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900	Grammar Learning by a Self-Organizing Network Michiro Negishi Dept. of Cognitive and Neural Systems, Boston University 111 Cummington Street Boston, MA 02215 email: negishi@cns.bu.edu Abstract This paper presents the design and simulation results of a selforganizing neural network which induces a grammar from example sentences. Input sentences are generated from a simple phrase structure grammar including number agreement, verb transitivity, and recursive noun phrase construction rules. The network induces a grammar explicitly in the form of symbol categorization rules and phrase structure rules. 1 Purpose and related works The purpose of this research is to show that a self-organizing network with a certain structure can acquire syntactic knowledge from only positive (i.e. grammatical) data, without requiring any initial knowledge or external teachers that correct errors. There has been research on supervised neural network models of language acquisition tasks [Elman, 1991, Miikkulainen and Dyer, 1988, John and McClelland, 1988]. Unlike these supervised models, the current model self-organizes word and phrasal categories and phrase construction rules through mere exposure to input sentences, without any artificially defined task goals. There also have been self-organizing models of language acquisition tasks [Ritter and Kohonen, 1990, Scholtes, 1991]. Compared to these models, the current model acquires phrase structure rules in more explicit forms, and it learns wider and more structured contexts, as will be explained below. 2 Network Structure and Algorithm The design of the current network is motivated by the observation that humans have the ability to handle a frequently occurring sequence of symbols (chunk) as an unit of information [Grossberg, 1978, Mannes, 1993]. The network consists of two parts: classification networks and production networks (Figure 1). The classification networks categorize words and phrases, and the production networks 28 Michiro Negishi evaluate how it is likely for a pair of categories to form a phrase. A pair of combined categories is given its own symbol, and fed back to the classifiers. After weights are formed, the network parses a sentence as follows. Input words are incrementally added to the neural sequence memory called the Gradient Field [Grossberg, 1978] (GF hereafter). The top (i.e. most recent) two symbols and the lookahead token are classified by three classification networks. Here a symbol is either a word or a phrase, and the lookahead token is the word which will be read in next. Then the lookahead token and the top symbol in the GF are sent to the right production network, and the top and the second ones are sent to the left production network. If the latter pair is judged to be more likely to form a phrase, the symbol pair reduces to a phrase, and the phrase is fed back to the GF after removing the top two symbols. Otherwise, the lookahead token is added to the sequence memory, causing a shift in the sequence memory. If the input sentence is grammatical, the repetition of this process reduces the whole sentence to a single "5" (sentence) symbol. The sequence of shifts and reductions (annoted with the resultant symbols) amounts to a parse of the sentence. During learning, the operations stated above are carried out as weights are gradually formed. In classification networks, the weights record a distribution pattern with respect to each symbol. That is, the weights record the co-occurrence of up to three adjacent symbols in the corpus. An symbol is classified in terms of this distribution in the classification networks. The production networks keep track of the categories of adjacent symbols. If the occurrence of one category reliably predicts the next or the previous one, the pair of categories forms a phrase, and is given the status of an symbol which is treated just like a word in the sentence. Because the symbols include phrases, the learned context is wider and more structured than the mere bigram, as well as the contexts utilized in [Ritter and Kohonen, 1990, Scholtes, 1991]. 3 Simulation 3.1 The Simulation Task The grammar used to generate input sentences (Table 3) is identical to that used in [Elman,1991], except that it does not include optionally transitive verbs and proper nouns. Lengths of the input sentences are limited to 16 words. To determine the completion of learning, after accepting 200 consecutive sentences with learning, learning is suppressed and other 200 sentences are processed to see if all are accepted. In addition, the network was tested for 44 ungrammatical sentences to see that they are correctly rejected. Ungrammatical sentences are derived by hand from randomly generated grammatical sentences. Parameters used in the simulation are : number of symbol nodes = 30 (words) + 250 (phrases), number of category nodes = ISO, f. = 10-9, 'Y = 0.25, p = 0.65, 0'1 = 0.00005, /31 = 0.005, f32 = 0.2, 0'3 = 0.0001, /33 = 0.001, and T = 4.0. Grammar Learning by a Self-Organizing NeMork 29 3.2 Acquired Syntax Rules Learning was completed after learning 19800 grammatical sentences. Tables 1 and 2 show the acquired syntax rules extracted from the connection weights. Note that category names such as Ns, VPp, are not given a priori, but assigned by the author for the exposition. Only rules that eventually may reach the "S"(sentence) node are shown. There were a small number of uninterpretable rules, which are marked I/?". These rules might disturb normal parsing for some sentences, but they were not activated while testing for 200 sentences after learning. 3.3 Discussion Recursive noun phrase structures should be learned by finding equivalences of distribution between noun phrases and nouns. However, nouns and noun phrases have the same contextual features only when they are in certain contexts. An examination of the acquired grammar reveals that the network finds equivalence of features not of ''N'' and ''N RC" (where RC is a relative clause) but of "N V" and ''N RC V" (when ''N RC" is subjective), or "V N" and /IV NRC" (when ''N RC" is objective). As an example, let us examine the parsing of the sentence [19912] below. The rule used to reduce FEEDS CATS WHO UVE (''V NRC") is PO, which is classified as category C4, which includes P121 (''V N") where V are the singular forms of transitive verbs, and also includes the ''V'' where V are singular forms of intransitive verbs. Thus, GIRL WHO FEEDS CATS WHO UVE is reduced to GIRL WHO "VPsingle". *[19912}*******··******··***··******** •• ***.** +---141---+ I +---88------+ I I +---206------+ I I I +----0----+ I I I I +-219-+ I I +-41-+ I +-36-+ I BOYS CHASE GIRL WHO FEEDS CATS WHO LIVE «Accepted» Top symbol was 77 4 Conclusion and Future Direction In this paper, a self-organizing neural network model of grammar learning was presented. A basic principle of the network is that all words and phrases are categorized by the contexts in which they appear, and that familiar sequence of categories are chunked. As it stands, the scope of the grammar used in the simulation is extremely limited. Also, considering the poverty of the actual learning environment, the learning of syntax should also be guided by the cognitive competence to comprehend the utterance situations and conversational contexts. However, being a self-organizing network, the current model offers a plausible model of natural language acquisition through mere exposures to only grammatical sentences, not requiring any external teacher or an explicit goal. 30 S := C4 :a C13 .C16 := C18 .C20 .C26 := C29 .C30 := C32 .Miclziro Negislzi Table 1. Acquired categorization rules C29 r NPs VPs ., I C52 .P41 rNsR·, C30r?·,1 C56 .P36 rNpR·, enr~vpp·, C58 := P28 r Ns vrs ., I LIVES I ALKS I P34 r Np vrVi I POrvrsNpRC·' I P68 r Ns RC so, I P74 rvrsNs RC·, I P147 rNpRCvrp·, = rNvr·, P121 rvrsNs·' I C69 .P206 rNs R VPs·' = rNsRCs·' P157 rvrsNpo, .. rVps·' P238 rNsRNvr·, GIRL I DOG I C74 .P219 rNpRVpp 0' I = rNpRCp·' CAT I BOY = /"Ns·, P249 rNpRNvr·, CHASE I FEED = rvrp·' C77 .P141 rNpVPp·' I WHO =rR·, P217 rNpRCVpp·, = rNPpVPp·, CHASES I FEEDS = rvrs·' C1l9 := Pl48 =/"vrsNvr·, BOYS I CATS I C122 .P243 = rNsR vrsNvr·, DOGS I GIRLS =rNp·' C139 .PIO rvrs NPs VPs·' I = /" VPs' VPp's ?, P93 r Ns RC VPs·' I P32 r vrs NPp VPp • , P138 rNs VPs·' = rNPsvps·, wnere P2 r vrp NP\.iPp ., I RCs = RVPs I RNvr P94 r vrp N ., I RCp = RVPp I RNvr P137 r?·, = r? 0, NPp = Np I NpRCp WALK I LIVE I NPs = Ns I NsRCs PI r~NpRC·' I P61 r pNpo, I P88rvrpNs RC·, I P122 r vrp Ns·' = rvpp·' Table 2. Acquired production rules PO P1 P2 PI0 P28 P32 P34 P36 P41 P61 P68 P74 P88 P93 P94 P121 P122 P137 P138 P141 P147 P148 P157 P206 P217 P219 P238 P243 P249 :=C20 rvrs·' :=C16 rvrp·' := C16 rvrp·, := C20 rvrs·' :=C13 rNs·' :=C20 rvrs·' :=C26 rNp·' := C26 ,. Np ., :=C13 rNs·' := C16 rvrp·' :=C69 rNsRCs·, :=C20 rvrs·' :=C16 rVTp·' := C69 rNsRCs·, := C16 rvrp·' := C20 r vrs • , :=C16 rVTp·' := C122 r NsR vrsNVT·, :=C13 rNs·' := C26 rNp·' :=C74 rNp RCs·, := C20 rvrs 0' :=C20 rvrs·' := C52 r Ns R ., := C74 rNp RCs·, := C56 r Np R ., :=C52rNsR·, := C52 r Ns R 0' :=C56 rNpR·, C74 rNpRCp 0' C74rNpRCp·' en /"NPpVpp·' C29 rNPs VPs·' C20 rvrso, C77/"NPpVPp·' C16/"vrp·' C18 /"R·I C18 /"R·, C26 /"Np·1 C20 /"vrs·' C69 rNs RCs·, C69 r Ns RCs ., C4 r VPs·' C58/"Nvr·, C13/"Ns·, C13 rNs·' C32 /"VPp 0' C4rVps·' C32 /"Vpp·, C16 rvrp·' C58/"Nvr·1 C26/"Np·' C4 rVPs·1 C32r Vpp·' C32r Vpp·' C58 rNvrol C119/"vrsNvr·, C58/"Nvr·, = r vrs Np RCp ., = rvrpNpRCp·' = rvrpNPp VPp·' .. rvrsNPs VPs·' = rNsvrs·' = /"vrsNPpVPp·' = /"Npvrp·' =rNpR·, =rNsR·, =/"vrpNp·' = /"NsRCs vrs·' = r vrs Ns RCs ., = rvrpNsRCs·' = r Ns RCs VPs ., = rvrpNvr·, = /"vrs Ns·1 = rvrpNs·, = /"?·I = rNsvps, =/"NpVPp·' = rNp RCs vrp·' = rvrsNvr·, = rvrsNp·, = /"NsRVPs·' = r Np RCs VPp ., = /"NpRVPp·' =/"NsRNvr·, = /" (Ns R vrs N) vr ·1 = /"Np RNvr·, Grammar Learning by a Self-Organizing Network 3/ Acknowledgements The author wishes to thank Prof. Dan Bullock, Prof. Cathy Harris, Prof. Mike Cohen, and Chris Myers of Boston University for valuable discussions. This work was supported in part by the Air Force Office of Scientific Research (AFOSR F49620-92-J-0225). References [Elman, 1991] Elman, J. (1991). Distributed representations, simple recurrent networks, and grammatical structure. Machine Learning, 7. [Grossberg, 1978] Grossberg, S. (1978). A theory of human memory: Selforganization and performance of sensory-motor codes, maps, and plans. Progress in Theoretical Biology,S. (John and McClelland, 1988] John, M. F. S. and McClelland, J. L. (1988). Applying contextual constraints in sentence comprehension. In Touretzky, D. 5., Hinton, G. E., and Sejnowsky, T. J., editors, Proceedings of the Second Connectionist Models Summer School 1988, Los Altos, CA. Morgan Kaufmann Publisher, Inc. [Mannes, 1993] Mannes, C. (1993). Self-organizing grammar induction using a neural network model. In Mitra, J., Cabestany, J., and Prieto, A., editors, New Trends in Neural Computation: Lecture Notes in Computer Science 686. Springer Verlag, New York. [Miikkulainen and Dyer, 1988] Miikkulainen, R. and Dyer, M. G. (1988). Encoding input/output representations in connectionist cognitive systems. In Touretzky, D. D., Hinton, G. E., and Senowsky, T. J., editors, Proceedings of the Second Connectionist Models Summer School 1988, Los Altos, CA. Morgan Kauffman Publisher, Inc. [Ritter and Kohonen, 1990] Ritter, H. and Kohonen, T. (1990). Learning seman totopic maps from context. Proceedings of. ITCNN 90, Washington D.C., 1. [Scholtes, 1991] Scholtes, J. C. (1991). Unsupervised context learning in natural language processing. Proceedings of I[CNN Seattle 1991. Appendix A. Activation and learning equations A.l Classification Network Activities eGradient Field where t is a discrete time, i is the symbol id. and Ii(t) is an input symbol. eInput Layer (1 ) X1Ai(t) = O(2(XOi(t)-O(XOi(t)))), X1Bi(t) = O(XOi(t)), X1ci(t) = Ii(t+1) Where the suffix A ,B, and C the most recent, the next to most recent, and the lookahead symbols, respectively'. Weights in networks A, B, and C are identical. O(x) = J 1 if x > 1.-2- M 1. 0 otherwIse 32 Michiro Negishi Here M is the maximum number of symbols on the gradient field . • Feature Layer X2;i = L Xl"j Wl"ji, X2;{1 = I(Xl;d(a+ 2: X2;j)), X2"i = X2;{ /(a+ 2: X2;f) j j j I(x) = 2/(1 + exp( -Tx)) - 1 where s is a suffix which is either A, B, or C and T is the steepness of the sigmoid function and a is a small positive constant. Table 4 shows the meaning of above suffix i . • Category Layer { I if i = min{jl2:k8X2"kW2"kj > p}, or X3pi = 0 if ¢ = min{jl2:k" X2"k W2"kj > p} & unreli =ja:r: {unrefJ} (2) otherwise Where p is the least match score required and ure Ii is an unreferenced count. A.2 Classification Learning .Feature Weights LlWl"ij = -alWl"ij +,81Xli(X2"j - Wl"ij) where al is the forgetting rate, and ,81 is the learning rate . • Categorization Weights { Ll W2"ij = /32X3"i(X2"i - W2"ij) if the node is selected by the first line of (2) W2"ij = X2"i if the node is selected by the second line of (2) where /32 is the learning rate. A.3 Production Network Activities .Mutual predictiveness = X3Ai W3ij, = X3Bi W3ij, = X3B j W4ji , = X3cjW4ji , = X4ij XSji = X7ij X8ji The phrase identification number for a category pair (i, j) is given algOrithmically in the current version by a cash function cash(i, j). (i) Case in which 'Y 2:ij X6ij ~ 2:ij X9ij Reduce { 1 if i = cash(I, J) where X6IJ =i]a:r: (X6ij ) X10 j = 0 otherwise XOi(t + 1) = 0.5 * pop(pOp(XOi(t))) + XlO, pop(x) = 2(x - 8(x)) (ii) Case in which 'Y Eij X6ij < 2:ij X9ij Shift The next input symbol is added on the gradient field, as was expressed in (1). Grammar Learning by a Self-Organizing Network 33 A.4 Production Learning where X3 Ai and X3 Bj are nodes that receive the next to the most recent symbol i and the most recent symbol j, respectively. Shift I Reduce Controller Production Networks (predlctlveness evaluators) Classification ~ ~ ) eedback hrases Networks ... ~ 0 (~ 00 171 o~_70v'ovO~I--+----.J Neural Sequence Me ory Lookahead Token Input words Figure 1. Block diagram of the network NP N I NRC VP V[NP] RC whoNPV I whoVP N boy I girl I cat I dog I boys I girls I cats 1 dogs V _ chase I feed I work I live I chases I feeds I works I lives urn agreement - Agreements between N and V within clause - Agreements between head Nand subordinate V (where a ro riate) er arguments - chase, feed -> require a direct object - walk, live -> preclude a direct object (Observed also for head/verb relations in relative clauses) Table 3. Grammar for generated sentences Table 4. Subfields in a feature layer 34 Michiro Negishi Category (The MOst RlI08nt) Category (The "Next to To the Most Recent) Production r---~------------~~------------------------------"'Ne~k X 3B Xo '----A--""I .. "'::. Gradient Field C (X,o> From the Production Network Terminals(lnputs) ... ABC ... -----.... Classification Path ---.. ~ Copy 'or Learning Lookahead Terminals c===J terminals E:::;:;:@:§:~:@ nonterminals _ category o Category (Lookahead) Nonterm inal s (Reduction Results) prllYious next next to next Figure 2. Classification Network x '0 Nonterminal The ext to Most Recent Category The Most Recent Category Figure 3. Production Network X 3C Xa V Xg + X7 Lookahead Category	1994	138
901	Coarse-to-Fine Image Search Using Neural Networks Clay D. Spence, John C. Pearson, and Jim Bergen Nationallnfonnation Display Laboratory P.O. Box 8619 Princeton, NJ 08543-8619 cspence@sarnoff.com John_Pearson@maca.sarnoff.com jbergen@sarnoff.com Abstract The efficiency of image search can be greatly improved by using a coarse-to-fine search strategy with a multi-resolution image representation. However, if the resolution is so low that the objects have few distinguishing features, search becomes difficult. We show that the performance of search at such low resolutions can be improved by using context information, i.e., objects visible at low-resolution which are not the objects of interest but are associated with them. The networks can be given explicit context information as inputs, or they can learn to detect the context objects, in which case the user does not have to be aware of their existence. We also use Integrated Feature Pyramids, which represent high-frequency information at low resolutions. The use of multiresolution search techniques allows us to combine information about the appearance of the objects on many scales in an efficient way. A natural fOlm of exemplar selection also arises from these techniques. We illustrate these ideas by training hierarchical systems of neural networks to find clusters of buildings in aerial photographs of farmland. 1 INTRODUCTION Coarse-to-fine image search is a general technique for improving the efficiency of any search algorithm. (Burt, 1988a and b; Burt et al., 1989). One begins by searching a lowresolution (coarse-scale) version of the image, typically obtained by constructing an image 982 Clay D. Spence, John C. Pearson, Jim Bergen pyramid (Burt and Adelson, 1983). Since this version is smaller than the original, there are far fewer pixels to be processed and the search is correspondingly faster. To improve the certainty of detection and refine the location estimates, the search process is repeated at higher resolution (finer scale), but only in those regions-of-interest (ROls) which were identified at lower resolution as likely to contain one of the objects to be found, thus greatly reducing the actual area to be searched. This process is repeated at successively higher resolutions until sufficient certainty and precision is achieved or the original image has been searched. 1 These pyramid techniques scale well with image size and can quickly process large images, but the relatively simple, hand-tuned pattern recognition components that are typically used limit their accuracy. Neural networks provide a complementary set of techniques, since they can learn complex patterns, but they don't scale well with image size. We have developed a Hybrid PyramidlNeural Network (HPNN) system that combines the two techniques so as to leverage their strengths and compensate for their weaknesses. A novel benefit of combining pyramids and neural networks is the potential for automatically learning to use context information, by which we mean any visible characteristics of the object's surroundings which tend to distinguish it from other objects. Examples of context information which might be useful in searching aerial photographs for man-made objects are the proximity of roads and terrain type. Pyramids provide the ability to detect context in a low-resolution image representation, which is necessary for efficient processing of large regions of the image. Neural networks provide the ability to discover context of which we may be unaware, and to learn the relevance of the context to detection, the dependence of detection probability on distance from a context object, etc. Context can help to narrow the search region at low resolutions and improve the detection performance since it is, by definition, relevant information. Of course, the usefulness of the idea will be different for each problem. Context can be explicitly provided, e.g., in the form of a road map. As mentioned above, a neural net may also learn to exploit some context which is not explicitly provided, but must be inferred from the image data. If there is such context and the network's architecture and input features are capable of exploiting it, it should learn to do so even if we use ordinary training methods that do not explicitly take context into account. We currently implement these functions in a hierarchical sequence of ordinary feed-forward networks with sigmoidal units, one network at each resolution in an image pyramid (Fig. 1). Each network receives some features extracted from a window in the image, and its output is interpreted as the probability of finding one of the searched-for objects in the center of the window. The network is scanned over all ROIs in this manner. (At the lowest resolution there is one ROI, which is the entire image.) The networks are trained one at a time, starting at the lowest resolution. Context is made available to networks at higher resolutions by giving them information from the lower-resolution networks as additional inputs, which we will call context inputs. These could be I Several authors have investigated multi-scale processing of one-dimensional signals with neural networks, e.g., Mozer (1994) and Burr and Miyata (1993) studied music composition. Burr and Miyata use sub-sampling as in a pyramid decomposition. Images differ somewhat from music in that there are primitive features at all scales (limited by the sampling rate and image size), whereas the average frequency spectrum of music over a long time span doesn't seem likely to be meaningful. The original paper on pyramid image representation is (Burt and Adelson, 1983). Coarse-To-Fine Image Search Using Neural Networks High-Resolution Features Target Probability Context Information Control of Search Region Features Figure 1. A Hierarchical Search System that Exploits Context. 983 taken from either the hidden units or output units of the lower-resolution networks. By training the networks in sequence starting at low resolution, we can choose to train the higher-resolution networks in the ROls selected by the lower-resolution networks, thus providing a simple means of exemplar selection. This coarse-to-fine training is often useful, since many problems have relatively few positive examples per image, but because of the size of the image there are an enormous number of more-or-Iess redundant negative examples. 2 ANEXAMPLEPROBLEM To demonstrate these ideas we applied them to the problem of finding clusters of buildings in aerial photographs of a rural area. Since buildings are almost always near a road, roads are context objects associated with buildings. We approached the problem in two ways to demonstrate different capabilities. First, we trained systems of networks with explicitl~ provided context in the form of road maps, in order to demonstrate the possible benefits. Second, we trained systems without explicitly-provided context, in order to show that the context could be learned. 2.1 EXPLICIT CONTEXT For comparison purposes, we trained one network to search a high-resolution image directly, with no explicit context inputs and no inputs from other networks. To demonstrate the effect of learned context on search we trained a second system with no explicit 2 It would be surprising if this extra infonnation didn't help, since we know it is relevant. For many applications digitized maps will be available, so demonstrating the possible perfonnance benefit is still worth-while. 984 Clay D. Spence, John C. Pearson, Jim Bergen context provided to the nets, but each received inputs from all of the networks at lower resolutions. These inputs were simply the outputs of those lower-resolution networks. To demonstrate the benefit of explicitly provided context, we trained a third system with both context inputs from lower-resolution networks and explicit context in the form of a road map of the area, which was registered with the image. 2.1.1 Features To preserve some distinguishing features at a given low resolution, we extracted simple features at various resolutions and represented them at the lower resolution. The low-resolution representations were constructed by reducing the feature images with the usual blur-and-sub-sample procedure used to construct a Gaussian pyramid (Burt and Adelson, 1983). The features should not be too computationally expensive to extract, otherwise the efficiency benefit of coarse-to-fine search would be canceled. The features used as inputs to the neural nets in the building-search systems were simple measures of the spatial image energy in several frequency bands. We constructed these feature images by building the Laplacian pyramid of the image3 and then taking the absolute values of the pixels in each image in the pyramid. We then constructed a Gaussian pyramid of each of these images, to provide versions of them at different resolutions. A neural net searching a given resolution received input from each energy image derived from the same resolution and all higher resolutions.4 Binary road-map images were constructed from the digitized aerial photographs. They were reduced in resolution by first perfonning a binary blur of the image and then subsampling it by two in each dimension. In the binary blur procedure each pixel is set to one if any of its nearest neighbors were road pixels before blurring. This is repeated to get road maps at each resolution. To give the networks a rough measure of distance to a road, we gave the nets inputs from linearly-blurred versions of the road maps, which were made by expanding even lower-resolution versions of the road map with the same linear expand operation used in constructing the Laplacian pyramid. These blurred road maps are therefore not binary. The networks at the fifth and third pyramid levels received inputs from the road-maps at their resolution and from the two lower resolutions, while the network at the first pyramid level received input from the road map at its resolution and the next lower resolution. 2.1.2 Training the Networks To estimate the probability of finding a building cluster at or around a given location in the image, each network received a single pixel from the same location in all of its input images. This should be adequate for search, since a single pixel in a feature image contains information about an extended region of the original. With the features we used, it also makes the system invariant to rotations, so the networks do not have to learn this invariance Also for simplicity, we did not train nets at all resolutions, but only at the fifth, third and first pyramid levels, i.e., on images which were one-thirty-second, one-eighth, 3 The Laplacian pyramid is usually constructed by expanding the lower-resolution levels of a Gaussian pyramid and subtracting each from the next-higher-resolution level. This gives a set of images which are band-passed with one-octave spacing between the bands. 4 There are many examples of more sophisticated features, e.g., Lane, et aI., 1992, Ballard and Wixson, 1993, and Greenspan, et aI., 1994. Simple features were adequate for demonstrating the ideas of this paper. Coarse-To-Fine Image Search Using Neural Networks No Road Maps or context Context 100 ~ 95 .~ ..... . iii o a.. 90 Q) :::l ... t~ 85 , I I r , , 985 r ,'- -1 - -:' ,- . ~ 80~~~~~~~~~~~~~~~~ a 5 10 15 20 % Folse Positives Figure 2. ROC Curves for the Three Network Search Systems. and one-half the width and height of the original image. A typical building cluster has a linear extent of about 30 pixels in the original images, so in the fifth Gaussian pyramid level they have few distinguishing features. The networks were the usual feed-forward nets with sum-and-sigmoid units and one hidden layer. They were trained using the crossentropy error measure, with a desired output of one in a hand-chosen polygonal region about each building cluster. The standard quadratic weight-decay term was added to this to get the objective function, and only one regularization constant was used. This was adjusted to give lowest error on a test set. We often found that the weights to two or more hidden units would become identical and/or very small as the regularization term was increased, and in these cases we pruned the extraneous units and began the search for the optimal regularization parameter again. We usually ended up with very small networks, with from two to five hidden units. Mter training a net at low-resolution, we expanded the image of that net's output in order to use it as a context input for the networks at higher resolution. 2.1.3 Performance To compare the performance of the three systems, we chose a threshold for the networks' outputs, and considered a building cluster to be detected by a network if the network's output was above the threshold at any pixel within the cluster. The number of detected clusters divided by the total number is the true-positive rate. The false-positive rate is more difficult to define, since we had in mind an application in which the detection system would draw a user's attention to objects which are likely to be of interest. The procedure we used attempts to measure the amount of effort a human would expend in searching regions with false detections by the network. See (Spence, et al., 1994) for details. The performance figures presented here were measured on a validation set, i.e., an image on which the network was not trained and which was not used to set the regularization parameter. The results presented in Tables 1 and 2 are for a single threshold, chosen for each network so that the true-positive rate was 90%. Figure 2 compares the ROC curves of the three systems, i.e. the parametric curves of the true and false-positive rates as the threshold on the network's output is varied. From Table 1, the features we used would seem adequate for search at very low resolution, although the performance could be better. 986 Clay D. Spence, lo/m C. Pearson, Jim Bergen Table 1: False-Positive Rates vs. Resolution. These are results for the system with both road-map and context inputs. PYRAMID LEVEL 5 3 1 FALSE-POSITIVE RATE 16% 4.6% 3.6% Table 2: False-Positive Rates of the Search Systems at 90% True-Positive Rate. NETWORK SYSTEM No Context or Road-Map Inputs Context Inputs, No Road Maps Context Inputs and Road Maps FALSE-POSITIVE RATE 7.6% 5.8% 3.6% Table 2 and Figure 2 clearly show the benefits of using the road map and context inputs, although the statistics are somewhat poor at the higher true-positive rates because of the small number of building clusters which are being missed 2.2 LEARNING CONTEXT Two things were changed to demonstrate that the context could be learned. First, the unoriented spatial energy features are not well suited for distinguishing between roads and other objects with a size similar to a road's width, so we used oriented energies instead. These were the oriented energies derived using steerable filters (Freeman and Adelson, 1991) at four orientations. To force orientation imariance, we sorted the four oriented energies at each pixel for each frequency band. These sorted oriented energy images were then reduced in size as appropriate for the resolution being searched by a network. In this case, we extracted energies only from the first, third, and fifth pyramid levels. The second change is the use of hidden unit outputs for context inputs, instead of the output unit's outputs. The output units estimate the probability of finding a building cluster. Although this may reflect information about roads, it is very indirect information about the roads. It is more likely to carry some information about the coarse-scale appearance of the potential building clusters. The hidden unit outputs should contain a richer description of the image at a coarser scale. 2.2.1 Performance The networks were trained in the same way as the networks described in Section 2.1. We trained three networks to search levels five, three, and one. For comparison purposes, we also trained a single-network to search in level one, with the same input features that were used for all of the networks of the hierarchical search system. These include, for example, three versions of each of the oriented energy images from level one (the second highest frequency band). Two of these versions were reduced in size to levels three and five, and then re-expanded to level one, so that they are simply blurred versions of the original energy images. This gives the network a direct source of information on the coarse-scale Coarse-To-Fine Image Search Using Neural Networks Non-hierarchical Hierarchical 100 ID 95 .~ "iii o a. 90 .-- ••••••••• --.. CIl ~ .= ~ 85 80~~~~~~ __ ~~~ a 5 10 15 20 % False Positives Figure 3. Performance of Hierarchical and Non-hierarchical Algorithms that Learn Context. 987 Figure 4. Image and Context Inputs to Highest-Resolution Network. appearance of the image. although it is an expensive representation in terms of computation and memory. and the network has more inputs so that it takes longer to train. The ROC curves for the two detection algorithms are shown in Figure 3. Their performance is about the same, suggesting that the coarser-scale networks in the hierarchical system are not performing computations that are useful for the finest-scale network, rather they simply pass on information about the coarse-scale appearance. In this example. the advantage of the hierarchical system is in the efficiency of both the training of the algorithm and its use after training. Figure 4 shows the level-one image and the outputs of several hidden units from the level-three network, expanded to the same size. These outputs suggest that road information is being extracted at coarse scale and passed to the high-resolution network. 3 DISCUSSION We have performed "proof-of-concept" simulations on a realistic search problem of two of the key concepts of our Hybrid Pyramid/Neural Network (HPNN) approach. We have shown that: 1. The HPNN can learn to utilize explicitly-provided context data to improve object detection; 2. the networks can learn to use context without being explicitly taught to do so. This second point implies that the person training the system does not need to know of the existence of the context objects. The networks at the different scales are trained in sequence. starting at low resolution. A benefit of this coarse-to-fine training is a simple form of exemplar selection. since we can choose to train a network only in the regions of 988 Clay D. Spence. John C. Pearson. Jim Bergen interest as detected by the already-trained networks at lower resolution. We could also train all of the networks simultaneously, so that the lower-resolution networks learn to extract the most helpful information from the low-resolution image. The resulting system would probably perform better, but it would also be more expensive to train. This approach should work quite well for many automatic target recognition problems, since the targets are frequently quite small. For more extended objects like clusters of buildings, our method is an efficient way of examining the objects on several length scales, but the detection at the finest scale is based on a small part of the potential object's appearance. Extended images of real objects typically have many features at each of several resolutions. We are currently working on techniques for discovering several characteristic features at each resolution by training several networks at each resolution, and integrating their responses to compute an overall probability of detection. Acknowledgments We would like to thank Peter Burt, P. Anandan, and Paul Sajda for many helpful discussions. This was work was funded by the National Information Display Laboratory and under ARPA contract No. NOOOI4-93-C-0202. References D.H. Ballard and L.E. Wixson (1993) Object recognition using steerable filters at multiple scales, Proceedings of the IEEE Workshop on Qualitative Vision, New York, NY. D. Burr and Y. Miyata (1993) Hierarchical recurrent networks for learning musical structure, Proceedings of the IEEE Conference on Neural Networks for Signal Processing Ill, C. Kamm, G. Kuhn, B. Yoon, S.Y. Kung, and R. Chellappa, eds., Piscataway, NJ, pp. 216-225. P. J. Burt and E. H. Adelson. (1983) The Laplacian pyramid as a compact image code. IEEE Transactions, Vol. COM-31:4, April, pp. 532-540. P.J. Burt (1988a) Smart sensing with a pyramid vision machine, Proceedings of the IEEE Vol. 76, pp. 1006-1015. PJ. Burt (1988b) Attention mechanisms for vision in a dynamic world, Proceedings of the 9th International Conference on Pattern Recognition, pp. 977-987. PJ. Burt, J.R. Bergen, R. Kolczynski, R. Hingorani, W.A. Lee, A. Leung, J. Lubin, and H. Shvayster (1989) Object tracking with a moving camera, Proceedings of the IEEE Workshop on Motion, Irvine. W.T. Freeman and E.H. Adelson (1991) The design and use of steerable filters, IEEE Trans. PAMI, 12:9, pp. 891-906. S.H. Lane, J.C. Pearson, and R. Sverdlove (1992) Neural networks for classifying image textures, Proceedings of the Government Applications of Neural Networks Conference, Dayton, Ohio. M.e. Mozer (1994) Neural network music composition by prediction: Exploring the benefits of psychoacoustic constraints and multiscale processing, to appear in Connection Science. PART IX APPLICATIONS	1994	139
902	Predicting the Risk of Complications in Coronary Artery Bypass Operations using Neural Networks Richard P. Lippmann, Linda Kukolich MIT Lincoln Laboratory 244 Wood Street Lexington, MA 02173-0073 Abstract Dr. David Shahian Lahey Clinic Burlington, MA 01805 Experiments demonstrated that sigmoid multilayer perceptron (MLP) networks provide slightly better risk prediction than conventional logistic regression when used to predict the risk of death, stroke, and renal failure on 1257 patients who underwent coronary artery bypass operations at the Lahey Clinic. MLP networks with no hidden layer and networks with one hidden layer were trained using stochastic gradient descent with early stopping. MLP networks and logistic regression used the same input features and were evaluated using bootstrap sampling with 50 replications. ROC areas for predicting mortality using preoperative input features were 70.5% for logistic regression and 76.0% for MLP networks. Regularization provided by early stopping was an important component of improved perfonnance. A simplified approach to generating confidence intervals for MLP risk predictions using an auxiliary "confidence MLP" was developed. The confidence MLP is trained to reproduce confidence intervals that were generated during training using the outputs of 50 MLP networks trained with different bootstrap samples. 1 INTRODUCTION In 1992 there were roughly 300,000 coronary artery bypass operations perfonned in the United States at a cost of roughly $44,000 per operation. The $13.2 billion total cost of these operations is a significant fraction of health care spending in the United States. This has led to recent interest in comparing the quality of cardiac surgery across hospitals using risk-adjusted procedures and large patient populations. It has also led to interest in better assessing risks for individual patients and in obtaining improved understanding of the patient and procedural characteristics that affect cardiac surgery outcomes. 1056 Richard P. Lippmann, Linda Kukolich, David Shahian INPUT FE ATURES ~ ... SELECT I-CLASSIFY ... FEATURES ~ AND I-.. REPLACE ... MISSING ~ FEATURES r-CONFIDENCE NETWORK r-Figure 1. Block diagram of a medical risk predictor. ... ... ~ RISK PROBABILITY CONFIDENCE INTERVAL This paper describes a experiments that explore the use of neural networks to predict the risk of complications in coronary artery bypass graft (CABO) surgery. Previous approaches to risk prediction for bypass surgery used linear or logistic regression or a Bayesian approach which assumes input features used for risk prediction are independent (e.g. Edwards, 1994; Marshall, 1994; Higgins, 1992; O'Conner, 1992). Neural networks have the potential advantages of modeling complex interactions among input features, of allowing both categorical and continuous input features, and of allowing more flexibility in fitting the expected risk than a simple linear or logistic function. 2 RISK PREDICTION AND FEATURE SELECTION A block diagram of the medical risk prediction system used in these experiments is shown in Figure 1. Input features from a patient's medical record are provided as 105 raw inputs, a smaller subset of these features is selected, missing features in this subset are replaced with their most likely values from training data, and a reduced input feature vector is fed to a classifier and to a "confidence network". The classifier provides outputs that estimate the probability or risk of one type of complication. The confidence network provides upper and lower bounds on these risk estimates. Both logistic regression and multilayer sigmoid neural network (MLP) classifiers were evaluated in this study. Logistic regression is the most common approach to risk prediction. It is structurally equivalent to a feed-forward network with linear inputs and one output unit with a sigmoidal nonlinearity. Weights and offsets are estimated using a maximum likelihood criterion and iterative "batch" training. The reference logistic regression classifier used in these experiments was implemented with the SPlus glm function (Mathsoft, 1993) which uses iteratively reweighted least squares for training and no extra regularization such as weight decay. Multilayer feed-forward neural networks with no hidden nodes (denoted single-layer MLPs) and with one hidden layer and from 1 to 10 hidden nodes were also evaluated as implemented using LNKnet pattern classification software (Lippmann, 1993). An MLP committee classifier containing eight members trained using different initial random weights was also evaluated. All classifiers were evaluated using a data base of 1257 patients who underwent coronary artery bypass surgery from 1990 to 1994. Classifiers were used to predict mortality, postoperative strokes, and renal failure. Predictions were made after a patient's medical history was obtained (History), after pre-surgical tests had been performed (Post-test), immediately before the operation (preop), and immediately after the operation (Postop). Bootstrap sampling (Efron, 1993) was used to assess risk prediction accuracy because there were so few Predicting the Risk of Complications in Coronary Artery Bypass Operations 1057 patients with complications in this data base. The number of patients with complications was 33 or 2.6% for mortality, 25 or 2.0% for stroke, and 21 or 1.7% for renal failure. All experiments were performed using 50 bootstrap training sets where a risk prediction technique is trained with a bootstrap training set and evaluated using left-out patterns. mSTORY Age COPD (Chronic Obs. Pul. Disease) POST·TEST Pulmonary Ventricular Congestion X-ray Cardiomegaly X-ray Pulmonary Edema PREOP NTG (Nitroglycerin) IABP (Intraaortic Balloon Pump) Urgency Status MI When POSTOP Blood Used (Packed Cells) Perfusion Time NComplications NHigh 27/674 71126 8nl 61105 6/21 211447 111115 10/127 7/64 121113 91184 Figure 2. Features selected to predict mortality. % True Hits 4.0% 5.6% 11.3% 5.7% 26.6% 4.7% 6.6% 7.9% 10.9% 10.6% 4.9% The initial set of 105 raw input features included binary (e.g. MalelFemale), categorical (e.g. MI When: none, old, recent, evolving), and continuous valued features (e.g. Perfusion Time, Age). There were many missing and irrelevant features and all features were only weakly predictive. Small sets of features were selected for each complication using the following procedures: (1) Select those 10 to 40 features experience and previous studies indicate are related to each complication, (2) Omit features if a univariate contingency table analysis shows the feature is not important, (3) Omit features that are missing for more than 5% of patients, (4) Order features by number of true positives, (5) Omit features that are similar to other features keeping the most predictive, and (7) Add features incrementally as a patient's hospital interaction progresses. This resulted in sets of from 3 to 11 features for the three complications. Figure 2 shows the 11 features selected to predict mortality. The first column lists the features, the second column presents a fraction equal to the number of complications when the feature was "high" divided by the number of times this feature was "high" (A threshold was assigned for continuous and categorical features that provided good separation), and the last column is the second column expressed as a percentage. Classifiers were provided identical sets of input features for all experiments. Continuous inputs to all classifiers were normalized to have zero mean and unit variance, categorical inputs ranged from -(D-1)/2 to (D-1)/2 in steps of 1.0, where D is the number of categories, and binary inputs were -0.5 or 0.5. 3 PERFORMANCE COMPARISONS Risk prediction was evaluated by plotting and computing the area under receiver operating characteristic (ROC) curves and also by using chi-square tests to determine how accurately classifiers could stratify subjects into three risk categories. Automated experiments were performed using bootstrap sampling to explore the effect of varying the training step size J058 100 ~ 80 :~ ~ (/) 60 c Q) en ~ 40 I ?fl. 20 o Richard P. Lippmann. Linda Kukolich, David Shahian " HISTORY (68.6%) o 20 40 60 80 100 0 20 40 60 80 100 % FALSE ALARMS (100 - Specificity) Figure 3. Fifty preoperative bootstrap ROCs predicting mortality using an MLP classifier with two hidden nodes and the average ROC (left), and average ROCS for mortality using history, preoperative, and postoperative features (right). from 0.005 to 0.1; of using squared-error, cross-entropy, and maximum likelihood cost functions; of varying the number of hidden nodes from 1 to 8; and of stopping training after from 5 to 40 epochs. ROC areas varied little as parameters were varied. Risk stratification, which measures how well classifier outputs approximate posterior probabilities, improved substantially with a cross-entropy cost function (instead of squared error), with a smaller stepsize (0.01 instead of 0.05 or 0.1) and with more training epochs (20 versus 5 or 10). An MLP classifier with two hidden nodes provided good overall performance across complications and patient stages with a cross-entropy cost function, a stepsize of 0.01, momentum of 0.6, and stochastic gradient descent stopping after 20 epochs. A single-layer MLP provided good performance with similar settings, but stopping after 5 epochs. These settings were used for all experiments. The left side of Figure 3 shows the 50 bootstrap ROCs created using these settings for a two-hidden-node MLP when predicting mortality with preoperative features and the ROC created by averaging these curves. There is a large variability in these ROes due to the small amount of training data. The ROC area varies from 67% to 85% (cr=4.7) and the sensitivity with 20% false alarms varies from 30% to 79%. Similar variability occurs for other complications. The right side of Figure 3 shows average ROCs for mortality created using this MLP with history, preoperative, and postoperative features. As can be seen, the ROC area and prediction accuracy increases from 68.6% to 79.2% as more input features become available. Figure 4 shows ROC areas across all complications and patient stages. Only three and two patient stages are shown for stroke and renal failure because no extra features were added at the missing stages for these complications. ROC areas are low for all complications and range from 62% to 80%. ROC areas are highest using postoperative features, lowest using only history features, and increase as more features are added. ROC areas are highest for mortality (68 to 80%) and lower for stroke (62 to 71 %) and renal failure (62 to 67% ).The MLP classifier with two hidden nodes (MLP) always provided slightly higher ROC areas than logistic regression. The average increase with the MLP classifier was 2.7 percentage Predicting the Risk of Complications in Coronary Artery Bypass Operations 1059 100 90 MORTALITY III Logistic -80 C [3 Single·Layer MLP ~ 70 • MLP ~ 60 • MLP·Commillee 8 a: 50 40 30 HISTORY POSTTEST PRE·OP POST·OP PATIENT STAGE 100 100 90 STROKE 90 RENAL FAILURE 80 80 I=2a C I=2a « 70 70 w ~ 60 60 8 a: 50 50 40 40 30 30 HISTORY POSTTEST POST·OP HISTORY POST·OP PATIENT STAGE PATIENT STAGE Figure 4. ROC areas across all complications and patient stages for logistic regression, single-layer MLP classifier, two-layer MLP classifier with two hidden nodes, and a committee classifier containing eight two-layer MLP classifiers trained using different random starting weights. points (the increase ranged from 0.3 to 5.5 points). The single-layer MLPclassifier also provided good performance. The average ROC area with the single-layer MLP was only 0.6 percentage points below that of the MLP with two hidden nodes. The committee using eight two-layer MLP classifiers performed no better than an individual two-layer MLP classifier. Classifier outputs were used to bin or stratify each patient into one of four risk levels (05%, 5-10%, and 10-100%) by treating the output as an estimate of the complication posterior probability. Figure 5 shows the accuracy of risk stratification for the MLP classifier for all complications. Each curve was obtained by averaging 50 individual curves obtained using bootstrap sampling as with the ROC curves. Individual curves were obtained by placing each patient into one of the three risk bins based on the MLP output. The x's represent the average MLP output for all patients in each bin. Open squares are the true percentage of patients in each bin who experienced a complication. The bars represent ±2 binomial deviations about the true patient percentages. Risk prediction is accurate if the x's are close to the squares and within the confidence intervals. As can be seen, risk prediction is accurate and close to the actual number of patients who experienced complications. It is difficult, however, to assess risk prediction given the limited numbers of patients in the two highest bins. For example, in Figure 5, the median number of patients with complications was only 2 out of 20 in the middle bin and 2 out of 13 in the upper bin. Good and similar risk stratification, as measured by a chi-square test, was provided by all classifiers. Differences between classifier predictions and true patient percentages were small and not statistically significant. 1060 Richard P. Lippmann, Linda Kukolich, David Shahian ~ ~------------------------------~ 30 10 MORTALITY 0- PATIENT COUNT X - MLP OUTPUT o ____ .M __________________________ __ 40 ~ ~--------------------------r_--_, RENAL FAILURE 30 20 10 o ~ __ _w~ ________ ~ __________ ~ __ ~ 0-5 5·10 10-100 BIN PROBABILITY RANGE ('¥o) Figure 5. Accuracy of MLP risk stratification for three complications using preoperative features. Open squares are true percentages of patients in each bin with a complication, x's are MLP predictions, bars represent ±2 binomial standard deviation confidence intervals. 4 CONFIDENCE MLP NETWORKS Estimating the confidence in the classification decision produced by a neural network is a critical issue that has received relatively little study. Not being able to provide a confidence measure makes it difficult for physicians and other professionals to accept the use of complex networks. Bootstrap sampling (Efron, 1993) was selected as an approach to generate confidence intervals for medical risk prediction because 1) It can be applied to any type of classifier, 2) It measures variability due to training algorithms, implementation differences, and limited training data, and 3) It is simple to implement and apply. As shown in the top half of Figure 6, 50 bootstrap sets of training data are created from the original training data by resampling with replacement. These bootstrap training sets are used to train 50 bootstrap MLP classifiers using the same architecture and training procedures that were selected for the risk prediction MLP. When a pattern is fed into these classifiers, their outputs provide an estimate of the distribution of the output of the risk prediction MLP. Lower and upper confidence bounds for any input are obtained by sorting these outputs and selecting the 10% and 90% cumulative levels. It is computationally expensive to have to maintain and query 50 bootstrap MLPs whenever confidence bounds are desired. A simpler approach is to train a single confidence MLP to replicate the confidence bounds predicted by the 50 bootstrap MLPs, as shown in the botPredicting the Risk of Complications in Coronary Artery Bypass Operations / INPUT PATTERN RISK PREDICTION MLP OUTPUT STATISTICS CONFIDENCE MLP UPPER LIMIT LOWER LIMIT Figure 6. A confidence MLP trained using 50 bootstrap MLPs produces upper and lower confidence bounds for a risk prediction MLP. 1061 tom half of Figure 6. The the confidence MLP is fed the input pattern and the output of the risk prediction MLP and produces at its output the confidence intervals that would have been produced by 50 bootstrap MLPs. The confidence MLP is a mapping or regression network that replaces the 50 bootstrap networks. It was found that confidence networks with one hidden layer, two hidden nodes, and a linear output could accurately reproduce the upper and lower confidence intervals created by 50 bootstrap two-layer MLP networks. The confidence network outputs were almost always within ±15% of the actual bootstrap bounds. Upper and lower bounds produced by these confidence networks for all patients using preoperative features predicting mortality are show in Figure 7. Bounds are high (± 1 0 percentage points) when the complication risk is near 20% and drop to lower values (±0.4 percentage points) when the risk is near 1 %. This relatively simple approach makes it possible to create and replicate confidence intervals for many types of classifiers. 5 SUMMARY AND FUTURE PLANS MLP networks provided slightly better risk prediction than conventional logistic regression when used to predict the risk of death, stroke, and renal failure on 1257 patients who underwent coronary artery bypass operations. Bootstrap sampling was required to compare approaches and regularization provided by early stopping was an important component of improved performance. A simplified approach to generating confidence intervals for MLP risk predictions using an auxiliary "confidence MLP" was also developed. The confidence MLP is trained to reproduce the confidence bounds that were generated during training by 50 MLP networks trained using bootstrap samples. Current research is validating these results using larger data sets, exploring approaches to detect outlier patients who are so different from any training patient that accurate risk prediction is suspect, developing approaches to explaining which input features are important for an individual patient, and determining why MLP networks provide improved performance. 1062 Richard P. Lippmann, Linda Kukolich, David Shahian ~r------.------.-------r------' 30 '# 025 I-5 :J20 W (,) Z 15 W Q u:: 10 Z o (,) 5 UPPER ... T TT ... J;.... T T COMPLICATION RISK% T T .. Figure 7. Upper and lower confidence bounds for all patients and preoperative mortality risk predictions calculated using two MLP confidence networks. ACKNOWLEDGMENT This work was sponsored by the Department of the Air Force. The views expressed are those of the authors and do not reflect the official policy or position of the U.S. Government. We wish to thank Stephanie Moisakis and Anne Nilson at the Lahey Clinic and Yuchun Lee at Lincoln Laboratory for assistance in organizing and preprocessing the data. BIBLIOGRAPHY F. Edwards, R. Clark, and M. Schwartz. (1994) Coronary Artery Bypass Grafting: The Society of Thoracic Surgeons National Database Experience. In Annals Thoracic Surgery, Vol. 57, 12-19. Bradley Efron and Robert J. Tibshirani. (1993) An Introduction to the Bootstrap. Monographs on Statistics and Applied Probability 57, New York: Chapman and Hall (1993). T. Higgins, F. Estafanous, et. al. (1992) Stratification of Morbidity and Mortality Outcome by Preoperative Risk Factors in Coronary Artery Bypass Patients. In Journal of the American Medical Society, Vol. 267, No. 17,2344-2348. R. Lippmann, L. Kukolich, and E. Singer. (1993) LNKnet: Neural Network, Machine Learning, and Statistical Software for Pattern Classification. In Lincoln Laboratory Journal, Vol. 6, No.2, 249-268. Marshall Guillenno, Laurie W. Shroyer, et al. (1994) Bayesian-Logit Model for Risk Assessment in Coronary Artery Bypass Grafting, In Annals Thoracic Surgery, Vol. 57, 1492-5000. G. O'Conner, S. Plume, et. al. (1992) Multivariate Prediction of In-Hospital Mortality Associated with Coronary Artery Bypass Surgery. In Circulation, Vol. 85, No.6, 21102118. Statistical Sciences. (1993) S-PLUS Guide to Statistical and Mathematical Analyses, Version 3.2, Seattle: StatSci, a division of MathSoft, Inc.	1994	14
903	ICEG Morphology Classification using an Analogue VLSI Neural Network Richard Coggins, Marwan Jabri, Barry Flower and Stephen Pickard Systems Engineering and Design Automation Laboratory Department of Electrical Engineering J03, University of Sydney, 2006, Australia. Email: richardc@sedal.su.oz.au Abstract An analogue VLSI neural network has been designed and tested to perform cardiac morphology classification tasks. Analogue techniques were chosen to meet the strict power and area requirements of an Implantable Cardioverter Defibrillator (ICD) system. The robustness of the neural network architecture reduces the impact of noise, drift and offsets inherent in analogue approaches. The network is a 10:6:3 multi-layer percept ron with on chip digital weight storage, a bucket brigade input to feed the Intracardiac Electrogram (ICEG) to the network and has a winner take all circuit at the output. The network was trained in loop and included a commercial ICD in the signal processing path. The system has successfully distinguished arrhythmia for different patients with better than 90% true positive and true negative detections for dangerous rhythms which cannot be detected by present ICDs. The chip was implemented in 1.2um CMOS and consumes less than 200n W maximum average power in an area of 2.2 x 2.2mm2. 1 INTRODUCTION To the present time, most ICDs have used timing information from ventricular leads only to classify rhythms which has meant some dangerous rhythms can not be distinguished from safe ones, limiting the use of the device. Even two lead 732 Richard Coggins, Marwan Jabri, Barry Flower, Stephen Pickard 4.00 HO 3.00 2.00 I.SO ___ :::::::! 1.00 Q O.SO Figure 1: The Morphology of ST and VT retrograde 1:1. atrial/ventricular systems fail to distinguish some rhythms when timing information alone is used [Leong and Jabri, 1992]. A case in point is the separation of Sinus Tachycardia (ST) from Ventricular Tachycardia with 1:1 retrograde conduction. ST is a safe arrhythmia which may occur during vigorous exercise and is characterised by a heart rate of approximately 120 beats/minute. VT retrograde 1:1 also occurs at the same low rate but can be a potentially fatal condition. False negative detections can cause serious heart muscle injury while false positive detections deplete the batteries, cause patient suffering and may lead to costly transplantation of the device. Figure 1 shows however, the way in which the morphology changes on the ventricular lead for these rhythms. Note, that the morphology change is predominantly in the "QRS complex" where the letters QRS are the conventional labels for the different points in the conduction cycle during which the heart is actually pumping blood. For a number of years, researchers have studied template matching schemes in order to try and detect such morphology changes. However, techniques such as correlation waveform analysis [Lin et. al., 1988], though quite successful are too computationally intensive to meet power requirements. In this paper, we demonstrate that an analogue VLSI neural network can detect such morphology changes while still meeting the strict power and area requirements of an implantable system. The advantages of an analogue approach are born out when one considers that an energy efficient analogue to digital converter such as [Kusumoto et. al., 1993] uses 1.5nJ per conversion implying 375n W power consumption for analogue to digital conversion of the ICEG alone. Hence, the integration of a bucket brigade device and analogue neural network provides a very efficient way of interfacing to the analogue domain. Further, since the network is trained in loop with the ICD in real time, the effects of device offsets, noise, QRS detection jitter and signal distortion in the analogue circuits are largely alleviated. The next section discusses the chip circuit designs. Section 3 describes the method ICEG Morphology Classification Using an Analogue VLSI Neural Network "Column AoIcIr.AowAcId ... 1axl Syna .... AIRy I o.ta Reglsl... I I WTAI Bu1I ... IClkcMmux 10 DOD DO Figure 2: Floor Plan and Photomicrograph of the chip 733 used to train the network for the morphology classification task. Section 4 describes the classifier performance on seven patients with arrhythmia which can not be distinguished using the heart rate only. Section 5 summarises the results, remaining problems and future directions for the work. 2 ARCHITECTURE The neural network chip consists of a 10:6:3 multilayer perceptron, an input bucket brigade device (BBD) and a winner take all (WTA) circuit at the output. A floor plan and photomicrograph of the chip appears in figure 2. The BBD samples the incoming ICEG at a rate of 250Hz. For three class problems, the winner take all circuit converts the winning class to a digital signal. For the two class problem considered in this paper, a simple thresholding function suffices. The following subsections briefly describe the functional elements of the chip. The circuit diagrams for the chip building blocks appear in figure 3. 2.1 BUCKET BRIGADE DEVICE One stage of the bucket brigade circuit is shown in figure 3. The BBD uses a two phase clock to shift charge from cell to cell and is based on a design by Leong [Leong, 1992]. The BBD operates by transferring charge deficits from S to D in each of the cells. PHIl and PHI2 are two phase non-overlapping clocks. The cell is buffered from the synapse array to maintain high charge transfer efficiency. A sample and hold facility is provided to store the input on the gates of the synapses. The BBD clocks are generated off chip and are controlled by the QRS complex detector in the lCD. 2.2 SYNAPSE This synapse has been used on a number of neural network chips previously. e.g. [Coggins et. al., 1994]. The synapse has five bits plus sign weight storage which 734 Richard Coggins, Marwan Jabri, Barry Flower, Stephen Pickard NEURON .---------------------------------------- --------------------, , , , , ~ ! BUJIOIII' 00 " BUCKET BRIGADE ClLL Figure 3: Neuron, Bucket Brigade and Synapse Circuit Diagrams. sets the bias to a differential pair which performs the multiplication. The bias references for the weights are derived from a weighted current source in the corner of the chip. A four quadrant multiplication is achieved by the four switches at the top of the differential pair. 2.3 NEURON Due to the low power requirements, the bias currents of the synapse arrays are of the order of hundreds of nano amps, hence the neurons must provide an effective resistance of many mega ohms to feed the next synapse layer while also providing gain control. Without special high resistance polysilicon, simple resistive neurons use prohibitive area, However, for larger networks with fan-in much greater than ten, an additional problem of common mode cancellation is encountered, That is, as the fan-in increases, a larger common mode range is required or a cancellation scheme using common mode feedback is needed. The neuron of figure 3 implements such a cancellation scheme, The mirrors MO/M2 and Ml/M3 divide the input current and facilitate the sum at the drain of M7. M7/M8 mirrors the sum so that it may be split into two equal currents by the mirrors formed by M4, M5 and M6 which are then subtracted from the input currents. Thus, the differential voltage vp - Vm is a function of the transistor transconductances, the common mode input current and the feedback factor, The gain of the neuron can be controlled by varying the width to length ratio of the mirror transistors MO and Ml. The implementation in this case allows seven gain combinations, using a three bit RAM cell to store the gain, ICEG Morphology Classification Using an Analogue VLSI Neural Network RunnngMUME Implantable C.cio¥erlor DefibrillalOr Ne ..... 1 Nelwa'1< Chip Figure 4: Block Diagram of the Training and Testing System. 735 The importance of a common mode cancellation scheme for large networks can be seen when compared to the straight forward approach of resistive or switched capacitor neurons. This may be illustrated by considering the energy usage of the two approaches. Firstly, we need to define the required gain of the neuron as a function of its fan-in. If we assume that useful inputs to the network are mostly sparse, i.e. with a small fraction of non-zero values, then the gain is largely independent of the fan-in, yet the common mode signal increases linearly with fanin. For the case of a neuron which does not cancel the common mode, the power supply voltage must be increased to accommodate the common mode signal, thus leading to a quadratic increase in energy use with fan-in. A common mode cancelling neuron on the other hand, suffers only a linear increase in energy use with fan-in since extra voltage range is not required and the increased energy use arises only due to the linear increase in common mode current. 3 TRAINING SYSTEM The system used to train and test the neural network is shown in figure 4. Control of training and testing takes place on the PC. The PC uses a PC-LAB card to provide analogue and digital I/O. The PC plays the ICEG signal to the input of the commercial ICD in real time. Note, that the PC is only required for initially training the network and in this case as a source of the heart signal. The commercial ICD performs the function of QRS complex detection using analogue circuits. The QRS complex detection signal is then used to freeze the BBD clocks of the chip, so that a classification can take place. When training, a number of examples of the arrhythmia to be classified are selected from a single patient data base recorded during an electrophysiological study and previously classified by a cardiologist. Since most of the morphological information is in the QRS complex, only these segments of the data are repeatedly presented to 736 Richard Coggins. Marwan Jabri. Barry Flower. Stephen Pickard Patient % Training Attempts Converged Average Run 1 Run ~ Iterations H=3 H = 6 H=3 H=6 1 80 10 60 60 62 2 80 100 0 10 86 3 0 0 0 10 101 4 60 10 40 40 77 5 100 80 0 60 44 6 100 40 60 60 46 7 80 100 40 100 17 Table 1: Training Performance of the system on seven patients. the network. The weights are adjusted according to the training algorithm running on the PC using the analogue outputs of the network to reduce the output error. The PC writes weights to the chip via the digital I/Os of the PC-LAB card and the serial weight bus of network. The software package implementing the training and testing, called MUME [Jabri et. al., 1992], provides a suite of training algorithms and control options. Online training was used due to its success in training small networks and because the presentation of the QRS complexes to the network was the slowest part of the training procedure. The algorithm used for weight updates in this paper was summed weight node perturbation [Flower and Jabri, 1993]. The system was trained on seven different patients separately all of whom had VT with 1: 1 retrograde conduction. Note, that patient independent training has been tried but with mixed results [Tinker, 1992]. Table 1 summarises the training statistics for the seven patients. For each patient and each architecture, five training runs were performed starting from a different random initial weight set. Each of the patients was trained with eight of each class of arrhythmia. The network architecture used was 10:H:1, where H is the number of hidden layer neurons and the unused neurons being disabled by setting their input weights to zero. Two sets of data were collected denoted Run 1 and Run 2. Run 1 corresponded to output target values of ±0.6V within margin 0.45V and Run 2 to output target values of ±0.2V within margin 0.05V. A training attempt was considered to have converged when the training set was correctly classified within two hundred training iterations. Once the morphologies to be distinguished have been learned for a given patient, the remainder of the patient data base is played back in a continuous stream and the outputs of the classifier at each QRS complex are logged and may be compared to the classifications of a cardiologist. The resulting generalisation performance is discussed in the next section. 4 MORPHOLOGY CLASSIFIER GENERALISATION PERFORMANCE Table 2 summarises the generalisation performance of the system on the seven patients for the training attempts which converged. Most of the patients show a correct classification rate better than 90% for at least one architecture on one of the ICEG Morphology Classification Using an Analogue VLSI Neural Network 737 Patient No. of % Correct Classifications Run 1 Complexes H i3 H = 6 ST VT ST VT ST VT 1 440 61 89±10 89±3 58±0 99±0 2 94 57 99±1 99±1 100±0 99±1 3 67 146 4 166 65 66±44 76±37 99±1 50±3 5 61 96 82±1 75±13 94±6 89±9 6 61 99 84±8 97±1 90±5 99±1 7 28 80 98±5 97±3 99±1 99±1 % Correct Classifications Run 2 1 440 61 88±2 99±1 86±14 99±1 2 94 57 94±6 94±3 3 67 146 84±2 99±1 4 166 65 76±18 59±2 87±7 100±0 5 61 96 88±2 49±5 84±1 82±5 6 61 99 92±6 90±10 99±1 99±1 7 28 80 94±3 99±0 94±3 92±3 Table 2: Generalisation Performance of the system on seven patients. runs, whereas, a timing based classifier can not separate these arrhythmia at all. For each convergent weight set the network classified the test set five times. Thus, the "% Correct" columns denote the mean and standard deviation of the classifier performance with respect to both training and testing variations. By duty cycling the bias to the network and buffers, the chip dissipates less than 200n W power for a nominal heart rate of 120 beats/minute during generalisation. 5 DISCUSSION Referring to table 1 we see that the patient 3 data was relatively difficult to train. However, for the one occasion when training converged generalisation performance was quite acceptable. Inspection of this patients data showed that typically, the morphologies of the two rhythms were very similar. The choice of output targets, margins and architecture appear to be patient dependent and possibly interacting factors. Although larger margins make training easier for some patients they appear to also introduce more variability in generalisation performance. This may be due to the non-linearity of the neuron circuit. Further experiments are required to optimise the architecture for a given patient and to clarify the effect of varying targets, margins and neuron gain. Penalty terms could also be added to the error function to minimise the possibility of missed detections of the dangerous rhythm. The relatively slow rate of the heart results in the best power consumption being obtained by duty cycling the bias currents to the synapses and the buffers. Hence, the bias settling time of the weighted current source is the limiting factor for reducing power consumption further for this design. By modifying the connection of the current source to the synapses using a bypassing technique to reduce transients in 738 Riclulrd Coggins, Marwan Jabri, Barry Flower, Stephen Pickard the weighted currents, still lower power consumption could be achieved. 6 CONCLUSION The successful classification of a difficult cardiac arrhythmia problem has been demonstrated using. an analogue VLSI neural network approach. Furthermore, the chip developed has shown very low power consumption of less than 200n W, meeting the requirements of an implantable system. The chip has performed well, with over 90% classification performance for most patients studied and has proved to be robust when the real world influence of analogue QRS detection jitter is introduced by a commercial implantable cardioverter defibrillator placed in the signal path to the classifier. Acknowledgements The authors acknowledge the funding for the work in this paper provided under Australian Generic Technology Grant Agreement No. 16029 and thank Dr. Phillip Leong of the University of Sydney and Dr. Peter Nickolls of Telectronics Pacing Systems Ltd., Australia for their helpful suggestions and advice. References [Castro et. al., 1993] H.A. Castro, S.M. Tam, M.A. Holler, "Implementation and Performance of an analogue Nonvolatile Neural Network," Analogue Integrated Circuits and Signal Processing, vol. 4(2), pp. 97-113, September 1993. [Lin et. al., 1988] D. Lin, L.A. Dicarlo, and J .M. Jenkins, "Identification of Ventricular Tachycardia using Intracavitary Electrograms: analysis of time and frequency domain patterns," Pacing (3 Clinical Electrophysiology, pp. 1592-1606, November 1988. [Leong, 1992] P.H.W. Leong, Arrhythmia Classification Using Low Power VLSI, PhD Thesis, University of Sydney, Appendix B, 1992. [ Kusumoto et. al., 1993] K. Kusumoto et. al., "A lObit 20Mhz 30mW Pipelined Interpolating ADC," ISSCC, Digest of Technical Papers, pp. 62-63, 1993. [Leong and Jabri, 1992] P.H.W. Leong and M. Jabri, "MATIC - An Intracardiac Tachycardia Classification System", Pacing (3 Clinical Electrophysiology, September 1992. [Coggins et. al., 1994] R.J. Coggins and M.A. Jabri, "WATTLE: A Trainable Gain Analogue VLSI Neural Network", NIPS6, Morgan Kauffmann Publishers, 1994. [Jabri et. al., 1992] M.A. Jabri, E.A. Tinker and L. Leerink, "MUME- A MultiNet-Multi-Architecture Neural Simulation Environment", Neural Network Simulation Environments, Kluwer Academic Publications, January, 1994. [Flower and Jabri, 1993] B. Flower and M. Jabri, "Summed Weight Neuron Perturbation: an O(N) improvement over Weight Perturbation," NIPS5, Morgan Kauffmann Publishers, pp. 212-219, 1993. [Tinker, 1992] E.A. Tinker, "The SPASM Algorithm for Ventricular Lead Timing and Morphology Classification," SEDAL ICEG-RPT-016-92, Department of Electrical Engineering, University of Sydney, 1992.	1994	140
904	Spatial Representations in the Parietal Cortex May Use Basis Functions Alexandre Pouget alex@salk.edu Terrence J. Sejnowski terry@salk.edu Howard Hughes Medical Institute The Salk Institute La Jolla, CA 92037 and Department of Biology University of California, San Diego Abstract The parietal cortex is thought to represent the egocentric positions of objects in particular coordinate systems. We propose an alternative approach to spatial perception of objects in the parietal cortex from the perspective of sensorimotor transformations. The responses of single parietal neurons can be modeled as a gaussian function of retinal position multiplied by a sigmoid function of eye position, which form a set of basis functions. We show here how these basis functions can be used to generate receptive fields in either retinotopic or head-centered coordinates by simple linear transformations. This raises the possibility that the parietal cortex does not attempt to compute the positions of objects in a particular frame of reference but instead computes a general purpose representation of the retinal location and eye position from which any transformation can be synthesized by direct projection. This representation predicts that hemineglect, a neurological syndrome produced by parietal lesions, should not be confined to egocentric coordinates, but should be observed in multiple frames of reference in single patients, a prediction supported by several experiments. 158 Alexandre Pouget, Terrence J. Sejnowski 1 Introduction The temporo-parietal junction in the human cortex and its equivalent in monkeys, the inferior parietal lobule, are thought to playa critical role in spatial perception. Lesions in these regions typically result in a neurological syndrome, called hemineglect, characterized by a lack of motor exploration toward the hemispace contralateral to the site of the lesion. As demonstrated by Zipser and Andersen [11), the responses of single cells in the monkey parietal cortex are also consistent with this presumed role in spatial perception. In the general case, recovering the egocentric position of an object from its multiple sensory inputs is difficult because of the multiple reference frames that must be integrated. In this paper, we consider a simpler situation in which there is only visual input and all body parts are fixed but the eyes, a condition which has been extensively used for neurophysiological studies in monkeys. In this situation, the head-centered position of an object, X, can be readily recovered from the retinal location,R, and current eye position, E, by vector addition: (1) If the parietal cortex contains a representation of the egocentric position of objects, then one would expect to find a representation of the vectors, X, associated with these objects. There is an extensive literature on how to encode a vector with a population of neurons, and we first present two schemes that have been or are used as working hypothesis to study the parietal cortex. The first scheme involves what is typically called a computational map, whereas the second uses a vectorial representation [9]. This paper shows that none of these encoding schemes accurately accounts for all the response properties of single cells in the parietal cortex. Instead, we propose an alternative hypothesis which does not aim at representing X per se; instead, the inputs Rand E are represented in a particular basis function representation. We show that this scheme is consistent with the way parietal neurons respond to the retinal position of objects and eye position, and we give computational arguments for why this might be an efficient strategy for the cortex. 2 Maps and Vectorial Representations One way to encode a two-dimensional vector is to use a lookup table for this vector which, in the case of a two-dimensional vector, would take the form of a twodimensional neuronal map. The parietal cortex may represent the egocentric location of object, X, in a similar fashion. This predicts that the visual receptive field of parietal neurons have a fixed position with respect to the head (figure IB). The work of Andersen et al. (1985) have clearly shown that this is not the case. As illustrated in figure 2A, parietal neurons have retinotopic receptive fields. In a vectorial representation, a vector is encoded by N units, each of them coding for the projection of the vector along its preferred direction. This entails that the activity, h, of a neuron is given by: Spatial Representations in the Parietal Cortex May Use Basis Functions A B Map Representation ~ V v x C Vectorial Representation °io o 0 o 0 j[ ·1110 -'JO 0 90 180 a (I)'gr") 159 Figure 1: Two neural representations of a vector. A) A vector if in cartesian and polar coordinates. B) In a map representation, units have a narrow gaussian tuning to the horizontal and vertical components of if. Moreover, the position of the peak response is directly related to the position of the units on the map. C) In a vectorial representation, each unit encodes the projection of if along its preferred direction ( central arrows) . This results in a cosine tuning to the vector angle, () . (2) Wa is usually called the preferred direction of the cells because the activity is maximum whenever () = 0; that is, when A points in the same direction as Wa. Such neurons have a cosine tuning to the direction of the egocentric location of objects, as shown also in figure lC. Cosine tuning curves have been reported in the motor cortex by Georgopoulos et al. (1982) , suggesting that the motor cortex uses a vectorial code for the direction of hand movement in extrapersonal space. The same scheme has been also used by Goodman and Andersen (1990), and Touretzski et al. (1993) to model the encoding of egocentric position of objects in the parietal cortex. Touretzski et al. (1993) called their representation a sinusoidal array instead of a vectorial representation. Using Eq. 1, we can rewrite Eq. 2: (3) This second equation is linear in Ii and if and uses the same vectors, Wa , in both dot products. This leads to three important predictions: 1) The visual receptive fields of parietal neurons should be planar. 2) The eye position receptive fields of parietal neurons should also be planar; that is, for a given retinal positions, the response of parietal neuron should be a linear function of eye position. 160 A o -101'--'---'--~-'--'--'---'---' -40 -20 0 20 40 Retinal Position (Deg) Alexandre Pouget, Terrence J. Sejnowski B @ • Figure 2: Typical response of a neuron in the parietal cortex of a monkey. A) Visual receptive field has a fixed position on the retina, but the gain of the response is modulated by eye position (ex). (Adpated from Andersen et al., 1985) B) Example of an eye position receptive field, also called gain field, for a parietal cell. The nine circles indicate the amplitude of the response to an identical retinal stimulation for nine different eye positions. Outer circles show the total activity, whereas black circles correspond to the total response minus spontaneous activity prior to visual stimulation. (Adpated from Zipser et al., 1988) 3) The preferred direction for retinal location and eye position should be identical. For example, if the receptive field is on the right side of the visual field, the gain field should also increase with eye positon to the right side. The visual receptive fields and the eye position gain fields of single parietal neurons have been extensively studied by Andersen et al. [2]. In most cases, the visual receptive fields were bell-shaped with one or several peaks and an average radius of 22 degrees of visual angle [1], a result that is clearly not consistent with the first prediction above. We show in figure 2A an idealized visual receptive field of a parietal neuron. The effect of eye position on the visual receptive field is also illustrated. The eye position clearly modulates the gain of the visual response. The prediction regarding the receptive field for eye position has been borne out by statistical analysis. The gain fields of 80% of the cells had a planar component [1, 11]. One such gain field is shown in figure 2B. There is not enough data available to determine whether or not the third prediction is valid. However, indirect evidence suggests that if such a correlation exists between preferred direction for retinal location and for eye position, it is probably not strong. Cells with opposite preferred directions [2, 3] have been observed. Furthermore, although each hemisphere represents all possible preferred eye position directions, there is a clear tendency to overrepresent the contralateral retinal hemifield [1]. In conclusion, the experimental data are not fully consistent with the predictions of the vectorial code. The visual receptive fields, in particular, are strongly nonlinear. If these nonlinearities are computationally neutral, that is, they are averaged out in subsequent stages of processing in the cortex, then the vectorial code could capture Spatial Representations in the Parietal Cortex May Use Basis Functions 161 the essence of what the parietal cortex computes and, as such, would provide a valid approximation of the neurophysiological data. We argue in the next section that the nonlinearities cannot be disregarded and we present a representational scheme in which they have a central computational function. 3 Basis Function Representation 3.1 Sensorimotor Coordination and Nonlinear Function Approximation The function which specified the pattern of muscle activities required to move a limb, or the body, to a specific spatial location is a highly nonlinear function of the sensory inputs. The cortex is not believed to specify patterns of muscle activation, but the intermediate transformations which are handled by the cortex are often themselves nonlinear. Even if the transformations are actually linear, the nature of cortical representations often makes the problem a nonlinear mapping. For example, there exists in the putamen and premotor cortex cells with gaussian head-centered visual receptive fields [7J which means that these cells compute gaussians of A or, equivalently, gaussians of R + E, which is nonlinear in Rand E. There are many other examples of sensory remappings involving similar computations. If the parietal cortex is to have a role in these remappings, the cells should respond to the sensory inputs in a way that can be used to approximate the nonlinear responses observed elsewhere. One possibility would be for parietal neurons to represent input signals such as eye position and retinal location with basis functions. A basis function decomposition is a well-known method for approximating nonlinear functions which is, in addition, biologically plausible [8J. In such a representation, neurons do not encode the head-centered locations of objects, A; instead, they compute functions of the input variables, such as Rand E, which can be used subsequently to approximate any functions of these variables. 3.2 Predictions of the Basis Function Representation Not all functions are basis functions. Linear functions do not qualify, nor do sums of functions which, individually, would be basis functions, such as gaussian functions of retinal location plus a sigmoidal functions of eye position. If the parietal cortex uses a basis function representation, two conditions have to be met: 1) The visual and the eye position receptive fields should be smooth nonlinear function of Rand E. 2) The selectivities to Rand E should interact nonlinearly The visual receptive fields of parietal neurons are typically smooth and nonlinear. Gaussian or sum of gaussians appear to provide good models of their response profiles [2]. The eye position receptive field on the other hand, which is represented by the gain field, appears to be approximately linear. We believe, however, that the published data only demonstrate that the eye position receptive field is monotonic, 162 Alexandre Pouget, Terrence J. Sejnowski Head-Centered Retinotopic o o Figure 3: Approximation of a gaussian head-centered (top-left) and a retinotopic (top-right) receptive field, by a linear combination of basis function neurons. The bottom 3-D plots show the response to all possible horizontal retinal position, r x , and horizontal eye positions, ex, of four typical basis function units. These units are meant to model actual parietal neurons but not necessarily linear. In published experiments, eye position receptive fields (gain fields) were sampled at only nine points, which makes it difficult to distinguish between a plane and other functions such as a sigmoidal function or a piecewise linear function. The hallmark of a nonlinearity would be evidence for saturation of activity within working range of eye position. Several published gain fields show such saturations [3, 11], but a rigorous statistical analysis would be desirable. Andersen et al. (1985) have have shown that the responses of parietal neurons are best modeled by a multiplication between the retinal and eye position selectivities which is consistent with the requirements for basis functions. Therefore, the experimental data are consistent with our hypothesis that the parietal cortex uses a basis function representation. The response of most gain-modulated neurons in the parietal cortex could be modeled by multiplying a gaussian tuning to retinal position by a sigmoid of eye position, a function which qualifies as a basis function. 3.3 Simulations We simulated the response of 121 parietal gain-modulated neurons modeled by multiplying a gaussian of retinal position, rx, with a sigmoid of eye position, ex : Spatial Representations in the Parietal Cortex May Use Basis Functions (rz-rra):il e~ .. ~ h,o=----e,,;-e,l'j 1 +et 163 (4) where the centers of the gaussians for retinalloction rxi and the positions of the sigmoids for eye postions exi were uniformly distributredo The widths of the gaussian (T and the sigmoid t were fixed. Four of these functions are shown at the bottom of figure 3. We used these basis functions as a hidden layer to approximate two kinds of output functions: a gaussian head-centered receptive field and a gaussian retinotopic receptive field. Neurons with these response properties are found downstream of the parietal cortex in the premotor cortex [7] and superior colliculus, two structures believed to be involved in the control of, respectively, arm and eye movements. The weights for a particular output were obtained by using the delta rule. Weights were adjusted until the mean error was below 5% of the maximum output value. Figure 3 shows our best approximations for both the head-centered and retinotopic receptive fields. This demonstrates that the same pool of neurons can be used to approximate several diffferent nonlinear functions. 4 Discussion Neurophysiological data support our hypothesis that the parietal cortex represents its inputs, such as the retinal location of objects and eye position, in a format suitable to non-linear function approximation, an operation central to sensorimotor coordination. Neurons have gaussian visual receptive fields modulated by monotonic function of eye position leading to response function that can be modeled by product of gaussian and sigmoids. Since the product of gaussian and sigmoids forms basis functions, this representation is good for approximating nonlinear functions of the input variables. Previous attempts to characterize spatial representations have emphasized linear encoding schemes in which the location of objects is represented in egocentric coordinates. These codes cannot be used for nonlinear function approximation and, as such, may not be adequate for sensorimotor coordination [6, 10]. On the other hand, such representations are computationally interesting for certain operations, like addition or rotation. Some part of the brain more specialized in navigation like the hippocampus might be using such a scheme [10]. In figure 3, a head-centered or a retinotopic receptive field can be computed from the same pool of neurons. It would be arbitrary to say that these neurons encode the positions of objects in egocentric coordinates. Instead, these units encode a position in several frames of reference simultaneously. If the parietal cortex uses this basis function representation, we predict that hemineglect, the neurological syndrome which results from lesions in the parietal cortex, should not be confined to any particular frame of reference. This is precisely the conclusion that has emerged from recent studies of parietal patients [4]. Whether the behavior of parietal patients can be fully explained by lesions of a basis function representation remains to be investigated. 164 Alexandre Pouget, Terrence J. Sejnowski Acknowledgments We thank Richard Andersen for helpful conversations and with access to unpublished data. References [1] R.A. Andersen, C. Asanuma, G. Essick, and R.M. Siegel. Corticocortical connections of anatomically and physiologically defined subdivisions within the inferior parietal lobule. Journal of Comparative Neurology, 296(1):65-113, 1990. [2] R.A. Andersen, G.K. Essick, and R.M. Siegel. Encoding of spatial location by posterior parietal neurons. Science, 230:456-458, 1985. [3] R.A. Andersen and D. Zipser. The role of the posterior parietal cortex in coordinate transformations for visual-motor integration. Canadian Journal of Physiology and Pharmacology, 66:488-501, 1988. [4] M. Behrmann and M. Moscovitch. Object-centered neglect in patient with unilateral neglect: effect of left-right coordinates of objects. Journal of Cognitive Neuroscience, 6:1-16, 1994. [5] A.P. Georgopoulos, J.F. Kalaska, R. Caminiti, and J.T. Massey. On the relations between the direction of two-dimensional arm movements and cell discharge in primate motor cortex. Journal of Neuroscience, 2(11):1527-1537, 1982. [6] S.J. Goodman and R.A. Andersen. Algorithm programmed by a neural model for coordinate transformation. In International Joint Conference on Neural Networks, San Diego, 1990. [7] M.S. Graziano, G.s. Yap, and C.G. Gross. Coding of visual space by premotor neurons. Science, 266:1054-1057, 1994. [8] T. Poggio. A theory of how the brain might work. Cold Spring Harbor Symposium on Quantitative Biology, 55:899-910, 1990. [9] J .F. Soechting and M. Flanders. Moving in three-dimensional space: frames of reference, vectors and coordinate systems. Annual Review in Neuroscience, 15:167-91, 1992. [10] D.S. Touretzky, A.D. Redish, and H.S. Wan. Neural representation of space using sinusoidal arrays. Neural Computation, 5:869-884, 1993. [11] D. Zipser and R.A. Andersen. A back-propagation programmed network that stimulates reponse properties of a subset of posterior parietal neurons. Nature, 331:679- 684, 1988.	1994	15
905	Reinforcement Learning Algorithm for Partially Observable Markov Decision Problems Tommi Jaakkola tommi@psyche.mit.edu Satinder P. Singh singh@psyche.mit.edu Michael I. Jordan jordan@psyche.mit.edu Department of Brain and Cognitive Sciences, BId. E10 Massachusetts Institute of Technology Cambridge, MA 02139 Abstract Increasing attention has been paid to reinforcement learning algorithms in recent years, partly due to successes in the theoretical analysis of their behavior in Markov environments. If the Markov assumption is removed, however, neither generally the algorithms nor the analyses continue to be usable. We propose and analyze a new learning algorithm to solve a certain class of non-Markov decision problems. Our algorithm applies to problems in which the environment is Markov, but the learner has restricted access to state information. The algorithm involves a Monte-Carlo policy evaluation combined with a policy improvement method that is similar to that of Markov decision problems and is guaranteed to converge to a local maximum. The algorithm operates in the space of stochastic policies, a space which can yield a policy that performs considerably better than any deterministic policy. Although the space of stochastic policies is continuous-even for a discrete action space-our algorithm is computationally tractable. 346 Tommi Jaakkola, Satinder P. Singh, Michaell. Jordan 1 INTRODUCTION Reinforcement learning provides a sound framework for credit assignment in unknown stochastic dynamic environments. For Markov environments a variety of different reinforcement learning algorithms have been devised to predict and control the environment (e.g., the TD(A) algorithm of Sutton, 1988, and the Q-Iearning algorithm of Watkins, 1989). Ties to the theory of dynamic programming (DP) and the theory of stochastic approximation have been exploited, providing tools that have allowed these algorithms to be analyzed theoretically (Dayan, 1992; Tsitsiklis, 1994; Jaakkola, Jordan, & Singh, 1994; Watkins & Dayan, 1992). Although current reinforcement learning algorithms are based on the assumption that the learning problem can be cast as Markov decision problem (MDP), many practical problems resist being treated as an MDP. Unfortunately, if the Markov assumption is removed examples can be found where current algorithms cease to perform well (Singh, Jaakkola, & Jordan, 1994). Moreover, the theoretical analyses rely heavily on the Markov assumption. The non-Markov nature of the environment can arise in many ways. The most direct extension of MDP's is to deprive the learner of perfect information about the state of the environment. Much as in the case of Hidden Markov Models (HMM's), the underlying environment is assumed to be Markov, but the data do not appear to be Markovian to the learner. This extension not only allows for a tractable theoretical analysis, but is also appealing for practical purposes. The decision problems we consider here are of this type. The analog of the HMM for control problems is the partially observable Markov decision process (POMDP; see e.g., Monahan, 1982). Unlike HMM's, however, there is no known computationally tractable procedure for POMDP's. The problem is that once the state estimates have been obtained, DP must be performed in the continuous space of probabilities of state occupancies, and this DP process is computationally infeasible except for small state spaces. In this paper we describe an alternative approach for POMDP's that avoids the state estimation problem and works directly in the space of (stochastic) control policies. (See Singh, et al., 1994, for additional material on stochastic policies.) 2 PARTIAL OBSERVABILITY A Markov decision problem can be generalized to a POMDP by restricting the state information available to the learner. Accordingly, we define the learning problem as follows. There is an underlying MDP with states S = {SI, S2, ... , SN} and transition probability pO " the probability of jumping from state S to state s' when action a is II! taken in state s. For every state and every action a (random) reward is provided to the learner. In the POMDP setting, the learner is not allowed to observe the state directly but only via messages containing information about the state. At each time step t an observable message mt is drawn from a finite set of messages according to an unknown probability distribution P(mlst) 1. We assume that the learner does 1 For simplicity we assume that this distribution depends only on the current state. The analyses go through also with distributions dependent on the past states and actions Reinforcement Learning Algorithm for Markov Decision Problems 347 not possess any prior information about the underlying MDP beyond the number of messages and actions. The goal for the learner is to come up with a policy-a mapping from messages to actions-that gives the highest expected reward. As discussed in Singh et al. (1994), stochastic policies can yield considerably higher expected rewards than deterministic policies in the case of POMDP's. To make this statement precise requires an appropriate technical definition of "expected reward," because in general it is impossible to find a policy, stochastic or not, that maximizes the expected reward for each observable message separately. We take the timeaverage reward as a measure of performance, that is, the total accrued reward per number of steps taken (Bertsekas, 1987; Schwartz, 1993). This approach requires the assumption that every state of the underlying controllable Markov chain is reachable. In this paper we focus on a direct approach to solving the learning problem. Direct approaches are to be compared to indirect approaches, in which the learner first identifies the parameters of the underlying MDP, and then utilizes DP to obtain the policy. As we noted earlier, indirect approaches lead to computationally intractable algorithms. Our approach can be viewed as providing a generalization of the direct approach to MDP's to the case of POMDP's. 3 A MONTE-CARLO POLICY EVALUATION Advantages of Monte-Carlo methods for policy evaluation in MDP's have been reviewed recently (Barto and Duff, 1994). Here we present a method for calculating the value of a stochastic policy that has the flavor of a Monte-Carlo algorithm. To motivate such an approach let us first consider a simple case where the average reward is known and generalize the well-defined MDP value function to the POMDP setting. In the Markov case the value function can be written as (cf. Bertsekas, 1987): N V(s) = lim ~ E{R(st, Ut) - Risl = s} N-oo~ t=l (1) where St and at refer to the state and the action taken at the tth step respectively. This form generalizes easily to the level of messages by taking an additional expectation: V(m) = E {V(s)ls -+ m} (2) where s -+ m refers to all the instances where m is observed in sand E{·ls -+ m} is a Monte-Carlo expectation. This generalization yields a POMDP value function given by V(m) = I: P(slm)V(s) (3) ,Em in which P(slm) define the limit occupancy probabilities over the underlying states for each message m. As is seen in the next section value functions of this type can be used to refine the currently followed control policy to yield a higher average reward. Let us now consider how the generalized value functions can be computed based on the observations. We propose a recursive Monte-Carlo algorithm to effectively compute the averages involved in the definition of the value function. In the simple 348 Tommi Jaakkola, Satinder P. Singh, Michael I. Jordan case when the average payoff is known this algorithm is given by Xt(m) Xt(m) f3t(m) (1 - Kt(m) htf3t-l(m) + Kt(m) (4) Xt(m) lIt(m) (1- Kt(m))lIt-l(m) + f3t(m)[R(st, at) - R] (5) where Xt(m) is the indicator function for message m, Kt(m) is the number of times m has occurred, and 'Yt is a discount factor converging to one in the limit. This algorithm can be viewed as recursive averaging of (discounted) sample sequences of different lengths each of which has been started at a different occurrence of message m. This can be seen by unfolding the recursion, yielding an explicit expression for lit (m). To this end, let tk denote the time step corresponding to the ph occurrence of message m and for clarity let Rt = R(st, Ut) - R for every t. Using these the recursion yields: 1 lIt(m) = Kt(m) [ Rtl + rl,l Rt1+1 + ... + r1,t-tl Rt (6) where we have for simplicity used rk ,T to indicate the discounting at the Tth step in the kth sequence. Comparing the above expression to equation 1 indicates that the discount factor has to converge to one in the limit since the averages in V(s) or V(m) involve no discounting. To address the question of convergence of this algorithm let us first assume a constant discounting (that is, 'Yt = 'Y < 1). In this case, the algorithm produces at best an approximation to the value function. For large K(m) the convergence rate by which this approximate solution is found can be characterized in terms of the bias and variance. This gives Bias{V(m)} Q( (1 - r)-l / K(m) and Var{V(m)} Q( (1 r)-2 / K(m) where r = Ehtk-tk-l} is the expected effective discounting between observations. Now, in order to find the correct value function we need an appropriate way of letting 'Yt 1 in the limit. However, not all such schedules lead to convergent algorithms; setting 'Yt = 1 for all t, for example, would not. By making use of the above bounds a feasible schedule guaranteeing a vanishing bias and variance can be found. For instance, since 'Y > 7 we can choose 'Yk(m) = 1 - K(m)I/4. Much faster schedules are possible to obtain by estimating r. Let us now revise the algorithm to take into account the fact that the learner in fact has no prior knowledge of the average reward. In this case the true average reward appearing in the above algorithm needs to be replaced with an incrementally updated estimate Rt- l . To improve the effect this changing estimate has on the values we transform the value function whenever the estimate is updated. This transformation is given by Xt(m) (1 - Kt(m) )Ct-1(m) + f3t(m) lIt(m) - Ct(m)(Rt - Rt-l) (7) (8) and, as a result, the new values are as if they had been computed using the current estimate of the average reward. Reinforcement Learning Algorithm for Markov Decision Problems 349 To carry these results to the control setting and assign a figure of merit to stochastic policies we need a quantity related to the actions for each observed message. As in the case of MDP's, this is readily achieved by replacing m in the algorithm just described by (m, a). In terms of equation 6, for example, this means that the sequences started from m are classified according to the actions taken when m is observed. The above analysis goes through when m is replaced by (m, a), yielding "Q-values" on the level of messages: (9) In the next section we show how these values can be used to search efficiently for a better policy. 4 POLICY IMPROVEMENT THEOREM Here we present a policy improvement theorem that enables the learner to search efficiently for a better policy in the continuous policy space using the "Q-values" Q(m, a) described in the previous section. The theorem allows the policy refinement to be done in a way that is similar to policy improvement in a MDP setting. Theorem 1 Let the current stochastic policy 1I"(alm) lead to Q-values Q1r(m, a) on the level of messages. For any policy 11"1 (aim) define J1r 1 (m) = L: 11"1 (alm)[Q1r(m, a) - V 1r(m)] a The change in the average reward resulting from changing the current policy according to 1I"(alm) -+ (1- {)1I"(alm) + {1I"1(alm) is given by ~R1r = {L: p1r(m)J1r 1 (m) + O({2) m where p1r (m) are the occupancy probabilities for messages associated with the current policy. The proof is given in Appendix. In terms of policy improvement the theorem can be interpreted as follows. Choose the policy 1I"1(alm) such that 1 J1r (m) = max[Q1r(m, a) - V 1r(m)] (10) a If now J1r 1 (m) > 0 for some m then we can change the current policy towards 11"1 and expect an increase in the average reward as shown by the theorem. The { factor suggests local changes in the policy space and the policy can be refined until m~l J1r\ m) = 0 for all m which constitutes a local maximum for this policy improvement method. Note that the new direction 1I"1(alm) in the policy space can be chosen separately for each m. 5 THE ALGORITHM Based on the theoretical analysis presented above we can construct an algorithm that performs well in a POMDP setting. The algorithm is composed of two parts: First, 350 Tommi Jaakkola, Satinder P. Singh, Michael I. Jordan Q(m, a) values-analogous to the Q-values in MDP-are calculated via a MonteCarlo approach. This is followed by a policy improvement step which is achieved by increasing the probability of taking the best action as defined by Q(m,a). The new policy is guaranteed to yield a higher average reward (see Theorem 1) as long as for somem max[Q(m, a) - V(m)] > 0 a (11) This condition being false constitutes a local maximum for the algorithm. Examples illustrating that this indeed is a local maximum can be found fairly easily. In practice, it is not feasible to wait for the Monte-Carlo policy evaluation to converge but to try to improve the policy before the convergence. The policy can be refined concurrently with the Monte-Carlo method according to 1I"(almn) -+ 1I"(almn) + €[Qn(mn, a) - Vn(mn)] (12) with normalization. Other asynchronous or synchronous on-online updating schemes can also be used. Note that if Qn(m, a) = Q(m, a) then this change would be statistically equivalent to that of the batch version with the concomitant guarantees of giving a higher average reward. 6 CONCLUSIONS In this paper we have proposed and theoretically analyzed an algorithm that solves a reinforcement learning problem in a POMDP setting, where the learner has restricted access to the state of the environment. As the underlying MDP is not known the problem appears to the learner to have a non-Markov nature. The average reward was chosen as the figure of merit for the learning problem and stochastic policies were used to provide higher average rewards than can be achieved with deterministic policies. This extension from MDP's to POMDP's greatly increases the domain of potential applications of reinforcement learning methods. The simplicity of the algorithm stems partly from a Monte-Carlo approach to obtaining action-dependent values for each message. These new "Q-values" were shown to give rise to a simple policy improvement result that enables the learner to gradually improve the policy in the continuous space of probabilistic policies. The batch version of the algorithm was shown to converge to a local maximum. We also proposed an on-line version of the algorithm in which the policy is changed concurrently with the calculation of the "Q-values." The policy improvement of the on-line version resembles that of learning automata. APPENDIX Let us denote the policy after the change by 11"!. Assume first that we have access to Q1I"(s, a), the Q-values for the underlying MDP, and to p1l"< (slm), the occupancy probabilities after the policy refinement. Define J(m, 11"!, 11"!, 11") = L 1I"!(alm) L p1l"< (slm)[Q1I"(s, a) - V1I"(s)] (13) where we have used the notation that the policies on the left hand side correspond to the policies on the right respectively. To show how the average reward depends Reinforcement Learning Algorithm for Markov Decision Problems 351 on this quantity we need to make use of the following facts. The Q-values for the underlying MDP satisfy (Bellman's equation) Q1r(s, a) = R(s, a) - R1r + L:P~3' V1r(s') (14) " In addition, 2:0 7r(alm)Q1r(s, a) = V1r(s), implying that J(m, 7r!, 7r!, 7r f ) = 0 (see eq. 13). These facts allow us to write J(m, 7r f , 7r f , 7r) - J(m, 7r f , 7r f , 7rf ) L: 7r f (alm) L: p 1r«slm)[Q1r(s, a) - V1r (s) - Q1r< (s, a) + V1r< (s)] (15) 3 By weighting this result for each class by p 1r< (m) and summing over the messages the probability weightings for the last two terms become equal and the terms cancel. This procedure gives us R1r< - R1r = L:P1r«m)J(m, 7r f ,7r f , 7r) (16) m This result does not allow the learner to assess the effect of the policy refinement on the average reward since the JO term contains information not available to the learner. However, making use of the fact that the policy has been changed only slightly this problem can be avoided. As 7r! is a policy satisfying maxmo l7r f (alm) -7r(alm)1 ::; (, it can then be shown that there exists a constant C such that the maximum change in P(slm), pes), P(m) is bounded by Cf. Using these bounds and indicating the difference between 7r! and 7r dependent quantities by ~ we get L:[7r(alm) + ~7r(alm)] L)P1r(slm) + ~p1r(slm)][Q1r(s, a) - V1r (s)] o o 3 o 3 where the second equality follows from 2:0 7r(alm)[Q1r(s, a) - V1r(s)] = 0 and the third from the bounds stated earlier. The equation characterizing the change in the average reward due to the policy change (eq. 16) can be now rewritten as follows: R1r< - R1r = L: p 1r< (m)J(m, 7r f , 7r, 7r) + 0«(2) m 352 Tommi Jaakkola, Satinder P. Singh, Michael I. Jordan m a where the bounds (see above) have been used for Vir' (m) - p1r(m). This completes the proof. 0 Acknowledgments The authors thank Rich Sutton for pointing out errors at early stages of this work. This project was supported in part by a grant from the McDonnell-Pew Foundation, by a grant from ATR Human Information Processing Research Laboratories, by a grant from Siemens Corporation and by grant NOOOI4-94-1-0777 from the Office of Naval Research. Michael I. Jordan is a NSF Presidential Young Investigator. References Barto, A., and Duff, M. (1994). Monte-Carlo matrix inversion and reinforcement learning. In Advances of Neural Information Processing Systems 6, San Mateo, CA, 1994. Morgan Kaufmann. Bertsekas, D. P. (1987). Dynamic Programming: Deterministic and Stochastic Models. Englewood Cliffs, N J: Prentice-Hall. Dayan, P. (1992). The convergence of TD(A) for general A. Machine Learning, 8, 341-362. Jaakkola, T., Jordan M.I., and Singh, S. P. (1994). On the convergence of stochastic iterative Dynamic Programming algorithms. Neural Computation 6, 1185-1201. Monahan, G. (1982). A survey of partially observable Markov decision processes. Management Science, 28, 1-16. Singh, S. P., Jaakkola, T., Jordan, M.1. (1994). Learning without state estimation in partially observable environments. In Proceedings of the Eleventh Machine Learning Conference. Sutton, R. S. (1988). Learning to predict by the methods of temporal differences. Machine Learning, 3, 9-44. Schwartz, A. (1993). A reinforcement learning method for maximizing un discounted rewards. In Proceedings of the Tenth Machine Learning Conference. Tsitsiklis J. N. (1994). Asynchronous stochastic approximation and Q-Iearning. Machine Learning 16, 185-202. Watkins, C.J.C.H. (1989). Learning from delayed rewards. PhD Thesis, University of Cambridge, England. Watkins, C.J .C.H, & Dayan, P. (1992). Q-Iearning. Machine Learning, 8, 279-292.	1994	16
906	FINANCIAL APPLICATIONS OF LEARNING FROM HINTS Yaser s. Abu-Mostafa California Institute of Technology and NeuroDollars, Inc. e-mail: yaser@caltech.edu Abstract The basic paradigm for learning in neural networks is 'learning from examples' where a training set of input-output examples is used to teach the network the target function. Learning from hints is a generalization of learning from examples where additional information about the target function can be incorporated in the same learning process. Such information can come from common sense rules or special expertise. In financial market applications where the training data is very noisy, the use of such hints can have a decisive advantage. We demonstrate the use of hints in foreign-exchange trading of the U.S. Dollar versus the British Pound, the German Mark, the Japanese Yen, and the Swiss Franc, over a period of 32 months. We explain the general method of learning from hints and how it can be applied to other markets. The learning model for this method is not restricted to neural networks. 1 INTRODUCTION When a neural network learns its target function from examples (training data), it knows nothing about the function except what it sees in the data. In financial market applications, it is typical to have limited amount of relevant training data, with high noise levels in the data. The information content of such data is modest, and while the learning process can try to make the most of what it has, it cannot create new information on its own. This poses a fundamental limitation on the 412 Yaser S. Abu-Mostafa learning approach, not only for neural networks, but for all other models as well. It is not uncommon to see simple rules such as the moving average outperforming an elaborate learning-from-examples system. Learning from hints (Abu-Mostafa, 1990, 1993) is a value-added feature to learning from examples that boosts the information content in the data. The method allows us to use prior knowledge about the target function, that comes from common sense or expertise, along with the training data in the same learning process. Different types of hints that may be available in a given application can be used simultaneously. In this paper, we give experimental evidence of the impact of hints on learning performance, and explain the method in some detail to enable the readers to try their own hints in different markets. Even simple hints can result in significant improvement in the learning performance. Figure 1 shows the learning performance for foreign exchange (FX) trading with and without the symmetry hint (see section 3), using only the closing price history. The plots are the Annualized Percentage Returns (cumulative daily, unleveraged, transaction cost included), for a sliding one-year test window in the period from April 1988 to November 1990, averaged over the four major FX markets with more than 150 runs per currency. The error bar in the upper left corner is 3 standard deviations long (based on 253 trading days, assuming independence between different runs). The plots establish a statistically significant differential in performance due to the use of hints. This differential holds for all four currencies. 10 r---------~--------_r--------~--------~r_--------~ average: without hint -;-I" average: with hint;..':':.--, " I ~ 8 30 /"_,,, " 6 / " ,-' iI'·.' " ,4 J _",'''.-~''''''''.'-'' 2 ,~_" .. " .. ,,-" _A' ,-,,-, .. '" ,.1 o '.~:<.~.~ .......................................... . .. , .. I , ..... r';' ,-' " " " -2 ~--------~--------~--______ ~ ________ L-________ ~ o 50 100 150 200 250 Test Day Number Figure 1: Learning performance with and without hint Since the goal of hints is to add information to the training data, the differential in performance is likely to be less dramatic if we start out with more informative training data. Similarly, an additional hint may not have a pronounced effect if Financial Applications of Learning from Hints 413 we have already used a few hints in the same application. There is a saturation in performance in any market that reflects how well the future can be forecast from the past. (Believers in the Efficient Market Hypothesis consider this saturation to be at zero performance). Hints will not make us forecast a market better than whatever that saturation level may be. They will, however, enable us to approach that level through learning. This paper is organized as follows. Section 2 characterizes the notion of very noisy data by defining the '50% performance range'. We argue that the need for extra information in financial market applications is more pronounced than in other pattern recognition applications. In section 3, we discuss our method for learning from hints. We give examples of different types of hints, and explain how to represent hints to the learning process. Section 4 gives result details on the use of the symmetry hint in the four major FX markets. Section 5 provides experimental evidence that it is indeed the information content of the hint, rather than the incidental regularization effect, that results in the performance differential that we observe. 2 FINANCIAL DATA This section provides a characterization of very noisy data that applies to the financial markets. For a broad treatment of neural-network applications to the financial markets, the reader is referred to (Abu-Mostafa et al, 1994). Input X Input X --Other Information MARKET NEURAL NETWORK Target Ouput ... y Forecast y Figure 2: Illustration of the nature of noise in financial markets Consider the market as a system that takes in a lot of information (fundamentals, news events, rumors, who bought what when, etc.) and produces an output y (say up/down price movement for simplicity). A model, e.g., a neural network, attempts 414 Yaser S. Abu-Mostafa to simulate the market (figure 2), but it takes an input x which is only a small subset of the information. The 'other information' cannot be modeled and plays the role of noise as far as x is concerned. The network cannot determine the target output y based on x alone, so it approximates it with its output y. It is typical that this approximation will be correct only slightly more than half the time. What makes us consider x 'very noisy' is that y and y agree only! + f. of the time (50% performance range). This is in contrast to the typical pattern recognition application, such as optical character recognition, where y and y agree 1 f. of the time (100% performance range). It is not the poor performance per se that poses a problem in the 50% range, but rather the additional difficulty of learning in this range. Here is why. In the 50% range, a performance of ! + f. is good, while a performance of ! f. is disastrous. During learning, we need to distinguish between good and ~ad hypotheses based on a limited set of N examples. The problem with the 50% range is that the number of bad hypotheses that look good on N points is huge. This is in contrast to the 100% range where a good performance is as high as 1 f.. The number of bad hypotheses that look good here is limited. Therefore, one can have much more confidence in a hypothesis that was learned in the 100% range than one learned in the 50% range. It is not uncommon to see a random trading policy making good money for a few weeks, but it is very unlikely that a random character recognition system will read a paragraph correctly. Of course this problem would diminish if we used a very large set of examples, because the law of large numbers would make it less and less likely that y and y can agree! + f. of the time just by 'coincidence'. However, financial data has the other problem of non-stationarity. Because of the continuous evolution in the markets, old data may represent patterns of behavior that no longer hold. Thus, the relevant data for training purposes is limited to fairly recent times. Put together, noise and non-stationarity mean that the training data will not contain enough information for the network to learn the function. More information is needed, and hints can be the means of providing it. 3 HINTS In this section, we give examples of different types of hints and discuss how to represent them to the learning process. We describe a simple way to use hints that allows the reader to try the method with minimal effort. For a more detailed treatment, please see (Abu-Mostafa, 1993). As far as our method is concerned, a hint is any property that the target function is known to have. For instance, consider the symmetry hint in FX markets as it applies to the U.S. Dollar versus the German Mark (figure 3). This simple hint asserts that if a pattern in the price history implies a certain move in the market, then this implication holds whether you are looking at the market from the U.S. Dollar viewpoint or the German Mark viewpoint. Formally, in terms of normalized prices, the hint translates to invariance under inversion of these prices. Is the symmetry hint valid? The ultimate test for this is how the learning performance is affected by the introduction of the hint. The formulation of hints is an art. Financial Applications of Learning from Hints 415 We use our experience, common sense, and analysis of the market to come up with a list of what we believe to be valid properties of this market. We then represent these hints in a canonical form as we will see shortly, and proceed to incorporate them in the learning process. The improvement in performance will only be as good as the hints we put in. u.s. DOLLAR ? • GERMAN MARK Figure 3: Illustration of the symmetry hint in FX markets The canonical representation of hints is a more systematic task. The first step in representing a hint is to choose a way of generating 'virtual examples' of the hint. For illustration, suppose that the hint asserts that the target function y is an odd function of the input. An example of this hint would have the form y( -x) = -y(x) for a particular input x. One can generate as many virtual examples as needed by picking different inputs. After a hint is represented by virtual examples, it is ready to be incorporated in the learning process along with the examples of the target function itself. Notice that an example of the function is learned by minimizing an error measure, say (y(x) - y(x))2, as a way of ultimately enforcing the condition y(x) = y(x). In the same way, a virtual example of the oddness hint can be learned by minimizing (y(x) + y(-x))2 as a way of ultimately enforcing the condition y(-x) = -y(x). This involves inputting both x and -x to the network and minimizing the difference between the two outputs. It is easy to show that this can be done using backpropagation (Rumelhart et al, 1986) twice. The generation of a virtual example of the hint does not require knowing the value of the target function; neither y(x) nor y( -x) is needed to compute the error for the oddness hint. In fact, x and -x can be artificial inputs. The fact that we do not need the value of the target function is crucial, since it was the limited resource of examples for which we know the value of the target function that got us interested in hints in the first place. On the other hand, for some hints, we can take the examples of the target function that we have, and employ the hint to duplicate these examples. For instance, an example y(x) = 1 can be used to infer a second example y( -x) = -1 using the oddness hint. Representing the hint by duplicate examples is an easy way to try simple hints using the same software that we use for learning from examples. 416 Yaser S. Abu-Mostafa Let us illustrate how to represent two common types of hints. Perhaps the most common type is the invariance hint. This hint asserts that i)(x) = i)(x/) for certain pairs x, x'. For instance, "i) is shift-invariant" is formalized by the pairs x, x' that are shifted versions of each other. To represent the invariance hint, an invariant pair (x, x') is picked as a virtual example. The error associated with this example is (y(x) - y(x/»2. Another related type of hint is the monotonicity hint. The hint asserts for certain pairs x, x' that i)(x) :5 i)(x/). For instance, "i) is monotonically non decreasing in x" is formalized by the pairs x, x' such that x < x'. One application where the monotonicity hint occurs is the extension of personal credit. If person A is identical to person B except that A makes less money than B, then the approved credit line for A cannot exceed that of B. To represent the monotonicity hint, a monotonic pair (X,X/) is picked as a virtual example. The error associated with this example is given by (y(x) - Y(X/»2 if y(x) > y(x/) and zero if y(x) :5 y(x'). 4 FX TRADING We applied the symmetry hint in the four FX markets of the U.S. Dollar versus the British Pound, the German Mark, the Japanese Yen, and the Swiss Franc. In each case, only the closing prices for the preceding 21 days were used for inputs. The objective (fitness) function we chose was the total return on the training set, and we used simple filtering methods on the inputs and outputs of the networks. In each run, the training set consisted of 500 days, and the test was done on the following 253 days. All four currencies show an improved performance when the symmetry hint is used. Roughly speaking, we are in the market half the time, each trade takes 4 days, the hit rate is close to 50%, and the A.P.R. without hint is 5% and with hint is 10% (the returns are annualized, unleveraged, and include the transaction cost; spread and average slippage). Notice that having the return as the objective function resulted in a fairly good return with a modest hit rate. 5 CROSS CHECKS In this final section, we report more experimental results aimed at validating our claim that the information content of the hint is the reason behind the improved performance. Why is this debatable? A hint plays an incidental role as a constraint on the neural network during learning, since it restricts the solutions the network may settle in. Because overfitting is a common problem in learning from examples, any restriction may improve the out-of-sample performance by reducing overfitting (Akaike, 1969, Moody, 1992). This is the idea behind regularization. To isolate the informative role from the regularizing role of the symmetry hint, we ran two experiments. In the first experiment, we used an uninformative hint, or 'noise' hint, which provides a random target output for the same inputs used in the examples of the symmetry hint. Figure 4 contrasts the performance of the noise hint with that of the real symmetry hint, averaged over the four currencies. Notice that the performance with the noise hint is close to that without any hint (figure 1), which is consistent with the notion of uninformative hint. The regularization effect seems to be negligible. Financial Applications of Learning from Hints c ~ " +> Q) IX Q) "" f\1 +> C Q) tl ~ Q) .,. "0 Q) N .... .-< f\1 " C ~ 10 ~--------~~--------~----------~----------~-----------n 8 6 4 2 o -2 ~ __________ ~ ________ ~ __________ ~ __________ ~ __________ -u o 50 100 150 200 250 Test Day Number Figure 4: Performance of the real hint versus a noise hint 10 .------------r------------.------------r----------~r_----------~ 5 0 -5 -10 -15 with fa 1 s e ):l+tlt--:::=':with JJi!trI' hint ----. I 3 a ,,-.,,-•• I .. '~ .. ~ .. -J .. - .. .. "'"~~.~.~.~.~.~.~.~~~~~~~.~.~.~ .•. ~~~-.~~-.-. •. -.-----.-'-.~-.-.~'.~ .• ~-------.-.-.-.-. ....... ... . .......... ..................................... . -20 ~--________ ~ __________ ~ __________ -L __________ ~ __________ --u o 50 100 150 200 250 Test Day Number Figure 5: Performance of the real hint versus a false hint 417 418 Yaser S. Abu-Mostafa In the second experiment, we used a harmful hint, or 'false' hint, in place of the symmetry hint. The hint takes the same examples used in the symmetry hint and asserts antisymmetry instead. Figure 5 contrasts the performance of the false hint with that of the real symmetry hint. As we see, the false hint had a detrimental effect on the performance. This is consistent with the hypothesis that the symmetry hint is valid, since its negation results in worse performance than no hint at all. Notice that the transaction cost is taken into consideration in all of these plots, which works as a negative bias and amplifies the losses of bad trading policies. CONCLUSION We have explained learning from hints, a systematic method for combining rules and data in the same learning process, and reported experimental results of a statistically significant improvement in performance in the four major FX markets that resulted from using a simple symmetry hint. We have described different types of hints and simple ways of using them in learning, to enable the readers to try their own hints in different markets. Acknowledgements I would like to acknowledge Dr. Amir Atiya for his valuable input. I am grateful to Dr. Ayman Abu-Mostafa for his expert remarks. References Abu-Mostafa, Y. S. (1990), Learning from hints in neural networks, Journal of Complexity 6, pp. 192-198. Abu-Mostafa, Y. S. (1993), A method for learning from hints, Advances in Neural Information Processing Systems 5, S. Hanson et al (eds), pp. 73-80, MorganKaufmann. Abu-Mostafa, Y. S. et al (eds) (1994), Proceedings of Neural Networks in the Capital Markets, Pasadena, California, November 1994. Akaike, H. (l969), Fitting autoregressive models for prediction, Ann. Inst. Stat. Math. 21, pp. 243-247. Moody, J. (1992), The effective number of parameters: An analysis of generalization and regularization in nonlinear learning systems, in Advances in Neural Information Processing Systems 4, J. Moody et al (eds), pp. 847-854, Morgan Kaufmann. Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986), Learning internal representations by error propagation, Parallel Distributed Processing 1, D. Rumelhart et al, pp. 318-362, MIT Press. Weigend, A., Rumelhart, D., and Huberman, B. (1991), Generalization by weight elimination with application to forecasting, in Advances in Neural Information Processing Systems 3, R. Lippmann et al (eds), pp. 875-882, Morgan Kaufmann.	1994	17
