elementary proofs are sketched, while the main technical results will be proved in detail in the full version of the paper. Consider a stationary stochastic process X = { ... ,X -1, X 0, X 1, ... }, where Xi is a random variable defined over a compact domain in R and such that IXil ::; B with probability 1, for some positive constant B. The problem of one-step prediction, in the mean square sense, can then be phrased as that of finding a function f (.) of the infinite past, such that E IXo - f(X=~) 12 is minimal, where we use the notation xf = (Xi, Xi ti, ... ,Xj ), ·This work was supported in part by the a grant from the Israel Science Foundation Structural Risk Minimization/or Nonparametric Time Series Prediction 309 j ~ i. It is well known that the optimal predictor in this case is given by the conditional mean, E[XoIX:!J While this solution, in principle, settles the issue of optimal prediction, it does not settle the issue of actually computing the optimal predictor. First of all, note that ~o compute the conditional mean, the probabilistic law generating the stochastic process X must be known. Furthermore, the requirement of knowing the full past, X=-~, is of course rather stringent. In this work we consider the more practical situation, where a finite sub-sequence Xi" = (Xl, X 2,··· ,XN) is observed, and an optimal prediction is needed, conditioned on this data. Moreover, for each finite sample size N we allow the pre.dictors to be based only on a finite lag vector of size d. Ultimately, in order to achieve full generality one may let d -+ 00 when N -+ 00 in order to obtain the optimal predictor. We first consider the problem of selecting an empirical estimator from a class of functions Fd,n : Rd -+ R, where n is a complexity index of the class (for example, the number of computational nodes in a feedforward neural network with a single hidden layer), and If I ::; B for f E Fd,n. Consider then an empirical predictor fd,n,N(Xi=~), i > N, for Xi based on the finite data set Xi" and depending on the d-dimensional lag vector Xi=~, where fd,n,N E Fd,n. It is possible to split the error incurred by this predictor into three terms, each possessing a rather intuitive meaning. It is the competition between these terms which determines the optimal solution, for a fixed amount of data. First, define the loss of a functional predictor f : Rd -+ R as L(f) = E IXi f(xi=~) 12 , and let fd,n be the optimal function in Fd,n minimizing this loss. Furthermore, denote the optimal lag d predictor by fd' and its associated loss by L'd. We are then able to split the loss of the empirical predictor fd,n,N into three basic components, L(fd,n,N) = (Ld,n,N - L'd,n) + (L'd,n - L'd) + L'd, (I) where Ld,n,N = L(fd,n,N). The third term, L'd, is related to the error incurred in using a finite memory model (of lag size d) to predict a process with potentially infinite memory. We do not at present have any useful upper bounds for this term, which is related to the rate of convergence in the martingale convergence theorem, which to the best of our knowledge is unknown for the type of mixing processes we study in this work. The second term in (1) , is related to the so-called approximation error, given by Elfei (X:=-~) - fel,n (Xf=~) 12 to which it can be immediately related through the inequality IIalP - IblPI ::; pia - bll max( a, b) Ip-l . This term measures the excess error incurred by selecting a function f from a class of limited complexity Fd,n, while the optimal lag d predictor fei may be arbitrarily complex. Of course, in order to bound this term we will have to make some regularity assumptions about the latter function. Finally, the first term in (1) r~resents the so called estimation error, and is the only term which depends on the data Xl . Similarly to the problem of regression for i.i.d. data, we expect that the approximation and estimation terms lead to conflicting demands on the choice of the the complexity, n, of the functional class Fd,n. Clearly, in order to minimize the approximation error the complexity should be made as large as possible. However, doing this will cause the estimation error to increase, because of the larger freedom in choosing a specific function in Fd,n to fit the data. However, in the case of time series there is an additional complication resulting from the fact that the misspecification error L'd is minimized by choosing d to be as large as possible, while this has the effect of increasing both the approximation as well as the estimation errors. We thus expect that sOrhe optimal values of d and n exist for each sample size N. Up to this point we have not specified how to select the empirical estimator f d,n,N. In this work we follow the ideas of Vapnik [8], which have been studied extensively in the context of i.i.d observations, and restrict our selection to that hypothesis which minimizes the empirical error, given by LN(f) = N~d 2::~d+l IXi f(x:=~)12 . For this function it is easy to establish (see for example [8]) that (Ld,n,N - L'd,n) ::; 2 sUP!E.rd,n IL(f) - LN(f)I· The main distinction from the i.i.d case, of course, is that random variables appearing in 310 R. Meir the empirical error, LN(f), are no longer independent. It is clear at this point that some assumptions are needed regarding the stochastic process X, in order that a law of large numbers may be established. In any event, it is obvious that the standard approach of using randomization and symmetrization as in the i.i.d case [3] will not work here. To circumvent this problem, two approaches have been proposed. The first makes use of the so-called method of sieves together with extensions of the Bernstein inequality to dependent data [6]. The second approach, to be pursued here, is based on mapping the problem onto one characterized by an i.i.d process [10], and the utilization of the standard results for the latter case. In order to have some control of the estimation error discussed above, we will restrict ourselves in this work to the class of so-called mixing processes. These are processes for which the 'future' depends only weakly on the 'past', in a sense that will now be made precise. Following the definitions and notation of Yu [10], which will be utilized in the sequel, let (7t = (7(Xf) and (7:+m = (7(Xt~m)' be the sigma-algebras of events generated by the random variables Xf = (X1,X2 , ••• ,Xt) and Xi1.m = (X1+m,Xl+m+1 , •• . ), respectively. We then define 13m, the coefficient of absolute regularity, as 13m = SUPt>l Esup {IP(BI(7I) - P(B)I : BE (7:+m} , where the expectation is taken with respect-to (71 = (7(XD. A stochastic process is said to be 13-mixing if (3m -t 0 as m -t 00. We note that there exist many other definitions of mixing (see [2] for details). The motivation for using the 13-mixing coefficient is that it is the weakest form of mixing for which uniform laws of large numbers can be established. In this work we consider two type of processes for which this coefficient decays to zero, namely algebraically decaying processes for which 13m ~ /3m- r , /3, r > 0, and exponentially mixing processes for which 13m ~ /3 exp{ -bm K }, jJ, b, I\, > O. Note that for Markov processes mixing implies exponential mixing, so that at least in this case, there is no loss of generality in assuming that the process is exponentially mixing. Note also that the usual i.i.d process may be obtained from either the exponentially or algebraically mixing process, by taking the limit I\, -t 00 or r -t 00, respectively. In this section we follow the approach taken by Yu [10] in deriving uniform laws of large numbers for mixing processes, extending her mainly asymptotic results to finite sample behavior, and somewhat broadening the class of processes considered by her. The basic idea in [10], as in many related approaches, involves the construction of an independentblock sequence, which is shown to be 'close' to the original process in a well-defined probabilistic sense. We briefly recapitulate the construction, slightly modifying the notation in [10] to fit in with the present paper. Divide the sequence xi' into 2J-lN blocks, each of size aN; we assume for simplicity that N = 2J-lNaN. The blocks are then numbered according to their order in the block-sequence. For 1 ~ j ~ J-lN define H j = {i : 2(j - l)aN + 1 ~ i ~ (2j - l)aN} and Tj = {i : (2j - l)aN + 1 ~ i ~ (2j)aN}. Denote the random variables corresponding to the H j and Tj indices as X(j) = {Xi : i E Hj } and X' (j) = {Xi : i E Tj }. The sequence of H-blocks is then denoted by X aN = {X(j)}j:l. Now, construct a sequence of independent and identically distributed (i.i.d.) blocks {3(j) )}j:l' where 3(j) = {~i : i E H j }, such that the sequence is independent of Xi" and each block has the same distribution as the block X(j) from the original sequence. Because the process is stationary, the blocks 3(j) are not only independent but also identically distributed. The basic idea in the construction of the independent block sequence is that it is 'close', in a well-defined sense to the original blocked sequence X aN . Moreover, by appropriately selecting the number of blocks, J-lN, depending on the mixing nature of the sequence, one may relate properties of the original sequence X f", to those of the independent block sequence 3 aN (see Lemma 4.1 in [10)). Let F be a class of bounded functions, such that 0 ~ f ~ B for any f E F. In order to Structural Risk Minimizationfor Nonparametric Time Series Prediction 311 relate the uniform deviations (with respect to F) of the original sequence Xi' to those of the independent-block sequence BaN' use is made of Lemma 4.1 from [10]. We also utilize Lemma 4.2 from [10] and modify it so that it holds for finite sample size. Consider the block-independent sequence BaN and define EJ.LN 1 = J.L1N 'E~:1 f(B(j)) where f(=.