907	An Auditory Localization and Coordinate Transform Chip Timothy K. Horiuchi timmer@cns.caltech.edu Computation and Neural Systems Program California Institute of Technology Pasadena, CA 91125 Abstract The localization and orientation to various novel or interesting events in the environment is a critical sensorimotor ability in all animals, predator or prey. In mammals, the superior colliculus (SC) plays a major role in this behavior, the deeper layers exhibiting topographically mapped responses to visual, auditory, and somatosensory stimuli. Sensory information arriving from different modalities should then be represented in the same coordinate frame. Auditory cues, in particular, are thought to be computed in head-based coordinates which must then be transformed to retinal coordinates. In this paper, an analog VLSI implementation for auditory localization in the azimuthal plane is described which extends the architecture proposed for the barn owl to a primate eye movement system where further transformation is required. This transformation is intended to model the projection in primates from auditory cortical areas to the deeper layers of the primate superior colliculus. This system is interfaced with an analog VLSI-based saccadic eye movement system also being constructed in our laboratory. Introduction Auditory localization has been studied in many animals, particularly the barn owl. Most birds have a resolution of only 10 to 20 degrees, but owls are able to orient 788 Timothy Horiuchi to sound with an accuracy of 1 to 2 degrees which is comparable with humans. One important cue for localizing sounds is the relative time of arrival of a sound to two spatially separated ears. A neural architecture first described by Jeffress (1948) for measuring this time difference has been shown to exist in the barn owl auditory localization system (Konishi 1986). An analog VLSI implementation of the barn owl system constructed by Lazzaro (1990) is extended here to include a transformation from head coordinates to retinal coordinates. In comparison to the barn owl, the neurophysiology of auditory localization in cats and primates is not as well understood and a clear map of auditory space does not appear to be present in the inferior colliculus as it is in the owl. It has been suggested that cortical auditory regions may provide the head-based map of auditory space (Groh and Sparks 1992). In primates, where much of the oculomotor system is based in retinotopic coordinates, head-based information must ultimately be transformed in order to be used. While other models of coordinate transformation have been proposed for visual information (e.g. Zipser and Andersen 1988, Krommenhoek et al. 1993) and for auditory information (Groh and Sparks 1992), the model of coordinate transformation used in this system is a switching network which shifts the entire projection of the head-based map of auditory space onto a retinotopic "colliculus" circuit. This particular model is similar to a basis function approach where intermediate units have compact receptive fields in an eye-position / head-based azimuth space and the output units sum the outputs of a subset of these units. The auditory localization system described here provides acoustic target information to an analog VLSI-based saccadic eye movement system (Horiuchi, Bishofberger, and Koch 1994) being developed in our laboratory for multimodal operation. Bandpass _::::::10-,.,... ....... Filtering @3.2kHz electret condenser microphones Bandpass Filtering @3.2kHz Thresholded Zero-crossings Thresholded Zero-crossings Localization and Coordinate Transform Chip ~--Eye Position Sound Source Location (in retinal coordinates) Figure 1: Block diagram of the auditory localization system. The analog front end consists of external discrete analog electronics. An Auditory Localization and Coordinate Transform Chip i f 0 ·1 M¥ Snop II 45 degrees R of Center .' ... .. . " ~+-----~-----+----~~----+-----~----~ i I ·1 o M¥ Snop II CerRr POIIition ~+-----~-----+------~----+-----~----~ I f o ·1 Finga' Snop 0145 degrees LofCCDtcr . . . . ' . .. : : ". ~ : ~ : : ~ ~ : ~ ~~----~-----+----~~----+-----~----~ o 789 Figure 2: Filtered signals of the left and right microphones from 3 different angles. 790 Timothy Horiuchi The Localization System The analog front-end of the system (see figure 1) consists of three basic components, the microphones, the filter stage, and the thresholded, zero-crossing stage. Two microphones are placed with their centers about 2 inches apart. For any given time difference in arrival of acoustic stimuli, there are many possible locations from which the sound could have originated. These points describe a hyperbola with the two microphones as the two foci. If the sound source is distant enough, we can estimate the angle since the hyperbola approaches an asymptote. The current system operates on a single frequency and the inter-microphone distance has been chosen to be just under one wavelength apart at the filter frequency. The filter frequency chosen was 3.2 kHz because the author's finger snap, used extensively during development contained a large component at that frequency. The next step in the computation consists of triggering a digital pulse at the moment of zerocrossing if the acoustic signal is large enough. Thn::sb.olded Zct<Hlrossing Detection Pulses 6 7.5 5 5.0 4 25 '-- '-- '-- '-- '-- '-- '-0.0 ~ 3 ~ .e ·25 .§ } 2 .!! ! 1\ ·5.0 ~ II I!' < flA~ 'R ] ·75 II it 0 ..... Ii J'L ~ ! ..... '" ",... VV....., V V V '" ·10.0 ·1 V V ·125 ·2 -15.0 ·3 ·175 0 2 3 4 5 6 1imo (aeconda) (10") Figure 3: Example of output pulses from the external circuitry. Zero phase is chosen to be the positive-slope zero-crossing. Top: Digital pulses are generated at the time of zero phase for signals whose derivative is larger than a preset threshold. Bottom: 3.2 kHz Bandpass filtered signal for a finger snap. Phase Detection and Coordinate Transform in Analog VLSI The analog VLSI component of the system consists of two axon delay lines (Mead 1988) which propagate the left and right microphone pulse signals in opposing directions in order to compute the cross correlation (see Fig 4.) The location of the peak in this correlation technique represents the relative phase of the two signals. An Auditory Localization and Coordinate Transform Chip 791 This technique is described in more detail and with more biological justification by Lazzaro (1990). The current implementation contains 15 axon circuits in each delay line. This is shown in figure 4. At each position in the correlation delay line are logical AND circuits which output a logic one when there are two active axon units at that location. Since these units only turn on for specific time delays, they define auditory "receptive fields". The output of this subsystem are 15 digital lines which are passed on to the coordinate transform. Left Channel JlllL Right Channel Jllll Figure 4: Diagram of the double axon delay line which accepts digital spikes on the inputs and propagates them across the array. Whenever two spikes meet, a pulse is generated on the output AND units. The position of the AND circuit which gets activated indicates the relative time of arrival of the left and right inputs. NOTE: the actual circuit contains 15 axon units. A Head-based Auditory Units left ear-.... IIIUI=llI1l~ .... .- right ear f3 'S ::> s:: .g • .-4 A (I) ~ ~ ~ eye position Retinotopic Auditory Units Figure 5: For the one-dimensional case described in this project, the appropriate transform from head to retinal coordinates is a rotation which subtracts the eye position. The eye position information on the chip is represented as a voltage which activates one of the eye position units. The spatial pattern of activation from the auditory units is then "steered" to the output stage with the appropriate shift. (See figure 5). This 792 Timothy Horiuchi is similar to a shift scheme proposed by Pitts and McCulloch (1947) for obtaining pitch invariance for chord recognition. The eye position units are constructed from an array of "bump" circuits (Delbriick 1993) which compare the eye position voltage with its local voltage reference. The two dimensional array of intermediate units take the digital signal from the auditory units and switch the "bump" currents onto the output lines. The output current lines drive the inputs of a centroid circuit. The current implementation of the shift can be viewed as a basis function approach where a population of intermediate units respond to limited "ball-like" regions in the two-dimensional space of horizontal eye position and sound source azimuth (head-coordinates). The output units then sum the outputs of only those intermediate units which represent the same retinal location. It should be noted that this coordinate transformation is closely related to the "dendrite model" proposed for the projection of cortical auditory information to the deep SC by Groh and Sparks (1992). The final output stage converts this spatial array of current carrying lines into a single output voltage which represents the centroid of the stimulus in retinal coordinates. This centroid circuit (DeWeerth 1991) is intended to represent the primate SC where a similar computation is believed to occur. Results and Conclusions Figure 6 shows three plots of the chip's output voltage as a function of the interpulse time interval. Figure 7 shows three plots of the full system's output voltage for different eye position voltages. The output is roughly linear with azimuth and linear with eye position voltage. In operation, the system input consists of a sound entering the two microphones and the output consists of an analog voltage representing the position of the sound source and a digital signal indicating that the analog data is valid. The auditory localization system described here is currently in use with an analog VLSI-based model of the primate saccadic system to expand its operation into the auditory domain (Horiuchi, Bishofberger, & Koch 1994). In addition to the effort of our laboratory to model and understand biological computing structures in real-time systems, we are exploring the use of these low power integrated sensors in portable applications such as mobile robotics. Analog VLSI provides a compact and efficient implementation for many neuromorphic computing architectures which can potentially be used to provide, small, fast, low power sensors for a wide variety of applications. Acknowledgements The author would like to acknowledge Prof. Christof Koch for his academic support and use of laboratory facilities for this project, Brooks Bishofberger for his assistance in constructing some of the discrete electronics and Prof. Carver Mead for running the CNS184 course under which this chip was fabricated. The author is supported by an AASERT grant from the Office of Naval Research. An Auditory Localization and Coordinate Transform Chip 793 Output vs. Arrival Time Difference (3 eye positions) 3.1 3.0 2.9 6' ..; 2.8 old ..j N d 2.7 N II 0 £ 2.6 0 1 2.5 0 > <;; g 2.4 2.3 2.2 2.1 -600 -200 o 200 400 600 R to L delay (microseconda) Figure 6: Chip output vs. input pulse timing: The chip was driven with a signal generator and the output voltage was plotted for three different eye position voltages. Due to the discretized nature of the axon, there are only 15 axon locations at which pulses can meet. This creates the staircase response. References T. Delbriick (1993) Investigations of Analog VLSI Visual Transduction and Motion Processing, Ph.D. Thesis, California Institute of Technology J. Groh and D. Sparks (1992) 2 Models for Transforming Auditory Signals from Head-Centered to Eye-Centered Coordinates Bioi. Cybern. 67(4) 291-302. T. Horiuchi, B. Bishofberger, & C. Koch, (1994) An Analog VLSI-based Saccadic System, In (ed.), Advances in Neural Information Processing Systems 6 San Mateo, CA: Morgan Kaufman L. A. Jeffress (1948) A Place Theory of Sound Localization J. Comp_ Physiol. Psychol. 41: 35-39. M. Konishi (1986) Centrally Synthesized Maps of Sensory Space. TINS April, pp. 163-168. K. P. Krommenhoek, A. J. Van Opstal, C. C. A. M. Gielen, J. A. ,M. Van Gisbergen. (1993) Remapping of Neural Activity in the Motor Colliculus: A Neural Network Study. Vision Research 33(9):1287-1298. 794 Timothy Horiuchi J. Lazzaro. (1990) Silicon Models of Early Audition, Ph.D. Thesis, California Institute of Technology C. Mead, (1988) Analog VLSI and Neural Systems Menlo Park: Addison-Wesley W. Pitts and W. S. McCulloch, (1947) How we know universals: the perception of auditory and visual forms. Bulletin of Mathematical Biophysics 9:127-147. D. Zipser and R. A. Andersen (1988) A back-propagation programmed network that simulates response properties of a subset of posterior parietal neurons. Nature 331:679-684. Localization Output vs. Sound Source Azimuth 3.6 3.4 3.2 3.0 II -e ,. 28 on ..; 'Ij N 26 'Ij II 24 t S 22 -e ~ 20 1.8 1.6 1.4 ·20 0 Figure 7: Performance of the full system on continuous input (sinusoidal) delivered by a speaker from different angles. Note that 90 degrees denotes the center position. The three plots are the outputs for three different settings of the eye position input voltage.	1994	18
908	Limits on Learning Machine Accuracy Imposed by Data Quality Corinna Cortes, L. D. Jackel, and Wan-Ping Chiang AT&T Bell Laboratories Holmdel, NJ 07733 Abstract Random errors and insufficiencies in databases limit the performance of any classifier trained from and applied to the database. In this paper we propose a method to estimate the limiting performance of classifiers imposed by the database. We demonstrate this technique on the task of predicting failure in telecommunication paths. 1 Introduction Data collection for a classification or regression task is prone to random errors, e.g. inaccuracies in the measurements of the input or mis-labeling of the output. Missing or insufficient data are other sources that may complicate a learning task and hinder accurate performance of the trained machine. These insufficiencies of the data limit the performance of any learning machine or other statistical tool constructed from and applied to the data collection no matter how complex the machine or how much data is used to train it. In this paper we propose a method for estimating the limiting performance of learning machines imposed by the quality of the database used for the task. The method involves a series of learning experiments. The extracted result is, however, independent of the choice of learning machine used for these experiments since the estimated limiting performance expresses a characteristic of the data. The only requirements on the learning machines are that their capacity (VC-dimension) can be varied and can be made large, and that the learning machines with increasing capacity become capable of implementing any function. 240 Corinna Cortes. L. D. Jackel. Wan-Ping Chiang We have applied the technique to data collected for the purpose of predicting failures in telecommunication channels of the AT&T network. We extracted information from one of AT&T's large databases that continuously logs performance parameters of the network. The character and amount of data comes to more material than humans can survey. The processing of the extracted information is therefore automated by learning machines. We conjecture that the quality of the data imposes a limiting error rate on any learning machine of,... 25%, so that even with an unlimited amount of data, and an arbitrarily complex learning machine, the performance for this task will not exceed ,... 75% correct. This conjecture is supported by experiments. The relatively high noise-level of the data, which carries over to a poor performance of the trained classifier, is typical for many applications: the data collection was not designed for the task at hand and proved inadequate for constructing high performance classifiers. 2 Basic Concepts of Machine Learning We can picture a learning machine as a device that takes an unknown input vector and produces an output value. More formally, it performs some mapping from an input space to an output space. The particular mapping it implements depends of the setting of the internal parameters of the learning machine. These parameters are adjusted during a learning phase so that the labels produced on the training set match, as well as possible, the labels provided. The number of patterns that the machine can match is loosely called the "capacity" of the machine. Generally, the capacity of a machine increases with the number of free parameters. After training is complete, the generalization ability of of the machine is estimated by its performance on a test set which the machine has never seen before. The test and training error depend on both the the number of training examples I, the capacity h of the machine, and, of course, how well suited the machine is to implement the task at hand. Let us first discuss the typical behavior of the test and training error for a noise corrupted task as we vary h but keep the amount I of training data fixed. This scenario can, e.g., be obtained by increasing the number of hidden units in a neural network or increasing the number of codebook vectors in a Learning Vector Quantization algorithm [6]. Figure la) shows typical training and test error as a function of the capacity of the learning machine. For h < I we have many fewer free parameters than training examples and the machine is over constrained. It does not have enough complexity to model the regularities of the training data, so both the training and test error are large (underfitting). As we increase h the machine can begin to fit the general trends in the data which carries over to the test set, so both error measures decline. Because the performance of the machine is optimized on only part of the full pattern space the test error will always be larger than the training error. As we continue to increase the capacity of the learning machine the error on the training set continues to decline, and eventually it reaches zero as we get enough free parameters to completely model the training set. The behavior of the error on the test set is different. Initially it decreases, but at some capacity, h, it starts to rise. The rise occurs because the now ample resources of the training machine are applied to learning vagaries of the training Limits on Learnillg Machine Accuracy Imposed by Data Quality % error a) capacity ( h) % error b) . •. ~ . ..,..---- .. Ie " training training set size ( I) 241 % error c) • .. .. intrinsic noise level capacity ( h) Figure 1: Errors as function of capacity and training set size. Figure la) shows characteristic plots of training and test error as a function of the learning machine capacity for fixed training set size. The test error reaches a minimum at h = h while the training error decreases as h increases. Figure Ib) shows the training and test errors at fixed h for varying I. The dotted line marks the asymptotic error Eoo for infinite I. Figure lc) shows the asymptotic error as a function of h. This error is limited from below by the intrinsic noise in the data. set, which are not reproduced in the test set (overfitting). Notice how in Figure la) the optimal test error is achieved at a capacity h* that is smaller than the capacity for which zero error is achieved on the training set. The learning machine with capacity h* will typically commit errors on misclassified or outlying patterns of the training set. We can alternatively discuss the error on the test and training set as a function of the training set size I for fixed capacity h of the learning machine. Typical behavior is sketched in Figure Ib). For small I we have enough free parameters to completely model the training set, so the training error is zero. Excess capacity is used by the learning machine to model details in the training set, leading to a large test error. As we increase the training set size I we train on more and more patterns so the test error declines. For some critical size of the training set, Ie, the machine can no longer model all the training patterns and the training error starts to rise. As we further increase I the irregularities of the individual training patterns smooth out and the parameters of the learning machine is more and more used to model the true underlying function. The test error declines, and asymptotically the training and test error reach the same error value Eoo . This error value is the limiting performance of the given learning machine to the task. In practice we never have the infinite amount of training data needed to achieve Eoo. However, recent theoretical calculations [8, 1, 2, 7, 5] and experimental results [3] have shown that we can estimate Eoo by averaging the training and test errors for I> Ie. This means we can predict the optimal performance of a given machine. For a given type of learning machine the value of the asymptotic error Eoo of the machine depends on the quality of the data and the set of functions it can implement. The set of available functions increases with the capacity of the machine: 242 Corinna Cortes, L. D. Jackel, Wall-Ping Chiang low capacity machines will typically exhibit a high asymptotic error due to a big difference between the true noise-free function of the patterns and the function implemented by the learning machine, but as we increase h this difference decreases. If the learning machine with increasing h becomes a universal machine capable of modeling any function the difference eventually reaches zero, so the asymptotic error Eoo only measures the intrinsic noise level of the data. Once a capacity of the machine has been reached that matches the complexity of the true function no further improvement in Eoo can be achieved. This is illustrated in Figure lc). The intrinsic noise level of the data or the limiting performance of any learning machine may hence be estimated as the asymptotic value of Eoo as obtained for asymptotically universal learning machines with increasing capacity applied to the task. This technique will be illustrated in the following section. 3 Experimental Results In this section we estimate the limiting performance imposed by the data of any learning machine applied to the particular prediction task. 3.1 Task Description To ensure the highest possible quality of service, the performance parameters of the AT&T network are constantly monitored. Due to the high complexity of the network this performance surveillance is mainly corrective: when certain measures exceed preset thresholds action is taken to maintain reliable, high quality service. These reorganizations can lead to short, minor impairments of the quality of the communication path. In contrast, the work reported here is preventive: our objective is to make use of the performance parameters to form predictions that are sufficiently accurate that preemptive repairs of the channels can be made during periods of low traffic. In our study we have examined the characteristics of long-distance, 45 Mbitsfs communication paths in the domestic AT&T network. The paths are specified from one city to another and may include different kinds of physical links to complete the paths. A path from New York City to Los Angeles might include both optical fiber and coaxial cable. To maintain high-quality service, particular links in a path may be switched out and replaced by other, redundant links. There are two primary ways in which performance degradation is manifested in the path. First is the simple bit-error rate, the fraction of transmitted bits that are not correctly received at the termination of the path. Barring catastrophic failure (like a cable being cut), this error rate can be measured by examining the error-checking bits that are transmitted along with the data. The second instance of degradation, ''framing error" , is the failure of synchronization between the transmitter and receiver in a path. A framing error implies a high count of errored bits. In order to better characterize the distribution of bit errors, several measures are historically used to quantify the path performance in a 15 minutes interval. These measures are: Low-Rate The number of seconds with exactly 1 error. Limits Oil Learning Machine Accllracy Imposed by Data Quality 243 "No-Trouble" patterns: Frame-Error In Rate-High .. I Rate-Medlum I J Rate-Low I , o time (days) 21 "Trouble" pattems: • II I ,,' • , I l"e Figure 2: Errors as function of time. The 3 top patterns are members of the "No-Trouble" class. The 3 bottom ones are members of the "Trouble" class. Errors are here plotted as mean values over hours. Medium-Rate The number of seconds with more than one but less than 45 errors. High-Rate The number of seconds with 45 or more errors, corresponding to a bit error rate of at least 10-6 • Frame-Error The number of seconds with a framing error. A second with a frameerror is always accompanied by a second of High-Rate error. Although the number of seconds with the errors described above in principle could be as high as 900, any value greater than 255 is automatically clipped back to 255 so that each error measure value can be stored in 8 bits. Daily data that include these measures are continuously logged in an AT&T database that we call Perf(ormance)Mon(itor). Since a channel is error free most of the time, an entry in the database is only made if its error measures for a 15 minute period exceed fixed low thresholds, e.g. 4 Low-Rate seconds, 1 Medium- or HighRate second, or 1 Frame-Error. In our research we "mined" PerfMon to formulate a prediction strategy. We extracted examples of path histories 28 days long where the path at day 21 had at least 1 entry in the PerfMon database. We labeled the examples according to the error-measures over the next 7 days. If the channel exhibited a 15-minute period with at least 5 High-Rate seconds we labeled it as belonging to the class "Trouble". Otherwise we labeled it as member of "No-Trouble" . The length of the history- and future-windows are set somewhat arbitrarily. The history has to be long enough to capture the state of the path but short enough that our learning machine will run in a reasonable time. Also the longer the history the more likely the physical implementation of the path was modified so the error measures correspond to different media. Such error histories could in principle be eliminated from the extracted examples using the record of the repairs and changes 244 Corinna Cortes, L. D. lackel, Wan-Ping Chiang of the network. The complexity of this database, however, hinders this filtering of examples. The future-window of7 days was set as a design criterion by the network system engineers. Examples of histories drawn from PerfMon are shown in Figure 2. Each group of traces in the figure includes plots of the 4 error measures previously described. The 3 groups at the top are examples that resulted in No-Trouble while the examples at the bottom resulted in Trouble. Notice how bursty and irregular the errors are, and how the overall level of Frame- and High-Rate errors for the Trouble class seems only slightly higher than for the No-Trouble class, indicating the difficulty of the classification task as defined from the database PerfMon. PerfMon constitutes, however, the only stored information about the state of a given channel in its entirety and thus all the knowledge on which one can base channel end-to-end predictions: it is impossible to install extra monitoring equipment to provide other than the 4 mentioned end-to-end error measures. The above criteria for constructing examples and labels for 3 months of PerfMon data resulted in 16325 examples from about 900 different paths with 33.2% of the examples in the class Trouble. This means, that always guessing the label of the largest class, No-Trouble, would produce an error rate of about 33%. 3.2 Estimating Limiting Performance The 16325 path examples were randomly divided into a training set of 14512 examples and a test set of 1813 examples. Care was taken to ensure that a path only contributes to one of the sets so the two sets were independent, and that the two sets had similar statistical properties. Our input data has a time-resolution of 15 minutes. For the results reported here the 4 error measures of the patterns were subsampled to mean values over days yielding an input dimensionality of 4 x 21. We performed two sets of independent experiments. In one experiment we used fully connected neural networks with one layer of hidden units. In the other we used LVQ learning machines with an increasing number of codebook vectors. Both choices of machine have two advantages: the capacity of the machine can easily be increased by adding more hidden units, and by increasing the number of hidden units or number of codebook vectors we can eventually model any mapping [4]. We first discuss the results with neural networks. Baseline performance was obtained from a threshold classifier by averaging all the input signals and thresholding the result. The training data was used to adjust the single threshold parameter. With this classifier we obtained 32% error on the training set and 33% error on the test set. The small difference between the two error measures indicate statistically induced differences in the difficulty of the training and test sets. An analysis of the errors committed revealed that the performance of this classifier is almost identical to always guessing the label of the largest class "No-Trouble": close to 100% of the errors are false negative. A linear classifier with about 200 weights (the network has two output units) obtained 28% error on the training set and 32% error on the test set. Limits on Learning Machine Accuracy Imposed by Data Quality 245 40 classification error. % 40 classification error. % test 30 30 20 20 training 10~ ____ -. __________ ~~ 3 4 weights (log10) 10 ~-------------~ 2 3 codebook vectors (log 1 (} Figure 3: a) Measured classification errors for neural networks with increasing number of weights (capacity). The mean value between the test and training error estimates the performance of the given classifier trained with unlimited data. b) Measured classification errors for LVQ classifiers with increasing number of codebook vectors. Further experiments exploited neural nets with one layer of respectively 3, 5, 7, 10, 15, 20, 30, and 40 hidden units. All our results are summarized in Figure 3a). This figure illustrates several points mentioned in the text above. As the complexity of the network increases, the training error decreases because the networks get more free parameters to memorize the data. Compare to Figure 1a). The test error also decreases at first, going through a minimum of 29% at the network with 5 hidden units. This network apparently has a capacity that best matches the amount and character of the available training data. For higher capacity the networks overfit the data at the expense of increased error on the test set. Figure 3a) should also be compared to Figure 1c). In Figure 3a) we plotted approximate values of Eoo for the various networks the minimal error of the network to the given task. The values of Eoo are estimated as the mean of the training and test errors. The value of Eoo appears to flatten out around the network with 30 units, asymptotically reaching a value of 24% error. An asymptotic Eoo-value of 25% was obtained from LVQ-experiments with increasing number of codebook vectors. These results are summarized in Figure 3b). We therefore conjecture that the intrinsic noise level of the task is about 25%, and this number is the limiting error rate imposed by the quality of the data on any learning machine applied to the task. 246 Corinna Cortes, L. D. Jackel, Wan-Ping Chiang 4 Conclusion In this paper we have proposed a method for estimating the limits on performance imposed by the quality of the database on which a task is defined. The method involves a series of learning experiments. The extracted result is, however, independent of the choice of learning machine used for these experiments since the estimated limiting performance expresses a characteristic of the data. The only requirements on the learning machines are that their capacity can be varied and be made large, and that the machines with increasing capacity become capable of implementing any function. In this paper we have demonstrated the robustness of our method to the choice of classifiers: the result obtained with neural networks is in statistical agreement with the result obtained for LVQ classifiers. Using the proposed method we have investigated how well prediction of upcoming trouble in a telecommunication path can be performed based on information extracted from a given database. The analysis has revealed a very high intrinsic noise level of the extracted information and demonstrated the inadequacy of the data to construct high performance classifiers. This study is typical for many applications where the data collection was not necessarily designed for the problem at hand. Acknowledgments We gratefully acknowledge Vladimir Vapnik who brought this application to the attention of the Holmdel authors. One of the authors (CC) would also like to thank Walter Dziama, Charlene Paul, Susan Blackwood, Eric Noel, and Harris Drucker for lengthy explanations and helpful discussions of the AT&T transport system. References [1] s. Bos, W. Kinzel, and M. Opper. The generalization ability of perceptrons with continuous output. Physical Review E, 47:1384-1391, 1993. [2] Corinna Cortes. Prediction of Generalization Ability in Learning Machines. PhD thesis, University of Rochester, NY, 1993. [3] Corinna Cortes, L. D. Jackel, Sara A. So1la, V. Vapnik, and John S. Denker. Learning curves: Asymptotic value and rate of convergence. In Advances in Neural Information Processing Systems, volume 6. Morgan Kaufman, 1994. [4] G. Cybenko, K. Hornik, M. Stinchomb, and H. White. Multilayer feedforward neural networks are universal approximators. Neural Networks, 2:359-366, 1989. [5] T. L. Fine. Statistical generalization and learning. Technical Report EE577, Cornell University, 1993. [6] Teuvo Kohonen, Gyorgy Barna, and Ronald Chrisley. Statistical pattern recognition with neural networks: Benchmarking studies. In Proc. IEEE Int. Con! on Neural Networks, IJCNN-88, volume 1, pages 1-61-1-68, 1988. [7] N. Murata, S. Yoshizawa, and S. Amari. Learning curves, model selection, and complexity of neural networks. In Advances in Neural Information Processing Systems, volume 5, pages 607-614. Morgan Kaufman, 1992. [8] H. S. Seung, H. Sompolinsky, and N. Tishby. Statistical mechanics of learning from examples. Physical Review A, 45:6056-6091, 1992.	1994	19
909	A Model of the Neural Basis of the Rat's Sense of Direction William E. Skaggs James J. Knierim bill@nsma.arizona. edu jim@nsma.arizona. edu Hemant S. Kudrimoti hemant@nsma. arizona. edu Bruce L. McNaughton bruce@nsma. arizona. edu ARL Division of Neural Systems, Memory, And Aging 344 Life Sciences North, University of Arizona, 'IUcson AZ 85724 Abstract In the last decade the outlines of the neural structures subserving the sense of direction have begun to emerge. Several investigations have shed light on the effects of vestibular input and visual input on the head direction representation. In this paper, a model is formulated of the neural mechanisms underlying the head direction system. The model is built out of simple ingredients, depending on nothing more complicated than connectional specificity, attractor dynamics, Hebbian learning, and sigmoidal nonlinearities, but it behaves in a sophisticated way and is consistent with most of the observed properties ofreal head direction cells. In addition it makes a number of predictions that ought to be testable by reasonably straightforward experiments. 1 Head Direction Cells in the Rat There is quite a bit of behavioral evidence for an intrinsic sense of direction in many species of mammals, including rats and humans (e.g., Gallistel, 1990). The first specific information regarding the neural basis of this "sense" came with the discovery by Ranck (1984) of a population of "head direction" cells in the dorsal presubiculum (also known as the "postsubiculum") of the rat. A head direction cell 174 William Skaggs, James J. Knierim, Hemant S. Kudrimoti, Bruce L. McNaughton fires at a high rate if and only if the rat's head is oriented in a specific direction. Many things could potentially cause a cell to fire in a head-direction dependent manner: what made the postsubicular cells particularly interesting was that when their directionality was tested with the rat at different locations, the head directions corresponding to maximal firing were consistently parallel, within the experimental resolution. This is difficult to explain with a simple sensory-based mechanism; it implies something more sophisticated.1 The postsubicular head direction cells were studied in depth by Taube et al. (1990a,b), and, more recently, head direction cells have also been found in other parts of the rat brain, in particular the anterior nuclei of the thalamus (Mizumori and Williams, 1993) and the retrosplenial (posterior cingulate) cortex (Chen et al., 1994a,b). Interestingly, all of these areas are intimately associated with the hippocampal formation, which in the rat contains large numbers of "place" cells. Thus, the brain contains separate but neighboring populations of cells coding for location and cells coding for direction, which taken together represent much of the information needed for navigation. Figure 1 shows directional tuning curves for three typical head direction cells from the anterior thalamus. In each of them the breadth of tuning is on the order of 90 degrees. This value is also typical for head direction cells in the postsubiculum and retrosplenial cortex, though in each of the three areas individual cells may show considerable variability. Figure 1: Polar plots of directional tuning (mean firing rate as a function of head direction) for three typical head direction cells from the anterior thalamus of a rat. Every study to date has indicated that the head direction cells constitute a unitary system, together with the place cells of the hippocampus. Whenever two head direction cells have been recorded simultaneously, any manipulation that caused one of them to shift its directional alignment caused the other to shift by the same amount; and when head direction cells have been recorded simultaneously with place cells, any manipulation that caused the head direction cells to realign either caused the hippocampal place fields to rotate correspondingly or to "remap" into a different pattern (Knierim et al., 1995). Head direction cells maintain their directional tuning for some time when the lights in the recording room are turned off, leaving an animal in complete darkness; the directionality tends to gradually drift, though, especially if the animal moves around (Mizumori and Williams, 1993). Directional tuning is preserved to some degree even 1 Sensitivity to the Earth's geomagnetic field has been ruled out as an explanation of head-directional firing. A Model of the Neural Basis of the Rat's Sense of Direction 175 if an animal is passively rotated in the dark, which indicates strongly that the head direction system receives information (possibly indirect) from the vestibular system. Visual input influences but does not dictate the behavior of head direction cells. The nature of this influence is quite interesting. In a recent series of experiments (Knierim et al., 1995), rats were trained to forage for food pellets in a gray cylinder with a single salient directional cue, a white card covering 90 degrees of the wall. During training, half of the rats were disoriented before being placed in the cylinder, in order to disrupt the relation between their internal sense of direction and the location of the cue card; the other half of the rats were not disoriented. Presumably, the rats that were not disoriented during training experienced the same initial relationship between their internal direction sense and the CUe card each time they were placed in the cylinder; this would not have been true of the disoriented rats. Head direction cells in the thalamus were subsequently recorded from both groups of rats as they moved in the cylinder. All rats were disoriented before each recording session. Under these conditions, the cue card had much weaker control over the head direction cells in the rats that had been disoriented during training than in the rats that had not been disoriented. For all rats the influence of the cue card upon the head direction system weakened gradually over the course of multiple recording sessions, and eventually they broke free, but this happened much sooner in the rats that had been disoriented during training. The authors concluded that a visual cue could only develop a strong influence upon the head direction system if the rat experienced it as stable. Figure 2 illustrates the shifts in alignment during a typical recording session. When the rat is initially placed in the cylinder, the cell's tuning curve is aligned to the west. Over the first few minutes of recording it gradually rotates to SSW, and there it stays. Note the "tail" of the curve. This comes from spikes belonging to another, neighboring head direction cell, which could not be perfectly isolated from the first. Note that, even though they come from different cells, both portions shift alignment synchronously. Figure 2: Shifts in alignment of a head direction cell over the course of a single recording session (one minute intervals). 2 The Model As reviewed above, the most important facts to be accounted for by any model of the head direction system are (1) the shape of the tuning curves for head direction cells, (2) the control of head direction cells by vestibular input, and (3) the stabilitydependent influence of visual cues on head direction cells. We introduce here a 176 William Skaggs, James J. Knierim, Hemant S. Kudrimoti, Bruce L McNaughton model that accounts for these facts. It is a refinement of a model proposed earlier by McNaughton et al. (1991), the main addition being a more specific account of neural connections and dynamics. The aim of this effort is to develop the simplest possible architecture consistent with the available data. The reality is sure to be more complicated than this model. Figure 3 schematically illustrates the architecture of the model. There are four groups of cells in the model: head direction cells, rotation cells (left and right), vestibular cells (left and right), and visual feature detectors. For expository purposes it is helpful to think of the network as a set of circular layers; this does not reflect the anatomical organization of the corresponding cells in the brain. @ @ V.oIIbul .. col (rtght) 00 °°00 0 ° 0 ° 0 ° 0 0 00 @ @ 00 @ @ ~ROt.lOftC~I~etQ 66 @ ® ·00---- Rotadon cell (rtght) ° 0 @ @ 0°0 0°0 @ ® 0 00 ° ° ° ° ° 00 0 ® ® 00000 ® @ @ @ ® ~H~ eM.ocUon coM Figure 3: Architecture of the head direction cell model. The head direction cell group has intrinsic connections that are stronger than any other connections in the model, and dominate their dynamics, so that other inputs only provide relatively small perturbations. The connections between them are set up so that the only possible stable state of the system is a single localized cluster of active cells, with all other cells virtually silent. This will occur if there are strong excitatory connections between neighboring cells, and strong inhibitory connections between distant cells. It is assumed that the network of interconnections has rotation and reflection symmetry. Small deviations from symmetry will not impair the model too much; large deviations may cause it to have strong attractors at a few points on the circle, which would cause problems. The crucial property of this network is the following. Suppose it is in a stable state, with a single cluster of activated cells at one point on the circle, and suppose an external input is applied that excites the cells selectively on one side (left or right) A Model of the Neural Basis of the Rat' s Sense of Direction 177 of the peak. Then the peak will rotate toward the side at which the input is applied, and the rate of rotation will increase with the strength of the input. This feature is exploited by the mechanisms for vestibular and visual control of the system. The vestibular mechanism operates via a layer of "rotation" cells, corresponding to the circle of head direction cells (Units with a similar role were referred to as "H x H'" cells in the McNaughton et al. (1991) model). There are two groups of rotation cells, for left and right rotations. Each rotation cell receives excitatory input from the head direction cell at the same point on the circle, and from the vestibular system. The activation function of the rotation cell is sigmoidal or threshold linear, so that the cell does not become active unless it receives input simultaneously from both sources. Each right rotation cell sends excitatory projections to head direction cells neighboring it on the right, but not to those that neighbor it on the left, and contrariwise for left rotation cells. It is easy to see how the mechanism works. When the animal turns to the right, the right vestibular cells are activated, and then the right rotation cells at the current peak of the head direction system are activated. These add to the excitation of the head direction cells to the right of the peak, thereby causing the peak to shift rightward. This in turn causes a new set of rotation cells to become active (and the old ones inactive), and thence a further shift of the peak, and so on. The peak will continue to move around the circle as long as the vestibular input is active, and the stronger the vestibular input, the more rapidly the peak will move. If the gain of this mechanism is correct (but weak compared to the gain of the intrinsic connections of the head direction cells), then the peak will move around the circle at the same rate that the animal turns, and the location of the peak will function as an allocentric compass. This can only be expected to work over a limited range of turning rates, but the firing rates of cells in the vestibular nuclei are linearly proportional to angular velocity over a surprisingly broad range, so there is no reason why the mechanism cannot perform adequately. Of course the mechanism is intrinsically error-prone, and without some sort of external correction, deviations are sure to build up over time. But this is an inevitable feature of any plausible model, and in any case does not conflict with the available data, which, while sketchy, suggests that passive rotation of animals in the dark can cause quite erratic behavior in head direction cells (E. J. Markus, J. J . Knierim, unpublished observations). The final ingredient of the model is a set of visual feature detectors, each of which responds if and only if a particular visual feature is located at a particular angle with respect to the axis of the rat's head. Thus, these cells are feature specific and direction specific, but direction specific in the head-centered frame, not in the world frame. It is assumed that each visual feature detector projects weakly to all of the head direction cells, and that these connections are modifiable according to a Hebbian rule, specifically, ~W = a(Wmaxt(Aposd W)Apre, where W is the connection weight, Wmax is its maximum possible value, Apost is the firing rate of the postsynaptic cell, Apre is the firing rate of the presynaptic cell, and the function to has the shape shown in figure 4. (Actually, the rule is modified slightly to prevent any of the weights from becoming negative.) The net effect of 178 William Skaggs, James J. Knierim, Hemant S. Kudrimoti, Bruce L. McNaughton this rule is that the weight will only change when the presynaptic cell (the visual feature detector) is active, and the weight will increase if the postsynaptic cell is strongly active, but decrease ifit is weakly active or silent. Modification rules of this form have previously been proposed in theories of the development of topography in the neocortex (e.g., Bienenstock et al., 1982), and there is considerable evidence for such an effect in the control of LTP /LTD (Singer and Artola, 1994). 1(rate) rate Figure 4: Dependence of synaptic weight change on postsynaptic firing rate for connections from visual feature detectors to head direction cells in the model. To understand how this works, suppose we have a feature detecting cell that responds to a cue card whenever the cue card is directly in front of the rat. Suppose the rat's motion is restricted to a small area, and the cue card is far away, so that it is always at approximately the same absolute bearing (say, 30 degrees), and suppose the rat's head direction system is working correctly, i.e., functioning as an absolute compass. Then the cell will only be active at moments when the head direction cells corresponding to 30 degrees are active, and the Hebbian learning process will cause the feature detecting cell to be linked by strong weights to these cells, but by vanishing weights to other head direction cells. If the absolute bearing of the cue card were more variable, then the connection strengths from the feature detecting cell would be weaker and more broadly dispersed. In the limit where the bearing of the cue card was completely random, all connections would be weak and equal. Thus the influence of a visual cue is determined by the amount of training and by the variability in its bearing (with respect to the head direction system). It can be seen that the model implements a competition between visual inputs and vestibular inputs for control of the head direction cells. If the visual cues are rotated while the rat is left stationary, then the head direction cells may either rotate to follow the visual cues, or stick with the inertial frame, depending on parameter values and, importantly, on the training regimen imposed on the network. Both of these outcomes have been observed in anterior thalamic head direction cells (McNaughton et al., 1993). 3 Discussion Do the necessary types of cells exist in the brain? Cells in the brainstem vestibular nuclei are known to have the properties required by the model (Precht, 1978). The "rotation" cells would be recognizable from the fact that they would fire only when A Model of the Neural Basis of the Rat's Sense of Direction 179 the rat is facing in a particular direction and turning in a particular direction, with rate at least roughly proportional to the speed ofturning. Cells with these properties have been recorded in the postsubiculum (Markus et al., 1990) and retrosplenial cortex (Chen et ai., 1994a). The visual cells would be recognizeable from the fact that they would respond to visual stimuli only when they come from a particular direction with respect to the animal's head axis. Cells with these properties have been recorded in the inferior parietal cortex, the internal medullary lamina of the thalamus, and the superior colliculus (e.g., Sparks, 1986). The superior colliculus also contains cells that respond in a direction-dependent manner to auditory inputs, thus allowing a possiblility of control of the head direction system by sound sources. There do not seem to be any strong direct projections from the superior colli cui us to the components of the head direction system, but there are numerous indirect pathways. The most general prediction of the model is that the influence of vestibular input upon head direction cells is not susceptible to experience-dependent modification, whereas the influence of visual input is plastic, and is enhanced by the duration of experience, the richness of the visual cue array, and the distance of visual cues from the rat's region of travel. The "rotation" cells should be responsive to stimulation of the vestibular system. It is possible to activate the vestibular system by applying hot or cold water to the ears: if this is done in the dark, and head direction cells are simultaneously recorded, the model predicts that they will show periodic bursts of activity, with a frequency related to the intensity of the stimulus. For another prediction, suppose we train two groups of rats to forage in a cylinder containing a single landmark. For one group, the landmark is placed at the edge of the cylinder; for the other group, the same landmark is placed halfway between the center and the edge. The model predicts that in both cases the landmark will influence the head direction sytem, but the influence will be stronger and more tightly focused when the landmark is at the edge. In some respects the model is flexible, and may be extended without compromising its essence. For example, there is no intrinsic necessity that the vestibular system be the sole input to the rotation cells (other than the head direction cells). The performance of the system might be improved in some ways by sending the rotation cells input about optokinetic flow, or certain types of motor efference copy. But there is as yet no clear evidence for these things. On a more abstract level, the mechanism used by the model for vestibular control may be thought of as a special case of a general-purpose method for integration with neurons. As such, it has significant advantages over some previously proposed neural integrators, in particular, better stability properties. It might be worth considering whether the method is applicable in other situations where integrators are known to exist, for example the control of eye position. 180 William Skaggs, James J. Knierim, Hemant S. Kudrimoti, Bruce L. McNaughton Supported by MH46823 and O.N .R. References 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. J. Neurosci., 2:32-48. Chen, L. L., Lin, L., Green, E. J., Barnes, C. A., and McNaughton, B. L. (1994b). Head-direction cells in the rat posterior cortex. 1. Anatomical distribution and behavioral modulation. Exp. Brain Res., 101:8-23. Chen, L. L., Lin, L., Barnes, C. A., and McNaughton, B. L. (1994a). Head-direction cells in the rat posterior cortex. II. Contributions of visual and ideothetic information to the directional firing. Exp. Brain Res., 101:24-34. Gallistel, C. R. (1990). The Organization oj Learning. MIT Press, Cambridge, Massachusetts. Knierim, J. J ., Kudrimoti, H. S., and McNaughton, B. L. (1995). Place cells, head direction cells, and the learning of landmark stability. J. Neurosci. (in press). Markus, E. J., McNaughton, B. L., Barnes, C. A., Green, J. C., and Meltzer, J. (1990). Head direction cells in the dorsal presubiculum integrate both visual and angular velocity information. Soc. Neurosci. Abs., 16:44l. McNaughton, B. L., Chen, L. L., and Markus, E. J. (1991). ((Dead reckoning," landmark learning, and the sense of direction: A neurophysiological and computational hypothesis. J. Cognit. Neurosci., 3: 190-202. McNaughton, B. L., Markus, E. J., Wilson, M. A., and Knierim, J . J. (1993). Familiar landmarks can correct for cumulative error in the inertially based dead-reckoning system. Soc. Neurosci. Abs., 19:795. Mizumori, S. J. and Williams, J. D. (1993). Directionally selective mnemonic properties of neurons in the lateral dorsal nucleus of the thalamus of rats. J. Neurosci., 13:4015-4028. Precht, W. (1978). Neuronal operations in the vestibular system. Springer, New York. Ranck, Jr., J. B. (1984). Head direction cells in the deep cell layer of dorsal presubiculum in freely moving rats. Soc. Neurosci. Abs., 10:599. Singer, W. and Artola, A. (1994). Plasticity of the mature neocortex. In Selverston, A. I. and Ascher, P., editors, Cellular and molecular mechanisms underlying higher neural junctions, pages 49-69. Wiley. Sparks, D. L. (1986). Translation of sensory signals into commands for control of saccadic eye movements: role of primate superior colliculus. Physiol. Rev., 66:118-17l. Taube, J. S., Muller, R. V., and Ranck, Jr., J. B. (1990a). Head direction cells recorded from the postsubiculum in freely moving rats. I. Description and quantitative analysis. J. Neurosci., 10:420-435. Taube, J. S., Muller, R. V., and Ranck, Jr., J. B. (1990b) . Head direction cells recorded from the postsubiculum in freely moving rats. II. Effects of environmental manipulations. J. Neurosci., 10:436-447. PARTm LEARNiNGTHEORYANDDYNANUCS	1994	2
910	Interference in Learning Internal Models of Inverse Dynamics in Humans Reza Shadmehr; Tom Brashers-Krug, and Ferdinando Mussa-lvaldit Dept. of Brain and Cognitive Sciences M. I. T., Cambridge, MA 02139 reza@bme.jhu.edu, tbk@ai.mit.edu, sandro@parker.physio.nwu.edu Abstract Experiments were performed to reveal some of the computational properties of the human motor memory system. We show that as humans practice reaching movements while interacting with a novel mechanical environment, they learn an internal model of the inverse dynamics of that environment. Subjects show recall of this model at testing sessions 24 hours after the initial practice. The representation of the internal model in memory is such that there is interference when there is an attempt to learn a new inverse dynamics map immediately after an anticorrelated mapping was learned. We suggest that this interference is an indication that the same computational elements used to encode the first inverse dynamics map are being used to learn the second mapping. We predict that this leads to a forgetting of the initially learned skill. 1 Introduction In tasks where we use our hands to interact with a tool, our motor system develops a model of the dynamics of that tool and uses this model to control the coupled dynamics of our arm and the tool (Shadmehr and Mussa-Ivaldi 1994). In physical systems theory, the tool is a mechanical analogue of an admittance, mapping a force as input onto a change in state as output (Hogan 1985). In this framework, the ·Currently at Dept. Biomedical Eng, Johns Hopkins Univ, Baltimore, MD 21205 tCurrently at Dept. Physiology, Northwestern Univ Med Sch (M211), Chicago, IL 60611 1118 Reza Shadmehr, Tom Brashers-Krug, Ferdinando Mussa-Ivaldi Figure 1: The experimental setup. The robot is a very low friction planar mechanism powered by two torque motors that act on the shoulder and elbow joints. Subject grips the end-point of the robot which houses a force transducer and moves the hand to a series of targets displayed on a monitor facing the subject (not shown). The function of the robot is to produce novel force fields that the subject learns to compensate for during reaching movements. model developed by the motor control system during the learning process needs to approximate an inverse of this mapping. This inverse dynamics map is called an internal model of the tool. We have been interested in understanding the representations that the nervous system uses in learning and storing such internal models. In a previous work we measured the way a learned internal model extrapolated beyond the training data (Shadmehr and Mussa-Ivaldi 1994). The results suggested that the coordinate system of the learned map was in intrinsic (e.g., joint or muscles based) rather than in extrinsic (e.g., hand based) coordinates. Here we present a mathematical technique to estimate the input-output properties of the learned map. We then explore the issue of how the motor memory might store two maps which have similar inputs but different outputs. 2 Quantifying the internal model In our paradigm, subjects learn to control an artificial tool: the tool is a robot manipulandum which has torque motors that can be programmed to produce a variety of dynamical environments (Fig. 1). The task for the subject is to grasp the end-effector and make point to point reaching movements to a series of targets. The environments are represented as force fields acting on the subject's hand, and a typical case is shown in Fig. 2A. A typical experiment begins with the robot motors turned off. In this "null" environment subjects move their hand to the targets in a smooth, straight line fashion. When the force field is introduced, the dynamics of the task change and the hand trajectory is significantly altered (Shadmehr and MussaIvaldi 1994). With practice (typically hundreds of movements), hand trajectories return to their straight line path. We have suggested that practice leads to formation of an internal model which functions as an inverse dynamics mapping, i.e., from a desired trajectory (presumably in terms of hand position and velocity, Wolpert et al. 1995) to a prediction of forces that will be encountered along the trajectory. We designed a method to quantify these forces and estimate the output properties of the internal model. If we position a force transducer at the interaction point between the robot and the subject, we can write the dynamics of the four link system in Fig. 1 in terms of the Interference in Learning Internal Models of Inverse Dynamics in Humans following coupled vector differential equations: Ir(P)P + Gr(p,p)p = E(p,p) + J'{ F III (q)q + GII(q, q)q = C(q, q, q(t» - f; F 1119 (1) (2) where I and G are inertial and Corriolis/centripetal matrix functions, E is the torque field produced by the robot's motors, i.e., the environment, F is the force measured at the handle of the robot, C is the controller implemented by the motor system of the subject, q(t) is the reference trajectory planned by the motor system of the subject, J is the Jacobian matrix describing the differential transformation of coordinates from endpoint to joints, q and p are joint positions of the subject and the robot, and the subscripts sand r denote subject or robot matrices. In the null environment, i.e., E = ° in Eq. (1), a solution to this coupled system is q = q(t) and the arm follows the reference trajectory (typically a straight hand path with a Gaussian tangential velocity profile). Let us name the controller which accomplishes this task C = Co in Eq. (2). When the robot motors are producing a force field E # 0, it can be shown that the solution is q = q(t) if and only if the new controller in Eq. (2) is C = C1 = Co + f[ J;T E. The internal model composed by the subject is C1 - Co, i.e., the change in the controller after some training period. We can estimate this quantity by measuring the change in the interaction force along a given trajectory before and after training. If we call these functions Fo and FI, then we have: Fo(q, q, ij, q(t» J;T(Co - IlIq - Gllq) (3) FI(q,q,ij,q(t» JII-T(Co+f;J;TE-Illq-Gllq) (4) The functions Fo and FI are impedances of the subject's arm as viewed from the interaction port. Therefore, by approximating the difference FI - Fo, we have an estimate of the change in the controller. The crucial assumption is that the reference trajectory q*(t) does not change during the training process. In order to measure Fo, we had the subjects make movements in a series of environments. The environments were unpredictable (no opportunity to learn) and their purpose was to perturb the controller about the reference trajectory so we could measure Fo at neighboring states. Next, the environment in Fig. 2A was presented and the subject given a practice period to adapt. After training, FI was estimated in a similar fashion as Fo. The difference between these two functions was calculated along all measured arm trajectories and the results were projected onto the hand velocity space. Due to computer limitations, only 9 trajectories for each target direction were used for this approximation. The resulting pattern of forces were interpolated via a sum of Gaussian radial basis functions, and are shown in Fig. 2B. This is the change in the impedance of the arm and estimates the inputoutput property of the internal model that was learned by this subject. We found that this subject, which provided some of best results in the test group, learned to change the effective impedance of his arm in a way that approximated the imposed force field. This would be a sufficient condition for the arm to compensate for the force field and allow the hand to follow the desired trajectory. An alternate strategy might have been to simply co-contract arm muscles: this would lead to an increased stiffness and an ability to resist arbitrary environmental forces. Figure 2B suggests that practice led to formation of an internal model specific to the dynamics of the imposed force field. 1120 Reza Shadmehr, Tom Brashers-Krug, Ferdinando Mussa-Ivaldi A B ",",,-<...-) -200 0 200 ... ..... _<...-) Figure 2: Quantification of the change in impedance of a subject's arm after learning a force field. A: The force field produced by the robot during the training period. B: The change in the subject's arm impedance after the training period, i.e., the internal model. 2.1 Formation of the internal model in long-term memory Here we wished to determine whether subjects retained the internal model in longterm motor memory. We tested 16 naive subjects. They were instructed to move the handle of the robot to a sequence of targets in the null environment. Each movement was to last 500 ± 50 msec. They were given visual feedback on the timing of each movement. After 600 movements, subjects were able to consistently reach the targets in proper time. These trajectories constituted a baseline set. Subjects returned the next day and were re-familiarized with the timing of the task. At this point a force field was introduced and subjects attempted to perform the exact task as before: get to the target in proper time. A sequence of 600 targets was given. When first introduced, the forces perturbed the subject's trajectories, causing them to deviate from the straight line path. As noted in previous work (Shadmehr and Mussa-Ivaldi 1994), these deviations decreased with practice. Eventually, subject's trajectories in the presence of the force field came to resemble those of the baseline, when no forces were present. The convergence of the trajectories to those performed at baseline is shown for all 16 subjects in Fig. 3A. The timing performance of the subjects while moving in the field is shown in Fig. 3B. In order to determine whether subjects retained the internal model of the force field in long-term memory, we had them return the next day (24 to 30 hours later) and once again be tested on a force field. In half of the subjects, the force field presented was one that they had trained on in the previous day (call this field 1). In the other half, it was a force field which was novel to the subjects, field 2. Field 2 had a correlation value of -1 with respect to field 1 (i.e., each force vector in field 2 was a 180 degree rotation of the respective vector in field 1). Subjects who were tested on a field that they had trained on before performed significantly better (p < 0.01) than their initial performance (Fig. 4A), signifying retention. However, those who were given a field that was novel performed at naive levels (Fig. 4B). This result suggested that the internal model formed after practice in a given field was (1) specific to that field: performance on the untrained field was no better than Interference in Learning Internal Models of Inverse Dynamics in Humans 1121 0.9 0.85 ¥ .~ 0.8 :§ ~ 0.75 8 0.7 -; 0.9 ~ i 08 ~ 0.7 0.85 0.6 A 0 100 200 300 400 500 600 B 0 100 200 300 400 500 600 Movemen1 N!mber Movement Number Figure 3: Measures of performance during the training period (600 movements) for 16 naive subjects. Short breaks (2 minutes) were given at intervals of 200 movements. A : Mean ± standard error (SE) of the correlation coefficient between hand trajectory in a null environment (called baseline trajectories, measured before exposure to the field) , and trajectory in the force field. Hand trajectories in the field converge to that in the null field (i.e., become straight, with a bell shaped velocity profile). B: Mean ± SE of the movement period to reach a target. The goal was to reach the target in 0.5 ± 0.05 seconds. I 1 ., 0.9 E i= I:: \ ~iIJIJ ,l,ll,lI" :::;; 0.6 ~ , ~ ' , ; 0) "T' I,.", 1f11T'1 A Y!1y~ o 100 200 300 400 500 600 Movement Number B 0.8 10.75 ., E 0.7 i= ~ 0.65 E ~ 0.8 :::;; 0.55 0 1 00 200 300 400 500 600 Movement Number Figure 4: Subjects learned an internal model specific to the field and retained it in longterm memory. A: Mean ± standard error (SE) of the movement period in the force field (called field 1) during initial practice session (upper trace) and during a second session 24-30 hours after the initial practice (lower trace). B: Movement period in a different group of subjects during initial training (dark line) in field 1 and test in an anti-correlated field (called field 2) 24-30 hours later (gray line). performance recorded in a separate set of naive subjects who were given than field in their initial training day; and (2) could be retained, as evidenced by performance in the following day. 2.2 Interference effects of the motor memory In our experiment the "tool" that subjects learn to control is rather unusual, nevertheless, subjects learn its inverse dynamics and the memory is used to enhance performance 24 hours after its initial acquisition. We next asked how formation of this memory affected formation of subsequent internal models. In the previous section we showed that when a subject returns a day after the initial training, although the memory of the learned internal model is present, there is no interference (or decrement in performance) in learning a new, anti-correlated field. Here we show that when this temporal distance is significantly reduced, the just learned 1122 Reza Shadmehr, Tom Brashers-Krug, Ferdinanda Mussa-Ivaldi 200 300 400 Movement Number Figure 5: Interference in sequential learning of two uncorrelated force fields: The lower trace is the mean and standard error of the movement periods of a naive group of subjects during initial practice in a force field (called field 1). The upper trace is the movement period of another group of naive subjects in field 1, 5 minutes after practicing 400 movements in field 2, which was anti-correlated with field 1. model interferes with learning of a new field. Seven new subjects were recruited. They learned the timing of the task in a null environment and in the following day were given 400 targets in a force field (called field 1). They showed improvement in performance as before. After a short break (5-10 minutes in which they walked about the lab or read a magazine), they were given a new field: this field was called field 2 and was anti-correlated with respect to field 1. We found a significant reduction (p < 0.01) in their ability to learn field 2 (Fig. 5) when compared to a subject group which had not initially trained in field 1. In other words, performance in field 2 shortly after having learned field 1 was significantly worse than that of naives. Subjects seemed surprised by their inability to master the task in field 2. In order to demonstrate that field 2 in isolation was no more difficult to learn than field 1, we had a new set of subjects (n = 5) initially learn field 2, then field 1. Now we found a very large decrement in learn ability of field 1. One way to explain the decrement in performance shown in Fig. 5 is to assume that the same "computational elements" that represented the internal model of the first field were being used to learn the second field.! In other words, when the second field was given, because the forces were opposite to the first field, the internal model was badly biased against representing this second field: muscle torque patterns predicted for movement to a given target were in the wrong direction. In the connectionist literature this is a phenomenon called temporal interference (Sutton 1986). As a network is trained, some of its elements acquire large weights and begin to dominate the input-output transformation. When a second task is presented with a new and conflicting map (mapping similar inputs to different outputs), there are large errors and the network performs more poorly than a "naive" network. As the network attempts to learn the new task, the errors are fed to each element (i.e., pre-synaptic input). This causes most activity in those elements that 1 Examples of computational elements used by the nervous system to model inverse dynamics of a mechanical system were found by Shidara et al. (1993), where it was shown that the firing patterns of a set of Purkinje cells in the cerebellum could be reconstructed by an inverse dynamic representation of the eye. Interference in Learning Internal Models of Inverse Dynamics in Humans 1123 had the largest synaptic weight. If the learning algorithm is Hebbian, i.e., weights change in proportion to co-activation of the pre- and the post-synaptic element, then the largest weights are changed the most, effectively causing a loss of what was learned in the first task. Therefore, from a computational stand point, we would expect that the internal model of field 1 as learned by our subjects should be destroyed by learning of field 2. Evidence for "catastrophic interference" in these subjects is presented elsewhere in this volume (Brashers-Krug et al. 1995). The phenomenon of interference in sequential learning of two stimulus-response maps has been termed proactive interference or negative transfer in the psychological literature. In humans, interference has been observed extensively in verbal tasks involving short-term declarative memory (e.g., tasks involving recognition of words in a list or pairing of non-sense syllables, Bruce 1933, Melton and Irwin 1940, Sears and Hovland 1941). It has been found that interference is a function of the similarity of the stimulus-response maps in the two tasks: if the stimulus in the new learning task requires a response very different than what was recently learned, then there is significant interference. Interestingly, it has been shown that the amount of interference decreases with increased learning (or practice) on the first map (Siipola and Israel 1933). In tasks involving procedural memory (which includes motor learning, Squire 1986), the question of interference has been controversial: Although Lewis et al. (1949) reported interference in sequential learning of two motor tasks which involved moving levers in response to a set of lights, it has been suggested that the interference that they observed might have been due to cognitive confusion (Schmidt 1988). In another study, Ross (1974) reported little interference in subjects learning her motor tasks. We designed a task that had little or no cognitive components. We found that shortly after the acquisition of a motor memory, that memory strongly interfered with learning of a new, anti-correlated input-output mapping. However, this interference was not significant 24 hours after the memory was initially acquired. One possible explanation is that the initial learning has taken place in a temporary and vulnerable memory system. With time and/or practice, the information in this memory had transferred to long-term storage (Brashers-Krug et al. 1995). Brain imaging studies during motor learning suggest that as subjects become more proficient in a motor task, neural fields in the motor cortex display increases in activity (Grafton et al. 1992) and new fields are recruited (Kawashima et al. 1994). It has been reported that when a subject attempts to learn two new motor tasks successively (in this case the tasks consisted of two sequences of finger movements), the neural activity in the motor cortex is lower for the second task, even when the order ofthe tasks is reversed (Jezzard et al. 1994). It remains to be seen whether this decrement in neural activity in the motor cortex is correlated with the interference observed when subjects attempt to learn two different input-output mappings in succession (Gandolfo et al. 1994). References Brashers-Krug T , Shadmehr R, Todorov E (1995) Catastrophic interference in human motor learning. Adv Neural Inform Proc Syst, vol 7, in press. 1124 Reza Shadmehr, Tom Brashers-Krug, Ferdinando Mussa-Ivaldi Bruce RW (1933) Conditions of transfer of training. J Exp Psychol 16:343-361. French, R. (1992) Semi-distributed Representations and Catastrophic Forgetting in Connectionist Networks, Connection Science 4:365-377. Grafton ST et al. (1992) Functional anatomy of human procedural learning determined with regional cerebral blood flow and PET. J Neurosci 12:2542-2548. Gandolfo F, Shadmehr R, Benda B, Bizzi E (1994) Adaptive behavior ofthe monkey motor system to virtual environments. Soc Neurosci Abs 20(2):1411. Hogan N (1985) Impedance control: An approach for manipulation: Theory. J Dynam Sys Meas Cont 107:1-7. Jezzard P et al. (1994) Practice makes perfect: A functional MRI study oflong term motor cortex plasticity. 2nd Ann Soc. Magnetic Res., p. 330. Kawashima R, Roland PE, O'Sullivan BT (1994) Fields in human motor areas involved in preparation for reaching, actual reaching, and visuomotor learning: A PET study. J Neurosci 14:3462-3474. Lewis D, Shephard AH, Adams JA (1949) Evidences of associative interference in psychomotor performance. Science 110:271-273. Melton A W, Irwin JM (1940) The influence of degree of interpolated learning on retroactive inhibition and the overt transfer of specific responses. Amer J Psychol 53:173-203. Ross D (1974) Interference in discrete motor tasks: A test of the theory. PhD dissertation, Dept. Psychology, Univ. Michigan, Ann Arbor. Schmidt RA (1988) Motor Control and Learning: A Behavioral Emphasis. Human Kinetics Books, Champaign IL, pp. 409-411. Sears RR, Hovland CI (1941) Experiments on motor conflict. J Exp Psychol 28:280-286. Shadmehr R, Mussa-Ivaldi FA (1994) Adaptive representation of dynamics during learning of a motor task. J Neuroscience, 14(5):3208- 3224. Shidara M, Kawano K, Gomi H, Kawato M (1993) Inverse dynamics model eye movement control by Purkinje cells in the cerebellum. Nature 365:50-52. Siipola EM, Israel HE (1933) Habit interference as dependent upon stage oftraining. Amer J Psychol 45:205-227. Squire LR (1986) Mechanisms of memory. Science 232:1612-1619. Sutton RS (1986) Two problems with backpropagation and other steepest-descent learning procedures for networks. Proc 8th Cognitive Sci Soc, pp. 823-831. Wolpert DM, Ghahramani Z, Jordan MI (1995) Are arm trajectories planned in kimenatic or dynamic coordinates? An adaptation stUdy. Exp Brain Res, in press.	1994	20
911	Optimal Movement Primitives Terence D. Sanger Jet Propulsion Laboratory MS 303-310 4800 Oak Grove Drive Pasadena, CA 91109 (818) 354-9127 tds@ai.mit.edu Abstract The theory of Optimal Unsupervised Motor Learning shows how a network can discover a reduced-order controller for an unknown nonlinear system by representing only the most significant modes. Here, I extend the theory to apply to command sequences, so that the most significant components discovered by the network correspond to motion "primitives". Combinations of these primitives can be used to produce a wide variety of different movements. I demonstrate applications to human handwriting decomposition and synthesis, as well as to the analysis of electrophysiological experiments on movements resulting from stimulation of the frog spinal cord. 1 INTRODUCTION There is much debate within the neuroscience community concerning the internal representation of movement, and current neurophysiological investigations are aimed at uncovering these representations. In this paper, I propose a different approach that attempts to define the optimal internal representation in terms of "movement primitives" , and I compare this representation with the observed behavior. In this way, we can make strong predictions about internal signal processing. Deviations from the predictions can indicate biological constraints or alternative goals that cause the biological system to be suboptimal. The concept of a motion primitive is not as well defined as that of a sensory primitive 1024 Terence Sanger u p y z Figure 1: Unsupervised Motor Learning: The plant P takes inputs u and produces outputs y. The sensory map C produces intermediate variables z, which are mapped onto the correct command inputs by the motor network N. within the visual system, for example. There is no direct equivalent to the "receptive field" concept that has allowed interpretation of sensory recordings. In this paper, I will propose an internal model that involves both motor receptive fields and a set of movement primitives which are combined using a weighted sum to produce a large class of movements. In this way, a small number of well-designed primitives can generate the full range of desired behaviors. I have previously developed the concept of "optimal unsupervised motor learning" to investigate optimal internal representations for instantaneous motor commands. The optimal representations adaptively discover a reduced-order linearizing controller for an unknown nonlinear plant. The theorems give the optimal solution in general, and can be applied to special cases for which both linear and nonlinear adaptive algorithms exist (Sanger 1994b). In order to apply the theory to complete movements it needs to be extended slightly, since in general movements exist within an infinite-dimensional task space rather than a finite-dimensional control space. The goal is to derive a small number of primitives that optimally encode the full set of observed movements. Generation of the internal movement primitives then becomes a data-compression problem, and I will choose primitives that minimize the resultant mean-squared error. 2 OPTIMAL UNSUPERVISED MOTOR LEARNING Optimal Unsupervised Motor Learning is based on three principles: 1. Dimensionality Reduction 2. Accurate Reduced-order Control 3. Minimum Sensory error Consider the system shown in figure 1. At time t, the plant P takes motor inputs u and produces sensory outputs y. A sensory mapping C transforms the raw sensory data y to an intermediate representation z. A motor mapping takes desired values of z and computes the appropriate command u such that CPu = z. Note that the Optimal Movement Primitives 1025 loop in the figure is not a feedback-control loop, but is intended to indicate the flow of information. With this diagram in mind, we can write the three principles as: 1. dim[z] < dim[y] 2. GPNz=z 3. IIPNGy - yll is minimized We can prove the following theorems (Sanger 1994b): Theorem 1: For all G there exists an N such that G P N z = z. If G is linear and p- 1 is linear, then N is linear. Theorem 2: For any G, define an invertible map C such that GC-l =_1 on range[G]. Then liP NGy - yll is minimized when G is chosen such that Ily - G-1GII is minimized. If G and P are linear and the singular value decomposition of P is given by LT SR, then the optimal maps are G = Land N = RT S-I. For the discussion of movement, the linear case will be the most important since in the nonlinear case we can use unsupervised motor learning to perform dimensionality reduction and linearization of the plant at each time t. The movement problem then becomes an infinite-dimensional linear problem. Previously, I have developed two iterative algorithms for computing the singular value decomposition from input/output samples (Sanger 1994a). The algorithms are called the "Double Generalized Hebbian Algorithm" (DGHA) and the "Orthogonal Asymmetric Encoder" (OAE). DGHA is given by 8G 'Y(zyT - LT[zzT]G) 8NT 'Y(zuT - LT[zzT]NT ) while OAE is described by: 'Y(iyT - LT[ZiT]G) 'Y(Gy - LT[GGT]z)uT where LT[ ] is an operator that sets the above diagonal elements of its matrix argument to zero, y = Pu, z = Gy, z = NT u, and 'Y is a learning rate constant. Both algorithms cause G to converge to the matrix of left singular vectors of P, and N to converge to the matrix of right singular vectors of P (multiplied by a diagonal matrix for DGHA). DGHA is used in the examples below. 3 MOVEMENT In order to extend the above discussion to allow adaptive discovery of movement primitives, we now consider the plant P to be a mapping from command sequences u(t) to sensory sequences y(t). We will assume that the plant has been feedback linearized (perhaps by unsupervised motor learning). We also assume that the sensory network G is constrained to be linear. In this case, the optimal motor network N will also be linear. The intermediate variables z will be represented by a vector. The sensory mapping consists of a set of sensory "receptive fields" gi(t) 1026 Terence Sanger A .. Motor Map Sensory Map I n1(t)zl +- Zl +---lEt): : \ nn(~)~ ~ ~n +Figure 2: Extension of unsupervised motor learning to the case of trajectories. Plant input and output are time-sequences u(t) and y(t). The sensory and motor maps now consist of sensory primitives gi(t) and motor primitives ni(t). such that Zi = J gj(t)y(t)dt =< gily > and the motor mapping consists of a set of "motor primitives" ni(t) such that u(t) = L ni(t)zi i as in figure 2. If the plant is equal to the identity (complete feedback linearization), then gi(t) = ni(t). In this case, the optimal sensory-motor primitives are given by the eigenfunctions of the autocorrelation function of y(t). If the autocorrelation is stationary, then the infinite-window eigenfunctions will be sinusoids. Note that the optimal primitives depend both on the plant P as well as the statistical distribution of outputs y(t). In practice, both u(t) and y(t) are sampled at discrete time-points {tic} over a finite time-window, so that the plant input and output is in actuality a long vector. Since the plant is linear, the optimal solution is given by the singular value decomposition, and either the DGHA or OAE algorithms can be used directly. The resulting sensory primitives map the sensory information y(t) onto the finite-dimensional z, which is usually a significant data compression. The motor primitives map Z onto the sequence u(t), and the resulting y(t) = P[u(t)] will be a linear projection of y(t) onto the space spanned by the set {Pni(t)}. 4 EXAMPLE 1: HANDWRITING As a simple illustration, I examine the case of human handwriting. We can consider the plant to be the identity mapping from pen position to pen position, and the Optimal Movement Primitives 1027 1. 5. 2. 6. 3. 7. 4. 8. Figure 3: Movement primitives for sampled human handwriting. 1028 Terence Sanger human to be taking desired sensory values of pen position and converting them into motor commands to move the pen. The sensory statistics then reflect the set of trajectories used in producing handwritten letters. An optimal reduced-order control system can be designed based on the observed statistics, and its performance can be compared to human performance. For this example, I chose sampled data from 87 different examples of lower-case letters written by a single person, and represented as horizontal and vertical pen position at each point in time. Blocks of 128 sequential points were used for training, and 8 internal variables Zi were used for each of the two components of pen position. The training set consisted of 5000 randomly chosen samples. Since the plant is the identity, the sensory and motor primitives are the same, and these are shown as "strokes" in figure 3. Linear combinations of these strokes can be used to generate pen paths for drawing lowercase letters. This is shown in figure 4, where the word "hello" (not present in the training set) is written and projected using increasing numbers of intermediate variables Zi. The bottom of figure 4 shows the sequence of values of Zi that was used (horizontal component only). Good reproduction of the test word was achieved with 5 movement primitives. A total of 7 128-point segments was projected, and these were recombined using smooth 50% overlap. Each segment was encoded by 5 coefficients for each of the horizontal and vertical components, giving a total of 70 coefficients to represent 1792 data points (896 horizontal and vertical components) , for a compression ratio of 25:1. 5 EXAMPLE 2: FROG SPINAL CORD The second example models some interesting and unexplained neurophysiological results from microstimulation of the frog spinal cord. (Bizzi et al. 1991) measured the pattern of forces produced by the frog hindlimb at various positions in the workspace during stimulation of spinal interneurons. The resulting force-fields often have a stable" equilibrium point", and in some cases this equilibrium point follows a smooth closed trajectory during tonic stimulation of the interneuron. However, only a small number of different force field shapes have been found, and an even smaller number of different trajectory types. A hypothesis to explain this result is that larger classes of different trajectories can be formed by combining the patterns produced by these cells. This hypothesis can be modelled using the optimal movement primitives described above. Figure 5a shows a simulation of the frog leg. To train the network, random smooth planar movements were made for 5000 time points. The plant output was considered to be 32 successive cartesian endpoint positions, and the plant input was the timevarying force vector field. Two hidden units Z were used. In figure 5b we see an example of the two equilibrium point trajectories (movement primitives) that were learned by DG HA. Linear combinations of these trajectories account for over 96% of the variance of the training data, and they can approximate a large class of smooth movements. Note that many other pairs of orthogonal trajectories can accomplish this, and different trials often produced different orthogonal trajectory shapes. Optimal Movement Primitives 1029 Original 1. 5.~ 2 3. 8.~~ Coefficients Figure 4: Projection of test-word "hello" using increasing numbers of intermediate variables Zi. 1030 Terence Sanger WorkSpace o. b. Figure 5: a. Simulation of frog leg configuration. b. An example of learned optimal movement primitives. 6 CONCLUSION The examples are not meant to provide detailed models of internal processing for human or frog motor control. Rather, they are intended to illustrate the concept of optimal primitives and perhaps guide the search for neurophysiological and psychophysical correlates of these primitives. The first example shows that generation of the lower-case alphabet can be accomplished with approximately 10 coefficients per letter, and that this covers a considerable range of variability in character production. The second example demonstrates that an adaptive algorithm allows the possibility for the frog spinal cord to control movement using a very small number of internal variables. Optimal unsupervised motor learning thus provides a descriptive model for the generation of a large class of movements using a compressed internal description. A set of fixed movement primitives can be combined linearly to produce the necessary motor commands, and the optimal choice of these primitives assures that the error in the resulting movement will be minimized. References Bizzi E., Mussa-Ivaldi F. A., Giszter S., 1991, Computations underlying the execution of movement: A biological perspective, Science, 253:287-29l. Sanger T. D., 1994a, Two algorithms for iterative computation of the singular value decomposition from input/output samples, In Touretzky D., ed., Advances in Neural Information Processing 6, Morgan Kaufmann, San Mateo, CA, in press. Sanger T. D., 1994b, Optimal unsupervised motor learning, IEEE Trans. Neural Networks, in press.	1994	21
912	Using a Saliency Map for Active Spatial Selective Attention: Implementation & Initial Results Shumeet Baluja baluja@cs.cmu.edu School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Abstract Dean A. Pomerleau pomerleau@cs.cmu.edu School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 In many vision based tasks, the ability to focus attention on the important portions of a scene is crucial for good performance on the tasks. In this paper we present a simple method of achieving spatial selective attention through the use of a saliency map. The saliency map indicates which regions of the input retina are important for performing the task. The saliency map is created through predictive auto-encoding. The performance of this method is demonstrated on two simple tasks which have multiple very strong distracting features in the input retina. Architectural extensions and application directions for this model are presented. 1 MOTIVATION Many real world tasks have the property that only a small fraction of the available input is important at any particular time. On some tasks this extra input can easily be ignored. Nonetheless, often the similarity between the important input features and the irrelevant features is great enough to interfere with task performance. Two examples of this phenomena are the famous "cocktail party effect", otherwise known as speech recognition in a noisy environment, and image processing of a cluttered scene. In both cases, the extraneous information in the input signal can be easily confused with the important features, making the task much more difficult. The concrete real world task which motivates this work is vision-based road following. In this domain, the goal is to control a robot vehicle by analyzing the scene ahead, and choosing a direction to travel based on the location of important features like lane marking and road edges. This is a difficult task, since the scene ahead is often cluttered with extraneous features such as other vehicle, pedestrians, trees, guardrails, crosswalks, road signs and many other objects that can appear on or around a roadway. 1 While we have had significant success on the road following task using simple feed-forward neural networks to transform images of the road ahead into steering commands for the vehicle [pomerleau, 1993b], these methods fail when presented with cluttered environments like those encounI. For the gl:neral task of autonomous navigation, these extra features are extremely important, but for restricted task of road following. which is the focus of this paper. these features are merely distractions. Although we are addressing the more general task using the techniques described here in combination with other methods, a description of these efforts is beyond the scope of this paper. 452 Shumeet Baluja, Dean A. Pomerleall tered when driving in heavy traffic, or on city streets. The obvious solution to this difficulty is to focus the attention of the processing system on only the relevant features by masking out the "noise". Because of the high degree of simi1arity between the relevant features and the noise, this filtering is often extremely difficult. Simultaneously learning to perform a task like road following and filtering out clutter in a scene is doubly difficult because of a chicken-and-egg problem. It is hard to learn which features to attend to before knowing how to perform the task, and it is hard to learn how to perform the task before knowing which features to attend to. This paper describes a technique designed to solve this problem. It involves deriving a "saliency map" of the image from a neural network's internal representation which highlights regions of the scene considered to be important. This saliency map is used as feedback to focus the attention of the network's processing on subsequent images. This technique overcomes the chicken-and-egg problem by simultaneously learning to identify which aspects of the scene are important, and how to use them to perform a task. 2 THE SALIENCY MAP A saliency map is designed to indicate which portions of the image are important for completing the required task. The trained network should be able to accomplish two goals with the presentation of each image. The first is to perform the given task using the inputs and the saliency map derived from the previous image, and the second is to predict the salient portions of the next image. 2.1 Implementation The creation of the saliency map is similar to the technique of Input Reconstruction Reliability Estimation (IRRE) by [Pomerleau, 1993]. IRRE attempts to predict the reliability of a network's output. The prediction is made by reconstructing the input image from linear transformations of the activations in the hidden layer, and comparing it with the actual image. IRRE works on the premise that the greater the similarity between the input image and the reconstructed input image, the more the internal representation has captured the important input features, and therefore the more reliable the network's response. A similar method to IRRE can be used to create a saliency map. The saliency map should be determined by the important features in the current image for the task to be performed. Because compressed representations of the important features in the current image are represented in the activations of the hidden units, the saliency map is derived from these, as shown in Figure 1. It should be noted that the hidden units, from which the saliency map is derived, do not necessarily contain information similar to principal components (as is achieved through auto-encoder networks), as the relevant task may only require information on a small portion of the image. In the simple architecture depicted in Figure 1, the internal representation must contain information which can be transformed by a single layer of adjustable weights into a saliency map for the next image. If such a transformation is not possible, separate hidden layers, with input from the task-specific internal representations could be employed to create the saliency map. The saliency map is trained by using the next image, of a time-sequential data set, as the target image for the prediction, and applying standard error backpropagation on the differences between the next image and the predicted next image. The weights from the hidden Using a Saliency Map for Active Spatial Selective Attention 453 Output Units Hidden Units Input Retina (delayed I time step) Predicted L------! ~':~e ,....--=--., erived aliency p x X ----+--- Relevant Portion of Input Figure 1: A simple architecture for using a saliency map. The dashed line represents "chilled connections", i.e. errors from these connections do not propagate back further to impact the activations of the hidden units. This architecture assumes that the target task contains information which will help determine the salient portions of the next frame. units to the saliency map are adjusted using standard backpropagation, but the error terms are not propagated to the weights from the inputs to the hidden units. This ensures that the hidden representation developed by the network is determined only by the target task, and not by the task of prediction. In the implementation used here, the feedback is to the input layer. The saliency map is created to either be the same size as the input layer, or is scaled to the same size, so that it can influence the representation in a straight-forward manner. The saliency map's values are scaled between 0.0 and 1.0, where 1.0 represents the areas in which the prediction matched the next image exactly. The value of 1.0 does not alter the activation of the input unit, a value of 0.0 turns off the activation. The exact construction of the saliency map is described in the next section, with the description of the experiments. The entire network is trained by standard backpropagation; in the experiments presented, no modifications to the standard training algorithm were needed to account for the feedback connections. The training process for prediction is complicated by the potential presence of noise in the next image. The saliency map cannot "reconstruct" the noise in the next image, because it can only construct the portions of the next image which can be derived from the activation of the hidden units, which are task-specific. There/ore, the noise in the next image will not be constructed, and thereby will be de-emphasized in the next time step by the saliency map. The saliency map serves as a filter, which channels the processing to the important portions of the scene [Mozer, 1988]. One of the key differences between the filtering employed in this study, and that used in other focus of attention systems, is that this filtering is based on expectations from multiple frames, rather than on the retinal activations from a single frame. An alternative neural model of visual attention which was explored by [Olshausen et al., 1992] achieved focus of attention in single frames by using control neurons to dynamically modify synaptic strengths. The saliency map may be used in two ways. It can either be used to highlight portions of the input retina or, when the hidden layer is connected in a retinal fashion using weight sharing, as in [LeCun et al., 1990], it can be used to highlight important spatial locations within the hidden layer itself. The difference is between highlighting individual pixels from which the features are developed or highlighting developed features. Discussion of the psychological evidence for both of these types of highlighting (in single-frame retinal activation based context), is given in [pashler and Badgio, 1985]. This network architecture shares several characteristics with a Jordan-style recurrent network [Jordan, 1986], in which the output from the network is used to influence the pro454 Shumeet Baluja, Dean A. Pomerleau cessing of subsequent input patterns in a temporal sequence. One important distinction is that the feedback in this architecture is spatially constrained. The saliency map represents the importance of local patches of the input, and can influence only the network's processing of corresponding regions of the input. The second distinction is that the outputs are not general task outputs, rather they are specially constructed to predict the next image. The third distinction is in the form of this influence. Instead of treating the feedback as additional input units, which contribute to the weighted sum for the network's hidden units, this architecture uses the saliency map as a gating mechanism, suppressing or emphasizing the processing of various regions of the layer to which it is fed-back. In some respects, the influence of the saliency map is comparable to the gating network in the mixture of experts architecture [Jacobs et al., 1991]. Instead of gating between the outputs of multiple expert networks, in this architecture the saliency map is used to gate the activations of the input units within the same network. 3 THE EXPERIMENTS In order to study the feasibility of the saliency map without introducing other extraneous factors, we have conducted experiments with two simple tasks described below. Extensions of these ideas to larger problems are discussed in sections 4 & 5. The first experiment is representative of a typical machine vision task, in which the relevant features move very little in consecutive frames. With the method used here, the relevant features are automatically determined and tracked. However, if the relevant features were known a priori, a more traditional vision feature tracker which begins the search for features within the vicinity of the location of the features in the previous frame, could also perform well. The second task is one in which the feature of interest moves in a discontinuous manner. A traditional feature tracker without exact knowledge the feature's transition rules would be unable to track this feature, in the presence of the noise introduced. The transition rules of the feature of interest are learned automatically through the use of the saliency map. In the first task, there is a slightly tilted vertical line of high activation in a 30x32 input unit grid. The width of the line is approximately 9 units, with the activation decaying with distance from the center of the line. The rest of the image does not have any activation. The task is to center a gaussian of activation around the center of the x-intercept of the line, 5 pixels above the top of the image. The output layer contains 50 units. In consecutive images, the line can have a small translational move and/or a small rotational move. Sample training examples are given in Figure 2. This task can be easily learned in the presence of no noise. The task is made much harder when lines, which have the same visual appearance as the real line (in everything except for location and tilt) randomly appear in the image. In this case, it is vital that the network is able to distinguish between the real line and noise line by using information gathered from previous image(s). In the second task, a cross ("+") of size 5x5 appears in a 20x20 grid. There are 16 positions in which the cross can appear, as shown in Figure 2c. The locations in which the cross appears is set according to the transition rules shown in Figure 2c. The object of this problem is to reproduce the cross in a smaller lOx 1 0 grid, with the edges of the cross extended to the edges of the grid, as shown in Figure 2b. The task is complicated by the presence of randomly appearing noise. The noise is in the form of another cross which appears exactly similar to the cross of interest. Again, in this task, it is vital for the network to be able to distinguish between the real cross, and crosses which appear as noise. As in the first task, this is only possible with knowledge of the previous image(s). Using a Saliency Map for Active Spatial Selective Attention 455 At 12 5 9 2 7 1 II 16 10 15 4 \3 3 14 6 8 A2 Bt B2 c Figure 2: (A) The first task, image (AI) with no distractions, image (A2) with one distracting feature. (8) The second task, image (8 I) with no distractions, image (82) with two distractions. (C) Transition rules for the second task. 3.1 Results The two problems described above were attempted with networks trained both with and without noise. Each of the training sessions were also tested with and without the saliency map. Each type of network was trained for the same number of pattern presentations with the same training examples. The results are shown in Table 1. The results reported in Table I represent the error accrued over 10,000 testing examples. For task 1, errors are reported in terms of the absolute difference between the peak's of the Gaussians produced in the output layer, summed for all of the testing examples (the max error per image is 49). In task 2, the errors are the sum of the absolute difference between the network's output and the target output, summed across all of the outputs and all of the testing examples. When noise was added to the examples, it was added in the following manner (for both training and testing sets): In task 1, '1 noise' guarantees a noise element, similarly, '2 noise' guarantees two noise elements. However, in task 2, '1 noise' means that there is a 50% chance of a noise element occurring in the image, '2 noise' means that there is a 50% chance of another noise element occurring, independently of the appearance of the first noise element. The positions of the noise elements are determined randomly. The best performance. in task 1, came from the cases in which there was no noise in testing or training. and no saliency map was used. This is expected, as this task is not difficult when no noise is present. Surprisingly, in task 2, the best case was found with the saliency map, when training with noise and testing without noise. This performed even better than training without noise. Investigation into this result is currently underway. In task 1, when training and testing without noise. the saliency map can hurt performance. If the predictions made by the saliency map are not correct, the inputs appear slightly distorted; therefore, the task to be learned by the network becomes more difficult. Nevertheless. the benefit of using a saliency map is apparent when the test set contains noise. In task 2. the network without the saliency map, trained with noise, and tested without noise cannot perform well; the performance further suffers when noise is introduced into the testing set. The noise in the training prevents accurate learning. This is not the case when the saliency map is used (Table 1, task 2). When the training set contains noise, the network with the saliency map works better when tested with and without noise. 456 Sizumeet Baluja. Dean A. Pomerleau Table 1: Summed Error of 10,000 Testing Examples Testing Set Training Set Task 1 Task 2 o Noise 1 Noise 2 Noise o Noise I Noise 2 Noise o Noise (Saliency) 12672 60926 82282 7174 94333 178883 o Noise (No Saliency) 10241 91812 103605 7104 133496 216884 1 Noise (Saliency) 18696 26178 52668 4843 10427 94422 1 Noise (No Saliency) 14336 80668 97433 31673 150078 227650 When the noise increased beyond the level of training, to 2 noise elements per image, the performances of networks trained both with and without the saliency map declined. It is suspected that training the networks with increased noise will improve performance in the network trained with the saliency map. Nonetheless, due to the amount of noise compared to the small size of the input layer, improvements in results may not be dramatic. In Figure 3, a typical test run of the second task is shown. In the figure, the inputs, the predicted and actual outputs, and the predicted and actual saliency maps, are shown. The actual saliency map is just a smaller version of the unfiltered next input image. The input size is 20x20, the outputs are 10xlO, and the saliency map is 10xlO. The saliency map is scaled to 20x20 when it is applied to the next inputs. Note that in the inputs to the network, one cross appears much brighter than the other; this is due to the suppression of the distracting cross by the saliency map. The particular use of the saliency map which is employed in this study, proceeded as follows: the difference between the saliency map (derived from input imagei) and the input imagei+l was calculated. This difference image was scaled to the range of 0.0 to 1.0. Each pixel was then passed through a sigmoid; alternatively, a hard-step function could have been used. This is the saliency map. The saliency map was multiplied by input imagei+ 1; this was used as the input into the network. If the sigmoid is not used, features, such as incomplete crosses, sometimes appear in the input image. This happens because different portions of the cross may have slightly different saliency values associated with them -due to errors in prediction coupled with the scaling of the saliency map. Although the sigmoid helps to alleviate the problem, it does not eliminate it. This explains why training with no noise with a saliency map sometimes does not perform as well as training without a saliency map. 4 ALTERNATIVE IMPLEMENTATIONS An alternative method of implementing the saliency map is with standard additive connections. However, these connection types have several drawbacks in comparison with the multiplicative ones use in this study. First, the additive connections can drastically change the meaning of the hidden unit's activation by changing the sign of the activation. The saliency map is designed to indicate which regions are important for accomplishing the task based upon the features in the hidden representations; as little alteration of the important features as possible is desired. Second, if the saliency map is incorrect, and suggests an area of which is not actually important, the additive connections will cause 'ghost' images to appear. These are activations which are caused only by the influence of the addiUsing a Saliency Map for Active Spatial Selective Attention predicted saliency target saliency predicted output target output inputs 457 1 4 5 6 7 Figure 3: A typical sequence of inputs and outputs in the second task. Note that when two crosses are in the inputs. one is much brighter than the other. The "noise" cross is de-emphasized. tive saliency map. The multiplicative saliency map, as is implemented here, does not have either of these problems. A second alternative, which is more closely related to a standard recurrent network [Jordan, 1986], is to use the saliency map as extra inputs into the network. The extra inputs serve to indicate the regions which are expected to be important. Rather than hand-coding the method to represent the importance of the regions to the network, as was done in this study, the network learns to use the extra inputs when necessary. Further, the saliency map serves as the predicted next input. This is especially useful when the features of interest may have momentarily been partially obscured or have entirely disappeared from the image. This implementation is currently being explored by the authors for use in a autonomous road lane-tracking system in which the lane markers are not always present in the input image. 5 CONCLUSIONS & FUTURE DIRECTIONS These experiments have demonstrated that an artificial neural network can learn to both identify the portions of a visual scene which are important, and to use these important features to perform a task. The selective attention architecture we have develop uses two simple mechanisms, predictive auto-encoding to form a saliency map, and a constrained form of feedback to allow this saliency map to focus processing in the subsequent image. There are at least four broad directions in which this research should be extended. The first is, as described here, related to using the saliency map as a method for automatically actively focusing attention to important portions of the scene. Because of the spatial dependence of the task described in this paper, with the appropriate transformations, the output units could be directly fed back to the input layer to indicate saliency. Although this does not weaken the result, in terms of the benefit of using a saliency map, future work should also focus on problems which do not have this property to determine how easily a saliency map can be constructed. Will the use of weight sharing be enough to develop the necessary spatially oriented feature detectors? Harder problems are those with target tasks which does not explicitly contain spatial saliency information. An implementation problem which needs to be resolved is in networks which contain more than a single hidden layer: from which layer should the saliency map be con458 Shumeet Baluja, Dean A. Pomerleau structed? The trade-off is that at the higher layers, the information contained is more task specific. However, the higher layers may effectively extract the information required to perform the task, without maintaining the information required for saliency map creation. The opposite case is true in the lower layers; these may contain all of the information required, but may not provide enough discrimination to narrow the focus effectively. The third area for research is an alternative use for the saliency map. ANNs have often been criticized for their uninterpretability, and lack of mechanism to explain performance. The saliency map provides a method for understanding, at a high level, what portions of the inputs the ANN finds the most important. Finally, the fourth direction for research is the incorporation of higher level, or symbolic knowledge. The saliency map provides a very intuitive and direct method for focusing the network's attention to specific portions of the image. The saliency map may prove to be a useful mechanism to allow other processes, including human users, to simply "point at" the portion of the image to which the network should be paying attention. The next step in our research is to test the effectiveness of this technique on the main task of interest, autonomous road following. Fortunately, the first demonstration task employed in this paper shares several characteristics with road following. Both tasks require the network to track features which move over time in a cluttered image. Both tasks also require the network to produce an output that depends on the positions of the important features in the image. Because of these shared characteristics, we believe that similar performance improvements should be possible in the autonomous driving domain. Acknowledgments Shumeet Baluja is supported by a National Science Foundation Graduate Fellowship. Support has come from "Perception for Outdoor Navigation" (contract number DACA76-89-C-0014, monitored by the US Army Topographic Engineering Center) and "Unmanned Ground Vehicle System" (contract number DAAE07-90-C-R059, monitored by TACOM). Support has also come from the National Highway Traffic Safety Administration under contract number DTNH22-93-C-07023. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing official policies, either expressed or implied, of the National Science Foundation, ARPA, or the U.S. Government. References Cottrell, G.W. & Munro, P. (1988) Principal Component Analysis ofImages via back-propagation. Proc Soc. of Photo-Opticallnstr. Eng., Cambridge, MA. Jordan, M.I., (1989). Serial Order: A Parallel, Distributed Processing Approach. In Advances in Connectionist Theory: Speech, eds. J.L. Elman and D.E. Rumerlhart. Hillsdale: Erlbaum. Jacobs, R.A., Jordan, M.I., Nowlan, S.J. & Hinton, G.E. (1991). Adaptive Mixtures of Local Experts. Neural Computation, 3:1. LeCun, Y., Boser, B., Denker, J.S., Henderson, D. Howard, R.E., Hurnmand W., and Jackel, L.D. (1989) Backpropagation Applied to Handwritten Zip Code Recognition. Neural Computation 1,541-551. MIT, 1989. Mozer, M.C. (1988) A Connectionist Model of Selective Attention in Visual Perception. Technical Report, University of Toronto, CRG-TR-88-4. Pashler, H. & Badgio, P. (1985). Visual Attention and Stimulus Identification. Journal of Experimental Psychology: Human Perception and Performance, II 105-121. Pomerleau, D.A. (1993) Input Reconstruction Reliability Estimation. In Giles, C.L. Hanson, S.J. and Cowan, J.D. (eds). Advances in Neurallnfol71Ultion Processing Systems 5, CA: Morgan Kaufmann Publishers. Pomerleau, D.A. (1993b) Neural Network Perception for Mobile Robot Guidance, Kluwer Academic Publishing. Olshausen, B., Anderson, C., & Van Essen; D. (1992) A Neural Model of Visual Attention and Invariant Pattern Recognition. California Institute of Technology, CNS Program, memo-18.	1994	22
913	Factorial Learning and the EM Algorithm Zoubin Ghahramani zoubin@psyche.mit.edu Department of Brain & Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 Abstract Many real world learning problems are best characterized by an interaction of multiple independent causes or factors. Discovering such causal structure from the data is the focus of this paper. Based on Zemel and Hinton's cooperative vector quantizer (CVQ) architecture, an unsupervised learning algorithm is derived from the Expectation-Maximization (EM) framework. Due to the combinatorial nature of the data generation process, the exact E-step is computationally intractable. Two alternative methods for computing the E-step are proposed: Gibbs sampling and mean-field approximation, and some promising empirical results are presented. 1 Introduction Many unsupervised learning problems fall under the rubric of factorial learning-that is, the goal of the learning algorithm is to discover multiple independent causes, or factors, that can well characterize the observed data (Barlow, 1989; Redlich, 1993; Hinton and Zemel, 1994; Saund, 1995). Such learning problems often arise naturally in response to the actual process by which the data have been generated. For instance, images may be generated by combining multiple objects, or varying colors, locations, and poses, with different light sources. Similarly, speech signals may result from an interaction of factors such as the tongue position, lip aperture, glottal state, communication line, and background noises. The goal of factorial learning is to invert this data generation process, discovering a representation that will both parsimoniously describe the data and reflect its underlying causes. A recent approach to factorial learning uses the Minimum Description Length (MDL) principle (Rissanen, 1989) to extract a compact representation of the input (Zemel, 1993; Hinton and Zemel, 1994). This has resulted in a learning architecture 618 Zoubin Ghahramani called Cooperative Vector Quantization (CVQ), in which a set of vector quantizers cooperates to reproduce the input. Within each vector quantizer a competitive learning mechanism operates to select an appropriate vector code to describe the input. The CVQ is related to algorithms based on mixture models, such as soft competitive clustering, mixtures of experts (Jordan and Jacobs, 1994), and hidden Markov models (Baum et al., 1970), in that each vector quantizer in the CVQ is itself a mixture model. However, it generalizes this notion by allowing the mixture models to cooperate in describing features in the data set, thereby creating a distributed representations of the mixture components. The learning algorithm for the CVQ uses MDL to derive a cost function composed of a reconstruction cost (e.g. sum squared error), representation cost (negative entropy of the vector code), and model complexity (description length of the network weights), which is minimized by gradient descent. In this paper we first formulate the factorial learning problem in the framework of statistical physics (section 2). Through this formalism, we derive a novel learning algorithm for the CVQ based on the Expectation-Maximization (EM) algorithm (Dempster et al., 1977) (section 3). The exact EM algorithm is intractable for this and related factorial learning problems-however, a tractable mean-field approximation can be derived. Empirical results on Gibbs sampling and the mean-field approximation are presented in section 4. 2 Statistical Physics Formulation The CVQ architecture, shown in Figure 1, is composed of hidden and observable units, where the observable units, y, are real-valued, and the hidden units are discrete and organized into vectors Si, i = 1, ... , d. The network models a data generation process which is assumed to proceed in two stages. First, a factor is independently sampled from each hidden unit vector, Sj, according to its prior probability density, ?ri. Within each vector the factors are mutually exclusive, i.e. if Sij = 1 for some j, then Sik = 0, Vk -# j. The observable is then generated from a Gaussian distribution with mean 2:1=1 WiSi. Notation: d number of vectors 0 0 0 k number of hidden units per vector 0 0 0 p number of outputs 0 51 0 52 ••• 0 Sd N number of patterns 0 0 0 Sij hidden unit j in vector i 0 0 0 Si vector i of units (Si = [Si1, ... , Sik]) Wi weight matrix from Si to output VOl V0 2 VOd Y network output (observable) Figure 1. The factorial learning architecture. Defining the energy of a particular configuration of hidden states and outputs as 1 d d k H(s, y) = "2lly - L Wi sill 2 - L L Sij log 7rij, i=l i=l j=l (1) Factorial Learning and the EM Algorithm 619 the Boltzmann distribution 1 p(s, y) = -Z exp{-11.(s,y)}, free (2) exactly recovers the probability model for the CVQ. The causes or factors are represented in the multinomial variables Si and the observable in the multivariate Gaussian y. The undamped partition function, Zjree, can be evaluated by summing and integrating over all the possible configurations of the system to obtain Zjree = ~ 1 exp{ -11.(s, y)}dy = (21l")P/2, $ Y (3) which is constant, independent of the weights. This constant partition function results in desirable properties, such as the lack of a Boltzmann machine-like sleep phase (Neal, 1992), which we will exploit in the learning algorithm. The system described by equation (1)1 can be thought of as a special form of the Boltzmann machine (Ackley et al., 1985). Expanding out the quadratic term we see that there are pairwise interaction terms between every unit. The evaluation of the partition function (3) tells us that when y is unclamped the quadratic term can be integrated out and therefore all Si are independent. However, when y is clamped all the Si become dependent. 3 The EM Algorithm Given a set of observable vectors, the goal of the unsupervised learning algorithm is to find weight matrices such that the network is most likely to have generated the data. If the hidden causes for each observable where known, then the weight matrices could be easily estimated. However, the hidden causes cannot be inferred unless these weight matrices are known. This chicken-and-egg problem can be solved by iterating between computing the expectation of the hidden causes given the current weights and maximizing the likelihood of the weights given these expected causes-the two steps forming the basis of the Expectation-Maximization (EM) algorithm (Dempster et al., 1977). Formally, from (2) we obtain the expected log likelihood of the parameters ¢/: Q(¢,¢/) = (-11.(s,y) -logZjree)c,q, (4) where ¢ denotes the current parameters, ¢ = {Wi}?=1, and (-)c,q, denotes expectation given ¢ and the damped observables. The E-step of EM consists of computing this expected log likelihood. As the only random variables are the hidden causes, this simplifies to computing the (Si)c and (SiS])c terms appearing in the quadratic expansion of 11.. Once these terms have been computed, the M-step consists of maximizing Q with respect to the parameters. Setting the derivatives to zero we obtain a linear system, 1 For the remainder of the paper we will ignore the second term in (1), thereby assuming equal priors on the hidden states. Relaxing this assumption and estimating priors from the data is straightforward. 620 Zoubin Ghahramani which can be solved via the normal equations, where s is the vector of concatenated Si and the subscripts denote matrix size. For models in which the observable is a monotonic differentiable function ofLi WiSi, i.e. generalized linear models, least squares estimates of the weights for the M-step can be obtained iteratively by the method of scoring (McCullagh and NeIder, 1989). 3.1 E-step: Exact The difficulty arises in the E-step of the algorithm. The expectation of hidden unit j in vector i given pattern y is: P(Sij = 11Y; W) <x P(ylSij = 1;W)1l"ij Ii; Ii; Ii; <X L·· L .. LP(ylsij = 1, slh = 1, .. . ,Sdjd = 1; W)1l"ij jl=l ih,ti=l jd=l To compute this expectation it is necessary to sum over all possible configurations of the other hidden units. If each vector quantizer has k hidden units, each expectation has time complexity of O( kd - l ), i.e. O( N kd ) for a full E-step. The exponential time is due inherently to the cooperative nature of the model-the setting of one vector only determines the observable if all the other vectors are fixed. 3.2 E-step: Gibbs sampling Rather than summing over all possible hidden unit patterns to compute the exact expectations, a natural approach is to approximate them through a Monte Carlo method. As with Boltzmann machines, the CVQ architecture lends itself well to Gibbs sampling (Geman and Geman, 1984). Starting from a clamped observable y and a random setting of the hidden units {Sj}, the setting of each vector is updated in turn stochastically according to its conditional distribution Si '" p( sdY, {Sj h;ti; W). Each conditional distribution calculation requires k forward passes through the network, one for each possible state of the vector being updated, and k Gaussian distance calculations between the resulting predicted and clamped observables. If all the probabilities are bounded away from zero this process is guaranteed to converge to the equilibrium distribution of the hidden units given the observable. The first and second-order statistics, for (Si)c and (SiS])c respectively, can be collected using the Sij'S visited and p( Si Iy, {Sj h;ti; W) calculated during this sampling process. These estimated expectations are then used in the E-step. 3.3 E-step: Mean-field approximation Although Gibbs sampling is generally much more efficient than exact calculations, it too can be computationally demanding. A more promising approach is to approximate the intractable system with a tractable mean-field approximation (Parisi, 1988), and perform the E-step calculation on this approximation. We can write the Factorial Learning and the EM Algorithm 621 negative log likelihood minimized by the original system as a difference between the damped and undamped free energies: Cost -logp(y/W) = -log ~p(y, s/W) s -log~exp{-'J-l(y,s)} + log~ [ exp{-'J-l(y,s)}dy s s Jy Fcl - FJree The mean-field approximation allows us to replace each free energy in this cost with an upper bound approximation CostM F = F:t F - Ffrt'e. Unfortunately, a difference of two upper bounds is not generally an upper bound, and therefore minimizing CostM F in, for example, mean-field Boltzmann machines does not guarantee that we are minimizing an upper bound on Cost. However, for the factorial learning architectures described in this paper we have the property that FJree is constant, and therefore the mean-field approximation of the cost is an upper bound on the exact cost. The mean-field approximation can be obtained by approximating the probability density given by (1) and (2) by a completely factorized probability density: p(s, y) = (21r~P/2 exp{ -~/IY -1L/12} n m:y ',J In this approximation all units are independent: the observables are Gaussian distributed with mean IL and each hidden unit is binomially distributed with mean mij. To obtain the mean-field approximation we solve for the mean values that minimize the Kullback-Leibler divergence KC(p,p) == Ep[logp] - Ep[logp]. Noting that: Ep[Sij] = mij, Ep[S[j] = mij, Ep[SijSkd = mijmkl, and Ep[SijSik] = 0, we obtain the mean-field fixed point equations (5) where y == l:i WiIDi. The softmax function is the exponential normalized over the k hidden units in each IDi vector. The first term inside the softmax has an intuitive interpretation as the projection of the error in the observable onto the weights of the hidden unit vector i. The more a hidden unit can reduce this error, the higher its mean. The second term arises from the fact that Ep[s[j] = mij and not Ep[s[j] = m[j. The means obtained by iterating equation (5) are used in the E-step by substituting mi for (Si)c and IDiID] for (sisJk 4 Empirical Results Two methods, Gibbs sampling and mean-field, have been provided for computing the E-step of the factorial learning algorithm. There is a key empirical question that needs to be answered to determine the efficiency and accuracy of each method. For Gibbs sampling it is important to know how many samples will provide robust estimates of the expectations required for the E-step. It is well known that for stochastic Boltzmann machines the number of samples needed to obtain good 622 Zoubin Ghahramani estimates of the gradients is generally large and renders the learning algorithm prohibitively slow. Will this architecture suffer from the same problem? For mean-field it is important to know the loss incurred by approximating the true likelihood. We explore these questions by presenting empirical results on two small unsupervised learning problems. The first benchmark problem consists of a data set of 4 x 4 greyscale images generated by a combination of two factors: one producing a single horizontal line and the other, a vertical line (Figure 2a; cf. Zemel, 1993). Using a network with 2 vectors of 4 hidden units each, both the Gibbs sampling and mean-field EM algorithms converge on a solution after about a dozen steps (Figure 2b). The solutions found resemble the generative model of the data (Figure 2c & d). a) c) d) b) 70 . 0 o 5 :'i 0 ." ~ 40 ~ l O ; i 2 0 10 _= s ~.r-r--- _--~-="-~oo-~_~_~_~_~ 10 J'i 20 Iteration Figure 2. Lines Problem. a) Complete data set of 160 patterns. b) Learning curves for Gibbs (solid) and mean-field (dashed) forms of the algorithm. c) A sample output weight matrix after learning (MSE=1.20). The top vector of hidden units has come to represent horizontal lines, and the bottom, vertical lines. d) Another typical output weight matrix (MSE=1.78). The second problem consists of a data set of 6 x 6 images generated by a combination of three shapes-a cross, a diagonal line, and an empty square-each of which can appear in one of 16 locations (Figure 3a). The data set of 300 out of 4096 possible images was presented to a network with the architecture shown in Figure 3b. After 30 steps of EM, each consisting of 5 Gibbs samples of each hidden unit, the network reconstructed a representation that approximated the three underlying causes of the data-dedicating one vector mostly to diagonal lines, one to hollow squares, and one to crosses (Figure 3c). To assess how many Gibbs samples are required to obtain accurate estimates of the expectations for the E-step we repeated the lines problem varying the number of samples. Clearly, as the number of samples becomes large the Gibbs E-step becomes exact. Therefore we expect performance to asymptote at the performance of the exact E-step. The results indicate that, for this problem, 3 samples are sufficient to achieve ceiling performance (Figure 4). Surprisingly, a single iteration of the Factorial Learning and the EM Algorithm a) ;. ~+ .,. a:,. lSi- -~ , -:JI ~_- t rIl 1f --.~ • .p- rj~_- :'I,," D.... ~ ~ If :m' ~1r- ~.t ti 1I-. .q. at- ..:.p "t:I:I If !ill........ :11:1 ti+ .ttt~ ... c) b) 36 16 16 16 o 0 o 0 Figure 3. Shapes Problem. a) Sample images from the data set. b) Learning architecture used. c) Output weight matrix after learning. mean-field equations also performs quite well. 5 Discussion 623 The factorial learning problem for cooperative vector quantizers has been formulated in the EM framework, and two learning algorithms, based on Gibbs sampling and mean-field approximation, have been derived. Unlike the Boltzmann machine, Gibbs sampling for this architecture seems to require very few samples for adequate performance. This may be due to the fact that, whereas the Boltzmann machine relies on differences of noisy estimates for its weight changes, due to the constant partition function the factorial learning algorithm does not. The mean-field approximation also seems to perform quite well on all problems tested to date. This may also be a consequence of the constant partition function which guarantees that the mean-field cost is an upper bound on the exact cost. The framework can be extended to hidden Markov models (HMMs), showing that simple HMMs are a special case of dynamical CVQs, with the general case corresponding to parallel, factorial HMMs. The two principal advantages of such architectures are (1) unlike the traditional HMM, the state space can be represented as a combination of features, and (2) time series generated by multiple sources can be modeled. Simulation results on the Gibbs and mean-field EM algorithms for factorial HMMs are also promising (Ghahramani, 1995). 624 2.8 1 2 3 4 Gibbs samples or mean-field lIerarions Acknowledgements Zoubin Ghahramani Figure 4. Comparison of the Gibbs and mean-field EM algorithms for the lines data. Each data point shows the mean squared training error averaged over 10 runs of 20 EM steps, with standard error bars. For the Gibbs curve the abscissa is the number of samples per vector of hidden units; for the mean-field curve it is the number of iterations of equation (5). The author wishes to thank Lawrence Saul and Michael Jordan for invaluable discussions. This project was supported in part by a grant from the McDonnell-Pew Foundation, by a grant from ATR Human Information Processing Research Laboratories, by a grant from Siemens Corporation, and by grant N00014-94-1-0777 from the Office of Naval Research. References Ackley, D., Hinton, G., and Sejnowski, T. (1985). A learning algorithm for Boltzmann machines. Cognitive Science, 9:147-169. Barlow, H. (1989). Unsupervised learning. Neural Computation, 1:295-31l. Baum, L., Petrie, T., Soules, G., and Weiss, N. (1970). A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. The Annals of Mathematical Statistics, 41:164-17l. Dempster, A., Laird, N., and Rubin, D. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Royal Statistical Society Series B, 39:1-38. Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence,6:721-74l. Ghahramani, Z. (1995). Factorial learning and the EM algorithm. MIT Computational Cognitive Science TR 9501. Hinton, G. and Zemel, R. (1994). Autoencoders, minimum description length, and Helmholtz free energy. In Cowan, J., Tesauro, G., and Alspector, J., editors, Advances in Neural Information Processing Systems 6. Morgan Kaufmanm Publishers, San Francisco, CA. Jordan, M. and Jacobs, R. (1994). Hierarchical mixtures of experts and the EM algorithm. Neural Computation, 6:181-214. McCullagh, P .. and NeIder, J. (1989). Generalized Linear Models. Chapman & Hall, London. Neal, R. (1992). Connectionist learning of belief networks. Artificial Intelligence, 56:71113. Parisi, G. (1988). Statistical Field Theory. Addison-Wesley, Redwood City, CA. Redlich, A. (1993). Supervised factorial learning. Neural Computation, 5:750-766. Rissanen, J. (1989). Stochastic Complexity in Statistical Inquiry. World Scientific, Singapore. Saund, E. (1995). A multiple cause mixture model for unsupervised learning. Neural Computation,7(1):51-7l. Zemel, R. (1993). A minimum description length framework for unsupervised learning. Ph.D. Thesis, Dept. of Computer Science, University of Toronto, Toronto, Canada.	1994	23
914	Pairwise Neural Network Classifiers with Probabilistic Outputs David Price A2iA and ESPCI 3 Rue de l'Arrivee, BP 59 75749 Paris Cedex 15, France a2ia@dialup.francenet.fr Stefan Knerr ESPCI and CNRS (UPR AOOO5) 10, Rue Vauquelin, 75005 Paris, France knerr@neurones.espci.fr Leon Personnaz, Gerard Dreyfus ESPeI, Laboratoire d'Electronique 10, Rue Vauquelin, 75005 Paris, France dreyfus@neurones.espci.fr Abstract Multi-class classification problems can be efficiently solved by partitioning the original problem into sub-problems involving only two classes: for each pair of classes, a (potentially small) neural network is trained using only the data of these two classes. We show how to combine the outputs of the two-class neural networks in order to obtain posterior probabilities for the class decisions. The resulting probabilistic pairwise classifier is part of a handwriting recognition system which is currently applied to check reading. We present results on real world data bases and show that, from a practical point of view, these results compare favorably to other neural network approaches. 1 Introduction Generally, a pattern classifier consists of two main parts: a feature extractor and a classification algorithm. Both parts have the same ultimate goal, namely to transform a given input pattern into a representation that is easily interpretable as a class decision. In the case of feedforward neural networks, the interpretation is particularly easy if each class is represented by one output unit. For many pattern recognition problems, it suffices that the classifier compute the class of the input pattern, in which case it is common practice to associate the pattern to the class corresponding to the maximum output of the classifier. Other problems require graded (soft) decisions, such as probabilities, at the output of the 1110 David Price, Stefan Knerr, Leon Personnaz, Gerard Dreyfus classifier for further use in higher context levels: in speech or character recognition for instance, the probabilistic outputs of the phoneme (character) recognizer are often used by a Hidden-Markov-Model algorithm or by some other dynamic programming algorithm to compute the most probable word hypothesis. In the context of classification, it has been shown that the minimization of the Mean Square Error (MSE) yields estimates of a posteriori class probabilities [Bourlard & Wellekens, 1990; Duda & Hart, 1973]. The minimization can be performed by a feedforward multilayer perceptrons (MLP's) using the backpropagation algorithm, which is one of the reasons why MLP's are widely used for pattern recognition tasks. However, MLPs have well-known limitations when coping with real-world problems, namely long training times and unknown architecture. In the present paper, we show that the estimation of posterior probabilities for a K-class problem can be performed efficiently using estimates of posterior probabilities for K(K -1 )/2 two-class sub-problems. Since the number of sub-problems increases as K2, this procedure was originally intended for applications involving a relatively small number of classes, such as the 10 classes for the recognition of handwritten digits [Knerr et aI., 1992]. In this paper we show that this approach is also viable for applications with K» 10. The probabilistic pairwise classifier presented in this paper is part of a handwriting recognition system, discussed elsewhere [Simon, 1992], which is currently applied to check reading. The purpose of our character recognizer is to classify pre-segmented characters from cursive handwriting. The probabilistic outputs of the recognizer are used to estimate word probabilities. We present results on real world data involving 27 classes, compare these results to other neural network approaches, and show that our probabilistic pairwise classifier is a powerful tool for computing posterior class probabilities in pattern recognition problems. 2 Probabilistic Outputs from Two-class Classifiers Multi-class classification problems can be efficiently solved by "divide and conquer" strategies which partition the original problem into a set of K(K-l)/2 two-class problems. For each pair of classes (OJ and (OJ, a (potentially small) neural network with a single output unit is trained on the data of the two classes [Knerr et aI., 1990, and references therein]. In this section, we show how to obtain probabilistic outputs from each of the two-class classifiers in the pairwise neural network classifier (Figure 1). K(K-I)12 two-class networks inputs Figure 1: Pairwise neural network classifier. Pairwise Neural Network Classifiers with Probabilistic Outputs 1111 It has been shown that the \llinimization of the MSE cost function (or likewise a cost function based on an entropy measure, [Bridle, 1990]) leads to estimates of posterior probabilities. Of course, the quality of the estimates depends on the number and distribution of examples in the training set and on the minimization method used. In the theoretical case of two classes <01 and <02, each Gaussian distributed, with means m 1 and m2, a priori probabilities Pq and Pr2, and equal covariance matrices ~, the posterior probability of class <01 given the pattern x is: Pr(class=<o\ I X=x) = __________ --'1'----_________ _ 1 + Pr2 exp( _ !-(2xT~-$m\-m2) + m!~-\m2 - m T~-lm$) PrJ 2 (1) Thus a single neuron with a sigmoidal transfer function can compute the posterior probabilities for the two classes. However, in the case of real world data bases, classes are not necessarily Gaussian distributed, and therefore the transformation of the K(K-l )/2 outputs of our pairwise neural network classifier to posterior probabilities proceeds in two steps. In the first step, a class-conditional probability density estimation is performed on the linear output of each two-class neural network: for both classes <OJ and <OJ of a given twoclass neural network, we fit the probability density over Vjj (the weighted sum of the inputs of the output neuron) to a function. We denote by <Ojj the union of classes <OJ and <OJ. The resulting class-conditional densities p(vij I <OJ) and p(Vjj I <OJ) can be transformed to probabilities Pr(<Oj I <OJ' /\ (Vij=Vjj» and Pr(<Oj I <Ojj /\ (Vjj=Vjj» via the Bayes rule (note that Pr(<Ojj /\ (Vij=Vjj) 1 <OJ) = Pr«Vij=Vjj) I <OJ)): p( VjJ" I <OJ) Pr( <OJ) Pr(<Oj I <Ojj/\(Vij=Vij» = -----"-----L p(Vjj I <Ok) Pr(<Ok) ke{j,j} (2) It is a central assumption of our approach that the linear classifier output Vij is as informative as the input vector x. Hence, we approximate Prij = Pr(<Oj I <Ojj /\ (X=x» by Pr(<Oi I <Ojj /\ (V=Vjj». Note that Pji = I-Pjj. In the second step, the probabilities Prij are combined to obtain posterior probabilities Pr(<Oj I (X=x» for all classes <Oi given a pattern x. Thus, the network can be considered as generating an intermediate data representation in the recognition chain, subject to further processing [Denker & LeCun, 1991]. In other words, the neural network becomes part of the preprocessing and contributes to dimensionality reduction. 3 Combining the Probabilities Prij of the Two-class Classifiers to a posteriori Probabilities The set of two-class neural network classifiers discussed in the previous section results in probabilities Prjj for all pairs (i, j) with i * j. Here, the task is to express the posterior probabilities Pr(<Oj I (X=x» as functions of the Prjj1112 David Price, Stefan Knerr, Leon Personnaz, Gerard Dreyfus We assume that each pattern belongs to only one class: K Pr( U Olj I (X=x» = 1 j=1 From the definition of Olij. it follows for any given i: K K Pr(U Olj I (X=x» = Pr( U Olij I (X=x» = 1 J=I j=l,ji U sing the closed form expression for the probability of the union of N events Ei: N N N (3) (4) Pr(U Ei) = L Pr(Ej) + ... + (_I)k.1 L Pr(EhA ... AEh) + ... + (-I)N.lpr(EIA ... AEN) i= 1 i= 1 i}< ... <ik it follows from (4): K L Pr(Olij I (X=x» - (K-2) Pr(Oli I (X=x» = 1 (5) j=l,ji With Pr(OliAOli'A(X=X» Pr(Oli I (X=x» Prij = Pr(Oli I OlijA(X=X» = J = --'----~ Pr(OlijA(X=X» Pr(Olij I (X=x» (6) one obtains the final expression for the K posterior probabilities given the K(K-l)12 twoclass probabilities Prji : Pr(Oli I (X=x» = __ --"-1 __ _ f _1 __ (K-2) j=I,#i Prij (7) In [Refregier et aI., 1991], a method was derived which allows to compute the K posterior probabilities from only (K-l) two-class probabilities using the following relation between posterior probabilities and two-class probabilities: Prij = Pr(Oli I (X=x» Prji Pr(Olj I (X=x» (8) However, this approach has several practical drawbacks. For instance, in practice, the quality of the estimation of the posterior probabilities depends critically on the choice of the set of (K-l) two-class probabilities, and finding the optimal subset of (K-l) Prij is costly, since it has to be performed for each pattern at recognition time. Pairwise Neural Network Classifiers with Probabilistic Outputs 1113 4 Application to Cursive Handwriting Recognition We applied the concepts described in the previous sections to the classification of presegmented characters from cursive words originating from real-world French postal checks. For cursive word recognition it is important to obtain probabilities at the output of the character classifier since it is necessary to establish an ordered list of hypotheses along with a confidence value for further processing at the word recognition level: the probabilities can be passed to an Edit Distance algorithm [Wagner et at, 1974] or to a Hidden-Markov-Model algorithm [Kundu et aI., 1989] in order to compute recognition scores for words. For the recognition of the amounts on French postal checks we used an Edit Distance algorithm and made extensive use of the fact that we are dealing with a limited vocabulary (28 words). The 27 character classes are particularly chosen for this task and include pairs of letters such as "fr", "gttl, and "tr" because these combinations of letters are often difficult to presegment. Other characters, such as tlk" and "y" are not included because they do not appear in the given 28 word vocabulary. 0~)(" C.A~ t:; ~"U::l._ r;.., ~ \\:..~~~ (~tSl\h~~ \J'~ ~~~ ~~~~~~ &.nl' tL'i.upr rm'A Figure 2: Some examples of literal amounts from live French postal checks. A data base of about 3,300 literal amounts from postal checks (approximately 16,000 words) was annotated and, based on this annotation, segmented into words and letters using heuristic methods [Simon et aI., 1994]. Figure 2 shows some examples of literal amounts. The writing styles vary strongly throughout the data base and many checks are difficult to read even for humans. Note that the images of the pre-segmented letters may still contain some of the ligatures or other extraneous parts and do not in general resemble hand-printed letters. The total of about 55,000 characters was divided into three sets: training set (20,000), validation set (20,000), and test set (15,000). All three sets were used without any further data base cleaning. Therefore, many letters are not only of very bad quality, but they are truly ambiguous: it is not possible to recognize them uniquely without word context. Figure 3: Reference lines indicating upper and lower limit of lower case letters. Before segmentation, two reference lines were detected for each check (Figure 3). They indicate an estimated upper and lower limit of the lower case letters and are used for 1114 David Price, Stefan Knerr, Leon Personnaz, Gerard Dreyfus nonnalization of the pre-segmented characters (Figure 4) to 10 by 24 pixel matrices with 16 gray values (Figure 5). This is the representation used as input to the classifiers. Figure 4: Segmentation of words into isolated letters (ligatures are removed later). Figure 5: Size nonnalized letters: 10 by 24 pixel matrices with 16 gray values. The simplest two-class classifier is a single neuron; thus, 351 neurons of the resulting pairwise classifier were trained on the training data using the generalized delta rule (sigmoidal transfer function). In order to avoid overfitting, training was stopped at the minimum of MSE on the validation set. The probability densities P(Vij I IDi) were estimated on the validation set: for both classes IDi and IDj of a given neuron, we fitted the probability densities over the linear output Vij to a Gaussian. The two-class probabilities Prij and Prji were then obtained via Bayes rule. The 351 probabilities Prij were combined using equation (7) in order to obtain a posteriori probabilities Pr(IDi I (X=x», i E {1, .. ,27}. However, the a priori probabilities for letters as given by the training set are different from the prior probabilities in a given word context [Bourlard & Morgan, 1994]. Therefore, we computed the posterior probabilities either by using, in Bayes rule, the prior probabilities of the letters in the training set, or by assuming that the prior probabilities are equal. In the first case, many infonnative letters, for instance those having ascenders or descenders, have little chance to be recognized at all due to small a priori probabilities. Table 1 gives the recognition perfonnances on the test set for classes assumed to have equal a priori probabilities as well as for the true a priori probabilities of the test set. For each pattern, an ordered list (in descending order) of posterior class probabilities was generated; the recognition perfonnance is given (i) in tenns of percentage of true classes found in first position, and (ii) in tenns of average position of the true class in the ordered list. As mentioned above, the results of the first column are the most relevant ones, since the classifier outputs are subsequently used for word recognition. Note that the recognition rate (first position) of isolated letters without context for a human reader can be estimated to be around 70% to 80%. We compared the results of the pairwise classifier to a number of other neural network classification algorithms. First, we trained MLPs with one and two hidden layers and various numbers of hidden units using stochastic backpropagation. Here again, training was stopped based on the minimum MSE on the validation set. Second, we trained MLPs with a single hidden layer using the Softmax training algorithm [Bridle, 1990]. As a third approach, we trained 27 MLPs with 10 hidden units each, each MLP separating one class from all others. Table 1 gives the recognition perfonnances on the test set. The Softmax Pairwise Neural Network Classifiers with Probabilistic Outputs 1115 training algorithm clearly gives the best results in terms of recognition performance. However, the pairwise classifier has three very attractive features for classifier design: (i) training is faster than for MLP's by more than one order of magnitude; therefore, many different designs (changing pattern representations for instance) can be tested at a small computational cost; (ii) in the same spirit, adding a new class or modifying the training set of an existing one can be done without retraining all two-class classifiers; (iii) at least as importantly, the procedure gives more insight into the classification problem than MLP's do. Classifier A veragePosition First Position A veragePosition First Position equal prior probs equal prior probs true prior probs true prior probs Pairwise 2.9 48.9 % 2.6 52.2 % Classifier MLP 3.6 48.9 % 2.7 60.0 % (100 hid. units) Softmax 2.6 54.9 % 2.2 61.9 % (100 hid. units) 27 MLPs 3.2 41.6 % 2.4 55.8 % Table I: Recognition performances on the test set in terms of average position and recognition rate (first position) for the various neural networks used. Our pairwise classifier is part of a handwriting recognition system which is currently applied to check reading. The complete system also incorporates other character recognition algorithms as well as a word recognizer which operates without pre-segmentation. The result of the complete check recognition chain on a set of test checks is the following: (i) at the word level, 83.3% of true words are found in first position; (ii) 64.1 % of well recognized literal amounts are found in first position [Simon et al., 1994]. Recognizing also the numeral amount, we obtained 80% well recognized checks for 1 % error. 5 Conclusion We have shown how to obtain posterior class probabilities from a set of pairwise classifiers by (i) performing class density estimations on the network outputs and using Bayes rule, and (ii) combining the resulting two-class probabilities. The application of our pairwise classifier to the recognition of real world French postal checks shows that the procedure is a valuable tool for designing a recognizer, experimenting with various data representations at a small computational cost and, generally, getting insight into the classification problem. Acknowledgments The authors wish to thank J.C. Simon, N. Gorsky, O. Baret, and J.C. Deledicq for many informative and stimulating discussions. 1116 David Price, Stefan Knerr, Leon Personnaz, Gerard Dreyfus References H.A. Bourlard, N. Morgan (1994). Connectionist Speech Recognition. Kluwer Academic Publishers. H.A. Bourlard, C. Wellekens (1990). Links between Markov Models and Multilayer Perceptrons. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 12, No. 12, 1167-1178. J.S. Bridle (1990). Probabilistic Interpretation of Feedforward Classification Network Outputs, with Relationships to Statistical Pattern Recognition. In Neurocomputing: Algorithms, Architectures and Applications, Fogelman-Soulie, and Herault (eds.). NATO ASI Series, Springer. J.S. Denker, Y.LeCun (1991). Transforming Neural-Net Output Levels to Probability Distributions. In Advances in Neural Information Processing Systems 3, Lippmann, Moody, Touretzky (eds.). Morgan Kaufman. R.O. Duda, P.E. Hart (1973). Pattern Classification and Scene Analysis. Wiley. S. Knerr, L. Personnaz, G. Dreyfus (1990). Single-Layer Learning Revisited: A Stepwise Procedure for Building and Training a Neural Network. In Neurocomputing: Algorithms, Architectures and Applications, Fogelman-Soulie and Herault (eds.). NATO AS! Series, Springer. S. Knerr, L. Personnaz, G. Dreyfus (1992). Handwritten Digit Recognition by Neural Networks with Single-Layer Training. IEEE Transactions on Neural Networks, Vol. 3, No.6, 962-968. A. Kundu, Y. He, P. Bahl (1989). Recognition of Handwritten Words: First and Second Order Hidden Markov Model Based Approach. Pattern Recognition, Vol. 22, No.3. lC. Simon (1992). Off-Line Cursive Word Recognition. Proceedings of the IEEE, Vol. 80, No.7, 1150-1161. Ph. Refregier, F. Vallet (1991). Probabilistic Approach for Multiclass Classification with Neural Networks. Int. Conference on Artificial Networks, Vol. 2, 1003-1007. J.C. Simon, O. Baret, N. Gorski (1994). Reconnaisance d'ecriture manuscrite. Compte Rendu Academie des Sciences, Paris, t. 318, Serie II, 745-752. R.A. Wagner, M.J. Fisher (1974). The String to String Correction Problem. J.A.C.M. Vol. 21, No.5, 168-173. .	1994	24
915	Real-Time Control of a Tokamak Plasma Using Neural Networks Chris M Bishop Neural Computing Research Group Department of Computer Science Aston University Birmingham, B4 7ET, U.K. c.m.bishop@aston.ac.uk Paul S Haynes, Mike E U Smith, Tom N Todd, David L Trotman and Colin G Windsor AEA Technology, Culham Laboratory, Oxfordshire OX14 3DB (Euratom/UKAEA Fusion Association) Abstract This paper presents results from the first use of neural networks for the real-time feedback control of high temperature plasmas in a tokamak fusion experiment. The tokamak is currently the principal experimental device for research into the magnetic confinement approach to controlled fusion. In the tokamak, hydrogen plasmas, at temperatures of up to 100 Million K, are confined by strong magnetic fields. Accurate control of the position and shape of the plasma boundary requires real-time feedback control of the magnetic field structure on a time-scale of a few tens of microseconds. Software simulations have demonstrated that a neural network approach can give significantly better performance than the linear technique currently used on most tokamak experiments. The practical application of the neural network approach requires high-speed hardware, for which a fully parallel implementation of the multilayer perceptron, using a hybrid of digital and analogue technology, has been developed. 1008 C. Bishop, P. Haynes, M. Smith, T. Todd, D. Trotman, C. Windsor 1 INTRODUCTION Fusion of the nuclei of hydrogen provides the energy source which powers the sun. It also offers the possibility of a practically limitless terrestrial source of energy. However, the harnessing of this power has proved to be a highly challenging problem. One of the most promising approaches is based on magnetic confinement of a high temperature (107 - 108 Kelvin) plasma in a device called a tokamak (from the Russian for 'toroidal magnetic chamber') as illustrated schematically in Figure 1. At these temperatures the highly ionized plasma is an excellent electrical conductor, and can be confined and shaped by strong magnetic fields. Early tokamaks had plasmas with circular cross-sections, for which feedback control of the plasma position and shape is relatively straightforward. However, recent tokamaks, such as the COMPASS experiment at Culham Laboratory, as well as most next-generation tokamaks, are designed to produce plasmas whose cross-sections are strongly noncircular. Figure 2 illustrates some of the plasma shapes which COMPASS is designed to explore. These novel cross-sections provide substantially improved energy confinement properties and thereby significantly enhance the performance of the tokamak. z R Figure 1: Schematic cross-section of a tokamak experiment showing the toroidal vacuum vessel (outer D-shaped curve) and plasma (shown shaded). Also shown are the radial (R) and vertical (Z) coordinates. To a good approximation, the tokamak can be regarded as axisymmetric about the Z-axis, and so the plasma boundary can be described by its cross-sectional shape at one particular toroidal location. Unlike circular cross-section plasmas, highly non-circular shapes are more difficult to produce and to control accurately, since currents through several control coils must be adjusted simultaneously. Furthermore, during a typical plasma pulse, the shape must evolve, usually from some initial near-circular shape. Due to uncertainties in the current and pressure distributions within the plasma, the desired accuracy for plasma control can only be achieved by making real-time measurements of the position and shape of the boundary, and using error feedback to adjust the currents in the control coils. The physics of the plasma equilibrium is determined by force balance between the Real-Time Control of Tokamak Plasma Using Neural Networks circle ellipse O-shape bean Figure 2: Cross-sections of the COMPASS vacuum vessel showing some examples of potential plasma shapes. The solid curve is the boundary of the vacuum vessel, and the plasma is shown by the shaded regions. 1009 thermal pressure of the plasma and the pressure of the magnetic field, and is relatively well understood. Particular plasma configurations are described in terms of solutions of a non-linear partial differential equation called the Grad-Shafranov (GS) equation. Due to the non-linear nature of this equation, a general analytic solution is not possible. However, the GS equation can be solved by iterative numerical methods, with boundary conditions determined by currents flowing in the external control coils which surround the vacuum vessel. On the tokamak itself it is changes in these currents which are used to alter the position and cross-sectional shape of the plasma. Numerical solution of the GS equation represents the standard technique for post-shot analysis of the plasma, and is also the method used to generate the training dataset for the neural network, as described in the next section. However, this approach is computationally very intensive and is therefore unsuitable for feedback control purposes. For real-time control it is necessary to have a fast (typically:::; 50J.lsec.) determination of the plasma boundary shape. This information can be extracted from a variety of diagnostic systems, the most important being local magnetic measurements taken at a number of points around the perimeter of the vacuum vessel. Most tokamaks have several tens or hundreds of small pick up coils located at carefully optimized points around the torus for this purpose. We shall represent these magnetic signals collectively as a vector m. For a large class of equilibria, the plasma boundary can be reasonably well represented in terms of a simple parameterization, governed by an angle-like variable B, given by R(B) Z(B) Ro + a cos(B + 8 sinB) Zo + a/\,sinB where we have defined the following parameters (1) 1010 C. Bishop, P. Haynes, M. Smith, T. Todd, D. Trotman, C. Windsor Ro radial distance of the plasma center from the major axis of the torus, Zo vertical distance of the plasma center from the torus midplane, a minor radius measured in the plane Z = Zo, K elongation, 6 triangularity. We denote these parameters collectively by Yk. The basic problem which has to be addressed, therefore, is to find a representation for the (non-linear) mapping from the magnetic signals m to the values of the geometrical parameters Yk, which can be implemented in suitable hardware for real-time control. The conventional approach presently in use on many tokamaks involves approximating the mapping between the measured magnetic signals and the geometrical parameters by a single linear transformation. However, the intrinsic non-linearity of the mappings suggests that a representation in terms of feedforward neural networks should give significantly improved results (Lister and Schnurrenberger, 1991; Bishop et a/., 1992; Lagin et at., 1993). Figure 3 shows a block diagram of the control loop for the neural network approach to tokamak equilibrium control. Neural Network Figure 3: Block diagram of the control loop used for real-time feedback control of plasma position and shape. 2 SOFTWARE SIMULATION RESULTS The dataset for training and testing the network was generated by numerical solution of the GS equation using a free-boundary equilibrium code. The data base currently consists of over 2,000 equilibria spanning the wide range of plasma positions and shapes available in COMPASS. Each equilibrium configuration takes several minutes to generate on a fast workstation. The boundary of each configuration is then fitted using the form in equation 1, so that the equilibria are labelled with the appropriate values of the shape parameters. Of the 120 magnetic signals available on COMPASS which could be used to provide inputs to the network, a Real-Time Control o/Tokamak PLasma Using Neural Networks 1011 subset of 16 has been chosen using sequential forward selection based on a linear representation for the mapping (discussed below). It is important to note that the transformation from magnetic signals to flux surface parameters involves an exact linear invariance. This follows from the fact that, if all of the currents are scaled by a constant factor, then the magnetic fields will be scaled by this factor, and the geometry of the plasma boundary will be unchanged. It is important to take advantage of this prior knowledge and to build it into the network structure, rather than force the network to learn it by example. We therefore normalize the vector m of input signals to the network by dividing by a quantity proportional to the total plasma current. Note that this normalization has to be incorporated into the hardware implementation of the network, as will be discussed in Section 3. 4 01 2 c .5. 0- ° CIS :E 1iI ~ -2 ::J -4 4 ~ 2 ~ ° CD Z ~ -2 :::I CD z -4 Database Database 01 c .5. 2 ~ :E ° 1iI CD c ::J -2 Database Database 1.2 go.8 .5. 0CIS :E 1iI0.4 CD c ::J ° Database 1.2 ~O.8 ~ CD z ~O.4 :::I CD Z ° Database Figure 4: Plots of the values from the test set versus the values predicted by the linear mapping for the 3 equilibrium parameters, together with the corresponding plots for a neural network with 4 hidden units. .2 .2 The results presented in this paper are based on a multilayer perceptron architecture having a single layer of hidden units with 'tanh' activation functions, and linear output units. Networks are trained by minimization of a sum-of-squares error using a standard conjugate gradients optimization algorithm, and the number of hidden J012 C. Bishop, P. Haynes, M. Smith, T. Todd, D. Trotman, C. Windsor units is optimized by measuring performance with respect to an independent test set. Results from the neural network mapping are compared with those from the optimal linear mapping, that is the single linear transformation which minimizes the same sum-of-squares error as is used in the neural network training algorithm, as this represents the method currently used on a number of present day tokamaks. Initial results were obtained on networks having 3 output units, corresponding to the values of vertical position ZQ, major radius RQ, and elongation K; these being parameters which are of interest for real-time feedback control. The smallest normalized test set error of 11.7 is obtained from the network having 16 hidden units. By comparison, the optimal linear mapping gave a normalized test set error of 18.3. This represents a reduction in error of about 30% in going from the linear mapping to the neural network. Such an improvement, in the context of this application, is very significant. For the experiments on real-time feedback control described in Section 4 the currently available hardware only permitted networks having 4 hidden units, and so we consider the results from this network in more detail. Figure 4 shows plots of the network predictions for various parameters versus the corresponding values from the test set portion of the database. Analogous plots for the optimal linear map predictions versus the database values are also shown. Comparison of the corresponding figures shows the improved predictive capability of the neural network, even for this sub-optimal network topology. 3 HARDWARE IMPLEMENTATION The hardware implementation of the neural network must have a bandwidth of 2: 20 kHz in order to cope with the fast timescales of the plasma evolution. It must also have an output precision of at least (the the analogue equivalent of) 8 bits in order to ensure that the final accuracy which is attainable will not be limited by the hardware system. We have chosen to develop a fully parallel custom implementation of the multilayer perceptron, based on analogue signal paths with digitally stored synaptic weights (Bishop et al., 1993). A VME-based modular construction has been chosen as this allows flexibility in changing the network architecture, ease of loading network weights, and simplicity of data acquisition. Three separate types of card have been developed as follows: • Combined 16-input buffer and signal normalizer. This provides an analogue hardware implementation of the input normalization described earlier. • 16 x 4 matrix multiplier The synaptic weights are produced using 12 bit frequency-compensated multiplying DACs (digital to analogue converters) which can be configured to allow 4-quadrant multiplication of analogue signals by a digitally stored number. • 4-channel sigmoid module There are many ways to produce a sigmoidal non-linearity, and we have opted for a solution using two transistors configured as along-tailed-pair, Real-Time Control of Tokamak Plasma Using Neural Networks 1013 to generate a 'tanh' sigmoidal transfer characteristic. The principal drawback of such an approach is the strong temperature sensitivity due to the appearance of temperature in the denominator of the exponential transistor transfer characteristic. An elegant solution to this problem has been found by exploiting a chip containing 5 transistors in close thermal contact. Two of the transistors form the long-tailed pair, one of the transistors is used as a heat source, and the remaining two transistors are used to measure temperature. External circuitry provides active thermal feedback control, and stability to changes in ambient temperature over the range O°C to 50°C is found to be well within the acceptable range. The complete network is constructed by mounting the appropriate combination of cards in a VME rack and configuring the network topology using front panel interconnections. The system includes extensive diagnostics, allowing voltages at all key points within the network to be monitored as a function of time via a series of multiplexed output channels. 4 RESULTS FROM REAL-TIME FEEDBACK CONTROL Figure 5 shows the first results obtained from real-time control of the plasma in the COMPASS tokamak using neural networks. The evolution of the plasma elongation, under the control of the neural network, is plotted as a function of time during a plasma pulse. Here the desired elongation has been preprogrammed to follow a series of steps as a function of time. The remaining 2 network outputs (radial position Ro and vertical position Zo) were digitized for post-shot diagnosis, but were not used for real-time control. The solid curve shows the value of elongation given by the corresponding network output, and the dashed curve shows the post-shot reconstruction of the elongation obtained from a simple 'filament' code, which gives relatively rapid post-shot plasma shape reconstruction but with limited accuracy. The circles denote the elongation values given by the much more accurate reconstructions obtained from the full equilibrium code. The graph clearly shows the network generating the required elongation signal in close agreement with the reconstructed values. The typical residual error is of order 0.07 on elongation values up to around 1.5. Part of this error is attributable to residual offset in the integrators used to extract magnetic field information from the pick-up coils, and this is currently being corrected through modifications to the integrator design. An additional contribution to the error arises from the restricted number of hidden units available with the initial hardware configuration. While these results represent the first obtained using closed loop control, it is clear from earlier software modelling of larger network architectures (such as 32- 16-4) that residual errors of order a few % should be attainable. The implementation of such larger networks is being persued, following the successes with the smaller system. Acknowledgements We would like to thank Peter Cox, Jo Lister and Colin Roach for many useful discussions and technical contributions. This work was partially supported by the UK Department of Trade and Industry. 1014 c: o 1.8 ~ 14 C) • c: o as 1.0 C. Bishop, P. Haynes, M. Smith, T. Todd, D. Trotman, C. Windsor shot 9576 0.0 0.1 0.2 time (sec.) Figure 5: Plot of the plasma elongation K. as a function of time during shot no. 9576 on the COMPASS tokamak, during which the elongation was being controlled in real-time by the neural network. References Bishop C M, Cox P, Haynes P S, Roach C M, Smith M E U, Todd T N and Trotman D L, 1992. A neural network approach to tokamak equilibrium control. In Neural Network Applications, Ed. J G Taylor, Springer Verlag, 114-128. Bishop C M, Haynes P S, Roach C M, Smith ME U, Todd T N, and Trotman D L. 1993. Hardware implementation of a neural network for plasma position control in COMPASS-D. In Proceedings of the 17th. Symposium on Fusion Technology, Rome, Italy. 2 997-1001. Lagin L, Bell R, Davis S, Eck T, Jardin S, Kessel C, Mcenerney J, Okabayashi M, Popyack J and Sauthoff N. 1993. Application of neural networks for real-time calculations of plasma equilibrium parameters for PBX-M, In Proceedings of the 17th. Symposium on Fusion Technology, Rome, Italy. 21057-106l. Lister J Band Schnurrenberger H. 1991. Fast non-linear extraction of plasma parameters using a neural network mapping. Nuclear Fusion. 31, 1291-1300.	1994	25
916	An Actor/Critic Algorithm that Equivalent to Q-Learning • IS Robert H. Crites Computer Science Department University of Massachusetts Amherst, MA 01003 crites~cs.umass.edu Andrew G. Barto Computer Science Department University of Massachusetts Amherst, MA 01003 barto~cs.umass.edu Abstract We prove the convergence of an actor/critic algorithm that is equivalent to Q-Iearning by construction. Its equivalence is achieved by encoding Q-values within the policy and value function of the actor and critic. The resultant actor/critic algorithm is novel in two ways: it updates the critic only when the most probable action is executed from any given state, and it rewards the actor using criteria that depend on the relative probability of the action that was executed. 1 INTRODUCTION In actor/critic learning systems, the actor implements a stochastic policy that maps states to action probability vectors, and the critic attempts to estimate the value of each state in order to provide more useful reinforcement feedback to the actor. The result is two interacting adaptive processes: the actor adapts to the critic, while the critic adapts to the actor. The foundations of actor/critic learning systems date back at least to Samuel's checker program in the late 1950s (Samuel,1963). Examples of actor/critic systems include Barto, Sutton, & Anderson's (1983) ASE/ ACE architecture and Sutton's (1990) Dyna-PI architecture. Sutton (1988) notes that the critic in these systems performs temporal credit assignment using what he calls temporal difference (TD) methods. Barto, Sutton, & Watkins (1990) note a relationship between actor/critic 402 Robert Crites, Andrew G. Barto architectures and a dynamic programming (DP) algorithm known as policy iteration. Although DP is a collection of general methods for solving Markov decision processes (MDPs), these algorithms are computationally infeasible for problems with very large state sets. Indeed, classical DP algorithms require multiple complete sweeps of the entire state set. However, progress has been made recently in developing asynchronous, incremental versions of DP that can be run online concurrently with control (Watkins, 1989; Barto et ai, 1993). Most of the theoretical results for incremental DP have been for algorithms based on a DP algorithm known as value iteration. Examples include Watkins' (1989) Q-Iearning algorithm (motivated by a desire for on-line learning), and Bertsekas & Tsitsiklis' (1989) results on asynchronous DP (motivated by a desire for parallel implementations). Convergence proofs for incremental algorithms based on policy iteration (such as actor/critic algorithms) have been slower in coming. Williams & Baird (1993) provide a valuable analysis of the convergence of certain actor/critic learning systems that use deterministic policies. They assume that a model of the MDP (including all the transition probabilities and expected rewards) is available, allowing the use of operations that look ahead to all possible next states. When a model is not available for the evaluation of alternative actions, one must resort to other methods for exploration, such as the use of stochastic policies. We prove convergence for an actor/critic algorithm that uses stochastic policies and does not require a model of the MDP. The key idea behind our proof is to construct an actor/critic algorithm that is equivalent to Q-Iearning. It achieves this equivalence by encoding Q-values within the policy and value function of the actor and critic. By illustrating the way Qlearning appears as an actor/critic algorithm, the construction sheds light on two significant differences between Q-Iearning and traditional actor/critic algorithms. Traditionally, the critic attempts to provide feedback to the actor by estimating V1I', the value function corresponding to the current policy'll". In our construction, instead of estimating ~, the critic directly estimates the optimal value function V"'. In practice, this means that the value function estimate V is updated only when the most probable action is executed from any given state. In addition, our actor is provided with more discriminating feedback, based not only on the TD error, but also on the relative probability of the action that was executed. By adding these modifications, we can show that this algorithm behaves exactly like Q-Iearning constrained by a particular exploration strategy. Since a number of proofs of the convergence of Q-Iearning already exist (Tsitsiklis, 1994; Jaakkola et ai, 1993; Watkins & Dayan, 1992), the fact that this algorithm behaves exactly like Q-Iearning implies that it too converges to the optimal value function with probability one. 2 MARKOV DECISION PROCESSES Actor/critic and Q-Iearning algorithms are usually studied within the Markov decision process framework. In a finite MDP, at each discrete time step, an agent observes the state :z: from a finite set X, and selects an action a from a finite set Ax by using a stochastic policy'll" that assigns a probability to each action in Ax. The agent receives a reward with expected value R(:z:, a), and the state at the next An Actor/Critic Algorithm That Is Equivalent to Q-Learning 403 time step is y with probability pll(:z:,y). For any policy 7£' and :z: E X, let V""(:Z:) denote the ezpected infinite-horizon discounted return from :z: given that the agent uses policy 7£'. Letting rt denote the reward at time t, this is defined as: V""(:z:) = E7r [L:~o,trtl:z:o = :z:], (1) where :Z:o is the initial state, 0 :::; , < 1 is a factor used to discount future rewards, and E7r is the expectation assuming the agent always uses policy 7£'. It is usual to call V7r (:z:) the value of:z: under 7£'. The function V"" is the value function corresponding to 7£'. The objective is to find an optimal policy, i.e., a policy,7£', that maximizes the value of each state :z: defined by (1). The unique optimal value function, V, is the value function corresponding to any optimal policy. Additional details on this and other types of MOPs can be found in many references. 3 ACTOR/CRITIC ALGORITHMS A generic actor/critic algorithm is as follows: 1. Initialize the stochastic policy and the value function estimate. 2. From the current state :z:, execute action a randomly according to the current policy. Note the next state y, the reward r, and the TO error e = [r + ,V(y)] - V(:z:), where 0 :::; , < 1 is the discount factor. 3. Update the actor by adjusting the action probabilities for state :z: using the TO error. If e > 0, action a performed relatively well and its probability should be increased. If e < 0, action a performed relatively poorly and its probability should be decreased. 4. Update the critic by adjusting the estimated value of state :z: using the TO error: V(:z:) -- V(:z:) + a e where a is the learning rate. 5. :z: -- y. Go to step 2. There are a variety of implementations of this generic algorithm in the literature. They differ in the exact details of how the policy is stored and updated. Barto et al (1990) and Lin (1993) store the action probabilities indirectly using parameters w(:z:, a) that need not be positive, and need not sum to one. Increasing (or decreasing) the probability of action a in state :z: is accomplished by increasing (or decreasing) the value of the parameter w(:z:, a). Sutton (1990) modifies the generic algorithm so that these parameters can be interpreted as action value estimates. He redefines e in step 2 as follows: e = [r + ,V(y)] - w(:z:, a). For this reason, the Oyna-PI architecture (Sutton, 1990) and the modified actor/critic algorithm we present below both reward less probable actions more readily because of their lower estimated values. 404 Robert Crites, Andrew G. Barto Barto et al (1990) select actions by adding exponentially distributed random numbers to each parameter w(:z:, a) for the current state, and then executing the action with the maximum sum. Sutton (1990) and Lin (1993) convert the parameters w(:z:, a) into action probabilities using the Boltzmann distribution, where given a temperature T, the probability of selecting action i in state :z: is ew(x,i)/T '" ew(x,a)/T' L.JaEA .. In spite of the empirical success of these algorithms, their convergence has never been proven. 4 Q-LEARNING Rather than learning the values of states, the Q-Iearning algorithm learns the values of state/action pairs. Q(:z:, a) is the expected discounted return obtained by performing action a in state :z: and performing optimally thereafter. Once the Q function has been learned, an optimal action in state :z: is any action that maximizes Q(:z:, .). Whenever an action a is executed from state :z:, the Q-value estimate for that state/action pair is updated as follows: Q(:z:, a) +- Q(:z:, a) + O!xa(n) [r + "y maxbEAlI Q(y, b) - Q(:z:, a)), where O!xa (n) is the non-negative learning rate used the nth time action a is executed from state :z:. Q-Learning does not specify an exploration mechanism, but requires that all actions be tried infinitely often from all states. In actor/critic learning systems, exploration is fully determined by the action probabilities of the actor. 5 A MODIFIED ACTOR/CRITIC ALGORITHM For each value v E !R, the modified actor/critic algorithm presented below uses an invertible function, H.", that assigns a real number to each action probability ratio: H1J : (0,00) -+ !R. Each H." must be a continuous, strictly increasing function such that H.,,(l) = v, and HH .. (Z2)(i;) = H",(Zl) for all Zl,Z2 > o. One example of such a class of functions is H.,,(z) = T In(z) + v, v E !R, for some positive T. This class of functions corresponds to Boltzmann exploration in Qlearning. Thus, a kind of simulated annealing can be accomplished in the modified actor/critic algorithm (as is often done in Q-Iearning) by gradually lowering the "temperature" T and appropriately renormalizing the action probabilities. It is also possible to restrict the range of H." if bounds on the possible values for a given MDP are known a priori. For a state :z:, let Pa be the probability of action a, let Pmax be the probability of the most probable action, amax , and let Za = ~. An Actor/Critic Algorithm That Is Equivalent to Q-Leaming 405 The modified actor/critic algorithm is as follows: 1. Initialize the stochastic poliCj and the value function estimate. 2. From the current state :z:, execute an action randomly according to the current policy. Call it action i. Note the next state y and the immediate reward r, and let e = [r + -yV(y)] - Hy(X) (Zi). 3. Increase the probability of action i if e > 0, and decrease its probability if e < O. The precise probability update is as follows. First calculate zt = H~tX)[HY(x)(Zi) + aXi(n) e]. Then determine the new action probabilities by dividing by normalization factor N = zt + E#i Zj, as follows: a:~ a:' Pi +-:W, and Pj +- =jt, j =P i. 4. Update V(:z:) only if i = UomIlX' Since the action probabilities are updated after every action, the most probable action may be different before and after the update. If i = amllx both before and after step 3 above, then update the value function estimate as follows: V(:z:) +- V(:z:) + aXi(n) e Otherwise, if i = UomIlX before or after step 3: V(:z:) +- HY(x)(Npk), where action Ie is the most probable action after step 3. 5. :z: +- y. Go to step 2. 6 CONVERGENCE OF THE MODIFIED ALGORITHM Theorem: The modified actor/critic algorithm given above converge6 to the optimal value function V· with probability one if: 1. The 6tate and action 6et6 are finite. 2. E:=o axlI(n) = 00 and E:=o a!lI(n) < 00. Space does not permit us to supply the complete proof, which follows this outline: 1. The modified actor/critic algorithm behaves exactly the same as a Qlearning algorithm constrained by a particular exploration strategy. 2. Q-Iearning converges to V· with probability one, given the conditions above (Tsitsiklis, 1993; Jaakkola et aI, 1993; Watkins & Dayan, 1992). 3. Therefore, the modified actor/critic algorithm also converges to V· with probability one. 406 Robert Crites, Andrew G. Barto The commutative diagram below illustrates how the modified actor/critic algorithm behaves exactly like Q-Iearning constrained by a particular exploration strategy. The function H recovers Q-values from the policy ?r and value function V. H- 1 recovers (?r, V) from the Q-values, thus determining an exploration strategy. Given the ability to move back and forth between (?r, V) and Q, we can determine how to change (?r, V) by converting to Q, determining updated Q-values, and then converting back to obtain an updated (?r, V). The modified actor/critic algorithm simply collapses this process into one step, bypassing the explicit use of Q-values. ( VA) Modified Actor/Critic ( VA) 7r -------.. 7r , t , t+1 H H-l A A (Jt ------------Q---L-ea-r-ru-·n-g----------~~ (Jt+1 Following the diagram above, (?r, V) can be converted to Q-values as follows: Going the other direction, Q-values can be converted to (?r, V) as follows: and The only Q-value that should change at time t is the one corresponding to the state/action pair that was visited at time tj call it Q(:z:, i). In order to prove the convergence theorem, we must verify that after an iteration ofthe modified actor/critic algorithm, its encoded Q-values match the values produced by Q-Iearning: Qt+1(:Z:, a) = Qt(:Z:, i) + Qx.(n) [r + "y max Qt(Y, b) - Qt(:Z:, i)], a = i. (2) bEAli (3) In verifying this, it is necessary to consider the four cases where Q(:z:, i) is, or is not, the maximum Q-value for state :z: at times t and t + 1. Only enough space exists to present a detailed verification of one case. Case 1: Qt(:Z:, i) = ma:z: Qt(:Z:,·) and Qt+l(:Z:, i) = ma:z: Qt+l(:Z:, .). In this case, Jli(t) = Pmax(t) and P.(t + 1) = Pmax(t + 1), since Hyt(x) and Hyt+1(x) are strictly increasing. Therefore Zi (t) = 1 and Zi (t + 1) = 1. Therefore, Vi ( :z:) = HYt(x)[1] = HYt(x)[Zi(t)] = Qt(:Z:, i), and An Actor/Critic Algorithm That Is Equivalent to Q-Learning Hyt+1(x) [Zi(t + 1)] Hyt+1(x) [1] Vt+l(:C) Vt(:c) + O!xi(n) e Qt(:Z:, i) + O!xi(n) [r + "y max Qt(Y, b) - Qt(:Z:, i)]. bEAJI This establishes (2). To show that (3) holds, we have that Vt+l(:Z:) Vt(:c) + O!xi(n) e and Qt(:z:, i) + O!xi(n) e HYt(x)[Zi(t)] + O!xi(n) e HYt(x)[H~t~x)[HYt(x)[Zi(t)] + O!xi(n) e]] Hyt(x) [zt(t)] Hyt+1(x) [za(t + 1)] H [ Pa(t + 1) ] Yt+1(x) Pmax(t + 1) H [Za(t)/lV] f Yt+1(x) zt(t)/lV i a i= i H [Za(t)] Yt+1(x) zt(t) za(t) HHfrt( .. )[zt(t)][zt(t)1 by (4) Hyt(x) [za(t)] by a property of H Qt(:Z:, a). The other cases can be shown similarly. 7 CONCLUSIONS 407 (4) We have presented an actor/critic algorithm that is equivalent to Q-Iearning constrained by a particular exploration strategy. Like Q-Iearning, it estimates V" directly without a model of the underlying decision process. It uses exactly the same amount of storage as Q-Iearning: one location for every state/action pair. (For each state, IAI- 1 locations are needed to store the action probabilities, since they must sum to one. The remaining location can be used to store the value of that state.) One advantage of Q-Iearning is that its exploration is uncoupled from its value function estimates. In the modified actor/critic algorithm, the exploration strategy is more constrained. 408 Robert Crites, Andrew G. Barto It is still an open question whether other actor/critic algorithms are guaranteed to converge. One way to approach this question would be to investigate further the relationship between the modified actor/critic algorithm described here and the actor/critic algorithms that have been employed by others. Acknowledgements We thank Vijay GullapaUi and Rich Sutton for helpful discussions. This research was supported by Air Force Office of Scientific Research grant F49620-93-1-0269. References A. G. Barto, S. J. Bradtke &. S. P. Singh. (1993) Learning to act using real-time dynamic programming. Artificial Intelligence, Accepted. A. G. Barto, R. S. Sutton &. C. W. Anderson. (1983) Neuronlike adaptive elements that can solve difficult learning control problems. IEEE Transactions on Systems, Man, and Cybernetics 13:835-846. A. G. Barto, R. S. Sutton &. C. J. C. H. Watkins. (1990) Learning and sequential decision making. In M. Gabriel &. J. Moore, editors, Learning and Computational Neuroscience: Foundations of Adaptive Networks. MIT Press, Cambridge, MA. D. P. Bertsekas &. J. N. Tsitsiklis. (1989) Parallel and Distributed Computation: Numerical Metkods. Prentice-Hall, Englewood Cliffs, N J. T. Jaakkola, M. 1. Jordan &. S. P. Singh. (1993) On the convergence of stochastic iterative dynamic programming algorithms. MIT Computational Cognitive Science Technical Report 9307. L. Lin. (1993) Reinforcement Learning for Robots Using Neural Networks. PhD Thesis, Carnegie Mellon University, Pittsburgh, PA. A. L. Samuel. (1963) Some studies in machine learning using the game of checkers. In E. Feigenbaum &. J. Feldman, editors, Computers and Tkougkt. McGraw-Hill, New York, NY. R. S. Sutton. (1988) Learning to predict by the methods of temporal differences. Mackine Learning 3:9-44. R. S. Sutton. (1990) Integrated architectures for learning, planning, and reacting based on approximating dynamic programming. In Proceedings of tke Seventk International Conference on Mackine Learning. J. N. Tsitsiklis. (1994) Asynchronous stochastic approximation and Q-Iearning. Mackine Learning 16:185-202. C. J. C. H. Watkins. (1989) Learning from Delayed Rewards. PhD thesis, Cambridge University. C. J. C. H. Watkins &. P. Dayan. (1992) Q-Iearning. Mackine Learning 8:279-292. R. J. Williams &. L. C. Baird. (1993) Analysis ofsome incremental variants of policy iteration: first steps toward understanding actor-critic learning systems. Technical Report NU-CCS-93-11. Northeastern University College of Computer Science. PART V ALGORITHMS AND ARCIDTECTURES	1994	26
917	Template-Based Algorithms for Connectionist Rule Extraction Jay A. Alexander and Michael C. Mozer Department of Computer Science and Institute for Cognitive Science University of Colorado Boulder, CO 80309--0430 Abstract Casting neural network weights in symbolic terms is crucial for interpreting and explaining the behavior of a network. Additionally, in some domains, a symbolic description may lead to more robust generalization. We present a principled approach to symbolic rule extraction based on the notion of weight templates, parameterized regions of weight space corresponding to specific symbolic expressions. With an appropriate choice of representation, we show how template parameters may be efficiently identified and instantiated to yield the optimal match to a unit's actual weights. Depending on the requirements of the application domain, our method can accommodate arbitrary disjunctions and conjunctions with O(k) complexity, simple n-of-m expressions with O( k!) complexity, or a more general class of recursive n-of-m expressions with O(k!) complexity, where k is the number of inputs to a unit. Our method of rule extraction offers several benefits over alternative approaches in the literature, and simulation results on a variety of problems demonstrate its effectiveness. 1 INTRODUCTION The problem of understanding why a trained neural network makes a given decision has a long history in the field of connectionist modeling. One promising approach to this problem is to convert each unit's weights and/or activities from continuous numerical quantities into discrete, symbolic descriptions [2, 4, 8]. This type of reformulation, or rule extraction, can both explain network behavior and facilitate transfer of learning. Additionally, in intrinsically symbolic domains, there is evidence that a symbolic description can lead to more robust generalization [4]. 610 Jay A. Alexander, Michael C. Mozer We are interested in extracting symbolic rules on a unit-by-unit basis from connectionist nets that employ the conventional inner product activation and sigmoidal output functions. The basic language of description for our rules is that of n-of-m expressions. An n-of-m expression consists of a list of m subexpressions and a value n such that 1 ~ n ~ m. The overall expression is true when at least n of the m subexpressions are true. An example of an n-of-m expression stated using logical variables is the majority voter function X = 2 of (A, B, C). N-of-m expressions are interesting because they are able to model behaviors intermediate to standard Boolean OR (n = 1) and AND (n = m) functions. These intermediate behaviors reflect a limited form of two-level Boolean logic. (To see why this is true, note that the expression for X above is equivalent to AB + BC + AC.) In a later section we describe even more general behaviors that can be represented using recursive forms of these expressions. N-of-m expressions fit well with the activation behavior of sigmoidal units, and they are quite amenable to human comprehension. To extract an n-of-m rule from a unit's weights, we follow a three-step process. First we generate a minimal set of candidate templates, where each template is parameterized to represent a given n-of-m expression. Next we instantiate each template's parameters with optimal values. Finally we choose the symbolic expression whose instantiated template is nearest to the actual weights. Details on each of these steps are given below. 2 TEMPLATE-BASED RULE EXTRACTION 2.1 Background Following McMillan [4], we define a weight template as a parameterized region of weight space corresponding to a specific symbolic function. To see how weight templates can be used to represent symbolic functions, consider the weight vector for a sigmoidal unit with four inputs and a bias: w = WI w2 w3 w4 b Now consider the following two template vectors: t1 = -p P 0 -p J.5p t2 = P -p P P -D.5p These templates are parameterized by the variable p. Given a large positive value of p (say 5.0) and an input vector 1 (whose components are approximately 0 and 1), t1 describes the symbolic expression J of('lz, 12, 14), while t2 describes the symbolic expression 2 of(Ib 12• h. 14), A general description for n-of-m templates of this form is the following: 1. M of the weight values are set to ±p , P > 0; all others are set to O. (+p is used for normal subexpressions, -p for negated sUbexpressions) 2. The bias value is set to (0.5 + mneg - n)p, where mneg represents the number of negated subexpressions. When the inputs are Boolean with values -1 and + 1, the form of the templates is the same, except the template bias takes the value (J + m - 2n)p. This seemingly trivial difference turns out to have a significant effect on the efficiency of the extraction process. Template-Based Algorithms for Connectionist Rule Extraction 611 2.2 Basic extraction algorithm Generating candidate templates Given a sigmoidal unit with k inputs plus a bias, the total number of n-of-m expressions that unit may compute is an exponential function of k: /c m /c T = ~ ~ 2m(k) = ~ 2mk! = 2k3/C-1 £.J £.J m £.J (k-m)!(m-l)! m=ln=1 m=1 For example, T k=1O is 393,660, while T k=20 is over 46 billion. Fortunately we can apply knowledge of the unit's actual weights to explore this search space without generating a template for each possible n-of-m expression. Alexander [1] proves that when the -11+1 input representation is used, we need consider at most one template for each possible choice of n and m. For a given choice of nand m, a template is indicated when sign( 1 + m - 2n) = sign(b). A required template is formed by setting the template weights corresponding to the m highest absolute value actual weights to sp, where s represents the sign of the corresponding actual weight. The template bias is set to (1 + m - 2n)p. This reduces the number of templates required to a polynomial function of k: Values for T k=1O and T k=20 are now 30 and 110, respectively, making for a very efficient pruning of the search space. When 011 inputs are used, this simple procedure does not suffice and many more templates must be generated. For this reason, in the remainder of this paper we focus on the -11+ 1 case and assume the use of symmetric sigmoid functions. Instantiating template parameters Instantiating a weight template t requires finding a value for p such that the Euclidean distance d = lit - wl1 2 is minimized. Letting Uj = 1 if template weight tj is nonzero, Uj = 0 otherwise, the value of p that minimizes this distance for any -11+ 1 template is given by: /c L IWjIUj+ (1 +m-2n)b p* = I:....: · ==-I!...-_____ ~m + (1 + m - 2n) 2 Finding the nearest template and checking extraction validity Once each template is instantiated with its value of p, the distance between the template and the actual weight vector is calculated, and the minimal distance template is selected as the basis for rule extraction. Having found the nearest template t, we can use its values as part of a rudimentary check on extraction validity. For example, we can define the extraction error as 100% x Ilt-wI1 2/1IwI1 2 to measure how well the nearest symbolic rule fits the actual weights. We can also examine the value of p used in t. Small values of p translate into activation levels in the linear regime of the sigmoid functions, compromising the assumption of Boolean outputs propagating to subsequent inputs. 612 Jay A. Alexander, Michael C. Mozer 2.3 Extending expressiveness While the n-of-m expressions treated thus far are fairly powerful, there is an interesting class of symbolic behaviors that cannot be captured by simple n-of-m expressions. The simplest example of this type of behavior may be seen in the single hidden unit version of xor described in [6]. In this 2-1-1 network the hidden unit H learns the expression AND(h 12), while the output unit (which connects to the two inputs as well as to the hidden unit) learns the expression AND[OR(h 12),Hj. This latter expression may be viewed as a nested or recursive form of n-of-m expression, one where some of the m subexpressions may themselves be n-of-m expressions. The following two forms of recursive n-of-m expressions are linearly separable and are thus computable by a single sigmoidal unit: OR [Cn-of-m ' COR j AND [Cn-of-m , CAND j where Cn-of-m is a nested n-of-m expression (1 S; n S; m) COR is a nested OR expression (n = 1) CAND is a nested AND expression (n = m) These expressions may be seen to generalize simple n-of-m expressions in the same way that simple n-of-m expressions generalize basic disjunctions and conjunctions.1 We term the above forms augmented n-of-m expressions because they extend simple n-of-m expressions with additional disjuncts or conjuncts. Templates for these expressions (under the -1/+ 1 input representation) may be efficiently generated and instantiated using a procedure similar to that described for simple n-of-m expressions. When augmented expressions are included in the search, the total number of templates required becomes: This figure is O(k) worse than for simple n-of-m expressions, but it is still polynomial in k and is quite manageable for many problems. (Values for T k=1O and Tk=20 are 150 and 1250, respectively.) A more detailed treatment of augmented n-of-m expressions is given in [1]. 3 RELATED WORK Here we briefly consider two alternative systems for connectionist rule extraction. Many other methods have been developed; a recent summary and categorization appears in [2]. 3.1 McMillan McMillan described the projection of actual weights to simple weight templates in [4]. McMillan's parameter selection and instantiation procedures are inefficient compared to those described here, though they yield equivalent results for the classes of templates he used. McMillan treated only expressions with m S; 2 and no negated sUbexpressions. 1 In fact the nesting may continue beyond one level. Thus sigmoidal units can compute expressions like OR[AND(Cn_of_m ' CAND), COR]. We have not yet experimented with extensions of this sort. Template-Based Algorithms for Connectionist Rule Extraction 613 3.2 Towell and Shavlik Towell and Shavlik [8] use a domain theory to initialize a connectionist network, train the network on a set of labeled examples, and then extract rules that describe the network's behavior. To perform rule extraction, Towell and Shavlik first group weights using an iterative clustering algorithm. After applying additional training, they typically check each training pattern against each weight group and eliminate groups that do not affect the classification of any pattern. Finally, they scan remaining groups and attempt to express a rule in purely symbolic n-of-m form. However, in many cases the extracted rules take the form of a linear inequality involving multiple numeric quantities. For example, the following rule was extracted from part of a network trained on the promoter recognition task [5] from molecular biology: Minus35 " -10 < + 5.0 * nt(@-37 '--T-G--A' ) + 3.1 * nt(@-37 '---GT---') + 1.9 * nt(@-37 '----C-CT' ) + 1.5 * nt(@-37 '---C--A-' ) - 1.5 * nt(@-37 '------GC' ) - 1.9 * nt(@-37 '--CAW---' ) - 3.1 * nt(@-37 '--A----C' ) where nt() returns the number of true subexpressions, @-37 locates the subexpressions on the DNA strand, and "_N indicates a don't-care subexpression. Towell and Shavlik's method can be expected to give more accurate results than our approach, but at a cost. Their method is very compute intensive and relies substantially on access to a fixed set of training patterns. Additionally, it is not clear that their rules are completely symbolic. While numeric expressions were convenient for the domains they studied, in applications where one is interested in more abstract descriptions, such expressions may be viewed as providing too much detail, and may be difficult for people to interpret and reason about. Sometimes one wants to determine the nearest symbolic interpretation of unit behavior rather than a precise mathematical description. Our method offers a simpler paradigm for doing this. Given these differences, we conclude that both methods have their place in rule extraction tool kits. 4 SIMULATIONS 4.1 Simple logic problems We used a group of simple logic problems to verify that our extraction algorithms could produce a correct set of rules for networks trained on the complete pattern space of each function. Table 1 summarizes the results.2 The rule-plus-exception problem is defined as /= AB + 1\B CD; xor-l is the 2-1-1 version of xordescribed in Section 2.3; and xor-2 is a strictly layered (2-2-1) version of xor [6]. The negation problem is also described in [6]; in this problem one of the four inputs controls whether the other inputs appear normally or negated at the outputs. (As with xor-l, the network for negation makes use of direct inputJ output connections.) In addition to the perfect classification performance of the rules, the large values of p* and small values of extraction error (as defined in Section 2.2) provide evidence that the extraction process is very accurate. 614 Jay A. Alexander, MichaeL C. Mozer Hidden Averagep· Extraction Error Patterns Unit Correctly Network Penalty Hidden Output Hidden Output Classified Problem Topology Term Unit(s) Unit(s) Unit(s) Unit(s) by Rules rule-plus-exception 4-2-1 2.72 6.15 0.8% 1.3% 100.0% xor-l 2-1-1 5.68 4.40 0.1 % 0.1 % 100.0 % xor-2 2-2-1 4.34 5.68 0.4% 1.0% 100.0 % negation 4-3-4 activation 5.40 5.17 0.2% 2.2% 100.0 % Table 1: Simulation summary for simple logic problems Symbolic solutions for these problems often come in fonns different from the canonical fonn of the function. For example, the following rules for the rule-pLus-exception problem show a level of negation within the network: H1 OR (A, B, c, D) H2 = AND (A, B) o OR (H1' H2 ) Example results on xor-J show the expected use of an augmented n-of-m expression: H OR (1 1 , 1 2 ) o OR [AND(I1 , 1 2), H! 4.2 The MONK's problems We tested generalization perfonnance using the MONK's problems [5,7], a set of three problems used to compare a variety of symbolic and connectionist learning algorithms. A summary of these tests appears in Table 2. Our perfonnance was equal to or better than all of the systems tested in [7] for the monks-J and monks-2 problems. Moreover, the rules extracted by our algorithm were very concise and easy to understand, in contrast to those produced by several of the symbolic systems. (The two connectionist systems reported in [7] were opaque, Le., no rules were extracted.) As an example, consider the following output for the monks-2 problem: HI 2 of (head_shape round, body_shape round, is_smiling yes, holding sword, jacket_color red, has_tie yes) H2 3 of (head_shape round, body_shape round, is_smiling yes, holding sword, jacket_color red, has_tie not no) o AND (HI' H2 ) The target concept for this problem is exactly 2 of the attributes have their first value. These rules demonstrate an elegant use of n-of-m expressions to describe the idea of "exactly 2" as "at least 2 but not 3". The monks-3 problem is difficult due to (intentional) training set noise, but our results are comparable to the other systems tested in [7]. 2 All results in this paper are for networks trained using batch-mode back propagation on the cross-entropy error function. Training was stopped when outputs were within 0.05 of their target values for each pattern or a fixed number of epochs (typically 10(0) was reached. Where indicated, a penalty term for nonBoolean hidden activations or hidden weight decay was added to the main error function. For the breast cancer problem shown in Table 4.3, hidden rules were extracted first and the output units were retrained briefly before extracting their rules. Results for the problems in Table 4.3 used leave-one-out testing or 100fold cross-validation (with 10 different initial orderings) as indicated. All results are averages over 10 replications with different initial weights. Template-Based Algorithms for Connectionist Rule Extraction 615 Hidden Training Set Test Set Unit Network Penalty #of Perf. of Perf. of #of Perf. of Perf. of Problem Topology Term Patterns Network Rules Patterns Network Rules monies-I 17-3--1 decay 124 100.0% 100.0% 432 100.0% 100.0% monles-2 17-2-1 decay 169 100.0% 100.0% 432 100.0% 100.0% monks-3 l7"'{)""l 122 93.4% 93.4% 432 97.2% 97.2% Table 2: Simulation summary for the MONK's problems 4.3 VCI repository problems The final set of simulations addresses extraction performance on three real-world databases from the UCI repository [5]. Table 3 shows that good results were achieved. For the promoters task, we achieved generalization performance of nearly 88%, compared to 93-96% reported by Towell and Shavlik [8]. However, our results are impressive when viewed in light of the simplicity and comprehensibility of the extracted output. While Towell and Shavlik's results for this task included 5 rules like the one shown in Section 3.2, our single rule is quite simple: promoter = 5 of (@-45 'AA-------TTGA-A-----T------T-----AAA----C') Results for the house-votes-84 and breast-eaneer-wise problems are especially noteworthy since the generalization performance of the rules is virtually identical to that of the raw networks. This indicates that the rules are capturing a significant portion of the computation being performed by the networks. The following rule was the one most frequently extracted for the house-votes-84 problem, where the task is to predict party affiliation: Democrat OR [ 5 of (V3 , V7 , V9, V1O ' Vll , V12 ) , v4 1 where V3 voted for adoption-of-the-budget-resolution bill V4 voted for physician-fee-freeze bill V7 voted for anti-satellite-test-ban bill V9 = voted for rnx-missile bill V10 = voted for immigration bill Vll voted for synfuels-corporation-cutback bill V12 = voted for education-spending bill Shown below is a typical rule set extracted for the breast-eaneer-wise problem. Here the goal is to diagnose a tumor as benign or malignant based on nine clinical attributes. Malignant = AND (H1 , H2 ) Hl = 4 of (thickness> 3, size> 1, adhesion> 1, epithelial> 5, nuclei> 3, chromatin> 1, normal> 2, mi toses > 1) H2 = 30f(thickness>6,size>1,shape>1,epithelial>1, nuclei> 8, normal> 9) H3 = not used As suggested by the rules, we used a thermometer (cumulative) coding of the nominally valued attributes so that less-than or greater-than subexpressions could be efficiently represented in the hidden weights. Such a representation is often useful in diagnosis tasks. We also limited the hidden weights to positive values due to the nature of the attributes. 616 Jay A. Alexander, Michael C. Mozer Training Set Test Set Network #I of Perf. of Perf. of #I of Perf. of Perf. of Problem Topology Patterns Network Rules Patterns Network Rules promoters 2284-1 105 100.0% 95.9% I 94.2% 87.6% house-votes-84 164-1 387 97.3 % 96.2% 43 95.7% 95.9% breast-cancer-wisc 81-3-1 630 98.5 % 96.3 % 70 95.8% 95.2% Table 3: Simulation summary for uel repository problems Taken as a whole our simulation results are encouraging, and we are conducting further research on rule extraction for more complex tasks. 5 CONCLUSION We have described a general approach for extracting various types of n-of-m symbolic rules from trained networks of sigmoidal units, assuming approximately Boolean activation behavior. While other methods for interpretation of this sort exist, ours represents a valuable price/performance point, offering easily-understood rules and good extraction performance with computational complexity that scales well with the expressiveness desired. The basic principles behind our approach may be flexibly applied to a wide variety of problems. References [1] Alexander, J. A. (1994). Template-based procedures for neural network interpretation. MS Thesis. Department of Computer Science, University of Colorado, Boulder, CO. [2] Andrews, R., Diederich, l, and Tickle, A. B. (1995). A survey and critique of techniques for extracting rules from trained artificial neural networks. To appear in Fu, L. M. (Ed.), Knowledge-Based Systems, Special Issue on Knowledge-Based Neural Networks. [3] Mangasarian, O. L. and Wolberg, W. H. (1990). Cancer diagnosis via linear programming. SIAM News 23:5, pages 1 & 18. [4] McMillan, C. (1992). Rule induction in a neural network through integrated symbolic and subsymbolic processing. PhD Thesis. Department of Computer Science, University of Colorado, Boulder, CO. [5] Murphy, P. M. and Aha, D. W. (1994). UCI repository of machine learning databases. [Machine-readable data repository]. Irvine, CA: University of California, Department of Information and Computer Science. Monks data courtesy of Sebastian Thrun, promoters data courtesy of M. Noordewier and J. Shavlik, congressional voting data courtesy of Jeff Schlimmer, breast cancer data courtesy of Dr. William H. Wolberg (see also [3] above). [6] Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986). Learning internal representations by error propagation. In Rumelhart, D. E., McClelland, l L., and the PDP Research Group, Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Volume 1: Foundations, pages 318-362. Cambridge, MA: MIT Press. [7] Thrun, S. B., and 23 other authors (1991). The MONK's problems - A performance comparison of different learning algorithms. Technical Report CS-CMU-91-197. Carnegie Mellon University, Pittsburgh, PA. [8] Towell, G. and Shavlik, J. W. (1992). Interpretation of artificial neural networks: Mapping knowledge-based neural networks into rules. In Moody, J. E., Hanson, S. J., and Lippmann, R. P. (Eds.), Advances in Neurallnfonnation Processing Systems, 4:977-984. San Mateo, CA: Morgan Kaufmann.	1994	27
918	Reinforcement Learning with Soft State Aggregation Satinder P. Singh Tommi Jaakkola Michael I. Jordan singh@psyche.mit.edu tommi@psyche.mit.edu jordan@psyche.mit.edu Dept. of Brain & Cognitive Sciences (E-lO) M.I.T. Cambridge, MA 02139 Abstract It is widely accepted that the use of more compact representations than lookup tables is crucial to scaling reinforcement learning (RL) algorithms to real-world problems. Unfortunately almost all of the theory of reinforcement learning assumes lookup table representations. In this paper we address the pressing issue of combining function approximation and RL, and present 1) a function approximator based on a simple extension to state aggregation (a commonly used form of compact representation), namely soft state aggregation, 2) a theory of convergence for RL with arbitrary, but fixed, soft state aggregation, 3) a novel intuitive understanding of the effect of state aggregation on online RL, and 4) a new heuristic adaptive state aggregation algorithm that finds improved compact representations by exploiting the non-discrete nature of soft state aggregation. Preliminary empirical results are also presented. 1 INTRODUCTION The strong theory of convergence available for reinforcement learning algorithms (e.g., Dayan & Sejnowski, 1994; Watkins & Dayan, 1992; Jaakkola, Jordan & Singh, 1994; Tsitsiklis, 1994) makes them attractive as a basis for building learning control architectures to solve a wide variety of search, planning, and control problems. Unfortunately, almost all of the convergence results assume lookup table representa362 Satinder Singh, Tommi Jaakkola, Michael I. Jordan tions for value functions (see Sutton, 1988; Dayan, 1992; Bradtke, 1993; and Vanroy & Tsitsiklis, personal communication; for exceptions). It is widely accepted that the use of more compact representations than lookup tables is crucial to scaling RL algorithms to real-world problems. In this paper we address the pressing issue of combining function approximation and RL, and present 1) a function approximator based on a simple extension to state aggregation (a commonly used form of compact representation, e.g., Moore, 1991), namely soft state aggregation, 2) a theory of convergence for RL with arbitrary, but fixed, soft state aggregation, 3) a novel intuitive understanding of the effect of state aggregation on online RL, and 4) a new heuristic adaptive state aggregation algorithm that finds improved compact representations by exploiting the non-discrete nature of soft state aggregations. Preliminary empirical results are also presented. Problem Definition and Notation: We consider the problem of solving large Markovian decision processes (MDPs) using RL algorithms and compact function approximation. We use the following notation: 8 for state space, A for action space, pa(s, s') for transition probability, Ra(s) for payoff, and 'Y for discount factor. The objective is to maximize the expected, infinite horizon, discounted sum of payoffs. 1.1 FUNCTION APPROXIMATION: SOFT STATE CLUSTERS In this section we describe a new function approximator (FA) for RL. In section 3 we will analyze it theoretically and present convergence results. The FA maps the state space 8 into M > 0 aggregates or clusters from cluster space X. Typically, M < < 181. We allow soft clustering, where each state s belongs to cluster x with probability P(xls), called the clustering probabilities. This allows each state s to belong to several clusters. An interesting special case is that of the usual state aggregation where each state belongs only to one cluster. The theoretical model is that the agent can observe the underlying state but can only update a value function for the clusters. The value of a cluster generalizes to all states in proportion to the clustering probabilities. Throughout we use the symbols x and y to represent individual clusters and the symbols sand s' to represent individual states. 2 A GENERAL CONVERGENCE THEOREM An online RL algorithm essentially sees a sequence of quadruples, < St, at, StH, rt >, representing a transition from current state St to next state StH on current action at with an associated payoff rt. We will first prove a general convergence theorem for Q-learning (Watkins & Dayan, 1992) applied to a sequence of quadruples that mayor may not be generated by a Markov process (Bertsekas, 1987). This is required because the RL problem at the level of the clusters may be non-Markovian. Conceptually, the sequence of quadruples can be thought of as being produced by some process that is allowed to modify the sequence of quadruples produced by a Markov process, e.g., by mapping states to clusters. In Section 3 we will specialize the following theorem to provide specific results for our function approximator. Consider any stochastic process that generates a sequence of random quadruples, \]! = {< Xi, ai, Yi, ri > h, where Xi, Yi E Y, ai E A, and ri is a bounded real number. Note that Xi+l does not have to be equal to Yi. Let IYI and IAI be finite, and define Reinforcement Learning with Soft State Aggregation indicator variables and Define ( ) { 1 when Wi =< x,a,y,. > (for any r) Xi x, a, Y = ° th . o erwlse, when Wi =< x, a, .,. > (for any y, and any r) otherwise. P,a ,( ) _ 2:t-iXk(x,a,y) IJx,y , , 2:1=i n(x, a) and 363 Theorem 1: If V( > 0, 3ME < 00, such that for all i ~ 0, for all x, y E Y, and for all a E A, the following conditions characterize the infinite sequence W: with probability 1 - (, IPi~i+M.(X,y)-pa(x,y)1 < ( IRi,i+MJX) - Ra(x)1 < (, and (1) where for all x, a, and y, with probability one Pg,oo(x, y) = pa(x, y), and Rg,oo(x) = Ra(x). Then, online Q-learning applied to such a sequence will converge with probability one to the solution of the following system of equations: Vx E Y, and Va E A, Q(x, a) = Ra(x) + 'Y " pa(x, y) maxQ(y, a') L.J a'EA yEY (2) Proof: Consider the semi-batch version of Q-learning that collects the changes to the value function for M steps before making the change. By assumption, for any (, making ME large enough will ensure that with probability 1 - (, the sample quantities for the ith batch, Pti+M (x, y) and Ri i+M (i) (x) are within ( of the ,«.) ,< • asymptotic quantities. In Appendix A we prove that the semi-batch version of Qlearning outlined above converges to the solution of Equation 2 with probability one. The semi-batch proof can be extended to online Q-learning by using the analysis developed in Theorem 3 of Jaakkola et al. (1994). In brief, it can be shown that the difference caused by the online updating vanishes in the limit thereby forcing semi-batch Q-learning and online Q-learning to be equal asymptotically. The use of the analysis in Theorem 3 from Jaakkola et al. (1994) requires that the learning rate parameters CY are such that alex) () _ 1 uniformly w.p.l.; ME(k) is maxlEM.(Ic)al x the kth batch of size ME' If CYt(x) is non-increasing in addition to satisfying the conventional Q-learning conditions, then it will also meet the above requirement. o Theorem 1 provides the most general convergence result available for Q-learning (and TD(O»; it shows that for an arbitrary quadruple sequence satisfying the ergodicity conditions given in Equations 1, Q-learning will converge to the solution of the MDP constructed with the limiting probabilities (Po,oo) and payoffs (Ro,oo). Theorem 1 combines and generalizes the results on hard state aggregation and value iteration presented in Vanroy & Tsitsiklis (personal communication), and on partially observable MDPs in Singh et al. (1994). 364 Satinder Singh, Tommi Jaakkola, Michael I. Jordan 3 RL AND SOFT STATE AGGREGATION In this section we apply Theorem 1 to provide convergence results for two cases: 1) using Q-Iearning and our FA to solve MDPs, and 2) using Sutton's (1988) TD(O) and our FA to determine the value function for a fixed policy. As is usual in online RL, we continue to assume that the transition probabilities and the payoff function of the MDP are unknown to the learning agent. Furthermore, being online such algorithms cannot sample states in arbitrary order. In this section, the clustering probabilities P(xls) are assumed to be fixed. Case 1: Q-learning and Fixed Soft State Aggregation Because of function approximation, the domain of the learned Q-value function is constrained to be X x A (X is cluster space). This section develops a "Bellman equation" (e.g., Bertsekas, 1987) for Q-Iearning at the level of the cluster space. We assume that the agent follows a stationary stochastic policy 7r that assigns to each state a non-zero probability of executing every action in every state. Furthermore, we assume that the Markov chain under policy 7r is ergodic. Such a policy 7r is a persistently exciting policy. Under the above conditions p'II"(slx) = :, ;C;,:;p$"C$')' where for all s, p'll"(s) is the steady-state probability of being in state s. Corollary 1: Q-Iearning with soft state aggregation applied to an MDP while following a persistently exciting policy 7r will converge with probability one to the solution of the following system of equations: Vex, a) E (X x A), and pa(s, y) = E$' pa(s, s')P(yls'). The Q-value function for the state space can then be constructed via Q(s, a) = Ex P(xls)Q(x, a) for all (s, a). Proof: It can be shown that the sequence of quadruples produced by following policy 7r and independently mapping the current state s to a cluster x with probability P(xls) satisfies the conditions of Theorem 1. Also, it can be shown that Note that the Q-values found by clustering are dependent on the sampling policy 7r, unlike the lookup table case. Case 2: TD(O) and Fixed Soft State Aggregation We present separate results for TD(O) because it forms the basis for policy-iterationlike methods for solving Markov control problems (e.g., Barto, Sutton & Anderson, 1983) a fact that we will use in the next section to derive adaptive state aggregation methods. As before, because of function approximation, the domain of the learned value function is constrained to be the cluster space X. Corollary 2: TD(O) with soft state aggregation applied to an MDP while following a policy 7r will converge with probability one to the solution of the following system Reinforcement Learning with Soft State Aggregation 365 of equations: Vx EX, V(x) (4) where again as in Q-Iearning the value function for the state space can be constructed via V(s) = Lx P(xls)V(x) for all s. Proof: Corollary 1 implies Corollary 2 because TD(O) is a special case of Q-Iearning for MDPs with a single (possibly randomized) action in each state. Equation 4 provides a "Bellman equation" for TD(O) at the level of the cluster space. 0 4 ADAPTIVE STATE AGGREGATION In previous sections we restricted attention to a function approximator that had a fixed compact representation. How might one adapt the compact representation online in order to get better approximations of value functions? This section presents a novel heuristic adaptive algorithm that improves the compact representation by finding good clustering probabilities given an a priori fixed number of clusters. Note that for arbitrary clustering, while Corollaries 1 and 2 show that RL will find solutions with zero Bellman error in cluster space, the associated Bellman error in the state space will not be zero in general. Good clustering is therefore naturally defined in terms of reducing the Bellman error for the states of the MDP. Let the clustering probabilities be parametrized as follows P(xls; 0) = L:~~:' ;~/ .. ), where o( x , s) is the weight between state s and cluster x. Then the Bellman error at .state s given parameter ° (a matrix) is, J(sIO) V( slO) - [R' (s) + r ~ P' (8, s')V( S'IO)] ~ P(xls; O)V(xIO) - [w(s) + r ~ P'(s, s') ~ P(xls'; O)V(XIO)] Adaptive State Aggregation (ASA) Algorithm: Step 1: Compute V(xIO) for all x E X using the TD(O) algorithm. S . oJ 2(9) tep 2. Let !:l.0 - -a 89 . Go to step 1. where Step 2 tries to minimize the Bellman error for the states by holding the cluster values fixed to those computed in Step 1. We have 8J 2(sI0) 80(y, s) 2J(sI0) [P(yls; 0)(1 - ,p7r(s, s»(V(yIO) - V(sIO»]. The Bellman error J(sIO) cannot be computed directly because the transition probabilities P( s, s') are unknown. However, it can be estimated by averaging the sample 366 Sa tinder Singh, Tommi Jaakkola, Michael I. Jordan Bellman error. P(yls; B) is known, and (1 - ,P7r (s, s)) is always positive, and independent of y, and can therefore be absorbed into the step-size c¥. The quantities V(yIB) and V(sIB) are available at the end of Step 1. In practice, Step 1 is only carried out partially before Step 2 is implemented. Partial evaluation works well because the changes in the clustering probabilities at Step 2 are small, and because the final V(xIB) at the previous Step 1 is used to initialize the computation of V(xIB) at the next Step 1. ------ 2 Clusters --------- 4 Clusters to Clusters -- 20 Clusters 4 .. ,'. .. ",' " ....... --... " . ... " .... . ' . . ". Iterations of ASA Figure 1: Adaptive State Clustering. See text for explanation. Figure 1 presents preliminary empirical results for the ASA algorithm. It plots the squared Bellman error summed over the state space as a function of the number of iterations of the ASA algorithm with constant step-size c¥. It shows error curves for 2, 4, 10 and 20 clusters averaged over ten runs of randomly constructed 20 state Markov chains. Figure 4 shows that ASA is able to adapt the clustering probabilities to reduce the Bellman error in state space, and as expected the more clusters the smaller the asymptotic Bellman error. In future work we plan to test the policy iteration version of the adaptive soft aggregation algorithm on Markov control problems. 5 SUMMARY AND FUTURE WORK Doing RL on aggregated states is potentially very advantageous because the value of each cluster generalizes across all states in proportion to the clustering probabilities. The same generalization is also potentially perilous because it can interfere with the contraction-based convergence of RL algorithms (see Vee, 1992; for a discussion). This paper resolves this debate for the case of soft state aggregation by defining a set of Bellman Equations (3 and 4) for the control and policy evaluation problems in the non-Markovian cluster space, and by proving that Q-Iearning and TD(O) solve them respectively with probability one. Theorem 1 presents a general convergence result that was applied to state aggregation in this paper, but is also a generalization of the results on hidden state presented in Singh et al. (1994), and may be applicable Reinforcement Learning with Soft State Aggregation 367 to other novel problems. It supports the intuitive picture that if a non-Markovian sequence of state transitions and payoffs is ergodic in the sense of Equation 1, then RL algorithms will converge w.p.l. to the solution of an MDP constructed with the limiting transition probabilities and payoffs. We also presented a new algorithm, ASA, for adapting compact representations, that takes advantage of the soft state aggregation proposed here to do gradient descent in clustering probability space to minimize squared Bellman error in the state space. We demonstrated on simple examples that ASA is able to adapt the clustering probabilities to dramatically reduce the Bellman error in state space. In future work we plan to extend the convergence theory presented here to discretizations of continuous state MDPs, and to further test the ASA algorithms. A Convergence of semi-batch Q-Iearning (Theorem 1) Consider a semi-batch algorithm that collects the changes to the Q-value function for M steps before making the change to the Q-value function. Let and kM R%(x) = L riXi(x, a); i=(k-l)M kM Mk(X, a) = L Xi(X, a) i=(k-l)M kM Mk(x,a,y) = L Xi(x,a,y) i=(k-l)M Then the Q-value of (x, a) after the kth batch is given by: Qk+l(X, a) = (1- Mk(X, a)ak(x, a))Qk(x, a) ) [ Rk(x) ~ Mk(X, a, y) I ] +Mk(x, a ak(x, a) M ( ) +, ~ M ( ) m;pcQk(Y, a) kx,a kx,a a yEY Let Q be the solution to Equation 2. Define, Rk(x) ~ Mk(x, a, y) ,Fk(x, a) = M ( ) +, ~ M ( ) m~xQk(Y, a) - Q(x, a), kx,a kx,a a yEY then, if Vk(X) = maxa Qk(X, a) and Vex) = maxa Q(x, a), ~ Mk(X, a, y) Rk(x) a Fk(x, a) = ,~ M ( ) (Vk(y) - V(y)] + (M ( ) - Ro,oo(x)) kx,a kx,a y ~ [(Mk(x,a,y) a ( ))-()] +, ~ Mk(X,a) - PO,oo x,y V y , y The quantity Fk(x, a) can be bounded by IIFk(x, a) II :s; ,llVk - VII + II(~~~~L - Rg,oo(x))11 +,11 Ly(~tX~~~y - Po,oo(x, y))V(y)11 :s; ,llVk - VII + Cfr, h M· th I f I R&(x) R a ()I d I ~ (Mk(x,a,y) na ( ))1 B were fk 1S e arger 0 Mk(x,a) 0,00 x , an 'l..Jy Mk(x,a) .ro,oo x, y . y 368 Sa tinder Singh, Tommi Jaakkola, Michael I. Jordan assumption for any (. > 0, 3Mf < 00 such that (.~< < (. with probability 1 - c The variance of Fk(X, a) can also be shown to be bounded because the variance of the sample probabilities is bounded (everything else is similar to standard Q-learning for MDPs). Therefore by Theorem 1 of Jaakkola et al. (1994), f<2r any (. > 0, with probability (1 - f), Qk(X, a) Qoo(X, a), where IQoo(x, a) - Q(x, a)1 ~ Cc Therefore, semi-batch Q-learning converges with probability one. 0 Acknowledgements This project was supported in part by a grant from the McDonnell-Pew Foundation, by a grant from ATR Human Information Processing Research Laboratories, and by a grant from Siemens Corporation. Michael I. Jordan is a NSF Presidential Young Investigator. References A. G. Barto, R. S. Sutton, & C. W. Anderson. (1983) Neuronlike elements that can solve difficult learning control problems. IEEE SMC, 13:835-846. D. P. Bertsekas. (1987) Dynamic Programming: Deterministic and Stochastic Models, Prentice-Hall. S. J. Bradtke. (1993) Reinforcement learning applied to linear quadratic regulation. In Advances in Neural Information Processing Systems 5, pages 295-302. P. Dayan. (1992) The convergence of TD(A) for general A. Machine Learning, 8(3/4):341-362. P. Dayan & T.J. Sejnowski. (1994) TD(A) converges with probability 1. Machine Learning, 13(3). T. Jaakkola, M. I. Jordan, & S. P. Singh. (1994) On the convergence of stochastic iterative dynamic programming algorithms. Neural Computation, 6(6):11851201. A. W. Moore. (1991) Variable resolution dynamic programming: Efficiently learning action maps in multivariate real-valued state-spaces. In M aching Learning: Proceedings of the Eighth International Workshop, pages 333- 337. S. P. Singh, T. Jaakkola, & M. I. Jordan. (1994) Learning without state-estimation in partially observable markovian decision processes. In Machine Learning: Proceedings of the Eleventh International Conference, pages 284-292. R. S. Sutton. (1988) Learning to predict by the methods of temporal differences. Machine Learning, 3:9-44. J. Tsitsiklis. (1994) Asynchronous stochastic approximation and Q-learning. Machine Learning, 16(3):185-202. B. Vanroy & J. Tsitsiklis. (personal communication) C. J. C. H. Watkins & P. Dayan. (1992) Q-learning. Machine Learning, 8(3/4):279292. R. C. Yee. (1992) Abstraction in control learning. Technical Report COINS Technical Report 92-16, Department of Computer and Information Science, University of Massachusetts, Amherst, MA 01003. A dissertation proposal.	1994	28
919	A Connectionist Technique for Accelerated Textual Input: Letting a Network Do the Typing Dean A. Pomerleau pomerlea@cs.cmu.edu School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Abstract Each year people spend a huge amount of time typing. The text people type typically contains a tremendous amount of redundancy due to predictable word usage patterns and the text's structure. This paper describes a neural network system call AutoTypist that monitors a person's typing and predicts what will be entered next. AutoTypist displays the most likely subsequent word to the typist, who can accept it with a single keystroke, instead of typing it in its entirety. The multi-layer perceptron at the heart of Auto'JYpist adapts its predictions of likely subsequent text to the user's word usage pattern, and to the characteristics of the text currently being typed. Increases in typing speed of 2-3% when typing English prose and 10-20% when typing C code have been demonstrated using the system, suggesting a potential time savings of more than 20 hours per user per year. In addition to increasing typing speed, AutoTypist reduces the number of keystrokes a user must type by a similar amount (2-3% for English, 1020% for computer programs). This keystroke savings has the potential to significantly reduce the frequency and severity of repeated stress injuries caused by typing, which are the most common injury suffered in today's office environment. 1 Introduction People in general, and computer professionals in particular, spend a huge amount of time typing. Most of this typing is done sitting in front of a computer display using a keyboard as the primary input device. There are a number of efforts using artificial neural networks and other techniques to improve the comfort and efficiency of human-computer communication using alternative modalities. Speech recognition [Waibel et al., 1988], handwritten character recognition [LeCun et al., 1989], and even gaze tracking [Baluja & Pomerleau, 1993] have 1040 Dean Pomerleau the potential to facilitate this communication. But these technologies are still in their infancy, and at this point cannot approach the speed and accuracy of even a moderately skilled typist for textual input. Is there some way to improve the efficiency of standard keyboard-based human-computer communication? The answer is yes, there are several ways to make typing more efficient. The first, called the Dvorak keyboard, has been around for over 60 years. The Dvorak keyboard has a different arrangement of keys, in which the most common letters, E, T, S, etc., are on the home row right under the typist's fingers. This improved layout requires the typist's fingers to travel1116th as far, resulting in an average of20% increase in typing speed. Unfortunately, the de facto standard in keyboards is the inefficient QWERTY configuration, and people are reluctant to learn a new layout. This paper describes another approach to improving typing efficiency, which can be used with either the QWERTY or DVORAK keyboards. It takes advantage of the hundreds of thousands of computer cycles between the typist's keystrokes which are typically wasted while the computer idly waits for additional input. By spending those cycles trying to predict what the user will type next, and allowing the typist to accept the prediction with a single keystroke, substantial time and effort can be saved over typing the entire text manUally. There are actually several such systems available today, including a package called "Autocompletion" developed for gnu-emacs by the author, and an application called "Magic Typist" developed for the Apple Macintosh by Olduvai Software. Each of these maintains a database of previously typed words, and suggests completions for the word the user is currently in the middle of typing, which can be accepted with a single keystroke. While reasonable useful, both have substantial drawbacks. These systems use a very naive technique for calculating the best completion, simply the one that was typed most recently. In fact, experiments conducted for this paper indicated that this "most recently used" heuristic is correct only about 40% of the time. In addition, these two systems are annoyingly verbose, always suggesting a completion if a word has been typed previously which matches the prefix typed so far. They interrupt the user's typing to suggest a completion even if the word they suggest hasn't been typed in many days, and there are many other alternative completions for the prefix, making it unlikely that the suggestion will be correct. These drawbacks are so severe that these systems frequently decrease the user's typing speed, rather than increase it. The Auto'JYpist system described in this paper employs an artificial neural network during the spare cycles between keystrokes to make more intelligent decisions about which completions to display, and when to display them. 2 The Prediction Task To operationalize the goal of making more intelligent decisions about which completions to display, we have defined the neural networks task to be the following: Given a list of candidate completions for the word currently being typed, estimate the likelihood that the user is actually typing each of them. For example, if the user has already types the prefix "aut", the word he is trying to typing could anyone of a large number of possibilities, including "autonomous", "automatic", "automobile" etc. Given a list of these possibilities taken from a dictionary, the neural network's task is to estimate the probability that each of these is the word the user will type. A neural network cannot be expected to accurately estimate the probability for a particular completion based on a unique representation for each word, since there are so many words A Connectionist Technique for Accelerated Textual Input 1041 ATTRIBUTE DESCRIPTION absolute age time since word was last typed relative age ratio of the words age to age of the most recently typed alternative absolute frequency number of times word has been typed in the past relative frequency ratio of the words frequency to that of the most often typed alternative typed previous 1 if user has typed word previously, o otherwise total length the word's length, in characters remaining length the number of characters left after the prefix to be typed for this word special character match the percentage of "special characters" (Le. not a-z) in this word relative to the percentage of special characters typed recently capitalization match 1 if the capitalization of the prefix the user has already typed matches the word's usual capitalization, 0 otherwise. Table 1: Word attributes used as input to the neural network for predicting word probabilities. in the English language, and there is only very sparse data available to characterize an individual's usage pattern for any single word. Instead, we have chosen to use an input representation that contains only those characteristics of a word that could conceivably have an impact on its probability of being typed. The attributes we employed to characterize each completion are listed in Table 1. These are not the only possible attributes that could be used to estimate the probability of the user typing a particular word. An additional characteristic that could be helpful is the word's part of speech (i.e. noun, verb, adjective, etc.). However this attribute is not typically available or even meaningful in many typing situations, for instance when typing computer programs. Also, to effectively exploit information regarding a word's part of speech would require the network to have knowledge about the context of the current text. In effect, it would require at least an approximate parse tree of the current sentence. While there are techniques, including connectionist methods [Jain, 1991], for generating parse trees, they are prone to errors and computationally expensive. Since word probability predictions in our system must occur many times between each key the user types, we have chosen to utilize only the easy to compute attributes shown in Table 1 to characterize each completion. 3 Network Processing The network architecture employed for this system is a feedforward multi-layer perceptron. Each of the networks investigated has nine input units, one for each of the attributes listed in Table 1, and a single output unit. As the user is typing a word, the prefix he has typed so far is used to find candidate completions from a dictionary, which contains 20,000 English words plus all words the user has typed previously. For each of these candidate completions, the nine attributes in Table 1 are calculated, and scaled to the range of 0.0 to 1.0. These values become the activations of the nine units in the input layer. Activation is propagated through the network to produce an activation for the single output unit, representing the 1042 Dean Pomerleau probability that this particular candidate completion is the one the user is actually typing. These candidate probabilities are then used to determine which (if any) of the candidates should be displayed to the typist, using a technique described in a later section. To train the network, the user's typing is again monitored. After the user finishes typing a word, for each prefix of the word a list of candidate completions, and their corresponding attributes, is calculated. These form the input training patterns. The target activation for the single output unit on a pattern is set to 1.0 if the candidate completion represented by that pattern is the word the user was actually typing, and 0.0 if the candidate is incorrect. Note that the target output activation is binary. As will be seen below, the actual output the network learns to produce is an accurate estimate of the completion's probability. Currently, training of the network is conducted off-line, using a fixed training set collected while a user types normally. Training is performed using the standard backpropagation learning algorithm. 4 Experiments Several tests were conducted to determine the ability of multi-layer perceptrons to perform the mapping from completion attributes to completion probability. In each of the tests, networks were trained on a set of inputJoutputexemplars collected over one week of a single subject's typing. During the training data collection phase, the subject's primary text editing activities involved writing technical papers and composing email, so the training patterns represent the word choice and frequency distributions associated with these activities. This training set contained of 14,302 patterns of the form described above. The first experiment was designed to determine the most appropriate network architecture for the prediction task. Four architecture were trained on a 10,000 pattern subset of the training data, and the remaining 4,302 patterns were used for cross validation. The first of the four architectures was a perceptron, with the input units connected directly to the single output unit. The remaining three architectures had a single hidden layer, with three, six or twelve hidden units. The networks with hidden units were fully connected without skip connections from inputs to output. Networks of three and six hidden units which included skip connections were tested, but did not exhibit improved performance over the networks without skip connections, so they are not reported. Each of the network architectures were trained four times, with different initial random weights. The results reported are those produced by the best set of weights from these trials. Note that the variations between trials with a single architecture were small relative to the variations between architectures. The trained networks were tested on a disjoint set of 10,040 collected while the same subject was typing another technical paper. Three different performance metrics were employed to evaluate the performance of these architectures on the test set. The first was the standard mean squared error (MSE) metric, depicted in Figure 1. The MSE results indicate that the architectures with six and twelve hidden units were better able to learn the task than either the perceptron, or the network with only three hidden units. However the difference appears to be relatively small, on the order of about 10%. MSE is not a very informative error metric, since the target output is binary (1 if the completion is the one the user was typing, 0 otherwise), but the real goal is to predict the probability that the completion is correct. A more useful measure of performance is shown in Figure 2. For each of the four architectures, it depicts the predicted probability that a completion is correct, as measured by the network's output activation value, vs. the A Connectionist Technique for Accelerated Textual Input 0.095 0.070 ....... __ Perceptron 3 Hidden Units 6 Hidden Units 12 Hidden Units 1043 Figure 1: Mean squared error for four networks on the task of predicting completion probability. actual probability that a completion is correct. The lines for each of the four networks were generated in the following manner. The network's output response on each of the 10,040 test patterns was used to group the test patterns into 10 categories. All the patterns which represented completions that the network predicted to have a probability of between o and 10% of being correct (output activations of 0.0-0.1) were placed in one category. Completions that the network predicted to have a 10-20% change of being right were placed in the second category, etc. For each of these 10 categories, the actual likelihood that a completion classified within the category is correct was calculated by determining the percent of the completions within that category that were actually correct. As a concrete example, the network with 6 hidden units produced an output activation between 0.2 and 0.3 on 861 of the 10,040 test patterns, indicating that on these patterns it considered there to be a 20-30% chance that the completion each pattern represented was the word the user was typing. On 209 of these 861 patterns in this category, the completion was actually the one the user was typing, for a probability of 24.2%. Ideally, the actual probability should be 25%, half way between the minimum and maximum predicted probability thresholds for this category. This ideal classification performance is depicted as the solid 45° line labeled "Target" in Figure 2. The closer the line for a given network matches this 45° line, the more the network's predicted probability matches the actual probability for a completion. Again, the networks with six and twelve hidden units outperformed the networks with zero and three hidden units, as illustrated by their much smaller deviations from the 45° line in Figure 2. The output activations produced by the networks with six and twelve hidden units reflect the actual probability that the completion is correct quite accurately. However prediction accuracy is only half of what is required to perform the final system goal, which recall was to identify as many high probability completions as possible, so they can be suggested to the user without requiring him to manually type them. If overall accuracy of the probability predictions were the only requirement, a network could score quite highly by classifying 1044 Dean Pomerleau 1.00 n;;get 0 Perceptron ..... ..... 0.80 3 Hidden Units .D ~ 6 Hidden Units .D e 0.60 12 Hidden Units ~ ~ 0.40 a u < 0.20 0.00 Figure 2: Predicted vs. actual probability of a completion being correct for the four architectures tested. every pattern into the 10-20% category, since about 15% of the 10,040 completions in the test set represent the word the user was typing at the time. But a constant prediction of 10-20% probability on every alternative completion would not allow the system to identify and suggest to the user those individual completions that are much more likely than the other alternatives. To achieve the overall system goal, the network must be able to accurately identify as many high probability completions as possible. The ability of each of the four networks to achieve this goal is shown in Figure 3. This figures shows the percent of the 10,040 test patterns each of the four networks classified as having more than a 60% probability of being correct. The 60% probability threshold was selected because it represents a level of support for a single completion that is significantly higher than the support for all the others. As can be seen in Figure 3, the networks with hidden units again significantly outperformed the perceptron, which was able to correctly identify fewer than half as many completions as highly likely. 5 Auto1)rpist System Architecture and Performance The networks with six and twelve hidden units are able to accurately identify individual completions that have a high probability of being the word the user is typing. In order to exploit this prediction ability and speed up typing, we have build an X-window based application called AutoTypist around the smaller of the two networks. The application serves as the front end for the network, monitoring the user's typing and identifying likely completions for the current word between each keystroke. If the network at the core of AutoTypist identifies a single completion that it is both significantly more probably than all the rest, and also longer than a couple characters, it will momentarily display the completion after the current cursor location in whatever application the user is currently typing1• If the displayed completion is the word the user is typing, he can accept it with a single keystroke (The criterion for displaying a completion, and the human interface for AutoTypist, are somewhat more sophisticated than this description. However for the purposes of this paper, a high level description is sufficient. A Connectionist Technique for Accelerated Textual Input Percent of 6.0 Patterns Classified5.0 as over 60% Probable 4.0 3.0 2.0 1.0 Perceptron 3 Hidden Units 6 Hidden Units 1045 12 Hidden Units Figure 3: Percent of candidate completions classified as having more than a 60% chance of being correct for the four architectures tested. and move on to typing the next word. If the displayed completion is incorrect, he can continue typing and the completion will disappear. Quantitative results with the fully integrated Auto1Ypist system, while still preliminary, are very encouraging. In a two week trial with two subjects, who could type at 40 and 60 wpm without AutoTypists, their typings speeds were improved by 2.37% and 2.21 % respectively when typing English text. Accuracy improvements during these trials were even larger, since spelling mistakes become rare when AutoTypist is doing a significant part of the typing automatically. When writing computer programs, speed improvements of 12.93% and 18.47% were achieved by the two test subjects. This larger speedup was due to the frequent repetition of variable and function names in computer programs, which Auto1Ypist was able to expedite. Not only is computer code faster to produce with AutoTypist, it is also easier to understand. AutoTypist encourages the programmer to use long, descriptive variable and function names, by making him type them in their entirety only once. On subsequent instances of the same name, the user need only type the first few characters and then exploitAutoTypist's completion mechanism to type the rest. These speed improvements were achieved by subjects who are already relatively proficient typists. Larger gains can be expected for less skilled typists, since typing an entire word with a single keystroke will save more time when each keystroke takes longer. Perhaps an even more significant benefit results from the reduced number of keystrokes Auto1Ypist requires the user to type. During the test trials described above, the two test subjects had to strike an average of 2.89% fewer keys on the English text, and 16.42% fewer keys on the computer code than would have been required to type the text out in its entirety. Clearly this keystroke savings has the potential to benefit typists who suffer from repeated stress injuries brought on by typing. Unfortunately it is impossible to quantitatively compare these results with those of the other completion-based typing aids described in the introduction, since the other systems have not been quantitatively evaluated. Subjectively, Auto1Ypist is far less disturbing than the 1046 Dean Pomerleau alternatives, since it only displays a completion when there is a very good chance it is the correct one. 6 Future Work Further experiments are required to verify the typing speed improvements possible with AutoTypist, and to compare it with alternative typing improvement systems. Preliminary experiments suggest a network trained on the word usage patterns of one user can generalize to that of other users, but it may be necessary to train a new network for each individual typist. Also, the experiments conducted for this paper indicate that a network trained on one type of text, English prose, can generalize to text with quite different word frequency patterns, C language computer programs. However substantial prediction improvements, and therefore typing speedup, may be possible by training separate networks for different types of text. The question of how to rapidly adapt a single network, or perhaps a mixture of expert networks, to new text types is one which should be investigated. Even without these extensions, AutoTypist has the potential to greatly improve the comfort and efficiency of the typing tasks. For people who type English text two hours per workday, even the conservative estimate of a 2% speedup translates into 10 hours of savings per year. The potential time savings for computer programming is even more dramatic. A programmer who types code two hours per workday could potentially save between 52 and 104 hours in a single year by using AutoTypist. With such large potential benefits, commercial development of the AutoTypist system is also being investigated. Acknowledgements I would like to thank David Simon and Martial Hebert for their helpful suggestions, and for acting as willing test subjects during the development of this system. References [Baluja & Pomerleau, 1993] Baluja, S. and Pomerleau, D.A. (1993) Non-Intrusive Gaze Tracking Using Artificial Neural Networks. In Advances in Neural Information Processing Systems 6, San Mateo, CA: Morgan Kaufmann Publishers. [Jain,1991] Jain, A.N. (1991) PARSEC: A connectionist learning architecture for parsing spoken language. Carnegie Mellon University School of Computer Science Technical Report CMU-CS-91-208. [LeCun et al., 1989] LeCun, Y., Boser, B., Denker, 1.S., Henderson, D., Howard, R.E., Hubbard, W., and Jackel, L.D. (1989) Backpropagation applied to handwritten zip code recognition. Neural Computation 1(4). [Waibel et al., 1988] Waibel, A., Hanazawa, T., Hinton, G., Shikano, K., Lang, K. (1988) Phoneme recognition: Neural Networks vs. Hidden Markov Models. Proceedings from Int. Conf on Acoustics, Speech and Signal Processing, New York, New York.	1994	29