(j») = 'EiEHj f(~i)' j = 1,2, ... , J-lN, is a sequence of independent random variables such that 111 ~ aNB. In the remainder of the paper we use variables with a tilde above them to denote quantities related to the transformed block sequence. Finally, we use the symbol EN to denote the empirical average with respect to the original sequence, namely EN f = (N - d)-1 'E~d+1 f(Xi). The following result can be proved by a simple extension of Lemma 4.2 in [10]. Lemma 1.1 Suppose F is a permissible class of boundedfunctions, If I ~ B for f E :F. Then p {sup lEN f - Efl > t:} ~ 2P {sup IEJ.LN 1 - Ell> aNt:} + 2J-lNf3aN' (2) fE';: fE';: The main merit of Lemma 1.1 is in the transformation of the problem from the domain of dependent processes, implicit in the quantity lEN f - Efl, to one characterized by independent processes, implicit in the term EJ.LN 1 - Ell corresponding to the independent blocks. The price paid for this transformation is the extra term 2J-lN f3aN which appears on the r.h.s of the inequality appearing in Lemma 1.1. 2 Error Bounds The development in Section 1 was concerned with a scalar stochastic process X. In order to use the results in the context of time series, we first define a new vector-valued pro........ ........ d+l cess X' = { ... ,X-1,XO,X1 , ... } where Xi = (Xi,Xi - 1, . :.... ,Xi-d) E ~ . For this sequence the f3-mixing coefficients obey the inequality f3m(X') ~ f3m-d(X). Let F be a space of functions mapping Rd -7 R, and for each f E F let the loss function be given by ff(Xf-d) = IXi f(X:~~W· The loss space is given by L,;: = {ff: f E F}. It is well known in the theory of empirical processes (see [7] for example), that in order to obtain upper bounds on uniform deviations of i.i.d sequences, use must be made of the so-called covering number of the function class F, with respect to the empirical it,N norm, given by it,N(f, g) = N- 1 'E~1 If(Xd - g(Xi)l· Similarly, we denote the empirical norm with respect to the independent block sequence by [1 ,J.LN' where [1,J.LN(f,g) = J-l,'/ 'E~:1 11(x(j)) - g(X(j) I, and where f(X(j») = 'EiEHj Xi and similarly for g. Following common practice we denote the t:-covering number of the functional space F using the metric p by N(t:, F, p). Definition 1 Let L';: be a class of real-valued functions from RD --t R, D = d + 1. For eachff E L,;:andx = (Xl,X2, .. . ,XaN ), Xi E R D , let if (x) = 'E~:lff(Xi)' Then define £,;: = {if: if E L';:} , where if : RaND -7 R+. In order to obtain results in terms of the covering numbers of the space L';: rather than £,;:, which corresponds to the transformed sequence, we need the following lemma, which is not hard to prove. Lemma 2.1 For any t: > 0 N (t:, £,;:, [1 ,J.LN) ~ N (t:jaN, L';:, h,N). 312 R. Meir PROOF The result follows by sequence of simple inequalities, showing that ll.J1.N (j, g) ~ aNh,N(f, g). I We now present the main result of this section, namely an upper bound for the uniform deviations of mixing processes, which in turn yield upper bounds on the error incurred by the empirically optimal predictor fd ,n.N. Theorem 2.1 Let X = { . .. ,Xl' X o, Xl, ... } be a bounded stationary (3-mixing stochastic process, with IXil ~ B, and let F be a class of bo unded functions, f : Rd ~ [0, B]. For each sample size N, let f~ be the function in :F which minimizes the empirical error, and 1* is the function in F minimizing the true error L(f). Then, where c' = c/128B. PROOF The theorem is established by making use of Lemma 1.1, and the basic results from the theory of uniform convergence for i.i.d. processes, together with Lemma 2.1 relating the covering numbers of the spaces iF and LF. The covering numbers of LF and Fare easily related using N(c, LF, Ll (P)) ~ N(c/2B, F, Ll (P)) . I Up to this point we have not specified J..tN and aN, and the result is therefore quite general. In order to obtain weak consistency we require that that the r.h.s. of (3) converge to zero for each c > O. This immediately yields the following conditions on J..tN (and thus also on aN through the condition 2aNJ..tN = N). Corollary 2.1 Under the conditions of Theorem 2.1, and the added requirements that d = o(aN) and N(c, F, h,N) < 00, the following choices of J..tN are sufficient to guarantee the weak consistency of the empirical predictor f N: J..tN ,..", N/t/(1+/t) J..tN""" N s/{1+s), 0 < s < r (exponential mixing), (algebraic mixing), where the notation aN ,..", bN implies that O(bN) ~ aN ~ O(bN ). (4) (5) PROOF Consider first the case of exponential mixing. In this case the r.h.s. of (3) clearly converges to zero because of the finiteness of the covering number. The fastest