920	A Mixture Model System for Medical and Machine Diagnosis Magnus Stensmo Terrence J. Sejnowski Computational Neurobiology Laboratory The Salk Institute for Biological Studies 10010 North Torrey Pines Road La Jolla, CA 92037, U.S.A. {magnus,terry}~salk.edu Abstract Diagnosis of human disease or machine fault is a missing data problem since many variables are initially unknown. Additional information needs to be obtained. The j oint probability distribution of the data can be used to solve this problem. We model this with mixture models whose parameters are estimated by the EM algorithm. This gives the benefit that missing data in the database itself can also be handled correctly. The request for new information to refine the diagnosis is performed using the maximum utility principle. Since the system is based on learning it is domain independent and less labor intensive than expert systems or probabilistic networks. An example using a heart disease database is presented. 1 INTRODUCTION Diagnosis is the process of identifying diseases in patients or disorders in machines by considering history, symptoms and other signs through examination. Diagnosis is a common and important problem that has proven hard to automate and formalize. A procedural description is often hard to attain since experts do not know exactly how they solve a problem. In this paper we use the information about a specific problem that exists in a database 1078 Magnus Stensmo. Terrence J. Sejnowski of cases. The disorders or diseases are determined by variables from observations and the goal is to find the probability distribution over the disorders, conditioned on what has been observed. The diagnosis is strong when one or a few of the possible outcomes are differentiated from the others. More information is needed if it is inconclusive. Initially there are only a few clues and the rest of the variables are unknown. Additional information is obtained by asking questions and doing tests. Since tests may be dangerous, time consuming and expensive, it is generally not possible or desirable to find the answer to every question. Unnecessary tests should be avoided . . There have been many attempts to automate diagnosis. Early work [Ledley & Lusted, 1959] realized that the problem is not always tractable due to the large number of influences that can exist between symptoms and diseases. Expert systems, e.g. the INTERNIST system for internal medicine [Miller et al., 1982], have rule-bases which are very hard and time consuming to build. Inconsistencies may arise when new rules are added to an existing database. There is also a strong domain dependence so knowledge bases can rarely be reused for new applications. Bayesian or probabilistic networks [Pearl, 1988] are a way to model a joint probability distribution by factoring using the chain rule in probability theory. Although the models are very powerful when built, there are presently no general learning methods for their construction. A considerable effort is needed. In the Pathfinder system for lymph node pathology [Heckerman et al., 1992] about 14,000 conditional probabilities had to be assessed by an expert pathologist. It is inevitable that errors will occur when such large numbers of manual assessments are involved. Approaches to diagnosis that are based on domain-independent machine learning alleviate some of the problems with knowledge engineering. For decision trees [Quinlan, 1986], a piece of information can only be used if the appropriate question comes up when traversing the tree. This means that irrelevant questions can not be avoided. Feedforward multilayer perceptrons for diagnosis [Baxt, 1990] can classify very well, but they need full information about a case. None of these these methods have adequate ways to handle missing data during learning or classification. The exponentially growing number of probabilities involved can make exact diagnosis intractable. Simple approximations such as independence between all variables and conditional independence given the disease (naive Bayes) introduce errors since there usually are dependencies between the symptoms. Even though systems based on these assumptions work surprisingly well, correct diagnosis is not guaranteed. This paper will avoid these assumptions by using mixture models. 2 MIXTURE MODELS Diagnosis can be formulated as a probability estimation problem with missing inputs. The probabilities of the disorders are conditioned on what has currently been observed. If we model the joint probability distribution it is easy to marginalize to get any conditional probability. This is necessary in order to be able to handle missing data in a principled way [Ahmad & Tresp, 1993]. Using mixture models [McLachlan & Basford, 1988], a simple closed form solution to optimal regression with missing data can be formulated. The EM algorithm, a method from parametric statistics for parameter estimation, is especially interesting in this context since it can also be formulated to handle missing data in the A Mixture Model System for Medical and Machine Diagnosis 1079 training examples [Dempster et al., 1977; Ghahramani & Jordan, 1994]. 2.1 THE EM ALGORITHM The data underlying the model is assumed to be a set of N D-dimensional vectors X = { Z I, . . . , Z N }. Each data point is assumed to have been generated independently from a mixture density with M components M M p(z) = LP(z,Wj;Oj) = LP(Wj)p(zlwj;Oj), (1) j=l j=l where each mixture component is denoted by Wj. p(Wj), the a priori probability for mixturewj, and 8 = (01 , ... , OM) are the model parameters. To estimate the parameters for the different mixtures so that it is likely that the linear combination of them generated the set of data points, we use maximum likelihood estimation. A good method is the iterative Expectation-Maximization, or EM, algorithm [Dempster et al., 1977]. Two steps are repeated. First a likelihood is formulated and its expectation is computed in the E-step. For the type of models that we will use, this step will calculate the probability that a certain mixture component generated the data point in question. The second step is the M-step where the parameters that maximize the expectation are found. This can be found analytically for models that can be written in an exponential form, e.g. Gaussian functions. Equations can be derived for both batch and on-line learning. Update equations for Gaussian distributions with and without missing data will be given here, other distributions are possible, e.g. binomial or multinomial [Stensmo & Sejnowski, 1994]. Details and derivations can be found in [Dempster et al., 1977; Nowlan, 1991; Ghahramani & Jordan, 1994; Stensmo & Sejnowski, 1994]. From (1) we form the log likelihood of the data N N M L(8IX) = L logp(zi; OJ) = L log LP(Wj )P(Zi IWj; OJ). j=l There is unfortunately no analytic solution to the logarithm of the sum in the right hand side of the equation. However, if we were to know which of the mixtures generated which data point we could compute it. The EM algorithm solves this by introducing a set of binary indicator variables Z = {Zij}. Zij = 1 if and only if the data point Zi was generated by mixture component j. The log likelihood can then be manipulated to a form that does not contain the log of a sum. The expectation of %i using the current parameter values 8k is used since %i is not known directly. This is the E-step of the EM algorithm. The expected value is then maximized in theM-step. The two steps are iterated until convergence. The likelihood will never decrease after an iteration [Dempster et al., 1977]. Convergence is fast compared to gradient descent. One of the main motivations for the EM-algorithm was to be able to handle missing values for variables in a data set in a principled way. In the complete data case we introduced missing indicator variables that helped us solve the problem. With missing data we add the missing components to the Z already missing [Dempster et aI., 1977; Ghahramani & Jordan, 1994]. 1080 Magnus Stensmo, Terrence J. Sejnowski 2.2 GAUSSIAN MIXTURES We specialize here the EM algorithm to the case where the mixture components are radial Gaussian distributions. For mixture component j with mean I-'j and covariance matrix 1:j this is The form of the covariance matrix is often constrained to be diagonal or to have the same values on the diagonal, 1:j = o} I. This corresponds to axis-parallel oval-shaped and radially symmetric Gaussians, respectively. Radial and diagonal basis functions can function well in applications [Nowlan, 1991], since several Gaussians together can form complex shapes in the space. With fewer parameters over-fitting is minimized. In the radial case, with variance o} In the E-step the expected value of the likelihood is computed. For the Gaussian case this becomes the probability that Gaussian j generated the data point Pj(Z) = !(Wj)Gj(z) . l:k=l P(Wk)Gk(Z) The M-step finds the parameters that maximize the likelihood from the E-step. For complete data the new estimates are (2) N where Sj = I:Pj(Zi). i=l When input variables are missing the Gj(z) is only evaluated over the set of observed dimensions O. Missing (unobserved) dimensions are denoted by U. The update equation for p(Wj) is unchanged. To estimate itj we set zf = itY and use (2). The variance becomes A least squares regression was used to fill in missing data values during classification. For missing variables and Gaussian mixtures this becomes the same approach used by [Ahmad & Tresp, 1993]. The result of the regression when the outcome variables are missing is a probability distribution over the disorders. This can be reduced to a classification for comparison with other systems by picking the outcome with the maximum of the estimated probabilities. A Mixture Model System for Medical and Machine Diagnosis 1081 3 REQUESTING MORE INFORMATION During the diagnosis process, the outcome probabilities are refined at each step based on newly acquired knowledge. It it important to select the questions that lead to the minimal number of necessary tests. There is generally a cost associated with each test and the goal is to minimize the total cost. Early work on automated diagnosis [Ledley & Lusted, 1959] acknowledged the problem of asking as few questions as possible and suggested the use of decision analysis for the solution. An important idea from the field of decision theory is the maximum expected utility principle [von Neuman & Morgenstern, 1947]: A decision maker should always choose the alternative that maximizes some expected utility of the decision. For diagnosis it is the cost of misclassification. Each pair of outcomes has a utility u(x, y) when the correct diagnosis is x but y has been incorrectly determined. The expectation can be computed when we know the probabilities of the outcomes. The utility values have to be assessed manually in what can be a lengthy and complicated process. For this reason a simplification of this function has been suggested by [Heckerman et al., 1992]: The utility u(x, y) is 1 when both x and y are benign or both are malign, and 0 otherwise. This simplification has been found to work well in practice. Another complication with maximum expected utility principle can also make it intractable. In the ideal case we would evaluate every possible sequence of future choices to see which is the best. Since the size of the search tree of possibilities grows exponentially this is often not possible. A simplification is to 100k ahead only one or a few steps at a time. This nearsighted or myopic approach has been tested in practice with good results [Gorry & Barnett, 1967; Heckerman et al., 1992]. 4 THE DIAGNOSIS SYSTEM The system we have developed has two phases. First there is a learning phase where a probabilistic model is built. This model is then used for inference in the diagnosis phase. In the learning phase, the joint probability distribution of the data is modeled using mixture models. Parameters are determined from a database of cases by the EM algorithm. The k-means algorithm is used for initialization. Input and output variables for each case are combined into one vector per case to form the set of training patterns. The outcomes and other nominal variables are coded as J of N . Continuous variables are interval coded. In the diagnosis phase, myopic one-step look-ahead was used and utilities were simplified as above. The following steps were performed: 1. Initial observations were entered. 2. Conditional expectation regression was used to fill in unknown variables. 3. The maximum expected utility principle was used to recommend the next observation to make. Stop if nothing would be gained by further observations. 4. The user was asked to determine the correct value for the recommended observation. Any other observations could be made, instead of or in addition to this. 5. Continue with step 2. 1082 Magnus Stensmo, Terrence J. Sejnowski Table 1: The Cleveland Heart Disease database. Observation Description Values 1 age Age in years continuous 2 sex Sex of subject male/female 3 cp Chest pain four types 4 trestbps Resting blood pressure continuous 5 chol Serum cholesterol continuous 6 fbs Fasting blood sugar It or gt 120 mg/dl 7 restecg Resting electrocardiogr. five values 8 thalach Max heart rate achieved continuous 9 exang Exercise induced angina yes/no 10 oldpeak ST depr. induced by continuous exercise relative to rest 11 slope Slope of peak exercise up/flat/down STsegment 12 ca # major vess. col. ftourosc. 0-3 13 thaI Defect type normal/fixed/reversible Disorder Description Values 14 num Heart disease Not present/4 types 5 EXAMPLE The Cleveland heart disease data set from UC, Irvine has been used to test the system. It contains 303 examples of four types of heart disease and its absence. There are thirteen continuous- or nominally-valued variables (Table 1). The continuous variables were interval coded with one unit per standard deviation away from the mean value. This was chosen since they were approximately normally distributed. Nominal variables were coded with one unit per value. In total the 14 variables were coded with 55 units. The EM steps were repeated until convergence ( 60-150 iterations). A varying number of mixture components (20-120) were tried. Previously reported results have used only presence or absence of the heart disease. The best of these has been a classification rate of 78.9% using a system that incrementally built prototypes [Gennari et al., 1989]. We have obtained 78.6% correct classification with 60 radial Gaussian mixtures as described above. Performance increased with the number of mixture components. It was not sensitive to a varying number of mixture components during training unless there were too few of them. Previous investigators have pointed out that there is not enough information in the thirteen variables in this data set to reach 100% [Gennari et al., 1989]. An annotated transcript of a diagnosis session is shown in Figure 1. 6 CONCLUSIONS AND FURTHER WORK Several properties of this model remain to be investigated. It should be tested on several more databases. Unfortunately databases are typically proprietary and difficult to obtain. Future prospects for medical databases should be good since some hospitals are now using computerized record systems instead of traditional paper-based. It should be fairly easy to A Mixture Model System for Medical and Machine Diagnosis The leftmost number of the five numbers in a line is the estimated probability for no heart disease, followed by the probabilities for the four types of heart disease. The entropy, defined as - l:. Pi log Pi' of the diagnoses are given at the same time as a measure of how decisive the current conclusion is. A completely detennined diagnosis has entropy O. Initially all of the variables are unknown and starting diagnoses are the unconditional prior probabilities. Disorders (entropy = 1.85): 0.541254 0.181518 0.118812 0.115512 0.042904 What is cp ? 3 The first question is chest pain, and the answer changes the estimated probabilities. This variable is continuous. The answer is to be interpreted how far from the mean the observation is in standard deviations. As the decision becomes more conclusive, the entropy decreases. Disorders (entropy = 0.69): 0.888209 0.060963 0.017322 0.021657 0.011848 What is age ? 0 Disorders (entropy = 0.57): 0.91307619 0.00081289 0.02495360 0.03832095 0.02283637 What is oldpeak ? -2 Disorders (entropy = 0.38): 0.94438718 0.00089016 0.02539957 0.02691099 0.00241210 What is chol? -1 Disorders (entropy = 0.11): 0.98848758 0.00028553 0.00321580 0.00507073 0 . 00294036 We have now detennined that the probability of no heart disease in this case is 98.8%. The remaining 0.2% is spread out over the other possibilities. Figure 1: Diagnosis example. generate data for machine diagnosis. 1083 An alternative way to choose a new question is to evaluate the variance change in the output variables when a variable is changed from missing to observed. The idea is that a variable known with certainty has zero variance. The variable with the largest resulting conditional variance could be selected as the query, similar to [Cohn et aI., 1995]. One important aspect of automated diagnosis is the accompanying explanation for the conclusion, a factor that is important for user acceptance. Since the basis functions have local support and since we have estimates for the probability of each basis function having generated the observed data, explanations for the conclusions could be generated. Instead of using the simplified utilities with values 0 and 1 for the expected utility calculations they could be learned by reinforcement learning. A trained expert would evaluate the quality of the diagnosis performed by the system, followed by adjustment of the utilities. The 0 and 1 values can be used as starting values. 1084 Magnus Stensmo. Terrence J. Sejnowski Acknowledgements The heart disease database is from the University of California, Irvine Repository of Machine Learning Databases and originates from R. Detrano, Cleveland Clinic Foundation. Peter Dayan provided helpful comments on an earlier version of this paper. References Ahmad, S. & Tresp, V. (1993). Some solutions to the missing feature problem in vision. In Advances in Neural Information Processing Systems, vol. 5, pp 393-400. Morgan Kaufmann, San Mateo, CA. Baxt, W. (1990). Use of an artificial neural network for data analysis in clinical decisionmaking: The diagnosis of acute coronary occlusion. Neural Computation, 2(4),480489. Cohn, D. A., Ghahramani, Z. & Jordan, M.1. (1995). Active learning with statistical models. 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McLachlan, G. J. & Basford, K. E. (1988). Mixture Models: Inference and Applications to Clustering. Marcel Dekker, Inc., New York, NY. Miller, R. A., Pople, H. E. & Myers, 1 D. (1982). Internist-I: An experimental computerbased diagnostic consultant for general internal medicine. New England Journal of Medicine, 307, 468-476. Nowlan, S. J. (1991). Soft Competitive Adaptation: Neural Network Learning Algorithms based on Fitting Statistical Mixtures. PhD thesis, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA. Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo, CA. Quinlan, 1 R. (1986). Induction of decision trees. Machine Learning, 1,81-106. Stensmo, M. & Sejnowski, T. J. (1994). A mixture model diagnosis system. Tech. Rep. INC-9401, Institute for Neural Computation, University of California, San Diego. von Neuman, J. & Morgenstern, O. (1947). Theory of Games and Economic Behavior. Princeton University Press, Princeton, NJ.	1994	3
921	Advantage Updating Applied to a Differential Game Mance E. Harmon Wright Laboratory WL/AAAT Bldg. 635 2185 Avionics Circle Wright-Patterson Air Force Base, OH 45433-7301 harmonme@aa.wpafb.mil A. Harry Klopr Wright Laboratory klopfah@aa.wpafb.mil Leemon C. Baird III· Wright Laboratory baird@cs.usafa.af.mil Category: Control, Navigation, and Planning Keywords: Reinforcement Learning, Advantage Updating, Dynamic Programming, Differential Games Abstract An application of reinforcement learning to a linear-quadratic, differential game is presented. The reinforcement learning system uses a recently developed algorithm, the residual gradient form of advantage updating. The game is a Markov Decision Process (MDP) with continuous time, states, and actions, linear dynamics, and a quadratic cost function. The game consists of two players, a missile and a plane; the missile pursues the plane and the plane evades the missile. The reinforcement learning algorithm for optimal control is modified for differential games in order to find the minimax point, rather than the maximum. Simulation results are compared to the optimal solution, demonstrating that the simulated reinforcement learning system converges to the optimal answer. The performance of both the residual gradient and non-residual gradient forms of advantage updating and Q-learning are compared. The results show that advantage updating converges faster than Q-learning in all simulations. The results also show advantage updating converges regardless of the time step duration; Q-learning is unable to converge as the time step duration ~rows small. U.S.A.F. Academy, 2354 Fairchild Dr. Suite 6K4l, USAFA, CO 80840-6234 354 Mance E. Hannon, Leemon C. Baird ll/, A. Harry Klopf 1 ADVANTAGE UPDATING The advantage updating algorithm (Baird, 1993) is a reinforcement learning algorithm in which two types of information are stored. For each state x, the value V(x) is stored, representing an estimate of the total discounted return expected when starting in state x and performing optimal actions. For each state x and action u, the advantage, A(x,u), is stored, representing an estimate of the degree to which the expected total discounted reinforcement is increased by performing action u rather than the action currently considered best. The optimal value function V* (x) represents the true value of each state. The optimal advantage function A * (x,u) will be zero if u is the optimal action (because u confers no advantage relative to itself) and A * (x,u) will be negative for any suboptimal u (because a suboptimal action has a negative advantage relative to the best action). The optimal advantage function A * can be defined in terms of the optimal value function v: A(x,u) = ~[RN(X,U)- V(x)+ rNV(x')] bat (1) The definition of an advantage includes a l/flt term to ensure that, for small time step duration flt, the advantages will not all go to zero. Both the value function and the advantage function are needed during learning, but after convergence to optimality, the policy can be extracted from the advantage function alone. The optimal policy for state x is any u that maximizes A * (x,u). The notation ~ax (x) = max A(x,u) " (2) defines Amax(x). If Amax converges to zero in every state, the advantage function is said to be normalized. Advantage updating has been shown to learn faster than Q-Iearning (Watkins, 1989), especially for continuous-time problems (Baird, 1993). If advantage updating (Baird, 1993) is used to control a deterministic system, there are two equations that are the equivalent of the Bellman equation in value iteration (Bertsekas, 1987). These are a pair of two simultaneous equations (Baird, 1993): A(x,u)-maxA(x,u') =(R+ y l1lV(x')- V(X»)_l w ~t (3) maxA(x,u)=O (4) " where a time step is of duration L1t, and performing action u in state x results in a reinforcement of R and a transition to state Xt+flt. The optimal advantage and value functions will satisfy these equations. For a given A and V function, the Bellman residual errors, E, as used in Williams and Baird (1993) and defined here as equations (5) and (6).are the degrees to which the two equations are not satisfied: E1 (xl,u) = (R(x"u)+ y l1lV(xt+l1I)- V(XI»)~- A(x"u)+ max A(x,,u' ) (5) ~t w E2(x,u)=-maxA(x,u) (6) " Advantage Updating Applied to a Differential Game 355 2 RESIDUAL GRADIENT ALGORITHMS Dynamic programming algorithms can be guaranteed to converge to optimality when used with look-up tables, yet be completely unstable when combined with functionapproximation systems (Baird & Harmon, In preparation). It is possible to derive an algorithm that has guaranteed convergence for a quadratic function approximation system (Bradtke, 1993), but that algorithm is specific to quadratic systems. One solution to this problem is to derive a learning algorithm to perfonn gradient descent on the mean squared Bellman residuals given in (5) and (6). This is called the residual gradient form of an algorithm. There are two Bellman residuals, (5) and (6), so the residual gradient algorithm must perfonn gradient descent on the sum of the two squared Bellman residuals. It has been found to be useful to combine reinforcement learning algorithms with function approximation systems (Tesauro, 1990 & 1992). If function approximation systems are used for the advantage and value functions, and if the function approximation systems are parameterized by a set of adjustable weights, and if the system being controlled is deterministic, then, for incremental learning, a given weight W in the functionapproximation system could be changed according to equation (7) on each time step: dW = _ a a[E;(x"u,) + E;(x"u,)] 2 aw _ _ E ( ) aE1 (x, ,u,) _ E ( ) aE2 (x" u, ) a 1 x"u, aw a 2 x"u, aw = _a(_l (R + yMV(X'+M) - V (x) ) - A(x"u,) + max A(X"U)) dt u _(_I (yfJJ aV(x,+fJJ) _ av(x,))_ aA(x"u,) + am~XA(xt'U)J dt aw aw aw aw (7) amaxA(x"u) -amaxA(x"u) U a U W As a simple, gradient-descent algorithm, equation (7) is guaranteed to converge to the correct answer for a deterministic system, in the same sense that backpropagation (Rumelhart, Hinton, Williams, 1986) is guaranteed to converge. However, if the system is nondetenninistic, then it is necessary to independently generate two different possible "next states" Xt+L1t for a given action Ut perfonned in a given state Xt. One Xt+L1t must be used to evaluate V(Xt+L1t), and the other must be used to evaluate %w V(xt+fJJ)' This ensures that the weight change is an unbiased estimator of the true Bellman-residual gradient, but requires a system such as in Dyna (Sutton, 1990) to generate the second Xt+L1t. The differential game in this paper was detenninistic, so this was not needed here. 356 Mance E. Harmon, Leemon C. Baird /11, A. Harry KLopf 3 THE SIMULATION 3.1 GAME DEFINITION We employed a linear-quadratic, differential game (Isaacs, 1965) for comparing Q-learning to advantage updating, and for comparing the algorithms in their residual gradient forms. The game has two players, a missile and a plane, as in games described by Rajan, Prasad, and Rao (1980) and Millington (1991). The state x is a vector (xm,xp) composed of the state of the missile and the state of the plane, each of which are composed of the poSition and velocity of the player in two-dimensional space. The action u is a vector (um,up) composed of the action performed by the missile and the action performed by the plane, each of which are the acceleration of the player in two-dimensional space. The dynamics of the system are linear; the next state xt+ 1 is a linear function of the current state Xl and action Ul. The reinforcement function R is a quadratic function of the accelerations and the distance between the players. R(x,u)= [distance2 + (missile acceleration)2 - 2(plane acceleration)2]6t (8) R(X,U)=[(Xm -Xp)2 +U~-2U!]llt (9) In equation (9), squaring a vector is equivalent to taking the dot product of the vector with itself. The missile seeks to minimize the reinforcement, and the plane seeks to maximize reinforcement. The plane receives twice as much punishment for acceleration as does the missile, thus allowing the missile to accelerate twice as easily as the plane. The value function V is a quadratic function of the state. In equation (10), Dm and Dp are weight matrices that change during learning. (10) The advantage function A is a quadratic function of the state X and action u. The actions are accelerations of the missile and plane in two dimensions. A(x,u)=x~Amxm +x~BmCmum +u~Cmum + x~Apxp +x~BpCpup +u~Cpup (11) The matrices A, B, and C are the adjustable weights that change during learning. Equation (11) is the sum of two general quadratic functions. This would still be true if the second and fifth terms were xBu instead of xBCu. The latter form was used to simplify the calculation of the policy. Using the xBu form, the gradient is zero when u=-C-lBx!2. Using the xBCu form, the gradient of A(x,u) with respect to u is zero when u=-Bx!2, which avoids the need to invert a matrix while calculating the policy. 3.2 THE BELLMAN RESIDUAL AND UPDATE EQUATIONS Equations (5) and (6) define the Bellman residuals when maximizing the total discounted reinforcement for an optimal control problem; equations (12) and (13) modify the algorithm to solve differential games rather than optimal control problems. Advantage Updating Applied to a Differential Game 357 E1(x"u,) = (R(x"u,)+ r6tV(xl+6t)- V(X,»)..!...- A(x"u,)+ minimax A(x,) (12) tl.t E2(x"u,)=-minimax A(x,) (13) The resulting weight update equation is: tl.W = -aU R+ r 6tV(X'M')- V(x,») 1t - A(x"u,)+minimax A(X,») .((rt:., aV~6t) aV(X,»)_1 _ aA(x"u,) + aminimax A(X,») aW tl.t aw aw (14) " A() aminimax A(x,) -amzmmax x, aw For Q-leaming, the residual-gradient form of the weight update equation is: tl.W = -a( R+ r 6t minimax Q(Xl+dt)-Q(x"u,») .( r 6t -kminimax Q(x,+6t)--kQ(x"u,») 4 RESULTS 4.1 RESIDUAL GRADIENT ADVANTAGE UPDATING RESULTS (15) The optimal weight matrices A * , B , C , and D * were calculated numerically with Mathematica for comparison. The residual gradient form of advantage updating learned the correct policy weights, B, to three significant digits after extensive training. Very interesting behavior was exhibited by the plane under certain initial conditions. The plane learned that in some cases it is better to turn toward the missile in the short term to increase the distance between the two in the long term. A tactic sometimes used by pilots. Figure 1 gives an example. 10r-------...... ............................ ~ ....... i ... ·· \. ~/ ....................... ......... .' .' ...... I \. GO V C • .... til 0.01 .001 .0001 0 0.04 0.08 0.12 Time Figure 1: Simulation of a missile (dotted line) pursuing a plane (solid line), each having learned optimal behavior. The graph of distance vs. time show the effects of the plane's maneuver in turning toward the missile. 358 Mance E. Harmon. Leemon C. Baird III. A. Harry Klopf 4.2 COMPARATIVE RESULTS . The error in the policy of a learning system was defined to be the sum of the squared errors in the B matrix weights. The optimal policy weights in this problem are the same for both advantage updating and Q-learning, so this metric can be used to compare results for both algorithms. Four different learning algorithms were compared: advantage updating, Q-Iearning, Residual Gradient advantage updating, and Residual Gradient Qlearning. Advantage updating in the non-residual-gradient form was unstable to the point that no meaningful results could be obtained, so simulation results cannot be given for it. 4.2.1 Experiment Set 1 The learning rates for both forms of Q-Iearning were optimized to one significant digit for each simulation. A single learning rate was used for residual-gradient advantage updating in all four simulations. It is possible that advantage updating would have performed better with different learning rates. For each algorithm, the error was calculated after learning for 40,000 iterations. The process was repeated 10 times using different random number seeds and the results were averaged. This experiment was performed for four different time step durations, 0.05, 0.005, 0.0005, and 0.00005. The non-residualgradient form of Q-Iearning appeared to work better when the weights were initialized to small numbers. Therefore, the initial weights were chosen randomly between 0 and 1 for the residual-gradient forms of the algorithms, and between 0 and 10-8 for the non-residualgradient form of Q-learning. For small time steps, nonresidual-gradient Q-Iearning performed so poorly that the error was lower for a learning rate of zero (no learning) than it was for a learning rate of 10-8. Table 1 gives the learning rates used for each simulation, and figure 2 shows the resulting error after learning. 8 • 6 ---0 0-[J Final 4 -D--FQ Error 2 -·-RAU • • 0 0.05 0.005 0.0005 0.00005 TIme Step Duration Figure 2: Error vs. time step size comparison for Q-Learning (Q), residual-gradient Q-Learning(RQ), and residual-gradient advantage updating(RAU) using rates optimal to one significant figure for both forms of Q-Iearning, and not optimized for advantage updating. The final error is the sum of squared errors in the B matrix weights after 40,000 time steps of learning. The final error for advantage updating was lower than both forms of Q-learning in every case. The errors increased for Qlearning as the time step size decreased. Advantage Updating Applied to a Differential Game 359 Time step duration, III 5.10-2 5.10-3 5.10-4 5.10-5 Q 0.02 0.06 0.2 0.4 RQ 0.08 0.09 0 0 RAU 0.005 0.005 0.005 0.005 Table 1: Learning rates used for each simulation. Learning rates are optimal to one significant figure for both forms of Q-learning, but are not necessarily optimal for advantage updating. 4.2.2 Experiment Set 2 Figure 3 shows a comparison of the three algorithms' ability to converge to the correct policy. The figure shows the total squared error in each algorithms' policy weights as a function of learning time. This simulation ran for a much longer period than the simulations in table 1 and figure 2. The learning rates used for this simulation were identical to the rates that were found to be optimal for the shorter run. The weights for the non-Residual gradient form of Q-Iearning grew without bound in all of the long experiments, even after the learning rate was reduced by an order of magnitude. Residual gradient advantage updating was able to learn the correct policy, while Q-learning was unable to learn a policy that was better than the initial, random weights. Leorning Ability Comporison 10~------------------, Error .1 ---RAU - - - - -, RO, ,01 ,001 0 2 3 4 5 Time Steps in millions 5 Conclusion Figure 3 The experimental data shows residual-gradient advantage updating to be superior to the three other algorithms in all cases. As the time step grows small, Q-learning is unable to learn the correct policy. Future research will include the use of more general networks and implementation of the wire fitting algorithm proposed by Baird and Klopf (1994) to calculate the policy from a continuous choice of actions in more general networks. 360 Mance E. Hannon. Leemon C. Baird Ill. A. Harry Klopf Acknowledgments This research was supported under Task 2312Rl by the Life and Environmental Sciences Directorate of the United States Air Force Office of Scientific Research. References Baird, L.C. (1993). Advantage updating Wright-Patterson Air Force Base, OH. (Wright Laboratory Technical Report WL-TR-93-1146, available from the Defense Technical information Center, Cameron Station, Alexandria, VA 22304-6145). Baird, L.C., & Harmon, M. E. (In preparation). Residual gradient algorithms WrightPatterson Air Force Base, OH. (Wright Laboratory Technical report). Baird, L.C., & Klopf, A. H. (1993). Reinforcement learning with high-dimensional. continuous actions Wright-Patterson Air Force Base, OH. (Wright Laboratory technical report WL-TR-93-1147, available from the Defense Technical information Center, Cameron Station, Alexandria, VA 22304-6145). Bertsekas, D. P. (1987). Dynamic programming: Deterministic and stochastic models. Englewood Cliffs, NJ: Prentice-Hall. Bradtke, S. J. (1993). Reinforcement Learning Applied to Linear Quadratic Regulation. Proceedings of the 5th annual Conference on Neural Information Processing Systems. Isaacs, Rufus (1965). Differential games. New York: John Wiley and Sons, Inc. Millington, P. J. (1991). Associative reinforcement learning for optimal control. Unpublished master's thesis, Massachusetts Institute of Technology, Cambridge, MA. Rajan, N., Prasad, U. R., and Rao, N. J. (1980). Pursuit-evasion of two aircraft in a horizontal plane. Journal of Guidance and Control. 3(3). May-June, 261-267. Rumelhart, D., Hinton, G., & Williams, R. (1986). Learning representations by backpropagating errors. Nature. 323 .. 9 October, 533-536. Sutton, R. S. (1990). Integrated architectures for learning, planning, and reacting based on approximating dynamic programming. Proceedings of the Seventh International Conference on Machine Learning. Tesauro, G. (1990). Neurogammon: A neural-network backgammon program. Proceedings of the International Joint Conference on Neural Networks. 3 .. (pp. 33-40). San Diego, CA. Tesauro, G. (1992). Practical issues in temporal difference learning. Machine Learning, 8(3/4), 279-292. Watkins, C. J. C. H. (1989). Learningfrom delayed rewards. Doctoral thesis, Cambr~dge University, Cambridge, England.	1994	30
922	Pattern Playback in the '90s Malcolm Slaney Interval Research Corporation 180 l-C Page Mill Road, Palo Alto, CA 94304 malcolm@interval.com Abstract Deciding the appropriate representation to use for modeling human auditory processing is a critical issue in auditory science. While engineers have successfully performed many single-speaker tasks with LPC and spectrogram methods, more difficult problems will need a richer representation. This paper describes a powerful auditory representation known as the correlogram and shows how this non-linear representation can be converted back into sound, with no loss of perceptually important information. The correlogram is interesting because it is a neurophysiologically plausible representation of sound. This paper shows improved methods for spectrogram inversion (conventional pattern playback), inversion of a cochlear model, and inversion of the correlogram representation. 1 INTRODUCTIONl My interest in auditory models and perceptual displays [2] is motivated by the problem of sound understanding, especially the separation of speech from noisy backgrounds and interfering speakers. The correlogram and related representations are a pattern space within which sounds can be "understood" and "separated" [3][4]. I am therefore interested in resynthesizing sounds from these representations as a way to test and evaluate sound separation algorithms, and as a way to apply sound separation to problems such as speech enhancement. The conversion of sound to a correlogram involves the intermediate representation of a cochleagram, as shown in Figure 1, so cochlear-model inversion is addressed as one piece of the overall problem. 1. Much of this work was performed by Malcolm Slaney, Daniel Naar and Richard F. Lyon while all three were employed at Apple Computer. The mathematical details of this work were presented at the 1994ICASSP[I]. 828 Malcolm Slaney Waveform Cochleagram Correlogram j 1r--*T'T~"""I-~!"'T!-~~-'-ooI-'1 ::::S:ec::ti :n:: h .... t--'~~'--t.:lfga..~ l!!!Jnme ------------~ ------------~ Figure 1. Three stages in low-level auditory perception are shown here. Sound waves are converted into a detailed representation with broad spectral bands, known as cochleagrams. The correlogram then summarizes the periodicities in the cochleagram with short-time autocorrelation. The result is a perceptual movie synchronized to the acoustic signal. The two inversion problems addressed in this work are indicated with arrows from right to left There are three factors which can be used to judge the quality of an auditory model: psychoacoustic comparisons, neurophysiological plausibility, and does it represent the perceptually relevant information? First, the correlogram has been shown to simply and accurately predict human pitch perception [5]. The neurophysiological basis for the correlogram has not been found, but there are neural circuits performing the same calculation in the mustached bat's echolocation system [6]. Finally, from an information representation point of view, does the correlogram preserve the salient information? The results of this paper show that no information has been lost. Since the psychoacoustic, neurophysiological, and information representation measures are all positive, perhaps the correlogram is the basis of most auditory processing. The inversion techniques described here are important because they allow us to readily evaluate the results of sound separation models that "zero out" unwanted portions of the signal in the correlogram domain. This work extends the convex projection approach of Irino [7] and Yang [8] by considering a different cochlear model, and by including the correlogram inversion. The convex projection approach is well suited to "filling in" missing information. While this paper only describes the process for one particular auditory model, the techniques are equally useful for other models. This paper describes three aspects of the problem: cochleagram inversion, conversion of the correlogram into spectrograms, and spectrogram inversion. A number of reconstruction options are explored in this paper. Some are fast, while other techniques use time-consuming iterations to produce reconstructions perceptually equivalent to the original sound. Fast versions of these algorithms could allow us to separate a speaker's voice from the background noise in real time. 2 COCHLEAGRAM INVERSION Figure 2 shows a block diagram of the cochlear model [9] that is used in this work. The basis of the model is a bank of filters, implemented as a cascade of low-pass filters, that splits the input signal into broad spectral bands. The output from each filter in the bank is called a channel. The energy in each channel is detected and used to adjust the channel gain, implementing a simple model of auditory sensitivity adaptation, or automatic gain control (AGe). The half-wave rectifier (HWR) detection nonlinearity provides a waveform for each channel that roughly represents the instantaneous neural firing rate at each position along the cochlea. E ~---+ > ~ Cochlear Filterbank Detector (HWR) Adaptation or AGC Figure 2. Three stages of the simple cochlear model used in this paper are shown above. Pattern Playback in the '90s 829 The cochleagram is converted back into sound by reversing the three steps shown in Figure 2. First the AGe is divided out, then the negative portions of each cochlear channel are recovered by using the fact that each channel is spectrally limited. Finally, the cochlear filters are inverted by running the filters backwards, and then correcting the resulting spectral slope. The AGe stage in this cochlear model is controlled by its own output. It is a combination of a multiplicative gain and a simple first-order filter to track the history of the output signal. Since the controlling signal is directly available, the AGe can be inverted by tracking the output history and then dividing instead of multiplying. The performance of this algorithm is described by Naar [10] and will not be addressed here. It is worth noting that AGe inversion becomes more difficult as the level of the input signal is raised, resulting in more compression in the forward path. The next stage in the inversion process can be done in one of two ways. After AGC inversion, both the positive values of the signal and the spectral extant of the signal are known. Projections onto convex sets [11], in this case defined by the positive values of the detector output and the spectral extant of the cochlear filters, can be used to find the original signal. This is shown in the left half of Figure 3. Alternatively, the spectral projection filter can be combined with the next stage of processing to make the algorithm more efficient. The increased efficiency is due to better match between the spectral projection and the cochlear filterbank, and due to the simplified computations within each iteration. This is shown in the right half of Figure 3. The result is an algorithm that produces nearly perfect results with no iterations at all. Spectral Projection Filter Bank Inversion ~ ~ AGC Inversion Temporal Projection ~ Q) E -go ~Qi 0> L... __ ..... CI)~ C Other --110.. Inversion hannels~ 8 o Figure 3. There are two ways to use convex projections to recover the information lost by the detectors. The conventional approach is shown on the left. The right figure shows a more efficient approach where the spectral projection has been combined with the filterbank inversion Finally, the multiple outputs from the cochlear filterbank are converted back into a single waveform by correcting the phase and summing all channels. In the ideal case, each cochlear channel contains a unique portion of the spectral energy, but with a bit of phase delay and amplitude change. For example, if we run the signal through the same filter the spectral content does not change much but both the phase delay and amplitude change will be doubled. More interestingly, if we run the signal through the filter backwards, the forward and backward phase changes cancel out. After this phase correction, we can sum all channels and get back the Original waveform, with a bit of spectral coloration. The spectral coloration or tilt can be fixed with a simple filter. A more efficient approach to correct the spectral tilt is to scale each channel by an appropriate weight before summing, as shown in Figure 4. The result is a perfect reconstruction, over those frequencies where the cochlear filters are non-zero. Figure 5 shows results from the cochleagram inversion procedure. An impulse is shown on the left, before and after 10 iterations of the HWR inversion (using the algorithm on the right half of Figure 3). With no iterations the result is nearly perfect, except for a bit of noise near the center. The overall curvature of the baseline is due to the fact that informa830 Malcolm Slaney ~ Time IIR or FIR E ~ Time c:UI c:,a ~'S Reversed Filter to -0 .... as;:, Reversed ~ .... 0Filter for Correct ;:'0> €,e. Filter for .s5 0> 0>;:, iIO Inversion Tilt CJ)~ itO Inversion Figure 4. Two approaches are shown here to invert the filterbank. The left diagram shows the normal approach, the right figure shows a more efficient approach where the spectral-tilt filter is converted to a simple multiplication. E -0 .... c:.g ;:'0> 0> CJ)~ tion near DC has been lost as it travels through the auditory system and there is no way to recover it with the information that we have. A more interesting example is shown on the right. Here the word "tap" 1 has been reconstructed, with and without the AGC inversion. With the AGe inversion the result is nearly identical to the original. The auditory system is very sensitive to onsets and quickly adapts to steady state sounds like vowels. It is interesting to compare this to the reconstruction withoutAGC inversion. Without the AGC, the result is similar to what the ear hears, the onsets are more prominent and the vowels are deemphasized. This is shown in the right half of Figure 5. Impulse inversion with no iterations Impulse iteration with 10 iterations "Tap" reconstruction "Tap" Reconstruction with AGC Inversion without AGC Inversion Figure 5. The cochlear reconstructions of an impulse and the word "tap" are shown here. The first and second reconstructions show an impulse reconstruction with and without iterations. The third and fourth waveforms are the word "tap" with and without the AGe inversion. 3 CORRELOGRAM INVERSION The correlogram is an efficient way to capture the short-time periodicities in the auditory signal. Many mechanical measurements of the cochlea have shown that the response is highly non-linear. As the signal level changes there are large variations in the bandwidth and center frequency of the cochlear response. With these kinds of changes, it is difficult to imagine a system that can make sense of the spectral profile. This is especially true for decisions like pitch determination and sound separation. But through all these changes in the cochlear filters, the timing information in the signal is preserved. The spectral profile, as measured by the cochlea, might change, but the rate of glottal pulses is preserved. Thus I believe the auditory system is based on a representation of sound that makes short-time periodicities apparent. One such representation is the correlogram. The correlogram measures the temporal correlation within each channel, either using FFfs which are most efficient in computer implementations, or neural delay lines much like those found in the binaural system of the owl. 1. The syllable "tap", samples 14000 through 17000 of the "trainldr5/fcdfll sxl06/sx106.adc" utterance on the TIMIT Speech Database, is used in all voiced examples in this paper. Pattern Playback in the '90s 831 The process of inverting the correlogram is simplified by noting that each autocorrelation is related by the Fourier transform to a power spectrum. By combining many power spectrums into a picture, the result is a spectrogram. This process is shown in Figure 6. In this way, a separate spectrogram is created for each channel. There are known techniques for converting a spectrogram, which has amplitude information but no phase information, back into the original waveform. The process of converting from a spectrogram back into a waveform is described in Section 4. The correlogram inversion process consists of inverting many spectrograms to form an estimate of a cochleagram. The cochleagram is inverted using the techniques described in Section 2. Frame 42 of Correlogram Frame 43 of Correlogram Time IFFTto get of spectrogram Spectrogram Inversion ., One line of Cochleagram Time Figure 6. Correlogram inversion is possible by noting that each row of the correlogram contains the same information as a spectrogram of the same row of cochleagram output. By converting the correlogram into many spectrograms, the spectrogram inversion techniques described in Section 4 can be used. The lower horizontal stripe in the spectrogram is due to the narrow passband of the cochlear channel. Half-wave rectification of the cochlear filter output causes the upper horizontal stripes. One important improvement to the basic method is possible due to the special characteristics of the correlogram. The essence of the spectrogram inversion problem is to recover the phase information that has been thrown away. This is an iterative procedure and would be costly ifit had to be performed on each channel. Fortunately, there is quite a bit of overlap between cochlear channels. Thus the phase recovered from one channel can be used to initialize the spectrogram inversion for the next channel. A difficulty with spectrogram inversion is that the absolute phase is lost. By using the phase from one channel to initialize the next, a more consistent set of cochlear channel outputs is recovered. 4 SPECTROGRAM INVERSION While spectrograms are not an accurate model of human perception, an implementation of a correlogram includes the calculation of many spectrograms. Mathematically, an autocorrelation calculation is similar to a spectrogram or short-time power spectrum. One column of a conventional spectrogram is related to an autocorrelation of a portion of the original waveform ~y a Fourier transform (see Figure 6). Unfortunately, the final representation of both spectrograms and autocorrelations is missing the phase information. The main task of a spectrogram inversion algorithm is to recover a consistent estimate ofthe missing phase. This process is not magical, it can only recover a signal that has the same magnitude spectrum as the original spectrogram. But the consistency constraint on the time evolution of the signal power spectrum also constrains the time evolution of the spectral phase. 832 Malcolm Slaney The basic procedure in spectrogram inversion [12] consists of iterating between the time and the frequency domains. Starting from the frequency domain, the magnitude but not the phase is known. As an initial guess, any phase value can be used. The individual power spectra are inverse Fourier transformed and then summed to arrive at a single waveform. If the original spectrogram used overlapping windows of data, the information from adjacent windows either constructively or destructively interferes to estimate a waveform. A spectrogram of this new data is calculated, and the phase is now retained. We know the original magnitude was correct. Thus we can estimate a better spectrogram by combining the original magnitude information with the new phase information. It can be shown that each iteration will reduce the error. Figure 7 shows an outline of steps that can be used to improve the consistency of phase estimates during the first iteration. As each portion of the waveform is added to the estimated signal, it is possible to add a linear phase so that each waveform lines up with the proceedings segments. The algorithm described in the paragraph above assumes an initial phase of zero. A more likely phase guess is to choose a phase that is consistent with the existing data. The result with no iterations is a waveform that is often closer to the original than that calculated assuming zero initial phase and ten iterations. The total computational cost is minimized by combining these improvements with the initial phase estimates from adjacent channels of the correlogram. Thus when inverting the first channel of the correlogram, a cross-correlation is used to pick the initial phase and a few more iterations insure a consistent result. After the first channel, the phase of the proceeding channel is used to initialize the spectrogram inversion and only a few iterations are necessary to fine tune the waveform. Reconstructions from segments 1 through N o 400 SegmentN+1 b;&/\; I o 400 Cross Correlation Maximize fit by chOOSing peak Rotated Segment N+ 1 hi\;\; I o 400 New Reconstruction 300 o Figure 7. A procedure for adjusting the phase of new segments when inverting a spectrogram is shown above. As each new segment (bottom left) is converted from a power spectrum into a waveform, a linear phase is added to maximize the fit with the existing segments (top left.) The amount of rotation is determined by a cross correlation (middle). Adding the new segment with the proper rotation (top right) produces the new waveform (bottom right.) Pattern Playback in the '90s 833 5 PUTTING IT TOGETHER This paper has described two steps to convert a correlogram into a sound. These steps are detailed below: I) For each row of the correlogram: a) Convert the autocorrelation data into power spectrum (Section 3). b) Use spectrogram inversion (Section 4) to convert the spectrograms into an estimate of cochlear channel output. c) Assemble the results of spectrogram inversion into an estimate of the cochleagram. 2) Invert the cochleagram using the techniques described in Section 2. This process is diagrammed in Figures I and 6. 6 RESULTS Figure 8 shows the results of the complete reconstruction process for a 200Hz impulse train and the word "tap." In both cases, no iterations were performed for either the spectrogram or filterbank inversion. More iterations reduce the spectral error, but do not make the graphs look better or change the perceptual quality much. It is worth noting that the "tap" reconstruction from a correlogram looks similar to the cochleagram reconstruction without the AGC (see Figure 5.) Reducing the level of the input signal, thus reducing the amount of compression performed by the AGC, results in a correlogram reconstruction similar to the original waveform. 0.02 r---------. 0.01 o -0.01 -0.02"'--~~~~~ 50 100 150 200 0.05f"'"'"!~----"""" o -0.05 "'-------.... o 1000 2000 3000 Figure 8. Reconstructions from the correlogram representation of an impulse train and the word "tap" are shown above. Reducing the input signal level, thus minimizing the effect of errors when inverting the AGe, produces results identical to the original "tap." It is important to note that the algorithms described in this paper are designed to minimize the error in the mean-square sense. This is a convenient mathematical definition, but it doesn't always correlate with human perception. A trivial example of this is possible by comparing a waveform and a copy of the waveform delayed by lOms. Using the meansquared error, the numerical error is very high yet the two waveforms are perceptually equivalent. Despite this, the results of these algorithms based on mean-square error do sound good. 7 CONCLUSIONS This paper has described several techniques that allow several stages of an auditory model to be converted back into sound. By converting each row of the correlogram into a spectrogram, the spectrogram inversion techniques of Section 4 can be used. The special characteristics of a correlogram described in Section 3 are used to make the calculation more efficient. Finally, the cochlear filterbank can be inverted to recover the original waveform. The results are waveforms, perceptually identical to the original waveforms. 834 Malcolm Slaney These techniques will be especially useful as part of a sound separation system. I do not believe that our auditory system resynthesizes partial waveforms from the auditory scene. Yet, all research systems generate separated sounds so that we can more easily perceive their success. More work is still needed to fine-tune these algorithm and to investigate the ability to reconstruct sounds from partial correlograms. Acknowledgments I am grateful for the inspiration provided by Frank Cooper's work in the early 1950's on pattern playback[13][14]. His work demonstrated that it was possible to convert a spectrogram, painted onto clear plastic, into sound. This work in this paper was performed with Daniel Naar and Richard F. Lyon. We are grateful for the help we have received from Richard Duda (San Jose State), Shihab Shamma (U. of Maryland), Jim Boyles (The MathWorks) and Michele Covell (Interval Research). References [1] Malcolm Slaney, D. Naar, R. F. Lyon, "Auditory model inversion for sound separation," Proc. of IEEE ICASSP, Volume II, pp. 77-80, 1994. [2] M. Slaney and R. F. Lyon, "On the importance of time-A temporal representation of sound," in Visual Representations of Speech Signals, eds. M. Cooke, S. Beet, and M. Crawford, J. Wiley and Sons, Sussex, England, 1993. [3] R. F. Lyon, "A computational model of binaural localization and separation," Proc. of IEEE ICASSP, 1148-1151, 1983. [4] M. Weintraub, "The GRASP sound separation system," Proc. of IEEE ICASSP, pp. 18A.6.1-18A.6.4, 1984. [5] D. Hennes, "Pitch analysis," in Visual Representations of Speech Signals, eds. M. Cooke, S. Beet, and M. Crawford, J. Wiley and Sons, Sussex, England, 1993. [6] N. Suga, "Cortical computational maps for auditory imaging," Neural Networks, 3, 321, 1990. [7] T. lrino, H. Kawahara, "Signal reconstruction from modified auditory wavelet transfonn," IEEE Trans. on Signal Processing, 41,3549-3554, Dec. 1993. [8] x. Yang, K. Wang, and S. Sharnma, "Auditory representations of acoustic signals," IEEE Trans. on Information Theory, 38, 824-839, 1992. [9] R. F. Lyon, "A computational model of filtering, detection, and compression in the cochlea," Proc. of the IEEE ICASSP, 1282-1285,1982. [10] D. Naar, "Sound resynthesis from a correlogram," San Jose State University, Department of Electrical Engineering, Technical Report #3, May 1993. [11] R. W. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Sys., vol. 22, 735, 1975. [12] D. Griffin and J. Lim, "Signal estimation from modified short-time Fourier transfonn," IEEE Trans. on Acoustics, Speech, and Signal Processing, 32, 236-242, 1984. [13] F. S. Cooper, "Some Instrumental Aids to Research on Speech," Report on the Fourth Annual Round Table Meeting on Linguistics and Language Teaching, Georgetown University Press, 46-53, 1953. [14] F. S. Cooper, "Acoustics in human communications: Evolving ideas about the nature of speech," J. Acoust. Soc. Am., 68(1),18-21, July 1980.	1994	31
923	An Analog Neural Network Inspired by Fractal Block Coding Fernando J. Pineda The Applied Physics Laboratory The Johns Hopkins University Johns Hokins Road Laurel, MD 20723-6099 Andreas G. Andreou Dept. of Electrical & Computer Engineering The Johns Hopkins University 34th & Charles St. Baltimore, MD 21218 Abstract We consider the problem of decoding block coded data, using a physical dynamical system. We sketch out a decompression algorithm for fractal block codes and then show how to implement a recurrent neural network using physically simple but highly-nonlinear, analog circuit models of neurons and synapses. The nonlinear system has many fixed points, but we have at our disposal a procedure to choose the parameters in such a way that only one solution, the desired solution, is stable. As a partial proof of the concept, we present experimental data from a small system a 16-neuron analog CMOS chip fabricated in a 2m analog p-well process. This chip operates in the subthreshold regime and, for each choice of parameters, converges to a unique stable state. Each state exhibits a qualitatively fractal shape. 1. INTRODUCTION Sometimes, a nonlinear approach is the simplest way to solve a linear problem. This is true when computing with physical dynamical systems whose natural operations are nonlinear. In such cases it may be expensive, in terms of physical complexity, to linearize the dynamics. For example in neural computation active ion channels have highly non linear input-output behaviour (see Hille 1984). Another example is 796 Fernando Pineda. Andreas G. Andreou subthreshold CMOS VLSI technology 1. In both examples the physics that governs the operation of the active devices, gives rise to gain elements that have exponential transfer characteristics. These exponentials result in computing structures with non-linear dynamics. It is therefore worthwhile, from both scientific and engineering perspectives, to investigate the idea of analog computation by highly non-linear components. This paper, explores an approach for solving a specific linear problem with analog circuits that have nonlinear transfer functions. The computational task considered here is that of fractal block code decompression (see e.g. Jacquin, 1989). The conventional approach to decompressing fractal codes is essentially an excercise in solving a high-dimenional sparse linear system of equations by using a relaxation algorithm. The relaxation algorithm is performed by iteratively applying an affine transformation to a state vector. The iteration yields a sequence of state vectors that converges to a vector of decoded data. The approach taken in this paper is based on the observation that one can construct a physically-simple nonlinear dyanmical system whose unique stable fixed point coincides with the solution of the sparse linear system of equations. In the next section we briefly summarize the basic ideas behind fractal block coding. This is followed by a description of an analog circuit with physically-simple nonlinear neurons. We show how to set the input voltages for the network so that we can program the position of the stable fixed point. Finally , we present experimental results obtained from a test chip fabricated in a 2mm CMOS process. 2. FRACTAL BLOCK CODING IN A NUTSHELL Let the N-dimensional state vector I represent a one dimensional curve sampled on N points. An affine transformation of this vector is simply a transformation of the form I' = WI+B , where W is an NxN -element matrix and B is an N-component vector. This transformation can be iterated to produce a sequence of vectors I(O) ... . ,I(n). The sequence converges to a unique final state 1* that is independent of the initial state 1(0) if the maximum eigenvalue A.max of the matrix W satisfies Amax < 1. The uniqueness of the final state implies that to transmit the state r to a receiver, we can either transmit r directly, or we can transmit Wand B and let the receiver perform the iteration to generate r. In the latter case we say that Wand B constitute an encoding of the state 1. For this encoding to be useful, the amount of data needed to transmit Wand B must be less than the amount of data needed to transmit r This is the case when Wand B are sparse and parameterized and when the total number of bits needed to transmit these parameters is less than the total number of bits needed to transmit the uncompressed state r Fractal block coding is a special case of the above approach. It amounts to choosing a lWe consider subthreshold analog VLSI., (Mead 1989; Andreou and Boahen, 1994). A simple subthreshold model is ~ = I~nfet) exp(K'Vgb)( exp( -vsb) -exp( -Vdb») for NFETS, where 1C - 0.67 and I~ t) = 9.7 x 10-18 A. The voltage differences Vgb, ,vsb,and Vdb are in units of the thermal voltafje, Vth= 0.025V. We use a corresponding expression for PFETs of the from Ids = I~pfe exp( -K'Vgb)( exp(vsb) - exp(vdb») where I~Pfet ) =3.8xl0-18 A. An Analog Neural Network Inspired by Fractal Block Coding 797 blocked structure for the matrix W. This structure forces large-scale features to be mapped into small-scale features. The result is a steady state r that represents a curve with self similar (actually self affine) features. As a concrete example of such a structure, consider the following transformation of the state I. for O~ i ~ N -1 2 N r i = wR12i- N + bR for 2: ~ i ~ N-l (1) This transformation has two blocks. The transformation of the first N/2 components of I depend on the parameters W Land b L while the transformation of the second N/2 components depend on the parameters WR, and bR . Consequently just four parameters completely specify this transformation. This transformation can be expressed as a single affine transformation as follows: /' 0 wL 10 bL I'N/2-1 wL 1 bL = N/2-1 + (2) /' N12 WR 1 bR N/2 /' N-l wR I N- 1 bR The top and bottom halves of I I depend on the odd and even components of I respectively. This subsampling causes features of size I to be mapped into features of size 112. A subsampled copy of the state I with transformed intensities is copied into the top half of 1'. Similarly, a subsampled copy of the state I with transformed intensities is copied into the bottom half of 1'. If this transformation is iterated, the sequence of transformed vectors will converge provided the eigenvalues determined by WL and WR are all less than one (i.e. WL and WR < 1). Although this toy example has just four free parameters and is thus too trivial to be useful for actual compression applications, it does suffice to generate state vectors with fractal properties since at steady state, the top and bottom halves of I' differ from the entire curve by an affine transformation. In this paper we will not describe how to solve the inverse problem which consists of finding a parameterized affine transformation that produces a given final state T. We note, however, that it is a special (and simpler) case of the recurrent network training problem, since the problem is linear, has no hidden units and has only one fixed point. The reader is refered to (Pineda, 1988) or. for a least squares algorithm in the context of neural nets or to (Monroe and Dudbridge, 1992) for a least squares algorithm in the context of coding. 3. A CMOS NEURAL NETWORK MODEL Now that we have described the salient aspects of the fractal decompression problem, we tum to the problem of implementing an analog neural network whose nonlinear dynamics converges to the same fixed point as the linear system. Nonlinearity arises because we 798 Fernando Pineda, Andreas G. Andreou make no special effort to linearize the gain elements (controlled conductances and transconductances) of the implementation medium. In this section we first describe a simple neuron. Then we analyze the dynamics of a network composed of such neurons. Finally we describe how to program the fixed point in the actual physical network. 3.1 The analog Neuron \\(~t) woulgll) like to create a neuron model that calculates the transformation I = al + b . Consider the circuit shown in figure 1. This has three functional sections which compute by adding and subtracting currents and. where voltages are "log" coded; this is the essence of the "current-mode" aproach in circuit design (Andreou et.al. 1994). The first section, receives an input voltage from a presynaptic neuron, converts it into a current I(in), and multiplies it by a weight a. The second section adds and subtracts the bias current b. The last section converts the output current into an output voltage and transmits it to the next neuron in the network. Since the transistors have exponential transfer characteristics, this voltage is logarithmically coded. The parameters a and b are set by external voltages. Theyarameter a, is set by a single external voltage Va while the bias parameter b = br -) - b( + is set by two external voltages vb(+) and vbH . Two voltages are used for b to account for both positive and negative bIas values since b( -»0 and b( + »0 . '{,(+) r---------, r----, r-----I I I I I I I I I I I I I I I I I I I I I I I I (ou I (in) I I I I II I l II V I : aiin~ : I I I I ~ I I I I (out) I I~--~---+----~~r--;--~--~V I I I I I I I I I I I I L ________ .I I I I I I I L ______ -' ~-) I I I I I I I I -----Figure 1. The analog neuron has three sections. To derive the dynamical equations of the neuron, it is neccesary to add up all the currents and invoke Kirchoffs current law, which requires that lout) _alin) +b(+) -b(-) = Ic . (3) If we now assume a simple subthreshold model for the behavior of the FET's and PFETs in the neuron, we can obtain the following expression for the current across the capacitor: Q dlout) --=1 lout) dt c (4) An Analog Neural Network Inspired by Fractal Block Coding 799 where Q = Cl1cVth determines the characteristic time scale of the neuron2. It immediately follows from the last two expressions that the dynamics of a single neuron is determined by the equation dJCout) Q = _/(out) (I(out) _ al(in) _ b). dt (5) Where b = M-) - M+) . This equation appears to have a quadratic nonlinearity on the r.h.s. In fact, the noninearity is even more complicated since, the cooeficients a, M +) and b( -) are not constants, but depend on I(out) (through v(out). Application of the simple subthreshold model, results in a multiplier gain that is a function of v( out) (and hence tout)) as well as Va . It is given by a( va' v<out») = 2exp(- v~ {sinh( v~ - va) -sinh( v~ _v(out» J Similarly, the currents b(+) and bH are given by and b( +) = I~fpet) exp( KV b(+) )( 1- exp( _v(out») b(-) = I~nfet) exp(KVb(_) )(I_exp(_v(Out») respectively, where va == vdd - va . 3.2 Network dynamics and Stability considerations (6) (7.a) (7.b) With these results we conclude that, a network of neurons, in which each neuron receives input from only one other neuron, would have a dynamical equation of the form d[' ( ) Q _! = -1·(/· - a· I· 1·(·) - b·) dt !! ! I J! I (8) where the connectivity of the network is determined by the function j( i) . The fixed points of these highly nonlinear equations occur when the r.h.s. of (8) vanishes. This can only happen if either Ii = Oor if (Ii - aJj(i) - bi ) = 0 for each i. The local stability of each of these fixed points follows by examining the eigenvalues (A.) of the corresponding jacobian. The expression for the jacobian at a general point I is J'k = dFi = -Q[(/. - a.J.(·) - b·)8·k + [,(1- a~1 .(.) - b~)8'k - a.J.8 '(')k] (9) ! dlk ! I J I I I I I J I I I I I J I . Where the partial derivatives, a'i and b'j are with respect to Ii. At a fixed point the jacobian takes the form { bi8ik if Ii = 0 Jik = Q -/i [ (1- a[lj(i) - b[)8ik - ai8j(i)k] if (Ii - ailj(i) - bi ) = O· (10) 2C represents the total gate capacitance from all the transistors connected to the horizontal line of the neuron. For the 2J..l analog proc~s~, the gate capacitance is aprroximately 0.5 fF/J..l2 so a 10J..l x 10J..l FET has a charactenstlc charge of Q =2.959 x 10- 4 Coulombs at room temperature. 800 Fernando Pineda, Andreas G. Andreou There are two cases of interest. The first case is when no neurons have zero output. This is the "desired solution." In this case, the jacobian specializes to Jik =-QIi[(1-aiIj(i) -b[)Oik -aiOj(i)k]' (11) Where, from (6) and (7), it can be shown that the partial derivatives, a'i and b'i are both non-positive. It immediately follows, from Gerschgorin's theorem, that a sufficient condition that the eigenvalues be negative and that the fixed point be stable, is that lail <l. The second case is when at least one of the neurons has zero output. We call these fixed points the "spurious solutions." In this case some of the eigenvalues are very easy to calculate because terms of the form (bi -).) ,where Ii = 0, can be factored from the expression for det(J-.?J). Thus some eigenvalues can be made positive by making some of the bi positive. Accordingly, if all the bi satisfy bi >0 , some of the eigenvalues will necessarily be positive and the spurious solutions will be unstable. To summarize the above discussion, we have shown that by choosing bi >0 and lail <1 for all i, we can make the desired fixed point stable and the spurious fixed points unstable. Note that a sufficient condition for bi >0 is if b~ +) = O. It remains to show that the system must converge to the desired fixed point, i.e. that the system cannot oscillate or wander chaotically. To do this we consider the connectivity of the network we implemented in our test chip. This is shown schematically in figure 2. The first eight neurons receive input from the odd numbered neurons while the second eight neurons receive input from the even numbered neurons. The neurons on the lefthand side all share the weight, WL, while the neurons on the right share the weight WR. By tracing the connections, we find that there are two independent loops of neurons: loop #1 = {0,8,12,14,IS,7,3,1} and loop #2 = {2,9,4,1O,13,6,1l,S}. Figure 2. The connection topology for the test chip is determined by the matrix of equation (1). The neurons are labeled 0-15. By inspecting each loop, we see that it passes through either the left or right hand range an even number of times. Hence, if there are any inhibitory weights in a loop, there must be an even number of them. This is the "even loop criterion", and it suffices to prove that the network is globally asymptotically stable, (Hirsch, 1987). 3.3. Programming the fixed point The nonlinear circuit of the previous section converges to a fixed point which is the solution of the following system of transcendental equations * * (-) * Ii -ai(li ,va)Ij(i) -bi (Ii ,vb<-»-O (12) An Analog Neural Network Inspired by Fractal Block Coding 801 where the coefficients ai and bi are given by equations (6) and (7b) respectively. Similarly, the iterated affine transformations converge to the solution of the following linear equations * * Ii - A/j(j) - Bj = 0 (13) where the coefficients {Ai ,Bi } and the connectionsj(i) are obtained by solving the approximate inverse problem with the additional constraints that bi >0 and lail <1 for all i,. The requirement that the fixed points of the two systems be identical results in the conditions * Aj = aj(lj ,va) B - b(-)(I* ) j j j ,V b(-) (14) These equations can be solved for the required input voltages Va, and vb(-). Thus we are able to construct a nonlinear dynamical system that converges to the same fixed point as a linear system. For this programming method to work, of course, the subthreshold model we have used to characterize the network must accurately model the physical properties of the neural network. 4. PRELIMINARY RESULTS As a first step towards realizing a working system, we fabricated a Tiny chip containing 16 neurons arranged in two groups of eight. The topology is the same as shown in figure 2. The neurons are similar to those in figure 1 except that the bias term in each block of 8 neurons has the form b = kb( -) + (7 - k )b( -) , where O::;k::;7 is the label of a particular neuron within a block. This form increases the complexity of the neurons, but also allows us to represent ramps more easily (see figure 3). We fabricated the chip through MOSIS in a 2~m p-well CMOS process. A switching layer allows us to change the connection topology at run-time. One of the four possible configurations corresponds to the toplogy of figure 2. Six external voltages {Va ,V H ' Vi)H ' Va ,VbH ' Vi)H }parameterize the fixed points of the network. These are confrolfM blpote~tioIdeters~ There is multiplexing circuitry included on the chip that selects which neuron output is to be amplified by a sense-amp and routed off-chip. The neurons can be addressed individually by a 4-bit neuron address. The addressing and analog-to-digital conversion is performed by a Motorolla 68HCIIAI microprocessor. We have operated the chip at 5volts and at 2.6 volts. Figure 3. shows the scanned steady state output of one of the test chips for a particular choice of input parameters with v dd =5 volts. The curve in figure 3. exhibits the qualitatively self-similar features of a recursively generated object. We are able to see three generations of a ramp. At 2.5 volts we see a very similar curve. We find that the chip draws 16.3 ~ at 2.5 volts. This corresponds to a steady state power dissipation of 411lW. Simulations indicate that the chip is operating in the subthreshold regime when Vdd = 2.5 volts. Simulations also indicate that the chip settles in less than one millisecond. We are unable to perform quantitiative measurements with the first chip because of several layout errors. On the other hand, we have experimentally verified that the network is indeed stable and that network produces qualitative fractals. We explored the parameter space informatlly. At no time did we encounter anything but the desired solutions. 802 Fernando Pineda, Andreas G. Andreou O~--~~~------~--==:L----~----~----~o 2 4 6 8 10 12 14 Neuron label Figure 3 D/A output for chip #3 for a particular set of input voltages. We have already fabricated a larger design without the layout problems of the prototype. This second design has 32 pixeles and a richer set of permitted topologies. We expect to make quantitative measurements with this second design. In particular we hope to use it to decompress an actual block code. Acknowledgements The work described here is funded by APL !R&D as well as a grant from the National Science Foundation ECS9313934, Paul Werbos is the monitor. The authors would like to thank Robert Jenkins, Kim Strohbehn and Paul Furth for many useful conversations and suggestions. References Andreou, A.G. and Boahen, K.A. Neural Information Processing I: The Current-Mode approach, Analog VLSI: Signal and Information Processing, (eds: M Ismail and T. Fiez) MacGraw-Hill Inc., New York. Chapter 6 (1994). Hille, B., Ionic Channels of Excitable Membranes, Sunderland, MA, Sinauer Associates Inc. (1984). Hirsch, M. ,Convergence in Neural Nets, Proceedings of the IEEE ICNN, San Diego, CA, (1987). Jacquin, A. E., A Fractal Theory of iterated Markov operators with applications to digital image coding, Ph.D. Dissertation, Georgia Institute of Technology (1989). Mead, c., Analog VLSI and Neural Systems, Addison Wesley, (1989) Monroe, D.M. and Dudbridge, F. Fractal block coding of images, Electronics Letters, 28, pp. 1053-1055, (1992). Pineda, F.J., Dynamics and Architecuture for Neural Computation, Journal of Complexity, 4, 216-245 (1988).	1994	32
924	Phase-Space Learning Fu-Sheng Tsung Chung Tai Ch'an Temple 56, Yuon-fon Road, Yi-hsin Li, Pu-li Nan-tou County, Taiwan 545 Republic of China Garrison W. Cottrell· Institute for Neural Computation Computer Science & Engineering University of California, San Diego La Jolla, California 92093 Abstract Existing recurrent net learning algorithms are inadequate. We introduce the conceptual framework of viewing recurrent training as matching vector fields of dynamical systems in phase space. Phasespace reconstruction techniques make the hidden states explicit, reducing temporal learning to a feed-forward problem. In short, we propose viewing iterated prediction [LF88] as the best way of training recurrent networks on deterministic signals. Using this framework, we can train multiple trajectories, insure their stability, and design arbitrary dynamical systems. 1 INTRODUCTION Existing general-purpose recurrent algorithms are capable of rich dynamical behavior. Unfortunately, straightforward applications of these algorithms to training fully-recurrent networks on complex temporal tasks have had much less success than their feedforward counterparts. For example, to train a recurrent network to oscillate like a sine wave (the "hydrogen atom" of recurrent learning), existing techniques such as Real Time Recurrent Learning (RTRL) [WZ89] perform suboptimally. Williams & Zipser trained a two-unit network with RTRL, with one teacher signal. One unit of the resulting network showed a distorted waveform, the other only half the desired amplitude. [Pea89] needed four hidden units. However, our work demonstrates that a two-unit recurrent network with no hidden units can learn the sine wave very well [Tsu94]. Existing methods also have several other ·Correspondence should be addressed to the second author: gary@cs.ucsd.edu 482 Fu-Sheng Tsung, Garrison W. Cottrell limitations. For example, networks often fail to converge even though a solution is known to exist; teacher forcing is usually necessary to learn periodic signals; it is not clear how to train multiple trajectories at once, or how to insure that the trained trajectory is stable (an attractor). In this paper, we briefly analyze the algorithms to discover why they have such difficulties, and propose a general solution to the problem. Our solution is based on the simple idea of using the techniques of time series prediction as a methodology for recurrent network training. First, by way of introducing the appropriate concepts, consider a system of coupled autonomousl first order network equations: FI (Xl (t), X2(t), ... , Xn (t)) F2(XI(t), X2(t),· · ·, Xn(t)) or, in vector notation, X(t) = F(X) where XCt) = (XICt), X2(t),·· ., xn(t)) The phase space is the space of the dependent variables (X), it does not include t, while the state space incorporates t. The evolution of a trajectory X(t) traces out a phase curve, or orbit, in the n-dimensional phase space of X . For low dimensional systems (2- or 3-D), it is easy to visualize the limit sets in the phase space: a fixed point and a limit cycle become a single point and a closed orbit (closed curve), respectively. In the state space they become an infinite straight line and a spiral. F(X) defines the vector field of X, because it associates a vector with each point in the phase space of X whose direction and magnitude determines the movement of that point in the next instant of time (by definition, the tangent vector). 2 ANALYSIS OF CURRENT APPROACHES To get a better understanding of why recurrent algorithms have not been very effective, we look at what happens during training with two popular recurrent learning techniques: RTRL and back propagation through time (BPTT). With each, we illustrate a different problem, although the problems apply equally to each technique. RTRL is a forward-gradient algorithm that keeps a matrix of partial derivatives of the network activation values with respect to every weight. To train a periodic trajectory, it is necessary to teacher-force the visible units [WZ89], i.e., on every iteration, after the gradient has been calculated, the activations of the visible units are replaced by the teacher. To see why, consider learning a pair of sine waves offset by 90°. In phase space, this becomes a circle (Figure la). Initially the network 1 Autonomous means the right hand side of a differential equation does not explicitly reference t, e.g. dx/dt = 2x is autonomous, even though x is a function oft, but dx/dt = 2x+t is not. Continuous neural networks without inputs are autonomous. A nonautonomous system can always be turned into an autonomous system in a higher dimension. Phase-Space Learning 483 a b Figure 1: Learning a pair of sine waves with RTRL learning. (a) without teacher forcing, the network dynamics (solid arrows) take it far from where the teacher (dotted arrows) assumes it is, so the gradient is incorrect. (b) With teacher forcing, the network's visible units are returned to the trajectory. (thick arrows) is at position Xo and has arbitrary dynamics. After a few iterations, it wanders far away from where the teacher (dashed arrows) assumes it to be. The teacher then provides an incorrect next target from the network's current position. Teacher-forcing (Figure 1b), resets the network back on the circle, where the teacher again provides useful information. However, if the network has hidden units, then the phase space of the visible units is just a projection of the actual phase space of the network, and the teaching signal gives no information as to where the hidden units should be in this higherdimensional phase space. Hence the hidden unit states, unaltered by teacher forcing, may be entirely unrelated to what they should be. This leads to the moving targets problem. During training, every time the visible units re-visit a point, the hidden unit activations will differ, Thus the mapping changes during learning. (See [Pin88, WZ89] for other discussions of teacher forcing.) With BPTT, the network is unrolled in time (Figure 2). This unrolling reveals another problem: Suppose in the teaching signal, the visible units' next state is a non-linearly separable function of their current state. Then hidden units are needed between the visible unit layers, but there are no intermediate hidden units in the unrolled network. The network must thus take two time steps to get to the hidden units and back. One can deal with this by giving the teaching signal every other iteration, but clearly, this is not optimal (consider that the hidden units must "bide their time" on the alternate steps).2 The trajectories trained by RTRL and BPTT tend to be stable in simulations of simple tasks [Pea89, TCS90], but this stability is paradoxical. Using teacher forcing, the networks are trained to go from a point on the trajectory, to a point within the ball defined by the error criterion f (see Figure 4 (a)). However, after learning, the networks behave such that from a place near the trajectory, they head for the trajectory (Figure 4 (b)). Hence the paradox. Possible reasons are: 1) the hidden unit moving targets provide training off the desired trajectory, so that if the training is successful, the desired trajectory is stable; 2) we would never consider the training successful if the network "learns" an unstable trajectory; 3) the stable dynamics in typical situations have simpler equations than the unstable dynamics [N ak93]. To create an unstable periodic trajectory would imply the existence of stable regions both inside and outside the unstable trajectory; dynamically this is 2 At NIPS, 0 delay connections to the hidden units were suggested, which is essentially part of the solution given here. 484 Figure 2: A nonlinearly separable mapping must be computed by the hidden units (the leftmost unit here) every other time step. " a Fu-Sheng Tsung, Garrison W. Cottrell ~------------. Figure 3: The network used for iterated prediction training. Dashed connections are used after learning. b Figure 4: The paradox of attractor learning with teacher forcing. (a) During learning, the network learns to move from the trajectory to a point near the trajectory. (b) After learning, the network moves from nearby points towards the trajectory. more complicated than a simple periodic attractor. In dynamically complex tasks, a stable trajectory may no longer be the simplest solution, and stability could be a problem. In summary, we have pointed out several problems in the RTRL (forward-gradient) and BPTT (backward-gradient) classes of training algorithms: 1. Teacher forcing with hidden units is at best an approximation, and leads to the moving targets problem. 2. Hidden units are not placed properly for some tasks. 3. Stability is paradoxical. 3 PHASE-SPACE LEARNING The inspiration for our approach is prediction training [LF88], which at first appears similar to BPTT, but is subtly different. In the standard scheme, a feedforward network is given a time window, a set of previous points on the trajectory to be learned, as inputs. The output is the next point on the trajectory. Then, the inputs are shifted left and the network is trained on the next point (see Figure 3). Once the network has learned, it can be treated as recurrent by iterating on its own predictions. The prediction network differs from BPTT in two important ways. First, the visible units encode a selected temporal history of the trajectory (the time window). The point of this delay space embedding is to reconstruct the phase space of the underlying system. [Tak81] has shown that this can always be done for a deterministic system. Note that in the reconstructed phase space, the mapping from one Phase-Space Learning a YI+I r;.-.-.-. -------.. ,--... ... ... ... ... ... ... ... ... ... ... '" '" '" .. , '" ••• 485 b Figure 5: Phase-space learning. (a) The training set is a sample of the vector field. (b) Phase-space learning network. Dashed connections are used after learning. point to the next (based on the vector field) is deterministic. Hence what originally appeared to be a recurrent network problem can be converted into an entirely feed forward problem. Essentially, the delay-space reconstruction makes hidden states visible, and recurrent hidden units unnecessary. Crucially, dynamicists have developed excellent reconstruction algorithms that not only automate the choices of delay and embedding dimension but also filter out noise or get a good reconstruction despite noise [FS91, Dav92, KBA92]. On the other hand, we clearly cannot deal with non-deterministic systems by this method. The second difference from BPTT is that the hidden units are between the visible units, allowing the network to produce nonlinearly separable transformations of the visible units in a single iteration. In the recurrent network produced by iterated prediction, the sandwiched hidden units can be considered "fast" units with delays on the input/output links summing to 1. Since we are now lear~ing a mapping in phase space, stability is easily ensured by adding additional training examples that converge towards the desired orbit.3 We can also explicitly control convergence speed by the size and direction of the vectors. Thus, phase-space learning (Figure 5) consists of: (1) embedding the temporal signal to recover its phase space structure, (2) generating local approximations of the vector field near the desired trajectory, and (3) functional approximation of the vector field with a feedforward network. Existing methods developed for these three problems can be directly and independently applied to solve the problem. Since feedforward networks are universal approximators [HSW89], we are assured that at least locally, the trajectory can be represented. The trajectory is recovered from the iterated output of the pre-embedded portion of the visible units. Additionally, we may also extend the phase-space learning framework to also include inputs that affect the output of the system (see [Tsu94] for details).4 In this framework, training multiple attractors is just training orbits in different parts of the phase space, so they simply add more patterns to the training set. In fact, we can now create designer dynamical systems possessing the properties we want, e.g., with combinations of fixed point, periodic, or chaotic attractors. 3The common practice of adding noise to the input in prediction training is just a simple minded way of adding convergence information. 4Principe & Kuo(this volume) show that for chaotic attractors, it is better to treat this as a recurrent net and train using the predictions. 486 Fu-Sheng Tsung, Garrison W. Cottrell 0.5 Q -0.5 -0.5 0 0.5 Figure 6: Learning the van der Pol oscillator. ( a) the training set. (b) Phase space plot of network (solid curve) and teacher (dots). (c) State space plot. As an example, to store any number of arbitrary periodic attractors Zi(t) with periods 11 in a single recurrent network, create two new coordinates for each Zi(t), (Xi(t),Yi(t)) = (sin(t),cos(t)), where (Xi,Yi) and (Xj,Yj) are disjoint circles for i 'I j. Then (Xi, Yi, Zi) is a valid embedding of all the periodic attractors in phase space, and the network can be trained. In essence, the first two dimensions form "clocks" for their associated trajectories. 4 SIMULATION RESULTS In this section we illustrate the method by learning the van der Pol oscillator (a much more difficult problem than learning sine waves), learning four separate periodic attractors, and learning an attractor inside the basin of another attractor. 4.1 LEARNING THE VAN DER POL OSCILLATOR The van der Pol equation is defined by: We used the values 0.7, 1, 1 for the parameters a, b, and w, for which there is a global periodic attractor (Figure 6). We used a step size of 0.1, which discretizes the trajectory into 70 points. The network therefore has two visible units. We used two hidden layers with 20 units each, so that the unrolled, feedforward network has a 2-20-20-2 architecture. We generated 1500 training pairs using the vector field near the attractor. The learning rate was 0.01, scaled by the fan-in, momentum was 0.75, we trained for 15000 epochs. The order of the training pairs was randomized. The attractor learned by the network is shown in (Figure 6 (b)). Comparison of the temporal trajectories is shown in Figure 6 (c); there is a slight frequency difference. The average MSE is 0.000136. Results from a network with two layers of 5 hidden units each with 500 data pairs were similar (MSE=0.00034). 4.2 LEARNING MULTIPLE PERIODIC ATTRACTORS [Hop82] showed how to store multiple fixed-point at tractors in a recurrent net. [Bai91] can store periodic and chaotic at tractors by inverting the normal forms of these attractors into higher order recurrent networks. However, traditional recurrent training offers no obvious method of training multiple attractors. [DY89] were able Phase.Space Learning 'I'---:.ru"='"""--'O~--:O-=-" ---! A .I'--::.ru-;---;;--;;-:Oj---! 8 100 1OO 300 400 D .ru 0 OJ E 487 _.0.1. Il.6, 0.63. 0.7 '1'--::4U~-::-0 --;Oj~-! F 1.--------, ~~ .1 0 50 100 150 1OO ~ 300 H Figure 7: Learning mUltiple attractors. In each case, a 2-20-20-2 network using conjugate gradient is used. Learning 4 attractors: (A) Training set. (B) Eight trajectories of the trained network. (C) Induced vector field of the network. There are five unstable fixed points. (D) State space behavior as the network is "bumped" between attractors. Learning 2 attractors, one inside the other: (E) Training set. (F) Four trajectories ofthe trained network. (G) Induced vector field of the network. There is an unstable limit cycle between the two stable ones. (H) State space behavior with a "bump". to store two limit cycles by starting with fixed points stored in a Hopfield net, and training each fixed point locally to become a periodic attractor. Our approach has no difficulty with multiple attractors. Figure 7 (A-D) shows the result of training four coexisting periodic attractors, one in each quadrant of the two-dimensional phase space. The network will remain in one of the attractor basins until an external force pushes it into another attractor basin. Figure 7 (E-H) shows a network with two periodic attractors, this time one inside the other. This vector field possess an unstable limit cycle between the two stable limit cycles. This is a more difficult task, requiring 40 hidden units, whereas 20 suffice for the previous task (not shown). 5 SUMMARY We have presented a phase space view of the learning process in recurrent nets. This perspective has helped us to understand and overcome some of the problems of using traditional recurrent methods for learning periodic and chaotic attractors. Our method can learn multiple trajectories, explicitly insure their stability, and avoid overfitting; in short, we provide a practical approach to learning complicated temporal behaviors. The phase-space framework essentially breaks the problem into three sub-problems: (1) Embedding a temporal signal to recover its phase space structure, (2) generating local approximations of the vector field near the desired trajectory, and (3) functional approximation in feedforward networks. We have demonstrated that using this method, networks can learn complex oscillations and multiple periodic attractors. 488 Fu-Sheng Tsung, Garrison W. Cottrell Acknowledgements This work was supported by NIH grant R01 MH46899-01A3. Thanks for comments from Steve Biafore, Kenji Doya, Peter Rowat, Bill Hart, and especially Dave DeMers for his timely assistance with simulations. References [Bai91] [Dav92] [DY89] [FS91] W. Baird and F. Eeckman. Cam storage of analog patterns and continuous sequences with 3n2 weights. In R.P. Lippmann, J .E. Moody, and D.S. Touretzky, editors, Advances in Neural Information Processing Systems, volume 3, pages 91-97, 1991. Morgan Kaufmann, San Mateo. M. Davies. Noise reduction by gradient descent. International Journal of Bifurcation and Chaos, 3:113-118, 1992. K. Doya and S. Yoshizawa. Memorizing oscillatory patterns in the analog neuron network. In IJCNN, Washington D.C., 1989. IEEE. J.D. Farmer and J.J. Sidorowich. Optimal shadowing and noise reduction. Physica D, 47:373-392, 1991. [Hop82] J.J. Hopfield. Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences, USA, 79, 1982. [HSW89] K. Hornik, M. Stinchcombe, and H. White. Multilayer feedforward networks are universal approximators. Neural Networks, 2:359-366, 1989. [KBA92] M.B. Kennel, R. Brown, and H. Abarbanel. Determining embedding dimension for phase-space reconstruction using a geometrical construction. Physical Review A, 45:3403-3411, 1992. [LF88] A. Lapedes and R. Farber. How neural nets work. In D.Z. Anderson, editor, Neural Information Processing Systems, pages 442-456, Denver 1987, 1988. American Institute of Physics, New York. [N ak93] Hiroyuki Nakajima. A paradox in learning trajectories in neural networks. Working paper, Dept. of EE II, Kyoto U., Kyoto, JAPAN, 1993. [Pea89] B.A. Pearlmutter. Learning state space trajectories in recurrent neural networks. Neural Computation, 1:263-269, 1989. [Pin88] F.J. Pineda. Dynamics and architecture for neural computation. Journal of Complexity, 4:216-245, 1988. [Tak81] F. Takens. Detecting strange attractors in turbulence. In D.A. Rand and L.-S. Young, editors, Dynamical Systems and Turbulence, volume 898 of Lecture Notes in Mathematics, pages 366-381, Warwick 1980, 1981. Springer-Verlag, Berlin. [TCS90] F-S. Tsung, G. W. Cottrell, and A. I. Selverston. Some experiments on learning stable network oscillations. In IJCNN, San Diego, 1990. IEEE. [Tsu94] F-S. Tsung. Modelling Dynamical Systems with Recurrent Neural Networks. PhD thesis, University of California, San Diego, 1994. [WZ89] R.J. Williams and D. Zipser. A learning algorithm for continually running fully recurrent neural networks. Neural Computation, 1:270-280, 1989.	1994	33
925	An experimental comparison of recurrent neural networks Bill G. Horne and C. Lee Giles· NEe Research Institute 4 Independence Way Princeton, NJ 08540 {horne.giles}~research.nj.nec.com Abstract Many different discrete-time recurrent neural network architectures have been proposed. However, there has been virtually no effort to compare these arch:tectures experimentally. In this paper we review and categorize many of these architectures and compare how they perform on various classes of simple problems including grammatical inference and nonlinear system identification. 1 Introduction In the past few years several recurrent neural network architectures have emerged. In this paper we categorize various discrete-time recurrent neural network architectures, and perform a quantitative comparison of these architectures on two problems: grammatical inference and nonlinear system identification. 2 RNN Architectures We broadly divide these networks into two groups depending on whether or not the states of the network are guaranteed to be observable. A network with observable states has the property that the states of the system can always be determined from observations of the input and output alone. The archetypical model in this class .. Also with UMIACS, University of Maryland, College Park, MD 20742 698 Bill G. Horne, C. Lee Giles Table 1: Terms that are weighted in various single layer network architectures. Ui represents the ith input at the current time step, Zi represents the value of the lh node at the previous time step. Architecture bias Ui Zi UiUj ZiUj ZiZj First order x x x High order x Bilinear x x x Quadratic x x x x x x was proposed by N arendra and Parthasarathy [9]. In their most general model, the output of the network is computed by a multilayer perceptron (MLP) whose inputs are a window of past inputs and outputs, as shown in Figure la. A special case of this network is the Time Delay Neural Network (TDNN), which is simply a tapped delay line (TDL) followed by an MLP [7]. This network is not recurrent since there is no feedback; however, the TDL does provide a simple form of dynamics that gives the network the ability model a limited class of nonlinear dynamic systems. A variation on the TDNN, called the Gamma network, has been proposed in which the TDL is replaced by a set of cascaded filters [2]. Specifically, if the output of one of the filters is denoted xj(k), and the output of filter i connects to the input of filter j, the output of filter j is given by, xj(k + 1) = I-'xi(k) + (l-I-')xj(k). In this paper we only consider the case where I-' is fixed, although better results can be obtained if it is adaptive. Networks that have hidden dynamics have states which are not directly accessible to observation. In fact, it may be impossible to determine the states of a system from observations of it's inputs and outputs alone. We divide networks with hidden dynamics into three classes: single layer networks, multilayer networks, and networks with local feedback. Single layer networks are perhaps the most popular of the recurrent neural network models. In a single layer network, every node depends on the previous output of all of the other nodes. The function performed by each node distinguishes the types of recurrent networks in this class. In each of the networks, nodes can be characterized as a nonlinear function of a weighted sum of inputs, previous node outputs, or products of these values. A bias term may also be included. In this paper we consider first-order networks, high-order networks [5], bilinear networks, and Quadratic networks[12]. The terms that are weighted in each of these networks are summarized in Table 1. Multilayer networks consist of a feedforward network coupled with a finite set of delays as shown in Figure lb. One network in this class is an architecture proposed by Robinson and Fallside [11], in which the feedforward network is an MLP. Another popular networks that fits into this class is Elman's Simple Recurrent Network (SRN) [3]. An Elman network can be thought of as a single layer network with an extra layer of nodes that compute the output function, as shown in Figure lc. In locally recurrent networks the feedback is provided locally within each individual An Experimental Comparison of Recurrent Neural Networks 699 MLP Figure 1: Network architectures: (a) Narendra and Parthasarathy's Recurrent Neural Network, (b) Multilayer network and (c) an Elman network. node, but the nodes are connected together in a feed forward architecture. Specifically, we consider nodes that have local output feedback in which each node weights a window of its own past outputs and windows of node outputs from previous layers. Networks with local recurrence have been proposed in [1, 4, 10]. 3 Experimental Results 3.1 Experimental methodology In order to make the comparison as fair as possible we have adopted the following methodology. • Resources. We shall perform two fundamental comparisons. One in which the number of weights is roughly the same for all networks, another in which the number of states is equivalent. In either case, we shall make these numbers large enough that most of the networks can achieve interesting performance levels. Number of weights. For static networks it is well known that the generalization performance is related to the number of weights in the network. Although this theory has never been extended to recurrent neural networks, it seems reasonable that a similar result might apply. Therefore, in some experiments we shall try to keep the number of weights approximately equal across all networks. Number of states. It can be argued that for dynamic problems the size of the state space is a more relevant measure for comparison than the number of weights. Therefore, in some experiments we shall keep the number of states equal across all networks. • Vanilla learning. Several heuristics have been proposed to help speed learning and improve generalization of gradient descent learning algorithms. However, such heuristics may favor certain architectures. In order to avoid these issues, we have chosen simple gradient descent learning algorithms. • Number of simulations. Due to random initial conditions, the recurrent neural network solutions can vary widely. Thus, to try to achieve a statistically significant estimation of the generalization of these networks, a large number of experiments were run. 700 Bill G. Horne, C. Lee Giles o stan );::===:====,O'l+------ll o o Figure 2: A randomly generated six state finite state machine. 3.2 Finite state machines We chose two finite state machine (FSM) problems for a comparison of the ability of the various recurrent networks to perform grammatical inference. The first problem is to learn the minimal, randomly generated six state machine shown in Figure 2. The second problem is to infer a sixty-four state finite memory machine [6] described by the logic function y(k) = u(k - 3)u(k) + u(k - 3)y(k - 3) + u(k)u(k - 3)Y(k - 3) where u(k) and y(k) represent the input and output respectively at time k and x represents the complement of x. Two experiments were run. In the first experiment all of the networks were designed such that the number of weights was less than, but as close to 60 as possible. In the second experiment, each network was restricted to six state variables, and if possible, the networks were designed to have approximately 75 weights. Several alternative architectures were tried when it was possible to configure the architecture differently and yield the same number of weights, but those used gave the best results. A complete set of 254 strings consisting of all strings of length one through seven is sufficient to uniquely identify both ofthese FSMs. For each simulation, we randomly partitioned the data into a training and testing set consisting of 127 strings each. The strings were ordered lexographically in the training set. For each architecture 100 runs were performed on each problem. The on-line Back Propagation Through Time (BPTT) algorithm was used to train the networks. Vanilla learning was used with a learning rate of 0.5. Training was stopped at 1000 epochs. The weights of all networks were initialized to random values uniformly distributed in the range [-0.1,0.1]. All states were initialize to zeros at the beginning of each string except for the High Order net in which one state was arbitrarily initialized to a value of 1. Table 2 summarizes the statistics for each experiment. From these results we draw the following conclusions. • The bilinear and high-order networks do best on the small randomly generated machine, but poorly on the finite memory machine. Thus, it would appear that there is benefit to having second order terms in the network, at least for small finite state machine problems. • N arendra and Parthasarathy's model and the network with local recurrence do far better than the other networks on the problem of inferring the finite memory An Experimental Comparison of Recurrent Neural Networks 701 Table 2: Percentage classification error on the FSM experiment for (a) networks with approximately the same number of weights, (b) networks with the same number of state variables. %P = The percentage of trials in which the training set was learned perfectly, #W = the number of weights, and #S = the number of states. training error testing error F5M Architecture t mean ( std) mea.n (std) 'YoP #W #5 N&P 2.8 (M) 16.9 (8.6) 22 56 8 TDNN 12.5 (2.1) 33.8 (U) 0 56 8 Gamma 19.6 (H) 24.8 (3.2) 0 56 8 First Order 12.9 (6.9) 26.5 (9.0) 0 48 6 RND High Order 0.8 (1.5) 6.2 (6.1 ) 60 50 5 Bilinear 1.3 (2.7) 5.7 (6.1) 46 55 5 Quadratic 12.9 (13.4) 17.7 (14.1) 12 45 3 Mullilayer 19.4 (13.6) 23.4 ( 13.5) 6 54 4 Elman 3.5 ~5.~~ 12.7 ~9 . !~ 27 55 6 Local 2.8 1.5 26.7 7.6 4 60 20 N&P 0.0 ~0 . 2 ~ 0 .1 ~ 1 . ~ ~ 99 56 8 TDNN 6.9 (2.1 ) 15.8 (3.2) 0 56 8 Gamma 7.7 (2.2) 15.7 (3.3) 0 56 8 First Order 4.8 (3.0) 16.0 (6.5) 1 48 6 FMM High Order 5.3 (4.0) 26.0 ( 5.1 ) 1 50 5 Bilinear 9.5 (10.4) 25.8 (7.0) 0 55 5 Quadratic 32.5 (10.8) 40.5 (7.3) 0 45 3 Multilayer 36.7 (11.9) 43.5 (8.5) 0 54 4 Elman 12.0 (12.5) 24.9 (7.9) 5 55 6 Local 0.1 ' (0.3) 1.0 (3 .0) 97 60 20 (a) tra.lnlng error testIng error F5M Architecture tt mea.n ( std) mea.n ( std) 'YoP #W #5 N&P 4.6 ( 8.~~ 14.1 (11 .3) 38 73 6 TDNN 11 .7 ( 2.0) 34.3 ( 3 .9) 0 73 6 Gamma 19.0 (H) 25.2 (3.1) 0 H 6 First Order 12.9 ( 6.9) 26.5 (9.0) 0 48 6 RND High Order 0 .3 ( 0.5) 4 .6 ( 5.1) 79 H 6 Bilinear 0 .6 ( 0 .9) 4 .4 ( U) 55 78 6 Quadratic 0 .2 ( 0 .5) 3.2 ( 2.6) 83 216 6 Mullilayer 15.4 (14.1) 19.9 (lU) 16 76 6 Elman 3.5 ( 5.5) 12.7 ( 9 .1) 27 55 6 Local 13.9 ( 405) 20.2 ( 5.7) 0 26 6 N&P 0 .1 ( 0.8) 0.3 ( 1.4) 97 73 6 TDNN 6 .8 ( 1.7) 16.2 ( 2.9) 0 73 6 Gamma 9.0 (2.9) 14.9 (2.8) 0 73 6 First Order 4.8 (3.0) 16.0 (6.5) 1 48 6 FMM High Order 1.2 ( 1.7) 25.1 ( 5.1) 31 H 6 Bilinear 2.6 ( 402) 20.3 ( 7.2) 21 78 6 Quadratic 12.6 (17.3) 26.1 (12.8) 13 216 6 MullUayer 38.1 (12.6) 42.8 ( 9.2) 0 76 6 Elman 12.8 ~H.:~ 27.6 (10.7) 8 55 6 Local 15.3 3 .8 22.2 ( 409) 0 26 6 (b) tThe TDNN and Gamma network both had 8 input taps and 4 hidden layer nodes. For the Gamma network, I' = 0.3 (RND) and I' = 0.7 (FMM). Narendra and Parthasarathy's network had 4 input and output taps and 5 hidden layer nodes. The High-order network used a "one-hot" encoding of the input values [5]. The multilayer network had 4 hidden and output layer nodes. The locally recurrent net had 4 hidden layer nodes with 5 input and 3 output taps, and one output node with 3 input and output taps. ttThe TDNN, Gamma network, and N arendra and Parthasarathy's network all had 8 hidden layer nodes. For the Gamma network, I' = 0.3 (RND) and I' = 0.7 (FMM). The High-order network again used a "one-hot" encoding of the input values. The multilayer network had 5 hidden and 6 output layer nodes. The locally recurrent net had 3 hidden layer nodes and one output layer node, all with only one input and output tap. 702 Bill G. Horne, C. Lee Giles machine when the number of states is not constrained. It is not surprising that the former network did so well since the sequential machine implementation of a finite memory machine is similar to this architecture [6]. However, the result for the locally recurrent network was unexpected. • All of the recurrent networks do better than the TDNN on the small random machine. However, on the finite memory machine the TDNN does surprisingly well, perhaps because its structure is similiar to Narendra and Parthasarathy's network which was well suited for this problem. • Gradient-based learning algorithms are not adequate for many of these architectures. In many cases a network is capable of representing a solution to a problem that the algorithm was not able to find. This seems particularly true for the Multilayer network. • Not surprisingly, an increase in the number of weights typically leads to overtraining. Although, the quadratic network, which has 216 weights, can consistently find solutions for the random machine that generalize well even though there are only 127 training samples. • Although the performance on the training set is not always a good indicator of' generalization performance on the testing set, we find that if a network is able to frequently find perfect solutions for the training data, then it also does well on the testing data. 3.3 Nonlinear system identification In this problem, we train the network to learn the dynamics of the following set of equations proposed in [8] zl(k) + 2z2(k) (k) zl(k+l) l+z~(k) +u (k) zl(k)Z2(k) (k) Z2 + 1 = + u 1 + z~(k) y(k) zl(k) + z2(k) based on observations of u( k) and y( k) alone. The same networks that were used for the finite state machine problems were used here, except that the output node was changed to be linear instead of sigmoidal to allow the network to have an appropriate dynamic range. We found that this caused some stability problems in the quadratic and locally recurrent networks. For the fixed number of weights comparison, we added an extra node to the quadratic network, and dropped any second order terms involving the fed back output. This gave a network with 64 weights and 4 states. For the fixed state comparison, dropping the second order terms gave a network with 174 weights. The locally recurrent network presented stability problems only for the fixed number of weights comparison. Here, we used a network that had 6 hidden layer nodes and one output node with 2 taps on the inputs and outputs each, giving a network with 57 weights and 16 states. In the Gamma network a value of l' = 0.8 gave the best results. The networks were trained with 100 uniform random noise sequences of length 50. Each experiment used a different randomly generated training set. The noise was An Experimental Comparison of Recurrent Neural Networks 703 Table 3: Normalized mean squared error on a sinusoidal test signal for the nonlinear system identification experiment. Archi teet ure Fixed # weights Fixed # states N&P 0.101 0.067 TDNN 0.160 0.165 Gamma 0.157 0.151 First Order 0.105 0.105 High Order 1.034 1.050 Bilinear 0.118 0.111 Quadratic 0.108 0.096 Multilayer 0.096 0.084 Elman 0.115 0.115 Local 0.117 0.123 uniformly distributed in the range [-2.0,2.0], and each sequence started with an initial value of Xl(O) = X2(0) = O. The networks were tested on the response to a sine wave of frequency 0.04 radians/second. This is an interesting test signal because it is fundamentally different than the training data. Fifty runs were performed for each network. BPTT was used for 500 epochs with a learning rate of 0.002. The weights of all networks were initialized to random values uniformly distributed in the range [-0.1,0.1]. Table 3 shows the normalized mean squared error averaged over the 50 runs on the testing set. From these results we draw the following conclusions. • The high order network could not seem to match the dynamic range of its output to the target, as a result it performed much worse than the other networks. It is clear that there is benefit to adding first order terms since the bilinear network performed so much better. • Aside from the high order network, all of the other recurrent networks performed better than the TDNN, although in most cases not significantly better. • The multilayer network performed exceptionally well on this problem, unlike the finite state machine experiments. We speculate that the existence of target output at every point along the sequence (unlike the finite state machine problems) is important for the multilayer network to be successful. • Narendra and Parthasarathy's architecture did exceptionally well, even though it is not clear that its structure is well matched to the problem. 4 Conclusions We have reviewed many discrete-time recurrent neural network architectures and compared them on two different problem domains, although we make no claim that any of these results will necessarily extend to other problems. Narendra and Parthasarathy's model performed exceptionally well on the problems we explored. In general, single layer networks did fairly well, however it is important to include terms besides simple state/input products for nonlinear system identification. All of the recurrent networks usually did better than the TDNN except 704 Bill G. Home, C. Lee Giles on the finite memory machine problem. In these experiments, the use of averaging filters as a substitute for taps in the TDNN did not seem to offer any distinct advantages in performance, although better results might be obtained if the value of J.I. is adapted. We found that the relative comparison of the networks did not significantly change whether or not the number of weights or states were held constant. In fact, holding one of these values constant meant that in some networks the other value varied wildly, yet there appeared to be little correlation with generalization. Finally, it is interesting to note that though some are much better than others, many of these networks are capable of providing adequate solutions to two seemingly disparate problems. Acknowledgements We would like to thank Leon Personnaz and Isabelle Rivals for suggesting we perform the experiments with a fixed number of states. References [1] A.D. Back and A.C. Tsoi. FIR and IIR synapses, a new neural network architecture for time series modeling. Neural Computation, 3(3):375-385, 1991. [2] B. de Vries and J .C. Principe. The gamma model: A new neural model for temporal processing. Neural Networks, 5:565-576, 1992. [3] J .L. Elman. Finding structure in time. Cognitive Science, 14:179-211, 1990. [4] P. Frasconi, M. Gori, and G. Soda. Local feedback multilayered networks. Neural Computation, 4:120-130, 1992. [5] C.L. Giles, C.B. Miller, et al. Learning and extracting finite state automata with second-order recurrent neural networks. Neural Computation, 4:393-405, 1992. [6] Z. Kohavi. Switching and finite automata theory. McGraw-Hill, NY, 1978. [7] K.J. Lang, A.H. Waibel, and G.E. Hinton. A time-delay neural network architecture for isolated word recognition. Neural Networks, 3:23-44, 1990. [8] K.S. Narendra. Adaptive control of dynamical systems using neural networks. In Handbook of Intelligent Control, pages 141-183. Van Nostrand Reinhold, NY, 1992. [9] K.S. Narendra and K. Parthasarathy. Identification and control of dynamical systems using neural networks. IEEE Trans. on Neural Networks, 1:4-27, 1990. [10] P. Poddar and K.P. Unnikrishnan. Non-linear prediction of speech signals using memory neuron networks. In Proc. 1991 IEEE Work. Neural Networks for Sig. Proc., pages 1-10. IEEE Press, 1991. [11] A.J. Robinson and F. Fallside. Static and dynamic error propagation networks with application to speech coding. In NIPS, pages 632-641, NY, 1988. AlP. [12] R.L. Watrous and G.M. Kuhn. Induction of finite-state automata using second-order recurrent networks. In NIPS4, pages 309-316, 1992.	1994	34
926	Inferring Ground Truth from Subjective Labelling of Venus Images Padhraic Smyth, Usama Fayyad Jet Propulsion Laboratory 525-3660, Caltech, 4800 Oak Grove Drive, Pasadena, CA 91109 Michael Burl, Pietro Perona Department of Electrical Engineering Caltech, MS 116-81, Pasadena, CA 91125 Pierre Baldi* Jet Propulsion Laboratory 303-310, Caltech, 4800 Oak Grove Drive, Pasadena, CA 91109 Abstract In remote sensing applications "ground-truth" data is often used as the basis for training pattern recognition algorithms to generate thematic maps or to detect objects of interest. In practical situations, experts may visually examine the images and provide a subjective noisy estimate of the truth. Calibrating the reliability and bias of expert labellers is a non-trivial problem. In this paper we discuss some of our recent work on this topic in the context of detecting small volcanoes in Magellan SAR images of Venus. Empirical results (using the Expectation-Maximization procedure) suggest that accounting for subjective noise can be quite significant in terms of quantifying both human and algorithm detection performance. 1 Introduction In certain pattern recognition applications, particularly in remote-sensing and medical diagnosis, the standard assumption that the labelling of the data has been * and Division of Biology, California Institute of Technology 1086 Padhraic Smyth. Usama Fayyad. Michael Burl. Pietro Perona. Pierre Baldi carried out in a reasonably objective and reliable manner may not be appropriate. Instead of "ground truth" one may only have the subjective opinion(s) of one or more experts. For example, medical data or image data may be collected off-line and some time later a set of experts analyze the data and produce a set of class labels. The central problem is that of trying to infer the "ground truth" given the noisy subjective estimates of the experts. When one wishes to apply a supervised learning algorithm to the data, the problem is primarily twofold: (i) how to evaluate the relative performance of experts and algorithms, and (ii) how to train a pattern recognition system in the absence of absolute ground truth. In this paper we focus on problem (i), namely the performance evaluation issue, and in particular we discuss the application of a particular modelling technique to the problem of counting volcanoes on the surface of Venus. For problem (ii), in previous work we have shown that when the inferred labels have a probabilistic interpretation, a simple mixture model argument leads to straightforward modifications of various learning algorithms [1]. It should be noted that the issue of inferring ground truth from subjective labels has appeared in the literature under various guises. French [2] provides a Bayesian perspective on the problem of combining multiple opinions. In the field of medical diagnosis there is a significant body of work on latent variable models for inferring hidden "truth" from subjective diagnoses (e.g., see Uebersax [3]). More abstract theoretical models have also been developed under assumptions of specific labelling patterns (e.g., Lugosi [4] and references therein). The contribution of this paper is twofold: (i) this is the first application of latent-variable subjective-rating models to a large-scale image analysis problem as far as we are aware, and (ii) the focus of our work is on the pattern recognition aspect of the problem, i.e., comparing human and algorithmic performance as opposed to simply comparing humans to each other. 2 Background: Automated Detection of Volcanoes in Radar Images of Venus Although modern remote-sensing and sky-telescope technology has made rapid recent advances in terms of data collection capabilities, image analysis often remains a strictly manual process and much investigative work is carried out using hardcopy photographs. The Magellan Venus data set is a typical example: between 1991 and 1994 the Magellan spacecraft transmitted back to earth a data set consisting of over 30,000 high resolution (75m per pixel) synthetic aperture radar (SAR) images of the Venusian surface [5]. This data set is greater than that gathered by all previous planetary missions combined planetary scientists are literally swamped by data. There are estimated to be on the order of 106 small (less than 15km in diameter) vl.sible volcanoes scattered throughout the 30,000 images [6]. It has been estimated that manually locating all of these volcanoes would require on the order of 10 man-years of a planetary geologist's time to carry out our experience has been that even a few hours of image analysis severely taxes the concentration abilities of human labellers. From a scientific viewpoint the ability to accurately locate and characterize the Inferring Ground Truth from Subjective Labelling of Venus Images 1087 many volcanoes is a necessary requirement before more advanced planetary geology studies can be carried out: analysis of spatial clustering patterns, correlation with other geologic features, and so forth. From an engineering viewpoint, automation of the volcano detection task presents a significant challenge to current capabilities in computer vision and pattern recognition due to the variability of the volcanoes and the significant background "clutter" present in most of the images. Figure 1 shows a Magellan subimage of size 30km square containing at least 10 small volcanoes. Volcanoes on Venus Figure 1: A 30km x 30km region from the Magellan SAR data, which contains a number of small volcanoes. The purpose of this paper is not to describe pattern recognition methods for volcano detection but rather to discuss some of the issues involved in collecting and calibrating labelled training data from experts. Details of a volcano detection method using matched filtering, SVD projections and a Gaussian classifier are provided in [7]. 3 Volcano Labelling Training examples are collected by having the planetary geologists examine an image on the computer screen and then using a mouse to indicate where they think the volcanoes (if any) are located. Typically it can take from 15 minutes to 1 hour to label an image (depending on how many volcanoes are present), where each image represents a 75km square patch on the surface of Venus. An image may contain on the order of 100 volcanoes, although a more typical number is between 30 and 40. There can be considerable ambiguity in volcano labelling: for the same image, different scientists can produce different label lists, and even the same scientist can produce different lists over time. To address this problem we introduced the notion of having the scientists label training examples into quantized probability 1088 Padhraic Smyth, Usama Fayyad, Michael Burl, Pierro Perona, Pierre Baldi bins or "types", where the probability bins correspond to visually distinguishable sub-categories of volcanoes. In particular, we have used 5 types: (1) summit pits, bright-dark radar pair, and apparent topographic slope, all clearly visible, probability 0.98, (2) only 2 of the 3 criteria of type 1 are visible, probability 0.80, (3) no summit pit visible, evidence of flanks or circular outline, probability 0.60, (4) only a summit pit visible, probability 0.50, (5) no volcano-like features visible, probability 0.0. These subjective probabilities correspond to the mean probability that a volcano exists at a particular location given that it belongs to a particular type and were elicited after considerable discussions with the planetary geologists. Thus, the observed data for each RDI consists of labels l, which are noisy estimates of true "type" t, which in turn is probabilistically related to the hidden event of interest, v, the presence of a volcano: T p(vlD = LP(vlt)p(tID (1) t=1 where T is the number of types (and labels). The subjective probabilities described above correspond to p(vlt): to be able to infer the probability of a volcano given a set of labels l it remains to estimate the p(tlD terms. 4 Inferring the Label-Type Parameters via the EM Procedure We follow a general model for subjective labelling originally proposed by Dawid and Skene [8] and apply it to the image labelling problem: more details on this overall approach are provided in [9]. Let N be the number of local regions of interest (RDl's) in the database (these are 15 pixel square image patches for the volcano application). For simplicity we consider the case of just a single labeller who labels a given set of regions of interest (RDIs) a number of times the extension to multiple labellers is straightforward assuming conditional independence of the labellings given the true type. Let nil be the number of times that RDI i is labelled with labell. Let lit denote a binary variable which takes value 1 if the true type of volcano i is t* , and is 0 otherwise. We assume that labels are assigned independently to a given RDI from one labelling to the next, given that the type is known. If the true type t* is known then T p(observed labelslt, i) ex: II p(lltti!· 1=1 Thus, unconditionally, we have T (T )Yit p(observed labels, tli) ex: g pet) gP(llttil , (2) (3) where lit = 1 if t = t* and 0 otherwise. Assuming that each RDI is labelled independently of the others (no spatial correlation in terms of labels), N T (T )Yit p(observed labels, t;) ex: ~ g pet) gp(lltt'l (4) Inferring Ground Truth from Subjective Labelling of Venus Images 1089 Still assuming that the types t for each ROI are known (the Yit)' the maximum likelihood estimators of p(llt) and p(t) are and p(llt) = ~i Yitnil EI Ei Yitnil (5) (6) From Bayes' rule one can then show that 1 T p(Yit = llobserved data) = C IIp(llt)nilp(t) I (7) where C is a normalization constant. Thus, given the observed data nil and the parameters p(llt) and p(t), one can infer the posterior probabilities of each type via Equation 7. However, without knowing the Yit values we can not infer the parameters p(llt) and p(t). One can treat the Yit as hidden and thus apply the well-known ExpectationMaximization (EM) procedure to find a local maximum of the likelihood function: 1. Obtain some initial estimates of the expected values of Yit, e.g., E[Yitl = ~ (8) Elnil 2. M-step: choose the values of p(llt) and p(t) which maximize the likelihood function (according to Equations 5 and 6), using E[Yitl in place of Yit. 3. E-step: calculate the conditional expectation of Yit, E[Yitldatal = p(Yit = Ildata) (Equation 7). 4. Repeat Steps 2 and 3 until convergence. 5 Experimental Results 5.1 Combining Multiple Expert Opinions Labellings from 4 geologists on the 4 images resulted in 269 possible volcanoes (ROIs) being identified. Application of the EM procedure resulted in label-type probability matrices as shown in Table 1 for Labeller C. The diagonal elements provide an indication of the reliability of the labeller. There is significant miscalibration for label 3's: according to the model, a label 3 from Labeller C is most likely to correspond to type 2. The label-type matrices of all 4 labellers (not shown) indicated that the model placed more weight on the conservative labellers (C and D) than the aggressive ones (A and B). The determination of posterior probabilities for each of the ROIs is a fundamental step in any quantitative analysis of the volcano data: p(vlD = E;=1 p(vlt)p(tID where the p(tlD terms are the posterior probabilities of type given labels provided 1090 Padhraic Smyth, Usama Fayyad, Michael Burl, Pietro Perona, Pierre Baldi Table 1: Type-Label Probabilities for Individual Labellers as estimated via the EM Procedure Probability(typellabel), Labeller C Type 1 Type 2 Type 3 Type 4 Type 5 Labell 1.000 0.000 0.000 0.000 0.000 Label 2 0.019 0.977 0.004 0.000 0.000 Label 3 0.000 0.667 0.175 0.065 0.094 Label 4 0.000 0.000 0.042 0.725 0.233 Label 5 0.000 0.000 0.389 0.000 0.611 Table 2: 10 ROIs from the database: original scientist labels shown with posterior probabilities estimated via the EM procedure Scientist Labels JD Posterior Probabilities (EM), p(tlD ROI A B C D Type 1 Type 2 Type 3 Type 4 Type 5 p(vlO 1 4 4 4 5 0.000 0.000 0.000 0.816 0.184 0.408 2 1 4 4 2 0.000 0.000 0.000 0.991 0.009 0.496 3 1 1 2 2 0.023 0.977 0.000 0.000 0.000 0.804 4 3 1 5 3 0.000 0.000 1.000 0.000 0.000 0.600 5 3 1 3 3 0.000 0.536 0.452 0.012 0.000 0.706 6 2 2 2 4 0.000 1.000 0.000 0.000 0.000 0.800 7 3 1 5 5 0.000 0.000 1.000 0.000 0.000 0.600 8 2 1 4 4 0.000 0.000 0.000 0.999 0.000 0.500 9 3 2 5 3 0.000 0.000 0.992 0.000 0.008 0.595 10 4 4 4 4 0.000 0.000 0.000 0.996 0.004 0.498 by the EM procedure, and the p(vlt) terms are the subjective volcano-type probabilities discussed in Section 3.2. As shown in Table 2, posterior probabilities for the volcanoes generally are in agreement with intuition and often correspond to taking the majority vote or the "average" of the C and D labels (the conservative labellers). However some p(vlD estimates could not easily be derived by any simple averaging or voting scheme, e.g., see ROIs 3, 5 and 7 in the table. 5.2 Experiment on Comparing Human and Algorithm Performance The standard receiver operating characteristic (ROC) plots detections versus false alarms [10]. The ROCs shown here differ in two significant ways [11]: (1) the false alarm axis is normalized relative to the number of true positives (necessary since the total number of possible false alarms is not well defined for object detection in images), and (2) the reference labels used in scoring are probabilistic: a detection "scores" p( v) on the detection axis and 1 - p( v) on the false alarm axis. Inferring Ground Truth from Subjective Labelling of Venus Images 1091 100 Mean Detection Rate [%) 80 60 40 20 -SVD Algorithm -t( Scientist A -0 Scientist B -- Scientist C .+ ScientistD 0~ __ ~ __ ~~======7 o 20 40 60 80 Mean False Alarm Rate (expressed as % of total number of volcanoes) 100 Mean Detection Rate [%) 80 60 40 -SVD Algorithm -t( Scientist A -0 Scientist B -- Scientist C .+ ScientistD 20 40 60 80 Mean False Alarm Rate (expressed as % of total number of volcanoes) Figure 2: Modified ROCs for both scientists and algorithms: (a) without the labelling or type uncertainty, (b) with full uncertainty model factored in. As before, data came from 4 images, and there were 269 labelled local regions. The SVD-Gaussian algorithm was evaluated in cross-validation mode (train on 3 images, test on the 4th) and the results combined. The first ROC (Figure 2(a)) does not take into account either label-type or type-volcano probabilities, i.e., the reference list (for algorithm training and overall evaluation) is a consensus list (2 scientists working together) where labels 1 ,2,3,4 are ignored and all labelled items are counted equally as volcanoes. The individual labellers and algorithm are then scored in the standard "non-weighted" ROC fashion. This curve is optimistic in terms of depicting the accuracy of the detectors since it ignores the underlying probabilistic nature of the labels. Even with this optimistic curve, volcano labelling is relatively inaccurate by either man or machine. Figure 2(b) shows a weighted ROC: for each of 4 scientists the probabilistic "reference labels" were derived via the EM procedure as in Table 2 from the other 3 scientists, and the detections of each scientist were scored according to each such reference set. Performance of the algorithm (the SVD-Gaussian method) was evaluated relative to the EM-derived label estimates of all 4 scientists. Accounting for all of the uncertainty in the data results in a more realistic, if less flattering, set of performance characteristics. The algorithm's performance degrades more than the scientist's performance (for low false alarms rates compared to Figure 2(a)) when the full noise model is used. The algorithm is estimating the posterior probabilities of volcanoes rather poorly and the complete uncertainty model is more sensitive to this fact. This is a function of the SVD feature space rather than the Gaussian classification model. 1092 Padhraic Smyth, Usama Fayyad, Michael Burl, Pietro Perona, Pierre Baldi 6 Conclusion Ignoring subjective uncertainty in image labelling can lead to significant overconfidence in terms of performance estimation (for both humans and machines). For the volcano detection task a simple model for uncertainty in the class labels provided insight into the performance of both human and algorithmic detectors. An obvious extension of the maximum likelihood framework outlined here is a Bayesian approach [12]: accounting for parameter uncertainty in the model given the limited amount of training data available is worth investigating. Acknowledgements The research described in this paper was carried out by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration and was supported in part by ARPA under grant number NOOOl4-92-J-1860. References 1. P. Smyth, "Learning with probabilistic supervision," in Computational Learning Theory and Natural Learning Systems 3, T. Petcshe, M. Kearns, S. Hanson, R. Rivest (eds), Cambridge, MA: MIT Press, to appear. 2. S. French, "Group consensus probability distributions: a critical survey," in Bayesian Statistics 2, J. M. Bernardo, M. H. DeGroot, D. V. Lindley, A. F. M. Smith (eds.), Elsevier Science Publishers, North-Holland, pp.183-202, 1985. 3. J . S. Uebersax, "Statistical modeling of expert ratings on medical treatment appropriateness," J. Amer. Statist. Assoc., vol.88, no.422, pp.421-427, 1993. 4. G. Lugosi, "Learning with an unreliable teacher," Pattern Recognition, vol. 25, no.1, pp.79-87. 1992. 5. Science, special issue on Magellan data, April 12, 1991. 6. J. C. Aubele and E. N. Slyuta, "Small domes on Venus: characteristics and origins," in Earth, Moon and Planets, 50/51, 493-532, 1990. 7. M. C. Burl, U. M. Fayyad, P. Perona, P. Smyth, and M. P. Burl, "Automating the hunt for volcanoes on Venus," in Proceedings of the 1994 Computer Vision and Pattern Recognition Conference: CVPR-94, Los Alamitos, CA: IEEE Computer Society Press, pp.302-309, 1994. 8. A. P. Dawid and A. M. Skene, "Maximum likelihood estimation of observer error-rates using the EM algorithm," Applied Statistics, vol.28, no.1, pp.2G-28, 1979. 9. P. Smyth, M. C. Burl, U. M. Fayyad, P. Perona, 'Knowledge discovery in large image databases: dealing with uncertainties in ground truth,' in Knowledge Discovery in Databases 2, U. M. Fayyad, G. Piatetsky-Shapiro, P. Smyth, R. Uthurasamy (eds.), AAAI/MIT Press, to appear, 1995. 10. M. S. Chesters, "Human visual perception and ROC methodology in medical imaging," Phys. Med. BioI., vol.37, no.7, pp.1433-1476, 1992. 11. M. C. Burl, U. M. Fayyad, P. Perona, P. Smyth, "Automated analysis of radar imagery of Venus: handling lack of ground truth," in Proceedings of the IEEE Conference on Image Processing, Austin, November 1994. 12. W . Buntine, "Operations for learning with graphical models," Journal of Artificial Intelligence Research, 2, pp.159-225, 1994.	1994	35