rate of convergence is achieved by balancing the two terms in the equation, leading to the choice J..tN '" N/t/(1+/t). In the case of algebraic mixing, the second term on the r.h.s. of (3) is of the order O(J..tNa"i/) where we have used d = o(aN). Since J..tNaN '" N, a sufficient condition to guarantee that this term converge to zero is that J..tN ,..", Ns/(1+s), 0 < s < r, as was claimed. I In order to derive bounds on the expected error, we need to make an assumption concerning the covering number of the space F. In particular, we know from the work Haussler [4J that the covering number is upper bounded as follows (2 B) Pdim(F) N(c ,F, L 1(P)) ~ e(Pdim(F) + 1) -7' for any measure P. Thus, assuming the finiteness of the pseudo-dimension of F guarantees a finite covering number. Structural Risk Minimization/or Nonparametric Time Series Prediction 313 3 Structural Risk Minimization The results in Section 2 provide error bounds for estimators formed by minimizing the empirical error over a fixed class of d-dimensional functions. It is clear that the complexity of the class of functions plays a crucial role in the procedure. If the class is too rich, manifested by very large covering numbers, clearly the estimation error term will be very large. On the other hand, biasing the class of functions by restricting its complexity, leads to poor approximation rates. A well-known strategy for overcoming this dilemma is obtained by considering a hierarchy of functional classes with increasing complexity. For any given sample size, the optimal trade-off between estimation and approximation can then be determined by balancing the two terms. Such a procedure was developed in the late seventies by Vapnik [8], and termed by him structural risk minimization (SRM). Other more recent approaches, collectively termed complexity regularization, have been extensively studied in recent years (e.g. [1]). It should be borne in mind, however, that in the context of time series there is an added complexity, that does not exist in the case of regression. Recall that the results derived in Section 2 assumed some fixed lag vector d. In general the optimal value of d is unknown, and could in fact be infinite. In order to achieve optimal performance in a nonparametric setting, it is crucial that the size of the lag be chosen adaptively as well. This added complexity needs to be incorporated into the SRM framework, if optimal performance in the face of unknown memory size is to be achieved. Let Fd,n, d, n E N be a sequence of functions, and define F = U~l U~=l Fd,n ' For any Fd,n let which from [4] is upper bounded by cc-Pdim(Fd.n). We observe in passing that Lugosi and Nobel [5] have recently considered situations where the pseudo-dimension Pdim(Fd,n) is unknown, and the covering number is estimated empirically from the data. Although this line of thought is potentially very useful, we do not pursue it here, but rather assume that upper bounds on the pseudo-dimensions of Fd,n are known, as is the case for many classes of functions used in practice (see for example [9]). In line with the standard approach in [8] we introduce a new empirical function, which takes into account both the empirical error as well as the complexity costs penalizing overly complex models (large complexity index n and lag size d). Let (6) where LN(f) is the empirical error of the predictor f and the complexity penalties ~ are given by IogN1(c, Fd,n) + Cn J-lN /64(2B)4 /-LN /64(2B)4 . (7) (8) The specific form and constants in these definitions are chosen with hindsight, so as to achieve the optimal rates of convergence in Theorem 3.1 below. The constants Cn and Cd are positive constants obeying l:~=1 e- Cn :::; 1 and similarly for Cd . A possible choice is Cn = 210g n + 1 and Cd = 210g d + 1. The value of J-lN can be chosen in accordance with Corollary 2.1. Let id,n,N minimize the empirical error LN(f) within the class of functions Fd,n' ",!e assume that the classes Fd,n are compact, so that such a minimizer exists. Further, let IN 314 be the function in F minimizing the complexity penalized loss (6), namely Ld n N(1~) = min min Ld n N(1~ n N) , , d2: 1 n2: 1 " " R. Meir (9) The following basic result establishes the consistency of the structural risk minimization approach, and yields upper bounds on its performance. Theorem 3.1 Let Fd,n, d, n E N be sequence offunctional classes, where 1 E Fd,n is a mapping from Rd to R The expected loss of the function iN, selected according to the SRM principle, is upper bounded by EL(iN) ::; min {inf L(J) + Cl d,n d,n The main merit of Theorem 3.1 is the demonstration that the SRM procedure achieves an optimal balance between approximation and estimation, while retaining its non parametric attributes. In particular, if the optimal lag d predictor 1J belongs to Fd,no for some no, the SRM predictor would converge to it at the same rate as if no were known in advance. The same type of adaptivity is obtained with respect to the lag size d. The non parametric rates of convergence of the SRM predictor will be discussed in the full paper. References [1] A. Barron. Complexity Regularization with Application to Artificial Neural Networks. In G. Roussas, editor, Nonparametric Functional Estimation and Related Topics, pages 561-576. Kluwer Academic Press, 1991. [2] L. Gyorfi, W. HardIe, P. Sarda, and P. Vieu. Nonparametric Curve Estimation from Time Series. Springer Verlag, New York, 1989. [3] D. Haussler. Decision Theoretic Generalizations of the PAC Model for Neural Net and Other Learning Applications. Information and Computation, 100:78-150, 1992. [4] D. Haussler. Sphere Packing Numbers for Subsets of the Boolean n-Cube with Bounded Vapnik-Chervonenkis Dimesnion. J. Combinatorial Theory, Series A 69:217-232,1995. [5] G. Lugosi and A. Nobel. Adaptive Model Selection Using Empirical Complexities. Submitted to Annals Statis., 1996. [6] D. Modha and E. Masry. Memory Universal Prediction of Stationary Random Processes. IEEE Trans. Inj. Th., January, 1998. [7] D. Pollard. Convergence of Empirical Processes. Springer Verlag, New York, 1984. [8] V. N. Vapnik. Estimation of Dependences Based on Empirical Data. Springer Verlag, New York, 1992. [9] M. Vidyasagar. A Theory of Learning and Generalization. Springer Verlag, New York,1996. [10] B. Yu. Rates of convergence for empirical processes of stationary mixing sequences. Annals of Probability, 22:94-116, 1984.
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Selecting weighting factors in logarithmic opinion pools Tom Heskes Foundation for Neural Networks, University of Nijmegen Geert Grooteplein 21, 6525 EZ Nijmegen, The Netherlands tom@mbfys.kun.nl Abstract A simple linear averaging of the outputs of several networks as e.g. in bagging [3], seems to follow naturally from a bias/variance decomposition of the sum-squared error. The sum-squared error of the average model is a quadratic function of the weighting factors assigned to the networks in the ensemble [7], suggesting a quadratic programming algorithm for finding the "optimal" weighting factors. If we interpret the output of a network as a probability statement, the sum-squared error corresponds to minus the loglikelihood or the Kullback-Leibler divergence, and linear averaging of the outputs to logarithmic averaging of the probability statements: the logarithmic opinion pool. The crux of this paper is that this whole story about model averaging, bias/variance decompositions, and quadratic programming to find the optimal weighting factors, is not specific for the sumsquared error, but applies to the combination of probability statements of any kind in a logarithmic opinion pool, as long as the Kullback-Leibler divergence plays the role of the error measure. As examples we treat model averaging for classification models under a cross-entropy error measure and models for estimating variances. 1 INTRODUCTION In many simulation studies it has been shown that combining the outputs of several trained neural networks yields better results than relying on a single model. For regression problems, the most obvious combination seems to be a simple linear Selecting Weighting Factors in Logarithmic Opinion Pools 267 averaging of the network outputs. From a bias/variance decomposition of the sumsquared error it follows that the error of the so obtained average model is always smaller or equal than the average error of the individual models. In [7] simple linear averaging is generalized to weighted linear averaging, with different weighting factors for the different networks in the ensemble. A slightly more involved bias/variance decomposition suggests a rather straightforward procedure for finding "optimal" weighting factors. Minimizing the sum-squared error is equivalent to maximizing the loglikelihood of the training data under the assumption that a network output can be interpreted as an estimate of the mean of a Gaussian distribution with fixed variance. In these probabilistic terms, a linear averaging of network outputs corresponds to a logarithmic rather than linear averaging of probability statements. In this paper, we generalize the regression case to the combination of probability statements of any kind. Using the Kullback-Leibler divergence as the error measure, we naturally arrive at the so-called logarithmic opinion pool. A bias/variance decomposition similar to the one for sum-squared error then leads to an objective method for selecting weighting factors. Selecting weighting factors in any combination of probability statements is known to be a difficult problem for which several suggestions have been made. These suggestions range from rather involved supra-Bayesian methods to simple heuristics (see e.g. [1, 6] and references therein). The method that follows from our analysis is probably somewhere in the middle: easier to compute than the supra-Bayesian methods and more elegant than simple heuristics. To stress the generality of our results, the presentation in the next section will be rather formal. Some examples will be given in Section 3. Section 4 discusses how the theory can be transformed into a practical procedure. 