927	Learning Prototype Models for Tangent Distance Trevor Hastie· Statistics Department Sequoia Hall Stanford University Stanford, CA 94305 email: trevor@playfair .stanford .edu Patrice Simard AT&T Bell Laboratories Crawfords Corner Road Holmdel, NJ 07733 email: patrice@neural.att.com Eduard Siickinger AT &T Bell Laboratories Crawfords Corner Road Holmdel, NJ 07733 email: edi@neural.att.com Abstract Simard, LeCun & Denker (1993) showed that the performance of nearest-neighbor classification schemes for handwritten character recognition can be improved by incorporating invariance to specific transformations in the underlying distance metric the so called tangent distance. The resulting classifier, however, can be prohibitively slow and memory intensive due to the large amount of prototypes that need to be stored and used in the distance comparisons. In this paper we develop rich models for representing large subsets of the prototypes. These models are either used singly per class, or as basic building blocks in conjunction with the K-means clustering algorithm. *This work was performed while Trevor Hastie was a member of the Statistics and Data Analysis Research Group, AT&T Bell Laboratories, Murray Hill, NJ 07974. J 000 Trevor Hastie, Patrice Simard, Eduard Siickinger 1 INTRODUCTION Local algorithms such as K-nearest neighbor (NN) perform well in pattern recognition, even though they often assume the simplest distance on the pattern space. It has recently been shown (Simard et al. 1993) that the performance can be further improved by incorporating invariance to specific transformations in the underlying distance metric the so called tangent distance. The resulting classifier, however, can be prohibitively slow and memory intensive due to the large amount of prototypes that need to be stored and used in the distance comparisons. In this paper we address this problem for the tangent distance algorithm, by developing rich models for representing large subsets of the prototypes. Our leading example of prototype model is a low-dimensional (12) hyperplane defined by a point and a set of basis or tangent vectors. The components of these models are learned from the training set, chosen to minimize the average tangent distance from a subset of the training images as such they are similar in flavor to the Singular Value Decomposition (SVD), which finds closest hyperplanes in Euclidean distance. These models are either used singly per class, or as basic building blocks in conjunction with K-means and LVQ. Our results show that not only are the models effective, but they also have meaningful interpretations. In handwritten character recognition, for instance, the main tangent vector learned for the the digit "2" corresponds to addition/removal of the loop at the bottom left corner of the digit; for the 9 the fatness of the circle. We can therefore think of some of these learned tangent vectors as representing additional invariances derived from the training digits themselves. Each learned prototype model therefore represents very compactly a large number of prototypes of the training set. 2 OVERVIEW OF TANGENT DISTANCE When we look at handwritten characters, we are easily able to allow for simple transformations such as rotations, small scalings, location shifts, and character thickness w hen identifying the character. Any reasonable automatic scheme should similarly be insensitive to such changes. Simard et al. (1993) finessed this problem by generating a parametrized 7dimensional manifold for each image, where each parameter accounts for one such invariance. Consider a single invariance dimension: rotation. If we were to rotate the image by an angle B prior to digitization, we would see roughly the same picture, just slightly rotated. Our images are 16 x 16 grey-scale pixelmaps, which can be thought of as points in a 256-dimensional Euclidean space. The rotation operation traces out a smooth one-dimensional curve Xi(B) with Xi(O) = Xi, the image itself. Instead of measuring the distance between two images as D(Xi,Xj) = IIXi - Xjll (for any norm 11·11), the idea is to use instead the rotation-invariant DI (Xi, Xj) = minoi,oj IIX i(B;) - Xj(Bj )11. Simard et al. (1993) used 7 dimensions of invariance, accounting fo:: horizontal and vertical location and scale, rotation and shear and character thickness. Computing the manifold exactly is impossible, given a digitized image, and would be impractical anyway. They approximated the manifold instead by its tangent Learning Prototype Models for Tangent Distance 1001 plane at the image itself, leading to the tangent model Xi(B) = Xi + TiB, and the tangent distance DT(Xi,Xj) = minoi,oj IIXi(Bd -Xj(Bj)ll. Here we use B for the 7-dimensional parameter, and for convenience drop the tilde. The approximation is valid locally, and thus permits local transformations. Non-local transformations are not interesting anyway (we don't want to flip 6s into 9s; shrink all digits down to nothing.) See Sackinger (1992) for further details. If 11·11 is the Euclidean norm, computing the tangent distance is a simple least-squares problem, with solution the square-root of the residual sum-of-squares of the residuals in the regression with response Xi - Xj and predictors (-Ti : Tj ). Simard et al. (1993) used DT to drive a 1-NN classification rule, and achieved the best rates so far-2.6%-on the official test set (2007 examples) of the USPS data base. Unfortunately, 1-NN is expensive, especially when the distance function is non-trivial to compute; for each new image classified, one has to compute the tangent distance to each of the training images, and then classify as the class of the closest. Our goal in this paper is to reduce the training set dramatically to a small set of prototype models; classification is then performed by finding the closest prototype. 3 PROTOTYPE MODELS In this section we explore some ideas for generalizing the concept of a mean or centroid for a set of images, taking into account the tangent families. Such a centroid model can be used on its own, or else as a building block in a K-means or LVQ algorithm at a higher level. We will interchangeably refer to the images as points (in 256 space). The centroid of a set of N points in d dimensions minimizes the average squared norm from the points: (1) 3.1 TANGENT CENTROID One could generalize this definition and ask for the point M that minimizes the average squared tangent distance: N MT = argm,Jn LDT(Xi,M)2 (2) i=l This appears to be a difficult optimization problem, since computation of tangent distance requires not only the image M but also its tangent basis TM. Thus the criterion to be minimized is 1002 Trevor Hastie, Patrice Simard, Eduard Sackinger where T(M) produces the tangent basis from M. All but the location tangent vectors are nonlinear functionals of M, and even without this nonlinearity, the problem to be solved is a difficult inverse functional. Fortunately a simple iterative procedure is available where we iteratively average the closest points (in tangent distance) to the current guess. Tangent Centroid Algorithm Initialize: Set M = ~ 2:~1 Xi, let TM = T(M) be the derived set of tangent vectors, and D = 2:i DT(Xi' M). Denote the current tangent centroid (tangent family) by M(-y) = M +TM"I. Iterate: 1. For each i find a 1'. and 8i that solves 11M + TM"I - Xi(8)11 = min'Y.9 1 N A 2. Set M +- N 2:'=1 (Xi(8i) - TMi'i) and compute the new tangent subspace TM = T(M). 3. Compute D = 2:iDT(Xi,M) Until: D converges. Note that the first step in Iterate is available from the computations in the third step. The algorithm divides the parameters into two sets: M in the one, and then TM, "Ii and 8, for each i in the other. It alternates between the two sets, although the computation of TM given M is not the solution of an optimization problem. It seems very hard to say anything precise about the convergence or behavior of this algorithm, since the tangent vectors depend on each iterate in a nonlinear way. Our experience has always been that it converges fairly rapidly « 6 iterations). A potential drawback of this algorithm is that the TM are not learned, but are implicit in M. 3.2 TANGENT SUBSPACE Rather than define the model as a point and have it generate its own tangent subspace, we can include the subspace as part of the parametrization: M(-y) = M + V"I. Then we define this tangent subspace model as the minimizer of N MS(M, V) = L min 11M + V"Ii - Xi(8d1l 2 . 1 'Yi.9i t= (3) over M and V. Note that V can have an arbitrary number 0 ::; r ::; 256 of columns, although it does not make sense for r to be too large. An iterative algorithm similar to the tangent centroid algorithm is available, which hinges on the SVD decomposition for fitting affine subspaces to a set of points. We briefly review the SVD in this context. Let X be the N x 256 matrix with rows the vectors Xi - X where X = ~ 2:~1 Xi. Then SVD(X) = UDVT is a unique decomposition with UNxR and V256xR the Learning Prototype Models for Tangent Distance 1003 orthonormal left and right matrices of singular vectors, and R = rank( X). D Rx R is a diagonal matrix of decreasing positive singular values. A pertinent property of the SVD is: Consider finding the closest affine, rank-r subspace to a set of points, or N 2 min 2: IIXi - M - v(r)'hll M,v(r),{9i} i=1 where v(r) is 256 x r orthonormal. The solution is given by the SVD above, with M = X and v(r) the first r columns of V, and the total squared distance E;=1 D;j. The V( r) are also the largest r principal components or eigenvectors of the covariance matrix of the Xi. They give in sequence directions of maximum spread, and for a given digit class can be thought of as class specific invariances. We now present our Tangent subspace algorithm for solving (3); for convenience we assume V is rank r for some chosen r, and drop the superscript. Tangent subspace algorithm Initialize: Set M = ~ Ef:l Xi and let V correspond to the first r right singular vectors of X. Set D = E;=1 D;j, and let the current tangent subspace model be M(-y) = M + V-y. Iterate: 1. For each i find that (ji which solves IIM(-y) - Xi (8)11 = min N A 2. Set M +- ~ Ei=1 (Xi (8i )) and replace the rows of X by Xi({jd - M. Compute the SVD of X, and replace V by the first r right singular vectors. 3. Compute D = E;=l D;j Until: D converges. The algorithm alternates between i) finding the closest point in the tangent subspace for each image to the current tangent subspace model, and ii) computing the SVD for these closest points. Each step of the alternation decreases the criterion, which is positive and hence converges to a stationary point of the criterion. In all our examples we found that 12 complete iterations were sufficient to achieve a relative convergence ratio of 0.001. One advantage of this approach is that we need not restrict ourselves to a sevendimensional V indeed, we have found 12 dimensions has produced the best results. The basis vectors found for each class are interesting to view as images. Figure 1 shows some examples of the basis vectors found, and what kinds of invariances in the images they account for. These are digit specific features; for example, a prominent basis vector for the family of 2s accounts for big versus small loops. 1004 Trevor Hastie, Patrice Simard, Eduard Siickinger Each of the examples shown accounts for a similar digit specific invariance. None of these changes are accounted for by the 7-dimensional tangent models, which were chosen to be digit nonspecific. Figure 1: Each column corresponds to a particular tangent subspace basis vector for the given digit. The top image is the basis vector itself, and the remaining 3 images correspond to the 0.1, 0.5 and 0.9 quantiles for the projection indices for the training data for that basis vector, showing a range of image models for that basis, keeping all the others at o. 4 SUBSPACE MODELS AND K-MEANS CLUSTERING A natural and obvious extension of these single prototype-per-class models, is to use them as centroid modules in a K-means algorithm. The extension is obvious, and space permits only a rough description. Given an initial partition of the images in a class into K sets: 1. Fit a separate prototype model to each of the subsets; 2. Redefine the partition based on closest tangent distance to the prototypes found in step 1. In a similar way the tangent centroid or subspace models can be used to seed LVQ algorithms (Kohonen 1989), but so far we have not much experience with them. 5 RESULTS Table 1 summarizes the results for some of these models. The first two lines correspond to a SVD model for the images fit by ordinary least squares rather than least tangent squares. The first line classifies using Euclidean distance to this model, the second using tangent distance. Line 3 fits a single 12-dimensional tangent subspace model per class, while lines 4 and 5 use 12-dimensional tangent subspaces as cluster Learning Prototype Models for Tangent Distance 1005 Table 1: Test errors for a variety of situations. In all cases the training data were 7291 USPS handwritten digits, and the test data the "official" 2007 USPS test digits. Each entry describes the model used in each class, so for example in row 5 there are 5 models per class, hence 50 in all. Prototype Metric # Prototypes7Class Error Rate 0 1-NN Euclidean ~ 700 0.053 1 12 dim SVD subspace Euclidean 1 0.055 2 12 dim SVD subspace Tangent 1 0.045 3 12 dim Tangent subspace Tangent 1 0.041 4 12 dim Tangent subspace Tangent 3 0.038 5 12 dim Tangent subspace Tangent 5 0.038 6 Tangent centroid Tangent 20 0.038 7 (4) U (6) Tangent 23 0.034 8 1-NN Tangent ~ 700 0.026 centers within each class. We tried other dimensions in a variety of settings, but 12 seemed to be generally the best. Line 6 corresponds to the tangent centroid model used as the centroid in a 20-means cluster model per class; the performance compares with with K=3 for the subspace model. Line 7 combines 4 and 6, and reduces the error even further. These limited experiments suggest that the tangent subspace model is preferable, since it is more compact and the algorithm for fitting it is on firmer theoretical grounds. Figure 4 shows some of the misclassified examples in the test set. Despite all the matching, it seems that Euclidean distance still fails us in the end in some of these cases. 6 DISCUSSION Gold, Mjolsness & Rangarajan (1994) independently had the idea of using "domain specific" distance measures to seed K-means clustering algorithms. Their setting was slightly different from ours, and they did not use subspace models. The idea of classifying points to the closest subspace is found in the work of Oja (1989), but of course not in the context of tangent distance. We are using Euclidean distance in conjunction with tangent distance. Since neighboring pixels are correlated, one might expect that a metric that accounted for the correlation might do better. We tried several variants using Mahalanobis metrics in different ways, but with no success. We also tried to incorporate information about where the images project in the tangent subspace models into the classification rule. We thus computed two distances: 1) tangent distance to the subspace, and 2) Mahalanobis distance within the subspace to the centroid for the subspace. Again the best performance was attained by ignoring the latter distance. In conclusion, learning tangent centroid and subspace models is an effective way 1006 Trevor Hastie, Patrice Simard, Eduard Siickinger true: 6 true: 2 true: 5 true: 2 true: 9 true: 4 pred. pro). ( 0 ) prado proj. ( 0 ) pred. pro). ( 8 ) pred. proj. ( 0 ) prado proj. ( 4 ) prado pro). ( 7 ) Figure 2: Some of the errorS for the test set corresponding to line (3) of table 4. Each case is displayed as a column of three images. The top is the true image, the middle the tangent projection of the true image onto the subspace model of its class, the bottom image the tangent projection of the image onto the winning class. The models are sufficiently rich to allow distortions that can fool Euclidean distance. to reduce the number of prototypes (and thus the cost in speed and memory) at a slight expense in the performance. In the extreme case, as little as one 12 dimensional tangent subspace per class and the tangent distance is enough to outperform classification using ~ 700 prototypes per class and the Euclidean distance (4.1 % versus 5.3% on the test data). References Gold, S., Mjolsness, E. & Rangarajan, A. (1994), Clustering with a domain specific distance measure, in 'Advances in Neural Information Processing Systems', Morgan Kaufman, San Mateo, CA. Kohonen, T. (1989), Self-Organization and Associative Memory (3rd edition), Springer-Verlag, Berlin. Oja, E. (1989), 'Neural networks, principal components, and subspaces', International Journal Of Neural Systems 1(1), 61-68. Sackinger, E. (1992), Recurrent networks for elastic matching in pattern recognition, Technical report, AT&T Bell Laboratories. Simard, P. Y, LeCun, Y. & Denker, J. (1993), Efficient pattern recognition using a new transformation distance, in 'Advances in Neural Information Processing Systems', Morgan Kaufman, San Mateo, CA, pp. 50-58.	1994	36
928	Diffusion of Credit in Markovian Models Yoshua Bengio· Dept. I.R.O., Universite de Montreal, Montreal, Qc, Canada H3C-3J7 bengioyCIRO.UMontreal.CA Paolo Frasconi Dipartimento di Sistemi e Informatica Universita di Firenze, Italy paoloCmcculloch.ing.unifi.it Abstract This paper studies the problem of diffusion in Markovian models, such as hidden Markov models (HMMs) and how it makes very difficult the task of learning of long-term dependencies in sequences. Using results from Markov chain theory, we show that the problem of diffusion is reduced if the transition probabilities approach 0 or 1. Under this condition, standard HMMs have very limited modeling capabilities, but input/output HMMs can still perform interesting computations. 1 Introduction This paper presents an important new element in our research on the problem of learning long-term dependencies in sequences. In our previous work [4J we found theoretical reasons for the difficulty in training recurrent networks (or more generally parametric non-linear dynamical systems) to learn long-term dependencies. The main result stated that either long-term storing or gradient propagation would be harmed, depending on whether the norm of the Jacobian of the state to state function was greater or less than 1. In this paper we consider a special case in which the norm of the Jacobian of the state to state function is constrained to be exactly 1 because this matrix is a stochastic matrix. We consider both homogeneous and non-homogeneous Markovian models. Let n be the number of states and At be the transition matrices (constant in the homogeneous case): Aij(ut} = P(qt = j I qt-l = i, Ut; e) where Ut is an external input (constant in the homogeneous case) and e is a vector of parameters. In the homogeneous case (e.g., standard HMMs), such models can learn the distribution of output sequences by associating an output distribution to each state. In ·also, AT&T Bell Labs, Holmdel, NJ 07733 554 Yoshua Bengio. Paolo Frasconi the non-homogeneous case, transition and output distributions are conditional on the input sequences, allowing to model relationships between input and output sequences (e.g. to do sequence 'regression or classification as with recurrent networks). We thus called Input/Output HMM (IOHMM) this kind of non-homogeneous HMM . In [3, 2] we proposed a connectionist implementation of IOHMMs. In both cases, training requires propagating forward probabilities and backward probabilities, taking products with the transition probability matrix or its transpose. This paper studies in which conditions these products of matrices might gradually converge to lower rank, thus harming storage and learning of long-term context. However, we find in this paper that IOHMMs can better deal with this problem than homogeneous HMMs. 2 Mathematical Preliminaries 2.1 Definitions A matrix A is said to be non-negative, written A 2:: 0, if Aij 2:: 0 Vi, j . Positive matrices are defined similarly. A non-negative square matrix A E R nxn is called row stochastic (or simply stochastic in this paper) if 'L,'l=1 Aij = 1 Vi = 1 . . . n. A non-negative matrix is said to be row [column} allowable if every row [column] sum is positive. An allowable matrix is both row and column allowable. A nonnegative matrix can be associated to the directed transition graph 9 that constrains the Markov chain. An incidence matrix A corresponding to a given non-negative matrix A replaces all positive entries of A by 1. The incidence matrix of A is a connectivity matrix corresponding to the graph 9 (assumed to be connected here). Some algebraic properties of A are described in terms of the topology of g. Definition 1 (Irreducible Matrix) A non-negative n x n matrix A is said to be irreducible if for every pair i,j of indices, :3 m = m(i,j) positive integer s.t. (Amhj > O. A matrix A is irreducible if and only if the associated graph is strongly connected (i.e., there exists a path between any pair of states i,j) . If :3k s.t. (Ak)ii > 0, d(i) is called the period of index i ifit is the greatest common divisor (g.c.d.) of those k for which (Ak)ii > O. In an irreducible matrix all the indices have the same period d, which is called the period of the matrix. The period of a matrix is the g.c.d. of the lengths of all cycles in the associated transition graph. Definition 2 (Primitive matrix) A non-negative matrix A is said to be primitive if there exists a positive integer k S.t. Ak > O. An irreducible matrix is either periodic or primitive (i.e. of period 1). A primitive stochastic matrix is necessarily allowable. 2.2 The Perron-Frobenius Theorem Theorem 1 (See [6], Theorem 1.1.) Suppose A is an n x n non-negative primitive matrix. Then there exists an eigenvalue r such that: 1. r is real and positive; 2. with r can be associated strictly positive left and right eigenvectors; 3. r> 1>'1 for any eigenvalue>. 1= r; 4· the eigenvectors associated with r are unique to constant multiples. 5. If 0 S B s A and f3 is an eigenvalue of B, then 1f31 s r . Moreover, 1f31 = r implies B = A. Diffusion of Credit in Markovian Models 555 6. r is simple root of the characteristic equation of A. A simple consequence of the theorem for stochastic matrices is the following: Corollary 1 Suppose A is a primitive stochastic matrix. Then its largest eigehvalue is 1 and there is only one corresponding right eigenvector, which is 1 = [1, 1 .. ·1]'. Furthermore, all other eigenvalues < 1. Proof. A1 = 1 by definition of stochastic matrices. This eigenvector is unique and all other eigenvalues < 1 by the Perron-Frobenius Theorem. If A is stochastic but periodic with period d, then A has d eigenvalues of module 1 which are the d complex roots of 1. 3 Learning Long-Term Dependencies with HMMs In this section we analyze the case of a primitive transition matrix as well as the general case with a canonical re-ordering of the matrix indices. We discuss how ergodicity coefficients can be used to measure the difficulty in learning long-term dependencies. Finally, we find that in order to avoid all diffusion, the transitions should be deterministic (0 or 1 probability). 3.1 Training Standard HMMs Theorem 2 (See [6], Theorem 4.2.) If A is a primitive stochastic matrix, then as t -+ 00, At -+ 1V' where v' is the unique stationary distribution of the Markov chain. The rate of approach is geometric. Thus if A is primitive, then liIDt-+oo At converges to a matrix whose eigenvalues are all 0 except for ,\ = 1 (with eigenvector 1), i.e. the rank of this product converges to 1, i.e. its rows are equal. A consequence oftheorem 2 is that it is very difficult to train ordinary hidden Markov models, with a primitive transition matrix, to model long-term dependencies in observed sequences. The reason is that the distribution over the states at time t > to becomes gradually independent of the distribution over the states at time to as t increases. It means that states at time to become equally responsible for increasing the likelihood of an output at time t. This corresponds in the backward phase of the EM algorithm for trainin~ HMMs to a diffusion of credit over all the states. In practice we train HMMs WIth finite sequences. However, training will become more and more numerically ill-conditioned as one considers longer term dependencies. Consider two events eo (occurring at to) and et (occurring at t), and suppose there are also "interesting" events occurring in between. Let us consider the overall influence of states at times 1" < t upon the likelihood of the outputs at time t. Because of the phenomenon of diffusion of credit, and because gradients are added together, the influence of intervening events (especially those occurring shortly before t) will be much stronger than the influence of eo . Furthermore, this problem gets geometrically worse as t increases. Clearly a positive matrix is primitive. Thus in order to learn long-term dependencies, we would like to have many zeros in the matrix of transition probabilities. Unfortunately, this generally supposes prior knowledge of an appropriate connectivity graph. 3.2 Coefficients of ergodicity To study products of non-negative matrices and the loss of information about initial state in Markov chains (particularly in the non-homogeneous case), we introduce the projective distance between vectors x and y: x·y· d(x',y') = ~~ In(--.:..l.). I ,) XjYi Clearly, some contraction takes place when d(x'A,y'A) ::; d(x',y'). 556 Yoshua Bengio, Paolo Frasconi Definition 3 BirkhofJ's contraction coefficient TB(A), for a non-negative columnallowable matrix A, is defined in terms of the projective distance: d(x' A, y' A) TB(A) = sup d(x', y') x ,y> Ojx;t>.y Dobrushin's coefficient Tl(A), for a stochastic matrix A, is defined as follows: 1 Tl(A) = 2 s~p L laik - ajkl· I,) k Both are proper ergodicity coefficients: 0 ~ T(A) ~ 1 and T(A) = 0 if and only if A has identical rows. Furthermore, T(AIA2) ~ T(Al)T(A2)(see [6]). 3.3 Products of Stochastic Matrices Let A (1 ,t) = A 1A2··· At- 1At denote a forward product of stochastic matrices AI, A2, ... At. From the properties of TB and Tl, if T(At} < 1, t > 0 then limt-l-oo T(A(l,t») = 0, i.e. A(l,t) has rank 1 and identical rows. Weak ergodicity is then defined in terms of a proper ergodic coefficient T such as TB and Tl: Definition 4 (Weak Ergodicity) The products of stochastic matrices A(p,r) are weakly ergodic if and only if for all to ~ 0 as t -+ 00, T(A(to,t») -+ O. Theorem 3 (See [6], Lemma 3.3 and 3.4.) Let A(l,t) a forward product of non-negative and allowable matrices, then the products A(l,t) are weakly ergodic if and only if the following conditions both hold: 1. 3to S.t. A(to,t) > 0 Vt > to A(to,t) 2. A (;~,t) -+ Wij (t) > 0 as t -+ 00, i. e. rows of A (to,t) tend to proportionality. ),k For stochastic matrices, row-proportionality is equivalent to row-equality since rows sum to 1. limt-l-oo ACto,t) does not need to exist in order to have weak ergodicity. 3.4 Canonical Decomposition and Periodic Graphs Any non-negative matrix A can be rewritten by relabeling its indices in the following canonical decomposition [6], with diagonal blocks Bi , Ci and Q: ( Bl 0 0 ... 0 ) 0 B2 0 " . 0 ..... ...... . . . . . . . . A= 0 C'+ 1 0 0 (1 ) . . . . . . . . . . .... . .. 0 0 Cr 0 Ll L2 Lr Q where Bi and Ci are irreducible, Bi are primitive and Ci are periodic. Define the corresponding sets of states as SBi' Se" Sq. Q might be reducible, but the groups of states in Sq leak into the B or C blocks, i.e., Sq represents the transient part of the state space. This decomposition is illustrated in Figure 1a. For homogeneous and non-homogeneous Markov models (with constant incidence matrix At = Ao), because P(qt E Sqlqt-l E Sq) < 1, liIl1t-l-oo P(qt E Sqlqo E Sq) = O. Furthermore, because the Bi are primitive, we can apply Theorem 1, and starting from a state in SB" all information about an initial state at to is gradually lost. Diffusion of Credit in Markovian Models 557 (b) Figure 1: (a): Transition graph corresponding to the canonical decomposition. (b): Periodic graph 91 becomes primitive (period 1) 92 when adding loop with states 4,5. A more difficult case is the one of (A(to ,t))jk with initial state j ESc, . Let di be the period of the ith periodic block Cj. It can be shown r6] that taking d products of periodic matrices with the same incidence matrix and period d yields a blockdiagonal matrix whose d blocks are primitive. Thus C(to,t) retains information about the initial block in which qt was. However, for every such block of size > 1, information will be gradually lost about the exact identity of the state within that block. This is best demonstrated through a simple example. Consider the incidence matrix represented by the graph 91 of Figure lb. It has period 3 and the only non-deterministic transition is from state 1, which can yield into either one of two loops. When many stochastic matrices with this graph are multiplied together, information about the loop in which the initial state was is gradually lost (i.e. if the initial state was 2 or 3, this information is gradually lost). What is retained is the phase information, i.e. in which block ({O}, {I}, or {2,3}) of a cyclic chain was the initial state. This suggests that it will be easy to learn about the type of outputs associated to each block of a cyclic chain, but it will be hard to learn anything else. Suppose now that the sequences to be modeled are slightly more complicated, requiring an extra loop of period 4 instead of 3, as in Figure lb. In that case A is primitive: all information about the initial state will be gradually lost. 3.5 Learning Long-Term Dependencies: a Discrete Problem? We might wonder if, starting from a positive stochastic matrix, the learning algorithm could learn the topology, i.e. replace some transition probabilities by zeroes. Let us consider the update rule for transition probabilities in the EM algorithm: A oL A ij 8A;j (2) ij ~ " oL . wj Aij oA.j Starting from Aij > 0 we could obtain a new Aij = 0 only if O~~j = 0, i.e. on a local maximum of the likelihood L. Thus the EM training algorithm will not exactly obtain zero probabilities. Transition probabilities might however approach O. It is also interesting to ask in which conditions we are guaranteed that there will not be any diffusion (of influence in the forward phase, and credit in the backward phase of training). It requires that some of the eigenvalues other than Al = 1 have a norm that is also 1. This can be achieved with periodic matrices C (of period 558 5 Periodic_ -: -"~::::-~:-~,~~:.~~:-~:-~~:~--.-.-.~~~~~"'-"" Left-to-right-10 -15 ~ --20 -25 -30 / Full connected '-. '-, .. '/, .. , .. , Left-to-right (triangular) 5 10 15 20 25 30 T (a) Yoshua Bengio, PaoLo Frasconi %·.';:"'·.::1: .. ·· .. ·:.· .. · d·.:·· •.•.. ·.·.:.'.·· .. :·.·: .•... . . ~:;:~. l?:t :l:~; """ ill) .. : '''''I II . ".,'. I I t=3 t=4 (b) Figure 2: (a) Convergence of Dobrushin's coefficient (see Definition 3. (b) Evolution of products A(l,t) for fully connected graph. Matrix elements are visualized with gray levels. d), which have d eigenvalues that are the d roots of 1 on the complex unit circle. To avoid any loss of information also requires that Cd = I be the identity, since any diagonal block of Cd with size more than 1 will yield to a loss of information (because of diffusion in primitive matrices). This can be generalized to reducible matrices whose canonical form is composed of periodic blocks Ci with ct = I. The condition we are describing actually corresponds to a matrix with only 1 's and O's_ If At is fixed, it would mean that the Markov chain is also homogeneous. It appears that many interesting computations can not be achieved with such constraints (i.e. only allowing one or more cycles of the same period and a purely deterministic and homogeneous Markov chain). Furthermore, if the parameters of the system are the transition probabilities themselves (as in ordinary HMMs), such solutions correspond to a subset of the corners of the 0-1 hypercube in parameter space. Away from those solutions, learning is mostly influenced by short term dependencies, because of diffusion of credit. Furthermore, as seen in equation 2, algorithms like EM will tend to stay near a corner once it is approached. This suggests that discrete optimization algorithms, rather continuous local algorithms, may be more appropriate to explore the (legal) corners of this hypercube. 4 Experiments 4.1 Diffusion: Numerical Simulations Firstly, we wanted to measure how (and if) different kinds of products of stochastic matrices converged, for example to a matrix of equal rows. We ran 4 simulations, each with an 8 states non-homogeneous Markov chain but with different constraints on the transition graph: 1) 9 fully connected; 2) 9 is a left-to-right model (i.e. A is upper triangular); 3) 9 is left-to-right but only one-state skips are allowed (i.e. A is upper bidiagonal); 4) At are periodic with period 4. Results shown in Figure 2 confirm the convergence towards zero of the ergodicity coefficient 1 , at a rate that depends on the graph topology. In Figure 2, we represent visually the convergence of fully connected matrices, in only 4 time steps, towards equal columns. lexcept for the experiments with periodic matrices, as expected Diffusion of Credit in Markovian Models (a) 100,----~-.. -···-~/~···~~~\-.~-:-/~:--~-~-~--~~-~--yg-iV-en~ 80 /\ • .1 ~_. \ Randomly co",,..;ted. 20 __ -----'\ • \ \ 24" ... , \\ Fully connected, \. \ ./ ',\ 40stales •. / \\ .... \ , ' ..,\ \ .........~ ~ Fully <XlOIIeCIed, .. 'I 16 llitale.1II \ \ Fully conrect.:ted. \. \"--24 sl.ate." " \ \ , \ , -_ •• -yoo,.. ...... 559 Cb.1 10 Span 1000 (b) Figure 3: (a): Generating HMM. Numbers out of state circles denote output symbols. (b): Percentage of convergence to a good solution (over 20 trials) for various series o( experiments as the span of dependencies is increased. 4.2 Training Experiments To evaluate how diffusion impairs training, a set of controlled experiments were performed, in which the training sequences were generated by a simple homogeneous HMM with long-term dependencies, depicted in Figure 3a. Two branches generate similar sequences except for the first and last symbol. The extent of the longterm context is controlled by the self transition probabilities of states 2 and 5, A = P(qt = 2lqt-l - 2) = P(qt = 5lqt-l = 5). Span or "half-life" is log(.5)/ log(A), i.e. Aspan = .5). Following [4], data was generated for various span of long-term dependencies (0.1 to 1000). For each series of experiments, varying the span, 20 different training trials were run per span value, with 100 training sequences2 . Training was stopped either after a maximum number of epochs (200), of after the likelihood did not improve significantly, i.e., (L(t) - L(t - l))/IL(t)1 < 10- 5 , where L(t) is the logarithm of the likelihood of the training set at epoch t. If the HMM is fully connected (except for the final absorbing state) and has just the right number of states, trials almost never converge to a good solution (1 in 160 did). Increasing the number of states and randomly putting zeroes helps. The randomly connected HMMs had 3 times more states than the generating HMM and random connections were created with 20% probability. Figure 3b shows the average number of converged trials for these different types of HMM topology. A trial is considered successful when it yields a likelihood almost as good or better than the likelihood of the generating HMM on the same data. In all cases the number of successful trials rapidly drops to zero beyond some value of span. 5 Conclusion and Future Work In previous work on recurrent networks we had found that propagating credit over the long term was incompatible with storing information for the long term. For Markovian models, we found that when the transition probabilities are close to 1 and 0, information can be stored for the long term AND credit can be prop2it appeared sufficient since the likelihood of the generating HMM did not improve much when trained on this data 560 Yoshua Bengio, PaoLo Frasconi agated over the long term. However, like for recurrent networks, this makes the problem of learning long-term dependencies look more like a discrete optimization problem. Thus it appears difficult for local learning algorithm such as EM to learn optimal transition probabilities near 1 or 0, i.e. to learn the topology, while taking into account long-term dependencies. The arguments presented are essentially an application of established mathematical results on Markov chains to the problem of learning long term dependencies in homogeneous and non-homogeneous HMMs. These arguments were also supported by experiments on artificial data, studying the phenomenon of diffusion of credit and the corresponding difficulty in training HMMs to learn long-term dependencies. IOHMMs [1] introduce a reparameterization of the problem: instead of directly learning the transition probabilities, we learn parameters of a function of an input sequence. Even with a fully connected topology, transition probabilities computed at each time step might be very close to ° and 1. Because of the non-stationarity, more interestin~ computations can emerge than the simple cycles studied above. For example in l3] we found IOHMMs effective in grammar inference tasks. In [1] comparative experiments were performed with a preliminary version of IOHMMs and other algorithms such as recurrent networks, on artificial data on which the span of long-term dependencies was controlled. IOHMMs were found much better than the other algorithms at learning these tasks. Based on the analysis presented here, we are also exploring another approach to learning long-term dependencies that consists in building a hierarchical representation of the state. This can be achieved by introducing several sub-state variables whose Cartesian product corresponds to the system state. Each of these sub-state variables can operate at a different time scale, thus allowing credit to propagate over long temporal spans for some of these variables. Another interesting issue to be investigated is whether techniques of symbolic prior knowledge injection (such as in (5]) can be exploited to choose good topologies. One advantage, compared to traditIOnal neural network approaches, is that the model has an underlying finite state structure and is thus well suited to inject discrete transition rules. Acknowledgments We would like to thank Leon Bottou for his many useful comments and suggestions, and the NSERC and FCAR Canadian funding agencies for support. References [1] Y. Bengio and P. Frasconi. Credit assignment through time: Alternatives to backpropagation. In J. D. Cowan, et al., eds., Advances in Neural Information Processing Systems 6. Morgan Kaufmann, 1994. [2] Y. Bengio and P. Frasconi. An Input Output HMM Architecture. In this volume: J. D. Cowan, et al., eds., Advances in Neural Information Processing Systems 7. Morgan Kaufmann, 1994. [3] Y. Bengio and P. Frasconi. An EM approach to learning sequential behavior. Technical Report RT-DSI-ll/94, University of Florence, 1994. [4] Y. Bengio, P. Simard, and P. Frasconi. Learning long-term dependencies with gradient descent is difficult. IEEE Trans. Neural Networks, 5(2):157- 166, 1994. [5] P. Frasconi, M. Gori, M. Maggini, and G. Soda. Unified integration of explicit rules and learning by example in recurrent networks. IEEE Trans. on Knowledge and Data Engineering, 7(1), 1995. [6] E. Seneta. Nonnegative Matrices and Markov Chains. Springer, New York, 1981.	1994	37
929	The NilOOO: High Speed Parallel VLSI for Implementing Multilayer Perceptrons Leon N Cooper Michael P. Perrone Thomas J. Watson Research Center P.O. Box 704 Yorktown Heights, NY 10598 mppGwatson.ibm.com Institute for Brain and Neural Systems Brown University Providence, Ri 02912 IncGcns.brown.edu Abstract In this paper we present a new version of the standard multilayer perceptron (MLP) algorithm for the state-of-the-art in neural network VLSI implementations: the Intel Ni1000. This new version of the MLP uses a fundamental property of high dimensional spaces which allows the 12-norm to be accurately approximated by the It -norm. This approach enables the standard MLP to utilize the parallel architecture of the Ni1000 to achieve on the order of 40000, 256-dimensional classifications per second. 1 The Intel NilOOO VLSI Chip The Nestor/Intel radial basis function neural chip (Ni1000) contains the equivalent of 1024 256-dimensional artificial digital neurons and can perform at least 40000 classifications per second [Sullivan, 1993]. To attain this great speed, the Ni1000 was designed to calculate "city block" distances (Le. the II-norm) and thus to avoid the large number of multiplication units that would be required to calculate Euclidean dot products in parallel. Each neuron calculates the city block distance between its stored weights and the current input: neuron activity = L IWi - :eil (1) where w, is the neuron's stored weight for the ith input and :ei is the ith input. Thus the Nil000 is ideally suited to perform both the RCE [Reillyet al., 1982] and 748 Michael P. Perrone. Leon N. Cooper PRCE [Scofield et al., 1987] algorithms or any of the other commonly used radial basis function (RBF) algorithms. However, dot products are central in the calculations performed by most neural network algorithms (e.g. MLP, Cascade Correlation, etc.). Furthermore, for high dimensional data, the dot product becomes the computation bottleneck (i.e. most ofthe network's time is spent calculating dot products). If the dot product can not be performed in parallel there will be little advantage using the NilOOO for such algorithms. In this paper, we address this problem by showing that we can extend the NilOOO to many of the standard neural network algorithms by representing the Euclidean dot product as a function of Euclidean norms and by then using a city block norm approximation to the Euclidean norm. Section 2, introduces the approximate dot productj Section 3 describes the City Block MLP which uses the approximate dot productj and Section 4 presents experiments which demonstrate that the City Block MLP performs well on the NIST OCR data and on human face recognition data. 2 Approximate Dot Product Consider the following approximation [Perrone, 1993]: 1 11Z11 ~ y'n1Z1 (2) where z is some n-dimensional vector, II· II is the Euclidean length (i.e. the 12 norm) and I· I is the City Block length (i.e. the 11-norm). This approximation is motivated by the fact that in high dimensional spaces it is accurate for a majority of the points in the space. In Figure 1, we suggest an intuitive interpretation of why this approximation is reasonable. It is clear from Figure 1 that the approximation is reasonable for about 20% of the points on the arc in 2 dimensions. 1 As the dimensionality of the data space increases, the tangent region in Figure 1 expands asymptotically to fill the entire space and thus the approximation becomes more valid. Below we examine how accurate this approximation is and how we can use it with the NilOOO, particularly in the MLP context. Given a set of vectors, V, all with equal city block length, we measure the accuracy of the approximation by the ratio of the variance of the Euclidean lengths in V to the squared mean Euclidean lengths in V. If the ratio is low, then the approximation is good and all we must do is scale the city block length to the mean Euclidean length to get a good fit. 2 In particular, it can be shown that assuming all the vectors of the space are equally likely, the following equation holds [Perrone, 1993]: O'~ < (a~(!n+ 1) -1)ILfower, (3) where n is the dimension of the space; ILn is the average Euclidean length of the set of vectors with fixed city block length Sj O'~ is the variance about the average Euclidean length; ILlower is the lower bound for ILn and is given by ILlower == an S / Vnj 1 In fact, approximately 20% of the points are within 1% of each other and 40% of the points are within 5% of each other. 2 Note that in Equation 2 we scale by 1/ fo. For high dimensional spaces this is a good approximation to the ratio of the mean Euclidean length to the City Block length. VLSI for Implementing Multilayer Perceptrons 749 Figure 1: Two dimensional interpretation of the city block approximation. The circle corresponds to all of the vectors with the same Euclidean length. The inner square corresponds to all of the vectors with city block length equal the Euclidean length of the vectors in the circle. The outer square (tangent to the circle) corresponds to the set of vectors over which we will be making our approximation. In order to scale the outer square to the inner square, we multiple by 11 Vn where n is the dimensionality of the space. The outer square approximates the circle in the regions near the tangent points. In high dimensional spaces, these tangent regions approximate a large portion of the total hypersphere and thus the city block distance is a good approximation along most of the hypersphere. and an is defined by n -1 J en (n) ~ 2 a n =-1+ +-- n + 1 2?r( n - 1) 2 n + 1 . (4) From this equation we see that the ratio of O"~ to /-L~ower in the large n limit is bounded above by 0.4. This bound is not very tight due to the complexity of the calculations required; however Figure 3 suggests that a much tighter bound must exist. A better bound exists if we are willing to add a minor constraint to our high dimensional space [Perrone, 1993]. In the case in which each dimension of the vector is constrained such that the entire vector cannot lie along a single axis,3 we can show that 0"2 :::::: 2(n -1) ( ~ _ 1)2 /-L~ower n (n + 1) 2 V S a~ , (5) where S is the city block length of the vector in question. Thus in this case, the ratio of O"~ to /-L~ower decreases at least as fast as lin since nl S will be some fixed constant independent of n. 4 This dependency on nand S is shown in Figure 2. This result suggests that the approximation will be very accurate for many real-world pattern 3For example, when the axes are constrained to be in the range [D, 1] and the city block length of the vector is greater than 1. Note that this is true for the majority of the points in a n dimensional unit hypercube. ~Thus the accuracy improves as S increases towards its maximum value. 750 Michael P. Perrone, Leon N. Cooper recognition tasks such as speech and high resolution image recognition which can typically have thousand or even tens of thousands of dimensions. 1 0.8 0.6 0.4 0 .2 SIn 0.025 SIn 0.05 SIn 0.1 SIn 0 .2 SIn 0 .3 \.-,----.---------------------------o L-__ ~~:~·~~~::~~ :· ·~~:2~:~ ··--u -- ~--·_-·_··_-·_ · -~--_·-_ .. _-._-._ ... ~.-~ . -~-._ .. _ .. ~ .. -_ .. _ .. _._--_ .. ~.-._.-_ : :_~~:-~.:_~_~-_:;~~_ : :~~:~~~-:~~:~ --~7 o 100 200 300 400 500 Figure 2: Plot of unj I-'lower vs. n for constrained vectors with varying values of Sin. As S grows the ratio shrinks and consequently, accuracy improves. If we assume that all of the vectors are uniformly distributed in an n-dimensional unit hypercube, it is easy to show that the average city block length is nj2 and the variance of the city block length is n/12. Since Sjn will generally be within one standard deviation ofthe mean, we find that typically 0.2 < Sjn < 0.8. We can use the same analysis on binary valued vectors to derive similar results. We explore this phenomenon further by considering the following Monte Carlo simulation. We sampled 200000 points from a uniform distribution over an n-dimensional cube. The Euclidean distance of each of these points to a fixed corner of the cube was calculated and all the lengths were normalized by the largest possible length, ~. Histograms of the resulting lengths are shown in Figure 3 for four different values of n. Note that as the dimension increases the variance about the mean drops. From Figure 3 we see that for as few as 100 dimensions, the standard deviation is approximately 5% of the mean length. 3 The City Block MLP In this section we describe how the approximation explained in Section 2 can be used by the NilOOO to implement MLPs in parallel. Consider the following formula for the dot product (6) VLSI for Implementing Multilayer Perceptrons 751 0.45 0 . 4 0.35 0.3 g 0.25 ~ {} e 0.2 t>. 0 . 15 0.1 0.05 0.45 0.5 0.55 0.6 0.65 Norma1izad LenQth ~ooo D~mQg~ons ~ 100 D.1.mes.1.ons - ..... -10 D;l.mas1.ons -c:.--:2 D.1.mas1.ons ...... - .0.7 0.75 O.B Figure 3: Probability distributions for randomly draw lengths. Note that as the dimension increases the variance about the mean length drops. where II· II is the Euclidean length (i.e. 12-norm).5 Using Equation 2, we can approximation Equation 6 by ...... 1(1"';;'12 I'" ;;'12) :t! • Y ~ :t! + YI :t! YI 4n (7) where n is the dimension of the vectors and I . I is the city block length. The advantage to the approximation in Equation 7 is that it can be implemented in parallel on the Ni1000 while still behaving like a dot product. Thus we can use this approximation to implement MLP's on an Ni1000. The standard functional form for MLP's is given by [Rumelhart et al., 1986] N d h(:t!;a,f3) = u(aok + Lajku (!30j + Lf3ij:t!i») (8) j=1 i=1 were u is a fixed ridge function chosen to be u(:t!) = (1 + e -:t) \ N is the number of hidden units; k is the class label; d is the dimensionality of the data space; and a and f3 are adjustable parameters. The alternative which we propose, the City Block MLP, is given by [Perrone, 1993] N 1 d 1 d gk(:t!; a, f3) = u(aok + L ajku (f30j + 4(L lf3ij + :t!i1)2 - 4(L lf3ij -:t!i 1)2») (9) j=1 n i=1 n i=1 or riNote also that depending on the information available to us, we could use either i· y = }(IIi + yW -lIiW -11?7W) i· y= }<lIiIl2 + IIYlI2 -Iii - Y112). 752 Michael P. Perrone, Leon N. Cooper DATA SET HIDDEN STANDARD CITYBLOCK ENSEMBLE UNITS % CORRECT % CORRECT CITYBLOCK Faces 12 94.6±1.4 92.2±1.9 96.3 Numbers 10 98.4±0.17 97.3±0.26 98.3 Lowercase 20 88.9±0.31 84.0±0.48 88.6 Uppercase 20 90.5±0.39 85.6±O.89 90.7 Table 1: Comparison of MLPs classification performance with and with out the city block approximation to the dot product. The final column shows the effect of function space averaging. where the two city block calculation would be performed by neurons on the Nil000 chip. 6 The City Block MLP learns in the standard way by minimizing the mean square error (MSE), MSE = 2: (glc (Xi; 0:, (3) - tlci) 2 (10) ilc where tic; is the value of the data at Xi for a class k. The MSE is minimized using the backpropagation stochastic gradient descent learning rule [Werbos, 1974]: For a fixed stepsize f'/ and each k, randomly choose a data point Xi and change 'Y by the amount A _ o(MSEi) L.l.'Y -f'/ , o'Y (11) where 'Y is either 0: or {3 and MSEi is the contribution to the MSE of the ith data point. Note that although we have motivated the City Block MLP above as an approximation to the standard MLP, the City Block MLP can also be thought of as special case of radial basis function network. 4 Experimental Results This section describes experiments using the City Block MLP on a 120-dimensional representation of the NIST Handwritten Optical Character Recognition database and on a 2294-dimensional grayscale human face image database. The results indicate that the performance of networks using the approximation is as good as networks using the exact dot product [Perrone, 1993]. In order to test the performance of the City Block MLP, we simulated the behavior of the NilOOO on a SPARC station in serial. We used the approximation only on the first layer of weights (i.e. those connecting the inputs to the hidden units) where the dimensionality is highest and the approximation is most accurate. The approximation was not used in the second layer of weights (i.e. those connecting the hidden units to the output units were calculated in serial) since the number of hidden units was low and therefore do not correspond to a major computational bottleneck. It should be noted that for a 2 layer MLP in which the number of hidden units and output units are much lower than the input dimensionality, the 6The dot product between the hidden and the output layers may also be approximated in the same way but it is not shown here. In fact, the NilOOO could be used to perform all of the functions required by the network. VLSI for Implementing Multilayer Perceptrons 753 DATA SET HIDDEN STANDARD CITYBLOCK ENSEMBLE UNITS FOM FOM CITYBLOCK Numbers 10 92.1±0.57 87.4±0.83 92.5 Lowercase 20 59.7±1.7 44.4±2.0 62.7 Uppercase 20 60.0±1.8 44.6±4.5 66.4 Table 2: Comparison of MLPs FOM. The FOM is defined as the 100 minus the number rejected minus 10 time the number incorrect. majority of the computation is in the calculation of the dot products in the first weight layer. So even using the approximation only in the first layer will significantly accelerate the calculation. Also, the Nil000 on-chip math coprocessor can perform a low-dimensional, second layer dot product while the high-dimensional, first layer dot product is being approximated in parallel by the city block units. In practice, if the number of hidden units is large, the approximation to the dot product may also be used in the second weight layer. In the simulations, the networks used the approximation when calculating the dot product only in the feedforward phase of the algorithm. For the feedback ward phase (i.e. the error backpropagation phase), the algorithm was identical to the original backward propagation algorithm. In other words the approximation was used to calculate the network activity but the stochastic gradient term was calculated as if the network activity was generated with the real dot product. This simplification does not slow the calculation because all the terms needed for the backpropagation phase are calculated in the forward propagation phase In addition, it allows us to avoid altering the backpropagation algorithm to incorporate the derivative of the city block approximation. We are currently working on simulations which use city block calculations in both the forward and backward passes. Since these simulations will use the correct derivative for the functional form of the City Block MLP, we expect that they will have better performance. In practice, the price we pay for making the approximation is reduced performance. We can avoid this problem by increasing the number of hidden units and thereby allow more flexibility in the network. This increase in size will not significantly slow the algorithm since the hidden unit activities are calculated in parallel. In Table 1 and Table 2, we compare the performance of a standard MLP without the city block approximation to a MLP using the city block approximation to calculate network activity. In all cases, a population of 10 neural networks were trained from random initial weight configurations and the means and standard deviations were listed. The number of hidden units was chosen to give a reasonable size network while at the same time reasonably quick training. Training was halted by cross-validating on an independent hold-out set. From these results, one can see that the relative performances with and with out the approximation are similar although the City Block is slightly lower. We also perform ensemble averaging [Perrone, 1993, Perrone and Cooper, 1993] to further improve the performance of the approximate networks. These results are given in the last column of the table. From these data we see that by combining the city block approximation with the averaging method, we can generate networks which have comparable and sometimes better performance than the standard MLPs. In addition, because the Nil000 is running in parallel, there is minimal additional computational overhead for using 754 Michael P. Perrone, Leon N. Cooper the averaging. 7 5 Discussion We have described a new radial basis function network architecture which can be used in high dimensional spaces to approximate the learning characteristics of a standard MLP without using dot products. The absence of dot products allows us to implement this new architecture efficiently in parallel on an NilOOO; thus enabling us to take advantage of the Ni1000's extremely fast classification rates. We have also presented experimental results on real-world data which indicate that these high classifications rates can be achieved while maintaining or improving classification accuracy. These results illustrate that it is possible to use the inherent high dimensionality of real-world problems to our advantage. References [Perrone, 1993] Perrone, M. P. (1993). Improving Regression Estimation: Averaging Methods for Variance Reduction with Eztensions to General Convez Measure Optimization. PhD thesis, Brown University, Institute for Brain and Neural Systems; Dr. Leon N Cooper, Thesis Supervisor. [Perrone and Cooper, 1993] Perrone, M. P. and Cooper, L. N. (1993). When networks disagree: Ensemble method for neural networks. In Mammone, R. J., editor, Artificial Neural Networks for Speech and Vision. Chapman-Hall. Chapter 10. [Reillyet al., 1982] Reilly, D. L., Cooper, L. N., and Elbaum, C. (1982). A neural model for category learning. Biological Cybernetics, 45:35-41. [Rumelhart et al., 1986] Rumelhart, D. E., McClelland, J. L., and the PDP Research Group (1986). Parallel Distributed Processing, Volume 1: Foundations. MIT Press. [Scofield et al., 1987] Scofield, C. L., Reilly, D. L., Elbaum, C., and Cooper, L. N. (1987). Pattern class degeneracy in an unrestricted storage density memory. In Anderson, D. Z., editor, Neural Information Processing Systems. American Institute of Physics. [Sullivan, 1993] Sullivan, M. (1993). Intel and Nestor deliver second-generation neural network chip to DARPA: Companies launch beta site program. Intel Corporation News Release. Feb. 12. [Werbos, 1974] Werbos, P. (1974). Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences. PhD thesis, Harvard University. 7The averaging can also be applied to the standard MLPs with a corresponding improvement in performance. However, for serial machines averaging slows calculations by a factor equal to the number of averaging nets.	1994	38
930	Model of a Biological Neuron as a Temporal Neural Network Sean D. Murphy and Edward W. Kairiss Interdepartmental Neuroscience Program, Department of Psychology, and The Center for Theoretical and Applied Neuroscience, Yale University, Box 208205, New Haven, CT 06520 Abstract A biological neuron can be viewed as a device that maps a multidimensional temporal event signal (dendritic postsynaptic activations) into a unidimensional temporal event signal (action potentials). We have designed a network, the Spatio-Temporal Event Mapping (STEM) architecture, which can learn to perform this mapping for arbitrary biophysical models of neurons. Such a network appropriately trained, called a STEM cell, can be used in place of a conventional compartmental model in simulations where only the transfer function is important, such as network simulations. The STEM cell offers advantages over compartmental models in terms of computational efficiency, analytical tractabili1ty, and as a framework for VLSI implementations of biological neurons. 1 INTRODUCTION Discovery of the mechanisms by which the mammalian cerebral cortex processes and stores information is the greatest remaining challenge in the brain sciences. Numerous modeling studies have attempted to describe cortical information processing in frameworks as varied as holography, statistical physics, mass action, and nonlinear dynamics. Yet, despite these theoretical studies and extensive experimental efforts, the functional architecture of the cortex and its implementation by cortical neurons are largely a mystery. Our view is that the most promising approach involves the study of computational models with the following key properties: (1) Networks consist of large (> 103 ) numbers of neurons; (2) neurons are connected by modifiable synapses; and (3) the neurons themselves possess biologically-realistic dynamics. Property (1) arises from extensive experimental observations that information processing and storage is distributed over many neurons. Cortical networks are also characterized by sparse connectivity: the probability that any two local cortical neurons are synaptically connected is typically less than 0.1. These and other observations suggest that key features of cortical dynamics may not be apparent unless large, sparsely-connected networks are studied. Property (2) is suggested by the accumulated evidence that (a) memory formation is subserved by use-dependent synaptic modification, and (b) Hebbian synaptic plasticity is present in many areas of the brain thought to be important for memory. It is also well known that artificial networks composed of elements that are connected by Hebb-like synapses have powerful computational properties. 86 Sean D. Murphy, Edward W. Kairiss Property (3) is based on the assumption that biological neurons are computationally more complex than. for example. the processing elements that compose artificial (connectionist) neural networks. Although it has been difficult to infer the computational function of cortical neurons directly from experimental data, models of neurons that explicitly incorporate biophysical components (e.g. neuronal geometry, channel kinetics) suggest a complex, highly non-linear dynamical transfer function. Since the "testability" of a model depends on the ability to make predictions in terms of empirically measurable single-neuron firing behavior, a biologically-realistic nodal element is necessary in the network model. Biological network models with the above properties (e.g. Wilson & Bower, 1992; Traub and Wong, 1992) have been handicapped by the computationally expensive single-neuron representation. These "compartmental" models incorporate the neuron's morphology and membrane biophysics as a large (102 _104) set of coupled, non-linear differential equations. The resulting system is often stiff and requires higher-order numerical methods and small time-steps for accurate solution. Although the result is a realistic approximation of neuronal dynamics, the computational burden precludes exhaustive study of large networks for functionality such as learning and memory. The present study is an effort to develop a computationally efficient representation of a single neuron that does not compromise the biological dynamical behavior. We take the position that the "dynamical transfer function" of a neuron is essential to its computational abstraction, but that the underlying molecular implementation need not be explicitly represented unless it is a target of analysis. We propose that a biological neuron can be viewed as a device that performs a mapping from multidimensional spatio-temporal (synaptic) events to unidimensional temporal events (action potentials). This computational abstraction will be called a Spatio-Temporal Event Mapping (STEM) cell. We describe the architecture of the neural net that implements the neural transfer function, and the training procedure required to develop realistic dynamics. Finally, we discuss our preliminary analyses of the performance of the model when compared with the full biophysical representation. 2 STEM ARCHITECTURE The architecture of the STEM cell is similar to that found in neural nets for temporal sequence processing (e.g. review by Mozer, in press). In general, these networks have 2 components: (1) a short-term memory mechanism that acts as a preprocessor for (2) a nonlinear feedforward network. For example, de Vries & Principe (1992) describe the utility of the gamma net, a real-time neural net for temporal processing, in time series prediction. The preprocessor in the gamma net is the gamma memory structure, implemented as a network of adaptive dispersive elements (de Vries & Principe, 1991). The preprocessor in our model (the "tau layer", described below) is somewhat simpler, and is inspired by the temporal dynamics of membrane conductances found in biological neurons. The STEM architecture (diagrammed in Figure 1) works by building up a vectorial representation of the state of the neuron as it continuously receives incoming synaptic activations, and then labeling that vector space as either "FIRE" or "DON'T FIRE". This is accomplished with the use of four major components: (1) TAU LAYER: a layer of nodes that continuously maps incoming synaptic activations into a finite-dimensional vector space (2) FEEDBACK TAU NODE: a node that maintains a vectorial representation of the past activity of the cell itself (3) MLP: a multilayer perceptron that functions as a nonlinear spatial mapping network that performs the "FIRE" / "NO-FIRE" labeling on the tau layer output (4) OUTPUT FILTER: this adds a refractory period and threshold to the MLP output that contrains the format of the output to be discrete-time events. Model of Biological Neuron as a Temporal Neural Network 87 input (spike train at each synapse) " V I spatio-temporalto spatial projection layer ! _ u l nonlinear spatial mapping network I spike train output Q I output processor : Figure 1: Information Flow in The STEM Cell The tau layer (Fig. 2) consists of N + 1 tau nodes, where N is the number of synapses on the cell, and the extra node is used for feedback. Each tau node consists of M tau units. Each tau unit within a single tau node receives an identical input signal. Each tau unit within a tau node calculates a second-order rise-and-decay function with unique time constants. The tau units within a tau node translate arbitrary temporal events into a vector form, with each tau-unit corresponding to a different vector component. Taken as a whole, all of the tau unit outputs of the tau node layer comprise a high-dimensional vector that represents the overall state of the neuron. Functionally, the tau layer approximates a oneto-one mapping between the spatio-temporal input and the tau-unit vector space. The output of each tau unit in the tau layer is fed into the input layer of a multilayer perceptron (MLP) which, as will be explained in the next section, has been trained to label the tau-layer vector as either FIRE or NO-FIRE. The output of the MLP is then fed into an output filter with a refractory period and threshold. The STEM architecture is illustrated in Fig. 3. (A) (afferents) (feedback from action potentials) synapse # 1,* ~ 3~·" N~ I ~+TAU_NODE,a~r (all outputs go to MLP) (B) p~esynaptic mput (C) TAU-UNIT DYNAMICS d ' -xl dt I xi , I = a)-'to I Ti _d T,; , I ..\ = xl-dt I I 't. I ai(t) I I I x~(~ T~(t~ TAU NODE TAU UNITS tM....r"-.~~ Tlll __ Figure 2: Tau Layer Schematic. (A) the tau layer has an afferent section, with N tau nodes, and a single-node feedback section. (B) Each tau node contains M tau units, and therefore has 1 input and M outputs (C) Each of the M tau units in a tau node has a rise and decay function with different constants. The equations are given for the ith tau unit of the jth tau node. a is input activity, x an internal state variable, and T the output. 88 Sean D. Murphy, Edward W. Kairiss afferent inputs (axons) single tau node from output with B internal tau units & IaU node layer sN / hidden layer of MLP single output unit of MLP Schematic of output filter processin~ MLP ',))f\",AU M -n-;foutput filter I I I B units in 1st layer from feedback NxM units in axon subset of 1st layer of multilayer perceptron x = full forward connections between layers output axon Figure 3: STEM Architecture. Afferent activity enters the tau layer, where it is converted into a vectorial representation of past spaiotemporal activity. The MLP maps this vector into a FlRE/NO-FIRE output unit, the continuous value of which is converted to a discrete event signal by the refractory period and threshold of the output filter. 3 STEM TRAINING There are six phases to training the STEM cell: (1) Biology: anatomical and physiological data are collected on the cell to be modeled. (2) Compartmental Model: a compartmental model of the cell is designed, typically with a simulation environment such as GENESIS. As much biological detail as possible is incorporated into the model. (3) Transfer Function Trials: many random input sequences are generated for the compartmental model. The firing response of the model is recorded for each input sequence. (4) Samplin~ assi~nments; In the next step, sampling will need to be done on the affect of the input sequences on the STEM tay layer. The timing of the sampling is calculated by separating the response of the compartmental model on each trial into regions where no spikes occur, and regions surrounding spikes. High-rate sampling times are determined for spike regions, and lower rate times are determined for quiet regions. (5) Tau layer trials: the identical input sequences applied to the compartmental model in step #3 are applied to an isolated tau layer of the STEM cell. The spike events from the compartmental model are used as input for the feedback node. For each input sequence, the tay layer is sampled at the times calculated in step #4, and the vector is labeled as FIRE or NO-FIRE (0 or 1). (6) MLP training: conjugate-gradient and line-search methods are used to train the multilayer perceptron using the results of step #5 as training vectors. Model of Biological Neuron as a Temporal Neural Network 89 Training is continued until a minimal performance level is reached, as determined by comparing the response of the STEM cell to the original compartmental model on novel input sequences. 4 RESULTS The STEM cell has been initially evaluated using Roger Traub's (1991) compartmental model of a hippocampal CAl cell, implemented in GENESIS by Dave Beeman. This is a relatively simple model structurally, with 19 compartments connected in a linear segment, with the soma in the middle. Dynamically, however, it is one of the most accurate and sophisticated models published, with on the order of 100 voltage- and Ca++ sensitive membrane channel mechanisms. 94 synapses were placed on the model. Each synapse recevied a random spike train with average frequency 10Hz during training. A diagram of the model and the locations of synaptic input is given in Fig. 4. Inputs going to a single compartment were treated as members of a common synapse, so there were a total of 13 tau nodes, with 5 tau units per node, for a total of 65 tau units, plus 5 additional units from the feedback tau node. These fed into 70 units in the input layer of the MLP. Two STEM cells were trained, one on a passive shell of the CAl cell, and the other with all of the membrane channels included. Both used 70 units in the hidden layer 1200m 880m 6mlfllll_m 14m 8m mwmiil received 8 synaptic inputs Fig. 4 Structure of Traub's CAl cell compartmental model -42. -+4. -44'., -48. -150. -!5Z. -64. -116. -ee. 0.5 1.0 1.5 2.0 STEM cell 2.5 10.0<H'"----t------+-----t----+-----+-------,. B. 4. 2. 0.5 1.0 1.5 20 2.5 Figure 5: CAl Passive Cell Output. The somatic voltage for the passive compartmental model and the corresponding output-filtered spike events are given in the upper graph. The lower graph shows the repsonse of the STEM cell to the same input. The upper trace of the lower graph is the output filter response, the lower trace is the raw output of the MLP. Horizontal axes in seconds. Vertical axis: top, m V, and bottom, arbitrary units. 90 compartmental model 10, -0, -10, 000000,00-20, .00-30, .00-40, ,00-eo, ,00-60, STEM cell 0.5 Sean D. Murphy, Edward W. Kairiss r--r--r-I':: 1.0 1.5 2.0 2.5 10,i'lO"!'"----t-----+-----t-----t------t-------r 8, 0.5 1.0 1.5 2.0 2.5 Figure 6: Response of Active CAl Cell, The upper graph is the somatic voltage of the CA 1 cell in response to a random input The lower graph is the response of the STEM cell to the same input. The upper trace of the STEM graph is the output filter response, and the lower trace is the raw MLP output.Horizontal axes in seconds, Vertical axis: top, m V, and bottom, arbitrary units, of the MLP, Comparisons of the compartmental model vs. STEM are shown in Fig. 5 and , for the passive and active models, respectively, Fig. 6. For the active cell model, the STEM cell was approximately 10 times faster, and used 10% of the memory as the compartmental model. 5 DISCUSSION The STEM cell is a general spatio-temporal node architecture and is similar to many context networks that have been previously developed. Its role as a single node in a large meta-network is unexplored, albeit interesting because of its capacity to mimic the transfer functions of biological neurons. Between the complexity range of connectionist networks to biological networks, there may be a multitude of useful computational schemes for solving different types of problems. The STEM cell is an attempt to efficiently capture some elements of biological neural function in a form that can be scaled along this range. Characterization of the computational function of the neuron is a topic of considerable interest and debate. STEM cells may be useful as a rough measure of how complex the transfer function of a given biophysical model is. For example, it might be able to answer the question: Which instantiates a more sophisticated nonlinear spatio-temporal map, a single-compartment cell with complex somatic Ca++ dynamics, or a cell with active Na+ channels in a complex dendritic tree? STEM architecture may also be interesting for theoretical and applied ANN research as a connectionist representation of a biological neuron. The expanding body of work on focused network architectures (Mozer, 1989; Stometta, 1988) may be an avenue towards the formalization of biological neural transfer functions. Because a VLSI implementation Model of Biological Neuron as a Temporal Neural Network 91 of a STEM cell could be reprogrammed on the fly to assume the transfer function of any pre-trained biophysically-modeled cell, a VLSI STEM network chip is a more versatile approach to artifical implementations of biological neurons than reconstructing compartmental models in VLSI. Our future plans include using networks with delay lines and hebbian learning rules, both of which the STEM architecture is directly suited for, to investigate the capacity for STEM networks to perform real-time dynamic pattern tracking. The present implementation of the STEM cell is by no means an optimal one. We are experimenting with alternative components for the MLP, such as modular recurrent networks. Acknowledgements This work was supported by the Yale Center for Theoretical and Applied Neuroscience. References de Vries, B. and Principe, J.C. (1991) A theory for neural networks with time delays. In R.P. Lippmann, J. Moody, & D.S. Touretzlcy (Eds.), Advances in Neural Information Processing Systems 3 (pp. 162-168). San Mateo, CA: Morgan Kaufmann. de Vries, B. and Principe, J.C. (1992) The gamma model - A new neural net model for temporal processing. Neural Networks, 5, 565-576. Mozer, M.C. (1989). A focused back-propagation algorithm for temporal pattern recognition, Complex Systems, 3, 349-381. Mozer, M.C. (in press) Neural net architectures for temporal sequence processing. In A. Weigend & N. Gershenfeld (Eds.), Predicting the Future and Understanding the Past. Redwood City, CA: Addison-Wesley. Stornetta, W.S., Hogg, T., & Huberman, B.A. (1988). A dynamical approach to temporal pattern processing. In Anderson D.Z. (Ed.), Neural Information Processing Systems, 750759. Traub, R.D. and Wong, R. K. S. (1991). Neuronal Networks of the Hippocampus. Cambridge: Cambridge University Press. Wilson, M. and Bower, J.M. (1992) Cortical oscillations and temporal interactions in a computer simulation of piriform cortex. J. Neurophysiol. 67:981-95.	1994	39
931	Learning with Preknowledge: Clustering with Point and Graph Matching Distance Measures Steven Gold!, Anand Rangarajan1 and Eric Mjolsness2 Department of Computer Science Yale University New Haven, CT 06520-8285 Abstract Prior constraints are imposed upon a learning problem in the form of distance measures. Prototypical 2-D point sets and graphs are learned by clustering with point matching and graph matching distance measures. The point matching distance measure is approx. invariant under affine transformations - translation, rotation, scale and shear - and permutations. It operates between noisy images with missing and spurious points. The graph matching distance measure operates on weighted graphs and is invariant under permutations. Learning is formulated as an optimization problem. Large objectives so formulated ('" million variables) are efficiently minimized using a combination of optimization techniques - algebraic transformations, iterative projective scaling, clocked objectives, and deterministic annealing. 1 Introduction While few biologists today would subscribe to Locke's description of the nascent mind as a tabula rasa, the nature of the inherent constraints - Kant's preknowl1 E-mail address of authors: lastname-firstname@cs.yale.edu 2Department of Computer Science and Engineering, University of California at San Diego (UCSD), La Jolla, CA 92093-0114. E-mail: emj@cs.ucsd.edu 714 Steven Gold, Anand Rangarajan, Eric Mjolsness edge - that helps organize our perceptions remains much in doubt. Recently, the importance of such preknowledge for learning has been convincingly argued from a statistical framework [Geman et al., 1992]. Researchers have proposed that our brains may incorporate preknowledge in the form of distance measures [Shepard, 1989]. The neural network community has begun to explore this idea via tangent distance [Simard et al., 1993], model learning [Williams et al., 1993] and point matching distances [Gold et al., 1994]. However, only the point matching distances have been invariant under permutations. Here we extend that work by enhancing both the scope and function of those distance measures, significantly expanding the problem domains where learning may take place. We learn objects consisting of noisy 2-D point-sets or noisy weighted graphs by clustering with point matching and graph matching distance measures. The point matching measure is approx. invariant under permutations and affine transformations (separately decomposed into translation, rotation, scale and shear) and operates on point-sets with missing or spurious points. The graph matching measure is invariant under permutations. These distance measures and others like them may be constructed using Bayesian inference on a probabilistic model of the visual domain. Such models introduce a carefully designed bias into our learning, which reduces its generality outside the problem domain but increases its ability to generalize within the problem domain. (From a statistical viewpoint, outside the problem domain it increases bias while within the problem domain it decreases variance). The resulting distance measures are similar to some of those hypothesized for cognition. The distance measures and learning problem (clustering) are formulated as objective functions. Fast minimization of these objectives is achieved by a combination of optimization techniques - algebraic transformations, iterative projective scaling, clocked objectives, and deterministic annealing. Combining these techniques significantly increases the size of problems which may be solved with recurrent network architectures [Rangarajan et al., 1994]. Even on single-cpu workstations non-linear objectives with a million variables can routinely be minimized. With these methods we learn prototypical examples of 2-D points set and graphs from randomly generated experimental data. 2 Distance Measures in Unsupervised Learning 2.1 An Affine Invariant Point Matching Distance Measure The first distance measure quantifies the degree of dissimilarity between two unlabeled 2-D point images, irrespective of bounded affine transformations, i.e. differences in position, orientation, scale and shear. The two images may have different numbers of points. The measure is calculated with an objective that can be used to find correspondence and pose for unlabeled feature matching in vision. Given two sets of points {Xj} and {Yk}, one can minimize the following objective to find the affine transformation and permutation which best maps Y onto X : J K J K Epm(m, t,A) = L: L: mjkllXj - t - A· Ykll 2 + g(A) - a L: L: mjk j=lk=l j=lk=l with constraints: Yj Ef=l mjk ~ 1 , Yk Ef=l mjk ~ 1 , Yjk mjk 2:: O. Learning with Preknowledge 715 A is decomposed into scale, rotation, vertical shear and oblique shear components. g(A) regularizes our affine transformation - bounding the scale and shear components. m is a fuzzy correspondence matrix which matches points in one image with corresponding points in the other image. The inequality constraint on m allows for null matches - that is a given point in one image may match to no corresponding point in the other image. The a term biases the objective towards matches. Then given two sets of points {Xj} and {Yk} the distance between them is defined as: D({Xj}, {Yk}) = min (Epm(m,t, A) I constraints on m) m,t,A This measure is an example of a more general image distance measure derived in [Mjolsness, 1992]: d(z, y) = mind(z, T(y» E [0, (0) T where T is a set of transformation parameters introduced by a visual grammar. Using slack variables, and following the treatment in [Peterson and Soderberg, 1989; Yuille and Kosowsky, 1994] we employ Lagrange multipliers and an z logz barrier function to enforce the constraints with the following objective: J K J K Epm(m, t, A) = L: L: mjkllXj - t - A· Ykll 2 + g(A) - a L: L: mjk j=lk=l j=lk=1 1 J+1 K+1 J K+1 K J+1 +-p L: L: mjk(logmjk - 1) + L: J.'j (L: mjk - 1) + L: lIk(L: mjk - 1) (1) j=1 k=l j=l k=l k=1 j=l In this objective we are looking for a saddle point. (1) is minimized with respect to m, t, and A which are the correspondence matrix, translation, and affine transform, and is maximized with respect to l' and 11, the Lagrange multipliers that enforce the row and column constraints for m. The above can be used to define many different distance measures, since given the decomposition of A it is trivial to construct measures which are invariant only under some subset of the transformations (such as rotation and translation). The regularization and a terms may also be individually adjusted in an appropriate fashion for a specific problem domain. 2.2 Weighted Graph Matching Distance Measures The following distance measure quantifies the degree of dissimilarity between two unlabeled weighted graphs. Given two graphs, represented by adjacency matrices Gab and gij, one can minimize the objective below to find the permutation which best maps G onto g:' A I B J Egm(m) = L: L:(L: Gabmbi - L: majgji)2 a=l i=l b=1 ;=1 716 Steven Gold, Anand Rangarajan, Eric Mjolsness with constraints: 'Va 2::=1 mai = 1 , 'Vi 2::=1 mai = 1 , 'Vai mai ;::: O. These constraints are enforced in the same fashion as in (1). An algebraic fixed-point transformation and self-amplification term further transform the objective to: A I B J 1 Egm(m) = L L(J.'ai(L Gabmbi - z: majDji) 2J.'~i - ,lTaimai + ~lT~i) a=1 i=1 b=1 j=1 lAI A I I A +(j L z: mai(logmai - 1) + L lI:a(Z: mai - 1) + z: Ai(L mai - 1) (2) a=1 i=1 a=1 i=1 i=1 a=1 In this objective we are also looking for a saddle point. A second, functionally equivalent, graph matching objective is also used III the clustering problem: A B I J Egm/(m) = LZ:LZ:maim bj(Gab-Dji)2 (3) a=lb=li=lj=l with constraints: 'Va 2::=1 mai = 1 , 'Vi 2::=1 mai = 1 , 'Vai mai ;::: O. 2.3 The Clustering Objective The learning problem is formulated as follows: Given a set of I objects, {Xi} find a set of A cluster centers {Ya} and match variables {Mia} defined as M. _ {I if Xi is in Ya's cluster la 0 otherwise, such that each object is in only one cluster, and the total distance of all the objects from their respective cluster centers is minimized. To find {Ya} and {Mia} minimize the cost function, I A Eeltuter(Y,M) = LLMiaD(Xi' Ya) i=l a=l with the constraint that 'Vi 2:a Mia = 1 , 'Vai Mai ;::: O. D(Xi, Ya), the distance function, is a measure of dissimilarity between two objects. The constraints on M are enforced in a manner similar to that described for the distance measure, except that now only the rows of the matrix M need to add to one, instead of both the rows and the columns. I A 1 I A Z:Z:MiaD(Xi, Ya) + (j Z:Z: Mia(log Mia - 1) i=l a=1 i=l a=1 Eeituter(Y, M) I A + Z:Ai(LMia - 1) (4) i=l a=1 Learning with Pre knowledge 717 Here, the objects are point-sets or weighted graphs. If point-sets the distance measure D(Xi, Ya) is replaced by (1), if graphs it is replaced by (2) or (3). Therefore, given a set of objects, X, we construct Ecltuter and upon finding the appropriate saddle point of that objective, we will have Y, their cluster centers, and M, their cluster memberships. 3 The Algorithm The algorithm to minimize the clustering objectives consists of two loops - an inner loop to minimize the distance measure objective [either (1) or (2)] and an outer loop to minimize the clustering objective (4). Using coordinate descent in the outer loop results in dynamics similar to the EM algorithm [Jordan and Jacobs, 1994] for clustering. All variables occurring in the distance measure objective are held fixed during this phase. The inner loop uses coordinate ascent/descent which results in repeated row and column projections for m. The minimization of m, and the distance measure variables [either t, A of (1) or 1', (f of (2)], occurs in an incremental fashion, that is their values are saved after each inner loop call from within the outer loop and are then used as initial values for the next call to the inner loop. This tracking of the values of the distance measure variables in the inner loop is essential to the efficiency of the algorithm since it greatly speeds up each inner loop optimization. Most coordinate ascent/descent phases are computed analytically, further speeding up the algorithm. Some local minima are avoided, by deterministic annealing in both the outer and inner loops. The mUlti-phase dynamics maybe described as a clocked objective. Let {D} be the set of distance measure variables excluding m. The algorithm is as follows: Initialize {D} to the equivalent of an identity transform, Y to random values Begin Outer Loop Begin Inner Loop Initialize {D} with previous values Find m, {D} for each ia pair: Find m by softmax, projecting across j, then k, iteratively Find {D} by coordinate descent End Inner Loop Find M ,Y using fixed values of m, {D}, determined in inner loop: Find M by softmax, across i Find Y by coordinate descent Increase f3M, f3m End Outer Loop When analytic solutions are computed for Y the outer loop takes a form similar to fuzzy ISODATA clustering, with annealing on the fuzziness parameter. 4 Methods and Experimental Results Four series of experiments were ran with randomly generated data to evaluate the learning algorithms. Point sets were clustered in the first three experiments and weighted graphs were clustered in the fourth. In each experiment a set of object 718 Steven Gold, Anand Rangarajan, Eric Mjolsness models were randomly generated. Then from each object model a set of object instances were created by transforming the object model according to the problem domain assumed for that experiment. For example, an object represented by points in two dimensional space was translated, rotated, scaled, sheared, and permuted to form a new point set. A object represented by a weighted graph was permuted. Noise was added to further distort the object. Parts of the object were deleted and spurious features (points) were added. In this manner, from a set of object models, a larger number of object instances were created. Then with no knowledge of the original objects models or cluster memberships, we clustered the object instances using the algorithms described above. The results were evaluated by comparing the object prototypes (cluster centers) formed by each experimental run to the object models used to generate the object instances for that experiment. The distance measures used in the clustering were used for this comparison, i.e. to calculate the distance between the learned prototype and the original object. Note that this distance measure also incorporates the transformations used to create the object instances. The mean and standard deviations of these distances were plotted (Figure 1) over hundreds of experiments, varying the object instance generation noise. The straight line appearing on each graph displays the effect of the noise only. It is the expected object model-object prototype distance if no transformations were applied, no features were deleted or added, and the cluster memberships of the object instances were known. It serves as an absolute lower bound on our learning algorithm. The noise was increased in each series of experiments until the curve flattened - that is the object instances became so distorted by noise that no information about the original objects could be recovered by the algorithm. In the first series of experiments (Figure 1a), point set objects were translated, rotated, scaled, and permuted. Initial object models were created by selecting points with a uniform distribution within a unit square. The transformations to create the object instance were selected with a uniform distribution within the following bounds; translation: ±.5, rotation: ±27°, log(scale): ± log(.5). 100 object instances were generated from 10 object models. All objects contained 20 points.The standard deviation of the Gaussian noise was varied by .02 from .02 to .16. 15 experiments were run at each noise level. The data point at each error bar represents 150 distances (15 experiments times 10 model-prototype distances for each experiment). In the second and third series of experiments (Figures 1b and 1c), point set objects were translated, rotated, scaled, sheared (obliquely and vertically), and permuted. Each object point had a 10% probability of being deleted and a 5% probability of generating a spurious point. The point sets and transformations were randomly generated as in the first experiment, except for these bounds; log(scale): ± log(.7), log(verticalshear): ±log(.7), and log(obliqueshear): ±log(.7). In experiment 2, 64 object instances and 4 object models of 15 points each were used. In experiment 3, 256 object instances and 8 object models of 20 points each were used. Noise levels like experiment 1 were used, with 20 experiments run at each noise level in experiment 2 and 10 experiments run at each noise level in experiment 3. In experiment 4 (Figure 1d), object models were represented by fully connected weighted graphs. The link weights in the initial object models were selected with a uniform distribution between 0 and 1. The objects were then randomly permuted Learning with Preknowledge 719 (a) (b) 0.05 0.1 0.15 0.05 0.1 0.15 standard deviation standard deviation (c) (d) 3r---~----~----~, 0.05 0.1 0.15 ~ Ir-'~(" CD2 ("It-' ~ ~,/ .V "01 V or o 0.05 0.1 0.15 standard deviation standard deviation Figure 1: (a): 10 clusters, 100 point sets, 20 points each, scale ,rotation, translation, 120 experiments (b): 4 clusters, 64 point sets, 15 points each, affine, 10 % deleted, 5 % spurious, 140 experiments (c): 8 clusters, 256 point sets, 20 points each, affine, 10 % deleted, 5 % spurious, 70 experiments (d): 4 clusters, 64 graphs, 10 nodes each, 360 experiments to form the object instance and uniform noise was added to the link weights. 64 object instances were generated from 4 object models consisting of 10 node graphs with 100 links. The standard deviation of the noise was varied by .01 from .01 to .12. 30 experiments where run at each noise level. In most experiments at low noise levels (~ .06 for point sets, ~ .03 for graphs), the object prototypes learned were very similar to the object models. Even at higher noise levels object prototypes similar to the object models are formed, though less consistently. Results from about 700 experiments are plotted. The objective for experiment 3 contained close to one million variables and converged in about 4 hours on an SGI Indigo workstation. The convergence times of the objectives of experiments 1, 2 and 4 were 120, 10 and 10 minutes respectively. 5 Conclusions It has long been argued by many, that learning in complex domains typically associated with human intelligence requires some type of prior structure or knowledge. We have begun to develop a set of tools that will allow the incorporation of prior 720 Steven Gold, Anand Rangarajan, Eric Mjolsness structure within learning. Our models incorporate many features needed in complex domains like vision - noise, missing and spurious features, non-rigid transformations. They can learn objects with inherent structure, like graphs. Many experiments have been run on experimentally generated data sets. Several directions for future research hold promise. One might be the learning of OCR data [Gold et al., 1995]. Second a supervised learning stage could be added to our algorithms. Finally the power of the distance measures can be enhanced to operate on attributed relational graphs with deleted nodes and links [Gold and Rangarajan, 1995]. Acknowledgements ONR/DARPA: NOOOI4-92-J-4048, AFOSR: F49620-92-J-0465 and Yale CTAN. References S. Geman, E. Bienenstock, and R. Doursat. (1992) Neural networks and the bias/variance dilemma. Neural Computation 4:1-58. S. Gold, E. Mjolsness and A. Rangarajan. (1994) Clustering with a domain-specific distance measure. In J . Cowan et al., (eds.), NIPS 6. Morgan Kaufmann. S. Gold, C. P. Lu, A. Rangarajan, S. Pappu and E. Mjolsness. (1995) New algorithms for 2D and 3D point matching: pose estimation and correspondence. In G. Tesauro et al., (eds.), NIPS 7. San Francisco, CA: Morgan Kaufmann. S. Gold and A. Rangarajan (1995) A graduated assignment algorithm for graph matching. YALEU/DCS/TR-I062, Yale Univ., CS Dept. M. I. Jordan and R. A. Jacobs. (1994) Hierarchical mixtures of experts and the EM algorithm. Neural Computation, 6:181-214. E. Mjolsness. Visual grammars and their neural networks. (1992) SPIE Conference on the Science of Artificial Neural Networks, 1110:63-85. C. Peterson and B. Soderberg. A new method for mapping optimization problems onto neural networks. (1989) International Journal of Neural Systems,I(1):3-22. A. Rangarajan, S. Gold and E. Mjolsness. (1994) A novel optimizing network architecture with applications. YALEU /DCS/TR-I036, Yale Univ. , CS Dept. R. Shepard. (1989). Internal representation of universal regularities: A challenge for connectionism. In L. Nadel et al. , (eds.), Neural Connections, Mental Computation. Cambridge, MA, London, England: Bradford/MIT Press. P. Simard, Y. Le Cun, and J. Denker. (1993) Efficient pattern recognition using a transformation distance. In S. Hanson et ai., (eds.), NIPS 5. San Mateo, CA: Morgan Kaufmann. C. Williams, R. Zemel, and M. Mozer. (1993) Unsupervised learning of object models. AAAI Tech. Rep. FSS-93-04, Univ. of Toronto, CS Dept. A. L. Yuille and J .J. Kosowsky. (1994) . Statistical physics algorithms that converge. Neural Computation, 6:341-356.	1994	4
932	Non-linear Prediction of Acoustic Vectors Using Hierarchical Mixtures of Experts S.R.Waterhouse A.J.Robinson Cambridge University Engineering Department, Trumpington St., Cambridge, CB2 1PZ, England. Tel: [+44] 223 332800, Fax: [+44] 223 332662, Email: srwlO01.ajr@eng.cam.ac.uk URL: http://svr-www.eng.cam.ac.ukr srw1001 Abstract In this paper we consider speech coding as a problem of speech modelling. In particular, prediction of parameterised speech over short time segments is performed using the Hierarchical Mixture of Experts (HME) (Jordan & Jacobs 1994). The HME gives two advantages over traditional non-linear function approximators such as the Multi-Layer Percept ron (MLP); a statistical understanding of the operation of the predictor and provision of information about the performance of the predictor in the form of likelihood information and local error bars. These two issues are examined on both toy and real world problems of regression and time series prediction. In the speech coding context, we extend the principle of combining local predictions via the HME to a Vector Quantization scheme in which fixed local codebooks are combined on-line for each observation. 1 INTRODUCTION We are concerned in this paper with the application of multiple models, specifically the Hierarchical Mixtures of Experts, to time series prediction, specifically the problem of predicting acoustic vectors for use in speech coding. There have been a number of applications of multiple models in time series prediction. A classic example is the Threshold Autoregressive model (TAR) which was used by Tong & 836 S. R. Waterhouse, A. J. Robinson Lim (1980) to predict sunspot activity. More recently, Lewis, Kay and Stevens (in Weigend & Gershenfeld (1994)) describe the use of Multivariate and Regression Splines (MARS) to the prediction of future values of currency exchange rates. Finally, in speech prediction, Cuperman & Gersho (1985) describe the Switched Inter-frame Vector Prediction (SIVP) method which switches between separate linear predictors trained on different statistical classes of speech. The form of time series prediction we shall consider in this paper is the single step prediction fI(t) of a future quantity y(t) , by considering the previous c: samples. This may be viewed as a regression problem over input-output pairs {x t), y(t)}~ where x(t) is the lag vector (y(t-I), y(t-2), ... , y(t-p». We may perform this regression using standard linear models such as the Auto-Regressive (AR) model or via nonlinear models such as connectionist feed-forward or recurrent networks. The HME overcomes a number of problems associated with traditional connectionist models via its architecture and statistical framework. Recently, Jordan & Jacobs (1994) and Waterhouse & Robinson (1994) have shown that via the EM algorithm and a 2nd order optimization scheme known as Iteratively Reweighted Least Squares (IRLS), the HME is faster than standard Multilayer Perceptrons (MLP) by at least an order of magnitude on regression and classification tasks respectively. Jordan & Jacobs also describe various methods to visualise the learnt structure of the HME via 'deviance trees' and histograms of posterior probabilities. In this paper we provide further examples of the structural relationship of the trained HME and the input-output space in the form of expert activation plots. In addition we describe how the HME can be extended to give local error bars or measures of confidence in regression and time series prediction problems. Finally, we describe the extension of the HME to acoustic vector prediction, and a VQ coding scheme which utilises likelihood information from the HME. 2 HIERARCHICAL MIXTURES OF EXPERTS The HME architecture (Figure 1) is based on the principle of 'divide and conquer' in which a large, hard to solve problem is broken up into many, smaller, easier to solve problems. It consists of a series of 'expert networks' which are trained on different parts of the input space. The outputs of the experts are combined by a 'gating network' which is trained to stochastically select the expert which is performing best at solving a particular part of the problem. The operation of the HME is as follows: the gating networks receive the input vectors x(t) and produce as outputs probabilities P(mi/.x(t), 7'/j) for each local branch mj of assigning the current input to the different branches, where T/j are the gating network parameters. The expert networks sit at the leaves of the tree and each output a vector flJt) given input vector x(t) and parameters Bj . These outputs are combined in a weighted sum by P(mjlX<t), T/j) to give the overall output vector for this region. This procedure continues recursively upwards to the root node. In time series prediction, each expert j is a linear single layer network with the form: flY) = B; x (t) where B; is matrix and x(t) is the lag vector discussed earlier, which is identical in form to an AR model. Non-Linear Prediction of Acoustic Vectors Using Hierarchical Mixtures of Experts 837 x x x x Figure 1: The Hierarchical Mixture of Experts. 2.1 Error bars via HME Since each expert is an AR model, it follows that the output of each expert y(t) is the expected value of the observations y(t) at each time t. The conditional likelihood of yet) given the input and expert mj is P(y(t) I x (t), mj, Bj) = 12:Cj I exp ( - ~ (y - yy»)T Cj(y - yjt))) where Cj is the covariance matrix for expert mj which is updated during training as: C = _1_ "'" h(t)(y(r) _ y(t)l (y(t) _ y~t)) J '" h~t) L..J J ] ] L.Jt J t where hy) are the posterior probabilities I of each expert mj' Taking the moments of the overall likelihood of the HME gives the output of the HME as the conditional expected value of the target output yct), yet) = E(yct)lxct), 0, M) = 2: P(mjlxct), l1j)E(y(t)lxct), ej,mj) = 2: gY)iJ/t), j j Where M represents the overall HME model and e the overall set of parameters. Taking the second central moment of yct) gives, C = E«y(t) - yy»)2I xct), 0, M) = 2: P(mJlx(t), l1j)E«y(t) - yjt))2I x (t), ej, mj) j = 2: gjt)(Cj + yjt). iJj(t)T), j lSee (Jordan & Jacobs 1994) for a fuller discussion of posterior probabilities and likelihoods in the context of the HME. 838 S. R. Waterhouse, A. J. Robinson which gives, in a direct fashion, the covariance of the output given the input and the model. If we assume that the observations are generated by an underlying model, which generates according to some function f(x(t)) and corrupted by zero mean normally distributed noise n(x) with constant covariance 1:, then the covariance of y(t) is given by, V(y(t)) = V(to) + 1:, so that the covariance computed by the method above, V(y(t)), takes into account the modelling error as well as the uncertainty due to the noise. Weigend & Nix (1994) also calculate error bars using an MLP consisting of a set of tanh hidden units to estimate the conditional mean and an auxiliary set of tanh hidden units to estimate the variance, assuming normally distributed errors. Our work differs in that there is no assumption of normality in the error distribution, rather that the errors of the terminal experts are distributed normally, with the total error distribution being a mixture of normal distributions. 3 SIMULATIONS In order to demonstrate the utility of our approach to variance estimation we consider one toy regression problem and one time series prediction problem. 3.1 Toy Problem: Computer generated data 2,------..--...., 1.5 ~-0.5 -1 -0.2 -1.5 -0.4 -2 -O.S -2.5 -0.8 -3 '------'------' o 2 -1 '------'-----" o 2 0 . 08,------~-..., N O.OS ~ fO.04 ~ -0.02 0.8 Q) go.S .~ ~0.4 2 x 2 x x x Figure 2: Performance on the toy data set of a 5 level binary HME. (a) training set (dots) and underlying function f(x) (solid), (b) underlying function (solid) and prediction y(x) (dashed), (c) squared deviation of prediction from underlying function, (d) true noise variance (solid) and variance of prediction (dashed). By way of comparison, we used the same toy problem as Weigend & Nix (1994) which consists of 1000 training points and 10000 separate evaluation points from Non-Linear Prediction of Acoustic Vectors Using Hierarchical Mixtures of Experts 839 the function g(x) where g(x) consists of a known underlying function f(x) corrupted by normally distributed noise N(O, (J2(X)) , f(x) = sin(2.5x) x sin(l. 5x), (J2(x) = 0.01 + O. 25 x [1 - sin(2.5x)f. As can be seen by Figure 2, the HME has learnt to approximate both the underlying function and the additive noise variance. The deviation of the estimated variance from the "true" noise variance may be due to the actual noise variance being lower than the maximum denoted by the solid line at various points. 3.2 Sunspots 1920 1930 1940 8 • • 6 .. t 8.4 - • )( w .2 - . • • .. 1950 Year II 1960 • .. l• • • • • • 1970 1980 I • ... -• -.. • • OL-------~--------~--------~--------L-------~--------~ 1920 1930 1940 1950 1960 1970 1980 Year Figure 3: Performance on the Sunspots data set. (a) Actual Values (x) and predicted values (0) with error bars. (b) Activation of the expert networks; bars wide in the vertical axis indicate strong activation. Notice how expert 7 concentrates on the lulls in the series while expert 2 deals with the peaks. I METHOD I NMSE' Train Test 1700-1920 1921-1955 1956-1979 MLP 0.082 0.086 0.35 TAR 0.097 0.097 0.28 HME 0.061 0.089 0.27 Table 1: Results of single step prediction on the Sunspots data set using a mixture of 7 experts (104 parameters) and a lag vector of 12 years. NMSE' is the NMSE normalised by the variance of the entire record 1700 to 1979. 840 S. R. Waterhouse, A. J. Robinson The Sunspots2 time series consists of yearly sunspot activity from 1700 to 1979 and was first tackled using connectionist models by Weigend, Huberman & Rumelhart (1990) who used a 12-8-1 MLP (113 parameters) . Prior to this work, the TAR was used by Tong (1990). Our results, which were obtained using a random leave 10% out cross validation method, are shown in Table 1. We are considering only single step prediction on this problem, which involves prediction of the next value based on a set of previous values of the time series. Our results are evaluated in terms of Normalised Mean Squared Error (NMSE) (Weigend et al. 1990), which is defined as the ratio of the variance of the prediction on the test set to the variance of the test set itself. The HME outperforms both the TAR and the MLP on this problem, and additionally provides both information about the structure of the network after training via the expert activation plot and error bars of the predictions, as shown in Figure 3. Further improvements may be possible by using likelihood information during cross validation so that a joint optimisation of overall error and variance is achieved. 4 SPEECH CODING USING HME In the standard method of Linear Predictive Coding (LPC) (Makhoul 1975), speech is parametrised into a set of vectors of duration one frame (around 10 ms). Whilst simple scalar quantization of the LPC vectors can achieve bit rates of around 2400 bits per second (bps), Yong, Davidson & Gersho (1988) have shown that simple linear prediction of Line Spectral Pairs (LSP) (Soong & Juang 1984) vectors followed by Vector Quantization (VQ) (Abut, Gray & Rebolledo 1984) of the error vectors can yield bit rates of around 800 bps. In this paper we describe a speech coding framework which uses the HME in two stages. Firstly, the HME is used to perform prediction of the acoustic vectors. The error vectors are then quantized efficiently by using a VQ scheme which utilises the likelihood information derived from the HME. 4.1 Mixing VQ codebooks ia Gating networks In a VQ scheme using a Euclidean distance measure, there is an implicit assumption that the inputs follow a Gaussian probability density function (pdf). This is satisfied if we quantize the residuals from a linear predictor, but not the residuals from an HME which follow a mixture of Gaussians pdf. A more efficient method is therefore to generate separate VQ code books for each expert in the HME and combine them via the priors on each expert from the gating networks. The code book for the overall residual vectors on the test set is then generated at each time dynamically by choosing the first D x gjt) codes, where D is the size of the expert codebooks and gY) is the prior on each expert. 2 Available via anonymous ftp at fip.cs.colorado.edu III jpub jTime-Series as DataSunspots.Yearly Non-Linear Prediction of Acoustic Vectors Using Hierarchical Mixtures of Experts 841 4.2 Results of Speech Coding Evaluations Initial experiments were performed using 23 Mel scale log energy frequency bins as acoustic vectors and using single variances Cj = (J;I as expert network covariance matrices. The results of training over 100,000 frames and evaluation over a further 100,000 frames on the Resource Management (RM) corpus are shown in Table 2 and Figure 4 which shows the good specialisation of the HME in this problem. METHOD Prediction Gain (dB) Train Test Linear 12.07 10.95 1 level HME 18.1 15.55 2 level HME 20.20 16.39 Table 2: Prediction of Acoustic Vectors using linear prediction and binary branching HMEs with 1 and 2 levels. Prediction gain (Cuperman & Gersho 1985) is the ratio of the signal variance to prediction error variance. (a) Spectogram 20 !!! ::3 c;; 15 Cf ~ 10 ::3 . 8 < 5 o 10 20 30 40 50 Frame 60 (b) Gating Network Decisions 71--- --__.I 70 80 90 100 I 6~----.... ~.---~--__ --------_=--~~-- ~5~~---------1~---.I---1IHI~I~·--.... ~.~.----~4~~~----------------~~~~---------w3~~---------~-----.. ~ __ ------_.-----------------2.-.------- --11-1...---11---1- --11-1 •• I 1~--------------~~~~--------------------------------~ .. ...I • II 10 20 30 40 50 60 Frame 70 80 90 100 Figure 4: The behaviour of a mixture of 7 experts at predicting Mel-scale log energy frequency bins over 100 16ms frames. The top figure is a spectrogram of the speech and the lower figure is an expert activation plot, showmg the gating network decisions. We have conducted further experiments using LSPs and cepstrals as acoustic vectors, and using diagonal expert network covariance matrices, on a very large speech corpus. However, initial experiments show only a small improvement in gain over a single linear predictor and further investigation is underway. We have also coded acoustic vectors using 8 bits per frame with frame lengths of 12.5 ms, passing power, pitch and degree of voicing as side band information, without appreciable distortion over simple LPC coding. A full system will include prediction of all acoustic 842 S. R. Waterhouse, A. J. Robinson parameters and we anticipate further reductions on this initial figure with future developments. 5 CONCLUSION The aim of speech coding is the efficient coding of the speech signal with little perceptual loss. This paper has described the use of the HME for acoustic vector prediction. We have shown that the HME can provide improved performance over a linear predictor and in addition it provides a time varying variance for the prediction error. The decomposition of the linear prediction problem into a solution via a mixture of experts also allows us to construct a VQ codebook on the fly by mixing the codebooks of the various experts. We expect that the direct computation of the time varying nature of the prediction accuracy will find many applications. Within the acoustic vector prediction problem we would like to exploit this information by exploring the continuum between the fixed bit rate coder described here and a variable bit rate coder that produces constant spectral distortion. Acknowledgements This work was funded in part by Hewlett Packard Laboratories, UK. Steve Waterhouse is supported by an EPSRC Research Students hip and Tony Robinson was supported by a EPSRC Advanced Research Fellowship. References Abut, H., Gray, R. M. & Rebolledo, G. (1984), 'Vector quantization of speech and speechlike waveforms', IEEE Transactions on Acoustics, Speech, and Signal Processing. Cuperman, V. & Gersho, A. (1985), 'Vector predictive coding of speech at 16 kblt/s', IEEE Transactions on Communications COM-33, 685-696. Jordan, M. I. & Jacobs, R. A. (1994), 'Hierarchical Mixtures of Experts and the EM algorithm', Neural Computation 6, 181-214. Makhoiil, J. (1975), 'Linear prediction: A tutorial review', Proceedings of the IEEE 63(4) 561-580. Soong~ F. k. & Juang, B. H. (1984), Line spectrum pair (LSP) and speech data compressIon. Tong, H. (1990), Non-linear Time Series: a dynamical systems approach, Oxford Universi~y Press. Tong, H. & Lim, K. (1980), 'Threshold autoregression, limit cycles and cyclical data', Journal of Royal Statistical Society. Waterhouse, S. R. & Robinson, A. J. (1994), Classification using hierarchical mixtures of experts, in 'IEEE Workshop on Neural Networks for SigI!al Processing'. Weigend, A. S. & Gershenfeld, N. A. (1994), Time Series Prediction: Forecasting the Future and Understanding the Past) Addison-Wesley. Weigend, A. S. & Nix, D. A. (1994), Predictions with confidence intervals (local error bars), Technical Report CU-CS-724-94, Department of Computer Science and Institute of Coznitive Science, University of Colorado, Boulder, CO 80309-0439. Weigend, A. S., Huberman, B. A. & Rumelhart, D. E. (1990), 'Predicting the future: a connectionist approach', International Journal of Neural Systems 1, 193-209. Yong, M., Davidson, G. & Gersho, A. (1988), Encoding of LPC spectral parameters using switched-adaptive interframe vector prediction, in 'Proceedings of the IEEE International Conference on Acoustics Speech, and Signal Processing', pp. 402-405.	1994	40
933	JPMAX: Learning to Recognize Moving Objects as a Model-fitting Problem Suzanna Becker Department of Psychology, McMaster University Hamilton, Onto L8S 4K1 Abstract Unsupervised learning procedures have been successful at low-level feature extraction and preprocessing of raw sensor data. So far, however, they have had limited success in learning higher-order representations, e.g., of objects in visual images. A promising approach is to maximize some measure of agreement between the outputs of two groups of units which receive inputs physically separated in space, time or modality, as in (Becker and Hinton, 1992; Becker, 1993; de Sa, 1993). Using the same approach, a much simpler learning procedure is proposed here which discovers features in a single-layer network consisting of several populations of units, and can be applied to multi-layer networks trained one layer at a time. When trained with this algorithm on image sequences of moving geometric objects a two-layer network can learn to perform accurate position-invariant object classification. 1 LEARNING COHERENT CLASSIFICATIONS A powerful constraint in sensory data is coherence over time, in space, and across different sensory modalities. An unsupervised learning procedure which can capitalize on these constraints may be able to explain much of perceptual self-organization in the mammalian brain. The problem is to derive an appropriate cost function for unsupervised learning which will capture coherence constraints in sensory signals; we would also like it to be applicable to multi-layer nets to train hidden as well as output layers. Our ultimate goal is for the network to discover natural object classes based on these coherence assumptions. 934 Suzanna Becker 1.1 PREVIOUS WORK Successive images in continuous visual input are usually views of the same object; thus, although the image pixels may change considerably from frame to frame, the image usually can be described by a small set of consistent object descriptors, or lower-level feature descriptors. We refer to this type of continuity as temporal coherence. This sort of structure is ubiquitous in sensory signals, from vision as well as other senses, and can be used by a neural network to derive temporally coherent classifications. This idea has been used, for example, in temporal versions of the Hebbian learning rule to associate items over time (Weinshall, Edelman and B iilt hoff, 1990; FOldiak, 1991). To capitalize on temporal coherence for higher-order feature extraction and classification, we need a more powerful learning principle. A promising approach is to maximize some measure of agreement between the outputs of two groups of units which receive inputs physically separated in space, time or modality, as in (Becker and Hinton, 1992; Becker, 1993; de Sa, 1993). This forces the units to extract features which are coherent across the different input sources. Becker and Hinton's (1992) Imax algorithm maximizes the mutual information between the outputs of two modules, y~ and Yb, connected to different parts of the input, a and b. Becker (1993) extended this idea to the problem of classifying temporally varying patterns by applying the discrete case of the mutual information cost function to the outputs of a single module at successive time steps, y-;'(t) and y-;'(t + 1). However, the success of this method relied upon the back-propagation of derivatives to train the hidden layer and it was found to be extremely susceptible to local optima. de Sa's method (1993) is closely related, and minimizes the probability of disagreement between output classifications, y~(t) and y1(t), produced by two modules having different inputs, e.g., from different sensory modalities. The success of this method hinges upon bootstrapping the first layer by initializing the weights to randomly selected training patterns, so this method too is susceptible to the problem of local optima. If we had a more flexible cost function that could be applied to a multi-layer network, first to each hidden layer in turn, and finally to the~utput layer for classification, so that the two layers could discover genuinely different structure, we might be able to overcome the problem of getting trapped in local optima, yielding a more powerful and efficient learning procedure. We can analyze the optimal solutions for both de Sa's and Becker's cost functions (see Figure 1 a) and see that both cost functions are maximized by having perfect one-to-one agreement between the two groups of units over all cases, using a one-of-n encoding, i.e., having only a single output unit on for each case. A major limitation of these methods is that they strive for perfect classifications by the units. While this is desirable at the top layer of a network, it is an unsuitable goal for training intermediate layers to detect low-level features. For example, features like oriented edges would not be perfect predictors across spatially or temporally nearby image patches in images of translating and rotating objects. Instead, we might expect that an oriented edge at one location would predict a small range of similar orientations at nearby locations. So we would prefer a cost function whose optimal solution was more like those shown in Figure 1 b) or c). This would allow a feature i in group a to agree with any of several nearby features, e.g. i - 1, i, or i + 1 in group b. JPMAX 935 a) b) c) • II • • , === .!iii .. === ;;; ...... , •• 11 II' ill!! •••• I!!!!! •• • II =-.1. I' • • 11, ••• I' • • 11 n = • • • • ;;; = I!!!!! • I!!IJ!!' .. :== II' •• • == -= = • •• • •• • I ••• II' I • • II .. • • , • • • ii ••• II "'-= .i. ·1' ••• • J[., • •• • • II .'. ;;; •• .. . == .i. Ilii • • Figure 1: Three possible joint distributions for the probability that the i th and j th units in two sets of m classification units are both on. White is high density, black is low density. The optimal joint distribution for Becker's and de Sa's algorithms is a matrix with all its density either in the diagonal as in a), or any subset of the diagonal entries for de Sa's method, or a permutation of the diagonal matrix for Becker's algorithm. Alternative distributions are shown in b) and c). 1.2 THE JPMAX ALGORITHM One way to achieve an arbitrary configuration of agreement over time between two groups of units (as in Figure 1 b) or c» is to treat the desired configuration as a prior joint probability distribution over their outputs. We can obtain the actual distribution by observing the temporal correlations between pairs of units' outputs in the two groups over an ensemble of patterns. We can then optimize the actual distribution to fit the prior. We now derive two different cost functions which achieve this result. Interestingly, they result in very similar learning rules. Suppose we have two groups of m units as shown in Figure 2 a), receiving inputs, x"7t and xi" from the same or nearby parts of the input image. Let Ca(t) and Cb(t) be the classifications of the two input patches produced by the network at time step t; the outputs of the two groups of units, y"7t(t) and yi,(t), represent these classification probabilities: enetoi(t) Yai(t) P(Ca(t) = i) = Lj eneto;(t) enetbi (t) Ybi(t) = P(Cb(t) = i) = Lj enetb;(t) (1) (the usual "soft max" output function) where netai(t) and netbj(t) are the weighted net inputs to units. We could now observe the expected joint probability distribution qij = E [Yai(t)Ybj(t + 1)]t = E (P(Ca(t) = i, Cb(t + 1) = j)]t by computing the temporal covariances between the classification probabilities, averaged over the ensemble of training patterns; this joint probability is an m2-valued random variable. Given the above statistics, one possible cost function we could minimize is the -log probability of the observed temporal covariance between the two sets of units' outputs under some prior distribution (e.g. Figure 1 b) or c». If we knew the actual frequency counts for each (joint) classification k = kll ,·· ., kIm, k21 , ... ,kmm, 936 Suzanna Becker b) At b'~-#--#----------"'-" """-"''''''''''''''''''''':iIo. <t (I) t Figure 2: a) Two groups of 15 units receive inputs from a 2D retina. The groups are able to observe each other's outputs across lateral links with unit time delays. b) A second layer of two groups of 3 units is added to the architecture in a). rather than just the observed joint probabilities, qij = E [~] , then given our prior model, pu, ... ,Pmm, we could compute the probability of the observations under a multinomial distribution: (2) Using the de Moivre-Laplace approximation leads to the following: P(k) ~ 1 exp (_! L (kij n Pij)2) v(27rn)m 2 -1 IL,j Pij 2 i,j npij (3) Taking the derivative of the - log probability wrt kij leads to a very simple learning rule which depends only on the observed probabilities qij and priors Pij: fJ -logP(k) npij k ij fJkij npij ~ Pij qij (4) Pij To obtain the final weight update rule, we just multiply this by n %!i:l . One problem with the above formulation is that the priors Pij must not be too close to zero for the de Moivre-Laplace approximation to hold. In practice, this cost function works well if we simply ignore the derivative terms where the priors are zero. An alternative cost function (as suggested by Peter Dayan) which works equally well is the Kullback-Liebler divergence or G-error between the desired joint probabilities Pij and the observed probabilities qij: G(p,q) = - LL (Pij logpij - Pij lOgqij) j (5) JPMAX 937 Figure 3: 10 of the 1500 tmining patterns: geometric objects centered in 36 possible locations on a 12-by-12 pixel grid. Object location varied mndomly between patterns. The derivative of G wrt qij is: aG Pij aqij qij subject to Llij qij = 1 (enforced by the softmax output function). larity between the learning rules given by equations 4 and 6. 2 EXPERIMENTS (6) Note the simiThe network shown in Figure 2 a) was trained to minimize equation 5 on an ensemble of pattern trajectories of circles, squares and triangles (see Figure 3) for five runs starting from different random initial weights, using a gradient-based learning method. For ten successive frames, the same object would appear, but with the centre varying randomly within the central six-by-six patch of pixels. In the last two frames, another randomly selected object would appear, so that object trajectories overlapped by two frames. These images are meant to be a crude approximation to what a moving observer would see while watching multiple moving objects in a scene; at any given time a single object would be approximately centered on the retina but its exact location would always be jittering from moment to moment. In these simulations, the prior distribution for the temporal covariances between the two groups of units' outputs was a block-diagonal configuration as in Figure 1 c), but with three five-by-five blocks along the diagonal. Our choice of a blockdiagonal prior distribution with three blocks encodes the constraint that units in a given block in one group should try to agree with units in the same block in the other group; so each group should discover three classes of features. The number of units within a block was varied in preliminary experiments, and five units was found to be a reasonable number to capture all instances of each object class (although the performance of the algorithm seemed to be robust with respect to the number of units per block). The learning took about 3000 iterations of steepest descent with 938 Suzanna Becker .----.... . .. . .. . . . . . .. .... . .. .• .•• .•• .. ....... .• -1-· ... ... .. . I' ..•. . ... ... .. • ~~;. i1.=::::.: ;~. :~:" ' ~:. ~::.... :: . ...... .. . .. . .. .. . . ... .. . .. ,'. ... . ... .... . . ... • 1 ••• 1: ..... -.1 I' • ••• • ••• ••. :1" .. ....... . •• . . .... .. • · . .. ..... .' ..... .. . ... .; . .. .. .. • .. .. 'a_" _" •• .. ... I . _ _ _ _ _ , . .. .. e o _ .. .... __ ----,........ •••••• .. ... .. I •• . • . •• . '.. . • . . ....... ••••• II .... .... .1. .. . .... ..... . .•. . . ........ . ..... -i·" . .. . .. ..... • 1 .. • .. ·1 •••••••••• ~ .••..••••• :.. . ••• • 1 • .. • • •• ' .a 1 ' 1 •• 0 ; ' • .. ':. • •• .. i " " . . ' ;' • ·i ........... : " ._ , ' . • •• :... a • •••...• ~.~.: i .. ~ .. ~i~" ..• ::: ") ;: . :': :.~:~ji----.. ......... ·: .......... ::: .;~ I ... .; .. , .... .... .... . .. . . ..... i·· .. ··.. ..... .. .... .. ... I· .. .•. .... .. ...... .. ...... •• a •••• · .... . .. • .. I .. " ..... • • • • .. • ....... . · ..... . a • • • •• i....... . . . . .. . . ........ . :::' ":: \~. . ..... . . . Figure 4: Weights learned by one of the two groups of 15 units in the first layer. White weights are positive and black are negative. momentum to converge, but after about 1000 iterations only very small weight changes were made. Weights learned on a typical run for one of the two groups of fifteen units are shown in Figure 4. The units' weights are displayed in three rows corresponding to units in the three blocks in the block-diagonal joint prior matrix. Units within the same block each learned different instances of the same pattern class. For example, on this run units in the first block learned to detect circles in specific positions. Units in the second block tended to learn combinations of either horizontal or vertical lines, or sometimes mixtures of the two. In the third block, units learned blurred, roughly triangular shape detectors, which for this training set were adequate to respond specifically to triangles. In all five runs the network converged to equivalent solutions (only the groups' particular shape preferences varied across runs). Varying the number of units per block from three to five (Le. three three-by-three blocks versus three five-by-five blocks of units) produced similar results, except that with fewer units per block, each unit tended to capture multiple instances of a particular object class in different positions. A second layer of two groups of three units was added to the network, as shown in Figure 2 b). While keeping the first layer of weights frozen, this network was trained using exactly the same cost function as the first layer for about 30 iterations using a gradient-based learning method. This time the prior joint distribution for the two classifications was a three-by-three matrix with 80% of the density along the diagonal and 20% evenly distributed across the remainder of the distribution. Units in this layer learned to discriminate fairly well between the three object classes, as shown in Figure 5 a). On a test set with the ambiguous patterns removed (Le., patterns containing multiple objects), units in the second layer achieved very JPMAX a) 1. ~ 0.80 . . ~ 0.50 c ~ 1 '0 0.40 c i £ 0.20 0.00 b) 1.00 c 0 .. :;; 0.80 0 . . ; 0.60 i '0 0.40 c 0 0 1°,20 0.00 Loyer 2 Unit Responses on Training Set I Circles Squares Triangles Loyer 2 Unit Responses on Test Set (no overlaps) LIllI • • .... ----. UM3 Unit 6 939 Figure 5: Response probabilities for the six output units to each of the three shapes. accurate object discrimination as shown in Figure 5 b). On ambiguous patterns containing multiple objects, the network's performance was disappointing. The output units would sometimes produce the "correct" response, i.e., all the units representing the shapes present in the image would be partially active. Most often, however, only one of the correct shapes would be detected, and occasionally the network's response indicated the wrong shape altogether. It was hoped· that the diagonally dominant prior mixed with a uniform density would allow units to occasionally disagree, and they would therefore be able to represent cases of multiple objects. It may have helped to use a similar prior for the hidden layer; however, this would increase the complexity of the learning considerably. 3 DISCUSSION We have shown that the algorithm can learn 2D-translation-invariant shape recognition, but it should handle equally well other types of transformations, such as rotation, scaling or even non-linear transformations. In principle, the algorithm should be applicable to real moving images; this is currently being investigated. Although we have focused here on the temporal coherence constraint, the algorithm could be applied equally well using other types of coherence, such as coherence 940 Suzanna Becker across space or across different sensory modalities. Note that the units in the first layer of the network did not learn anything about the geometric transformations between translated versions of the same object; they simply learned to associate different views together. In this respect, the representation learned at the hidden layer is similar to that predicted by the "privileged views" theory of viewpoint-invariant object recognition advocated by Weinshall et al. (1990) (and others). Their algorithm learns a similar representation in a single layer of competing units with temporal Hebbian learning applied to the lateral connections between these units. However, the algorithm proposed here goes further in that it can be applied to subsequent stages of learning to discover higher-order object classes. Yuille et al. (1994) have previously proposed an algorithm based on similar principles, which also involves maximizing the log probability of the network outputs under a prior; in one special case it is equivalent to Becker and Hinton's Imax algorithm. The algorithm proposed here differs substantially, in that we are dealing with the ensemble-averaged joint probabilities of two populations of units, and fitting this quantity to a prior; further, Yuille et aI's scheme employs back-propagation. One challenge for future work is to train a network with smaller receptive fields for the first layer units, on images of objects with common low-level features, such as squares and rectangles. At least three layers of weights would be required to solve this task: units in the first layer would have to learn local object parts such as corners, while units in the next layer could group parts into viewpoint-specific whole objects and in the top layer viewpoint-invariance, in principle, could be achieved. Acknowledgements Helpful comments from Geoff Hinton, Peter Dayan, Tony Plate and Chris Williams are gratefully acknowledged. References Becker, S. (1993). Learning to categorize objects using temporal coherence. In Advances in Neural Information Processing Systems 5, pages 361- 368. Morgan Kaufmann. Becker, S. and Hinton, G. E. (1992). A self-organizing neural network that discovers surfaces in random-dot stereograms. Nature, 355: 161-163. de Sa, V. R. (1993). Minimizing disagreement for self-supervised classification. In Proceedings of the 1993 Connectionist Models Summer School, pages 300-307. Lawrence Erlbaum associates. F51diak, P. (1991). Learning invariance from transformation sequences. Neural Computation, 3(2):194-200. Weinshall, D., Edelman, S., and Biilthoff, H. H. (1990). A self-organizing multiple-view representation of 3D objects. In Advances in Neural Information Processing Systems 2, pages 274-282. Morgan Kaufmann. Yuille, A. L., Stelios, M. S., and Xu, L. (1994). Bayesian Self-Organization. Technical Report No. 92-10, Harvard Robotics Laboratory.	1994	41