2 LOGARITHMIC OPINION POOLS Let us consider the general problem of building a probability model of a variable y given a particular input x. The "output" y may be continuous, as for example in regression analysis, or discrete, as for example in classification. In the latter case integrals over y should be replaced by summations over all possible values of y. Both x and y may be vectors of several elements; the one-dimensional notation is chosen for convenience. We suppose that there is a "true" conditional probability model q(ylx) and have a whole ensemble (also called pool or committee) of experts, each supplying a probability model Pa(Ylx). p{x) is the unconditional probability distribution of inputs. An unsupervised scenario, as for example treated in [8], is obtained if we simply neglect the inputs x or consider them constant. We define the distance between the true probability q(ylx) and an estimate p(ylx) to be the Kullback-Leibler divergence , J J [p(Y1x)] J\. (q, p) == dx p(x) dy q(ylx)log q(ylx) If the densities p(x) and q(ylx) correspond to a data set containing a finite number p of combinations {xlJ ,ylJ}, minus the Kullback divergence is, up to an irrelevant 268 T. Heskes constant, equivalent to the loglikelihood defined as L(p, {i, Y}) == ~ L logp(y~ Ix~) . ~ The more formal use of the Kullback-Leibler divergence instead of the loglikelihood is convenient in the derivations that follow. Weighting factors Wa are introduced to indicate the reliability of each ofthe experts fr. In the following we will work with the constraints La Wa = 1, which is used in some of the proofs, and Wa ? 0 for all experts fr, which is not strictly necessary, but makes it easier to interpret the weighting factors and helps to prevent overfitting when weighting factors are optimized (see details below). We define the average model j)(y/x) to be the one that is closest to the given set of models: j)(y/x) == argmin L waJ{(P,Pa) . p(ylx) a Introducing a Lagrange mUltiplier for the constraint J dxp(y/x) = 1, we immediately find the solution (1) with normalization constant Z(x) = J dy II[Pa(Y/X)]W Q • a (2) This is the logarithmic opinion pool, to be contrasted with the linear opinion pool, which is a linear average'ofthe probabilities. In fact, logarithmic opinion pools have been proposed to overcome some of the weaknesses of the linear opinion pool. For example, the logarithmic opinion pool is "externally Bayesian" , i.e., can be derived from joint probabilities using Bayes' rule [2]. A drawback of the logarithmic opinion pool is that if any of the experts assigns probability zero to a particular outcome, the complete pool assigns probability zero, no matter what the other experts claim. This property of the logarithmic opinion pool, however, is only a drawback if the individual density functions are not carefully estimated. The main problem for both linear and logarithmic opinion pools is how to choose the weighting factors Wa. The Kullback-Leibler divergence of the opinion pool p(y/x) can be decomposed into a term containing the Kullback-Leibler divergences of individual models and an "ambiguity" term: (3) Proof: The first term in (3) follows immediately from the numerator in (1), the second term is minus the logarithm of the normalization constant Z (x) in (2) which can, using (1), be rewritten as Selecting Weighting Factors in Logarithmic Opinion Pools 269 for any choice of y' for which p(y'lx) is nonzero. Integration over y' with probability measure p(y'lx) then yields (3) . Since the ambiguity A is always larger than or equal to zero, we conclude that the Kullback-Leibler divergence of the logarithmic opinion pool is never larger than the average Kullback-Leibler divergences of individual experts. The larger the ambiguity, the larger the benefit of combining the experts' probability assessments. Note that by using Jensen's inequality, it is also possible to show that the Kullback-Leibler divergence of the linear opinion pool is smaller or equal to the average KullbackLeibler divergences of individual experts. The expression for the ambiguity, defined as the difference between these two, is much more involved and more difficult to interpret (see e.g. [10]). The ambiguity of the logarithmic opinion pool depends on the weighting factors Wa, not only directly as expressed in (3), but also through p(ylx). We can make this dependency somewhat more explicit by writing A = ~ ~ waw/3K(Pa ,P/3) + ~ ~ Wa [K(P,Pa) - K(Pa,p)] . (4) a/3 a Proof: Equation (3) is valid for any choice of q(ylx). Substitute q(ylx) = p/3(ylx), multiply left- and righthand side by w/3, and sum over {3. Simple manipulation of terms than yields the result. Alas, the Kullback-Leibler divergence is not necessarily symmetric, i.e., in general K (PI, P2) # K (P2, pd . However, the difference K (PI, P2) - K (p2, pd is an order of magnitude smaller than the divergence K(PI,P2) itself. More formally, writing PI (ylx) = [1 +€(ylx )]p2(ylx) with €(ylx) small, we can easily show that K (PI, P2) is of order (some integral over) €2(ylx) whereas K(PI,P2) - K(p2,pd is of order €3(ylx). Therefore, if we have reason to assume that the different models are reasonably close together, we can, in a first approximation, and will, to make things tractable, neglect the second term in (4) to arrive at K(q,p) ~ ~ waK(q,Pa) ~ L waw/3 [K(Pa,P/3) + K(P/3,Pa)] . (5) a a ,/3 The righthand side of this expression is quadratic in the weighting factors W a , a property which will be very convenient later on. 3 EXAMPLES Regression. The usual assumption in regression analysis is that the output functionally depends on the input x, but is blurred by Gaussian noise with standard deviation (j. In other words, the probability model of an expert a can be written j1 [-(y - fa(x))2] Pa(ylx) = V ~ exp 2(j2 . (6) The function fa(x) corresponds to the network's estimate of the "true" regression given input x. The logarithmic opinion pool (1) also leads to a normal distribution with the same standard deviation (j and with regression estimate 270 T. Heskes In this case the Kullback-Leibler divergence is symmetric, which makes (5) exact instead of an approximation. In [7], this has all been derived starting from a sum-squared error measure. Variance estimation. There has been some recent interest in using neural networks not only to estimate the mean of the target distribution, but also its variance (see e.g. [9] and references therein). In fact, one can use the probability density (6) with input-dependent <7( x). We will consider the simpler situation in which an input-dependent model is fitted to residuals y, after a regression model has been fitted to estimate the mean (see also [5]). The probability model of expert 0' can be written ( j ) _ ~ [za(x)y2] Pa Y X - V ~ exp 2 ' where l/za(x) is the experts' estimate of the residual variance given input x. The logarithmic opinion pool is of the same form with za(x) replaced by z(x) = L waza(x) . Here the Kullback-Leibler divergence " 1 J [ z(x) z(x)] I\ (P,Pa) = 2 dx p(x) za(x) -log za(x) - 1 is asymmetric. We can use (3) to write the Kullback-Leibler divergence of the opinion pool explicitly in terms of the weighting factors Wa. The approximation (5), with • • 1 J [za(x) - zp(x)]2 Ii(Pa,PP) + Ii(Pp,Pa) -"2 dx p(x) za(x)zp(x) , is much more appealing and easier to handle. Classification. In a two-class classification problem, we can treat y as a discrete variable having two possible realizations, e.g., y E {-I, I}. A convenient representation for a properly normalized probability distribution is 1 Pa(yjX) = 1 + exp[-2ha(x)y] . In this logistic representation, the logarithmic opinion pool has the same form with The Kullback-Leibler divergence is asymmetric, but yields the simpler form to be used in the approximation (5). For a finite set of patterns, minus the loglikelihood yields the well-known cross-entropy error. Selecting Weighting Factors in Logarithmic Opinion Pools 271 The probability models in these three examples are part of the exponential family. The mean f 0:, inverse variance zo:, and logit ho: are the canonical parameters. It is straightforward to show that, with constant dispersion across the various experts, the canonical parameter of the logarithmic opinion pool is always a weighted average of the canonical parameters of the individual experts. Slightly more complicated expressions arise when the experts are allowed to have different estimates for the dispersion or for probability models that do not belong to the exponential family. 