934	The Electrotonic Transformation: a Tool for Relating Neuronal Form to Function Nicholas T. Carnevale Department of Psychology Yale University New Haven, CT 06520 Brenda J. Claiborne Division of Life Sciences University of Texas San Antonio, TX 79285 Abstract Kenneth Y. Tsai Department of Psychology Yale University New Haven, CT 06520 Thomas H. Brown Department of Psychology Yale University New Haven, CT 06520 The spatial distribution and time course of electrical signals in neurons have important theoretical and practical consequences. Because it is difficult to infer how neuronal form affects electrical signaling, we have developed a quantitative yet intuitive approach to the analysis of electrotonus. This approach transforms the architecture of the cell from anatomical to electrotonic space, using the logarithm of voltage attenuation as the distance metric. We describe the theory behind this approach and illustrate its use. 1 INTRODUCTION The fields of computational neuroscience and artificial neural nets have enjoyed a mutually beneficial exchange of ideas. This has been most evident at the network level, where concepts such as massive parallelism, lateral inhibition, and recurrent excitation have inspired both the analysis of brain circuits and the design of artificial neural net architectures. Less attention has been given to how properties of the individual neurons or processing elements contribute to network function. Biological neurons and brain circuits have 70 Nicholas Carnevale, Kenneth Y. Tsai, Brenda J. Claiborne, Thomas H. Brown been simultaneously subject to eons of evolutionary pressure. This suggests an essential interdependence between neuronal form and function, on the one hand, and the overall architecture and operation of biological neural nets, on the other. Therefore reverseengineering the circuits of the brain appears likely to reveal design principles that rely upon neuronal properties. These principles may have maximum utility in the design of artificial neural nets that are constructed of processing elements with greater similarity to biological neurons than those which are used in contemporary designs. Spatiotemporal extent is perhaps the most obvious difference between real neurons and processing elements. The processing element of most artificial neural nets is essentially a point in time and space. Its activation level is the instantaneous sum of its synaptic inputs. Of particular relevance to Hebbian learning rules, all synapses are exposed to the same activation level. These simplifications may insure analytical and implementational simplicity, but they deviate sharply from biological reality. Membrane potential, the biological counterpart of activation level, is neither instantaneous nor spatially uniform. Every cell has finite membrane capacitance, and all ionic currents are finite, so membrane potential must lag behind synaptic inputs. Furthermore, membrane capacitance and cytoplasmic resistance dictate that membrane potential will almost never be uniform throughout a living neuron embedded in the circuitry of the brain. The combination of ever-changing synaptic inputs with cellular anatomical and biophysical properties guarantees the existence of fluctuating electrical gradients. Consider the task of building a massively parallel neural net from processing elements with such "nonideal" characteristics. Imagine moreover that the input surface of each processing element is an extensive, highly branched structure over which approximately 10,000 synaptic inputs are distributed. It might be tempting to try to minimize or work around the limitations imposed by device physics. However, a better strategy might be to exploit the computational consequences of these properties by making them part of the design, thereby turning these apparent weaknesses into strengths. To facilitate an understanding of the spatiotemporal dynamics of electrical signaling in neurons, we have developed a new theoretical approach to linear electrotonus and a new way to make practical use of this theory. We present this method and illustrate its application to the analysis of synaptic interactions in hippocampal pyramidal cells. 2 THEORETICAL BACKGROUND Our method draws upon and extends the results of two prior approaches: cable theory and two-port analysis. 2.1 CABLE THEORY The modern use of cable theory in neuroscience began almost four decades ago with the work of RaIl (1977). Much of the attraction of cable theory derives from the conceptual simplicity of the steady-state decay of voltage with distance along an infinite cylindrical cable: V(x) = Voe-xl). where x is physical distance and .4 is the length constant. This exponential relationship makes it useful to define the electrotonic distance X as the The Electronic Transfonnation: A Tool for Relating Neuronal Fonn to Function 7l logarithm of the signal attenuation: X = lnVo/V(x). In an infinite cylindrical cable, electrotonic distance is directly proportional to physical distance: X = x/2 . However, cable theory is difficult to apply to real neurons since dendritic trees are neither infinite nor cylindrical. Because of their anatomical complexity and irregular variations of branch diameter and length, attenuation in neurons is not an exponential function of distance. Even if a cell met the criteria that would allow its dendrites to be reduced to a finite equivalent cylinder (RaIl 1977), voltage attenuation would not bear a simple exponential relationship to X but instead would vary inversely with a hyperbolic function (Jack et a!. 1983). 2.2 TWO-PORT THEORY Because of the limitations and restrictions of cable theory, Carnevale and Johnston (1982) turned to two-port analysis. Among their conclusions, three are most relevant to this discussion. Figure 1: Attenuation is direction-dependent. The first is that signal attenuation depends on the direction of signal propagation. Suppose that i and J are two points in a cell where i is "upstream" from J (voltage is spreading from i to J), and define the voltage attenuation from i to j: A~ = ~ IV). Next suppose that the direction of signal propagation is reversed, so that j is now upstream from i, and define the voltage attenuation A ~ = Vj I~. In general these two attenuations will not be equal: A~ *- AV. 1) )1 They also showed that voltage attenuation in one direction is identical to current attenuation in the opposite direction (Carnevale and Johnston 1982). Suppose current Ii enters the cell at i, and the current that is captured by a voltage clamp at J is Ii' and define the current attenuation A; = Ii 11 j' Because of the directional reciprocity between current and voltage attenuation, A! = AV. Similarly, if we interchange the 1) j1 current entry and voltage clamp sites, the current attenuation ratio would be A l = A ~ . )1 1) Finally, they found that charge and DC current attenuation in the same direction are identical (Carnevale and Johnston 1982). Therefore the spread of electrical signals between any two points is completely characterized by the voltage attenuations in both directions. 72 Nicholas Carnevale, Kenneth Y. Tsai, Brenda 1. Claiborne, Thomas H. Brown 2.3 THE ELECTROTONIC TRANSFORMATION The basic idea of the electrotonic transformation is to remap the cell from anatomical space into "electrotonic space," where the distance between points reflects the attenuation of an electrical signal spreading between them. Because of the critical role of membrane potential in neuronal function, it is usually most appropriate to deal with voltage attenuations. 2.3.1 The Distance Metric We use the logarithm of attenuation between points as the distance metric in electrotonic space: Li; = InAij (Brown et a1. 1992, Zador et a1. 1991). To appreciate the utility of this definition, consider voltage spreading from point i to point j , and suppose that k is on the direct path between i and j. The voltage attenuations are AI~ = V,/~ , At = Vk I ~, and A& = V, IV; = A~ At . This last equation and our definition of L establish the additive property of electrotonic distance Llj = Lik + LIg" That is, electrotonic distances are additive over a path that has a consistent directIon of signal propagation. This justifies using the logarithm of attenuation as a metric for the electrical separation between points in a cell. At this point several important facts should be noted. First, unlike the electrotonic distance X of classical cable theOI)" our new definition of electrotonic distance L always bears a simple and direct logarithmic relationship to attenuation. Second, because of membrane capacitance, attenuation increases with frequency. Since both steady-state and transient signals are of interest, we evaluate attenuations at several different frequencies. Third, attenuation is direction-dependent and usually asymmetric. Therefore at every frequency of interest, each branch of the cell has two different representations in electrotonic space depending on the direction of signal flow. 2.3.2 Representing a Neuron in Electrotonic Space Since attenuation depends on direction, it is necessary to construct transforms in pairs for each frequency of interest, one for signal spread away from a reference point (Vout) and the other for spread toward it (Vin). The soma is often a good choice for the reference point, but any point in the cell could be used, and a different vantage point might be more appropriate for particular analyses. The only difference between using one point i as the reference instead of any other point j is in the direction of signal propagation along the direct path between i and j (dashed arrows in Figure 2). where Vout relative to i is the same as Vin relative to j and vice versa. The directions of signal flow and therefore the attenuations along all other branches of the cell are unchanged. Thus the transforms relative to i and j differ only along the direct path ij, and once the T/~ut and Vin transforms have been created for one reference i, it is easy to assemble the transforms with respect to any other reference j. The Electronic Transformation: A Tool for Relating Neuronal Form to Function .,. .,. Figure 2: Effect of reference point location on direction of signal propagation. 73 We have found two graphical representations of the transform to be particularly useful. "Neuromorphic figures," in which the cell is redrawn so that the relative orientation of branches is preserved (Figures 3 and 4), can be readily compared to the original anatomy for quick, ''big picture" insights regarding synaptic integration and interactions. For more quantitative analyses, it is helpful to plot electrotonic distance from the reference point as a function of anatomical distance (Tsai et al. 1993). 3 COMPUTATIONAL METHODS The voltage attenuations along each segment of the cell are calculated from detailed, accurate morphometric data and the best available experimental estimates of the biophysical properties of membrane and cytoplasm. Any neural simulator like NEURON (Hines 1989) could be used to find the attenuations for the DC Vout transform. The DC Vin attenuations are more time consuming because a separate run must be performed for each of the dendritic terminations. However, the AC attenuations impose a severe computational burden on time-domain simulators because many cycles are required for the response to settle. For example, calculating the DC Vout attenuations in a hippocampal pyramidal cell relative to the soma took only a few iterations on a SUN Sparc 10-40, but more than 20 hours were required for 40 Hz (Tsai et al. 1994). Finding the full set of attenuations for a Vin transform at 40 Hz would have taken almost three months. Therefore we designed an O(N) algorithm that achieves high computational efficiency by operating in the frequency domain and taking advantage of the branched architecture of the cell. In a series of recursive walks through the cell, the algorithm applies Kirchhoff's laws to compute the attenuations in each branch. The electrical characteristics of each segment of the cell are represented by an equivalent T circuit. Rather than "lump" the properties of cytoplasm and membrane into discrete resistances and capacitances, we determine the elements of these equivalent T circuits directly from complex impedance functions that we derived from the impulse response of a finite cable (Tsai et a1. 1994). Since each segment is treated as a cable rather than an isopotential compartment, the resolution of the spatial grid does not affect accuracy, and there is no need to increase the resolution of the spatial grid in order to preserve accuracy as frequency increases. This is an important consideration for hippocampal neurons, which have long membrane time constants and begin to show increased attenuations at frequencies as low as 2 - 5 Hz (Tsai et al. 1994). It also allows us to treat a long unbranched neurite of nearly constant diameter as a single cylinder. Thus runtimes scale linearly with the number of grid points, are independent of frequency, and we can even reduce the number of grid points if the diameters of adjacent 74 Nicholas Carnevale. Kenneth Y. Tsai. Brenda 1. Claiborne. Thomas H. Brown unbranched segments are similar enough. A benchmark of a program that uses our algorithm with NEURON showed a speedup of more than four orders of magnitude without sacrificing accuracy (2 seconds vs. 20 hours to calculate the Vout attenuations at 40 Hz in a CAl pyramidal neuron model with 2951 grid points) (Tsai et al. 1994). 4 RESULTS 4.1 DC TRANSFORMS OF A CAl PYRAMIDAL CELL Figure 3 shows a two-dimensional projection of the anatomy of a rat CA 1 pyramidal neuron (cell 524, left) with neuromorphic renderings of its DC Vout and Vin transforms (middle and right) relative to the soma. The three-dimensional anatomical data were obtained from HRP-filled cells with a computer microscope system as described elsewhere (Rihn and Claiborne 1992, Claiborne 1992). The passive electrical properties used to compute the attenuations were Ri = 200 Ocm, em = 1 J.lF/cm2 (for nonzero frequencies, not shown here) and Rm = 30 kncm2 (Spruston and Johnston 1992). 524 1 Figure 3: CAl pyramidal cell anatomy (cell 524, left) with neuromorphic renderings of Vout (middle) and Vin (right) transforms at DC. The Vout transform is very compact, indicating that voltage propagates from the soma into the dendrites with relatively little attenuation. The basilar dendrites and the terminal branches of the primary apical dendrite are almost invisible, since they are nearly isopotential along their lengths. Despite the fact that the primary apical dendrite has a larger diameter than any of its daughter branches, most of the voltage drop for somatofugal signaling is in the primary apical. Therefore it accounts for almost all of the electrotonic length of the cell in the Vout transform. The Vin transform is far more extensive, but most of the electrotonic length of the cell is now in the basilar and terminal apical branches. This reflects the loading effect of downstream membrane on these thin dendritic branches. 4.2 SYNAPTIC INTERACTIONS The transform can also give clues to possible effects of electrotonic architecture on voltage-dependent forms of associative synaptic plasticity and other kinds of synaptic interactions. Suppose the cell of Figure 3 receives a weak or "student" synaptic input The Electronic Transformation: A Tool for Relating Neuronal Form to Function 75 located 400 J.lm from the soma on the primary apical dendrite, and a strong or "teacher" input is situated 300 J.lm from the soma on the same dendrite. [!] student @ teacher A. cell 524 B. cell 503 Figure 4: Analysis of synaptic interactions. The anatomical arrangement is depicted on the left in Figure 4A ("student" = square, "teacher" = circle). The Vin transform with respect to the student (right figure of this pair) shows that voltage spreads from the teacher to the student synapse with little attenuation, which would favor voltage-dependent associative interactions. Figure 4B shows a different CAl pyramidal cell in which the apical dendrite bifurcates shortly after arising from the soma. Two teacher synapses are indicated, one on the same branch as the student and the other on the opposite branch. The Vin transform with respect to the student (right figure of this pair) shows clearly that the teacher synapse on the same branch is closely coupled to the student, but the other is electrically much more remote and less likely to influence the student synapse. 5. SUMMARY The electrotonic transformation is based on a logical, internally consistent conceptual approach to understanding the propagation of electrical signals in neurons. In this paper we described two methods for graphically presenting the results of the transformation: neuromorphic rendering, and plots of electrotonic distance vs. anatomical distance. Using neuromorphic renderings, we illustrated the electrotonic properties of a previously unreported hippocampal CAl pyramidal neuron as viewed from the soma (cell 524, Figure 3). We also extended the use of the transformation to the study of associative interactions between "teacher" and "student" synapses by analyzing this cell from the viewpoint of a "student" synapse located in the apical dendrites, contrasting this result with a different cell that had a bifurcated primary apical dendrite (cell 503, Figure 4). This demonstrates the versatility of the electrotonic transformation, and shows how it can convey the electrical signaling properties of neurons in ways that are quickly and easily comprehended. This understanding is important for several reasons. First, electrotonus affects the integration and interaction of synaptic inputs, regulates voltage-dependent mechanisms of synaptic plasticity, and influences the interpretation of intracellular recordings. In addition, phylogeny, development, aging, and response to injury and disease are all accompanied by alterations of neuronal morphology, some subtle and some profound. 76 Nicholas Carnevale, Kenneth Y. Tsai, Brenda J. Claiborne, Thomas H. Brown The significance of these changes for brain function becomes clear only if their effects on neuronal signaling are reckoned. Finally, there is good reason to expect that neuronal electrotonus is highly relevant to the design of artificial neural networks. Acknowledgments We thank R.B. Gonzales and M.P. O'Boyle for their contributions to the morphometric analysis, and Z.F. Mainen for assisting in the initial development of graphical rendering. This work was supported in part by ONR, ARPA, and the Yale Center for Theoretical and Applied Neuroscience (CTAN). References Brown, T.H., Zador, A.M., Mainen, Z.F. and Claiborne, BJ. Hebbian computations in hippocampal dendrites and spines. In: Single Neuron Computation, eds. McKenna, T., Davis, J. and Zornetzer, S.F., New York, Academic Press, 1992, pp. 81-116. Carnevale, N. T. and Johnston, D.. Electrophysiological characterization of remote chemical synapses. J. Neurophysiol. 47:606-621, 1982. Claiborne, BJ. The use of computers for the quantitative, three-dimensional analysis of dendritic trees. In: Methods in Neuroscience. Vol. 10: Computers and Computation in the Neurosciences, ed. Conn, P.M., New York, Academic Press, 1992, pp. 315-330. Hines, M. A program for simulation of nerve equations with branching geometries. Internat. J. Bio-Med Computat. 24:55-68, 1989. Rall, W.. Core conductor theory and cable properties of neurons. In: Handbook of Physiology, The Nervous System, ed. Kandel, E.R., Bethesda, MD, Am. Physiol. Soc., 1977, pp.39-98. Rihn, L.L. and Claiborne, BJ. Dendritic growth and regression in rat dentate granule cells during late postnatal development. Brain Res. Dev. 54(1): 115-24, 1990. Spruston, N. and Johnston, D. Perforated patch-clamp analysis of the passive membrane properties of three classes of hippocampal neurons. J. Neurophysiol. 67:508529, 1992. Tsai, K.Y., Carnevale, N.T., Claiborne, BJ. and Brown, T.H. Morphoelectrotonic transforms in three classes of rat hippocampal neurons. Soc. Neurosci. Abst. 19: 1522, 1993. Tsai, K.Y., Carnevale, N.T., Claiborne, BJ. and Brown, T.H. Efficient mapping from neuroanatomical to electrotonic space. Network 5:21-46, 1994. Zador, A.M., Claiborne, BJ. and Brown, T.H. Electrotonic transforms of hippocampal neurons. Soc. Neurosci. Abst. 17: 1515, 1991.	1994	42
935	Reinforcement Learning Predicts the Site of Plasticity for Auditory Remapping in the Barn Owl Alexandre Pougett alex@salk.edu Cedric Deffayett Terrence J. Sejnowskit cedric@salk.edu terry@salk.edu tHoward Hughes Medical Institute The Salk Institute La Jolla, CA 92037 Department of Biology University of California, San Diego and tEcole Normale Superieure 45 rue d'Ulm 75005 Paris, France Abstract The auditory system of the barn owl contains several spatial maps. In young barn owls raised with optical prisms over their eyes, these auditory maps are shifted to stay in register with the visual map, suggesting that the visual input imposes a frame of reference on the auditory maps. However, the optic tectum, the first site of convergence of visual with auditory information, is not the site of plasticity for the shift of the auditory maps; the plasticity occurs instead in the inferior colliculus, which contains an auditory map and projects into the optic tectum. We explored a model of the owl remapping in which a global reinforcement signal whose delivery is controlled by visual foveation. A hebb learning rule gated by reinforcement learned to appropriately adjust auditory maps. In addition, reinforcement learning preferentially adjusted the weights in the inferior colliculus, as in the owl brain, even though the weights were allowed to change throughout the auditory system. This observation raises the possibility that the site of learning does not have to be genetically specified, but could be determined by how the learning procedure interacts with the network architecture. 126 Alexandre Pouget, Cedric Deffayet, Te"ence J. Sejnowski c:::======:::::» • Visual System Optic Tectum c,an ~_m .t Inferior Colllc-ulus External nucleua Forebrain Field L U~ a~ Inferior Colltculu. Cenlnll Nucleus C1ec) t Cochlea t Ovold.H. Nucleull ·"bala:m.ic Relay Figure 1: Schematic view of the auditory pathways in the barn owl. 1 Introduction The barn owl relies primarily on sounds to localize prey [6] with an accuracy vastly superior to that of humans. Figure 1A illustrates some of the nuclei involved in processing auditory signals. The barn owl determines the location of sound sources by comparing the time and amplitude differences of the sound wave between the two ears. These two cues are combined together for the first time in the shell and core of the inferior colliculus (ICc) which is shown at the bottom of the diagram. Cells in the ICc are frequency tuned and subject to spatial aliasing. This prevents them from unambiguously encoding the position of objects. The first unambiguous auditory map is found at the next stage, in the external capsule of the inferior colliculus (ICx) which itself projects to the optic tectum (OT). The OT is the first subforebrain structure which contains a multimodal spatial map in which cells typically have spatially congruent visual and auditory receptive fields. In addition, these subforebrain auditory pathways send one major collateral toward the forebrain via a thalamic relay. These collaterals originate in the ICc and are thought to convey the spatial location of objects to the forebrain [3]. Within the forebrain, two major structures have been involved in auditory processing: the archistriatum and field L. The archistriatum sends a projection to both the inferior colliculus and the optic tectum. Knudsen and Knudsen (1985) have shown that these auditory maps can adapt to systematic changes in the sensory input. Furthermore, the adaptation appears to be under the control of visual input, which imposes a frame of reference on the incoming auditory signals. In owls raised with optical prisms, which introduce a systematic shift in part of the visual field, the visual map in the optic tectum was identical to that found in control animals, but the auditory map in the ICx was shifted by the amount of visual shift introduced by the prisms. This plasticity ensures that the visual and auditory maps stay in spatial register during growth Reinforcement Learning Predicts the Site of Plasticity for Auditory Remapping 127 and other perturbations to sensory mismatch. Since vision instructs audition, one might expect the auditory map to shift in the optic tectum, the first site of visual-auditory convergence. Surprisingly, Brainard and Knudsen (1993b) observed that the synaptic changes took place between the ICc and the ICx, one synapse before the site of convergence. These observations raise two important questions: First, how does the animal knows how to adapt the weights in the ICx in the absence of a visual teaching signal? Second, why does the change take place at this particular location and not in the aT where a teaching signal would be readily available? In a previous model [7], this shift was simulated using backpropagation to broadcast the error back through the layers and by constraining the weights changes to the projection from the ICc to ICx. There is, however, no evidence for a feedback projection between from the aT to the ICx that could transmit the error signal; nor is there evidence to exclude plasticity at other synapses in these pathways. In this paper, we suggest an alternative approach in which vision guides the remapping of auditory maps by controlling the delivery of a scalar reinforcement signal. This learning proceeds by generating random actions and increasing the probability of actions that are consistently reinforced [1, 5] . In addition, we show that reinforcement learning correctly predicts the site of learning in the barn owl, namely at the ICx-ICc synapse, whereas backpropagation [8] does not favor this location when plasticity is allowed at every synapse. This raises a general issue: the site of synaptic adjustment might be imposed by the combination of the architecture and learning rule, without having to restrict plasticity to a particular synapse. 2 Methods 2.1 Network Architecture The network architecture of the model based on the barn owl auditory system, shown in figure 2A, contains two parallel pathways. The input layer was an 8x21 map corresponding to the ICc in which units responded to frequency and interaural phase differences. These responses were pooled together to create auditory spatial maps at subsequent stages in both pathways. The rest of the network contained a series of similar auditory maps, which were connected topographically by receptive fields 13 units wide. We did not distinguish between field L and the archistriatum in the forebrain pathways and simply used two auditory maps, both called FBr. We used multiplicative (sigma-pi) units in the aT whose activities were determined according to: Yi = L,. w~Br yfBr WfkBr yfc:c (1) j The multiplicative interaction between ICx and FBr activities was an important assumption of our model. It forced the ICx and FBr to agree on a particular position before the aT was activated. As a result, if the ICx-aT synapses were modified during learning, the ICx-FBr synapses had to be changed accordingly. 128 Alexandre Pouget, Cedric Deffayet, Terrence J. Sejnowski Figure 2: Schematic diagram of weights (white blocks) in the barn owl auditory system. A) Diagram of the initial weights in the network. B) Pattern of weights after training with reinforcement learning on a prism-induced shift offour units. The remapping took place within the ICx and FBr. C) Pattern of weights after training with backpropagation. This time the ICx-OT and FBr-OT weights changed. Weights were clipped between 5.0 and 0.01, except for the FBr-ICx connections whose values were allowed to vary between 8.0 and 0.01. The minimum values were set to 0.01 instead of zero to prevent getting trapped in unstable local minima which are often associated with weights values of zero. The strong coupling between FBr and ICx was another important assumption of the model whose consequence will be discussed in the last section. Examples were generated by simply activating one unit in the ICc while keeping the others to zero, thereby simulating the pattern of activity that would be triggered by a single localized auditory stimulus. In all simulations, we modeled a prism-induced shift of four units. 2.2 Reinforcement learning We used stochastic units and trained the network using reinforcement learning [1]. The weighted sum of the inputs, neti, passed through a sigmoid, f(x) , is interpreted as the probability, Pi, that the unit will be active: Pi = f(neti) * 0.99 + 0.01 (2) were the output of the unit Yi was: . _ {a with probability 1 - Pi y, 1 with probability Pi (3) Reinforcement Learning Predicts the Site of Plasticity for Auditory Remapping 129 Because of the form of the equation for Pi, all units in the network had a small probability (0.01) of being spontaneously active in the absence of any inputs. This is what allowed the network to perform a stochastic search in action space to find which actions were consistently associated with positive reinforcement. We ensured that at most one unit was active per trial by using a winner-take-all competition in each layer. Adjustable weights in the network were updated after each training examples with hebb-like rule gated by reinforcement: (4) A trial consisted in choosing a random target location for auditory input (ICc) and the output of the OT was used to generate a head movement. The reinforcement, r , was then set to 1 for head movements resulting in the foveation of the stimulus and to -0.05 otherwise. 2.3 Backpropagation For the backpropagation network, we used deterministic units with sigmoid activation functions in which the output of a unit was given by: (5) where neti is the weighted sum of the inputs as before. The chain rule was used to compute the partial derivatives of the squared error, E , with respect to each weights and the weights were updated after each training example according to: (6) The target vectors were similar to the input vectors, namely only one OT units was required to be activated for a given pattern, but at a position displaced by 4 units compared to the input. 3 Results 3.1 Learning site with reinforcement In a first set of simulation we kept the ICc-ICx and ICc-FBr weights fixed. Plasticity was allowed at these site in later simulations. Figure 2A shows the initial set of weights before learning starts. The central diagonal lines in the weight diagrams illustrate the fact that each unit receives only one non-zero weight from the unit in the layer below at the same location. 130 Alexandre Pouget, Cedric Deffayet, Terrence J. Sejnowski There are two solutions to the remapping: either the weights change within the ICx and FBr, or from the ICx and the FBr to the ~T. As shown in figure 2B, reinforcement learning converged to the first solution. In contrast, the weights between the other layers were unaltered, even though they were allowed to change. To prove that the network could have actually learned the second solution, we trained a network in which the ICc-ICx weights were kept fixed. As we expected, the network shifted its maps simultaneously in both sets of weights converging onto the OT, and the resulting weights were similar to the ones illustrated in figure 2C. However, to reach this solution, three times as many training examples were needed. The reason why learning in the ICx and FBr were favored can be attributed to probabilistic nature of reinforcement learning. If the probability of finding one solution is p, the probability of finding it twice independently is p2. Learning in the ICx and FBR is not independent because of the strong connection from the FBr to the ICx. When the remapping is learned in the FBR this connection automatically remapped the activities in the ICx which in turn allows the ICx-ICx weights to remap appropriately. In the OT on the other hand, the multiplicative connection between the ICx and FBr weights prevent a cooperation between this two sets of weights. Consequently, they have to change independently, a process which took much more training. 3.2 Learning at the ICc-ICx and ICc-FBr synapses The aliasing and sharp frequency tuning in the response of ICc neurons greatly slows down learning at the ICc-ICx and ICc-FBr synapses. We found that when these synapses were free to change, the remapping still took place within the ICx or FBr (figure 3). 3.3 Learning site with backpropagation In contrast to reinforcement learning, backpropagation adjusted the weights in two locations: between the ICx and the OT and between the Fbr and OT (figure 2C). This is the consequence of the tendency of the backpropagation algorithm to first change the weights closest to where the error is injected. 3.4 Temporal evolution of weights Whether we used reinforcement or supervised learning, the map shifted in a very similar way. There was a simultaneous decrease of the original set of weights with a simultaneous increase of the new weights, such that both sets of weights coexisted half way through learning. This indicates that the map shifted directly from the original setting to the new configuration without going through intermediate shifts. This temporal evolution of the weights is consistent the findings of Brainard and Knudsen (1993a) who found that during the intermediate phase of the remapping, cells in the inferior colli cuI us typically have two receptive fields. More recent work however indicates that for some cells the remapping is more continuous(Brainard and Knudsen, personal communication), a behavior that was not reproduced by either of the learning rule. Reinforcement Learning Predicts the Site of Plasticity for Auditory Remapping 131 Figure 3: Even when the ICc-ICx weights are free to change, the network update the weights in the ICx first. A separate weight matrix is shown for each isofrequency map from the ICc to ICx. The final weight matrices were predominantly diagonal; in contrast, the weight matrix in ICx was shifted. 4 Discussion Our simulations suggest a biologically plausible mechanism by which vision can guide the remapping of auditory spatial maps in the owl's brain. Unlike previous approaches, which relied on visual signals as an explicit teacher in the optic tectum [7], our model uses a global reinforcement signal whose delivery is controlled by the foveal representation of the visual system. Other global reinforcement signals would work as well. For example, a part of the forebrain might compare auditory and visual patterns and report spatial mismatch between the two. This signal could be easily incorporated in our network and would also remap the auditory map in the inferior colli cuI us. Our model demonstrates that the site of synaptic plasticity can be constrained by the interaction between reinforcement learning and the network architecture. Reinforcement learning converged to the most probably solution through stochastic search. In the network, the strong lateral coupling between ICx and FBr and the multiplicative interaction in the OT favored a solution in which the remapping took place simultaneously in the ICx and FBr. A similar mechanism may be at work in the barn owl's brain. Colaterals from FBr to ICx are known to exist, but the multiplicative interaction has not been reported in the barn owl optic tectum. Learning mechanisms may also limit synaptic plasticity. NMDA receptors have been reported in the ICx, but they might not be expressed at other synapses. There may, however, be other mechanisms for plasticity. The site of remapping in our model was somewhat different from the existing observations. We found that the change took place within the ICx whereas Brainard and Knudsen [3] report that it is between the ICc and the ICx. A close examination of their data (figure 11 in [3]) reveals that cells at the bottom of ICx were not 132 Alexandre Pouget, Cedric Deffayet, Terrence J. Sejnowski remapped, as predicted by our model, but at the same time, there is little anatomical or physiological evidence for a functional and hierarchical organization within the ICx. Additional recordings are need to resolve this issue. We conclude that for the barn owl's brain, as well as for our model, synaptic plasticity within ICx was favored over changes between ICc and ICx. This supports the hypothesis that reinforcement learning is used for remapping in the barn owl auditory system. Acknowledgments We thank Eric Knudsen and Michael Brainard for helpful discussions on plasticity in the barn owl auditory system and the results of unpublished experiments. Peter Dayan and P. Read Montague helped with useful insights on the biological basis of reinforcement learning in the early stages of this project. References [1] A.G. Barto and M.1. Jordan. Gradient following without backpropagation in layered networks. Proc. IEEE Int. Conf. Neural Networks, 2:629-636, 1987. [2] M.S. Brainard and E.1. Knudsen. Dynamics of the visual calibration of the map of interaural time difference in the barn owl's optic tectum. In Society For Neuroscience Abstracts, volume 19, page 369.8, 1993. [3] M.S. Brainard and E.!. Knudsen. Experience-dependent plasticity in the inferior colliculus: a site for visual calibration of the neural representation of auditory space in the barn owl. The journal of Neuroscience, 13:4589-4608, 1993. [4] E. Knudsen and P. Knudsen. Vision guides the adjustment of auditory localization in the young barn owls. Science, 230:545-548, 1985. [5] P.R. Montague, P. Dayan, S.J. Nowlan, A. Pouget, and T.J. Sejnowski. U sing aperiodic reinforcement for directed self-organization during development. In S.J. Hanson, J.D. Cowan, and C.L. Giles, editors, Advances in Neural Information Processing Systems, volume 5. Morgan-Kaufmann, San Mateo, CA, 1993. [6] R.S. Payne. Acoustic location of prey by barn owls (tyto alba). Journal of Experimental Biology, 54:535-573, 1970. [7] D.J. Rosen, D.E. Rumelhart, and E.I. Knudsen. A connectionist model of the owl's sound localization system. In Advances in Neural Information Processing Systems, volume 6. Morgan-Kaufmann, San Mateo, CA, 1994. [8] D.E. Rumelhart, G.E. Hinton, and R.J . Williams. Learning internal representations by error propagation. In D. E. Rumelhart, J. L. McClelland, and the PDP Research Group, editors, Parallel Distributed Processing, volume 1, chapter 8, pages 318-362. MIT Press, Cambridge, MA, 1986. Dynamic Modelling of Chaotic Time Series with Neural Networks Jose C. Principe, Jyh-Ming Kuo Computational NeuroEngineering Laboratory University of Florida, Gainesville, FL32611 principe@synapse.ee.ufi.edu Abstract This paper discusses the use of artificial neural networks for dynamic modelling of time series. We argue that multistep prediction is more appropriate to capture the dynamics of the underlying dynamical system, because it constrains the iterated model. We show how this method can be implemented by a recurrent ANN trained with trajectory learning. We also show how to select the trajectory length to train the iterated predictor for the case of chaotic time series. Experimental results corroborate the proposed method. 1.0 Introduction The search for a model of an experimental time series has been an important problem in science. For a long time the linear model was almost exclusively used to describe the system that produced the time series [1], but recently nonlinear models have also been proposed to replace the linear ones [2]. Lapedes and Farber [3] showed how artificial neural networks (ANNs) can be used to identify the dynamics of the unknown system that produced the time series. He simply used a multilayer perceptron to predict the next point in state space, and trained this topology with backpropagation. This paper explores more complex neural topologies and training methods with the goal of improving the quality of the identification of the dynamical system, and to understand better the issues of dynamic modelling with neural networks which are far from being totally understood. According to Takens' embedding theorem, a map F: Jilm + 1 ~ Jilm + 1 exists that transforms the current reconstructed state y (t) to the next state y (t + 1) , i.e. y(t+ 1) = F(y(t» (1) 312 Jose Principe, Jyh-Ming Kuo or where m is the estimated dimension of the unknown dynamical system cI>. Note that the map contains several trivial (nonlinear) filters and a predictor. The predictive mapping r: Jilm + 1 ~ R can be expressed as x(t+ 1) = r(x(t» (2) where x (t) = [x (t - 2m) ... x (t - 1) x (t)] T. This is actually the estimated nonlinear autoregressive model of the input time series. The existence of this predictive model lays a theoretical basis for dynamic modelling in the sense that we can build from a vector time series a model to approximate the mapping r. If the conditions of Takens embedding theorem are met, this mapping captures some of the properties of the unknown dynamical system cI> that produced the time series [7]. Presently one still does not have a capable theory to guarantee if the predictor has successfully identified the original model cI>. The simple point by point comparison between the original and predicted time series used as goodness of fit for non-chaotic time series breaks down for chaotic ones [5]. Two chaotic time series can be very different pointwise but be produced by the same dynamical system (two trajectories around the same attractor). The dynamic invariants (correlation dimension, Lyapunov exponents) measure global properties of the attractor, so they should be used as the rule to decide about the success of dynamic modelling. Hence, a pragmatic approach in dynamic modelling is to seed the predictor with a point in state space, feed the output to its input as an autonomous system, and create a new time series. If the dynamic invariants computed from this time series match the ones from the original time series, then we say that dynamic modelling was successful [5]. The long term behavior of the autonomous predictive model seems to be the key factor to find out if the predictor identified the original model. This is the distinguishing factor between prediction of chaotic time series and dynamic modelling. The former only addresses the instantaneous prediction error, while the latter is interested in long term behavior. In order to use this theory, one needs to address the choices of predictor implementation. Due to the universal mapping characteristics of multilayer perceptrons (MLPs) and the existence of well established learning rules to adapt the MLP coefficients, this type of network appears as an appropriate choice [3]. However, one must realize that the MLP is a static mapper, and in dynamic modelling we are dealing with time varying signals, where the past of the signal contains vital information to describe the mapping. The design considerations to select the neural network topology are presented elsewhere [4]. Wejust would like to say that the MLP has to be enhanced with short term memory mechanisms, and that the estimation of the correlation dimension should be used to set the size of the memory layer. The main goal of the paper is to establish the methodology to efficiently train neural networks for dynamic modelling. Dynamic Modelling of Chaotic Time Series with Neural Networks 313 2. Iterated versus Single Step Prediction. From eqn. 2 it seems that the resulting dynamic model F can be obtained through single step prediction. This has been the conventional way to handle dynamic modelling [2],[3]. The predictor is adapted by minimizing the error L _.1 E = L dist (x (i + 1) -F (1 (i» ) (3) I = 2m + 1 _.1 where L is the length of the time series, x(i) is the itb data sample, F is the map developed by the predictor and dist() is a distance measure (normally the L2 norm). Notice that the training to obtain the mapping is done independently from sample to sample, i.e. _.1 x(i+ 1) = F (xU» +51 _.1 x (i + j) = F (1 (i + j -1» + 5j where 5j are the instantaneous prediction errors, which are minimized during training. Notice that the predictor is being optimized under the assumption that the previous point in state space is known without error. The problem with this approach can be observed when we iterate the predictor as an autonomous system to generate the time series samples. If one wants to produce two samples in the future from sample i the predicted sample i+ 1 needs to be utilized to generate sample i+2. The predictor was not optimized to do this job, because during training the true i+ 1 sample was assumed known. As long as 51 is nonzero (as will be always the case for nontrivial problems), errors will accumulate rapidly. Single step prediction is more associated with extrapolation than with dynamic modelling, which requires the identification of the unique mapping that produces the time series. When the autonomous system generates samples, past values are used as inputs to generate the following samples, which means that the training should constrain also the iterates of the predictive mapping. Putting it in a simple way, we should train the predictor in the same way we are going to use it for testing (Le. as an autonomous system). We propose multistep prediction (or trajectory learning) as the way to constrain the iterates of the mapping developed by the predictor. Let us define " E = L dist(x(i + I)-xU + 1» (4) I = 2m + 1 where k is the number of prediction steps (length ofthe trajectory) and x (i + 1) is an estimate of the predictive map _.1 x(i+l) = F (i(i-2m), ... ,i(i» (5) 314 Jose Principe, Jyh-Ming Kuo with [ X (i) 0 SiS 2m j (i) = _.1. F (x(i-2m-l), ... ,x(i-l» i>2m Equation (5) states that i (i) is the i-2m iterate of the predictive part of the map (for i>2m), i.e. i(i+l) _ .1. -.1. - .1. _.1. t - 2m = (F (F ( ... F (x(2m»») = (F (x(2m») (6) Hence, minimizing the criterion expressed by equation (4) an optimal multistep predictor is obtained. The number of constraints that are imposed during learning is associated with k, the number of prediction steps, which corresponds to the number of iterations of the map. The more iterations, the less likely a sub-optimal solution is found, but note that the training time is being proportionally increased. In a chaotic time series there is a more important consideration that must be brought into the picture, the divergence of nearby trajectories, as we are going to see in a following section. 3. Multistep prediction with neural networks Figure 1 shows the topology proposed in [4] to identify the nonlinear mapping. Notice that the proposed topology is a recurrent neural network. with a global feedback loop. This topology was selected to allow the training of the predictor in the same way as it will be used in testing, i.e. using the previous network outputs to predict the next point. This recurrent architecture should be trained with a mechanism that will constrain the iterates of the map as was discussed above. Single step prediction does not fit this requirement. With multistep prediction, the model system can be trained in the same way as it is used in testing. We seed the dynamic net with a set of input samples, disconnect the input and feed back the predicted sample to the input for k steps. The mean square error between the predicted and true sample at each step is used as the cost function (equation (4». If the network topology was feed forward , batch learning could be used to train the network, and static backpropagation applied to train the net. However, as a recurrent topology is utilized, a learning paradigm such as backpropagation through time (BPTT) or real time recurrent learning (RTRL) must be utilized [6]. The use of these training methods should not come as a surprise since we are in fact fitting a trajectory over time, so the gradients are time varying. This learning method is sometimes called "trajectory learning" in the recurrent learning literature [6]. A criterion to select the length of the trajectory k will be presented below. The procedure described above must be repeated for several different segments of the time series. For each new training segment, 2m+ 1 samples of the original time series are used to seed the predictor. To ease the training we suggest that successive training sequences of length k overlap by q samples (q<k). For chaotic time series we also suggest that the error be weighted according to the largest Lyapunov exponent. Hence Dynamic Modelling of Chaotic Time Series with Neural Networks 315 the cost function becomes r " E = L L h(i) ·dist(x(i+jq+l)-i(i+jq+l» (7) J = 01 = 2m+ I where r is the number of training sequences, and ~ A I -(i-2m-I) h (i) = (e III/IX ) (8) In this equation A.max is the largest Lyapunov exponent and L\t the sampling interval. With this weighting the errors for later iteration are given less credit, as they should since due to the divergence of trajectories a small error is magnified proportionally to the largest Lyapunov exponent [7]. 4. Finding the length of the trajectory From the point of view of dynamic modelling, each training sequences should preferably contain enough information to model the attractor. This means that each sequence should be no shorter than the orbital length around the attractor. We proposed to estimate the orbital length as the reciprocal of the median frequency of the spectrum of the time series [8]. Basically this quantity is the average time required for a point to return to the same neighborhood in the attractor. The length of the trajectory is also equivalent to the number of constraints we impose on the iterative map describing the dynamical model. However, in a chaotic time series there is another fundamental limitation imposed on the trajectory length - the natural divergence of trajectories which is controlled by A.max' the largest Lyapunov exponent. If the trajectory length is too long, then instabilities in the training can be expected. A full discussion of this topic is beyond the scope of this paper, and is presented elsewhere [8]. We just want to say that when A.max is positive there is an uncertainty region around each predicted point that is a function of the number of prediction steps (due to cummulative error). If the trajectory length is too long the uncertainty regions from two neighboring trajectories will overlap, creating conflicting requirements for training (the model is requried to develop a map to follow both segments A and B- Figure 2). It turns out that one can approximately find the number of iterations is that will guarantee no overlap of uncertainty regions [8]. The length of the principal axis of the uncertainty region around a signal trajectory at iteration i can be estimated as £; = toe ~III/lXiAI (9) where £0 is the initial separation. Now assuming that the two principal axis of nearby trajectories are colinear, we should choose the number of iterations is such that the distance dj between trajectories is larger than the uncertainty region, i.e. d; ~ 2£; . I I The estimate of is should be averaged over a number of neighboring training sequences (-50 depending on the signal dynamics). Hence, to apply this method three quantities must be estimated: the largest Lyapunov 316 Jose Principe, Jyh-Ming Kuo exponent, using one of the accepted algorithms. The initial separation can be estimated from the one-step predictor. And is by averaging local divergence. The computation time required to estimate these quantities is usually much less than setting by trial and error the length of the trajectory until a reasonable learning curve is achieved. We also developed a method to train predictors for chaotic signals with large A.max' but it will not be covered in this paper [8]. 5. Results We used this methodology to model the Mackey-Glass system (d=30, sampled at 116 Hz). A signal of 500 samples was obtained by 4th order Runge-Kutta integration and normalized between -1,1. The largest Lyapunov exponent for this signal is 0.0071 nats/sec. We selected a time delay neural network (TDNN) with topology 8-14-1. The output unit is linear, and the hidden layer has sigmoid nonlinearities. The number of taps in the delay line is 8. We trained a one-step predictor and the multistep predictor with the methodology developed in this paper to compare results. The single step predictor was trained with static backpropagation with no momentum and step size of 0.001. Trained was stopped after 500 iterations. The final MSE was 0.000288. After training, the predictor was seeded with the first 8 points of the time series and iterated for 3,000 times. Figure 3a shows the corresponding output. Notice that the waveform produced by the model is much more regular that the Mackey-Glass signal, showing that some fine detail of the attractor has not been captured. Next we trained the same TDNN with a global feedback loop (TDNNGF). The estimate of the is over the neighboring orbits provided an estimate of 14, and it is taken as the length of the trajectory. We displaced each training sequence by 3 samples (q=3 in eqn 7). BPTT was used to train the TDNNGF for 500 iterations over the same signal. The final MSE was 0.000648, higher than for the TDNN case. We could think that the resulting predictor was worse. The TDNNGF predictor was initialized with the same 8 samples of the time series and iterated for 3,000 times. Figure 3b shows the resulting waveform. It "looks" much closer to the original Mackey-Glass time series. We computed the average prediction error as a function of iteration for both predictors and also the theoretical rate of divergence of trajectories assuming an initial error EO (Casdagli conjecture, which is the square of eqn 9) [7]. As can be seen in Figure 4 the TDNNGF is much closer to the theoretical limit, which means a much better model. We also computed the correlation dimension and the Lyapunov exponent estimated from the generated time series, and the figures obtained from TDNNGF are closer to the original time series. Figure 5 shows the instability present in the training when the trajectory length is above the estimated value of 14. For this case the trajectory length is 20. As can be seen the MSE decreases but then fluctuates showing instability in the training. 6. Conclusions This paper addresses dynamic modelling with artificial neural networks. We showed Dynamic Modelling of Chaotic Time Series with Neural Networks 317 that the network topology should be recurrent such that the iterative map is constrained during learning. This is a necessity since dynamic modelling seeks to capture the long term behavior of the dynamical system. These models can also be used as a sample by sample predictors. Since the network topology is recurrent, backpropagation through time or real time recurrent learning should be used in training. In this paper we showed how to select the length of the trajectory to avoid instability during training. A lot more work needs to be done to reliably capture dynamical properties of time series and encapsulate them in artificial models. But we believe that the careful analysis of the dynamic characteristics and the study of its impact on the predictive model performance is much more promising than guess work. According to this (and others) studies, modelling of chaotic time series of low Amax seems a reality. We have extended some of this work for time series with larger Amax, and successfully captured the dynamics of the Lorenz system [8]. But there, the parameters for learning have to be much more carefully selected, and some of the choices are still arbitrary. The main issue is that the trajectories diverge so rapidly that predictors have a hard time to capture information regarding the global system dynamics. It is interesting to study the limit of predictability of this type of approach for high dimensional and high Amax chaos. Predictor Corr. Dim. Lyapunov MG30 2.70+/-0.05 0.0073+/-0.000 1 TDNNGF 2.65+/-0.03 0.0074+/-0.0001 TDNN 1.60+1-0.10 0.0063+/-0.0001 segment B Figure 1. Prop_osed recurrent architecture Figure 2. State space representation in (IDNNGF) training a model 7. Acknowledgments This work was partially supported by NSF grant #ECS-9208789, and ONR #1494-941-0858. 318 -OJ! ~ m ~ ~ _ ~ w ~ ~ _ _ Figure 3a. Generated sequence with the TDNN , m ! TDNNerror / 002 j 015 TDNNGF e~ ~ /1 ,/ 0.01 /~ .-' C asdagli conjecture / ~ .005 • •• • ••••••••• _._.:::~ o o 5 to IS ~ ~ » Figure 4. Comparison of predictors 8. References Jose Principe, Jyh-Ming Kuo -OJ! -I~~~~!:--=--=--:!~~~--::!=----= o ~ '110. 110 ICIO ~ W JIO ~ .. II1II Figure 3b. uenerated sequence with the TDNNGF 0.2 Figure 5. Instability in training [1] Box, G. E., and G. M. Jenkins, Time Series Analysis, Forecasting and Control, Holden Day, San Francisco, 1970. [2] Weigend, A. S., B. A. Huberman, and D. E. Rumelhart, "Predicting the future: a connectionist approach," International Journal of Neural Systems, vol. 1, pp. 193209, 1990. [3] Lapedes, R., and R. Farber, "Nonlinear signal processing using neural networks: prediction and system modelling," Technical Report LA-UR87-2662, Los Alamos National Laboratory, Los Alamos, New Mexico, 1987. [4] Kuo J-M., Principe J.C., "A systematic approach to chaotic time series modeling with neural networks", in IEEE Workshop on Neural Nets for Signal Processing, Ermioni, Greece, 1994. [5] Principe, J. C., A. Rathie, and J. M. Kuo, "Prediction of chaotic time series with neural networks and the issue of dynamic modeling," International Journal of Biburcation and Chaos, vol. 2, no. 4, pp. 989-996, 1992. [6] Hertz, J, A. Krogh, and R. G. Palmer, Introduction to the Theory of Neural Computation, Addison-Wesley, Redwood City, CA, 1991. [7] Casdagli, M., "Nonlinear prediction of chaotic time series," Physica D 35, pp.335-356, 1989. [8] Kuo, J.M., "Nonlinear Dynamic Modelling with Artificial neural networks", Ph.D. dissertation, University of FLorida, 1993.	1994	43
936	Boltzmann Chains and Hidden Markov Models Lawrence K. Saul and Michael I. Jordan lksaulOpsyche.mit.edu, jordanOpsyche.mit.edu Center for Biological and Computational Learning Massachusetts Institute of Technology 79 Amherst Street, E10-243 Cambridge, MA 02139 Abstract We propose a statistical mechanical framework for the modeling of discrete time series. Maximum likelihood estimation is done via Boltzmann learning in one-dimensional networks with tied weights. We call these networks Boltzmann chains and show that they contain hidden Markov models (HMMs) as a special case. Our framework also motivates new architectures that address particular shortcomings of HMMs. We look at two such architectures: parallel chains that model feature sets with disparate time scales, and looped networks that model long-term dependencies between hidden states. For these networks, we show how to implement the Boltzmann learning rule exactly, in polynomial time, without resort to simulated or mean-field annealing. The necessary computations are done by exact decimation procedures from statistical mechanics. 1 INTRODUCTION AND SUMMARY Statistical models of discrete time series have a wide range of applications, most notably to problems in speech recognition (Juang & Rabiner, 1991) and molecular biology (Baldi, Chauvin, Hunkapiller, & McClure, 1992). A common problem in these fields is to find a probabilistic model, and a set of model parameters, that 436 Lawrence K. Saul, Michael I. Jordan account for sequences of observed data. Hidden Markov models (HMMs) have been particularly successful at modeling discrete time series. One reason for this is the powerful learning rule (Baum) 1972») a special case of the Expectation-Maximization (EM) procedure for maximum likelihood estimation (Dempster) Laird) & Rubin) 1977). In this work) we develop a statistical mechanical framework for the modeling of discrete time series. The framework enables us to relate HMMs to a large family of exactly solvable models in statistical mechanics. The connection to statistical mechanics was first noticed by Sourlas (1989») who studied spin glass models of error-correcting codes. We view the estimation procedure for HMMs as a special (and particularly tractable) case of the Boltzmann learning rule (Ackley) Hinton) & Sejnowski) 1985; Byrne) 1992). The rest of this paper is organized as follows. In Section 2) we review the modeling problem for discrete time series and establish the connection between HMMs and Boltzmann machines. In Section 3) we show how to quickly determine whether or not a particular Boltzmann machine is tractable) and if so) how to efficiently compute the correlations in the Boltzmann learning rule. Finally) in Section 4) we look at two architectures that address particular weaknesses of HMMs: the modelling of disparate time scales and long-term dependencies. 2 MODELING DISCRETE TIME SERIES A discrete time series is a sequence of symbols {jdr=l in which each symbol belongs to a finite countable set) i.e. jl E {1) 2) .. . ) m}. Given one long sequence) or perhaps many shorter ones) the modeling task is to characterize the probability distribution from which the time series are generated. 2.1 HIDDEN MARKOV MODELS A first-order Hidden Markov Model (HMM) is characterized by a set of n hidden states) an alphabet of m symbols) a transmission matrix ajj') an emission matrix bjj ) and a prior distribution 7I'j over the initial hidden state. The sequence of states {idr=l and symbols {jdr=l is modeled to occur with probability (1) The modeling problem is to find the parameter values (ajj' , bij ) 7I'j) that maximize the likelihood of observed sequences of training data. We will elaborate on the learning rule in section 2.3) but first let us make the connection to a well-known family of stochastic neural networks, namely Boltzmann machines. 2.2 BOLTZMANN MACHINES Consider a Boltzmann machine with m-state visible units) n-state hidden units) tied weights) and the linear architecture shown in Figure 1. This example represents the simplest possible Boltzmann "chain))) one that is essentially equivalent to a firstorder HMM unfolded in time (MacKay) 1994). The transition weights Aii' connect adjacent hidden units) while the emission weights Bjj connect each hidden unit to Boltzmann Chains and Hidden Markov Models un~ts Bij Bij hidden 1J ViS~ble ••• Aii , Au' units 437 Figure 1: Boltzmann chain with n-state hidden units, m-state visible units, transition weights Aiil, emission weights Bij, and boundary weights IIi. its visible counterpart. In addition, boundary weights IIi model an extra bias on the first hidden unit. Each configuration of units represents a state of energy L-1 L 1t[{il' jd] = -Ilil - L Ailil+t - 2: Bitio (2) l=l l=l where {idf=l ({jl }f=l) is the sequence of states over the hidden (visible) units. The probability to find the network in a particular configuration is given by P({ ' '}) 1 -{31-l Zl,)l = Ze , (3) where f3 = I/T is the inverse temperature, and the partition function Z = L e-fJ'H. (4) {idd is the sum over states that normalizes the Boltzmann distribution, eq. (3). Comparing this to the HMM distribution, eq. (1), it is clear that any first-order HMM can be represented by the Boltzmann chain of figure 1, provided we take1 Aii' = TIn aij/, Bij = TIn bij , IIi = TIn 7ri· (5) Later, in Section 4, we will consider more complicated chains whose architectures address particular shortcomings of HMMs. For now, however, let us continue to develop the example of figure 1, making explicit the connection to HMMs. 2.3 LEARNING RULES In the framework of Boltzmann learning (Williams & Hinton, 1990), the data for our problem consist of sequences of states over the visible units; the goal is to find the weights (Ail, Bij , IIi) that maximize the likelihood of the observed data. The likelihood of a sequence {jd is given by the ratio . P({il,jd) e-{3'H./Z Zc P({Jd) = P({idl{jl}) = e-{3'H./Zc = Z' (6) 1 Note, however, that the reverse statement-that for any set of parameters, this Boltzmann chain can be represented as an HMM-is not true. The weights in the Boltzmann chain represent arbitrary energies between ±oo, whereas the HMM parameters represent probabilities that are constrained to obey sum rules, such as Lil aiil = 1. The Boltzmann chain of figure 1 therefore has slightly more degrees of freedom than a first-order HMM. An interpretation of these extra degrees of freedom is given by MacKay (1994). 438 Lawrence K. Saul, Michael I. Jordan where Zc is the clamped partition function Zc = L e-/31i . {it} (7) Note that the sum in Zc is only over the hidden states in the network, while the visible states are clamped to the observed values bt}. The Boltzmann learning rule adjusts the weights of the network by gradient-ascent on the log-likelihood. For the example of figure 1, this leads to weight updates L-l ~Aii' = 7J/3 L [(6iil6ilil+Jc - (6iil6ilil+l)] ; (8) l=1 L ~Bij 7J/3 L [(6iil6jjl)C - (6iil 6jjl)] , (9) l=1 ~ni 7J/3 [(6ii1 )c - (6ii1 )] , (10) where 6ij stands for the Kronecker delta function, 7J is a learning rate, and (-) and (-) c denote expectations over the free and clamped Boltzmann distributions. The Boltzmann learning rule may also be derived as an Expectation-Maximization (EM) algorithm. The EM procedure is an alternating two-step method for maximum likelihood estimation in probability models with hidden and observed variables. For Boltzmann machines in general, neither the E-step nor the M-step can be done exactly; one must estimate the necessary statistics by Monte Carlo simulation (Ackley et al., 1985) or mean-field theory (Peterson & Anderson, 1987). In certain special cases (e.g. trees and chains), however, the necessary statistics can be computed to perform an exact E-step (as shown below). While the Mstep in these Boltzmann machines cannot be done exactly, the weight updates can be approximated by gradient descent. This leads to learning rules in the form of eqs. (8-10). HMMs may be viewed as a special case of Boltzmann chains for which both the E-step and the M-step are analytically tractable. In this case, the maximization in the M-step is performed subject to the constraints 2:i e/3Il• = 1, 2:il e/3A ;;1 = 1, and 2:j e/3B ;i = 1. These constraints imply Z = 1 and lead to closed-form equations for the weight updates in HMMs. 3 EXACT METHODS FOR BOLTZMANN LEARNING The key technique to compute partition functions and correlations in Boltzmann chains is known as decimation. The idea behind decimation2 is the following. Consider three units connected in series, as shown in Figure 2a. Though not directly connected, the end units have an effective interaction that is mediated by the middle one. In fact, the two weights in series exert the same influence as a single effective weight, given by (11) jl 2 A related method, the transfer matrix, is described by Stolarz (1994). Boltzmann Chains and Hidden Markov Models 439 A~~)' 1.1. + A~~). 11. (a) (b) (c) Figure 2: Decimation, pruning, and joining in Boltzmann machines. Replacing the weights in this way amounts to integrating out, or decimating, the degree offreedom represented by the middle unit. An analogous rule may be derived for the situation shown in Figure 2b. Summing over the degrees of freedom of the dangling unit generates an effective bias on its parent, given by ef3B• = L:: ef3B•j • j (12) We call this the pruning rule. Another type of equivalence is shown in Figure 2c. The two weights in parallel have the same effect as the sum total weight Ajjl = A~P + A~i) . (13) We call this the joining rule. It holds trivially for biases as well as weights. The rules for decimating, pruning, and joining have simple analogs in other types of networks (e.g. the law for combining resistors in electric circuits), and the strategy for exploiting them is a familiar one. Starting with a complicated network, we iterate the rules until we have a simple network whose properties are easily computed. A network is tractable for Boltzmann learning if it can be reduced to any pair of connected units. In this case, we may use the rules to compute all the correlations required for Boltzmann learning. Clearly, the rules do not make all networks tractable; certain networks (e.g. trees and chains), however, lend themselves naturally to these types of operations. 4 DESIGNER NETS The rules in section 3 can be used to quickly assess whether or not a network is tractable for Boltzmann learning. Conversely, they can be used to design networks that are computationally tractable. This section looks at two networks designed to address particular shortcomings of HMMs. 4.1 PARALLEL CHAINS AND DISPARATE TIME SCALES An important problem in speech recognition (Juang et al., 1991) is how to "combine feature sets with fundamentally different time scales." Spectral parameters, such 440 Lawrence K. Saul, Michael I. Jordan fast features coupled hidden units slow features Figure 3: Coupled parallel chains for features with different time scales. as the cepstrum and delta-cepstrum, vary on a time scale of 10 msec; on the other hand, prosodic parameters, such as the signal energy and pitch, vary on a time scale of 100 msec. A model that takes into account this disparity should avoid two things. The first is redundancy-in particular, the rather lame solution of oversampling the nonspectral features. The second is overfitting. How might this arise? Suppose we have trained two separate HMMs on sequences of spectral and prosodic features, knowing that the different features "may not warrant a single, unified Markov chain" (Juang et al., 1991). To exploit the correlation between feature sets, we must now couple the two HMMs. A naive solution is to form the Cartesian product of their hidden state spaces and resume training. Unfortunately, this results in an explosion in the number of parameters that must be fit from the training data. The likely consequences are overfitting and poor generalization. Figure 3 shows a network for modeling feature sets with disparate time scales-in this case, a 2: 1 disparity. Two parallel Boltzmann chains are coupled by weights that connect their hidden units. Like the transition and emission weights within each chain, the coupling weights are tied across the length of the network. Note that coupling the time scales in this way introduces far fewer parameters than forming the Cartesian product of the hidden state spaces. Moreover, the network is tractable by the rules of section 3. Suppose, for example, that we wish to compute the correlation between two neighboring hidden units in the middle of the network. This is done by first pruning all the visible units, then repeatedly decimating hidden units from both ends of the network. Figure 4 shows typical results on a simple benchmark problem, with data generated by an artificially constructed HMM. We tested the parallel chains model on 10 training sets, with varying levels of built-in correlation between features. A twostep method was used to train the parallel chains. First, we set the coupling weights to zero and trained each chain by a separate Baum-Welch procedure. Then, after learning in this phase was complete, we lifted the zero constraints and resumed training with the full Boltzmann learning rule. The percent gain in this second phase was directly related to the degree of correlation built into the training data, suggesting that the coupling weights were indeed capturing the correlation between feature sets. We also compared the performance of this Boltzmann machine versus that of a simple Cartesian-product HMM trained by an additional Baum-Welch procedure. While in both cases the second phase of learning led to reduced training error, the Cartesian product HMMs were decidedly more prone to overfitting. Boltzmann Chains and Hidden Markov Models ·1500 -1700 I I-J"anv::luu::a:cnu:nnuI I IIJOU"'XJ:o:x:o:a:J::) /tfA='=u"na"m .... .,m"l 200 400 1-1rainl"O ! -- croaa-vaJdation eoo eoo epoch (a) ~ 20 '" '" 10 0.2 441 0.' 0.6 0.8 feature colT8latkwl (b) Figure 4: (a) Log-likelihood versus epoch for parallel chains with 4-state hidden units, 6-state visible units, and 100 hidden-visible unit pairs (per chain). The second jump in log-likelihood occurred at the onset of Boltzmann learning (see text). (b) Percent gain in log-likelihood versus built-in correlation between feature sets. 4.2 LOOPS AND LONG-TERM DEPENDENCIES Another shortcoming of first-order HMMs is that they cannot exhibit long-term dependencies between the hidden states (Juang et aL , 1991). Higher-order and duration-based HMMs have been used in this regard with varying degrees of success. The rules of section 3 suggest another approach-namely, designing tractable networks with limited long-range connectivity. As an example, Figure 5a shows a Boltzmann chain with an internal loop and a long-range connection between the first and last hidden units. These extra features could be used to enforce known periodicities in the time series. Though tractable for Boltzmann learning, the loops in this network do not fit naturally into the framework of HMMs. Figure 5b shows learning curves for a toy problem, with data generated by another looped network. Carefully chosen loops and long-range connections provide additional flexibility in the design of probabilistic models for time series. Can networks with these extra features capture the long-term dependencies exhibited by real data? This remains an important issue for future research. Acknowledgements We thank G. Hinton, D. MacKay, P. Stolorz, and C. Williams for useful discussions. This work was funded by ATR Human Information Processing Laboratories, Siemens Corporate Research, and NSF grant CDA-9404932. References D. H. Ackley, G. E. Hinton, and T . J. Sejnowski. (1985) A Learning Algorithm for Boltzmann Machines. Cog. Sci. 9: 147- 160. P. Baldi, Y. Chauvin, T . Hunkapiller, and M. A. McClure. (1992) Proc. Nat. Acad. Sci. (USA) 91: 1059-1063. 442 Lawrence K. Saul, Michael I. Jordan I-tralning I ~ crosa..validation ·700 o 8 10 12 1. epoch (a) (b) Figure 5: (a) Looped network. (b) Log-likelihood versus epoch for a looped network with 4-state hidden units, 6-state visible units, and 100 hidden-visible unit pairs. L. Baum. (1972) An Inequality and Associated Maximization Technique in Statistical Estimation of Probabilistic Functions of Markov Processes, Inequalities 3:1-8. Byrne, W. (1992) Alternating Minimization and Boltzmann Machine Learning. IEEE Trans. Neural Networks 3:612-620. A. P. Dempster, N. M. Laird, and D. B. Rubin. (1977) Maximum Likelihood from Incomplete Data via the EM Algorithm. J. Roy. Statist. Soc. B, 39:1-38. C. Itzykson and J . Drouffe. (1991) Statistical Field Theory, Cambridge: Cambridge University Press. B. H. Juang and L. R. Rabiner. (1991) Hidden Markov Models for Speech Recognition, Technometrics 33: 251-272. D. J. MacKay. (1994) Equivalence of Boltzmann Chains and Hidden Markov Models, submitted to Neural Compo C. Peterson and J. R. Anderson. (1987) A Mean Field Theory Learning Algorithm for Neural Networks, Complex Systems 1:995-1019. 1. Saul and M. Jordan. (1994) Learning in Boltzmann Trees. Neural Compo 6: 1174-1184. N. Sourlas. (1989) Spin Glass Models as Error Correcting Codes. Nature 339: 693-695. P. Stolorz. (1994) Links Between Dynamic Programming and Statistical Physics for Heterogeneous Systems, JPL/Caltech preprint. C. Williams and G. E. Hinton. (1990) Mean Field Networks That Learn To Discriminate Temporally Distorted Strings. Proc. Connectionist Models Summer School: 18-22.	1994	44
937	Comparing the prediction accuracy of artificial neural networks and other statistical models for breast cancer survival Harry B. Burke Department of Medicine New York Medical College Valhalla, NY 10595 David B. Rosen Department of Medicine New York Medical College Valhalla, NY 10595 Philip H. Goodman Department of Medicine University of Nevada School of Medicine Reno, Nevada 89520 Abstract The TNM staging system has been used since the early 1960's to predict breast cancer patient outcome. In an attempt to increase prognostic accuracy, many putative prognostic factors have been identified. Because the TNM stage model can not accommodate these new factors, the proliferation of factors in breast cancer has lead to clinical confusion. What is required is a new computerized prognostic system that can test putative prognostic factors and integrate the predictive factors with the TNM variables in order to increase prognostic accuracy. Using the area under the curve of the receiver operating characteristic, we compare the accuracy of the following predictive models in terms of five year breast cancer-specific survival: pTNM staging system, principal component analysis, classification and regression trees, logistic regression, cascade correlation neural network, conjugate gradient descent neural, probabilistic neural network, and backpropagation neural network. Several statistical models are significantly more ac1064 Harry B. Burke, David B. Rosen, Philip H. Goodman curate than the TNM staging system. Logistic regression and the backpropagation neural network are the most accurate prediction models for predicting five year breast cancer-specific survival 1 INTRODUCTION For over thirty years measuring cancer outcome has been based on the TNM staging system (tumor size, number of lymph nodes with metastatic disease, and distant metastases) (Beahr et. al., 1992). There are several problems with this model (Burke and Henson, 1993). First, it is not very accurate, for breast cancer it is 44% accurate. Second its accuracy can not be improved because predictive variables can not be added to the model. Third, it does not apply to all cancers. In this paper we compare computerized prediction models to determine if they can improve prognostic accuracy. Artificial neural networks (ANN) are a class of nonlinear regression and discrimination models. ANNs are being used in many areas of medicine, with several hundred articles published in the last year. Representative areas of research include anesthesiology (Westenskow et. al., 1992), radiology (Tourassi et. al., 1992), cardiology (Leong and Jabri, 1982), psychiatry (Palombo, 1992), and neurology (Gabor and Seyal, 1992). ANNs are being used in cancer research including image processing (Goldberg et. al., 1992) , analysis of laboratory data for breast cancer diagnosis (0 Leary et. al., 1992), and the discovery of chemotherapeutic agents (Weinstein et. al., 1992). It should be pointed out that the analyses in this paper rely upon previously collected prognostic factors. These factors were selected for collection because they were significant in a generalized linear model such as the linear or logistic models. There is no predictive model that can improve upon linear or logistic prediction models when the predictor variables meet the assumptions of these models and there are no interactions. Therefore he objective of this paper is not to outperform linear or logistic models on these data. Rather, our objective is to show that, with variables selected by generalized linear models, artificial neural networks can perform as well as the best traditional models . There is no a priori reason to believe that future prognostic factors will be binary or linear, and that there will not be complex interactions between prognostic factors. A further objective of this paper is to demonstrate that artificial neural networks are likely to outperform the conventional models when there are unanticipated nonmonotonic factors or complex interactions. 2 METHODS 2.1 DATA The Patient Care Evaluation (PCE) data set is collected by the Commission on Cancer of the American College of Surgeons (ACS). The ACS, in October 1992, requested cancer information from hospital tumor registries in the United States. The ACS asked for the first 25 cases of breast cancer seen at that institution in 1983, and it asked for follow up information on each of these 25 patients through the date of the request. These are only cases of first breast cancer. Follow-up information included known deaths. The PCE data set contains, at best, eight year follow-up. Prediction Accuracy of Models for Breast Cancer Survival 1065 We chose to use a five year survival end-point. This analysis is for death due to breast cancer, not all cause mortality. For this analysis cases with missing data, and cases censored before five years, are not included so that the prediction models can be compared without putting any prediction model at a disadvantage. We randomly divided the data set into training, hold-out, and testing subsets of 3,100, 2,069, and 3,102 cases, respectively. 2.2 MODELS The TMN stage model used in this analysis is the pathologic model (pTNM) based on the 1992 American Joint Committee on Cancer's Manual for the Staging of Cancer (Beahr et. al., 1992). The pathologic model relies upon pathologically determined tumor size and lymph nodes, this contrasts with clinical staging which relies upon the clinical examination to provide tumor size and lymph node information. To determine the overall accuracy of the TNM stage model we compared the model's prediction for each patient, where the individual patient's prediction is the fraction of all the patients in that stage who survive, to each patient's true outcome. Principal components analysis, is a data reduction technique based on the linear combinations of predictor variables that minimizes the variance across patients (Jollie, 1982). The logistic regression analysis is performed in a stepwise manner, without interaction terms, using the statistical language S-PLUS (S-PLUS, 1992), with the continuous variable age modeled with a restricted cubic spline to avoid assuming linearity (Harrell et. al., 1988). Two types of Classification and Regression Tree (CART) (Breiman et. al., 1984) analyses are performed using S-PLUS. The first was a 9-node pruned tree (with 10-fold cross validation on the deviance), and the second was a shrunk tree with 13.7 effective nodes. The multilayer perceptron neural network training in this paper is based on the maximum likelihood function unless otherwise stated, and backpropagation refers to gradient descent. Two neural networks that are not multilayer perceptrons are tested. They are the Fuzzy ARTMAP neural network (Carpenter et. al., 1991) and the probabilistic neural network (Specht, 1990). 2.3 ACCURACY The measure of comparative accuracy is the area under the curve of the receiver operating characteristic (Az). Generally, the Az is a nonparametric measure of discrimination. Square error summarizes how close each patient's predicted value is to its true outcome. The Az measures the relative goodness of the set of predictions as a whole by comparing the predicted probability of each patient with that of all other patients. The computational approach to the Az that employs the trapezoidal approximation to the area under the receiver operating characteristic curve for binary outcomes was first reported by Bamber (Bamber, 1975), and later in the medical literature by Hanley (Hanley and McNeil, 1982). This was extended by Harrell (Harrell et. al., 1988) to continuous outcomes. 1066 Harry B. Burke, David B. Rosen, Philip H. Goodman Table 1: PCE 1983 Breast Cancer Data: 5 Year Survival Prediction, 54 Variables. PREDICTION MODEL ACCURACY· SPECIFICATIONS pTNM Stages Principal Components Analysis CART, pruned CART, shrunk Stepwise Logistic regression Fuzzy ARTMAP ANN Cascade correlation ANN Conjugate gradient descent ANN Probabilistic ANN Backpropagation ANN .720 .714 .753 .762 .776 .738 .761 .774 .777 .784 O,I,I1A,I1B,IIIA,I1IB,IV one scaling iteration 9 nodes 13.7 nodes with cubic splines 54-F2a, 128-1 54-21-1 54-30-1 bandwidth = 16s 54-5-1 * The area under the curve of the receiver operating characteristic. 3 RESULTS All results are based on the independent variable sample not used for training (i.e., the testing data set), and all analyses employ the same testing data set. Using the PCE breast cancer data set, we can assess the accuracy of several prediction models using the most powerful of the predictor variables available in the data set (See Table 1). Principal components analysis is not expected to be a very accurate model; with one scaling iteration, its accuracy is .714. Two types of classification and regression trees (CART), pruned and shrunk, demonstrate accuracies of .753 and .762, respectively. Logistic regression with cubic splines for age has an accuracy of .776. In addition to the backpropagation neural network and the probabilistic neural network, three types of neural networks are tested. Fuzzy ARTMAP's accuracy is the poorest at .738. It was too computationally intensive to be a practical model. Cascade-correlation and conjugate gradient descent have the potential to do as well as backpropagation. The PNN accuracy is .777. The PNN has many interesting features, but it also has several drawbacks including its storage requirements. The backpropagation neural network's accuracy is .784.4. 4 DISCUSSION For predicting five year breast cancer-specific survival, several computerized prediction models are more accurate than the TNM stage system, and artificial neural networks are as good as the best traditional statistical models. References Bamber D (1975). The area above the ordinal dominance graph and the area below the receiver operating characteristic. J Math Psych 12:387-415. Beahrs OH, Henson DE, Hutter RVP, Kennedy BJ (1992). Manual for staging of Prediction Accuracy of Models for Breast Cancer Survival 1067 cancer, 4th ed. Philadelphia: JB Lippincott. Burke HB, Henson DE (1993). Criteria for prognostic factors and for an enhanced prognostic system. Cancer 72:3131-5. Breiman L, Friedman JH, Olshen RA (1984). Classification and Regression Trees. Pacific Grove, CA: Wadsworth and Brooks/Cole. Carpenter GA, Grossberg S, Rosen DB (1991). Fuzzy ART: Fast stable learning and categorization of analog patterns by an adaptive resonance system. Neural Networks 4:759-77l. Gabor AJ, M. Seyal M (1992) . Automated interictal EEG spike detection using artificial neural networks. Electroencephalogr Clin Neurophysiology 83:271-80. Goldberg V, Manduca A, Ewert DL (1992). Improvement in specificity of ultrasonography for diagnosis of breast tumors by means of artificial intelligence. Med Phys 19:1275-8l. Hanley J A, McNeil BJ (1982). The meaning of the use of the area under the receiver operating characteristic (ROC) curve. Radiology 143:29-36. Harrell FE, Lee KL, Pollock BG (1988). Regression models in clinical studies: determining relationships between predictors and response. J Natl Cancer Instit 80:1198-1202. Jollife IT (1986). Principal Component Analysis. New York: Springer-Verlag, 1986. Leong PH, J abri MA (1982). MATIC - an intracardiac tachycardia classification system. PACE 15:1317-31,1982. O'Leary TJ, Mikel UV, Becker RL (1992). Computer-assisted image interpretation: use of a neural network to differentiate tubular carcinoma from sclerosing adenosis. Modern Pathol 5:402-5. Palombo SR (1992). Connectivity and condensation in dreaming. JAm Psychoanal Assoc 40:1139-59. S-PLUS (1991), v 3.0. Seattle, WA; Statistical Sciences, Inc. Specht DF (1990). Probabilistic neural networks. Neural Networks 3:109-18. Tourassi GD, Floyd CE, Sostman HD, Coleman RE (1993). Acute pulmonary embolism: artificial neural network approach for diagnosis. Radiology 189:555-58. Weinstein IN, Kohn KW, Grever MR et. al. (1992) Neural computing in cancer drug development: predicting mechanism of action. Science 258:447-51. Westenskow DR, Orr JA, Simon FH (1992). Intelligent alarms reduce anesthesiologist's response time to critical faults. Anesthesiology 77:1074-9, 1992. Learning with Product Units Laurens R. Leerink Australian Gilt Securities LTD 37-49 Pitt Street NSW 2000, Australia laurens@sedal.su.oz.au Bill G. Horne NEC Research Institute 4 Independence Way Princeton, NJ 08540, USA horne@research.nj.nec.com C. Lee Giles NEC Research Institute 4 Independence Way Princeton, NJ 08540, USA giles@research.nj.nec.com Marwan A. Jabri Department of Electrical Engineering The University of Sydney NSW 2006, Australia marwan@sedal.su.oz.au Abstract Product units provide a method of automatically learning the higher-order input combinations required for efficient learning in neural networks. However, we show that problems are encountered when using backpropagation to train networks containing these units. This paper examines these problems, and proposes some atypical heuristics to improve learning. Using these heuristics a constructive method is introduced which solves well-researched problems with significantly less neurons than previously reported. Secondly, product units are implemented as candidate units in the Cascade Correlation (Fahlman & Lebiere, 1990) system. This resulted in smaller networks which trained faster than when using sigmoidal or Gaussian units. 1 Introduction It is well-known that supplementing the inputs to a neural network with higher-order combinations ofthe inputs both increases the capacity of the network (Cover, 1965) and the the ability to learn geometrically invariant properties (Giles & Maxwell, 538 Laurens Leerink, C. Lee Giles, Bill G. Home, Marwan A. Jabri 1987). However, there is a combinatorial explosion of higher order terms as the number of inputs to the network increases. Yet in order to implement a certain logical function, in most cases only a few of these higher order terms are required (Redding et al., 1993). The product units (PUs) introduced by (Durbin & Rumelhart, 1989) attempt to make use of this fact. These networks have the advantage that, given an appropriate training algorithm, the units can automatically learn the higher order terms that are required to implement a specific logical function. In these networks the hidden layer units compute the weighted product ofthe inputs, that is N N II X~i instead of 2:XiWi (1) i=l i=l as in standard networks. An additional advantage of PUs is the increased information capacity of these units compared to standard summation networks. It is approximately 3N (Durbin & Rumelhart, 1989), compared to 2N for a single threshold logic function (Cover, 1965), where N is the number of inputs to the unit. The larger capacity means that the same functions can be implemented by networks containing less units. This is important for certain applications such as speech recognition where the data bandwidth is high or if realtime implementations are desired. When PUs are used to process Boolean inputs, best performance is obtained (Durbin & Rumelhart, 1989) by using inputs of {+1, -I}. If the imaginary component is ignored, with these inputs, the activation function is equivalent to a cosine summation function with {-1,+1} inputs mapped {I,D} (Durbin & Rumelhart, 1989). In the remainder of this paper the terms product unit (PU) and cos{ine) unit will be used interchangeably as all the problems examined have Boolean inputs. 2 Learning with Product Units As the basic mechanism of a PU is multiplicative instead of additive, one would expect that standard neural network training methods and procedures cannot be directly applied when training these networks. This is indeed the case. If a neural network simulation environment is available the basic functionality of a PU can be obtained by simply adding the cos function cos( 1(" * input) to the existing list of transfer functions. This assumes that Boolean mappings are being implemented and the appropriate {-1,+1} {I,D} mapping has been performed on the input vectors. However, if we then attempt to train a network on on the parity-6 problem shown in (Durbin & Rumelhart, 1989), it is found that the standard backpropagat ion (BP) algorithm simply does not work. We have found two main reasons for this. The first is weight initialization. A typical first step in the backpropagation procedure is to initialize all weights to small random values. The main reason for this is to use the dynamic range of the sigmoid function and it's derivative. However, the dynamic range of a PU is unlimited. Initializing the weights to small random Learning with Product Units 539 values results in an input to the unit where the derivative is small. So apart from choosing small weights centered around ml" with n = ±1, ±2, ... this is the worst possible choice. In our simulations weights were initialized randomly in the range [-2,2]. In fact, learning seems insensitive to the size of the weights, as long as they are large enough. The second problem is local minima. Previous reports have mentioned this problem, (Lapedes & Farber, 1987) commented that "using sin's often leads to numerical problems, and nonglobal minima, whereas sigmoids seemed to avoid such problems". This comment summarizes our experience of training with PUs. For small problems (less than 3 inputs) backpropagation provides satisfactory training. However, when the number of inputs are increased beyond this number, even with th: .weight initialization in the correct range, training usually ends up in a local mInIma. 3 Training Algorithms With these aspects in mind, the following training algorithms were evaluated: online and batch versions of Backpropagation (BP), Simulated Annealing (SA), a Random Search Algorithm (RSA) and combinations of these algorithms. BP was used as a benchmark and for use in combination with the other algorithms. The Delta-Bar-Delta learning rate adaptation rule (Jacobs, 1988) was used along with the batch version of BP to accelerate convergence, with the parameters were set to 0 = 0.35, K, = 0.05 and ¢ = 0.90. RSA is a global search method (i.e. the whole weight space is explored during training). Weights are randomly chosen from a predefined distribution, and replaced if this results in an error decrease. SA (Kirkpatrick et aI., 1983) is a standard optimization method. The operation of SA is similar to RSA, with the difference that with a decreasing probability solutions are accepted which increase the training error. The combination of algorithms were chosen (BP & SA, BP & RSA) to combine the benefits of global and local search. Used in this manner, BP is used to find the local minima. If the training error at the minima is sufficiently low, training is terminated. Otherwise, the global method initializes the weights to another position in weight space from which local training can continue. The BP-RSA combination requires further explanation. Several BP-(R)SA combinations were evaluated, but best performance was obtained using a fixed number of iterations of BP (in this case 120) along with one initial iteration of RSA. In this manner BP is used to move to the local minima, and if the training error is still above the desired level the RSA algorithm generates a new set of random weights from which BP can start again. The algorithms were evaluated on two problems, the parity problem and learning all logical functions of 2 and 3 inputs. The infamous parity problem is (for the product unit at least) an appropriate task. As illustrated by (Durbin & Rumelhart, 1989), this problem can be solved by one product unit. The question is whether the training algorithms can find a solution. The target values are {-1, + 1}, and the output is taken to be correct if it has the correct sign. The simulation results are shown in Table 1. It should be noted that one epoch of both SA and RSA involves 540 Laurens Leerink, C. Lee Giles, Bill G. Home, Marwan A. Jahri relaxing the network across the training set for every weight, so in computational terms their nepoeh values should be multiplied by a factor of (N + 1). Parity Online BP Batch BP SA RSA N neonv nepoeh n eonv nepoeh neonv nepoeh neon v nepoeh 6 10 30.4 7 34 10 12.6 10 15.2 8 8 101.3 2 700 10 52.8 10 45.4 10 6 203.3 0 10 99.9 10 74.1 Table 1: The parity N problem: The table shows neon v the number of runs out of 10 that have converged and nepoeh' the average number of training epochs required when training converged. For the parity problem it is clear that local learning alone does not provide good convergence. For this problem, global search algorithms have the following advantages: (1) The search space is bounded (all weights are restricted to [-2, +2]) (2) The dimension of search space is low (maximum of 11 weights for the problems examined). (3) The fraction of the weight space which satisfies the parity problem relative to the total bounded weight space is high. In a second set of simulations, one product unit was trained to calculate all 2(2N ) logical functions of the N input variables. Unfortunately, this is only practical for N E {2,3} . For N = 2 there are only 16 functions, and a product unit has no problem learning all these functions rapidly with all four training algorithms. In comparison a single summation unit can learn 14 (not the XOR & XNOR functions). For N =3, a product unit is able to implement 208 of the 256 functions, while a single summation unit could only implement 104. The simulation results are displayed in Table 2. Online BP Batch BP BP-RSA Table 2: Learning all logical functions of 3 inputs: The rows display nlogie , the average number of logical functions implemented by a product unit and nepoeh, the number of epochs required for convergence. Ten simulations were performed for each of the 256 logical functions, each for a maximum of 1,000 iterations. 4 Constructive Learning with Product Units Selecting the optimal network architecture for a specific application is a nontrivial and time-consuming task, and several algorithms have been proposed to automate this process. These include pruning methods and growing algorithms. In this section a simple method is proposed for adding PUs to the hidden layer of a three layer network. The output layer contains a single sigmoidal unit. Several constructive algorithms proceed by freezing a subset of the weights and limiting training to the newly added units. As mentioned earlier, for PUs a global Learning with Product Units 541 300 Tiling AI orithm ~ Upstart AI orithm I-t--< 81M using Pr Units >S-t 250 i!: 0 200 ~ c .!; <II C e 150 ::l CI) c '15 ~ D E 100 ::l Z 50 200 400 600 800 1000 1200 Number of patterns (2"N) Figure 1: The number of units required for learning the random mapping problems by the 'Tiling', 'Upstart' and SIM algorithms. search is required to solve the local-minima problems. Freezing a subset of the weights restricts the new solution to an affine subset of the existing weight space, often resulting in non-minimal networks (Ash, 1989). For this reason a simple incremental method (SIM) was implemented which retains the global search for all weights during the whole training process. The method used in our simulations is as follows: • Train a network using the BP-RSA combination on a network with a specified minimum number of hidden PUs. • If there is no convergence within a specified number of epochs, add a PU to the network. Reinitialize weights and continue training with the BP-RSA combination. • Repeat process until a solution is found or the network has grown a predetermined maximum size. The method of (Ash, 1989) was also evaluated, where neurons with small weights were added to a network according to certain criteria. The SIM performed better, possibly because of the global search performed by the RSA step. The 'Upstart' (Frean, 1990) and 'Tiling' (Mezard & Nadal, 1989) constructive algorithms were chosen as benchmarks. A constructive PU network was trained on two problems described in these papers, namely the parity problem and the random mapping problem. In (Frean, 1990) it was reported that the Upstart 542 Laurens Leerink, C. Lee Giles, Bill G. Home, Marwan A. Jabri algorithm required N units for all parity N problems, and 1,000 training epochs were sufficient for all values of N except N = 10, which required 10,000. As seen earlier, one PU is able to perform any parity function, and SIM required an an average of 74.1 iterations for N = 6,8,10. The random mapping problem is defined by assigning each of the 2N patterns its target { -1, + I} with 50% probability. This is a difficult problem, due to the absence of correlations and structure in the input. As in (Frean, 1990; Mezard & Nadal, 1989) the average of 25 runs were performed, each on a different training set. The number of units required by SIM is plotted in Figure 1. The values for the Tiling and Upstart algorithms are approximate and were obtained through inspection from a similar graph in (Frean, 1990). 5 U sing Cosine Candidate Units in Cascade Correlation Initially we wanted to compare the performance of SIM with the well-known 'cascade-correlation' (CC) algorithm of (Fahlman & Lebiere, 1990). However, the network architectures differ and a direct comparison between the number of units in the respective architectures does not reflect the efficiency of the algorithms. Instead, it was decided to integrate PUs into the CC system as candidate units. For these simulations a public domain version of CC was used (White, 1993) which supports four different candidate types; the asymmetric sigmoid, symmetric sigmoid, variable sigmoid and gaussian units. Facilities exist for either constructing homogeneous networks by selecting one unit type, or training with a pool of different units allowing the construction of hybrid networks. It was thus relatively simple to add PU candidate units to the system. Table 3 displays the results when CC was trained on the random logic problem using three types of homogeneous candidate units. N CC Sigmoid CC Gauss CC PU nunih nepOCh6 nunih nepoch, nunih n epoch6 7 6.6 924.5 6.7 642.6 5.7 493.8 8 12.1 1630.9 11.5 1128.2 9.9 833.8 9 20.5 2738.3 18.4 1831.1 16.4 1481.8 10 32.9 4410.9 30.2 2967.6 26.6 2590.8 Table 3: Learning random logic functions of N inputs: The table shows nunih' the average number of units required and nepoch6' the average number of training epochs required for convergence of CC using sigmoidal, Gaussian and PU candidate units. Figures are based on 25 simulations. In a separate experiment the performance of hybrid networks were re-evaluated on the same random logic problem. To enable a fair competition between candidate units of different types, the simulations were run with 40 candidate units, 8 of each type. The simulations were evaluated on 25 trails for each of the random mapping problems (7,8,9 and 10 inputs, a total of 1920 input vectors). In total 1460 hidden units were allocated, and in all cases PU candidate units were chosen above units of the 4 other types during the competitive stage. During this comparison all Learning with Product Units 543 parameters were set to default values, i.e. the weights of the PU candidate units were random numbers initialized in the range of [-1, +1]. As discussed earlier, this puts the PUs at a slight disadvantage as their optimum range is [-2, +2]. 6 Discussion The BP-RSA combination is in effect equivalent to the 'local optimization with random restarts' process discussed by (Karmarkar & Karp, 1982), where the local optimization is this case is performed by the BP algorithm. They reported that for certain problems where the error surface was 'exceedingly mountainous', multiple random-start local optimization outperformed more sophisticated methods. We hypothesize that adding PUs to a network makes the error surface sufficiently mountainous so that a global search is required. As expected, the higher separating capacity of the PU enables the construction of networks with less neurons than those produced by the Tiling and Upstart algorithms. The fact that SIM works this well is mainly a result of the error surface; the surface is so irregular that even training a network of fixed architecture is best done by reinitializing the weights if convergence does not occur within certain bounds. This again is in accordance with the results of (Karmarkar & Karp, 1982) discussed above. When used in CC we hypothesize that there are three main reasons for the choice of PUs above any of the other types during the competitive learning phase. Firstly, the higher capacity (in a information capacity sense) of the PUs allows a better correlation with the error signal. Secondly, having N competing candidate units is equivalent to selecting the best of N random restarts, and performs the required global search. Thirdly, although the error surface of networks with PUs contains more local minima than when using standard transfer functions, the surface is locally smooth. This allows effective use of higher-order error derivatives, resulting in fast convergence by the quickprop algorithm. In (Dawson & Schopflocher, 1992) it was shown that networks with Gaussian units train faster and require less units than networks with standard sigmoidal units. This is supported by our results shown in Table 3. However, for the problem examined, PUs outperform Gaussian units by approximately the same margin as Gaussian units outperform sigmoidal units. It should also be noted that these problems where not chosen for their suitability for PUs. In fact, if the problems are symmetric/regular the difference in performance is expected to increase. 7 Conclusion Of the learning algorithms examined BP provides the fastest training, but is prone to nonglobal minima. On the other hand, global search methods are impractical for larger networks. For the problems examined, a combination of local and global search methods were found to perform best. Given a network containing PUs, there are some atypical heuristics that can be used: (a) correct weight initialization (b) reinitialization of the weights if convergence is not rapidly reached. In addition, the representational power of PUs have enabled us to solve standard problems 544 lAurens Leerink, C. Lee Giles, Bill G. Home, Marwan A. labri using significantly smaller networks than previously reported, using a very simple constructive method. When implemented in the CC architecture, for the problems examined PUs resulted in smaller networks which trained faster than other units. When included in a pool of competing candidate units, simulations showed that in all cases PU candidate units were preferred over candidate units of the other four types. References Ash, T. (1989). Dynamic node creation in backpropagation networks. Connection Science, 1 (4), 365-375. Cover, T. (1965) . Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition. IEEE Transactions on Electronic Computers, 14, 326-334. Dawson, M. & Schopflocher, D. (1992). Modifying the generalized delta rule to train networks of nonmonotonic processors for pattern classification. Connection Science, 4, 19-31. Durbin, R. & Rumelhart, D. (1989). Product units: A computationally powerful and biologically plausible extension to backpropagation networks. Neural Computation, 1, 133- 142. Fahlman, S. & Lebiere, C. (1990). The cascade-correlation learning architecture. In Touretzky, D. (Ed.), Advances in Neural Information Processing Systems, volume 2, (pp. 524-532)., San Mateo. (Denver 1989), Morgan Kaufmann. Frean, M. (1990). The upstart algorithm: A method for constructing and training feedforward neural networks. Neural Computation, 2, 198-209. Giles, C. & Maxwell, T . (1987). Learning, invariance, and generalization in highorder neural networks. Applied Optics, 26(23),4972-4978. Jacobs, R. (1988). Increased rates of convergence through learning rate adaptation. Neural Networks, 1, 295-307. Karmarkar, N. & Karp, R. (1982). The differencing method of set partitioning. Technical Report UCB/CSD 82/113, Computer Science Division, University of California, Berkeley, California. Kirkpatrick, S., Jr., C. G., , & Vecchi, M. (1983). Optimization by simulated annealing. Science, 220. Reprinted in (?). Lapedes, A. & Farber, R. (1987). Nonlinear signal processing using neural networks: Prediction and system modelling. Technical Report LA-UR-87-2662, Los Alamos National Laboratory, Los Alamos, NM. Mezard, M. & Nadal, J.-P. (1989). Learning in feedforward layered networks: The tiling algorithm. Journal of Physics A, 22, 2191-2204. Redding, N., Kowalczyk, A., & Downs, T . (1993). A constructive higher order network algorithm that is polynomial-time. Neural Networks, 6,997. White, M. (1993). A public domain C implement ion of the Cascade Correlation algorithm. Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA.	1994	45
938	Efficient Methods for Dealing with Missing Data in Supervised Learning Volker '!'resp· Siemens AG Central Research Otto-Hahn-Ring 6 81730 Miinchen Germany Ralph Neuneier Siemens AG Central Research Otto-Hahn-Ring 6 81730 Miinchen Germany Abstract Subutai Ahmad Interval Research Corporation 1801-C Page Mill R<;l. Palo Alto, CA 94304 We present efficient algorithms for dealing with the problem of missing inputs (incomplete feature vectors) during training and recall. Our approach is based on the approximation of the input data distribution using Parzen windows. For recall, we obtain closed form solutions for arbitrary feedforward networks. For training, we show how the backpropagation step for an incomplete pattern can be approximated by a weighted averaged backpropagation step. The complexity of the solutions for training and recall is independent of the number of missing features. We verify our theoretical results using one classification and one regression problem. 1 Introduction The problem of missing data (incomplete feature vectors) is of great practical and theoretical interest. In many applications it is important to know how to react if the available information is incomplete, if sensors fail or if sources of information become A.t the time of the research for this paper, a visiting researcher at the Center for Biological and Computational Learning, MIT. E-mail: Volker.Tresp@zfe.siemens.de 690 VoLker Tresp. RaLph Neuneier. Subutai Ahmad unavailable. As an example, when a sensor fails in a production process, it might not be necessary to stop everything if sufficient information is implicitly contained in the remaining sensor data. Furthermore, in economic forecasting, one might want to continue to use a predictor even when an input variable becomes meaningless (for example, due to political changes in a country). As we have elaborated in earlier papers, heuristics such as the substitution of the mean for an unknown feature can lead to solutions that are far from optimal (Ahmad and Tresp, 1993, Tresp, Ahmad, and Neuneier, 1994). Biological systems must deal continuously with the problem of unknown uncertain features and they are certainly extremely good at it. From a biological point of view it is therefore interesting which solutions to this problem can be derived from theory and if these solutions are in any way related to the way that biology deals with this problem (compare Brunelli and Poggio, 1991). Finally, having efficient methods for dealing with missing features allows a novel pruning strategy: if the quality of the prediction is not affected if an input is pruned, we can remove it and use our solutions for prediction with missing inputs or retrain the model without that input (Tresp, Hollatz and Ahmad, 1995). In Ahmad and Tresp (1993) and in Tresp, Ahmad and Neuneier (1994) equations for training and recall were derived using a probabilistic setting (compare also Buntine and Weigend, 1991, Ghahramani and Jordan, 1994). For general feedforward neural networks the solution was in the form of an integral which has to be approximated using numerical integration techniques. The computational complexity of these solutions grows exponentially with the number of missing features. In these two publications, we could only obtain efficient algorithms for networks of normalized Gaussian basis functions. It is of great practical interest to find efficient ways of dealing with missing inputs for general feedforward neural networks which are more commonly used in applications. In this paper we describe an efficient approximation for the problem of missing information that is applicable to a large class of learning algorithms, including feedforward networks. The main results are Equation 2 (recall) and Equation 3 (training). One major advantage of the proposed solution is that the complexity does not increase with an increasing number of missing inputs. The solutions can easily be generalized to the problem of uncertain (noisy) inputs. 2 Missing Information During Recall 2.1 Theory We assume that a neural network N N(x) has been trained to predict E(ylx), the expectation of y E !R given x E ~. During recall we would like to know the network's prediction based on an incomplete input vector x = (XC, XU) where XC denotes the known inputs and XU the unknown inputs. The optimal prediction given the known features can be written as (Ahmad and Tresp, 1993) Efficient Methods for Dealing with Missing Data in Supervised Learning u ~=X o . Yi x;~,a) • Xl: 00 : 000 o X c o o 691 Figure 1: The circles indicate 10 Gaussians approximating the input density distribution. XC = Xl indicates the known input, X2 = XU is unknown. Similarly, for a network trained to estimate class probabilities, N Ni(X) ~ P(classilx), simply substitute P(classdxC) for E(ylx C ) and N Ni(X C , XU) for N N (XC, XU) in the last equation. The integrals in the last equations Can be problematic. In the worst case they have to be approximated numerically (Tresp, Ahmad and Neuneier, 1994) which is costly, since the computation is exponential in the number of missing inputs. For networks of normalized Gaussians, there exist closed form solutions to the integrals (Ahmad and Tresp, 1993). The following section shows how to efficiently approximate the integral for a large class of algorithms. 2.2 An Efficient Approximation Parzen windows are commonly used to approximate densities. Given N training data {(xk, yk)lk = 1, ... , N}, we can approximate 1 N P(x) ~ N L:G(x;xk,O') k=l (1) where k 11k 2 G( X; X ,0') = (211'0'2)D /2 exp( - 20'211x - X II ) is a multidimensional properly normalized Gaussian centered at data xk with variance 0'2. It has been shown (Duda and Hart (1973)) that Parzen windows approximate densities for N 00 arbitrarily well, if 0' is appropriately scaled. 692 Volker Tresp, Ralph Neuneier, Subutai Ahmad Using Parzen windows we may write where we have used the fact that and where G( xc; xc,le, u) is a Gaussian projected onto the known input dimensions (by simply leaving out the unknown dimensions in the exponent and in the normalization, see Ahmad and Tresp, 1993). xc,le are the components of the training data corresponding to the known input (compare Figure 1). Now, if we assume that the network prediction is approximately constant over the "width" of the Gaussians, u, we can approximate J NN(xC, x") G(XC,x";xle,u) dx" ~ NN(xC,x",Ie) G(XC;xc,le,u) where N N(xC, x",Ie) is the network prediction which we obtain if we substituted the corresponding components of the training data for the unknown inputs. With this approximation, (2) Interestingly, we have obtained a network of normalized Gaussians which are centered at the known components of the data points. The" output weights" N N(xC, x",Ie) consist of the neural network predictions where for the unknown input the corresponding components of the training data points have been substituted. Note, that we have obtained an approximation which has the same structure as the solution for normalized Gaussian basis functions (Ahmad and Tresp, 1994). In many applications it might be easy to select a reasonable value for u using prior knowledge but there are also two simple ways to obtain a good estimate for u using leave-one-out methods. The first method consists of removing the k - th pattern from the training data and calculating P(xle ) ~ N:l L:f:l,l# G(xle ; xl, u). Then select the u for which the log likelihood L:1e log P(xle ) is maximum. The second method consists of treating an input of the k - th training pattern as missing and then testing how well our algorithm (Equation 2) can predict the target. Select the u which gives the best performance. In this way it would even be possible to select input-dimension-specific widths Ui leading to "elliptical", axis-parallel Gaussians (Ahmad and Tresp, 1993). Efficient Methods for Dealing with Missing Data in Supervised Learning 693 Note that the complexity of the solution is independent of the number of missing inputs! In contrast, the complexity of the solution for feedforward networks suggested in Tresp, Ahmad and Neuneier (1994) grows exponentially with the number of missing inputs. Although similar in character to the solution for normalized RBFs, here we have no restrictions on the network architecture which allows us to choose the network most appropriate for the application. If the amount of training data is large, one can use the following approximations: • Select only the K nearest data points. The distance is determined based on the known inputs. K can probably be reasonably small « 10). In the extreme case, K = 1 and we obtain a nearest-neighbor solution. Efficient tree-based algorithms exist for computing the K-nearest neighbors. • Use Gaussian mixtures instead of Parzen windows to estimate the input data distribution. Use the centers and variances of the components in Equation 2. • Use a clustering algorithm and use the cluster centers instead of the data points in Equation 2. Note that the solution which substitutes the components of the training data closest to the input seems biologically plausible. 2.3 Experimental Results We tested our algorithm using the same data as in Ahmad and Tresp, 1993. The task was to recognize a hand gesture based on its 2D projection. As input, the classifier is given the 2D polar coordinates of the five finger tip positions relative to the 2D center of mass of the hand (the input space is therefore 10-D). A multi-layer perceptron was trained on 4368 examples (624 poses for each gesture) and tested on a similar independent test set. The inputs were normalized to a variance of one and u was set to 0.1. (For a complete description of the task see (Ahmad and Tresp, 1993).) As in (Ahmad & Tresp, 1993) we defined a correct classification as one in which the correct class was either classified as the most probable or the second most probable. Figure 2 shows experimental results. On the horizontal axis, the number of randomly chosen missing inputs is shown. The continuous line shows the performance using Equation 2 where we used only the 10 nearest neighbors in the approximation. Even with 5 missing inputs we obtain a score of over 90 % which is slightly better than the solution we obtained in Ahmad and Tresp (1993) for normalized RBFs. We expect our new solution to perform very well in general since we can always choose the best network for prediction and are not restricted in the architecture. As a benchmark we also included the case where the mean of the missing input was substituted. With 5 missing inputs, the performance is less than 60 %. 694 Volker Tresp. Ralph Neuneier. Subutai Ahmad 3D - Hanel Geetu", R8CC91ftIOn : 10 inpuIa ----.---0.9 '. 0.8 '. !II. !.7 .... . •.. " . 0.8 " . .•... ". " . .•........ 0.5 .... , 2 3 4 5 8 7 runber 01 rristing Inputs Figure 2: Experimental results using a generalization data set. The continuous line indicates the performance using our proposed method. The dotted lines indicate the performance if the mean of the missing input variable is substituted. As a comparison, we included the results obtained in Ahmad and Tresp (1993) using the closed-form solution for RBF -networks (dashed). 3 Training (Backpropagation) For a complete pattern (x.l:, yk), the weight update of a backpropagation step for weight Wj is A • (k _ NN. ( k))8NNw (xk) U.wJ (X Y w x 8 . Wj Using the approximation of Equation 1, we obtain for an incomplete data point (compare Tresp, Ahmad and Neuneier, 1994) (3) Here, IE compl indicates the sum over complete patterns in the training set, and (111 is the standard deviation of the output noise. Note that the gradient is a network of normalized Gaussian basis functions where the "output-weight" is now Efficient Methods for Dealing with Missing Data in Supervised Learning 695 The derivation of the last equation can be found in the Appendix. Figure 3 shows experimental results. Boston housing data: 13 inputs 0.15r------,r-------,r------,--~--__r--__._--"""T""--...., 0.14 0.13 sub6titute mean 0.12 B 28 CQI11)Iete pattems 1 0.11 0.1 ..... --------------------~~-------------0.09 0.08 O·061'----2'------J3'------J4'-----L5--~6-----'-7----'-8 ----'9 number of missing inputs Figure 3: In the experiment, we used the Boston housing data set, which consists of 506 samples. The task is to predict the housing price from 13 variables which were thought to influence the housing price in a neighborhood. The network (multilayer perceptron) was trained with 28 complete patterns plus an additional 225 incomplete samples. The horizontal axis indicates how many inputs were missing in these 225 samples. The vertical axis shows the generalization performance. The continuous line indicates the performance of our approach and the dash-dotted line indicates the performance, if the mean is substituted for a missing variable. The dashed line indicates the performance of a network only trained with the 28 complete patterns. 4 Conclusions We have obtained efficient and robust solutions for the problem of recall and training with missing data. Experimental results verified our method. All of our results can easily be generalized to the case of noisy inputs. Acknowledgement Valuable discussions with Hans-Georg Zimmermann, Tomaso Poggio, Michael Jordan and Zoubin Ghahramani are greatfully acknowledged. The first author would like to thank the Center for Biological and Computational Learning (MIT) for providing and excellent research environment during the summer of 1994. 696 Volker Tresp, Ralph Neuneier, Subutai Ahmad 5 Appendix Assuming the standard signal-plus-Gaussian-noise model we obtain for a complete sample P(xk,ykl{wi}) = G(yk;NNw(xk),(1y) P(xk) where {Wi} is the set of weights in the network. For an incomplete sample Using the same approximation as in Section 2.2, p(xc,k, ykl{ W;}) ~ L G(yk; N Nw(xc,k, xu,I), (1y) G(xc,k; xc, I, (1) l€compl where I sums over all complete samples. As before, we substitute for the missing components the ones from the complete training data. The log-likelihood C (a function of the network weights {Wi}) can be calculated as (xk can be either complete or incomplete) C = Lf=llog p(xk, yk I{ Wi}). The maximum likelihood solution consists of finding weights {Wi} which maximize the log-likelihood. Using the approximation of Equation 1, we obtain for an incomplete sample as gradient Equation 3 (compare Tresp, Ahmad and Neuneier, 1994). References Ahmad, S. and Tresp, V. (1993). Some Solutions to the Missing Feature Problem in Vision. In S. J. Hanson, J. D. Cowan and C. L. Giles, (Eds.), Advances in Neural Information Processing Systems 5, San Mateo, CA: Morgan Kaufmann. Brunelli, R. and Poggio, T. (1991). HyperBF Networks for Real Object Recognition. IJCAL Buntine, W. L. and Weigend, A. S. (1991). Bayesian Back-Propagation. Complex systems, Vol. 5, pp. 605-643. Duda, R. O. and Hart, P. E. (1973). Pattern Classification and Scene Analysis. John Wiley and Sons, New York. Ghahramani, Z. and Jordan, M. 1. (1994). Supervised Learning from Incomplete Data via an EM approach. In: Cowan, J. D., Tesauro, G., and Alspector, J., eds., Advances in Neural Information Processing Systems 6, San Mateo, CA, Morgan Kaufman. Tresp, V., Ahmad, S. and Neuneier, R. (1994). Training Neural Networks with Deficient Data. In: Cowan, J. D., Tesauro, G., and Alspector, J., eds., Advances in Neural Information Processing Systems 6, San Mateo, CA, Morgan Kaufman. Tresp, V., Hollatz, J. and Ahmad, S. (1995). Representing Probabilistic Rules with Networks of Gaussian Basis Functions. Accepted for publication in Machine Learning.	1994	46
939	PREDICTIVE CODING WITH NEURAL NETS: APPLICATION TO TEXT COMPRESSION J iirgen Schmidhuber Fakultat fiir Informatik Technische Universitat Miinchen 80290 Miinchen, Germany Abstract Stefan Heil To compress text files, a neural predictor network P is used to approximate the conditional probability distribution of possible "next characters", given n previous characters. P's outputs are fed into standard coding algorithms that generate short codes for characters with high predicted probability and long codes for highly unpredictable characters. Tested on short German newspaper articles, our method outperforms widely used Lempel-Ziv algorithms (used in UNIX functions such as "compress" and "gzip"). 1048 liirgen Schmidhuber, Stefan Heil 1 INTRODUCTION The method presented in this paper is an instance of a strategy known as "predictive coding" or "model-based coding". To compress text files, a neural predictor network P approximates the conditional probability distribution of possible "next characters", given n previous characters. P's outputs are fed into algorithms that generate short codes for characters with low information content (characters with high predicted probability) and long codes for characters conveying a lot of information (highly unpredictable characters) [5]. Two such standard coding algorithms are employed: Huffman Coding (see e.g. [1]) and Arithmetic Coding (see e.g. [7]). With the off-line variant of the approach, P's training phase is based on a set F of training files. After training, the weights are frozen. Copies of P are installed at all machines functioning as message receivers or senders. From then on, P is used to encode and decode unknown files without being changed any more. The weights become part of the code of the compression algorithm. Note that the storage occupied by the network weights does not have to be taken into account to measure the performance on unknown files - just like the code for a conventional data compression algorithm does not have to be taken into account. The more sophisticated on-line variant of our approach will be addressed later. 2 A PREDICTOR OF CONDITIONAL PROBABILITIES Assume that the alphabet contains k possible characters Zl, Z2, •.• , Z1c. The (local) representation of Zi is a binary k-dimensional vector r( Zi) with exactly one non-zero component (at the i-th position). P has nk input units and k output units. n is called the "time-window size". We insert n default characters Zo at the beginning of each file. The representation of the default character, r(zo), is the k-dimensional zero-vector. The m-th character of file f (starting from the first default character) is called efn. For all f E F and all possible m > n, P receives as an input r(e;"_n) 0 r(e;"_n+l) 0 ... 0 r(c!n_l), (1) where 0 is the concatenation operator, for vectors. P produces as an output Pin, a k-dimensional output vector. Using back-propagation [6][2][3][4], P is trained to mmlIDlze ~ L L II r(c!n) - pin 112 . jEFm>n (2) Expression (2) is minimal if pin always equals E(r(efn) I e;"_n,·· ·,c!n-l), (3) the conditional expectation of r( efn), given r( e;"_n) ore e;"_n+1)o . . . or( c!n-l). Due to the local character representation, this is equivalent to (Pin); being equal to the Predictive Coding with Neural Nets 1049 conditional probability (4) for all / and for all appropriate m> n, where (P,{Jj denotes the j-th component of the vector P/n. In general, the (P/n)i will not quite match the corresponding conditional probabilities. For normalization purposes, we define PI (.) _ (P/n)i m 1 j: f· L:j=I(Pm)j No normalization is used during training, however. 3 HOW TO USE THE PREDICTOR FOR COMPRESSION (5) We use a standard procedure for predictive coding. With the help of a copy of P, an unknown file / can be compressed as follows: Again, n default characters are inserted at the beginning. For each character cfn (m> n), the predictor emits its output P/n based on the n previous characters. There will be a k such that cfn = Zj:. The estimate of P(cfn = Zj: I c!n-n, ... , Crn-l) is given by P/n(k). The code of cfn, code( cfn), is generated by feeding P/n (k) into the Huffman Coding algorithm (see below), or, alternatively, into the Arithmetic Coding algorithm (see below). code(cfn) is written into the compressed file. The basic ideas of both coding algorithms are described next. 3.1 HUFFMAN CODING With a given probability distribution on a set of possible characters, Huffman Coding (e.g. [1]) encodes characters by bitstrings as follows. Characters are terminal nodes of a binary tree to be built in an incremental fashion. The probability of a terminal node is defined as the probability of the corresponding character. The probability of a non-terminal node is defined as the sum of the probabilities of its sons. Starting from the terminal nodes, a binary tree is built as follows: Repeat as long as possible: Among those nodes that are not children 0/ any non-terminal nodes created earlier, pick two with lowest associated probabilities. Make them the two sons 0/ a newly generated non-terminal node. The branch to the "left" son of each non-terminal node is labeled by a O. The branch to its "right" son is labeled by a 1. The code of a character c, code(c), is the bitstring obtained by following the path from the root to the corresponding node. Obviously, if c #- d, then code(c) cannot be the prefix of code(d). This makes the code uniquely decipherable. 1050 Jurgen Schmidhuber, Stefan Heil Characters with high associated probability are encoded by short bitstrings. Characters with low associated probability are encoded by long bitstrings. Huffman Coding guarantees minimal expected code length, provided all character probabilities are integer powers of ~. 3.2 ARlTHMETIC CODING In general, Arithmetic Coding works slightly better than Huffman Coding. For sufficiently long messages, Arithmetic Coding achieves expected code lenghts arbitrarily close to the information-theoretic lower bound. This is true even if the character probabilities are not powers of ~ (see e.g. [7]). The basic idea of Arithmetic Coding is: a message is encoded by an interval of real numbers from the unit interval [0,1[. The output of Arithmetic Coding is a binary representation of the boundaries of the corresponding interval. This binary representation is incrementally generated during message processing. Starting with the unit interval, for each observed character the interval is made smaller, essentially in proportion to the probability of the character. A message with low information content (and high corresponding probability) is encoded by a comparatively large interval whose precise boundaries can be specified with comparatively few bits. A message with a lot of information content (and low corresponding probability) is encoded by a comparatively small interval whose boundaries require comparatively many bits to be specified. Although the basic idea is elegant and simple, additional technical considerations are necessary to make Arithmetic Coding practicable. See [7] for details. Neither Huffman Coding nor Arithmetic Coding requires that the probability distribution on the characters remains fixed. This allows for using "time-varying" conditional probability distributions as generated by the neural predictor. 3.3 HOW TO "UNCOMPRESS" DATA The information in the compressed file is sufficient to reconstruct the original file without loss of information. This is done with the "uncompress" algorithm, which works as follows: Again, for each character efn (m > n), the predictor (sequentially) emits its output pin based on the n previous characters, where the e{ with n < I < m were gained sequentially by feeding the approximations p/ (k) of the probabilities P(e{ = ZIc I e{-n,···, e{-l) into the inverse Huffman Coding procedure (see e.g. [1]), or, alternatively (depending on which coding procedure was used), into the inverse Arithmetic Coding procedure (e.g. [7]). Both variants allow for correct decoding of c{ from eode(c{). With both variants, to correctly decode some character, we first need to decode all previous characters. Both variants are guaranteed to restore the original file from the compressed file. WHY NOT USE A LOOK-UP TABLE INSTEAD OF A NETWORK? Because a look-up table would be extremely inefficient. A look-up table requires kn +1 entries for all the conditional probabilities corresponding to all possible comPredictive Coding with Neural Nets 1051 binations of n previous characters and possible next characters. In addition, a special procedure is required for dealing with previously unseen combinations of input characters. In contrast, the size of a neural net typically grows in proportion to n2 (assuming the number of hidden units grows in proportion to the number of input units), and its inherent "generalization capability" is going to take care of previously unseen combinations of input characters (hopefully by coming up with good predicted probabilities). 4 SIMULATIONS We implemented both alternative variants of the encoding and decoding procedure described above. Our current computing environment prohibits extensive experimental evaluations of the method. The predictor updates turn out to be quite time consuming, which makes special neural net hardware recommendable. The limited software simulations presented in this section, however, will show that the "neural" compression technique can achieve "excellent" compression ratios. Here the term "excellent" is defined by a statement from [1]: "In general, good algorithms can be expected to achieve an average compression ratio of 1.5, while excellent algorithms based upon sophisticated processing techniques will achieve an average compression ratio exceeding 2.0." Here the average compression ratio is the average ratio between the lengths of original and compressed files. The method was applied to German newspaper articles. The results were compared to those obtained with standard encoding techniques provided by the operating system UNIX, namely "pack", "compress", and "gzip" . The corresponding decoding algorithms are "unpack", "uncompress", and "gunzip", respectively. ''pack'' is based on Huffman-Coding (e.g. [1]), while "compress" and "gzip" are based on techniques developed by Lempel and Ziv (e.g. [9]). As the file size goes to infinity, Lempel-Ziv becomes asymptotically optimal in a certain information theoretic sense [8]. This does not necessarily mean, however, that Lempel-Ziv is optimal for finite file sizes. The training set for the predictor was given by a set of 40 articles from the newspaper Miinchner M erkur, each containing between 10000 and 20000 characters. The alphabet consisted of k = 80 possible characters, including upper case and lower case letters, digits, interpunction symbols, and special German letters like "0", "ii", "a.". P had 430 hidden units. A "true" unit with constant activation 1.0 was connected to all hidden and output units. The learning rate was 0.2. The training phase consisted of 25 sweeps through the training set. The test set consisted of newspaper articles excluded from the training set, each containing between 10000 and 20000 characters. Table 1 lists the average compression ratios. The "neural" method outperformed the strongest conventional competitor, the UNIX "gzip" function based on a Lempel-Ziv algorithm. 1052 Jurgen Schmidhuber, Stefan Heil Method Compression Ratio I Huffman Coding (UNIX: pack) 1.74 Lempel-Ziv Coding (UNIX: compress) 1.99 Improved Lempel-Ziv ( UNIX: gzip -9) 2.29 Neural predictor + Huffman Coding, n = 5 2.70 Neural predictor + Arithmetic Coding, n = 5 2.72 Table 1: Compression ratios of various compression algorithms for short German text files « 20000 Bytes) from the unknown test set. Method Compression Ratio I Huffman Coding (UNIX: pack) 1.67 Lempel-Ziv Coding (UNIX: compress) 1.71 Improved Lempel-Zlv ( UNIX: gzip -9) 2.03 Neural predictor + Huffman Coding, n = 5 2.25 Neural predictor + Arithmetic Coding, n = 5 2.20 Table 2: Compression ratios for articles from a different newspaper. The neural predictor was not retrained. How does a neural net trained on articles from Miinchner Merkurperform on articles from other sources? Without retraining the neural predictor, we applied all competing methods to 10 articles from another German newspaper (the Frankenpost). The results are given in table 2. The Frankenpost articles were harder to compress for all algorithms. But relative performance remained comparable. Note that the time-window was quite small (n = 5). In general, larger time windows will make more information available to the predictor. In turn, this will improve the prediction quality and increase the compression ratio. Therefore we expect to obtain even better results for n > 5 and for recurrent predictor networks. 5 DISCUSSION / OUTLOOK Our results show that neural networks are promising tools for loss-free data compression. It was demonstrated that even off-line methods based on small time windows can lead to excellent compression ratios - at least with small text files, they can outperform conventional standard algorithms. We have hardly begun, however, to exhaust the potential of the basic approach. 5.1 ON-LINE METHODS A disadvantage of the off-line technique above is that it is off-line: The predictor does not adapt to the specific text file it sees. This limitation is not essential, however. It is straight-forward to construct an on-line variant of the approach. Predictive Coding with Neural Nets 1053 With the on-line variant, the predictor continues to learn during compression. The on-line variant proceeds like this: Both the sender and the receiver start with exactly the same initial predictor. Whenever the sender sees a new character, it encodes it using its current predictor. The code is sent to the receiver who decodes it. Both the sender and the receiver use exactly the same learning protocol to modify their weights. This implies that the modified weights need not be sent from the sender to the receiver and do not have to be taken into account to compute the average compression ratio. Of course, the on-line method promises much higher compression ratios than the off-line method. 5.2 LIMITATIONS The main disadvantage of both on-line and off-line variants is their computational complexity. The current off-line implementation is clearly slower than conventional standard techniques, by about three orders of magnitude (but no attempt was made to optimize the code with respect to speed). And the complexity of the on-line method is even worse (the exact slow-down factor depends on the precise nature of the learning protocol, of course). For this reason, especially the promising on-line variants can be recommended only if special neural net hardware is available. Note, however, that there are many commercial data compression applications which rely on specialized electronic chips. 5.3 ONGOING RESEARCH There are a few obvious directions for ongoing experimental research: (1) Use larger time windows - they seem to be promising even for off-line methods (see the last paragraph of section 4). (2) Thoroughly test the potential of on-line methods. Both (1) and (2) should greatly benefit from fast hardware. (3) Compare performance of predictive coding based on neural predictors to the performance of predictive coding based on different kinds of predictors. 6 ACKNOWLEDGEMENTS Thanks to David MacKay for directing our attention towards Arithmetic Coding. Thanks to Margit Kinder, Martin Eldracher, and Gerhard Weiss for useful comments. 1054 Jiirgen Schmidhuber, Stefan Heil References [1] G. Held. Data Compression. Wiley and Sons LTD, New York, 1991. [2] Y. LeCun. Une procedure d'apprentissage pour reseau a. seuil asymetrique. Proceedings of Cognitiva 85, Paris, pages 599-604, 1985. [3] D. B. Parker. Learning-logic. Technical Report TR-47, Center for Compo Research in Economics and Management Sci., MIT, 1985. [4] D. E. Rumelhart, G. E. Hinton, and R. J. Williams. Learning internal representations by error propagation. In Parallel Distributed Processing, volume 1, pages 318-362. MIT Press, 1986. [5] J. H. Schmidhuber and S. Heil. Sequential neural text compression. IEEE Transactions on Neural Networks, 1994. Accepted for publication. [6] P. J. Werbos. Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences. PhD thesis, Harvard University, 1974. [7] I. H. Witten, R. M. Neal, and J. G. Cleary. Arithmetic coding for data compression. Communications of the ACM, 30(6):520-540, 1987. [8] A. Wyner and J. Ziv. Fixed data base version of the Lempel-Ziv data compression algorithm. IEEE Transactions In/ormation Theory, 37:878-880, 1991. [9] J. Ziv and A. Lempel. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory, IT-23(5):337-343, 1977.	1994	47
940	Computational structure of coordinate transformations: A generalization study Zoubin Ghahramani zoubin@psyche.mit.edu Daniel M. Wolpert wolpert@psyche.mit.edu Michael I. Jordan jordan@psyche.mit.edu Department of Brain & Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 Abstract One of the fundamental properties that both neural networks and the central nervous system share is the ability to learn and generalize from examples. While this property has been studied extensively in the neural network literature it has not been thoroughly explored in human perceptual and motor learning. We have chosen a coordinate transformation system-the visuomotor map which transforms visual coordinates into motor coordinates-to study the generalization effects of learning new input-output pairs. Using a paradigm of computer controlled altered visual feedback, we have studied the generalization of the visuomotor map subsequent to both local and context-dependent remappings. A local remapping of one or two input-output pairs induced a significant global, yet decaying, change in the visuomotor map, suggesting a representation for the map composed of units with large functional receptive fields. Our study of context-dependent remappings indicated that a single point in visual space can be mapped to two different finger locations depending on a context variable-the starting point of the movement. Furthermore, as the context is varied there is a gradual shift between the two remappings, consistent with two visuomotor modules being learned and gated smoothly with the context. 1 Introduction The human central nervous system (CNS) receives sensory inputs from a multitude of modalities, each tuned to extract different forms of information from the 1126 Zoubin Ghahramani, Daniel M. Wolpert, Michael 1. Jordan environment. These sensory signals are initially represented in disparate coordinate systems, for example visual information is represented retinotopically whereas auditory information is represented tonotopically. The ability to transform information between coordinate systems is necessary for both perception and action. When we reach to a visually perceived object in space, for example, the location of the object in visual coordinates must be converted into a representation appropriate for movement, such as the configuration of the arm required to reach the object. The computational structure of this coordinate transformation, known as the visuomotor map, is the focus of this paper. By examining the change in visuomotor coordination under prismatically induced displacement and rotation, Helmholtz (1867/1925) and Stratton (1897a,1897b) pioneered the systematic study of the representation and plasticity of the visuomotor map. Their studies demonstrate both the fine balance between the visual and motor coordinate systems, which is disrupted by such perturbations, and the ability of the visuomotor map to adapt to the displacements induced by the prisms. Subsequently, many studies have further demonstrated the remarkable plasticity of the map in response to a wide variety of alterations in the relationship between the visual and motor system (for reviews see Howard, 1982 and Welch, 1986)-the single prerequisite for adaptation seems to be that the remapping be stable (Welch, 1986). Much less is known, however, about the topological properties of this map. A coordinate transformation such as the visuomotor map can be regarded as a function relating one set of variables (inputs) to another (outputs). For the visuomotor map the inputs are visual coordinates of a desired target and the outputs are the corresponding motor coordinates representing the arm's configuration (e.g. joint angles). The problem of learning a sensorimotor remapping can then be regarded as a function approximation problem. Using the theory of function approximation one can make explicit the correspondence between the representation used and the patterns of generalization that will emerge. Function approximators can predict patterns of generalization ranging from local (look-up tables), through intermediate (CMACs, Albus, 1975; and radial basis functions, Moody and Darken, 1989 ) to global (parametric models). In this paper we examine the representational structure of the visuomotor map through the study of its spatial and contextual generalization properties. In the spatial generalization study we address the question of how pointing changes over the reaching workspace after exposure to a highly localized remapping. Previous work on spatial generalization, in a study restricted to one dimension, has led to the conclusion that the visuomotor map is constrained to generalize linearly (Bedford, 1989). We test this conclusion by mapping out the pattern of generalization induced by one and two remapped points in two dimensions. In the contextual generalization study we examine the question of whether a single point in visual space can be mapped into two different finger locations depending on the context of a movement-the start point. If this context-dependent remapping occurs, the question arises as to how the mapping will generalize as the context is varied. Studies of contextual remapping have previously shown that variables such as eye position (Kohler, 1950; Hay and Pick, 1966; Shelhamer et al., 1991), the feel of prisms (Kravitz, 1972; Welch, 1971) or an auditory tone (Kravitz and Yaffe, 1972), can induce context-dependent aftereffects. The question of how these contextComputational Structure of Coordinate Transformations 1127 dependent maps generalize-which has not been previously explored-reflects on the possible representation of multiple visuomotor maps and their mixing with a context variable. 2 Spatial Generalization To examine the spatial generalization of the visuomotor map we measured the change in pointing behavior subsequent to one- and two-point remappings. In order to measure pointing behavior and to confine subjects to learn limited input-output pairs we used a virtual visual feedback setup consisting of a digitizing tablet to record the finger position on-line and a projection/mirror system to generate a cursor spot image representing the finger position (Figure 1a). By controlling the presence of the cursor spot and its relationship to the finger position, we could both restrict visual feedback of finger position to localized regions of space and introduce perturbations of this feedback. a) VGAScreen PnljecIor Ii \ /, \ Angerf_ /, \ /, \ /0: \ image / ,/ ,l \) Rear profection acreen -, ' >'"'!'f'----Vi-rtual-lma-II"Seml-sliverad mirTOr Dl!Jtizjng Tablel Actual FIngerPosI1Ion b) P_ 0 0 0 Actual Finger Position Anger Position ~~ o ;(.. "'" o a .......... 150m '.. ~ .. .... '.. -" C)[J' , , ..... · . . d)Q' . • 1 • • T • e)lZj' , " ,t , , , , Figure 1. a) Experimental setup. The subjects view the reflected image of the rear projection screen by looking down at the mirror. By matching the screenmirror distance to the mirror-tablet distance all projected images appeared to be in the plane of the finger (when viewed in the mirror) independent of head position. b) The position of the grid of targets relative to the subject. Also shown, for the x-shift condition, is the perceived and actual finger position when pointing to the central training target. The visually perceived finger position is indicated by a cursor spot which is displaced from the actual finger position. c) A schematic showing the perturbation for the x-shift group. To see the cursor spot on the central target the subjects had to place their finger at the position indicated by the tip of the arrow. d) & e) Schematics similar to c) showing the perturbation for the y-shift and two point groups, respectively. In the tradition of adaptation studies (e.g. Welch, 1986), each experimental session consisted of three phases: pre-exposure, exposure, and post-exposure. During the pre- and post-exposure phases, designed to assess the visuomotor map, the subject pointed repeatedly, without visual feedback of his finger position, to a grid of targets over the workspace. As no visual input of finger location was given, no learning of the visuomotor map could occur. During the exposure phase subjects pointed repeatedly to one or two visual target locations, at which we introduced a discrep1128 Zoubin Ghahramani, Daniel M. Wolpert, Michael!. Jordan ancy between the actual and visually displayed finger location. No visual feedback of finger position was given except when within 0.5 cm of the target, thereby confining any learning to the chosen input-output pairs. Three local perturbations of the visuomotor map were examined: a 10 cm rightward displacement (x-shift group, Figure lc), 10 cm displacement towards the body (y-shift group, Figure Id), and a displacement at two points, one 10 cm away from, and one 10 cm towards the body (two point group, Figure Ie). For example, for the x-shift displacement the subject had to place his finger 10 cm to the right of the target to visually perceive his finger as being on target (Figure Ib). Separate control subjects, in which the relationship between the actual and visually displayed finger position was left unaltered, were run for both the one- and two-point displacements, resulting in a total of 5 groups with 8 subjects each. 50 -€) 45 40 ] JS >30 25 X «(,:m) 45 CJ Cl 41) 25 t9 20 -10 -5 10 15 20 X (I.:m ) 55 50 45 c 40 '0 e 0. ~ 35 ~ >30 E25 20 G 15 50 __ .......................... " 4' __ .......................... ..... 40 __ ............ _ ........ _ .. ..';1.' __ ............ __ ,. 30 ........ --..... ~~ .............. 20 10 1 ~ 20 X (Lm) 45 \ 40 \ I ~ 35 >- 30 \ 25 20 -10 ".'i 10 15 20 55 5(J 45 30 25 2(J 15 x (em) , , , " , \ I I' t ~ I t ~ tit I I I I I t t t t J f X (em) -10 () 10 2() X (em) -15 · 10 -5 0 5 10 15 20 2.' -15 -10·5 I) 5 11) 15 211 25 -10 () !() 2() x (em) X (em) X (em) Figure 2. Results of the spatial generalization study. The first column shows the mean change in pointing, along with 95% confidence ellipses, for the x-shift, y-shift and two point groups. The second column displays a vector field of changes obtained from the data by Gaussian kernel smoothing. The third column plots the proportion adaptation in the direction of the perturbation. Note that whereas for the x- and yshift groups the lighter shading corresponds to greater adaptation, for the two point group lighter shades correspond to adaptation in the positive y direction and darker shades in the negative y direction. Computational Structure of Coordinate Transfonnations 1129 The patterns of spatial generalization subsequent to exposure to the three local remappings are shown in Figure 2. All three perturbation groups displayed both significant adaptation at the trained points, and significant, through decremented, generalization of this learning to other targets. As expected, the control groups (not shown) did not show any significant changes. The extent of spatial generalization, best depicted by the shaded contour plots in Figure 2, shows a pattern of generalization that decreases with distance away from the trained points. Rather than inducing a single global change in the map, such as a rotation or shear, the two point exposure appears to induce two opposite fields of decaying generalization at the intersection of which there is no change in the visuomotor map. 3 Contextual Generalization The goal of this experiment was first to explore the possibility that multiple visuomotor maps, or modules, could be learned, and if so, to determine how the overall system behaves as the context used in training each module is varied. To achieve this goal, we exposed subjects to context-dependent remappings in which a single visual target location was mapped to two different finger positions depending on the start point of the movement. Pointing to the target from seven different starting points (Figure 3) was assessed before and after an exposure phase. During this exposure phase subjects made repeated movements to the target from starting points 2 and 6 with a different perturbation of the visual feedback depending on the starting point. The form of these context-dependent remappings is shown in Figure 3. For example, for the open x-shift group (Figure 3c), the visual feedback of the finger was displaced to the right for movements from point 2 and to the left from point 6. Therefore the same visual target was mapped to two different finger positions depending on the context of the movement. To test learning of the remapping and generalization to other start points we examined the change in pointing, without visual feedback, to the target from the 7 start points. a) control b) crossed x-shift ~ ~' e'-' • .. ' -' . .. ... • t t, o 1234567 1234567 c) open x-shift ~ .. ' . " ........... ...•.•. . .....• o dlYL\ :,., .... o 0 1234567 1234567 Figure 3. Schematic of the exposure phase in the contextual generalization experiment. Shown are the actual finger path (solid line), the visually displayed finger path (dotted line), the seven start points and the target used in the pre- and postexposure phases. The perturbation introduced depended on whether the movement started form start point 2 or 6. Note that for the three perturbation groups, although the subjects saw a triangle being traced out, the finger took a different path. 1130 Zoubin Ghahramani, Daniel M. Wolpert, Michael I. Jordan The results are shown in Figure 4. Whereas the controls did not show any significant pattern of change, the three other groups showed adaptive, start point dependent, changes in the direction opposite to the perturbation. Thus, for example, the xopen group displayed a pattern of change in the leftward (negative x) direction for movements from the left start points and rightwards for movements from the right start points. Furthermore, as the start point was varied, the change in pointing varied gradually. a) U control 0--0 b) 2.0 croned x·,bltt -a 0.' a !!. !!. 1.0 " " 0 .S! .;J -0.' ;; c. c. ... ~v~ .. ." ." 0.0 .. .. >< -u ... -2.' , -1.0 , 4 2 4 Start point Start point Figure 4. a) Adaptation in the x direction plotted as a function of starting point for the control, crossed x-shift and open x-shift groups (mean and 1 s.e.). b) Adaptation in the y direction for the control and y-shift groups. 4 Discussion Clearly, from the perspective of function approximation theory, the problem of relearning the visuomotor mapping from exposure to one or two input-output pairs is ill-posed. The mapping learned, as measured by the pattern of generalization to novel inputs, therefore reflects intrinsic constraints on the internal representation used. The results from the spatial generalization study suggest that the visuomotor coordinate transformation is internally represented with units with large but localized receptive fields. For example, a neural network model with Gaussian radial basis function units (Moody and Darken, 1989), which can be derived by assuming that the internal constraint in the visuomotor system is a smoothness constraint (Poggio and Girosi, 1989), predicts a pattern of generalization very similar the one experimentally observed (e.g. see Figure 5 for a simulation of the two point generalization experiment).1 In contrast, previously proposed models for the representation of the visuomotor map based on global parametric representations in terms of felt direction of gaze and head position (e.g. Harris, 1965) or linear constraints (Bedford, 1989) do not predict the decaying patterns of Cartesian generalization found. 1 See also Pouget & Sejnowski (this volume) who, based on a related analysis of neurophysiological data from parietal cortex, suggest that a basis function representation may be used in this visuomotor area. Computational Structure of Coordinate Transformations 1131 50 , \ 45 \ \ \ \ t t } ~ 40 \ \ \ t e ~ 35 \ ~ ! ~ >I t t >30 \ I I I 25 ~ ~ 20 I' I' · 10 · 5 0 10 15 20 X (em) X (em) Figure 5. Simulation of the two point spatial generalization experiment using a radial basis function network with 64 units with 5 cm Gaussian receptive fields. The inputs to the network were the visual coordinates of the target and the outputs were the joint angles for a two-link planar arm to reach the target. The network was first trained to point accurately to the targets, and then, after exposure to the perturbation, its pattern of generalization was assessed. The results from the second study suggest that multiple visuomotor maps can be learned and modulated by a context. A suggestive computational model for how such separate modules can be learned and combined is the mixture-of-experts neural network architecture (Jacobs et al. , 1991). Interpreted in this framework, the gradual effect of varying the context seen in Figure 4 could reflect the output of a gating network which uses context to modulate between two visuomotor maps. However, our results do not rule out models in which a single visuomotor map is parametrized by starting location, such as one based on the coding of locations via movement vectors (Georgopoulos, 1990). 5 Conclusions The goal of these studies has been to infer the internal constraints in the visuomotor system through the study of its patterns of generalization to local remappings. We have found that local perturbations of the visuomotor map produce global changes, suggesting a distributed representation with large receptive fields. Furthermore, context-dependent perturbations induce changes in pointing consistent with a model of visuomotor learning in which separate maps are learned and gated by the context. The approach taken in this paper provides a strong link between neural network theory and the study of learning in biological systems. Acknowledgements This project was supported in part by a grant from the McDonnell-Pew Foundation, by a grant from ATR Human Information Processing Research Laboratories, by a grant from Siemens Corporation, and by grant N00014-94-1-0777 from the Office of Naval Research. Zoubin Ghahramani and Daniel M. Wolpert are McDonnell-Pew Fellows in Cognitive Neuroscience. Michael I. Jordan is a NSF Presidential Young Investigator. 1132 Zoubin Ghahramani, Daniel M. Wolpert, Michael I. Jordan References Albus, J. (1975). A new approach to manipulator control: The cerebellar model articulation controller (CMAC). J. of Dynamic Systems, Measurement, and Control, 97:220-227. Bedford, F. (1989). Constraints on learning new mappings between perceptual dimensions. J. of Experimental Psychology: Human Perception and Performance, 15(2):232-248. Georgopoulos, A. (1990). Neurophysiology of reaching. In Jeannerod, M., editor, Attention and performance XIII, pages 227-263. Lawrence Erlbaum, Hillsdale. Harris, C. (1965). Perceptual adaptation to inverted, reversed, and displaced vision. Psychological Review, 72:419-444. Hay, J. and Pick, H. (1966). Gaze-contingent prism adaptation: Optical and motor factors. J. of Experimental Psychology, 72:640-648. Howard, 1. (1982). Human visual orientation. Wiley, Chichester, England. Jacobs, R., Jordan, M., Nowlan, S., and Hinton, G. (1991). Adaptive mixture of local experts. Neural Computation, 3:79-87. Kohler, 1. (1950). Development and alterations of the perceptual world: conditioned sensations. Proceedings of the Austrian Academy of Sciences, 227. Kravitz, J. (1972). Conditioned adaptation to prismatic displacement. Perception and Psychophysics, 11:38-42. Kravitz, J. and Yaffe, F. (1972). Conditioned adaptation to prismatic displacement with a tone as the conditional stimulus. Perception and Psychophysics, 12:305-308. Moody, J. and Darken, C. (1989). Fast learning in networks of locally-tuned processing units. Neural Computation, 1(2):281-294. Poggio, T. and Girosi, F. (1989). A theory of networks for approximation and learning. AI Lab Memo 1140, MIT. Pouget, A. and Sejnowski, T. (1994). Spatial representation in the parietal cortex may use basis functions. In Tesauro, G., Touretzky, D., and Alspector, J., editors, Advances in Neural Information Processing Systems 7. Morgan Kaufmann. Shelhamer, M., Robinson, D., and Tan, H. (1991). Context-specific gain switching in the human vestibuloocular reflex. In Cohen, B., Tomko, D., and Guedry, F., editors, Annals Of The New York Academy Of Sciences, volume 656, pages 889-891. New York Academy Of Sciences, New York. Stratton, G. (1897a). Upright vision and the retinal image. Psychological Review, 4:182187. Stra.tton, G. (1897b). Vision without inversion of the retinal image. Psychological Review, 4:341-360, 463-481. von Helmholtz, H. (1925). Treatise on physiological optics (1867). Optical Society of America, Rochester, New York. Welch, R. (1971). Discriminative conditioning of prism adaptation. Perception and Psychophysics, 10:90-92. Welch, R. (1986). Adaptation to space perception. In Boff, K., Kaufman, L., and Thomas, J., editors, Handbook of perception and performance, volume 1, pages 24- 1-24-45. Wiley-Interscience, New York.	1994	48
941	Recognizing Handwritten Digits Using Mixtures of Linear Models Geoffrey E Hinton Michael Revow Peter Dayan Deparbnent of Computer Science, University of Toronto Toronto, Ontario, Canada M5S lA4 Abstract We construct a mixture of locally linear generative models of a collection of pixel-based images of digits, and use them for recognition. Different models of a given digit are used to capture different styles of writing, and new images are classified by evaluating their log-likelihoods under each model. We use an EM-based algorithm in which the M-step is computationally straightforward principal components analysis (PCA). Incorporating tangent-plane information [12] about expected local deformations only requires adding tangent vectors into the sample covariance matrices for the PCA, and it demonstrably improves performance. 1 Introduction The usual way of using a neural network for digit recognition is to train it to output one of the ten classes. When the training data is limited to N examples equally distributed among the classes, there are only N log2 10 bits of constraint in the class labels so the number of free parameters that can be allowed in a discriminative neural net model is severely limited. An alternative approach, motivated by density estimation, is to train a separate autoencoder network on examples of each digit class and to recognise digits by seeing which autoencoder network gives the best reconstruction of the data. Because the output of each autoencoder has the same dimensionality as the image, each training example provides many more bits of constraint on the parameters. In the example we describe, 7000 training images are sufficient to fit 384, 000 parameters and the training procedure is fast enough to do the fitting overnight on an R4400-based machine. Auto-encoders can be viewed in terms of minimum description length descriptions of data in which the input-hidden weights produce a code for a particular case and 1016 Geoffrey E. Hinton, Michael Revow, Peter Dayan the hidden-output weights embody a generative model which turns this code back into a close approximation of the original example [14,7]. Code costs (under some prior) and reconstruction error (squared error assuming an isotropic Gaussian misfit model) sum to give the overall code length which can be viewed as a lower bound on the log probability density that the autoencoder assigns to the image. This properly places the emphasis on designing appropriate generative models for the data which will generalise well by assigning high log probabilities to new patterns from the same classes. We apply this idea to recognising handwritten digits from grey-level pixel images using linear auto-encoders. Linear hidden units for autoencoders are barely worse than non-linear ones when squared reconstruction error is used [I], but have the great computational advantage during training that input-hidden and hiddenoutput weights can be derived from principal components analysis (PCA) of the training data. In effect a PCA encoder approximates the entire N dimensional distribution of the data with a lower dimensional"Gaussian pancake" [13], choosing, for optimal data compression, to retain just a few of the PCs. One could build a single PCA model for each digit - however the many different styles of writing suggest that more than one Gaussian pancake should be used, by dividing the population of a single digit class into a number of sub-classes and approximating each class by its own model. A similar idea for data compression was used by [9] where vector quantization was used to define sub-classes and PCA was performed within each sub-class (see also [2]). We used an iterative method based on the Expectation Maximisation (EM) algorithm [4] to fit mixtures of linear models. The reductio of the local linear approach would have just one training pattern in each model. This approach would amount to a nearest neighbour method for recognition using a Euclidean metric for error, a technique which is known to be infelicitous. One reason for this poor performance is that characters do not only differ in writing styles, but also can undergo simple transformations such as translations and rotations. These give rise to somewhat non-linear changes as measured in pixel space. Nearest neighbour methods were dramatically improved methods by defining a metric in which locally linearised versions of these transformations cost nothing [12]. In their tangent distance method, each training or test point is represented by a h-dimensionallinear subspace, where each of these dimensions corresponds to a linear version of one of the transformations, and distances are measured between subspaces rather than points. The local linear autoencoder method can be seen just like this - variations along one of the h principal component directions are free, while variations along the remaining principal directions cost. However, rather than storing and testing each training pattern, the local models summarise the regularities over a number of patterns. This reduces storage and recognition time, and allows the directions of free variation to be averaged over numbers of patterns and also to be determined by the data rather than being pre-specified. A priori knowledge that particular transformations are important can be incorporated using a version of the tangent-prop procedure [12], which is equivalent in this case to adding in slightly transformed versions of the patterns. Also reconstruction error could be assessed either at a test pattern, or, more like tangent distance, between its transformation subspace and the models' principal components subs paces. Recognizing Handwritten Digits Using Mixtures of Linear Models ,a , , , 1017 Figure 1: Didactic example of tangent information and local linear models. See text for details. Figure 1 illustrates the idea. Imagine that the four points 1-4 portray in image space different examples of the same digit, subject to some smooth transformation. As in tangent distance, one could represent this curve using the points and their local tangents (thick lines). However one might do better splitting it into three local linear models rather than four - model 'a' Gust a line in this simple case) averages the upper part of the curvp more effectively than the combination of the two tangents at '1' and '2'. However, given just the points, one might c~nstruct model 'b' for '3' and '4', which would be unfortunate. Incorporating information about the tangents as well would encourage the separation of these segments. Care should be taken in generalising this picture to high dimensional spaces. The next section develops the theory behind variants of these systems (which is very similar to that in [5, 10]), and section 3 discusses how they perform. 2 Theory Linear auto-encoders embody a model in which variations from the mean of a population along certain directions are cheaper than along others, as measured by the log-unlikelihood of examples. Creating such a generative model is straightforward. Principal component analysis (PCA) is performed on the training data and the leading h principal components are retained, defining an h-dimensional subspace onto which the n-dimensional inputs are projected. We choose h using cross-validation, although a minimum description length criterion could also be used. We ignore the effect of different variances along the different principal components and use a model in which the code length for example i (the negative log-likelihood) is proportional to the reconstruction error - the squared Euclidean distance between the output of the autoencoder and the pattern itself. Rather than having just one autoencoder for each digit, there is a whole collection, and therefore we use EM to assign examples to n sub-classes, just as in clustering using a mixture of Gaussians generative model. During the E-step, the responsibility for each pattern is assigned amongst the sub-classes, and in the M-step PCA is performed, altering the parameters of a sub-class appropriately to minimise the 1018 Geoffrey E. Hinton, Michael Revow, Peter Dayan reconstruction cost of the data for which it is responsible. Formally, the algorithm is: 1. Choose initial autoencoder assignments for each example in the training set (typically using a K-means clustering algorithm). 2. Perform PCA separately for each autoencoder; 3. Reassign patterns to the autoencoder that reconstructs them the best; 4. Stop if no patterns have changed sub-class, otherwise return to step 2. There is a 'soft' version of the algorithm in which the responsibility of autoencoder q for example i is calculated as ri.q = e-IIE,q 112 /Za 2 /(L.r e-IIE'TII2 /Z( 2 ) where Ei.r is the reconstruction error. For this, in step 2, the examples are weighted for the PCA by the responsibilities, and convergence is assessed by examining the change in the log-likelihood of the data at each iteration. The soft version requires a choice of O'z, the assumed variance in the directions orthogonal to the pancake. The algorithm generates a set of local linear models for each digit. Given a test pattern we evaluate the code length (the log-likelihood) against all the models for all the digits. We use a hard method for classification - determining the identity of the pattern only by the model which reconstructs it best. The absolute quality of the best reconstruction and the relative qualities of slightly sub-optimal reconstructions are available to reject ambiguous cases. For a given linear model, not counting the code cost implies that deformations of the images along the principal components for the sub-class are free. This is like the metric used by [11] except that they explicitly specified the directions in which deformations should be free, rather than learning them from the data. We wished to incorporate information about the preferred directions without losing the summarisation capacity of the local models, and therefore turned to the tangent prop algorithm [12]. Tangent prop takes into account information about how the output of the system should vary locally with particular distortions of the input by penalising the system for having incorrect derivatives in the relevant directions. In our case, the overall classification process is highly non-linear, making the application of tangent-prop to it computationally unattractive, but it is easy to add a tangent constraint to the reconstruction step because it is linear. Imagine requiring the system f(p) = A.p to reconstruct x + .M and x -7-. t as well as x, for an input example x, and distortion t (the tangent vector), and where 7-. is a weight. Their contribution to the error is proportional to Ix - A.xlz + 7-.zlt - A.tI Z, where the second term is equivalent to the error term that tangent-prop would add. Incorporating this into the PCA is as simple as adding a weighted version of tt T to the covariance matrix - the tangent vectors are never added to the means of the sub-classes. 3 Results We have evaluated the performance of the system on data from the CEDAR CDROM 1 database containing handwritten digits lifted from mail pieces passing through a United States Post Office [8]. We divided the br training set of binary Recognizing Handwritten Digits Using Mixtures of Linear Models 1019 Table 1: Classification errors on the validation test when different weightings are used for the tangent vectors during clustering and recognition. No rejections were allowed. segmented digits into 7,000 training examples, 2,000 validation examples and 2,000 "internal test" examples. All digits were equally represented in these sets. The binary images in the database are all of different sizes, so they were scaled onto a 16 x 16 grid and then smoothed with a Gaussian filter. The validation set was used to investigate different choices for the Gaussian filter variances, the numbers of sub-classes per digit and principal components per model, and the different weightings on the tangent vectors. Clearly this is a large parameter space and we were only able to perform a very coarse search. In the results reported here all digits have the same number of sub-classes (10) and the number of principal components per model was picked so as to explain 95% of the training set variance assigned to that model. Once a reasonable set of parameter settings had been decided upon, we used all 11,000 images to train a final version which was tested on the official bs set (2711 images). There are two major steps to the algorithm; defining sub-classes within each digit and reconstructing for recognition. We found that the tangent vectors should be more heavily weighted in the sub-class clustering step than during ultimate recognition. Figure 2 shows the means of the 10 sub-classes for the digit two where the clustering has been done (a) without or (b) with tangent vectors. It is clear that the clusters defined in (b) capture different styles of 2s in a way that those in (a) do not - they are more diverse and less noisy. They also perform better. The raw error rate (no rejections allowed) on the validation set with different amounts of tangent vector weightings are shown in Table 1. The results on the official test set (2711 examples) are shown in Table 2. 4 Discussion and Conclusions A mixture of local linear models is an effective way to capture the underlying styles of handwritten digits. The first few principal components (less than 20 on 256-dimensional data) extract a significant proportion of the variance in the images within each sub-class. The resulting models classify surprisingly well without requiring large amounts of either storage or computation during recognition. Further, it is computationally easy and demonstrably useful to incorporate tangent information requiring reconstruction to be good for small transformations of the sample patterns in a handful of pre-specified directions. Adding tangent information is exactly equivalent to replacing each digit by a Gaussian cloud of digits perturbed along the tangent plane and is much more efficient than adding extra 1020 Geoffrey E. Hinton, Michael Revow, Peter Dayan a) b) Figure 2: Cluster means for images of twos. (a) Without tangent vectors. (b) Cluster means using translation, rotation and scaling tangent vectors. stochastically perturbed examples. The weightings applied to reconstruction of the tangent vectors are equivalent to the variances of the cloud. There is an interesting relationship between mixtures of linear models and mixtures of Gaussians. Fitting a mixture of full-covariance Gaussians is typically infeasible when the dimensionality of the input image space, n, is high and the number of training examples is limited. Each Gaussian requires n parameters to define its mean and n(n + 1 )/2 more parameters to define its covariance matrix. One way to reduce the number of parameters is to use a diagonal covariance matrix but this is a very poor approximation for images because neighbouring pixel intensities are very highly correlated. An alternative way to simplify the covariance matrix is to flatten the Gaussian into a pancake that has h dimensions within the pancake and n - h dimensions orthogonal to the pancake. Within this orthogonal subspace we assume an isotropic Gaussian distribution (ie a diagonal covariance matrix with equal, small variances along the diagonal), so we eliminate (n - h)(n - h + 1 )/2 degrees of freedom which is nearly all the degrees of freedom of the full covariance matrix if h is small compared with n. Not counting the mean, this leaves h(2n - h + 1) /2 degrees of freedom which is exactly the number of parameters required to define the first h principal components. l Thus we can view PCA as a way of fiercely constraining a full covariance Gaussian but nevertheless leaving it free to model lThe h th principal component only requires n - h + 1 parameters because it must be orthogonal to all previous components. Recognizing Handwritten Digits Using Mixtures of Linear Models • • . .. ' I··· .. ·'· ·1·· .. · .. · '''1 ...... , ~it:I; .. :~l ~ ,,~:I!:::';;.·; I;':~·~; .. i;;::~·: .••• f.... . ..... _._.... .,. ... . .......... . . ..... ..... " I'" •. ........ •• • roo r. • .. .::P.'a • • ~ ........ "..... ..... •• u' o. •. ........ , ..... a. ,.. .. . .•. .... .. .-. ..... . .o... .... :d. .. : .1.: f!~~, -. :rL !si Li. .:t-ir:~:,. 'ii: .......... .................. . ••. ~ .....••.. a:!:· .... a&,. "'lii!;~" :H:fiiiiP.~!· ·~HiiiH~ :-!'**!Hr:~:' 1021 Figure 3: Reconstruction of a 2 (left column) using 2 (upper) and 0 (lower) models. Linear 0 e Linear Models + Tangent Dist Tan ent Distance Table 2: Classification errors on the official test test when no rejections are allowed. Memory requirements are indicated in terms of the number of 16 x 16 images that need to be stored. important correlations. We have investigated a number of extensions to the basic scheme. As only one of these yielded improved results on the validation set, we will only briefly review them. If instead of assuming an isotropic Gaussian within the pancake we use an ellipsoidal subspace, then we can can take into account the different variances along each of the h principal directions. This is akin to incorporating a code cost [14]. Similarly, the squared reconstruction error used in the basic scheme also assumes an isotropic distribution. Again a diagonal covariance matrix may be substituted. We surmise that this was not successful because we had insufficient training data to estimate the variances along some dimensions reliably; for example some of the edge pixels were never turned on in the training data for some models. The Euclidean metric for the reconstruction error is convenient as it authorises the use of a powerful methods such as principal components; however as pointed out in the introduction it is deficient for situations like character recognition. We tried a scheme in which the models were trained as described above, but tangent distance [11] was used during testing. This method yielded marginally improved results on both the validation and test sets (Table 2). More adventurous tangent options, including using them in the clustering phase, were explored by [5, 10]. PCA models are not ideal as generative models of the data because they say nothing about how to generate values of components in the directions orthogonal to the pancake. To ameliorate this, the generative model may be formulated as f(p) = A.p + €, where the components of € are independent and the factors p have some prior covariance matrix. The generative weights of this autoencoder can 1022 Geoffrey E. Hinton, Michael Revow, Peter Dayan be obtained using the technique of maximum likelihood factor analysis (which is closely related to PCA) and the resulting architecture and hierarchical variants of it can be formulated as real valued versions of the Helmholtz machine [3,6]. The cost of coding the factors relative to their prior is implicitly included in this formulation, as is the possibility that different input pixels are subject to different amounts of noise. Unlike PCA, factor analysis privileges the particular input coordinates (rather than being invariant to rotations of the input covariance matrix). References [1] Bourlard, H & Kamp, Y (1988). Auto-association by Multilayer Perceptrons and Singular Value Decomposition. BioI. Cybernetics 59,291-294. [2] Bregler, C & Omohundro, SM (1995). Non-linear image interpolation using surface learning. This volume. [3] Dayan, P, Hinton, GE, Neal, RM & Zemel, RS (1995). The Helmholtz machine. Neural Computation, in press. [4] Dempster, AP, Laird, NM & Rubin, DB (1976). Maximum likelihood from incomplete data via the EM algorithm. Proceedings of the Royal Statistical Society, 1-38. [5] Hastie, T, Simard, P & Sackinger, E (1995). Learning prototype models for tangent distance. This volume. [6] Hinton, GE, Dayan, P, Frey, BJ, Neal, RM (1995). The wake-sleep algorithm for unsupervised neural networks. Submitted for publication. [7] Hinton, GE & Zemel, RS (1994). Autoencoders, minimum deSCription length and Helmholtz free energy. In JD Cowan, G Tesauro & J Alspector, editors, Advances in Neural Information Processing Systems 6. San Mateo, CA: Morgan Kaufmann. [8] Hull, JJ (1994). A database for handwritten text recognition research. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16, 550-554. [9] Kambhatla, N & Leen, TK (1994). Fast non-linear dimension reduction. In JD Cowan, G Tesauro & J Alspector, editors, Advances in Neural Information Processing Systems 6. San Mateo, CA: Morgan Kaufmann. [10] Schwenk, H & Milgram, M (1995). Transformation invariant autoassociation with application to handwritten character recognition. This volume. [11] Simard, P, Le Cun, Y & and Denker, J (1993). Efficient pattern recognition using a new transformation distance. In SJ Hanson, JD Cowan & CL Giles, editors, Advances in Neural Information Processing Systems 5, 50-58. San Mateo, CA: Morgan Kaufmann. [12] Simard, P, Victorri, B, LeCun, Y & Denker, J (1992). Tangent Prop - A formalism for specifying selected invariances in an adaptive network. In JE Moody, SJ Hanson & RP Lippmann, editors, Advances in Neural Information Processing Systems 4. San Mateo, CA: Morgan Kaufmann. [13] Williams, CKI, Zemel, RS & Mozer, MC (1993). Unsupervised learning of object models. In AAAI FalI1993 Symposium on Machine Learning in Computer Vision, 20-24. [14] Zemel, RS (1993). A Minimum Description Length Framework for Unsupervised Learning. PhD Dissertation, Computer Science, University of Toronto, Canada. IThis research was funded by the Ontario Information Technology Research Centre and NSERC. We thank Patrice Simard, Chris Williams, Rob Tibshirani and Yann Le Cun for helpful discussions. Geoffrey Hinton is the Noranda Fellow of the Canadian Institute for Advanced Research.	1994	49
942	Learning Local Error Bars for Nonlinear Regression David A.Nix Department of Computer Science and Institute of Cognitive Science University of Colorado Boulder, CO 80309-0430 dnix@cs.colorado.edu Andreas S. Weigend Department of Computer Science and Institute of Cognitive Science University of Colorado Boulder, CO 80309-0430 andreas@cs.colorado.edu· Abstract We present a new method for obtaining local error bars for nonlinear regression, i.e., estimates of the confidence in predicted values that depend on the input. We approach this problem by applying a maximumlikelihood framework to an assumed distribution of errors. We demonstrate our method first on computer-generated data with locally varying, normally distributed target noise. We then apply it to laser data from the Santa Fe Time Series Competition where the underlying system noise is known quantization error and the error bars give local estimates of model misspecification. In both cases, the method also provides a weightedregression effect that improves generalization performance. 1 Learning Local Error Bars Using a Maximum Likelihood Framework: Motivation, Concept, and Mechanics Feed-forward artificial neural networks used for nonlinear regression can be interpreted as predicting the mean of the target distribution as a function of (conditioned on) the input pattern (e.g., Buntine & Weigend, 1991; Bishop, 1994), typically using one linear output unit per output variable. If parameterized, this conditional target distribution (CID) may also be ·http://www.cs.colorado.edu/~andreas/Home.html. This paper is available with figures in colors as ftp://ftp.cs.colorado.edu/pub/ Time-Series/MyPapers/nix.weigenCLnips7.ps.Z . 490 David A. Nix, Andreas S. Weigend viewed as an error model (Rumelhart et al., 1995). Here, we present a simple method that provides higher-order information about the cm than simply the mean. Such additional information could come from attempting to estimate the entire cm with connectionist methods (e.g., "Mixture Density Networks," Bishop, 1994; "fractional binning, "Srivastava & Weigend, 1994) or with non-connectionist methods such as a Monte Carlo on a hidden Markov model (Fraser & Dimitriadis, 1994). While non-parametric estimates of the shape of a C1D require large quantities of data, our less data-hungry method (Weigend & Nix, 1994) assumes a specific parameterized form of the C1D (e.g., Gaussian) and gives us the value of the error bar (e.g., the width of the Gaussian) by finding those parameters which maximize the likelihood that the target data was generated by a particular network model. In this paper we derive the specific update rules for the Gaussian case. We would like to emphasize, however, that any parameterized unimodal distribution can be used for the em in the method presented here. j------------, I I------T-------, I A I A2,. ) I , o y(x) 0 cr IX I '. /\ i \ O'OOh k : I I I I I I I : l ,-----------_. Figure 1: Architecture of the network for estimating error bars using an auxiliary output unit. All weight layers have full connectivity. This architecture allows the conditional variance ~2 -unit access to both information in the input pattern itself and in the hidden unit representation formed while learning the conditional mean, y(x). We model the desired observed target value d as d(x) = y(x) + n(x), where y(x) is the underlying function we wish to approximate and n(x) is noise drawn from the assumed cm. Just as the conditional mean of this cm, y(x), is a function of the input, the variance (j2 of the em, the noise level, may also vary as a function of the input x (noise heterogeneity). Therefore, not only do we want the network to learn a function y(x) that estimates the conditional mean y(x) of the cm, but we also want it to learn a function a-2(x) that estimates the conditional variance (j2(x). We simply add an auxiliary output unit, the a-2-unit, to compute our estimate of (j2(x). Since (j2(x) must be positive, we choose an exponential activation function to naturally impose this bound: a-2(x) = exp [Lk Wq2khk (x) + ,8], where,8 is the offset (or "bias"), and Wq2k is the weight between hidden unit k and the a-2-unit. The particular connectivity of our architecture (Figure 1), in which the a-2-unit has a hidden layer of its own that receives connections from both the y-unit's hidden layer and the input pattern itself, allows great flexibility in learning a-2(x). In contrast, if the a-2-unit has no hidden layer of its own, the a-2-unit is constrained to approximate (j2 (x) using only the exponential of a linear combination of basis functions (hidden units) already tailored to represent y(x) (since learning the conditional variance a-2(x) before learning the conditional mean y(x) is troublesome at best). Such limited connectivity can be too constraining on the functional forms for a-2( x) and, in our experience, I The case of a single Gaussian to represent a unimodal distribution can also been generalized to a mixture of several Gaussians that allows the modeling of multimodal distributions (Bishop, 1994). Learning Local Error Bars for Nonlinear Regression 491 produce inferior results. This is a significant difference compared to Bishop's (1994) Gaussian mixture approach in which all output units are directly connected to one set of hidden units. The other extreme would be not to share any hidden units at all, i.e., to employ two completely separate sets of hidden units, one to the y(x)-unit, the other one to the a-2(x)-unit. This is the right thing to do if there is indeed no overlap in the mapping from the inputs to y and from the inputs to cr2• The two examples discussed in this paper are between these two extremes; this justifies the mixed architecture we use. Further discussion on shared vs. separate hidden units for the second example of the laser data is given by Kazlas & Weigend (1995, this volume). For one of our network outputs, the y-unit, the target is easily available-it is simply given by d. But what is the target for the a-2-unit? By maximizing the likelihood of our network ' model N given the data, P(Nlx, d), a target is "invented" as follows. Applying Bayes' rule and assuming statistical independence of the errors, we equivalently do gradient descent in the negative log likelihood of the targets d given the inputs and the network model, summed over all patterns i (see Rumelhart et at., 1995): C = - Li In P(dilxi, N). Traditionally, the resulting form of this cost function involves only the estimate Y(Xi) of the conditional mean; the variance of the CID is assumed to be constant for all Xi, and the constant terms drop out after differentiation. In contrast, we allow the conditional variance to depend on x and explicitly keep these terms in C, approximating the conditional variance for Xi by a-2(Xi). Given any network architecture and any parametric form for the ern (Le., any error model), the appropriate weight-update equations for gradient decent learning can be straightforwardly derived. Assuming normally distributed errors around y(x) corresponds to a em density function of P(dilxj) = [27rcr2(Xi)t 1/ 2 exp {- d~:Y~.) 2}. Using the network output Y(Xi) ~ y(Xi) to estimate the conditional mean and using the auxiliary output a-2(Xi) ~ cr2(xd to estimate the conditional variance, we obtain the monotonically related negative log lik lib d I P(d I Af\ II 2 A2() [di-y(Xi)]2 S . all e 00, n i Xi, n J 2" n 7rcr Xi + 2".2(X.) . ummatlon over patterns gives the total cost: C = ! ,,{ [di =- y(xd] 2 + Ina-2(Xi) + In27r} 2 ~ cr2(Xi) , (1) To write explicit weight-update equations, we must specify the network unit transfer functions. Here we choose a linear activation function for the y-unit, tanh functions for the hidden units, and an exponential function for the a-2 -unit. We can then take derivatives of the cost C with respect to the network weights. To update weights connected to the Y and a-2 -units we have: 11 a-2~i) [di - Y(Xi)] hj(Xi) (2) 11 2a-2~Xi) {[di - y(Xi)f - a-2 (Xi) } hk (Xi) (3) where 11 is the learning rate. For weights not connected to the output, the weight-update equations are derived using the chain rule in the same way as in standard backpropagation. Note that Eq. (3) is equivalent to training a separate function-approximation network for a-2(x) where the targets are the squared errors [di - y(Xi)]2]. Note also that if a-2(Xj) is 492 David A. Nix, Andreas S. Weigend constant, Eqs. (1)-(2) reduce to their familiar forms for standard backpropagation with a sum-squared error cost function. The 1/&2(X) term in Eqs. (2)-(3) can be interpreted as a form of "weighted regression," increasing the effective learning rate in low-noise regions and reducing it in high-noise regions. As a result, the network emphasizes obtaining small errors on those patterns where it can (low &2); it discounts learning patterns for which the expected error is going to be large anyway (large &2). This weighted-regression term can itself be highly beneficial where outliers (i.e., samples from high-noise regions) would ordinarily pull network resources away from fitting low-noise regions which would otherwise be well approximated. For simplicity, we use simple gradient descent learning for training. Other nonlinear minimization techniques could be applied, however, but only if the following problem is avoided. If the weighted-regression term described above is allowed a significant influence early in learning, local minima frequently result. This is because input patterns for which low errors are initially obtained are interpreted as "low noise" in Eqs. (2)-(3) and overemphasized in learning. Conversely, patterns for which large errors are initially obtained (because significant learning of y has not yet taken place) are erroneously discounted as being in "high-noise" regions and little subsequent learning takes place for these patterns, leading to highly-suboptimal solutions. This problem can be avoided if we separate training into the following three phases: Phase I (Initial estimate of the conditional mean): Randomly split the available data into equal halves, sets A and 8. Assuming u2(x) is constant, learn the estimate of the conditional mean y(x) using set A as the training set. This corresponds to "traditional" training using gradient descent on a simple squared-error cost function, i.e., Eqs. (1)-(2) without the 1/&2(X) terms. To reduce overfitting, training is considered complete at the minimum of the squared error on the cross-validation set 8, monitored at the end of each complete pass through the training data. Phase II (Initial estimate of the conditional variance): Attach a layer of hidden units connected to both the inputs and the hidden units of the network from Phase I (see Figure 1). Freeze the weights trained in Phase I, and train the &2-unit to predict the squared errors (see Eq. (3», again using simple gradient descent as in Phase I. The training set for this phase is set 8, with set A used for cross-validation. If set A were used as the training set in this phase as well, any overfitting in Phase I could result in seriously underestimating u2(x). To avoid this risk, we interchange the data sets. The initial value for the offset (3 of the &2-unit is the natural logarithm of the mean squared error (from Phase I) of set 8. Phase II stops when the squared error on set A levels off or starts to increase. Phase ill (Weighted regression): Re-split the available data into two new halves, A' and 8'. Unfreeze all weights and train all network parameters to minimize the full cost function C on set A'. Training is considered complete when C has reached its minimum on set 8'. 2 Examples Example #1: To demonstrate this method, we construct a one-dimensional example problem where y(x) and u2(x) are known. We take the equation y(x) = sin(wax) sin(w,Bx) withwa = 3 andw,B = 5. We then generate (x, d) pairs by picking x uniformly from the interval [0, 7r /2] and obtaining the corresponding target d by adding normally distributed noise n(x) = N[0,u2(x)] totheunderlyingy(x), whereu2(x) = 0.02+0.25 x [1-sin(w,Bx)j2. Learnillg Local Error Bars for Nonlinear Regression 493 Table 1: Results for Example #1. ENMS denotes the mean squared error divided by the overall variance of the target; "Mean cost" represents the cost function (Eq. (1)) averaged over all patterns. Row 4 lists these values for the ideal model (true y(x) and a2(x)) given the data generated. Row 5 gives the correlation coefficient between the network's predictions for the standard error (i.e., the square root of the &2 -unit's activation) and the actually occurring L1 residual errors, Id(Xi) - y(x;) I. Row 6 gives the correlation between the true a(x) and these residual errors. Rows 7-9 give the percentage of residuals smaller than one and two standard deviations for the obtained and ideal models as well as for an exact Gaussian. 1 Training (N 103) 1 Evaluation (N 105) 1 ENM ."< Mean cost Er.. IU .,,< Mean cost 1 Phase I 0.576 0.853 0.593 0.882 2 Phase " 0.576 0.542 0.593 0.566 3 Phase III 0.552 0.440 0.570 0.462 4 n( x ) (exact additive noise) 0.545 0.430 0.563 0.441 I P I p J 5 ~ ~~ ~ \: residual errors) 0.564 0.548 6 a( x ,reridual errors) 0.602 0.584 I sId 2 sId I sId 2 sId 7 % of errors < .,.~ xl; 2"'~ xl 64.8 95.4 67.0 94.6 8 % of errors < a-(x); 2a-(x) 66.6 96.0 68.4 95.4 9 (aact Gaussian) 68.3 95.4 68.3 95.4 We generate 1000 patterns for training and an additional 105 patterns for post-training evaluation. Training follows exactly the three phases described above with the following details:2 Phase I uses a network with one hidden layer of 10 tanh units and TJ = 10-2 • For Phase II we add an auxiliary layer of 10 tanh hidden units connected to the a.2-unit (see Figure 1) and use the same TJ. Finally, in Phase III the composite network is trained with TJ = 10- 4. At the end of Phase I (Figure 2a), the only available estimate of (1'2 (x ) is the global root-mean-squared error on the available data, and the model misspecification is roughly uniform over x-a typical solution were we training with only the traditional squared-error cost function. The corresponding error measures are listed in Table 1. At the end of Phase II, however, we have obtained an initial estimate of (1'2 (x ) (since the weights to the i)-unit are frozen during this phase, no modification of i) is made). Finally, at the end of Phase III, we have better estimates of both y{ x) and (1'2 (x). First we note that the correlations between the predicted errors and actual errors listed in Table 1 underscore the near-optimal prediction of local errors. We also see that these errors correspond, as expected, to the assumed Gaussian error model. Second, we note that not only has the value of the cost function dropped from Phase II to Phase III, but the generalization error has also dropped, indicating an improved estimate of y( x ). By comparing Phases I and III we see that the quality of i)(x) has improved significantly in the low-noise regions (roughly x < 0.6) at a minor sacrifice of accuracy in the high-noise region. Example #2: We now apply our method to a set of observed data, the 1000-point laser 2Purther details: all inputs are scaled to zero mean and unit variance. All initial weights feeding into hidden units are drawn from a uniform distribution between -1 j i and 1 j i where i is the number of incoming connections. All initial weights feeding into y or &2 are drawn from a uniform distribution between -sji and sji where s is the standard deviation of the (overall) target distribution. No momentum is used, and all weight updates are averaged over the forward passes of 20 patterns. 494 David A. Nix. Andreas S. Weigend (bl Phaao 1 Phaloll Phaoolll 1~ 0.5 : ~ 0 . . •. \ -0.5 .• -1 o 1 x x x Phalol _II _"I x x x 11(1-11 Figure 2: (a) Example #1: Results after each phase of training. The top row gives the true y(x) (solid line) and network estimate y(x) (dotted line); the bottom row gives the true oo2(x) (solid line) and network estimate o-2(x) (dotted line). (b) Example #2: state-space embedding of laser data (evaluation set) using linear grey-scaling of 0.50 (lightest) < o-(Xt) < 6.92 (darkest). See text for details. intensity series from the Santa Fe competition.3 Since our method is based on the network's observed errors, the predicted error a-2(x) actually represents the sum of the underlying system noise, characterized by 002 (x), and the model misspecification. Here, since we know the system noise is roughly uniform 8-bit sampling resolution quantization error, we can apply our method to evaluate the local quality of the manifold approximation.4 The prediction task is easier if we have more points that lie on the manifold, thus better constraining its shape. In the competition, Sauer (1994) upsampled the 1000 available data points with an FFf method by a factor of 32. This does not change the effective sampling rate, but it "fills in" more points, more precisely defining the manifold. We use the same upsampling trick (without filtered embedding), and obtain 31200 full (x, d) patterns for learning. We apply the three-phase approach described above for the simple network of Figure 1 with 25 inputs (corresponding to 25 past values), 12 hidden units feeding the y-unit, and a libera130 hidden units feeding the a-2 -unit (since we are uncertain as to the complexity of (j2(x) for this dataset). We use 11 = 10-7 for Phase I and 11 = 10- 10 for Phases II and III. Since we know the quantization error is ±O. 5, error estimates less than this are meaningless. Therefore, we enforce a minimum value of (j2(x) = 0.25 (the quantization error squared) on the squared errors in Phases II and III. 3The data set and several predictions and characterizations are described in the volume edited by Weigend & Gershenfeld (1994). The data is available by anonymous ftp at ftp.cs.colorado.edu in /pub/Time-Series/SantaFe as A.dat. See also http://www . cs. colorado. edu/Time-Series/TSWelcome. html for further analyses of this and other time series data sets. 4When we make a single-step prediction where the manifold approximation is poor, we have little confidence making iterated predictions based on that predicted value. However, if we know we are in a low-error region, we can have increased confidence in iterated predictions that involve our current prediction. Learning Local Error Bars for Nonlinear Regression 495 Table 2: Results for Example #2 (See Table 1 caption for definitions). row I I Training (N 975) I Evaluation (N 23 950) . EMMC: Mean <XlSt E1"Juc: Mean cost I Phase I 0.00125 1.941 0.0156 7.213 2 Phase II 0.00125 1.939 0.0156 5.628 3 Phase III 0.00132 1.725 0.0139 5.564 p P 4 P( a( x ). residual errors) 0.557 0.366 1 sid 2std 1 sid 2 std % of errors < a( x); 2a( x) 69A 94.9 63.1 88.0 (exacl Gaussian) 68.3 95.4 68.3 95.4 Results are given in Table 2 for patterns generated from the available 1000 points and 24,000 additional points used for evaluation. Even though we have used a Gaussian error model, we know the distribution of errors is not Gaussian. This is reflected in rows 5 and 6, where the training data is modeled as having a Gaussian em but the evaluation values become considerably distorted from an exact Gaussian. Again, however, not only do we obtain significant predictability of the errors, but the method also reduces the squared-error measure obtained in Phase I. We can use the estimated error to characterize the quality of the manifold approximation on 24,000 post-training evaluation points, as illustrated in Figure 2b. The manifold approximation is poorer (darker) for higher predicted values of Xt and for values nearer the edge of the manifold. Note the dark-grey (high-error) vertical streak leaving the origin and dark points to its left which represent patterns involving sudden changes in oscillation intensity. 3 Discussion Since we are in effect approximating two functions simultaneously, we can apply many of the existing variations for improving function approximation designed for networks learning only y(x). For example, when using limited amounts of data, especially if it is noisy, the particular split of data into training and cross-validation sets we use introduces significant variation in the resulting y(x) due to overfitting, as demonstrated on financial data by Weigend & LeBaron (1994). If we want to estimate local error bars, not only must we fear overfitting y(x), but we must also be concerned with overfitting &.2 (x). If the standard method of stopping at the minimum of an appropriate cross-validation set does not suffice for a given problem, it is straightforward to employ the usual anti-overfitting weaponry (smooth &2 as a function of x, pruning, weight-elimination, etc.). Furthermore, we can bootstrap over our available dataset and create multiple composite networks, averaging their predictions for both y(x) and &2(X). Additionally, to incorporate prior information in a Bayesian framework as a form of regularization, Wolpert (personal communication, 1994) suggests finding the maximum a posteriori (instead of maximum-likelihood) conditional mean and variance using the same interpretation of the network outputs. In summary, we start with the maximum-likelihood principle and arrive at error estimates that vary with location in the input space. These local error estimates incorporate both underlying system noise and model misspecification. We have provided a computergenerated example to demonstrate the ease with which accurate error bars can be learned. We have also provided an example with real-world data in which the underlying system noise is small, uniform quantization error to demonstrate how the method can be used 496 David A. Nix, Andreas S. Weigend to characterize the local quality of the regression model. A significant feature of this method is its weighted-regression effect, which complicates learning by introducing local minima but can be potentially beneficial in constructing a more robust model with improved generalization abilities. In the framework presented, for any problem we must assume a specific parameterized cm then add one auxiliary output unit for each higher moment of the cm we wish to estimate locally. Here we have demonstrated the Gaussian case with a location parameter (conditional mean) and a scale parameter (local error bar) for a scalar output variable. The extension to multiple output variables is clear and allows a full covariance matrix to be used for weighted regression, including the cross-correlation between multiple targets. Acknowledgments This work is supported by the National Science Foundation under Grant No. RIA ECS-9309786 and by a Graduate Fellowship from the Office of Naval Research. We would like to thank Chris Bishop, Wray BUntine, Don Hush, Steve Nowlan, Barak Pearlmutter, Dave Rumelhart, and Dave Wolpert for helpful discussions. References C. Bishop. (1994) "Mixture Density Networks." Neural Computing Research Group Report NCRG/4288, Department of Computer Science, Aston University, Birmingham, UK. w.L. Buntine and A.S. Weigend. (1991) "Bayesian Backpropagation." Complex Systems, 5: 603--643. A.M. Fraser and A. Dimitriadis. (1994) "Forecasting Probability Densities Using Hidden Markov Models with Mixed States." In Time Series Prediction: Forecasting the Future and Understanding the Past, A.S. Weigend and N.A. Gershenfeld. eds .. Addison-Wesley, pp. 265- 282. P.T. Kazlas and A.S. Weigend. (1995) "Direct Multi-Step TIme Series Prediction Using TD(>.)." In Advances in Neural Infonnation Processing Systems 7 (NIPS*94, this volume). San Francisco, CA: Morgan Kaufmann. D.E. Rumelhart, R. Durbin. R. Golden. and Y. Chauvin. (1995) "Backpropagation: The Basic Theory." In Backpropagation: Theory, Architectures and Applications, Y. Chauvin and D.E. Rumelhart, eds., Lawrence Erlbaum, pp. 1- 34. T. Sauer. (1994) 'T1l11.e Series Prediction by Using Delay Coordinate Embedding." In Time Series Prediction: Forecasting the Future and Understanding the Past, A.S. Weigend and N.A. Gershenfeld. eds., Addison-Wesley, pp. 175-193. A.N. Srivastava and A.S. Weigend. (1994) "Computing the Probability Density in Connectionist Regression." In Proceedings of the IEEE International Conference on Neural Networks (IEEEICNN'94), Orlando, FL, p. 3786--3789. IEEE-Press. A.S. Weigend and N.A. Gershenfeld, eds. (1994) Time Series Prediction: Forecasting the Future and Understanding the Past. Addison-Wesley. A.S. Weigend and B. LeBaron. (1994) "Evaluating Neural Network Predictors by Bootstrapping." In Proceedings of the International Conference on Neural Infonnation Processing (ICONIP'94), Seoul, }(orea,pp.1207-1212. A.S. Weigend and D.A. Nix. (1994) "Predictions with Confidence Intervals (Local Error Bars)." In Proceedings of the International Conference on Neural Information Processing (ICONIP'94), Seoul, }(orea, p. 847- 852.	1994	5
943	A Critical Comparison of Models for Orientation and Ocular Dominance Columns in the Striate Cortex E. Erwin Beckman Institute University of Illinois Urbana, IL 61801, USA K. Obermayer Technische Fakultat U niversitat Bielefeld 33615 Bielefeld, FRG Abstract K. Schulten Beckman Institute University of Illinois Urbana, IL 61801, USA More than ten of the most prominent models for the structure and for the activity dependent formation of orientation and ocular dominance columns in the striate cort(>x have been evaluated. We implemented those models on parallel machines, we extensively explored parameter space, and we quantitatively compared model predictions with experimental data which were recorded optically from macaque striate cortex. In our contribution we present a summary of our results to date. Briefly, we find that (i) despite apparent differences, many models are based on similar principles and, consequently, make similar predictions, (ii) certain "pattern models" as well as the developmental "correlation-based learning" models disagree with the experimental data, and (iii) of the models we have investigated, "competitive Hebbian" models and the recent model of Swindale provide the best match with experimental data. 1 Models and Data The models for the formation and structure of orientation and ocular dominance columns which we have investigated are summarized in table 1. Models fall into two categories: "Pattern models" whose aim is to achieve a concise description of the observed patterns and "developmental models" which are focussed on the pro94 E. Erwin, K. Obermayer, K. Schulten Class Type Model Reference Pattern Structural 1. Icecube Hubel and Wiesel 1977 [~I Models Models 2. Pinwheel Braitenberg and Braitenberg 1979 161 3. Gotz Gotz 1987 (8) 4. Baxter Baxter and Dow 1989 11) Spectral 5. ROJer ROJer and Schwartz 1990 J20) Models 6. Niebur Niebur and Worgotter 1993 (15) 7. Swindale Swindale 1992a (21) Develop. Correlation 8. Linsker Linsker 1986c J12] Models Based Learning 9. Miller Miller 1989, 1994 113, 14) Competl bve 10. ~UM-h Ubennayer, et. al. 1990 P~J Hebbian 11. SOM-I Obermayer, et. al. 1992(17) 12. EN Durbin and Mitchison 1990 (7) Other 13. Tanaka Tanaka 1991 [22J 14. Yuille Yuille, et. al. 1992 (23) Table 1: Models of visual cortical maps which have been evaluated. cesses underlying their formation. Pattern models come in two varieties, "structural models" and "spectral models", which describe orientation and ocular dominance maps in real and in Fourier space, respectively. Developmental models fall into the categories "correlations based learning", "competitive Hebbian" learning and a few miscellaneous models. Models are compared with data obtained from macaque striate cortex through optical imaging [2, 3, 4, 16]. Data were recorded from the representation of the parafovea from the superficial layers of cortex. In the following we will state that a particular model reproduces a particular feature of the experimental data (i) if there exists a parameter regime where the model generates appropriate patterns and (ii) if the phenomena are robust. We will state that a particular model does not reproduce a certain feature (i) if we have not found an appropriate parameter regime and (ii) if there exists either a proof or good intuitive reasons that a lllodel cannot reproduce this feature. One has to keep in mind, though, that model predictions are compared with a fairly special set of data. Ocular dominance patterns, e.g., are known to vary between species and even between different regions within area 17 of an individual. Consequently, a model which does not reproduce certain featurE'S of ocular dominance or orientation colulllns in the macaque may well describE' those patterns in other species. Interspecies differences, however, are not. the focus of this contribution; results of corresponding modelling studies will be reported E'lsewhere. 2 Examples of Organizing Principles and Model Predictions It has been suggested t.hat the most important principles underlying the pattern of orientation and ocular dominance are "continuity" and "diversity" [7. 19, 21]. Continuity, because early image processing is often local in fE'atnre space, and diversity, because, e.g., the visual system may want to avoid perceptual scotomata. The continuity and diversity principles underlie almost all dE'scriptive and developmental A Critical Comparison of Models for Orientation and Ocular Dominance Columns 95 Figure 1: Typical patterns of orientation preferences as they are predicted by six of the models list.ed in Table 1. Orientation preferences are coded by gray values, where black whit.e denotes preferences for vertical _ horizont.al vertical. Top row (left to right): Models 7, 11, 9. Bottom row (left to right) Models 5, 12, 8. models, but. maps which comply with t.hese principles often differ in qualitat.ive ways: The icecube model, e.g., obeys bot.h principles but. contains no singularities in the orient.ation preference map and no branching of ocular dominance bands. Figure 1 shows orientat.ion maps generated by six different. algorithms taken from Tab. 1. Although all pat.t.erns are consist.ent. wit.h the continuit.y and diversity const.raints, closer comparison reveals differences. Thus additional element.s of organization must be considered. It has been suggested that maps are characterized by local correlations and global disorder. Figure 2 (left) shows as an exam pIe two-point correlation functions of orientation maps. The autocorrelation function [17] of one of the Cartesian coordinat.es of t.he orientation vector is plotted as a function of cortical distance. The fact. that all correlation functions decay indicates that the orientation map exhibits global disorder. Global disorder is predicted by all models except. the early pattern models 6, 8 and 9. Figure 2 (right) shows the corresponding power spectra. Bandpass-like spectra which are typical for the experiment.al data [16] are well predicted by models 10- 12. Interestingly, they are not predicted by model 9, which also fails reproducing the Mexican-hat shaped correlation functions (bold lines), and model 13. Based on the fact that. experimental maps are characterized by a bandpass-like power spectrum it has been suggested that orientation maps may be organized 96 E. Erwin, K. Obermayer, K. Schulten 1.0 ..._-------------, 1,0 "'0 .~ 1,8 -§ 1,6 S 0,4 ... ~ & 0,2 -0.5 ~----:."_ ................. _ __....-_ _4 o 10 20 30 40 0,0 5 15 20 0 10 distance (normalized) distance (normalized) Figure 2: Left: Spatial autocorrelation functions for one of the cartesian coordinates of the orientat.ion vector. Aut.ocorrelation functions were averaged over all directions. Right: Complex power spectra of orientation maps. Power was averaged over all directions of the wave vector. Modelnumhers as in Tab. 1. according to four principles [15]: continuity, diversity, homogeneity and isotropy. If those principles are implemented using bandpass filtered noise the resulting maps [15, 21] indeed share many properties with the experimental data. Above principles alone, however, are not sufficient: (i) There are models such as model ·5 which are based on those principles but generate different patterns, (ii) homogeneity and isotropy are hardly ever fulfilled ([16] and next paragraph), and (iii) those principles cannot. account for correlations between maps of various response properties [16]. Maps of orientation and ocular dominance in the macaque are anisotropic, i.e., there exist preferred directions along which orientation and ocular dominance slabs align [16]. Those anisotropies can emerge due to different mechanisms: (i) spontaneous symmetry breaking, (ii) model equations, which are not rotation invariant, and (iii) appropriately chosen boundary conditions. Figure 3 illustrates mechanisms (ii) and (iii) for model 11. Bot.h mechanisms indeed predict anisotropic pat.terns, however, preferred directions of orientation and ocular dominance align in both cases (fig. 3, left and center). This is not true for the experimental data, where preferred directions tend to be orthogonal [16]. Ort.hogonal preferred directions can be generated by llsing different neighborhood funct.ions for different components of the feature vector (fig. 3, right). However, this is not a satisfactory solution, and the issue of anisotropies is still unsolved. The pattern of orientation preference in the area 17 of the macaque exhibits four local elements of organization: linear zones, singularit.ies, saddle point.s and fractures [16]. Those element.s are correctly predict.ed by most. of the pat.t,ern models, except models 1- 3, and they appear in the maps generated by models 10- 14. Interestingly' models 9 and 13 predict very few linear zones, which is related to the fact. that those models generate orientat.ion maps with lowpass-like power spect.ra. Another important property of orientation maps is that orientation preferences and their spatial layout across cortex are not correlated which each other. One conseA Critical Comparison of Models for Orientation and Ocular Dominance Columns 97 .~; + :.~ . + ' .~,!-. + ..4-.. + + Figure 3: Anisotropic orientation and ocular dominance maps generated by model 11. The figure shows Fourier spectra [17] of orientation (top row) and ocular dominance maps (bottom row). Left: Maps generated with an elliptic neighborhood function (case (ii), see text); Center: Maps generated using circular input layers and an elliptical cortical sheet (case (iii), see text), Right: Maps generated with different, elliptic neighborhood functions for orientation preference and ocular dominance. '+' symbols indicate the locations of the origin. quence is that there exist singularities, near which the curl of the orientation vector field does not vanish (fig. 4, left). This rules out a class of pattern models where the orientation map is derived from the gradient of a potential function, model 5. Figure 4 (right) shows another consequence of this property. In those figures cortical area is plotted against the angular difference between the iso-orientation lines and the local orientation preference. The even distribution found in the experimental data is correctly predicted by models 1,6, 7 and 10-12. Model 8, however, predicts preference for large difference angles while model 9 - over a wide range of parameters - predicts preference for small difference angles (bold lines). Finally, let us consider correlations between the patterns of orientation preference and ocular dominance. Among the more prominent relationships present in macaque data are [3, 16,21]: (i) Singularities are aligned with the centers of ocular dominance bands, (ii) fractures are either aligned or run perpendicular, and (iii) iso-orientation bands in linear zones intersect ocular dominance bands at approximately right angles. Those relationships are readily reproduced only by models 7 and 10- 12. For model 9 reasonable orientation and ocular dominance patterns have not been generated at the same time. It would seem as if the parameter regime where reasonable orientation columns emerge is incompatible with the parameter regime where ocular dominance patterns are formed. 98 E. Erwin, K. Obermayer, K. Schulten 0J.5 '" e '" C+-I 0.10 0 u bO '" = O. ~ 8. 0 .. +----.--__ -....-_--1 03060 90 difference angle ( degrees) Figure 4: Left: This singularity is an example of a feature in the experimental data which is not allowed by model 5. The arrows indicat.e orientation vectors, whose angular component is twice the value of the local orientation preference. Right: Percentage of area as a function of the angular difference bet.ween preferred orient.ation and t.he local orientation gradient vector. Model numbers as in Table 1. 3 The Current Status of the Model Comparison Project Lack of space prohibit.s a detailed discussion of our findings hut we have summarized the current status of our project in Tables 2 and 3. Given the models list.ed in Tab. 1 and given the properties of t.he orientation and ocular dominance patt.erns in macaque striate cortex listed in Tables 2 and 3 it is models 7 and 10-12 which currently are in best agreement with the data. Those models, however, are fairly abstract. and simplified, and they cannot easily be extended to predict receptive field structure. Biological realism and predictions about. receptive fields are the advantages of models 8 and 9. Those models, however, cannot account for the observed orientation patterns. It. would, therefore, be of high interest, if elements of both approaches could be combined to achieve a better description of the dat.a. The main conclusion, however, is that there are now enough data available to allow a better evaluation of model approaches than just by visual comparison of the generated pat.terns. It. is our hope, that future studies will address at least those propert.ies of t.he patterns which are known and well described, some of which are list.ed in Tables 2 and 3. In case of developmental models more stringent tests require experiments which (i) monitor the actual time-course of pattern formation, and which (ii) study pattern development under experimentally modified conditions (deprivation experiments). Currently there is not enough data available to constrain models but the experiments are under way [5, 10, 11, 18]. Acknowledgements VVe are very much indebted to Drs. Linsker, Tanaka and Yuille for sharing modelling data. E.E. thanks t.he Beckman Institute for support.. K.O. thanks ZiF (Universitat Bielefeld) for it.s hospitality. Computing time on a CM-2 and a CM-5 was made available by NCSA. A Critical Comparison of Models for Orientation and Ocular Dominance Columns 99 Properties of OR Maps no. disbandlinear saddle sing. fracto indep. high amsoURorder pass zones points ±1/2 coord. spec. tropy bias 1 + + + n + n 2 + + + n n n 3 + + + + n n n 4 +2 + + + +2 n + n 5 + + + + + +1 + n 6 + + + + + +1 + + n 7 + + + + + +1 + + + n 8 + + + + + n n n 9 + + + + -/+ + n n 10 + + + + +1 + + + + 11 + + + + + +1 + + + + 12 + + + + + +1 + + + + 13 + + + + +1 + + n n 14 + ? ? + + + ? n n n Table 2: Evaluation of orientation (OR) map models. Properties of the experimental maps include (left to right): global disorder; bandpass-like power spectra; the presence of linear zones in roughly 50% of the map area; the presence of saddle points, singularities (±1/2 with equal densities), and fractures; independence between cortical and orientation preference coordinates; a distribution favoring high values of orientation specificity; global anisotropy; and a possible orientation bias. Symbols: '+': There exists a parameter regime in which a model generates maps with this property; '-': The model cannot reproduce this property; "n': The model makes no predictions; "?': Not enough data available. 1 Models agree with the data only if one assumes that fractures are loci of rapid orientation change rather than real discontinuities. 20ne of several cases. References [1] W. T. Baxter and B. M. Dow. Bioi. Cybern., 61:171-182, 1989. [2] G. G. Blasdel. J. Neurosci., 12:3115-3138, 1992. [3] G. G. Blasdel. J. Neurosci., 12:3139-3161,1992. [4] G. G. Blasdel and G. Salama. Nature, 321:579- 585, 1986. [5] T. Bonhoeffer, D. Kim, and W. Singer. Soc. Neurosci. Abs., 19:1800, 1993. [6] V. Braitenberg and C. Braitenberg. Bioi. Cybern., 33:179- 186, 1979. [7] R. Durbin and G. Mitchison. Nature, 343:341-344, 1990. [8] K. G. Gotz. Bioi. Cybern., 56:107-109, 1987. [9] D. Rubel and T. N. Wiesel. Proc. Roy. Soc. Lond. B, 198:1-59, 1977. [10] D. Rubel, T. N. Wiesel, and S. LeVay. Phil. Trans. Roy. Soc. Lond. B, 278:377409, 1977. [11] D. Kim and T. Bonhoeffer. Soc. Neurosci. Abs., 19:1800, 1993. [12] R. Linsker. Proc. Nat. Acad. Sci., USA, 83:8779-8783, 1986. 100 E. Erwin, K. Obermayer, K. Schulten Properties of OD Maps Correlations Between OR and OD no. segredisani soODstralocal global sing. spec. gation order tropy bias bismus orthog. orthog. vs.OD vs.OD 1 + + + n +M +:.l n 2 n n n n n n n n n 3 + + n n + n +2 n 4 n n n n n n n n n 5 + + + n +1 _I +I,O! _I 6 n n n n n n n n n 7 + + + + n + + + 8 n n n n n n n n n 9 + + + + + ?1 ?1 ?1 ?1 10 + + + + + + n + + 11 + + + + + +2 n +2 +2 12 + + + + + +1,2 n +1,2 +1,2 13 + + + + + n n n n 14 + + + + n n n n n Table 3: Left: Evaluation of ocular dominance (OD) map models. Properties of the experimental maps include (left to right): Segregated bands of eye dominance; global disorder; bandpass-like power spectra; global anisotropy; a bias to the representation of one eye; and OD-patterns in animals with strabismus. Right: Evalu M ation of correlations between OD and OR. Experimental maps show (left to right): Local and global orthogonality between OR and OD slabs; singularities preferably in monocular regions, and lower OR specificity in monocular regions. 1 Authors treated OD and OR in independent models, but we consider a combined version. 2 Correlations are stronger than in the experimental data. [13] K. D. Miller. J. Neurosci., 14:409- 441, 1994. [14] K. D. Miller, J. B. Keller, and M. P. Stryker. Science, 245:605-615, 1989. [15] E. Niebur and F. Worgotter. In F. H. Eeckman and J. M. Bower, Computation and Neura.l Systems, pp. 409-413. Kluwer Academic Publishers, 1993. [16] K. Obermayer and G. G. Blasdel. J. Neurosci., 13:4114-4129, 1993. [17] K. Obermayer, G. G. Blasdel, and K. Schulten. Phys. Rev. A, 45:7568-7589, 1992. [18] K Obermayer, L. Kiorpes, and G. G. Blasdel. In J. D. Cowan at al., Advances in Neural Information Processing Systems 6. Morgan Kaufmann, 1994. 543550. [19] K. Obermayer, H. Ritter, and K. Schulten. Proc. Nat. Acad. Sci., USA, 87:8345- 8349, 1990. [20] A. S. Rojer and E. L. Schwartz. Bioi. Cybern., 62:381- 391, 1990. [21] N. V. Swindale. Bioi. Cybern., 66:217-230, 1992. [22] S. Tanaka. Bioi. Cybern., 65:91- 98, 1991. [23] A. L. Yuille, J. A. Kolodny, and C. W. Lee. TR 91-3, Harvard Robotics Laboratory, 1991.	1994	50
944	Classifying with Gaussian Mixtures and Clusters Nanda Kambhatla and Todd K. Leen Department of Computer Science and Engineering Oregon Graduate Institute of Science & Technology P.O. Box 91000 Portland, OR 97291-1000 nanda@cse.ogi.edu, tleen@cse.ogi.edu Abstract In this paper, we derive classifiers which are winner-take-all (WTA) approximations to a Bayes classifier with Gaussian mixtures for class conditional densities. The derived classifiers include clustering based algorithms like LVQ and k-Means. We propose a constrained rank Gaussian mixtures model and derive a WTA algorithm for it. Our experiments with two speech classification tasks indicate that the constrained rank model and the WTA approximations improve the performance over the unconstrained models. 1 Introduction A classifier assigns vectors from Rn (n dimensional feature space) to one of K classes, partitioning the feature space into a set of K disjoint regions. A Bayesian classifier builds the partition based on a model of the class conditional probability densities of the inputs (the partition is optimal for the given model). In this paper, we assume that the class conditional densities are modeled by mixtures of Gaussians. Based on Nowlan's work relating Gaussian mixtures and clustering (Nowlan 1991), we derive winner-take-all (WTA) algorithms which approximate a Gaussian mixtures Bayes classifier. We also show the relationship of these algorithms to non-Bayesian cluster-based techniques like LVQ and k-Means. The main problem with using Gaussian mixtures (or WTA algorithms thereof) is the explosion in the number of parameters with the input dimensionality. We propose 682 Nanda Kambhatla. Todd K. Leen a constrained rank Gaussian mixtures model for classification. Constraining the rank of the Gaussians reduces the effective number of model parameters thereby regularizing the model. We present the model and derive a WTA algorithm for it. Finally, we compare the performance of the different mixture models discussed in this paper for two speech classification tasks. 2 Gaussian Mixture Bayes (GMB) classifiers Let x denote the feature vector (x E 'Rn ), and {n I , I = 1, ... , K} denote the classes. Class priors are denoted p(nI) and the class-conditional densities are denoted p(x I nI). The discriminant function for the Bayes classifier is (1) An input feature vector x is assigned to class I if 6I(x) > 6J(x) 'VJ:F I . Given the class conditional densities, this choice minimizes the classification error rate (Duda and Hart 1973). We model each class conditional density by a mixture composed of QI component Gaussians. The Bayes discriminant function (see Figure 1) becomes QI I j'(x)=PW)L "j,foiTlexp[-~(X-~J{~J-\x-~J)], (2) ;=1 (21r)n/2 I~fl where ~J and ~f are the mean and the covariance matrix of the lh mixture component for nl. 0.12 0.1 0.08 0.06 0.04 0.02 5 10 20 25 --~)o X Fig. 1: Figure showing the decision rule of a GMB classifier for a two class problem with one input feature. The horizontal axis represents the feature and the vertical axis represents the Bayes discriminant functions. In this example, the class conditional densities are modelled as a mixture of two Gaussians and equal priors are assumed. To implement the Gaussian mixture Bayes classifier (GMB) we first separate the training data into the different classes. We then use the EM algorithm (Dempster Classifying with Gaussian Mixtures and Clusters 683 et al 1977, Nowlan 1991) to determine the parameters for the Gaussian mixture density for each class. 3 Winner-take-all approximations to G MB classifiers In this section, we derive winner-take-all (WTA) approximations to GMB classifiers. We also show the relationship of these algorithms to non-Bayesian cluster-based techniques like LVQ and k-Means. 3.1 The WTA model for GMB The WTA assumptions (relating hard clustering to Gaussian mixtures; see (Nowlan 1991)) are: • p(x 101) are mixtures of Gaussians as in (2). • The summation in (2) is dominated by the largest term. This is "equivalent to assigning all of the responsibility for an observation to the Gaussian with the highest probability of generating that observation" (Nowlan 1991). To draw the relation between GMB and cluster-based classifiers, we further assume that: • The mixing proportions (oj) are equal for a given class. • The number of mixture components QI is proportional to p(O/). Applying all the above assumptions to (2), taking logs and discarding the terms that are identical for each class, we get the discriminant function AI Q.1 [1 (I 1 ( I T I-I I ] -y (x) = ~f "2log IEj!) + "2 x - ILj) ~j (x - ILj) . (3) The discriminant function (3) suggests an algorithm that approximates the Bayes classifier. We segregate the feature vectors by class and then train a separate vector quantizer (VQ) for each class. We then compute the means ILl and the covariance matrices E1Jor each Voronoi cell of each quantizer, and use (3) for classifying new patterns. We call this algorithm VQ-Covariance. Note that this algorithm does not do a maximum likelihood estimation of its parameters based on the probability model used to derive (3). The probability model is only used to classify patterns. 3.2 The relation to LVQ and k-Means Further assume that for each class, the mixture components are spherically symmetric with covariance matrix El = 0"2 I, with 0"2 identical for all classes. We obtain the discriminant function, A QI 2 -y/(x) = - r¢n II x J.t)~ II . )=1 (4) 684 Nanda Kamblwtla, Todd K. Leen This is exactly the discriminant function used by the learning vector quantizer (LVQ; Kohonen 1989) algorithm. Though LVQ employs a discriminatory training procedure (i.e it directly learns the class boundaries and does not explicitly build a separate model for each class), the implicit model of the class conditional densities used by LVQ corresponds to a GMB model under all the assumptions listed above. This is also the implicit model underlying any classifier which makes its classification decision based on the Euclidean distance measure between a feature vector and a set of prototype vectors (e.g. a k-Means clustering followed by classification based on (4)). 4 Constrained rank GMB classifiers In the preceding sections, we have presented a GMB classifier and some WTA approximations to GMB. Mixture models such as GMB generally have too many parameters for small data sets. In this section, we propose a way of regularizing the mixture densities and derive a WTA classifier for the regularized model. 4.1 The constrained rank model In section 2, we assumed that the class conditional densities of the feature vectors x are mixtures of Gaussians (5) where p.J and EJ are the means and covariance matrices for the jth component Gaussian. eJi and AJi are the orthonormal eigenvectors and eigenvalues of I::Ji (ordered such that A}l ~ '" ~ AJn). In (5), we have written the Mahalanobis distance in terms of the eigenvectors. For a particular data point x, the Mahalanobis distance is very sensitive to changes in the squared projections onto the trailing eigen-directions, since the variances are very small in these directions. This is a potential problem with small data sets. When there are insufficient data points to estimate all the parameters of the mixture density accurately, the trailing eigen-directions and their associated eigenvalues are likely to be poorly estimated. Using the Mahalanobis distance in (5) can lead to erroneous results in such cases. We propose a method for regularizing Gaussian mixture classifiers based on the above ideas. We assume that the trailing n - m eigen-directions of each Gaussian component are inaccurate due to overfitting to the training set. We rewrite the class conditional densities (5) retaining only the leading m (0 < m ::; n) eigen-directions Classifying with Gaussian Mixtures and Clusters 685 in the determinants and the Mahalanobis distances p(x I Ill) = f m 2) m I exp [-~(x - ILJ{ (f e;~fiT) (x - ILJ)] j=1 (21T) / ft=1 \i i=l Jl (6) We choose the value of m (the reduced rank) by cross-validation over a separate validation set. Thus, our model can be considered to be regularizing or constraining the class conditional mixture densities. If we apply the above model and derive the Bayes discriminant functions (1), we get, 8I (x) = p(nI) ~ m 2 a} m I exp [-~(X - ILJ)T (f e}~f/) (x - ILJ)]. j=1 (21T) / y'ft=l \i i=1 Jl (7) We can implement a constrained rank Gaussian mixture Bayes (GMB-Reduced) classifier based on (7) using the EM algorithm to determine the parameters of the mixture density for each class. We segregate the data into different classes and use the EM algorithm to determine the parameters of the full mixture density (5). We then use (7) to classify patterns. 4.2 A constrained rank WTA algorithm We now derive a winner-take-all (WTA) approximation for the constrained rank mixture model described above. We assume (similar to section 3.1) that • p(x I nI) are constrained mixtures of Gaussians as in (6). • The summation in (6) is dominated by the largest term (the WTA assumption). . • The mixing proportions (a}) are equal for a given class and the number of components QI is proportional to p(nI). Applying these assumptions to (7), taking logs and discarding the terms that are identical for each class, we get the discriminant function (8) It is interesting to compare (8) with (3) . Our model postulates that the trailing n - m eigen-directions of each Gaussian represent overfitting to noise in the training set. The discriminant functions reflect this; (8) retains only those terms of (3) which are in the leading m eigen-directions of each Gaussian. We can generate an algorithm based on (8) that approximates the reduced rank Bayes classifier. We separate the data based on classes and train a separate vector quantizer (VQ) for each class. We then compute the means IL}, the covariance matrices ~} for each Voronoi cell of each quantizer and the orthonormal eigenvectors 686 Nanda Kambhatla. Todd K. Leen Table 1: The test set classification accuracies for the TIMIT vowels data for different algorithms. ALGORITHM MLP (40 nodes in hidden layer) GMB (1 component; full) GMB (1 component; diagonal) GMB-Reduced (1 component; 13-D) VQ-Covariance (1 component) VQ-Covariance-Reduced (1 component; 13-D) LVQ (48 cells) ACCURACY 46.8% 41.4% 46.3% 51.2% 41.4% 51.2% 41.4% eJi and eigenvalues AJ for each covariance matrix EJ. We use (8) for classifying new patterns. Notice that the algorithm described above is a reduced rank version of VQ-Covariance (described in section 3.1). We call this algorithm VQ-CovarianceReduced. 5 Experimental Results In this section we compare the different mixture models and a multi layer perceptron (MLP) for two speech phoneme classification tasks. The measure used is the classification accuracy. 5.1 TIMIT data The first task is the classification of 12 monothongal vowels from the TIMIT database (Fisher and Doddington 1986). Each feature vector consists of the lowest 32 DFT coefficients, time-averaged over the central third of the vowel. We partitioned the data into a training set (1200 vectors), a validation set (408 vectors) for model selection, and a test set (408 vectors). The training set contained 100 examples of each class. The values of the free parameters for the algorithms (the number of component densities, number of hidden nodes for the MLP etc.) were selected by maximizing the performance on the validation set. Table 1 shows the results obtained with different algorithms. The constrained rank models (GMB-Reduced and VQ-Covariance-Reduced1) perform much better than all the unconstrained ones and even beat a MLP for this task. This data set consists of very few data points per class, and hence is particularly susceptible to over fitting by algorithms with a large number of parameters (like GMB). It is not surprising that constraining the number of model parameters is a big win for this task. INote that since the best validation set performance is obtained with only one component for each mixture density, the WTA algorithms are identical to the GMB algorithms (for these results). Classifying with Gaussian Mixtures and Clusters 687 Table 2: The test set classification accuracies for the CENSUS data for different algorithms. ALGORITHM MLP (80 nodes in hidden layer) GMB (1 component; full) GMB (8 components; diagonal) GMB-Reduced (2 components; 35-D) VQ-Covariance (3 components) VQ-Covariance-Reduced (4 components; 38-D) LVQ (55 cells) 5.2 CENSUS data ACCURACY 88.2% 77.2% 70.9% 82.5% 77.5% 84.2% 67.3% The next task we experimented with was the classification of 9 vowels (found in the utterances ofthe days of the week). The data was drawn from the CENSUS speech corpus (Cole et alI994). Each feature vector was 70 dimensional (perceptual linear prediction (PLP) coefficients (Hermansky 1990) over the vowel and surrounding context}. We partitioned the data into a training set (8997 vectors), a validation set (1362 vectors) for model selection, and a test set (1638 vectors). The training set had close to a 1000 vectors per class. The values of the free parameters for the different algorithms were selected by maximizing the validation set performance. Table 2 gives a summary of the classification accuracies obtained using the different algorithms. This data set has a lot more data points per class than the TIMIT data set. The best accuracy is obtained by a MLP, though the constrained rank mixture models still greatly outperform the unconstrained ones. 6 Discussion We have derived WTA approximations to GMB classifiers and shown their relation to LVQ and k-Means algorithms. The main problem with Gaussian mixture models is the explosion in the number of model parameters with input dimensionality, resulting in poor generalization performance. We propose constrained rank Gaussian mixture models for classification. This approach ignores some directions ( "noise") locally in the input space, and thus reduces the effective number of model parameters. This can be considered as a way of regularizing the mixture models. Our results with speech vowel classification indicate that this approach works better than using full mixture models, especially when the data set size is small. The WTA algorithms proposed in this paper do not perform a maximum likelihood estimation of their parameters. The probability model is only used to classify data. We can potentially improve the performance of these algorithms by doing maximum likelihood training with respect to the models presented here. 688 Nanda Kambhatla, Todd K. Leen Acknowledgments This work was supported by grants from the Air Force Office of Scientific Research (F49620-93-1-0253), Electric Power Research Institute (RP8015-2) and the Office of Naval Research (NOOOI4-91-J-1482). We would like to thank Joachim Utans, OGI for several useful discussions and Zoubin Ghahramani, MIT for providing MATLAB code for the EM algorithm. We also thank our colleagues in the Center for Spoken Language Understanding at OGI for providing speech data. References R.A. Cole, D.G. Novick, D. Burnett, B. Hansen, S. Sutton, M. Fanty. (1994) Towards Automatic Collection of the U.S. Census. Proceedings of the International Conference on Acoustics, Speech and Signal Processing 1994. A.P. Dempster, N.M. Laird, and D.B. Rubin. (1977) Maximum Likelihood from Incomplete Data via the EM Algorithm. J. Royal Statistical Society Series B, vol. 39, pp. 1-38. R.O. Duda and P.E. Hart. (1973) Pattern Classification and Scene Analysis. John Wiley and Sons Inc. W.M Fisher and G.R Doddington. (1986) The DARPA speech recognition database: specification and status. In Proceedings of the DARPA Speech Recognition Workshop, p93-99, Palo Alto CA. H. Hermansky. (1990) Perceptual Linear Predictive (PLP) analysis of speech. J. Acoust. Soc. Am., 87(4):1738-1752. T. Kohonen. (1989) Self-Organization and Associative Memory (3rd edition). Berlin: Springer-Verlag. S.J. Nowlan. (1991) Soft Competitive Adaptation: Neural Network Learning Algorithms based on Fitting Statistical Mixtures. CMU-CS-91-126 PhD thesis, School of Computer Science, Carnegie Mellon University.	1994	51
945	Anatomical origin and computational role of diversity in the response properties of cortical neurons Kalanit Grill Spectort Shimon Edelmant Rafael Malacht Depts of t Applied Mathematics and Computer Science and tN eurobiology The Weizmann Institute of Science Rehovot 76100, Israel {kalanit.edelman. malach }~wisdom . weizmann .ac.il Abstract The maximization of diversity of neuronal response properties has been recently suggested as an organizing principle for the formation of such prominent features of the functional architecture of the brain as the cortical columns and the associated patchy projection patterns (Malach, 1994). We show that (1) maximal diversity is attained when the ratio of dendritic and axonal arbor sizes is equal to one, as found in many cortical areas and across species (Lund et al., 1993; Malach, 1994), and (2) that maximization of diversity leads to better performance in systems of receptive fields implementing steerable/shiftable filters, and in matching spatially distributed signals, a problem that arises in many high-level visual tasks. 1 Anatomical substrate for sampling diversity A fundamental feature of cortical architecture is its columnar organization, manifested in the tendency of neurons with similar properties to be organized in columns that run perpendicular to the cortical surface. This organization of the cortex was initially discovered by physiological experiments (Mouncastle, 1957; Hubel and Wiesel, 1962), and subsequently confirmed with the demonstration of histologically defined columns. Tracing experiments have shown that axonal projections throughout the cerebral cortex tend to be organized in vertically aligned clusters or patches. In particular, intrinsic horizontal connections linking neighboring cortical sites, which may extend up to 2 - 3 mm, have a striking tendency to arborize selectively in preferred sites, forming distinct axonal patches 200 - 300 J.lm in diameter. Recently, it has been observed that the size of these patches matches closely the average diameter of individual dendritic arbors of upper-layer pyramidal cells 118 .... 2&00 10(0 : : &00 , /'" ! ~._ . _ ._ ; ~ .. _._. r::~ ._. \ ~ ~ ~ ~ ~ ~ N ,,".",p.d hm pa~ Kalanit Grill Spector, Shimon Edelman, Rafael Malach BO 80 100 ) '0 0:2 0" 0 S 0.8 t 1 :2 1 .. 1 S t IS r •• o betiIJMn I'IWI'ot'I .. d ptIId! Figure 1: Left: histograms of the percentage of patch-originated input to the neurons, plotted for three values of the ratio r between the dendritic arbor and the patch diameter (0,5, 1.0, 2.0). The flattest histogram is obtained for r = 1.0 Right: the di versity of neuronal properties (as defined in section 1) vs. r. The maximum is attained for r = 1.0, a value compatible with the anatomical data. (see Malach, 1994, for a review). Determining the functional significance of this correlation, which is a fundamental property that holds throughout various cortical areas and across species (Lund et al., 1993), may shed light on the general principles of operation of the cortical architecture. One such driving principle may be the maximization of diversity of response properties in the neuronal population (Malach, 1994). According to this hypothesis, matching the sizes of the axonal patches and the dendritic arbors causes neighboring neurons to develop slightly different functional selectivity profiles, resulting in an even spread of response preferences across the cortical population, and in an improvement of the brain's ability to process the variety of stimuli likely to be encountered in the environment. 1 To test the effect of the ratio between axonal patch and dendritic arbor size on the diversity of the neuronal population, we conducted computer simulations based on anatomical data concerning patchy projections (Rockland and Lund, 1982; Lund et al., 1993; Malach, 1992; Malach et al., 1993). The patches were modeled by disks, placed at regular intervals of twice the patch diameter, as revealed by anatomical labeling. Dendritic arbors were also modeled by disks, whose radii were manipulated in different simulations. The arbors were placed randomly over the axonal patches, at a density of 10,000 neurons per patch. We then calculated the amount of patchrelated information sampled by each neuron, defined to be proportional to the area of overlap of the dendritic tree and the patch. The results of the calculations for three 1 Necessary conditions for obtaining dendritic sampling diversity are that dendritic arbors cross freely through column borders, and that dendrites which cross column borders sample with equal probability from patch and inter-patch compartments. These assumptions were shown to be valid in (Malach, 1992; Malach, 1994). Diversity in the Response Properties of Cortical Neurons 119 values of the ratio of patch and arbor diameters appear in Figure 1. The presence of two peaks in the histogram obtained with the arbor/patch ratio r = 0.5 indicates that two dominant groups are formed in the population, the first receiving most of its input from the patch, and the second - from the inter-patch sources. A value of r = 2.0, for which the dendritic arbors are larger than the axonal patch size, yields near uniformity of sampling properties, with most of the neurons receiving mostly patch-originated input, as apparent from the single large peak in the histogram. To quantify the notion of diversity, we defined it as diversity "'< I ~; I > -1, where n(p) is the number of neurons that receive p percent of their inputs from the patch, and < . > denotes average over p. Figure 1, right, shows that diversity is maximized when the size of the dendritic arbors matches that of the axonal patches, in accordance with the anatomical data. This result confirms the diversity maximization hypothesis stated in (Malach, 1994). 2 Orientation tuning: a functional manifestation of sam pIing diversity The orientation columns in VI are perhaps the best-known example of functional architecture found in the cortex (Bubel and Wiesel, 1962). Cortical maps obtained by optical imaging (Grinvald et al., 1986) reveal that orientation columns are patchy rather then slab-like: domains corresponding to a single orientation appear as a mosaic of round patches, which tend to form pinwheel-like structures. Incremental changes in the orientation of the stimulus lead to smooth shifts in the position of these domains. We hypothesized that this smooth variation in orientation selectivity found in VI originates in patchy projections, combined with diversity in the response properties of neurons sampling from these projections. The simulations described in the rest of this section substantiate this hypothesis. Computer simulations. The goal of the simulations was to demonstrate that a limited number of discretely tuned elements can give rise to a continuum of responses. We did not try to explain how the original set of discrete orientations can be formed by projections from the LGN to the striate cortex; several models for this step can be found in the literature (Bubel and Wiesel, 1962; Vidyasagar, 1985).2 In setting the size of the original discrete orientation columns we followed the notion of a point image (MacIlwain, 1986), defined as the minimal cortical separation of cells with non-overlapping RFs. Each column was tuned to a specific angle, and located at an approximately constant distance from another column with the same orientation tuning (we allowed some scatter in the location of the RFs). The RFs of adjacent units with the same orientation preference were overlapping, and the amount of overlap 2ln particular, it has been argued (Vidyasagar, 1985) that the receptive fields at the output of the LGN are already broadly tuned for a small number of discrete orientations (possibly just horizontal and vertical), and that at the cortical level the entire spectrum of orientations is generated from the discrete set present in the geniculate projection. 120 Kalanit Grill Spector, Shimon Edelman, Rafael Malach 007 3< 32 OOS 30 OOs II 0 28 00' ! ! 003 126 g .. .. 24 • 00' 22 001 20 0 0 .0 20 30 '" So SO 70 '·0 .0 20 30 '" so .. runber ~ shlttlng fIlters nunber cI shttng tlMlI'1 Figure 2: The effects of (independent) noise in the basis RFs and in the steering/shifting coefficients. Left: the approximation error vs. the number of basis RFs used in the linear combination. Right: the signal to noise ratio vs. the number of basis RFs. The SNR values were calculated as 10 loglO (signal energy/noise energy). Adding RFs to the basis increases the accuracy of the resultant interpolated RF. was determined by the number of RFs incorporated into the network. The preferred orientations were equally spaced at angles between 0 and 1r. The RFs used in the simulations were modeled by a product of a 2D Gaussian G1 , centered at rj, with orientation selectivity G2, and optimal angle Oi: G(r, rj, 0, Oi) = G1(r, rj)G2(O, Oi). According to the recent results on shiftable/steerable filters (Simon celli et al., 1992), a RF located at ro and tuned to the orientation ,po can be obtained by a linear combination of basis RFs, as follows: M-IN-I G(r, ro, 0, ,po) L L bj(ro)ki(,po)G(r,rj,O,Oi) j=O i=O M-I N-I 2: bj(ro)G1(r, rj) 2: ki(,po)G2(O,Oi) (1 ) j=O i=O From equation 1 it is clear that the linear combination is equivalent to an outer product of the shifted and the steered RFs, with {ki(,pO)}~~1 and {bj(ro)}~~l denoting the steering and shifting coefficients, respectively. Because orientation and localization are independent parameters, the steering coefficients can be calculated separately from the shifting coefficients. The number of steering coefficients depends on the polar Fourier bandwidth of the basis RF, while the number of steering filters is inversely proportional to the basis RF size (Grill-Spector et al., 1995). In the presence of noise this minimal basis has to be extended (see Figure 2). The results ofthe simulation for several RF sizes are shown in Figure 3, left. As expected, the number of basis RFs required to approximate a desired RF is inversely proportional to the Diversity in the Response Properties of Cortical Neurons 121 The dependency oI1he nlnber d RF9 a1 the venene. '. ,5 2 25 35 van.,08 Figure 3: Left: error of the steering/shifting approximation for several basis RF sizes. Right: the number of basis RFs required to achieve a given error for different sizes of the basis RFs. The dashed line is the hyperbola num RFs x size = const. size of the basis RFs (Figure 3, right). Steerability and biological considerations. The anatomical finding that the columnar "borders" are freely crossed by dendritic and axonal arbors (Malach, 1992), and the mathematical properties of shiftable/steerable filters outlined above suggest that the columnar architecture in VI provides a basis for creating a continuum of RF properties, rather that being a form of organizing RFs in discrete bins. Computationally, this may be possible if the input to neurons is a linear combination of outputs of several RFs, as in equation 1. The anatomical basis for this computation may be provided by intrinsic cortical connections. It is known that long-range (I"V 1 mm) connections tend to link cells with like orientation preference, while the short-range (I"V 400 J.lm) connections are made to cells of diverse orientation preferences (Malach et al., 1993). We suggest that the former provide the inputs necessary to shift the position of the desired RF, while the latter participate in steering the RF to an arbitrary angle (see Grill-Spector et al., 1995, for details). 3 Matching with patchy connections Many visual tasks require matching between images taken at different points in space (as in binocular stereopsis) or time (as in motion processing). The first and foremost problem faced by a biological system in solving these tasks is that the images to be compared are not represented as such anywhere in the system: instead of images, there are patterns of activities of neurons, with RFs that are overlapping, are not located on a precise grid, and are subject to mixing by patchy projections in each successive stage of processing. In this section, we show that a system composed of scattered RFs with smooth and overlapping tuning functions can, as a matter of 122 Kalanit Grill Spector, Shimon Edelman, Rafael Malach fact, perform matching precisely by allowing patchy connections between domains. Moreover, the weights that must be given to the various inputs that feed a RF carrying out the match are identical to the coefficients that would be generated by a learning algorithm required to capture a certain well-defined input-output relationship from pairs of examples. DOMAIN A Figure 4: Unit C receives patchy input from areas A and B which contain receptors with overlapping RFs. Consider a unit C, sampling two domains A and B through a Gaussian-profile dendritic patch equal in size to that of the axonal arbor of cells feeding A and B (Figure 4). The task faced by unit C is to determine the degree to which the activity patterns in domains A and B match. Let <Pjp be the response of the j'th unit in A to an input x-;': ( .. ")2 A. . = exp{ Xp Xj } 'l'JP 20'2 (2) where xj be the optimal pattern to which the j'th unit is tuned (the response Bjp of a unit in B is of similar form). If, for example, domains A and B contain orientation selective cells, then xj would be the optimal combination of orientation and location of a bar stimulus. For simplicity we assume that all the RFs are of the same size 0', that unit C samples the same number of neurons N from both domains, and that the input from each domain to unit C is a linear combination of the responses of the units in each area. The input to C from domain A, with x-;, presented to the system is then: N Ain = L aj<pjp j=1 (3) Diversity in the Response Properties of Cortical Neurons 123 The problem is to find coefficients {aj} and {bj } such that on a given set of inputs {x-;} the outputs of domains A and B will match. We define the matching error as follows: Em = t (~a,~,p -~b'8'p)' (4) Proposition 1 The desired coefficients, minimizing Em, can be generated by an algorithm trained to learn an input/output mapping from a set of examples. This proposition can be proved by taking the derivative of Em with respect to the coefficients (Grill-Spector et al., 1995). Learning here can be carried out by radial basis function (RBF) approximation (Poggio and Girosi, 1990), which is particularly suitable for our purpose, because its basis functions can be regarded as multidimensional Gaussian RFs. 4 Summary Our results show that maximal diversity of neuronal response properties is attained when the ratio of dendritic and axonal arbor sizes is equal to 1, a value found in many cortical areas and across species (Lund et al., 1993; Malach, 1994). Maximization of diversity also leads to better performance in systems of receptive fields implementing steerablejshiftable filters, which may be necessary for generating the seemingly continuous range of orientation selectivity found in VI, and in ma.tching spatially distributed signals. This cortical organization principle may, therefore, have the double advantage of accounting for the formation of the cortical columns and the associated patchy projection patterns, and of explaining how systems of receptive fields can support functions such as the generation of precise response tuning from imprecise distributed inputs, and the matching of distributed signals, a problem that arises in visual tasks such as stereopsis, motion processing, and recognition. References Grill-Spector, K. , Edelman, S., and Malach, R. (1995). Anatomical origin and computational role of diversity in the response properties of cortical neurons. In Aertsen, A., editor, Brain Theory: biological basis and computational theory of vision. Elsevier. in press. Grinvald, A., Lieke, T., Frostigand, R., Gilbert, C., and Wiesel, T. (1986). Functional architecture of the cortex as revealed by optical imaging of intrinsic signals. Nature, 324:361-364. Rubel, D. and Wiesel, T . (1962). Receptive fields, binocular interactions and functional architecture in the cat's visual cortex. Journal of Physiology, 160:106-154. 124 Kalanit Grill Spector, Shimon Edelman, Rafael Malach Lund, J., Yoshita, S., and Levitt, J. (1993). Comparison of intrinsic connections in different areas of macaque cerebral cortex. Cerebral Cortex, 3:148- 162. MacIlwain, J. (1986). Point images in the visual system: new interest in an old idea. Trends in Neurosciences, 9:354- 358. Malach, R. (1992). Dendritic sampling across processing streams in monkey striate cortex. Journal of Comparative Neurobiology, 315:305-312. Malach, R. (1994). Cortical columns as devices for maximizing neuronal diversity. Trends in Neurosciences, 17:101- 104. Malach, R., Amir, Y., Harel, M., and Grinvald, A. (1993). Relationship between intrinsic connections and functional architecture,revealed by optical imaging and in vivo targeted biocytine injections in primate striate cortex. Proceedings of the National Academy of Science, USA, 90:10469- 10473. Mouncastle, V. (1957). Modality and topographic properties of single neurons of cat's somatic sensory cortex. Journal of Neurophysiology, 20:408-434. Poggio, T. and Girosi, F. (1990). Regularization algorithms for learning that are equivalent to multilayer networks. Science, 247:978- 982. Rockland, K. and Lund, J. (1982). Widespread periodic intrinsic connections in the tree shrew visual cortex. Science, 215: 1532-1534. Simoncelli, E., Freeman, W., Adelson, E., and Heeger, D. (1992). Shiftable multiscale transformations. IEEE Transactions on Information Theory, 38:587-607. Vidyasagar, T. (1985). Geniculate orientation biases as cartesian coordinates for cortical orientation detectors. In Models for the visual cortex, pages 390- 395. Wiley, New York.	1994	52