4 SELECTING WEIGHTING FACTORS The decomposition (3) and approximation (5) suggest an objective method for selecting weighting factors in logarithmic opinion pools. We will sketch this method for an ensemble of models belonging to the same class, say feedforward neural networks with a fixed number of hidden units, where each model is optimized on a different bootstrap replicate of the available data set. Su ppose that we have available a data set consisting of P combinations {x/J, y/J }. As suggested in [3], we construct different models by training them on different bootstrap replicates of the available data set. Optimizing nonlinear models is often an unstable process: small differences in initial parameter settings or two almost equivalent bootstrap replicates can result in completely different models. Neural networks, for example, are notorious for local minima and plateaus in weight space where models might get stuck. Therefore, the incorporation of weighting factors, even when models are constructed using the same pro·cedure, can yield a better generalizing opinion pool. In [4] good results have been reported on several regression problems. Balancing clearly outperformed bagging, which corresponds to Wo: = lin with n the number of experts, and bumping, which proposes to keep a single expert. Each example in the available data set can be viewed as a realization of an unknown probability density characterized by p(x) and q(ylx). We would like to choose the weighting factors Wo: such as to minimize the Kullback-Leibler divergence K(q, p) of the opinion pool. If we accept the approximation (5), we can compute the optimal weighting factors once we know the individual Kullbacks K(q,po:) and the Kullbacks between different models K(po:, PI3). Of course, both q(ylx) and p(x) are unknown, and thus we have to settle for estimates. In an estimate for K(po:,PI3) we can simply replace the average over p(x) by an average over all inputs x/J observed in the data set: A similar straightforward replacement for q(ylx) in an estimate for K(q, Po:) is biased, since each expert has, at least to some extent, been overfitted on the data set. In [4] we suggest how to remove this bias for regression models minimizing sumsquared errors. Similar compensations can be found for other probability models. Having estimates for both the individual Kullback-Leibler divergences K(q,po:) and the cross terms K (Po:, PI3), we can optimize for the weighting factors Wo:. Under the constraints 2:0: Wo: = 1 and Wo: 2: 0 the approximation (5) leads to a quadratic programming problem. Without this approximation, optimizing the weighting factors becomes a nasty exercise in nonlinear programming. 272 T. Heskes The solution of the quadratic programming problem usually ends up at the edge of the unit cube with many weighting factors equal to zero. On the one hand, this is a beneficial property, since it implies that we only have to keep a relatively small number of models for later processing. On the other hand, the obtained weighting factors may depend too strongly on our estimates of the individual Kullbacks K (q, Pet). The following version prohibits this type of overfitting. Using simple statistics, we obtain a rough indication for the accuracy of our estimates K(q,pet). This we use to generate several, say on the order of 20, different samples with estimates {K (q, pI), .. . , K (q, Pn)}. For each of these samples we solve the corresponding quadratic programming problem and obtain a set of weighting factors. The final weighting factors are obtained by averaging. In the end, there are less experts with zero weighting factors, at the advantage of a more robust procedure. Acknowledgements I would like to thank David Tax, Bert Kappen, Pierre van de Laar, Wim Wiegerinck, and the anonymous referees for helpful suggestions. This research was supported by the Technology Foundation STW, applied science division of NWO and the technology programme of the Ministry of Economic Affairs. References [1] J. Benediktsson and P. Swain. Consensus theoretic classification methods. IEEE Transactions on Systems, Man, and Cybernetics, 22:688-704, 1992. [2] R. Bordley. A multiplicative formula for aggregating probability assessments. Management Science, 28:1137-1148,1982. [3] L. Breiman. Bagging predictors. Machine Learning, 24:123-140, 1996. [4] T. Heskes. Balancing between bagging and bumping. In M. Mozer, M. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Eystems 9, pages 466-472, Cambridge, 1997. MIT Press. [5] T. Heskes. Practical confidence and prediction intervals. In M. Mozer, M. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Eystems 9, pages 176-182, Cambridge, 1997. MIT Press. [6] R. Jacobs. Methods for combining experts' probability assessments. Neural Computation, 7:867-888, 1995. [7] A. Krogh and J. Vedelsby. Neural network ensembles, cross validation, and active learning. In G. Tesauro, D. Touretzky, and T. Leen, editors, Advances in Neural Information Processing Eystems 7, pages 231-238, Cambridge, 1995. MIT Press. [8] P. Smyth and D. Wolpert. Stacked density estimation. These proceedings, 1998. [9] P. Williams. Using neural networks to model conditional multivariate densities. Neural Computation, 8:843-854, 1996. [10] D. Wolpert. On bias plus variance. Neural Computation, 9:1211-1243, 1997.
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