946	Synchrony and Desynchrony in Neural Oscillator Networks DeLiang Wang Department of Computer and Information Science and Center for Cognitive Science The Ohio State University Columbus, Ohio 43210, USA dwang@cis.ohio-state.edu Abstract David Terman Department of Mathematics The Ohio State University Columbus, Ohio 43210, USA terman@math.ohio-state.edu An novel class of locally excitatory, globally inhibitory oscillator networks is proposed. The model of each oscillator corresponds to a standard relaxation oscillator with two time scales. The network exhibits a mechanism of selective gating, whereby an oscillator jumping up to its active phase rapidly recruits the oscillators stimulated by the same pattern, while preventing others from jumping up. We show analytically that with the selective gating mechanism the network rapidly achieves both synchronization within blocks of oscillators that are stimulated by connected regions and desynchronization between different blocks. Computer simulations demonstrate the network's promising ability for segmenting multiple input patterns in real time. This model lays a physical foundation for the oscillatory correlation theory of feature binding, and may provide an effective computational framework for scene segmentation and figure/ground segregation. 1 INTRODUCTION A basic attribute of perception is its ability to group elements of a perceived scene into coherent clusters (objects). This ability underlies perceptual processes such as figure/ground segregation, identification of objects, and separation of different objects, and it is generally known as scene segmentation or perceptual organization. Despite the fact 200 DeLiang Wang. David Tennan that humans perform it with apparent ease, the general problem of scene segmentation remains unsolved in the engineering of sensory processing, such as computer vision and auditory processing. Fundamental to scene segmentation is the grouping of similar sensory features and the segregation of dissimilar ones. Theoretical investigations of brain functions and feature binding point to the mechanism of temporal correlation as a representational framework (von der Malsburg, 1981; von der Malsburg and Schneider, 1986). In particular, the correlation theory of von der Malsburg (1981) asserts that an object is represented by the temporal correlation of the fIring activities of the scattered cells coding different features of the object. A natural way of encoding temporal correlation is to use neural oscillations, whereby each oscillator encodes some feature (maybe just a pixel) of an object. In this scheme, each segment (object) is represented by a group of oscillators that shows synchrony (phase-locking) of the oscillations, and different objects are represented by different groups whose oscillations are desynchronized from each other. Let us refer to this form of temporal correlation as oscillatory correlation. The theory of oscillatory correlation has received direct experimental support from the cell recordings in the cat visual cortex (Eckhorn et aI., 1988; Gray et aI., 1989) and other brain regions. The discovery of synchronous oscillations in the visual cortex has triggered much interest from the theoretical community in simulating the experimental results and in exploring oscillatory correlation to solve the problems of scene segmentation. While several demonstrate synchronization in a group of oscillators using local (lateral) connections (Konig and Schillen, 1991; Somers and Kopell, 1993; Wang, 1993, 1995), most of these models rely on long range connections to achieve phase synchrony. It has been pointed out that local connections in reaching synchrony may play a fundamental role in scene segmentation since long-range connections would lead to indiscriminate segmentation (Sporns et aI., 1991; Wang, 1993). There are two aspects in the theory of oscillatory correlation: (1) synchronization within the same object; and (2) desynchronization between different objects. Despite intensive studies on the subject, the question of desynchronization has been hardly addressed. The lack of an effIcient mechanism for de synchronization greatly limits the utility of oscillatory correlation to perceptual organization. In this paper, we propose a new class of oscillatory networks and show that it can rapidly achieve both synchronization within each object and desynchronization between a number of simultaneously presented objects. The network is composed of the following elements: (I) A new model of a basic oscillator; (2) Local excitatory connections to produce phase synchrony within each object; (3) A global inhibitor that receives inputs from the entire network and feeds back with inhibition to produce desynchronization of the oscillator groups representing different objects. In other words, the mechanism of the network consists of local cooperation and global competition. This surprisingly simple neural architecture may provide an elementary approach to scene segmentation and a computational framework for perceptual organization. 2 MODEL DESCRIPTION The building block of this network, a single oscillator i, is defined in the simplest form as a feedback loop between an excitatory unit Xi and an inhibitory unit y( Synchrony and Desynchrony in Neural Oscillator Networks 201 8 , , dxldt= 0 o . -2 " " " :' " . o x , , , , 2 Figure 1: Nullclines and periodic orbit of a single oscillator as shown in the phase plane. When the oscillator starts at a randomly generated point in the phase plane, it quickly converged to a stable trajectory of a limit cycle. dXi 3 -=3x·- X · +2-y · +p+I·+S· dt l I , " dy · dt' = e (y(1 + tanh(xl/3) - Yi) Figure 2: Architecture of a two dimensional network with nearest neighbor coupling. The global inhibitor is indicated by the black circle. (la) (lb) where p denotes the amplitude of a Gaussian noise term. Ii represents external stimulation to the oscillator, and Si denotes coupling from other oscillators in the network. The noise term is introduced both to test the robustness of the system and to actively desynchronize different input patterns. The parameter e is chosen to be small. In this case (1), without any coupling or noise, corresponds to a standard relaxation oscillator. The x-nullcline of (1) is a cubic curve, while the y-nullc1ine is a sigmoid function, as shown in Fig. 1. If I > 0, these curves intersect along the middle branch of the cubic, and (1) is oscillatory. The periodic solution alternates between the silent and active phases of near steady state behavior. The parameter yis introduced to control the relative times that the solution spends in these two phases. If I < 0, then the nullc1ines of (1) intersect at a stable fixed point along the left branch of the cubic. In this case the system produces no oscillation. The oscillator model (1) may be interpreted as a model of spiking behavior of a single neuron, or a mean field approximation to a network of excitatory and inhibitory neurons. The network we study here in particular is two dimensional. However, the results can easily be extended to other dimensions. Each oscillator in the network is connected to only its four nearest neighbors, thus forming a 2-D grid. This is the simplest form of local connections. The global inhibitor receives excitation from each oscillator of the grid, and in turn inhibits each oscillator. This architecture is shown in Fig. 2. The intuitive reason why the network gives rise to scene segmentation is the following. When multiple connected objects are mapped onto the grid, local connectivity on the grid will group together the oscillators covered by each object. This grouping will be reflected 202 DeLiang Wang, David Tennan by phase synchrony within each object. The global inhibitor is introduced for desynchronizing the oscillatory responses to different objects. We assume that the coupling term Si in (1) is given by Si = L Wik Soo(xk, 9x) - Wz Soo(z, 9xz) kEN(i) S (x 9' = __ -=-1 __ 00 ,J 1+ exp[-K(x-e,] (2) (3) where Wik is a connection (synaptic) weight from oscillator k to oscillator i, and N(i) is the set of the neighoring oscillators that connect to i. In this model, NO) is the four immediate neighbors on the 2-D grid, except on the boundaries where N(i) may be either 2 or 3 immediate neighbors. 9x is a threshold (see the sigmoid function of Eq. 3) above which an oscillator can affect its neighbors. Wz (positive) is the weight of inhibition from the global inhibitor z, whose activity is defined as (4) where Goo = 0 if Xi < 9zx for every oscillator, and Goo = 1 if Xi ~ 9zx for at least one oscillator i. Hence 9zx represents a threshold. If the activity of every oscillator is below this threshold, then the global inhibitor will not receive any input. In this case z ~ 0 and the oscillators will not receive any inhibition. If, on the other hand, the activity of at least one oscillator is above the threshold 9zx then, the global inhibitor will receive input. In this case z ~ 1, and each oscillator feels inhibition when z is above the threshold 9zx. The parameter l/J determines the rate at which the inhibitor reacts to such stimulation. In summary, once an oscillator is active, it triggers the global inhibitor. This then inhibits the entire network as described in Eq. 1. On the other hand, an active oscillator spreads its activation to its nearest neighbors, again through (1), and from them to its further neighbors. In the next section, we give a number of properties of this system. Besides boundaries, the oscillators on the grid are basically symmetrical. Boundary conditions may cause certain distortions to the stability of synchrous oscillations. Recently, Wang (1993) proposed a mechanism called dynamic normalization to ensure that each oscillator, whether it is in the interior or on a boundary, has equal overall connection weights from its neighbors. The dynamic normalization mechanism is adopted in the present model to form effective connections. For binary images (each pixel being either 0 or 1), the outcome of dynamic normalization is that an effective connection is established between two oscillators if and only if they are neighbors and both of them are activated by external stimulation. The network defined above can readily be applied for segmentation of binary images. For gray-level images (each pixel being in a certain value range), the following slight modification suffices to make the network applicable. An effective connection is established between two oscillators if and only if they are neighbors and the difference of their corresponding pixel values is below a certain threshold. Synchrony and Desynchrony in Neural Oscillator Networks 203 3 ANALYTICAL RESULTS We have formally analyzed the network. Due to space limitations, we can only list the major conclusions without proofs. The interested reader can find the details in Terman and Wang (1994). Let us refer to a pattern as a connected region, and a block be a subset of oscillators stimulated by a given pattern. The following results are about singular solutions in the sense that we formally set E = O. However, as shown in (Terman and Wang, 1994), the results extend to the case E> 0 sufficiently small. Theorem 1. (Synchronization). The parameters of the system can be chosen so that all of the oscillators in a block always jump up simultaneously (synchronize). Moreover, the rate of synchronization is exponential. Theorem 2. (Multiple Patterns) The parameters of the system and a constant T can be chosen to satisfy the following. If at the beginning all the oscillators of the same block synchronize with each other and the temporal distance between any two oscillators belonging to two different blocks is greater than T, then (1) Synchronization within each block is maintained; (2) The blocks activate with a fixed ordering; (3) At most one block is in its active phase at any time. Theorem 3. (Desynchronization) If at the beginning all the oscillators of the system lie not too far away from each other, then the condition of Theorem 2 will be satisfied after some time. Moreover, the time it takes to satisfy the condition is no greater than N cycles, where N is the number of patterns. The above results are true with arbitrary number of oscillators. In summary, the network exhibits a mechanism, referred to as selective gating, which can be intuitively interpreted as follows. An oscillator jumping to its active phase opens a gate to quickly recruit the oscillators of the same block due to local connections. At the same time, it closes the gate to the oscillators of different blocks. Moreover, segmentation of different patterns is achieved very rapidly in terms of oscillation cycles. 4 COMPUTER SIMULATION To illustrate how this network is used for scene segmentation, we have simulated a 2Ox20 oscillator network as defined by (1)-(4). We arbitrarily selected four objects (patterns): two O's, one H, and one I ; and they form the word OHIO. These patterns were simultaneously presented to the system as shown in Figure 3A. Each pattern is a connected region, but no two patterns are connected to each other. All the oscillators stimulated (covered) by the objects received an external input 1=0.2, while the others have 1=-0.02. The amplitude p of the Gaussian noise is set to 0.02. Thus, compared to the external input, a 10% noise is included in every oscillator. Dynamic normalization results in that only two neighboring oscillators stimulated by a single pattern have an effective connection. The differential equations were solved numerically with the following parameter values: E = 0.02, l/J = 3.0; Y= 6.0, f3 = 0.1, K = 50, Ox = -0.5, and 0zx = 0xz = 0.1. The total effective connections were normalized to 6.0. The results described below were robust to considerable changes in the parameters. The phases of all the oscillators on the grid were randomly initialized. 204 DeLiang Wang, David Terman Fig. 3B-3F shows the instantaneous activity (snapshot) of the network at various stages of dynamic evolution. The diameter of each black circle represents the normalized x activity of the corresponding oscillator. Fig. 3B shows a snapshot of the network a few steps after the beginning of the simulation. In Fig. 3B, the activities of the oscillators were largely random. Fig. 3C shows a snapshot after the system had evolved for a short time period. One can clearly seethe effect of grouping and segmentation: all the oscillators belonging to the left 0 were entrained and had large activities. At the same time, the oscillators stimulated by the other three patterns had very small activities. Thus the left 0 was segmented from the rest of the input. A short time later, as shown in Fig. 3D, the oscillators stimulated by the right 0 reached high values and were separated from the rest of the input. Fig. 3E shows another snapshot after Fig. 3D. At this time, pattern I had its turn to be activated and separated from the rest of the input. Finally in Fig. 3F, the oscillators representing H were active and the rest of the input remained silent. This successive "pop-out" of the objects continued in a stable periodic fashion. To provide a complete picture of dynamic evolution, Fig. 30 shows the temporal evolution of each oscillator. Since the oscillators receiving no external input were inactive during the entire simulation process, they were excluded from the display in Fig. 30. The activities of the oscillators stimulated by each object are combined together in the figure. Thus, if they are synchronized, they appear like a single oscillator. In Fig. 30, the four upper traces represent the activities of the four oscillator blocks, and the bottom trace represents the activity of the global inhibitor. The synchronized oscillations within each object are clearly shown within just three cycles of dynamic evolution. 5 DISCUSSION Besides neural plausibility, oscillatory correlation has a unique feature as an computational approach to the engineering of scene segmentation and figure/ground segregation. Due to the nature of oscillations, no single -object can dominate and suppress the perception of the rest of the image permanently. The current dominant object has to give way to other objects being suppressed, and let them have a chance to be spotted. Although at most one object can dominant at any time instant, due to rapid oscillations, a number of objects can be activated over a short time period. This intrinsic dynamic process provides a natural and reliable representation of multiple segmented patterns. The basic principles of selective gating are established for the network with lateral connections beyond nearest neighbors. Indeed, in terms of synchronization, more distant connections even help expedite phase entrainment. In this sense, synchronization with all-to-all connections is an extreme case of our system. With nearest-neighbor connectivity (Fig. 2), any isolated part of an image is considered as a segment. In an noisy image with many tiny regions, segmentation would result in too many small fragments. More distant connections would also provide a solution to this problem. Lateral connections typically take on the form of Gaussian distribution, with the connection strength between two oscillators falling off exponentially. Since global inhibition is superimposed to local excitation, two oscillators positively coupled may be desynchronized if global inhibition is strong enough. Thus, it is unlikely that all objects in an image form a single segment as the result of extended connections. Synchrony and Desynchrony in Neural Oscillator Networks 205 Due to its critical importance for computer vision, scene segmentation has been studied quite extensively. Many techniques have been proposed in the past (Haralick and Shapiro, 1985; Sarkar and Boyer, 1993). Despite these techniques, as pointed out by Haralick and Shapiro (1985), there is no underlying theory of image segmentation, and the techniques tend to be adhoc and emphasize some aspects while ignoring others. Compared to the traditional techniques for scene segmentation, the oscillatory correlation approach offers many unique advantages. The dynamical process is inherently parallel. While conventional computer vision algorithms are based on descriptive criteria and many adhoc heuristics, the network as exemplified in this paper performs computations based on only connections and oscillatory dynamics. The organizational simplicity renders the network particularly feasible for VLSI implementation. Also, continuous-time dynamics allows real time processing, desired by many engineering applications. Acknowledgments DLW is supported in part by the NSF grant IRI-9211419 and the ONR grant NOOOI4-931-0335. DT is supported in part by the NSF grant DMS-9203299LE. References R. Eckhorn, et aI., "Coherent oscillations: A mechanism of feature linking in the visual cortex?" Bioi. Cybem., vol. 60, pp. 121-130, 1988. C.M. Gray, P. Konig, A.K. Engel, and W. Singer, "Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties," Nature, vol. 338, pp. 334-337, 1989. R.M. Haralick and L.G. Shapiro, "Image segmentation techniques," Comput. Graphics Image Process., vol. 29, pp. 100-132, 1985. P. Konig and T.B. Schillen, "Stimulus-dependent assembly formation of oscillatory responses: I. Synchronization," Neural Comput., vol. 3, pp. 155-166, 1991. S. Sarkar and K.L. Boyer, "Perceptual organization in computer vision: a review and a proposal for a classificatory structure," IEEE Trans. Syst. Man Cybern., vol. 23, 382-399, 1993. D. Somers, and N. Kopell, "Rapid synchronization through fast threshold modulation," BioI. Cybern, vol. 68, pp. 393-407, 1993. O. Sporns, G. Tononi, and G.M. Edelman, "Modeling perceptual grouping and figureground segregation by means of active reentrant connections," Proc. Natl. A cad. Sci. USA, vol. 88, pp. 129-133, 1991. D. Terman and D.L. Wang, "Global competition and local cooperation in a network of neural oscillators," Physica D, in press, 1994. C. von der Malsburg, "The correlation theory of brain functions," Internal Report 81-2, Max-Planck-Institut for Biophysical Chemistry, Gottingen, FRG, 1981. • C. von der Malsburg and W. Schneider, "A neural cocktail-party processor," Bioi. Cybern., vol. 54, pp. 29-40, 1986. D.L. Wang, "Modeling global synchrony in the visual cortex by locally coupled neural oscillators," Proc. 15th Ann. Conf. Cognit. Sci. Soc., pp. 1058-1063, 1993. D.L. Wang, "Emergent synchrony in locally coupled neural oscillators," IEEE Trans. on Neural Networks, in press, 1995. 206 DeLiang Wang, David Terman A B · .... , . , ..... ••••••••••••• •••••• ••••••••••••••••••• •••••••••••••••••••• • ••••••••••••••••• .... ~ •....••.•.•.... ••••••••••••••••••• ••••••• ••••••••••••• ••••••••••••••••••• ••••••••••••••••••• · .•..........•...... • ••••••••••••••• ••• •. .•. •.•. • · ••• ··e· .• · .......•.......... . •••••••••••••••••••• •••••••• • ••••• ••• ••• ••••••••••••••••••• •••••••••••••••••••• . . . .•.. . . . ...•.. . . ... ...... .•.. c . .... ... .. .... ... ... D E F G . . . . " . ........ . ... . · ....... .. ... . ..... . . .. ....... .... . . .. . . Figure 3. A An image composed of four patterns which were presented (mapped) to a 20x20 grid of oscillators. B A snapshot of the activities of the oscillator grid at the beginning of dynamic evolution. C A snapshot taken shortly after the beginning. D Shortly after C. E Shortly after D. F Shortly after E. G The upper four traces show the combined temporal activities of the oscillator blocks representing the four patterns, respectively, and the bottom trace shows the temporal activity of the global inhibitor. The simulation took 8,000 integration steps.	1994	53
947	A Computational Model of Prefrontal Cortex Function Todd S. Braver Dept. of Psychology Carnegie Mellon Univ. Pittsburgh, PA 15213 Jonathan D. Cohen Dept. of Psychology Carnegie Mellon Univ. Pittsburgh, PA 15213 Abstract David Servan-Schreiber Dept. of Psychiatry Univ . of Pittsburgh Pittsburgh, PA 15232 Accumulating data from neurophysiology and neuropsychology have suggested two information processing roles for prefrontal cortex (PFC): 1) short-term active memory; and 2) inhibition. We present a new behavioral task and a computational model which were developed in parallel. The task was developed to probe both of these prefrontal functions simultaneously, and produces a rich set of behavioral data that act as constraints on the model. The model is implemented in continuous-time, thus providing a natural framework in which to study the temporal dynamics of processing in the task. We show how the model can be used to examine the behavioral consequences of neuromodulation in PFC. Specifically, we use the model to make novel and testable predictions regarding the behavioral performance of schizophrenics, who are hypothesized to suffer from reduced dopaminergic tone in this brain area. 1 Introduction Prefrontal cortex (PFC) is an area of the human brain which is significantly expanded relative to other animals. There is general consensus that the PFC is centrally involved in higher cognitive activities such as planning, problem solving and language. Recently, the PFC has been associated with two specific information processing mechanisms: short-term active memory and inhibition. Active memory is the capacity of the nervous system to maintain information in the form of sustained activation states (e.g. , cell firing) for short periods of time. This can be distinguished from forms of memory that are longer in duration and are instantiated as 142 Todd S. Braver, Jonathan D. Cohen, David Servan-Schreiber modified values of physiological parameters (e.g., synaptic strength). Over the last two decades, there have been a large number of neurophysiological studies focusing on the cellular basis of active memory in prefrontal cortex. These studies have revealed neurons in PFC that fire selectively to specific stimuli and response patterns, and that remain active during a delay between these. Investigators such as Fuster (1989) and Goldman-Rakic (1987) have argued from this data that PFC maintains temporary information needed to guide behavioral responses through sustained patterns of neural activity. This hypothesis is consistent with behavioral findings from both animal and human lesion studies, which suggest that PFC is required for tasks involving delayed responses to prior stimuli (Fuster, 1989; Stuss & Benson, 1986). In addition to its role in active memory, many investigators have focused on the inhibitory functions of PFC. It has been argued that PFC representations are required to overcome reflexive or previously reinforced response tendencies in order to mediate a contextually appropriate - but otherwise weaker - response (Cohen & Servan-Schreiber, 1992). Clinically, it has been observed that lesions to PFC are often associated with a syndrome of behavioral disinhibition, in which patients act in impulsive and often socially inappropriate ways (Stuss & Benson, 1986). This syndrome has often been cited as evidence that PFC plays an important role inhibiting behaviors which are compelling but socially inappropriate. While the involvement of PFC in both active memory and inhibition is generally agreed upon, computational models can play an important role in providing mechanisms by which to explain how these two information processing functions arise. There are several computational models now in the literature which have focused on either the active memory (Zipser, 1991), or inhibitory (Levine & Pruiett, 1989) functions of PFC, or both functions together (Dehaene & Changeux, 1989; Cohen & Servan-Schreiber, 1992). These models have been instrumental in explaining the role of PFC in a variety of behavioral tasks (e.g., the Wisconsin Card Sort and Stroop). However, these earlier models are limited by their inability to fully capture the dynamical processes underlying active memory and inhibition. Specifically, none of the simulations have been tightly constrained by the temporal parameters found in the behavioral tasks (e.g., durations of stimuli, delay periods, and response latencies). This limitation is not found solely in the models, but is also a feature of the behavioral tasks themselves. The tasks simulated were not structured in ways that could facilitate a dynamical analysis of processing. In this paper we address the limitations of the previous work by describing both a new behavioral task and a computational model of PFC. These have been developed in parallel and, together, provide a useful framework for exploring the temporal dynamics of active memory and inhibition and their consequences for behavior. We then go on to describe how this framework can be used to examine neuromodulatory effects in PFC, which are believed to playa critical role in both normal functioning and in psychiatric disorders, such as schizophrenia. 2 Behavioral Assessment of Human PFC Function We have developed a task paradigm which incorporates two components central to the function of prefrontal cortex - short-term active memory and inhibition - and that can be used to study the dynamics of processing. The task is a variant of the continuous performance test (CPT), which is commonly used to study attention in A Computational Model of Prefrontal Cortex Function 143 behavioral and clinical research. In a standard version of the task (the CPT-AX), letters are presented one at a time in the middle of a computer screen. Subjects are instructed to press the target button to the letter X (probe stimulus) but only when it is preceded by an A (the cue stimulus). In previous versions of the CPT, subjects only responded on target trials. In the present version of the task, a two response forced-choice procedure is employed; on non-A-X trials subjects are asked to press the non-target button. This procedure allows for response latencies to be evaluated on every trial, thus providing more information about the temporal dimensions of processing in the task. Two additional modifications were made to the standard paradigm in order to maximally engage PFC activity. The memory function of PFC is tapped by manipulating the delay between stimuli. In the CPT-AX, the prior stimulus (cue or non-cue) provides the context necessary to decide how to respond to the probe letter. However, with a short delay (750 msec.), there is little demand on memory for the prior stimulus. This is supported by evidence that PFC lesions have been shown to have no effect on performance when there is only a short delay (Stuss & Benson, 1986). With a longer delay (5000 msec.), however, it becomes necessary to maintain a representation of the prior stimulus in order for it to be used as context for responding to the current one. The ability of the PFC to sustain contextual representations over the delay period can be determined behaviorally by comparing performance on short delay trials (50%) against those with long delays (50%). The inhibitory function of PFC is probed by introducing a prepotent response tendency that must be overcome to respond correctly. This tendency is introduced into the task by increasing the frequency of target trials (A followed by X). In the remaining trials, there are three types of distractors: 1) a cue followed by a nontarget probe letter (e.g. , A-Y); 2) a non-cue followed by the target probe letter (e.g., B-X); and a non-cue followed by a non-target probe letter (e.g., B-Y). Target trials occur 70% of the time, while each type of distract or trial occurs only 10% of the time. The frequency of targets promotes the development of a strong tendency to respond to the target probe letter whenever it occurs, regardless of the identity of the cue (since a response to the X itself is correct 7 out of 8 times). The ability to inhibit this response tendency can be examined by comparing accuracy on trials when the target occurs in the absence of the cue (B-X trials), with those made when neither the cue nor target occurs (i.e., B-Y trials, which provide a measure of non-specific response bias and random responding). Trials in which the cue but not the target probe appears (A-Y trials) are also particularly interesting with respect to PFC function. These trials measure the cumulative influence of active representations of context in guiding responses. In a normally functioning system, context representations should stabilize and increase in strength as time progresses. Thus, it is expected that A-Y accuracy will tend to decrease for long delay trials relative to short ones. As mentioned above, the primary benefit of this paradigm is that it provides a framework in which to simultaneously probe the inhibitory and memory functions associated with PFC. This is supported by preliminary neuroimaging data from our laboratory (using PET) which suggests that PFC is, in fact, activated during performance of the task. Although it is simple in structure, the task also generates a rich set of behavioral data. There are four stimulus conditions crossed with two delay conditions for which both accuracy and reaction time performance can be 144 100 90 80 70 60 750 650 ! 550 " .Ii 1 450 ..: 350 250 Todd S. Braver, Jonathan D. Cohen, David Servan-Schreiber Accurac, (Short Delay) Accurac, (Long Delay) V V 1- MODEL (Ace) J .DATA (Ace) RT(ShortDelay) RT (Long Delay) I 09 ~ '''J!.' -~~-~ , , -~~ , AX AY BX BY AX AY BX Trial Condition Trial Condition MODEL (Correct) --- MODEL (Incorrect) .. DATA (Correct) 'V DATA (Incorrect) BY Figure 1: Subjecl beha.viora.1 da.la. with model performa.nce superimposed. Top Panels: Acc ura.cy a.c ross both dela.ys in a.1I four condilions. Bottom Panels: Rea.ction times for both correct a.nd incorrec t responses in a.1I conditions. Ba.rs represent sta.nda.rd error of mea.surement for the empirica.l da.ta.. measured. Figure 1 shows data gathered from 36 college-age subjects performing this task. In brief, we found that: 1) Accuracy was relatively unchanged in the long delays compared to the short, demonstrating that active memory was adequately supporting performance; 2) A-Y accuracy, however, did slightly decrease at long delays, reflecting the normal build-up of context representations over time; 3) Accuracy on B-X trials was relatively high, supporting the assumption that subjects could effectively use context representations to inhibit prepotent responses; 4) A distinct pattern emerged in the latencies of correct and incorrect responses, providing information on the temporal dynamics of processing (i.e., responses to A-Y trials are slow on correct trials and fast on incorrect ones; the pattern is reversed for B-X trials). Taken together, the data provides specific, detailed information about normal PFC functioning, which act as constraints on the development and evaluation of a computational model. 3 A Computational Model of the CPT-AX We have developed a recurrent network model which produces detailed information regarding the temporal course of processing in the CPT-AX task. The network is composed of three modules: an input module, a memory module, and an output module. The memory module implements the memory and inhibitory functions believed to be carried out by PFC. Figure 2 shows a diagram of the model. Each unit in the input module represents a different stimulus condition: A, B, X & A Computational Model of Prefrontal Cortex Function 145 OUTPUT LAYER ~~L0~ INPUT LAYER Figure 2: A diagram of the CPT·AX model. Y. Units in the input module make excitatory connections on the response module, both directly and indirectly through the memory module. Lateral inhibition within each layer produces competition for representations. Activity from the cue stimulus flows to the memory module, which is responsible for maintaining a trace of the relevant context in each trial. Units in the memory module have self-excitatory connections, which allow for the activity generated by the cue to be sustained in the absence of input. The recurrent connectivity utilized by each unit in this module is assumed to be a simpler, but formally equivalent analogue of a fully connected recurrent cell assembly. Further, Zipser (1991) has used this type of connectivity to produce temporal activity patterns which are highly similar to the firing patterns of neurons in memory-associated areas of cortex, such as PFC. Activity from the input and memory modules is integrated in the output module. The output of this module determines whether a target (T) or non-target (N) response is made. To simulate the CPT-AX task we have purposefully kept the network architecture and size as simple as possible in order to maximize the model's interpretability. We have therefore not attempted to simulate neural information processing in a neuronby-neuron manner. Rather, the populations of a few units are seen as capturing the information processing characteristics of much larger populations of real neurons. In this way, it is possible to capture the stochastic, distributed, and dynamical properties of real neural networks with small and analytically tractable simulations. The simulation is run in a temporally continuous framework in which processing is governed by the following difference equation: (1 ) where 1 (2) is the state of unit j, Ij is the total input to j , dt is the time-step of integration, 'Y is the gain and f3 is the bias. The continuous framework is preferable to a discrete event-based one in that it allows for a plausible way to scale events appropriately to the exact temporal specifications of the task (i.e., the duration of stimuli and the delay between cue and probe). In addition, the continuous character of the simulation naturally provides a framework for inferring the reaction times in the various conditions. 146 Todd S. Braver, Jonathan D. Cohen, David Servan-Schreiber 4 Simulations of Behavioral Performance We used a continuous recurrent generalization of backpropagation (Pearlmutter, 1989) to train the network to perform the CPT-AX. All of the connection weights were developed entirely by the training procedure, with the constraint that that all self and between layer weights were forced to be positive and all within layer weights were forced to be negative. Training consisted of repeated presentation of each of the 8 conditions in the task (A-X,A-Y,B-X,B-Y, at both long and short delays), with the presentation frequency of each condition matching that of the behavioral task. Weights were updated after the presentation of each trial, biases ({3) were fixed at -2.5, and dt was set at 0.1. The network was trained deterministically; completion of training occurred when network accuracy reached 100% for each condition. Following training, weights were fixed. Errors and reaction time distributions were then simulated by adding zero-mean Gaussian noise to the net input of each unit at every time step during trial presentation. A trial consisted of the presentation of the cue stimulus, a delay period and then the probe stimulus. As mentioned above, the duration of these events was appropriately scaled to match the temporal parameters of the task (e.g., 300 msec. duration for cue and probe presentation, 750 msec. for short delays, 5000 msec. for long delays). A time constant (1") of 50 msec. was used for simulation in the network. This scaling factor provided sufficient temporal resolution to capture the relationship between the two task delays while still permitting a tractable way of simulating the events. Responses were determined by noting which output unit reached a threshold value first following presentation of the probe stimulus. Response latency was determined by calculating the number of time steps taken by the model to reach threshold multiplied by the time constant 1". To facilitate comparisons with the experimental reaction times, a constant k was added to all values produced. This parameter might correspond to the time required to execute a motor response. The value of k was determined by a least mean squares fit to the data. 1000 trials of each condition were run in order to obtain a reliable estimate of performance under stochastic conditions. The standard deviation of the noise distribution (0') and the threshold (T) of the response units were adjusted to produce the best fit to the subject data. Figure 1 compares the results of the simulation against the behavioral data. As can be seen in the figure, the model provides a good fit to the behavioral data in both the pattern of accuracy and reaction times. The model not only matches the qualitative pattern of errors and reaction times but produces very similar quantitative results as well. The match between model and experimental results is particularly striking when it is considered that there are a total of 24 data points that this model is fitting, with only 4 free parameters (O',T,1",k). The model's ability to successfully account for the pattern of behavioral performance provides convincing evidence that it captures the essential principles of processing in the task. We can then feel confident in not only examining normal processing, but also in extending the model to explore the effects of specific disturbances to processing in PFC. 5 Behavioral Effects of Neuromodulation in PFC In a previous meeting of this conference a simulation of a simpler version of the CPT was discussed (Servan-Schreiber, Printz, & Cohen, 1990). In this simulation the A Computational Model of Prefrontal Cortex Function 100 .... CJ 90 ~ ... ... Q U .... == 80 ~ CJ ... ~ =70 60 Accuracy (Short Delay) Accuracy (Long Delay) I , , I I I , I , I , I ~ AX AY BX BY AX AY BX BY MODEL (Normal Gain) -- - MODEL (Reduced Gain) .DATA (Controls) 147 Figure 3: Comparision of of model performance with normal and redu ced gain . The graph illustrates ~he effec~ of reducing gain in the memory layer on task performance. In the baseline network "1=1 , in ~he reduced-gain network "1=0.8. effects of system-wide changes in catecholaminergic tone were captured by changing the gain (-r) parameter of network units. Changes in gain are thought correspond to the action of modulatory neurotransmitters in modifying the responsivity of neurons to input signals (Servan-Schreiber et aI. , 1990; Cohen & Servan-Schreiber, 1992). The current simulation of the CPT offers the opportunity to explore the effects of neuromodulation on the information processing functions specific to PFC. The transmitter dopamine is known to modulate activity in PFC, and manipulations to prefrontal dopamine have been shown to have effects on both memory-related neuronal activity and behavioral performance (Sawaguchi & Goldman-Rakic, 1991). Furthermore, it has been hypothesized that reductions of the neuromodulatory effects of dopamine in PFC are responsible for some of the information processing deficits seen in schizophrenia. To simulate the behavior of schizophrenic subjects, we therefore reduce the gain ('Y) of units in the memory module of the network. With reduced gain in the memory module, there are striking changes in the model's performance of the task. As can be seen in Figure 3, in the short delay conditions the performance of the reduced-gain model is relatively similar to that of control subjects (and the intact model). However, at long delays, the reduced-gain model produces a qualitatively different pattern of performance. In this condition, the model has a high B-X error rate but a low A-Y error rate, a pattern which is opposite to that seen in the control subjects. This double dissociation in performance is a robust effect of the reduced-gain simulation (i.e., it seems relatively uninfluenced by other parameter adjustments) . Thus, the model makes clear-cut predictions which are both novel and highly testable. Specifically, the model predicts that: 1) Differences in performance be148 Todd S. Braver, lonatMn D. Cohen, David Servan-Schreiber tween control and schizophrenic subjects will be most apparent at long delays; 2) Schizophrenics will perform significantly worse than control subjects on B-X trials at long delays; 3) Schizophrenics will perform significantly better than control subjects on A-Y trials at long delays. This last prediction is especially interesting given the fact that tasks in which schizophrenics show superior performance relative to controls are relatively rare in experimental research. Furthermore, the model not only makes predictions regarding schizophrenic behavioral performance, but also offers explanations as to their mechanisms. Analyses of the trajectories of activation states in the memory module reveals that both of the dissociations in performance are due to failures in maintaining representations of the context set up by the cue stimulus. Reducing gain in the memory module blurs the distinction between signal and noise, and causes the context representations to decay over time. As a result, in the long delay trials, there is a higher probability that the model will show both failures of inhibition (more B-X errors) and memory (less A-Y errors). 6 Conclusions The results of this paper show how a computational analysis of the temporal dynamics of PFC information processing can aid in understanding both normal and disturbed behavior. We have developed a behavioral task which simultaneously probes both the inhibitory and active memory functions of PFC. We have used this task in combination with a computational model to explore the effects of neuromodulatory dysfunction, making specific predictions regarding schizophrenic performance in the CPT-AX. Confirmation of these predictions now await further testing. References Cohen, J. & Servan-Schreiber, D. (1992). Context, cortex, and dopamine: A connectionist approach to behavior and biology in schizophrenia. Psychological Review, 99, 45- 77. Dehaene, S. & Changeux, J. (1989). A simple model of prefrontal cortex function III delayed-response tasks. Journal of Cognitive Neuroscience, 1 (3), 244- 261. Fuster, J. (1989). The prefrontal cortex. New York: Raven Press. Goldman-Rakic, P. (1987). Circuitry of primate prefrontal cortex and regulation of behavior by representational memory. In F. Plum (Ed.), Handbook of physiology-the nervous system, v. Bethesda, MD: American Physiological Society, 373-417. Levine, D. & Pruiett, P. (1989). Modeling some effects of frontal lobe damage: novelty and perseveration. Neural Networks, 2 , 103-116. Pearlmutter, B. (1989). Learning state space trajectories in recurrent neural networks. Neural Computation, 1, 263-269. Sawaguchi, T. & Goldman-Rakic, P. (1991). D1 dopamine receptors in prefrontal cortex: Involvement in working memory. Science, 251 , 947-950. Servan-Schreiber, D., Printz, H., & Cohen, J. (1990). The effect of catecholamines on performance: From unit to system behavior. In D. Touretzky (Ed.), Neural information processing systems 2. San Mateo, GA: Morgan Kaufman, 100-108. Stuss, D. & Benson, D. (1986). The frontal lobes. New York: Raven Press. Zipser, D. (1991). Recurrent network model of the neural mechanism of short-term active memory. Neural Computation, 3,179- 19.3.	1994	54
948	Combining Estimators Using Non-Constant Weighting Functions Volker Trespand Michiaki Taniguchi Siemens AG, Central Research Otto-Hahn-Ring 6 81730 Miinchen, Germany Abstract This paper discusses the linearly weighted combination of estimators in which the weighting functions are dependent on the input. We show that the weighting functions can be derived either by evaluating the input dependent variance of each estimator or by estimating how likely it is that a given estimator has seen data in the region of the input space close to the input pattern. The latter solution is closely related to the mixture of experts approach and we show how learning rules for the mixture of experts can be derived from the theory about learning with missing features. The presented approaches are modular since the weighting functions can easily be modified (no retraining) if more estimators are added. Furthermore, it is easy to incorporate estimators which were not derived from data such as expert systems or algorithms. 1 Introduction Instead of modeling the global dependency between input x E ~D and output y E ~ using a single estimator, it is often very useful to decompose a complex mapping -'\.t the time of the research for this paper, a visiting researcher at the Center for Biological and Computational Learning, MIT. Volker.Tresp@zfe.siemens.de 420 Volker Tresp, Michiaki Taniguchi into simpler mappings in the form l (1) M n(x) = L hi(X) i=l The weighting functions hi(X) act as soft switches for the modules N Ni(X). In the mixture of experts (Jacobs et al., 1991) the decomposition is learned in an unsupervised manner driven by the training data and the main goal is a system which learns quickly. In other cases, the individual modules are trained individually and then combined using Equation 1. We can distinguish two motivations: first, in the work on averaging estimators (Perrone, 1993, Meir, 1994, Breiman, 1992) the modules are trained using identical data and the weighting functions are constant and, in the simplest case, all equal to one. The goal is to achieve improved estimates by averaging the errors of the individual modules. Second, a decomposition as described in Equation 1 might represent some "natural" decomposition of the problem leading to more efficient representation and training (Hampshire and Waibel, 1989). A good example is a decomposition into analysis and action. hi(x) might be the probability of disease i given the symptoms x, the latter consisting of a few dozen variables. The amount of medication the patient should take given disease i on the other hand represented by the output of module N Ni (x) might only depend on a few inputs such as weight, gender and age.2 Similarly, we might consider hie x) as the IF-part of the rule, evaluating the weight of the rule given x, and as N Ni(X) the conclusion or action which should be taken under rule i (compare Tresp, Hollatz and Ahmad, 1993). Equation 1 might also be the basis for biological models considering for example the role of neural modulators in the brain. Nowlan and Sejnowsky (1994) recently presented a biologically motivated filter selection model for visual motion in which modules provide estimates of the direction and amount of motion and weighting functions select the most reliable module. In this paper we describe novel ways of designing the weighting functions. Intuitively, the weighting functions should represent the competence or the certainty of a module, given the available information x. One possible measure is related to the number of training data that a module has seen in the neighborhood of x. Therefore, P(xli), which is an estimate of the distribution of the input data which were used to train module i is an obvious candidate as weighting function. Alternatively, the certainty a module assigns to its own prediction, represented by the inverse of the variance 1/ var( N Ni ( x» is a plausible candidate for a weighting function. Both approaches seem to be the flip-sides of the same coin, and indeed, we can show that both approaches are extremes of a unified approach. IThe hat stands for an estimates value. 2Note, that we include the case that the weighting functions and the modules might explicitly only depend on different subsets of x. Combining Estimators Using Non-Constant Weighting Functions (b) 0.5r-r-----T"I -1 -1 o j :1 ~ I ' I,:' I: t ' l , (d) 0.5.----------" o -1 -1 o 1 421 (e) 1 0.8 I, 0.6 I \ / / I 0 .4 \ \ 0.2 \ I \ (f) -1 -1 o 1 Figure 1: (a): Two data sets (1:, 2:0) and the underlying function (continuous). (b) The approximations of the two neural networks trained on the data sets (continuous: 1, dashed: 2). Note, that the approximation of a network is only reliable in the regions of the input space in which it has "seen" data. (c) The weighting functions for variance-based weighting. (d) The approximation using variance-based weighting (continuous). The approximation is excellent, except to the very right. (e) The weighting functions for density-based weighting (Gaussian mixtures approximation). (f) The approximation using density-based weighting (continuous). In particular to the right, the extrapolation is better than in (d). 2 Variance-based Weighting Here, we assume that the different modules N Ni(x) were trained with different data sets {(xLY~)}:~l but that they model identical input-output relationships (see Figure 1 a,b ). To give a concrete example, this would correspond to the case that we trained two handwritten digit classifiers using different data sets and we want to use both for classifying new data. If the errors of the individual modules are uncorrelated and unbiased,3 the combined estimator is also unbiased and has the smallest variance if we select the weighting functions inversely proportional the the variance of the modules 1 hi(X) = var(NNi(x)) (2) This can be shown using var(2:::'l gi(x)N Ni(X)) = 2:::'1 gl(x)var(N Ni(X)) and using Lagrange mUltiplier to enforce the constraint that 2:i gi(X) = 1. Intuitively, 3The errors are un correlated since the modules were trained with different data; correlation and bias are discussed in Section 8.1. 422 Volker Tresp, Michiaki Taniguchi Equation 2 says that a module which is uncertain about its own prediction should also obtain a smaller weight. We estimate the variance from the training data as (NN.( » '" 8N Ni(xl H-:- 18N Ni(X) var * x "" 8w * 8w' Hi is the Hessian, which can be approximated as «(12 is the output-noise variance, Tibshirani, 1994) 3 Density-based Weighting In particular if the different modules were trained with data sets from different regions of the input space, it might be a reasonable assumption that the different modules represent different input-output relationships. In terms of our example, this corresponds to the problem, that we have two handwritten digit classifiers, one trained with American data and one with European data. If the classifiers are used in an international setting, confusions are possible, since, for example, an American seven might be confused with a European one. Formally, we introduce an additional variable which is equal to zero if the writer is American and is equal to one if the writer is European. During recall, we don't know the state of that variable and we are formally faced with the problem of estimation with missing inputs. From previous work (Ahmad and Tresp, 1993) we know that we have to integrate over the unknown input weighted by the conditional probability of the unknown input given the known variables. In this case, this translates into Equation 1, where the weighting function is In our example, P(ilx) would estimate the probability that the writer is American or European given the data. Depending on the problem P(ilx) might be estimated in different ways. If x represents continuous variables, we use a mixture of Gaussians model (3) where G(x; cij, Eij) is our notation for a normal density centered at cij and with covariance Eij. Note that we have obtained a mixture of experts network with P( ilx) as gating network. A novel feature of our approach is that we maintain an estimate of the input data distribution (Equation 3), which is not modeled in the original mixture of experts network. This is advantageous if we have training data which are not assigned Combining Estimators Using Non-Constant Weighting Functions 423 to a module (in the mixture of experts, no data are assigned) which corresponds to training with missing inputs (the missing input is the missing assignment), for which the solution is known (Tresp et ai., 1994). If we use Gaussian mixtures to approximate P(xli), we can use generalized EM learning rules for adaptation. The adaptation of the parameters in the "gating network" which models P(xli) is therefore somewhat simpler than in the original mixture of experts learning rules (see Section 8.2). 4 Unified Approach In reality, the modules will often represent different mappings, but these mappings are not completely independent. Let's assume that we have an excellent American handwritten digit classifier but our European handwritten digit classifier is still very poor, since we only had few training data. We might want to take into account the results of the American classifier, even if we know that the writer was European. Mathematically, we can introduce a coupling between the modules. Let's assume that the prediction ofthe i-th module NNi(X) = li(x)+{i is a noisy version of the true underlying relationship li(x) and that {i is independent Gaussian noise with variance var(N Ni(X». Furthermore, we assume that the true underlying functions are coupled through a prior distribution (for simplicity we only assume two modules) 1 ) 2 P(h(x), h(x)) ex: exp(--2 -(I1(x) - 12(x ) ). vare We obtain as best estimates A 1 h(x) = K(x) [(var(N N2(x)) + vare) N Nt(x) + var(N Nl(x)) N N2(X)] A 1 h(x) = K(x) [var(N N2(x» N Nl(X) + (var(N Nt (x» + vare) N N2(X)] where K(x) = var(N Nl (x» + var(N N2(x)) + vare. We use density-based weighting to combine the two estimates: y(x) = P(llx)it(x)+ P(2Ix)i2(X). Note, that if vare -- 00 (no coupling) we obtain the density-based solution and for vare -- 0 (the mappings are forced to be identical) we obtain the variance-based solution. A generalization to more complex couplings can be found in Section 8.2.1. 5 Experiments We tested our approaches using the Boston housing data set (13 inputs, one continuous output). The training data set consisted of 170 samples which were divided into 20 groups using k-means clustering. The clusters were then divided randomly into two groups and two multi-layer perceptrons (MLP) were trained using those two 424 Volker Tresp. Michiaki Taniguchi data sets. Table 1 shows that the performances of the individual networks are pretty bad which indicates that both networks have only acquired local knowledge with only limited extrapolation capability. Variance-based weighting gives considerably better performance, although density-based weighting and the unified approach are both slightly better. Considering the assumptions, variance-based weighting should be superior since the underlying mappings are identical. One problem might be that we assumed that the modules are unbiased which might not be true in regions were a given module has seen no data. Table 1: Generalization errors N N2 I variance-based I density-based I unified I 0.6948 1.188 I 0.4821 I 0.4472 I 0.4235 I 6 Error-based Weighting In most learning tasks only one data set is given and the task is to obtain optimal predictions. Perrone (1994) has shown that simply averaging the estimates of a small number (i. e. 10) of neural network estimators trained on the same training data set often gives better performance than the best estimator out of this ensemble. Alternatively, bootstrap samples of the original data set can be used for training (Breimann, personal communication). Instead of averaging, we propose that Equation 1, where might give superior results (error-based weighting). Res(N Ni(X)) stands for an estimate of the input dependent residual squared error at x. As a simple approximation, Res(N Ni(X)) can be estimated by training a neural network with the residual squared errors of N Ni. Error-based weighting should be superior to simple averaging in particular if the estimators in the pool have different complexity. A more complex system would obtain larger weights in regions where the mapping is complex, since an estimator which is locally too simple has a large residual error, whereas in regions, where the mapping is simple, both estimators have sufficient complexity, but the simpler one has less variance. In our experiments we only tried networks with the same complexity. Preliminary results indicate that variance-based weighting and error-based weighting are sometimes superior to simple averaging. The main reason seems to be that the local overfitting of a network is reflected in a large variance near that location in input space. The overfitting estimator therefore obtains a small weight in that region (compare the overfitting of network 1 in Figure Ib near x = 0 and the small weight of network 1 close to x = 0 in Figure lc). Combining Estimators Using Non-Constant Weighting Functions 425 7 Conclusions We have presented modular ways for combining estimators. The weighting functions of each module can be determined independently of the other modules such that additional modules can be added without retraining of the previous system. This can be a useful feature in the context of the problem of catastrophic forgetting: additional data can be used to train an additional module and the knowledge in the remaining modules is preserved. Also note that estimators which are not derived from data can be easily included if it is possible to estimate the input dependent certainty or competence of that estimator. Acknowledgements: Valuable discussions with David Cohn, Michael Duff and Cesare Alippi are greatfully acknowledged. The first author would like to thank the Center for Biological and Computational Learning (MIT) for providing and excellent research environment during the summer of 1994. 8 Appendix 8.1 Variance-based Weighting: Correlated Errors and Bias We maintain that Li gi(X) = 1. In general (Le. the modules have seen the same data, or partially the same data), we cannot assume that the errors in the individual modules are independent. Let the M x M matrix O( x) be the covariance between the predictions of the modules N Ni(X). With h(x) = (h1(x) .... hM(X)T, the optimal weighting vector becomes h(x) = 0-1(X) U n(x) = u' 0-1(X) U where u is the M-dimensional unit vector. If the individual modules are biased (biasi(x) = ED(N Ni(X)) EYI.r(ylx)),~ we form the M x M matrix B(x), with Bii(x) = biasi(x)biasj(x), and the minimum variance solution is found for h(x) = (O(x) + B(X))-1 u n(x) = u' (O(x) + B(X))-1 u. 8.2 Density-based Weighting: GEM-learning Let's assume a training pattern (Xlo' YIo) which is not associated with a particular module. If wi is a parameter in network N Ni the error gradient becomes 8errorlo __ ( -NNo( ))p"(OI )8NNi(XIo) 8 ° Ylo , Xlo S Xlo, Ylo 8 ° • w' w' This equation can be derived from the solution to the problem of training with missing features (here: the true i is unknown, see Tresp, Ahmad and Neuneier, 1994). This corresponds also to the M-step in a generalized EM algorithm, where the E-step calculates " . P(Ylolxlo , i)P(xloli)P(i) " ° 2 P(SIXIo, YIo) = '" p o " 0 " . P(Ylolxlo, s) = G(ylo; N Ni(XIo) , (1 ). L..Ji (Ylolxlo, s)P(xlols)P(s) ~ E stands for the expected value; the expectation ED is taken with respect to all data sets of the same size. 426 Volker Tresp, Michiaki Taniguchi using the current parameters. The M-step in the "gating network" P(xli) is particularly simple using the well known EM-rules for Gaussian mixtures. Note, that P(module = i, mixture component: ilxk' Yk) needs to be calculated. 8.2.1 Unified Approach: Correlated Errors and General Coupling Let's form the vectors N N(x) = (N N1(x), ... N NM(X))T and /(x) = (!t(x), ... , /M(x)f. In a more general case, the prior coupling between the underlying functions is described by P(f(x)) = G(f(x);g(x), ~g(x)) where g{x) = (g1{X), ... ,gM{x)f. Furthermore, in a more general case, the estimates are not independent, P{N N{x)l/{x)) = G{N N(x); I(x), ~N{X)). The minimum variance solution is now The equations in Section 4 are special cases with M = 2, g{x) = 0, ~;1(x) = l/varcc x (1, -1)(1, -If, ~N(X) = 1 (var{N N1{x)), var(N N2{X)))T (1 is the 2D-unit matrix). References Ahmad, S. and Tresp, V. (1993). Some Solutions to the Missing Feature Problem in Vision. In S. J. Hanson, J. D. Cowan and C. L. Giles, (Eds.), Advances in Neural Information Processing Systems 5. San Mateo, CA: Morgan Kaufmann. Breiman, L. (1992). Stacked Regression. Dept. of Statistics, Berkeley, TR No. 367. Hampshire, J. and Waibel, A. (1989). The meta-pi network: Building Distributed Knowledge Representations for Robust Pattern Recognition. TR CMU-CS-89-166, CMU, PA. Jacobs, R. A., Jordan, M. 1., Nowlan, S. J. and Hinton, J. E. (1991). Adaptive Mixtures of Local Experts. Neuml Computation, Vol. 3, pp. 79-87. Meir, R. (1994). Bias, Variance and the Combination of Estimators: The Case of Linear Least Squares. TR: Dept. of Electrical Engineering, Technion, Haifa. Nowlan, S. J and Sejnowski, T. J. (1994). Filter Selection Model for Motion Segmentation and Velocity Integration. J. Opt. Soc. Am. A, Vol. 11, No. 12, pp. 1-24. Perrone, M. P. (1993). Improving Regression Estimates: Averaging Methods for Variance Reduction with Extensions to General Convex Measure Optimization. PhD thesis. Brown University. Tibshirani, R. (1994). A Comparison of Some Error Estimates for Neural Network Models. TR Department of Statistics, University of Toronto. Tresp, V., Ahmad, S. and Neuneier, R. (1994). Training Neural Networks with Deficient Data. In: Cowan, J. D., Tesauro, G., and Alspector, J., eds., Advances in Neural Information Processing Systems 6, San Mateo, CA, Morgan Kaufman. Tresp, V., Hollatz J. and Ahmad, S. (1993). Network Structuring and Training Using Rule-based Knowledge. In S. J. Hanson, J. D. Cowan and C. L. Giles, (Eds.), Advances in Neural Information Processing Systems 5, San Mateo, CA: Morgan Kaufmann.	1994	55
949	Stochastic Dynamics of Three-State Neural Networks Toru Ohira Sony Computer Science Laboratory 3-14-13 Higashi-gotanda, Tokyo 141, Japan ohira@csl.sony.co.jp Jack D. Cowan Depts. of Mathematics and Neurology University of Chicago Chicago, IL 60637 cowan@synapse.uchicago.edu Abstract We present here an analysis of the stochastic neurodynamics of a neural network composed of three-state neurons described by a master equation. An outer-product representation of the master equation is employed. In this representation, an extension of the analysis from two to three-state neurons is easily performed. We apply this formalism with approximation schemes to a simple three-state network and compare the results with Monte Carlo simulations. 1 INTRODUCTION Studies of single neurons or networks under the influence of noise have been a continuing item in neural network modelling. In particular, the analogy with spin systems at finite temprature has produced many important results on networks of two-state neurons. However, studies of networks of three-state neurons have been rather limited (Meunier, Hansel and Verga, 1989). A master equation was introduced by Cowan (1991) to study stochastic neural networks. The equation uses the formalism of "second quantization" for classical many-body systems (Doi, 1976a; Grassberger and Scheunert, 1980), and was used to study networks of of two-state neurons (Ohira and Cowan, 1993, 1994). In this paper, we reformulate the master equation using an outer-product representation of operators and extend our previous analysis to networks of three-state neurons. A hierarchy of moment equations for such networks is derived and approximation schemes are used to obtain equa272 Toru Ohira, Jack D. Cowan Figure 1: Transition rates for a three-state neuron. tions for the macroscopic activities of model networks. We compare the behavior of the solutions of these equations with Monte Carlo simulations. 2 THE BASIC NEURAL MODEL We first introduce the network described by the master equation. In this network (Cowan, 1991), neurons at each site, say the ith site, are assumed to cycle through three states: "quiescent", "activated" and "refractory", labelled 'qi', 'aj', and 'ri' respectively. We consider four transitions: q --t a, l' --t a, a --t 1', and " --t q. Two of these, q --t a and r --t a, are functions of the neural input current. We assume these are smoothly increasing functions of the input current and denoted them by fh, and (}2. The other two transition rates, a --t r, and r --t q, are defined as constants a and /3. The resulting stochastic transition scheme is shown in Figure 1. We assume that these transition rates depend only on the current state of the network and not on past states, and that all neural state transitions are asynchronous. This Markovian assumption is essential to the master equation description of this model. We represent the state of each neuron by three-dimensional basis vectors using the Dirac notation lai >, Iri > and Iqi >. They correspond, in more standard vector notation, to: Iq; >= (D, I la;>= 0), I h >= ( ~ ) o . • (1) We define the inner product of these states as < adai >=< qdqj >=< 1'jlri >= 1, (2) < qjlaj >=< ajlqi >=< rdai >=< ad"i >=< 1'dqi >=< Qil"j >= O. (3) Let the states (or configurations) of a network be represented by {In>}, the direct product space of each neuron in the network. (4) Let p[n, t] be the probability of finding the network in a particular state n at time t. We introduce the "neural state vector" for N neurons in a network as I~(t) >= L p[n, t]l n >, {fl} (5) Stochastic Dynamics of Three-State Neural Networks 273 where the sum is taken over all possible network states. With these definitions, we can write the master equation for a network with the transition rates shown in Figure 1, using the outer-product representations of operators (Sakurai, 1985). For example: ( 0 0 1) lai >< q;j = 0 0 0 o 0 0 . • (6) The master equation then takes the form of an evolution equation: , 8 - 8t 1<I>(t) >= LI<I>(t) > (7) with the network "Liouvillian" L given by: N N 1 N L = a I)lai}(ad -Iri}(a;j) + L)h}(1'il-lai}(,'d)e2(= L wijlaj}(ajl) . . n . 1=1 .=1 )=1 +,6L:~1 (Iri}(ril-Iqi}(r;!) + L:~l (lqi}(qil-lai}(qiI)OlUi L:7=1 wijlaj}(ajl).(8) where n is an average number of connections to each neuron, and Wij is the "weight" from the jth to the ith neuron. Thus the weights are normalized with respect to the average number n of connections per neuron. The master equation given here is the same as the one introduced by Cowan using Gell-Mann matrices (Cowan, 1991). However, we note that with the outer-product representation, we can extend the description from two to three-state neurons simply by including one more basis vector. In analogy with the analysis of two-state neurons, we introduce the state vector: N (a r cJI = IT (qi(q;! + ri(rd + ai(ail). (9) i=1 where the product is taken as a direct product, and ai, ri, and qi are parameters. We also introduce the point moments «ai(t)>>, «qi(t)>>, and «ri(t)>> as the probability that the ith neuron is active, quiescent, and refract.ory respectively, at time t. Similarly, we can define the multiple moment, for example, «aiqp'k .. , (t)>> as the probability that the ith neuron is active, the jth neuron is quiescent, the kth neuron is refractory and so on at time t. Then, it can be shown that they are given by: «SiSjSk ... (t)>> = (a = r = q= 1lsi}(sd @ ISj)(Sjl @ ISk)(Skl ... 1<I>(t)}, S = a, " ,q (10) For example, «riqjak(t)>> = (a = r = q = 1I"i}(ril @ Iqj}(qjl @ lak)(akl<I>(t)} (11) We note the following relations, «ai(t)>> + «qi(t)>> + «ri(t)>> = 1 (12) and «aHt)>> = «ai(t)>>, «r;(t)>> = «"i(t)>>, «q;(t)>> = «qi(t)>>, (13) 274 Toru Ohira, Jack D. Cowan 3 THE HIERARCHY OF MOMENT EQUATIONS We can now obtain an equation of motion for the moments. As is typical in the case of many-body problems, we obtain an analogue of the BBGKY hierarchy of equations (Doi, 1976b). This can be done by using the definition of moments, the master equation, and the a-r-q state vector. We show the hierarchy up to the second order: a 1 N --a «airj» = -a( «aiaj» - «airj») + (3«ai1'j» + «airj(}2(= '"' Wikak)>> t n~ k=l -«rirj(}2(~'E~=1 Wikak)>> - «qjrj(}lUf'E:=l Wikak)>> (19) We note that since «ai» + «ri» + «qi» = 1, (20) one of the parameters can be eliminated. We also note that the equations are coupled into higher orders in this hierarchy. This leads to a need for approximation schemes which can terminate the hierarchy at an appropriate order. In the following, we introduce first and the second moment level approximation schemes. For simplicity, we consider the special case in which (}l and (}2 are linear and equal. With the above simplication the first moment (mean field) approximation leads to: Stochastic Dynamics of Three-State Neural Networks 275 where -:t «ai» = a«ai» - Wi( «ri» + ~qi») o j3_ - Ot «ri» = -a«ai» + «ri» + Wi«ri», o j3 _ - Ot «qi» = - «ri» + Wi«qi», 1 N WI = iiL wlk«ak», k=l (22) (23) (24) (25) We also obtain the second moment approximation as: o 1 N - Ot «ai» = a«ai» - ff~ Wij( «qiaj» + «riaj»), (26) )=1 o 1 N - Ot «ri» = -a«ai» + j3«ri» + ff~ Wij«riaj», (27) )=1 o 1 N - Ot «qi» = -j3«ri» + ff~ Wij«qiaj», (28) )=1 -:t «aiaj» = 2a«aiaj» - Wij( «riaj» + «qiaj») -Wji( «airj» + «aiqj»), (29) o _ - Ot «airj» = -a( «aiaj» - «airj») + j3«airj» + Wji«airj» -Wij(<<7'i7'j» + «qi7'j»), (31) where (32) We note that the first moment dynamics obtained via the first approximation differs from that obtained from the second moment approximation, In the next section, we briefly examine this difference by comparing these approximations with Monte Carlo simulations. 276 Toru Ohira, Jack D. Cowan 4 COMPARISON WITH SIMULATIONS In this section, we compare first and second moment approximations with Monte Carlo simulation of a one dimensional ring of three-state neurons. This was studied in a previous publication (Ohira and Cowan, 1993) for two-state neurons. As shown there, each three-state neuron in the ring interacts with its two neighbors. More precisely, the Liouville operator is N N L = a 2:)lai}(ail-/ri}(a;!) + f3 L(/ri}{ril-/qi}(ril) i=1 i=1 1 N +2W2 L(/ri}(ri/-lai}(r;I)(/ai+l}(ai+d + /ai-l}{ai-d) i=1 1 N +2W1 L(/qi}{q;/ - /ai}(qil)(/ai+l}{ai+d + /ai-J}(ai-d) i=1 (33) We now define the dynamical variables of interest as follows: 1 1 1 Xa = N L «aj», Xr = N L «ri», Xq = N L «qi», (34) i=1 ;=1 ;=1 1 N 1 N 1 N 1]aa = N L «aiai+l», 1]rr = N L «riri+l», 1]ar = N L «airi+l». (35) ;=1 i=1 i=1 Then, for this network, the first moment approximation is given by 8 - 8tXa ax - W2XaXr - WIXqXa, 8 - at Xr = -ax - f3Xr + W2XqXa, Xq = 1 - Xa - Xr. The second moment approximation is given by 8 - at Xa = 8 - 8t Xr = {) - 8t1]aa 8 - 8t1]ar = 2a1]aa - W21]ar(Xa + 1) - Wl(Xa + 1)(Xa -1]ar -1]aa), 1 -a('1aa -1]ar) - f31]ar + 2W2'1ar(Xa + 1), 1 +21]rrXa + WIXa(Xr -1]rr -1]ar)' (36) (37) Stochastic Dynamics of Three-State Neural Networks 277 Monte Carlo simulations of a ring of 10000 neurons were performed and compared with the first and second moment approximation predictions. We fixed the following parameters: a = 1.0, f3 = 0.2, Wl = 0.01· WO, W2 = 0.6· Wo (38) x. I.' x",. • .... -.................. __ .............. - ..... .. " --------_ ....... --.. _--x, •.• x" .• ., ... -.. --..... --.... -_ .. ------_. .. ........... _----_ ........ _---X.'" (A) (8) Figure 2: Comparison of Monte Carlo simulations (dots) with the first moment (dashed line) and the second moment (solid line) approximations for the three state case with the fraction of total active and refractory state variables Xa (A) and Xr (B). Each graph is labeled by the values of wo/a. We varied Wo and sampled the numerical dynamics of these parameters. Some comparisons are shown in Figure 2 for the time dependence of the total number of active and refractory state variables. We dearly see the improvement of the second over the first moment level approximation. More simulations with different parameter ranges remain to be explored. 5 CONCLUSION We have introduced here a neural network master equation using the outer-product representation. In this representation, the extension from two to three-state neurons is transparent. We have taken advantage of this natural extension to analyse three-state networks. Even though the calculations involved are more intricate, we 278 Torn Ohira, Jack D. Cowan have obtained results indicating that the second moment level approximation is significantly more accurate than the first moment level approximation. We also note that as in the two-state case, the first moment level approximation produces more activation than the simulation. FUrther analytical and theoretical investigations are needed to fully uncover the dynamics of three-state networks described by the master equation introduced above. Acknowledgements This work was supported in part by the Robert R. McCOlmick fellowship at the University of Chicago, and in part by grant No. N0014-89-J-I099 from the US Department of the Navy, Office of Naval Research. References Cowan JD (1991) Stochastic neurodynamics in Advances in Neural Information Processing Systems (D. S. Touretzky, R. P. Lippman, J. E. Moody, ed.), vol. 3, Morgan Kaufmann Publishers, San Mateo Doi M (1976a) Second quantization representation for classical many-particle system. J. Phys. A: Math Gen. 9:1465-1477. Doi M (1976b) Stochastic theory of diffusion-controlled reactions. J. Phys. A: Math. Gen. 9:1479. Grassberger P, Scheunert M (1980) Fock-space methods for identical classical objects. Fortschritte der Physik 28:547 Meunier C, Hansel D, Verga A (1989) Information processing in three-state neural networks. J. Stat. Phys. 55:859 Ohira T, Cowan JD (1993) Master-equation approach to stochastic neurodynamics. Phys. Rev. E 48:2259 Ohira T, Cowan JD (1994) Feynman Diagrams for Stochastic Neurodynamics. In Proceedings of Fifth Australian Conference of Neural Networks, pp218-221 Sakurai JJ (1985) Modern Quantum Mechanics. Benjamin/Cummings, Menlo Park	1994	56
950	On the Computational Utility of Consciousness Donald W. Mathis and Michael C. Mozer mathis@cs.colorado.edu, mozer@cs.colorado.edu Department of Computer Science and Institute of Cognitive Science University of Colorado, Boulder Boulder, CO 80309-0430 Abstract We propose a computational framework for understanding and modeling human consciousness. This framework integrates many existing theoretical perspectives, yet is sufficiently concrete to allow simulation experiments. We do not attempt to explain qualia (subjective experience), but instead ask what differences exist within the cognitive information processing system when a person is conscious of mentally-represented information versus when that information is unconscious. The central idea we explore is that the contents of consciousness correspond to temporally persistent states in a network of computational modules. Three simulations are described illustrating that the behavior of persistent states in the models corresponds roughly to the behavior of conscious states people experience when performing similar tasks. Our simulations show that periodic settling to persistent (i.e., conscious) states improves performance by cleaning up inaccuracies and noise, forcing decisions, and helping keep the system on track toward a solution. 1 INTRODUCTION We propose a computational framework for understanding and modeling consciousness. Though our ultimate goal is to explain psychological and brain imaging data with our theory, and to make testable predictions, here we simply present the framework in the context of previous experimental and theoretical work, and argue that 12 Donald Mathis, Michael C. Mozer it is sensible from a computational perspective. We do not attempt to explain qualia-subjective experience and feelings. It is not clear that qualia are amenable to scientific investigation. Rather, our aim is to understand the mechanisms underlying awareness, and their role in cognition. We address three key questions: • What are the preconditions for a mental representation to reach consciousness? • What are the computational consequences of a representation reaching consciousness? Does a conscious state affect processing differently than an unconscious state? • What is the computational utility of consciousness? That is, what is the computational role of the mechanism(s) underlying consciousness? 2 THEORETICAL FRAMEWORK Modular Cognitive Architecture. We propose that the human cognitive architecture consists of a set of functionally specialized computational modules (e.g., Fodor, 1983). We imagine the modules to be organized at a somewhat coarse level and to implement processes such as visual object recognition, visual word-form recognition, auditory word and sound recognition, computation of spatial relationships, activation of semantic representations of words, sentences, and visual objects, construction of motor plans, etc. Cognitive behaviors require the coordination of many modules. For example, functional brain imaging studies indicate that there are several brain areas used for different subtasks during cognitive tasks such as word recognition (Posner & Carr, 1992). Modules Have Mapping And Cleanup Processes. We propose that modules perform an associative memory function in their domain, and operate via a two-stage process: a fast, essentially feedforward input-output mapping1 followed by a slower relaxation search (Figure 1). The computational justification for this two stage process is as follows. We assume that, in general, the output space of a module can represent a large number of states relative to the number of states that are meaningful or well formed-i.e., states that are interpretable by other modules or (for output modules) that correspond to sensible motor primitives. If we know which representations are well-formed, we can tolerate an inaccurate feedforward mapping, and "clean up" noise in the output by constraining it to be one of the well-formed states. This is the purpose of the relaxation step: to clean up the output of the feedforward step, resulting in a well-formed state. The cleanup process knows nothing about which output state is the best response to the input; it acts solely to enforce well-formedness. Similar architectures have been used recently to model various neuropsychological data (Hinton & Shallice, 1991; Mozer & Behrmann, 1990; Plaut & Shallice, 1993). The empirical motivation for identifying consciousness with the results of relaxation search comes from studies indicating that the contents of consciousness tend to be coherent, or well-formed (e.g., Baars, 1988; Crick, 1994; Damasio, 1989). Persistent States Enter Consciousness. In our model, module outputs enter consciousness if they persist for a sufficiently long time. What counts as long enough lWe do not propose that this process is feedforward at the neural level. Rather, we mean that any iterative refinement of the output over time is unimportant and irrelevant. On the Computational Utility of Consciousness 13 relaxation search feedforward mapping .00 •• Figure 1: Modules consist of two components. is not yet determined, but in order to model specific psychological data, we will be required to make this issue precise. At that time a specific commitment will need to be made, and this commitment must be maintained when modeling further data. An important property of our model is that there is no hierarchy of modules with respect to awareness, in contrast to several existing theories that propose that access to some particular module (or neural processing area) is required for consciousness (e.g., Baars, 1988). Rather, information in any module reaches awareness simply by persisting long enough. The persistence hypothesis is consistent with the theoretical perspectives of Smolensky (1988), Rumelhart et al (1986), Damasio (1989), Crick and Koch (1990), and others. 2.1 WHEN ARE MENTAL STATES CONSCIOUS? In our framework, the output of any module will enter consciousness if it persists in time. The persistence of an output state of a module is assured if: (1) it is a point attractor of the relaxation search (i.e., a well-formed state), and (2) the inputs to the module are relatively constant, i.e., they continue to be mapped into the same attractor basin. While our framework appears to make strong claims about the necessary and sufficient conditions for consciousness, without an exact specification of the modules forming the cognitive architecture, it is lacking as a rigorous, testable theory. A complete theory will require not only a specification of the modules, but will also have to avoid arbitrariness in claiming that certain cognitive operations or brain regions are modules while other are not. Ultimately, one must identify the neurophysiological and neuroanatomical properties of the brain that determine the module boundaries (see Crick, 1994, for a promising step in this regard). 3 COMPUTATIONAL UTILITY OF CONSCIOUSNESS For the moment, suppose that our framework provides a sensible account of awareness phenomena (demonstrating this is the goal of ongoing work.) If one accepts this, and hence the notion that a cleanup process and the resulting persistent states are required for awareness, questions about the role of cleanup in the model become quite interesting because they are equivalent to questions about the role of the mechanism underlying awareness in cognition. One question one might ask is whether there is computational utility to achieving conscious states. That is, does a system that achieves persistent states perform better than a system that does not? 14 Donald Mathis, Michael C. Mozer Does a system that encourages settling to well-formed states perform better than a system that does not? We now show that the answer to this question is yes. 3.1 ADDITION SIMULATION To examine the utility of cleanup, we trained a module to perform a simple multistep cognitive task: adding a pair of two-digit numbers in three steps.2 We tested the system with and without cleanup and compared the generalization performance. The network architecture (Figure 2) consists of a single module. The inputs consist of the problem statement and the current partial solution-state. The output is an updated solution-state. The module's output feeds back into its input. The problem statement is represented by four pools of units, one for each digit of each operand, where each pool uses a local encoding of digits. Partial solution states are represented by five pools, one for each of the three result digits and one for each of the two carry digits. Projection Input-ot,rtpu mapping t ( ~ ~ jU (copy) ~ if) 1 result 11 Iresult21 I result 31 leany tI leany21 + I hidden units I .,/ "~[~E::~:::][:iIl[iI] Iresult111resutt 211result311eany 111eany 21 " Figure 2: Network architecture for the addition task .......... (copy) Each addition problem was decomposed into three steps (Figure 3), each describing a transformation from one partial solution state to the next, and the mapping net was trained perform each transformation individually. ? ? 48 + 62 ??? step 1 --.. ? 1 48 + 62 ??O step 2 --.. 1 1 48 + 62 ?10 step 3 --.. 1 1 48 + 62 11 0 Figure 3: The sequence of steps in an example addition problem Step 1 Given the problem statement, activate the rightmost result digit and rightmost carry digit (comprising the first partial solution). 2 Of course, we don't believe that there is a brain module dedicated to addition problems. This choice was made because addition is an intuitive example of a multistep task. On the Computational Utility of Consciousness 15 Step 2 Given the first partial solution, activate the next result and carry digits (second partial solution). Step 3 Given the second partial solution, activate the leftmost result digit (final solution). The set of well-formed states in this domain consists of all possible combinations of digits and "don't knows" among the pools ("don't knows" are denoted by question marks in Figure 3). Local representations of digits are used within each pool, and "don't knows" are represented by the state in which no unit is active. Thus, the set of well-formed states are those in which either one or no units are active in each pool. To make these states attractors of the cleanup net, the connections were hand-wired such that each pool was a winner-take-all pool with an additional attractor at the zero state. To run the net, a problem statement pattern is clamped on the input units, and the net is allowed to update for 200 iterations. Unit activities were updated using an incremental rule approximating continuous dynamics: ai(t) = TI(L: Wijaj(t - 1)) + (1 - T)ai(t - 1) j where ai(t) is the activity of unit i at time t, T is a time constant in the interval [0,1]' and 10 is the usual sigmoid squashing function. Figure 4 shows the average generalization performance of networks run with and without cleanup, as a function of training set size. Note that, in principle, it is not necessary for the system to have a cleanup process to learn the training set perfectly, or to generalize perfectly. Thus, it is not simply the case that no solutions exist without cleanup. The generalization results were that for any size training set, percent correct on the generalization set is always better with cleanup than without. This indicates that although the mapping network often generalizes incorrectly, the output pattern often falls within the correct attractor basin. This is especially beneficial in multistep tasks because cleanup can correct the inaccuracies introduced by the mapping network, preventing the system from gradually diverging from the desired trajectory. Projection = No projection training set size (% of all problems) Figure 4: Cleanup improves generalization performance. Figure 5 shows an example run of a trained network. There is one curve for each of the five result and carry pools, showing the degree of "activity" of the ultimate target pattern, t, for that pool as a function of time. Activity is defined to be e-lit-aIl2 where a is the current activity pattern and t is the target. The network /6 Donald Mathis, Michael C. Mozer solves the problem by passing though the correct sequence of intermediate states, each of which are temporarily persistent. This resembles the sequence of conscious states a person might experience while performing this task; each step of the problem is performed by an unconscious process, and the results of each of step appear in conscious awareness. 1.0r----=====~-----_""::--_~""-:-"""""'----------, Activation of target pattern .8 in each pool .6 ,:' :'" : .' .' " " " ... :' ,: ,'. result digit I, -- carry digit I result digit 2, ----- carrydigit2 .......... result digit 3 -------- -----".:!/ ~~~~IO~~~~2~O~--~3~O---~4~O---~50~--~ TIme Figure 5: Network solving the addition task in three steps 3.2 CHOICE POINT SIMULATION In many ordinary situations, people are required to make decisions, e.g., drive straight through an intersection or turn left, order macaroni or a sandwich for lunch. At these choice points, any of the alternative actions are reasonable a priori. Contextual information determines which action is correct, e.g., whether you are trying to drive to work or to the supermarket. Conscious decision making often occurs at these choice points, except when the task is overlearned (Mandler, 1975). We modeled a simple form of a choice point situation. We trained a module to output sequences of states, e.g., ABCD or EFGH, where states were represented by unique activity patterns over a set of units. If the sequences shared no elements, then presenting the first element of any sequence would be sufficient to regenerate the sequence. But when sequences overlap, choice points are created. For example, with the sequences ABCD and AEFG, state A can be followed by either B or E. We show that cleanup allows the module to make a decision and complete one of the two sequences. Figure 6 shows the operation of the module with and without cleanup following presentation of an A after training on the sequence pair ABCD and AEFG. There is one curve for each state, showing the activation of that state (defined as before), as a function of time. When the network is run with cleanup, although both states Band E are initially partially activated, the cleanup process maps this ill-formed state to state B, and the network then correctly completes the sequence ABCD. Without cleanup, the initial activation of states Band E causes a blending of the two sequences ABCD and AEFG and the state degenerates.3 Although the arithmetic and choice point tasks seem simple in part because we predefined the set of well-formed states. However, because the architecture segre3In this simulation, we are not modeling the role of context in helping to select one sequence or another; we are simply assuming that either sequence is valid in the current context. The nature of the model does not change when we consider context. Assuming that the domain is not highly overlearned, the context will not strongly evoke one alternative action or the other in the feedforward mapping, leading to partial activation of multiple states, and the cleanup process will be needed to force a decision. On the Computational Utility of Consciollsness Activity of state 0 .• o. stateB __ state C - - - -state D ........ state E Figure 6: Decision-point task with and without cleanup 17 gates knowledge of well-formed ness from knowledge of how to solve problems in the domain, well-formedness could be learned simultaneously with, or prior to learning the task. One could imagine training the cleanup network to autoassociate states it observes in the domain before or during training using an unsupervised or self-supervised procedure. 4 COMPUTATIONAL CONSEQUENCES OF PERSISTENT STATES In a network of modules, a persistent well-formed state in one module exerts larger influences on the state of other modules than do transient or ill-formed states. As a result the dynamics of the system tends to be dominated by well-formed persistent states. We show this in a final simulation. The network consisted of two modules, A and B, connected in a simple feedforward cascade. Each module's cleanup net was trained to have ten attractors, locally represented in a winner-take-all pool of units. The mapping network of module B was trained to map the attractors of module A to attractors in B in a one-to-one fashion. Thus, state a1 in module A is mapped to {31 in module A, a2 to {32, etc. Module B was initialized to a well-formed state {31, and the output state of module A was varied, creating three conditions. In the persistent well-formed condition, module A was clamped in the well-formed state a2 for 50 time steps. In the transient well-formed condition, module A was clamped in state a2 for only 30 time steps. And in the ill-formed condition, module A was clamped in an ill-formed state in which two states, a2 and a3, were both partially active. Figure 7 shows the subsequent activation of state {32 in module B as a function of time. Module B undergoes a transition from state {31 to state {32 only in the persistent well-formed condition. This indicates that the conjunction of well-formedness and persistence is required to effect a transition from one state to another. 5 CONCLUSIONS Our computational framework and simulation results suggest the following answers to our three key questions: 18 Donald Mathis, Michael C. Mozer Activity of state ~2 in module B well-formed, persistent state <X2 in module A 0' well-formed, transient state <X2 in module A ill-formed, persistent state in module A Figure 7: Well-formeclness and persistence are both required for attractor transitions. • In order to reach consciousness, the output of a module must be both persistent and semantically well-formed, and must not initiate an overlearned process. • The computational consequences of conscious (persistent) representations include exerting larger influences on the cognitive system, resulting in increased ability to drive response processes such as verbal report. • The computational utility of consciousness in our model lies in the ability of cleanup to "focus" cognition, by keeping the system close to states which are semantically meaningful. Because the system has learned to process such states, performance is improved. References Baars, B. J. (1988) A Cognitive Theory of Con8ciou&ne8ll, Cambridge University Press. Crick, F. (1994) The astonishing hypothesis: The scientific search for the soul. Scribner. Crick, F., & Koch, C. (1990) Towards a neurobiological theory of consciousness. Sem. Neuro., 2: 263-275 Damasio, A. (1989) The brain binds entities and events by multiregional activation from convergence zones. Neural Computation, 1, 123-132 Fodor, J. A. (1983) The modularity of mind: An e8llay on faculty p8ychology. Cambridge, MA: MIT Press. Hinton, G. E., & Shallice, T. (1991) Lesioning an attractor network: Investigations of acquired dyslexia., P8ych. Rev., 98: 74-95 Mandler, G. (1975) Consciousness: Respectable, useful and probably necessary. In Information Processing and Cognition, The Loyola Symposium, R. Solso (Ed.). Erlbaum. Mozer, M. C., & Behrmann, M. (1990). On the interaction of spatial attention and lexical knowledge: A connectionist account of neglect dyslexia. Cognitive Neuro8Cience, 2, 96-123. Plaut, D. C., & Shallice, T. (1993) Perseverative and semantic influences on visual object naming errors in optic aphasia: A connectionist account. J. Cog. Neuro.,5(1): 89-117 Posner, M. 1., & Carr, T. (1992) Lexical access and the brain: Anatomical constraints on cognitive models of word recognition. American Journal of P8ychology, 105(1): 1-26 Rumelhart, D. E., Smolensky, P., McClelland, J. L., & Hinton, G. E. (1986) Schemata and sequential thought in PDP models. In J. L. McClelland & D. E. Rumelhart (Eds.), Parallel Di8tributed Proceuing, Vol. 2. Cambridge, MA: MIT Press. Smolensky, P. (1988) On the proper treatment of connectionism. Brain Behav. Sci., 11: 1-74	1994	57
951	Ocular Dominance and Patterned Lateral Connections in a Self-Organizing Model of the Primary Visual Cortex Joseph Sirosh and Risto Miikkulainen Department of Computer Sciences University of Texas at Austin, Austin, 'IX 78712 email: sirosh.risto~cs.utexas.edu Abstract A neural network model for the self-organization of ocular dominance and lateral connections from binocular input is presented. The self-organizing process results in a network where (1) afferent weights of each neuron organize into smooth hill-shaped receptive fields primarily on one of the retinas, (2) neurons with common eye preference form connected, intertwined patches, and (3) lateral connections primarily link regions of the same eye preference. Similar self-organization of cortical structures has been observed experimentally in strabismic kittens. The model shows how patterned lateral connections in the cortex may develop based on correlated activity and explains why lateral connection patterns follow receptive field properties such as ocular dominance. 1 Introduction Lateral connections in the primary visual cortex have a patterned structure that closely matches the response properties of cortical cells (Gilbert and Wiesel 1989; Malach et al.1993). For example, in the normal visual cortex, long-range lateral connections link areas with similar orientation preference (Gilbert and Wiesel 1989). Like cortical response properties, the connectivity pattern is highly plastic in early development and can be altered by experience (Katz and Callaway 1992). In a cat that is brought up squint-eyed from birth, the lateral connections link areas with the same ocular dominance instead of orientation (Lowel and Singer 1992). Such patterned lateral connections develop at the same time as the orientation selectivity and ocular dominance itself (Burkhalter et al.1993; Katz and Callaway 1992). Together, 110 Joseph Sirosh, Risto Miikkulainen these observations suggest that the same experience-dependent process drives the development of both cortical response properties and lateral connectivity. Several computational models have been built to demonstrate how orientation preference, ocular dominance, and retinotopy can emerge from simple self-organizing processes (e.g. Goodhill1993; Miller 1994; Obermayer et al.1992; von der Malsburg 1973). These models assume that the neuronal response properties are primarily determined by the afferent connections, and concentrate only on the self-organization of the afferent synapses to the cortex. Lateral interactions between neurons are abstracted into simple mathematical functions (e.g. Gaussians) and assumed to be uniform throughout the network; lateral connectivity is not explicitly taken into account. Such models do not explicitly replicate the activity dynamics of the visual cortex, and therefore can make only limited predictions about cortical function. We have previously shown how Kohonen's self-organizing feature maps (Kohonen 1982) can be generalized to include self-organizing lateral connections and recurrent activity dynamics (the Laterally Interconnected Synergetically Self-Organizing Map (LISSOM); Sirosh and Miikkulainen 1993, 1994a), and how the algorithm can model the development of ocular dominance columns and patterned lateral connectivity with abstractions of visual input. LISSOM is a low-dimensional abstraction of cortical self-organizing processes and models a small region of the cortex where all neurons receive the same input vector. This paper shows how realistic, high-dimensional receptive fields develop as part of the self-organization, and scales up the LISSOM approach to large areas of the cortex where different parts of the cortical network receive inputs from different parts of the receptor surface. The new model shows how (1) afferent receptive fields and ocular dominance columns develop from simple retinal images, (2) input correlations affect the wavelength of the ocular dominance columns and (3) lateral connections self-organize cooperatively and simultaneously with ocular dominance properties. The model suggests new computational roles for lateral connections in the cortex, and suggests that the visual cortex maybe maintained in a continuously adapting equilibrium with the visual input by co adapting lateral and afferent connections. 2 The LISSOM Model of Receptive Fields and Ocular Dominance The LISSOM network is a sheet of interconnected neurons (figure 1). Through afferent connections, each neuron receives input from two "retinas". In addition, each neuron has reciprocal excitatory and inhibitory lateral connections with other neurons. Lateral excitatory connections are short-range, connecting only close neighbors. Lateral inhibitory connections run for long distances, and may even implement full connectivity between neurons in the network. Neurons receive afferent connections from broad overlapping patches on the retina called anatomical receptive fields, or RFs. The N x N network is projected on to each retina of R x R receptors, and each neuron is connected to receptors in a square area of side s around the projections. Thus, neurons receive afferents from corresponding regions of each retina. Depending on the location of the projection, the number of afferents to a neuron from each retina could vary from ts x ~s (at the comers) to s x s (at the center). The external and lateral weights are organized through an unsupervised learning process. At each training step, neurons start out with zero activity. The initial response TJij of neuron (i, j) Ocular Dominance and Patterned Lateral Connections 111 Loft _.. fllgIIl Roll .. Figure 1: The Receptive-Field LISSOM architecture. The afferent and lateral connectionsof a single neuron in the liSSOM network are shown. All connection weights are positive. is based on the scalar product TJij = (T (L eabJJij,ab + L eCdJJij,Cd) , a,b c,d (1) where eab and ecd are the activations of retinal receptors (a, b) and (c, d) within the receptive fields of the neuron in each retina, JJij,ab and JJij,cd are the corresponding afferent weights, and (T is a piecewise linear approximation of the familiar sigmoid activation function. The response evolves over time through lateral interaction. At each time step, the neuron combines the above afferent activation I:: eJJ with lateral excitation and inhibition: TJij(t) = (T (L eJJ + "Ie L Eij,kITJkl(t - 1) - "Ii L Iij,klTJkl(t - 1)) , (2) k,1 k,1 where Eij,kl is the excitatory lateral connection weight on the connection from neuron (k, l) to neuron (i, j), Iij,kl is the inhibitory connection weight, and TJkl (t 1) is the activity of neuron (k, I) during the previous time step. The constants "Ie and "Ii determine the relative strengths of excitatory and inhibitory lateral interactions. The activity pattern starts out diffuse and spread over a substantial part of the map, and converges iteratively into stable focused patches of activity, or activity bubbles. After the-activity has settled, typically in a few iterations of equation 2, the connection weights of each neuron are modified. Both afferent and lateral weights adapt according to the same mechanism: the Hebb rule, normalized so that the sum of the weights is constant: ( r ) _ Wij,mn(t) + CtTJijXmn Wij,mn t + vt '"" ( ) 1 ' wmn [Wij ,mn t + CtTJijXmn (3) where TJij stands for the activity of neuron (i, j) in the final activity bubble, Wij,mn is the afferent or lateral connection weight (JJ, E or I), Ct is the learning rate for each type of connection (Ct a for afferent weights, Ct E for excitatory, and Ct I for inhibitory) and X mn is the presynaptic activity (e for afferent, TJ for lateral). 112 Joseph Sirosh, Risto Miikkulainen " (a) Random Initial Weights (b) Monocular RF (c) Binocular RF Figure 2: Self-organization of the afferent input weights into receptive fields. The afferent weights of a neuron at position (42,39) in a 60 x 60 network are shown before (a) and after self-organization (b). This particular neuron becomes monocular with strong connections to the right eye, and weak connections to the left. A neuron at position (38, 23) becomes binocular with appoximately equal weights to both eyes (c). Both excitatory and inhibitory lateral connections follow the same Hebbian learning process and strengthen by correlated activity. The short-range excitation keeps the activity of neighboring neurons correlated, and as self-organization progresses, excitation and inhibition strengthen in the vicinity of each neuron. At longer distances, very few neurons have correlated activity and therefore most long-range connections become weak. Such weak connections are eliminated, and through weight normalization, inhibition concentrates in a closer neighborhood of each neuron. As a result, activity bubbles become more focused and local, weights change in smaller neighborhoods, and receptive fields become better tuned to local areas of each retina. The input to the model consists of gaussian spots of "light" on each retina: t _ ((x - xd 2 + (y - Yi)2) <"x,y exp u 2 (4) where ex,y is the activation of receptor (x, V), u 2 is a constant determining the width of the spot, and (Xi,Yi): 0 ~ xi, Yi < R its center. At each input presentation, one spot is randomly placed at (Xi ,Yi) in the left retina, and a second spot within a radius of p x RN of (Xi, yd in the right retina. The parameter p E [0, 1] specifies the spatial correlations between spots in the two retinas, and can be adjusted to simulate different degrees of correlations between images in the two eyes. 3 Simulation results To see how correlation between the input from the two eyes affects the columnar structures that develop, several simulations were run with different values of p. The afferent weights of all neurons were initially random (as shown in figure 2a), with the total strength to both eyes being equal. Figures 2b,c show the final afferent receptive fields of two typical neurons in a simulation with p = 1. In this case, the inputs were uncorrelated, simulating perfect strabismus. In the early stages of such simulation, some of the neurons randomly develop a preference for one eye or the other. Nearby neurons will tend to share the same preference because lateral Ocular Dominance and Patterned Lateral Connections 113 (a) Connections of a Monocular Neuron (b) Connections of a Binocular Neuron Figure 3: Ocular dominance and lateral connection patterns. The ocular dominance of a neuron is measured as the difference in total afferent synaptic weight from each eye to the neuron. Each neuron is labeled with a grey-scale value (black ~ white) that represents continuously changing eye preference from exclusive left through binocular to exclusive right. Small white dots indicate the lateral input connections to the neuron marked with a big white dot. (a) The surviving lateral connections of a left monocular neuron predominantly link areas of the same ocular dominance. (b) The lateral connections of a binocular neuron come from both eye regions. excitation keeps neural activity partially correlated over short distances. As self-organization progresses, such preferences are amplified, and groups of neurons develop strong weights to one eye. Figure 2b shows the afferent weights of a typical monocular neuron. The extent of activity correlations on the network detennines the size of the monocular neuronal groups. Farther on the map, where the activations are anticorrelated due to lateral inhibition, neurons will develop eye preferences to the opposite eye. As a result, alternating ocular dominance patches develop over the map, as shown in figure 3.1 In areas between ocular dominance patches, neurons will develop approximately equal strengths to both eyes and become binocular, like the one shown in figure 2e. The width and number of ocular dominance columns in the network (and therefore, the wavelength of ocular dominance) depends on the input correlations (figure 4). When inputs in the two eyes become more correlated (p < 1), the activations produced by the two inputs in the network overlap closely and activity correlations become shorter range. By Hebbian adaptation, lateral inhibition concentrates in the neighborhood of each neuron, and the distance at which activations becomes anticorrelated decreases. Therefore, smaller monocular patches develop, and the ocular dominance wavelength decreases. Similar dependence was very recently observed in the cat primary visual cortex (LoweI1994). The LISSOM model demonstrates that the adapting lateral interactions and recurrent activity dynamics regulate the wavelength, and suggests how these processes help the cortex develop feature detectors at a scale 1 For a thorough treatment of the mathematical principles underlying the development of ocular dominance columns, see (GoodhillI993; Miller et al.1989; von der Malsburg and Singer 1988). 114 Joseph Sirosh, Risto Miikkulainen - 0 -0 (a) Strabismic case (b) Normal case Figure 4: Ocular dominance wavelength in strabismic and normal models. In the strabismic case, there are no between-eye correlations (p = 1), and broad ocular dominance columns are produced (a). With normal, partial between-eye correlations (p = 0.45 in this example), narrower stripes are formed (b). As a result, there are more ocular dominance columns in the normal case and the ocular dominance wavelength is smaller. that matches the input correlations. As eye preferences develop, left or right eye input tends to cause activity only in the left or right ocular dominance patches. Activity patterns in areas of the network with the same ocular dominance tend to be highly correlated because they are caused by the same input spot. Therefore, the long-range lateral connections between similar eye preference areas become stronger, and those between opposite areas weaker. After the weak lateral connections are eliminated, the initially wide-ranging connections are pruned, and eventually only connect areas of similar ocular dominance as shown in figure 3. Binocular neurons between ocular dominance patches will see some correlated activity in both the neigbboring areas, and maintain connections to both ocular dominance columns (figure 3b). The lateral connection patterns shown above closely match observations in the primary visual cortex. Lowel and Singer (1992) observed that when between-eye correlations are abolished in kittens by surgically induced strabismus, long-range lateral connections primarily link areas of the same ocular dominance. However, binocular neurons, located between ocular dominance columns, retained connections to both eye regions. The receptive field model confinns that such patterned lateral connections develop based on correlated neuronal activity, and demonstrates that they can self-organize simultaneously with ocular dominance columns. The model also predicts that the long-range connections have an inhibitory function. 4 Discussion In LISSOM, evolving lateral interactions and dynamic activity patterns are explicitly modeled. Therefore, LISSOM has several novel properties that set it apart from other selforganizing models of the cortex. Previous models (e.g. Goodhill1993; Milleret al.1989; Obermayer et al.1992; von der Malsburg 1973) have concentrated only on forming ordered topographic maps where clusters of adjacent neurons assume similar response properties such as ocular dominance or orientation preference. The lateral connections in LISSOM, in addition, adapt to encode correlations beOcular Dominance and Patterned Lateral Connections 115 tween the responses.2 This property can be potentially very useful in models of cortical function. While afferent connections learn to detect the significant features in the input space (such as ocularity or orientation), the lateral connections can learn correlations between these features (such as Gestalt principles), and thereby form a basis for feature grouping. As an illustration, consider a single spot of light presented to the left eye. The spot causes disjoint activity patterns in the left-eye-dominant patches. How can these multiple activity patterns be recognized as representing the same spatially coherent entity? As proposed by Singer et al. (1990), the long-range lateral connections between similar ocular dominance columns could synchronize cortical activity, and form a coherently firing assembly of neurons. The spatial coherence of the spot will then be represented by temporal coherence of neural activity. LISSOM can be potentially extended to model such feature binding. Even after the network has self-organized, the lateral and afferent connections remain plastic and in a continuously-adapting dynamic equilibrium with the input. Therefore, the receptive field properties of neurons can dynamically readapt when the activity correlations in the network are forced to change. For example, when a small area of the cortex is set inactive (or lesioned), the sharply-tuned afferent weight profiles of the neurons surrounding that region expand in size, and neurons begin to respond to the stimuli that previously activated only the lesioned area (Sirosh and Miikkulainen 1994b, 1994c). This expansion of receptive fields is reversible, and when the lesion is repaired, neurons return to their original tuning. Similar changes occur in response to retinal lesions as well. Such dynamic expansions of receptive fields have been observed in the visual cortex (Pettet and Gilbert 1992). The LISSOM model demonstrates that such plasticity is a consequence of the same self-organizing mechanisms that drive the development of cortical maps. 5 Conclusion The LISSOM model shows how a single local and unsupervised self-organizing process can be responsible for the development of both afferent and lateral connection structures in the primary visual cortex. It suggests that this same developmental mechanism also encodes higherorder visual information such as feature correlations into the lateral connections. The model forms a framework for future computational study of cortical reorganization and plasticity, as well as dynamic perceptual processes such as feature grouping and binding. Acknowledgments This research was supported in part by National Science Foundation under grant #IRI9309273. Computer time for the simulations was provided by the Pittsburgh Supercomputing Center under grants IRI930005P and TRA940029P. References Burkhalter, A., Bernardo, K. L., and Charles, V. (1993). Development of local circuits in human visual cortex. Journalo/Neuroscience, 13:1916-1931. Gilbert, C. D., and Wiesel, T. N. (1989). Columnar specificity of intrinsic horizontal and corticocortical connections in cat visual cortex. Journal 0/ Neuroscience, 9:2432-2442. 2Tbe idea was conceived by von der Malsburg and Singer (1988), but not modeled. 116 Joseph Sirosh, Risto Miikkulainen Goodhill, G. (1993). Topography and ocular dominance: a model exploring positive correlations. Biological Cybernetics, 69:109-118. Katz, L. C., and Callaway, E. M. (1992). Development of local circuits in mammalian visual cortex. Annual Review o/Neuroscience, 15:31-56. Kohonen, T. (1982). Self-organized formation of topologically correct feature maps. Biological Cybernetics, 43:59-69. Lowel, S. (1994). Ocular dominance column development: Strabismus changes the spacing of adjacent columns in cat visual cortex. Journal 0/ Neuroscience, 14(12):7451-7468. Lowel, S., and Singer, W. (1992). Selection of intrinsic horizontal connections in the visual cortex by correlated neuronal activity. Science, 255:209-212. Malach, R., Amir, Y., Harel, M., and Grinvald, A (1993). 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952	Effects of Noise on Convergence and Generalization in Recurrent Networks Kam Jim Bill G. Horne c. Lee Giles* NEC Research Institute, Inc., 4 Independence Way, Princeton, NJ 08540 {kamjim,horne,giles}~research.nj.nec.com *Also with UMIACS, University of Maryland, College Park, MD 20742 Abstract We introduce and study methods of inserting synaptic noise into dynamically-driven recurrent neural networks and show that applying a controlled amount of noise during training may improve convergence and generalization. In addition, we analyze the effects of each noise parameter (additive vs. multiplicative, cumulative vs. non-cumulative, per time step vs. per string) and predict that best overall performance can be achieved by injecting additive noise at each time step. Extensive simulations on learning the dual parity grammar from temporal strings substantiate these predictions. 1 INTRODUCTION There has been much research in applying noise to neural networks to improve network performance. It has been shown that using noisy hidden nodes during training can result in error-correcting codes which increase the tolerance of feedforward nets to unreliable nodes (Judd and Munro, 1992). Also, randomly disabling hidden nodes during the training phase increases the tolerance of MLP's to node failures (Sequin and Clay, 1990). Bishop showed that training with noisy data is equivalent to Tikhonov Regularization and suggested directly minimizing the regularized error function as a practical alternative (Bishop, 1994). Hanson developed a stochastic version of the delta rule which adapt weight means and standard deviations instead 650 Kam Jim, Bill G. Home, C. Lee Giles of clean weight values (Hanson, 1990). (Mpitsos and Burton, 1992) demonstrated faster learning rates by adding noise to the weight updates and adapting the magnitude of such noise to the output error. Most relevant to this paper, synaptic noise has been applied to MLP's during training to improve fault tolerance and training quality. (Murray and Edwards, 1993) In this paper, we extend these results by introducing several methods of inserting synaptic noise into recurrent networks, and demonstrate that these methods can improve both convergence and generalization. Previous work on improving these two performance measures have focused on ways of simplifying the network and methods of searching the coarse regions of state space before the fine regions. Our work shows that synaptic noise can improve convergence by searching for promising regions of state space, and enhance generalization by enforcing saturated states. 2 NOISE INJECTION IN RECURRENT NETWORKS In this paper, we inject noise into a High Order recurrent network (Giles et al., 1992) consisting of N recurrent state neurons Sj, L non-recurrent input neurons lie, and N 2 L weights Wijle. (For justification ofits use see Section 4.) The recurrent network operation is defined by the state process S:+1 = g(Lj,1e WijleSJlD, where g(.) is a sigmoid discriminant function. During training, an error function is computed as Ep = !f;, where fp = Sb -dp, Sb is the output neuron, and dp is the target output value tor pattern p. Synaptic noise has been simulated on Multi-Layered-Perceptrons by inserting noise to the weights of each layer during training (Murray et al., 1993). Applying this method to recurrent networks is not straightforward because effectively the same weights are propagated forward in time. This can be seen by recalling the BPTT representation of unrolling a recurrent network in time into T layers with identical weights, where T is the length of the input string. In Tables 2 and 3, we introduce the noise injection steps for eight recurrent network noise models representing all combinations of the following noise parameters: additive vs. multiplicative, cumulative vs. non-cumulative, per time step vs. per string. As their name imply, additive and multiplicative noise add or multiply the weights by a small noise term. In cumulative noise, the injected noise is accumulated, while in non-cumulative noise the noise from the current step is removed before more noise is injected on the next step. Per time step and per string noise refer to when the noise is inserted: either at each time step or only once for each training string respectively. Table 1 illustrates noise accumulation examples for all additive models (the multiplicative case is analogous). 3 ANALYSIS ON THE EFFECTS OF SYNAPTIC NOISE The effects of each noise model is analyzed by taking the Taylor expansion on the error function around the noise-free weight set. By truncating this expansion to second and lower order terms, we can interpret the effect of noise as a set of regularization terms applied to the error function. From these terms predictions can be made about the effects on generalization and convergence. A similar analysis was Effects of Noise on Convergence and Generalization in Recurrent Networks 651 performed on MLP's to demonstrate the effects of synaptic noise on fault tolerance and training quality (Murray et. al., 1993). Tables 2 and 3 list the noise injection step and the resulting first and second brder Taylor expansion terms for all noise models. These results are derived by assuming the noise to be zero-mean white with variance (F2 and uncorrelated in time. 3.1 Predictions on Generalization One common cause of bad generalization in recurrent networks is the presence of unsaturated state representations. Typically, a network cannot revisit the exact same point in state space, but tends to wander away from its learned state representation. One approach to alleviate this problem is to encourage state nodes to operate in the saturated regions of the sigmoid. The first order error expansion terms of most noise models considered are capable of encouraging the network to achieve saturated states. This can be shown by applying the chain rule to the partial derivative in the first order expansion terms: (1) where e~ is the net input to state node i at time step t. The partial derivatives g~ favor internal representations such that the effects of perturbations to the net inputs e: are minimized. Multiplicative noise implements a form of weight decay because the error expansion terms include the weight products Wt~ijk or Wt ,ijk Wu,ijk' Although weight decay has been shown to improve generalization on feedforward networks (Krogh and Hertz, 1992) we hypothesize this may not be the case for recurrent networks that are learning to solve FSA problems. Large weights are necessary to saturate the state nodes to the upper and lower limits of the sigmoid discriminant function. Therefore, we predict additive noise will allow better generalization because of its absence of weight decay. Noise models whose first order error term contain the expression a~rb . attSb i , l,k ",'mn. will favor saturated states for those partials whose sign correspond to the sign of a majority of the partials. It will favor unsaturated states, operating in the linear region of the sigmoid, for partials whose sign is the minority. Such sign-dependent enforcement is not optimal. The error terms for cumulative per time step noises sum a product with the expression v a:.rSb . a:Sb ,where v = min(t + 1, U + 1). The effect of cumulative noise ,.". .tlm" increases more rapidly because of v and thus optimal generalization and detrimental noise effects will occur at lower amplitudes than non-cumulative noise. For cumulative per string noise models, the products (t+ l)(u+ 1) and Wt,ijk Wu,lmn in the expansion terms rapidly overwhelm the raw error term. Generalization improvement is not expected for these models. We also reason that all generalization enhancements will be valid only for a range of noise values, above which noise overwhelms the raw error information. 652 Kam Jim, Bill G. Horne, C. Lee Giles 3.2 Predictions on Convergence Synaptic noise can improve convergence by favoring promising weights in the beginning stages of training. This can be demonstrated by examining the second order error expansion term for non-cumulative, multiplicative, per time step noise: When fp is negative, solutions with a negative second order state-weight partial derivative will be de-stabilized. In other words, when the output Sb is too small the network will favor updating in a direction such that the first order partial derivative is increasing. A corresponding relationship can be observed for the case when fp is positive. Thus the second order term of the error function will allow a higher raw error fp to be favored if such an update will place the weights in a more promising area, i.e. a region where weight changes are likely to move Sb in a direction to reduce the raw error. The anticipatory effect of this term is more important in the beginning stages of training where fp is large, and will become insignificant in the finishing stages of training as fp approaches zero. Similar to arguments in Section 3.1, the absence of weight decay will make the learning task easier and improve convergence. From this discussion it can be inferred that additive per time step noise models should yield the best generalization and convergence performance because of their sign-independent favoring of saturated states and the absence of weight decay. Furthermore, convergence and generalization performance is more sensitive to cumulative noise, i.e. optimal performance and detrimental effects will occur at lower amplitudes than in non-cumulative noise. 4 SIMULATION RESULTS In order to perform many experiments in a reasonable amount of computation time, we attempt to learn the simple "hidden-state" dual parity automata from sample strings encoded as temporal sequences. (Dual parity is a 4-state automata that recognizes binary strings containing an even number of ones and zeroes.) We choose a second-order recurrent network since such networks have demonstrated good performance on such problems (Giles et. al., 1992). Thus our experiments consist of 500 simulations for each data point and achieve useful (90%) confidence levels. Experiments are performed with both 3 and 4 state networks, both of which are adequate to learn the automata. The learning rate and momentum are set to 0.5, and the weights are initialized to random values between [-1.0, 1.0]. The data consists of 8191 strings of lengths 0 to 12. The networks are trained on a subset of the training set, called the working set, which gradually increases in size until the entire training set is classified correctly. Strings from the working set are presented in alphabetical order. The training set consists of the first 1023 strings of lengths 0 to 9, while the initial working set consists of 31 strings of lengths 0 to 4. During testing no noise is added to the weights of the network. Effects of Noise on Convergence and Generalization in Recurrent Networks 653 300rT-----,--~--rT----,_----. e· ·· ···· · · b 2150 1200 . • S; i ~150 ~ ~OO . 150 . ~ 2 3 o 0 0_15 ~ ~.15 2 TraininG Nole. Std Oev Traln'nG Nol_. Std Oev Figure 1: Best Convergence/Generalization for Additive and Multiplicative Noises. (a) multiplicative non-cumulative per time step; (b) additive cumulative per time step. 4.1 Convergence and Generalization Performance Simulated performance closely mirror our predictions. Improvements were observed for all noise models except for cumulative per string noises which failed to converge for all runs. Generalization improvement was more emphasized on networks with 4 states, while convergence enhancement was more noticeable on 3-state networks. The simulations show the following results: • Additive noise is better tolerated than multiplicative noise, and achieves better convergence and generalization (Figure 1). • Cumulative noise achieves optimal generalization and convergence at lower amplitudes than non-cumulative noise. Cumulative noise also has a narrower range of beneficial noise, which is defined as the range of noise amplitudes which yields better performance than that of a noiseless network (Figure 2a illustrates this for generalization). • Per time step noise achieves better convergence/generalization and has a wider range of beneficial values than per string noise (Figure 2b). Overall, the best performance is obtained by applying cumulative and noncumulative additive noise at each time step. These results closely match the predictions of section 3.1. The only exceptions are that all multiplicative noise models seem to yield equivalent performance. This discrepancy between prediction and simulation may be due to the detrimental effects of weight decay in multiplicative noise, which can conflict with the advantages of cumulative and per time step noise. 4.2 The Payoff Picture: Generalization vs. Convergence Generalization vs. Convergence results are plotted in Figure 3. Increasing noise amplitudes proceed from the left end-point of each curve to the right end-point. 654 Kam Jim, Bill G. Horne, C. Lee Giles Table 1: Examples: Additive Noise Accumulation. ~i is the noise at time step ti NOISE MODEL t1 per time step non-cumulative W+~l per time step cumulative W+~l per sequence non-cumulative W+~l per sequence cumulative W+~l 1 00 1 2 Training Not_. S'td D_v TIME STEPS t2 t3 ... W+~2 W+~3 ... W +~1 +~2 W +~1 +~2 + ~3 ... W+~l W+~l ... W+2~1 W+3~1 ... 300rT------r---r-~r---~----_. 2150 j200 .5 , 1150 ~ 100 Figure 2: (I) Best Generalization for Cumulative and Non-Cumulative Noises: a) cumulative additive per time step; b) non-cumulative additive per time step. (II) Best Generalization for Per Time Step and Per String Noises: a) non-cumulative per string additive; b) non-cumulative per time step additive. II e e 15.15 5.5 a. .... 5 .) 15 .. 4 .5 ~ ~ 4 ~ .... ~ 3.15 i 3 .5 ~ 'l\J ~ 3 1>5 3 Iii CJ 2.5 2 .5 2 2 c 1.15 1 .5 ~oo 200 300 400 ~OO 200 300 400 Convergence In epocha Convergence 'n epochs Figure 3: Payoff: Mean Generalization vs. Convergence for 4-state (I) and 3state(lI) recurrent network. (I a) Worst i-state - non-cumulative multiplicative per string; (Ib, Ic) Best 4-state - cumulative and non-cumulative additive per time step, respectively; (lIa) Worst 3-state - non-cumulative multiplicative per string; (lib) Best 3-state - cumulative additive per time step. Effects of Noise on Convergence and Generalization in Recurrent Networks 655 Table 2: Noise injection step and error expansion terms for per time step nOIse models. v = min(t + 1, U + 1). W'" is the noise-free weight set. Noise step l.tt order 2nd order Noise step l.tt order 2nd order Noise step 1st order 2nd order Noise step 1st order 2nd order Addlhve Additive t Wt,ijk = wtjl< + L AT,iJI< T-O AddItIve Additive TIT T -~ ws--NON-CUMULATIVE MU bpncahve ClIM~LATIVE M ulhplicati ve t Wt,ijl< = wtjl< II (1 + AThl<) T-O T-1 ~ T ~~ ~2 L ~ "W . . W . _ So 2 P Lt ,.JI< u,.JI< 8W .. 8W .. . . t,.JI< u,.JI< t ,u=O .JI< NON-CUMULATIVE Multlpllcahve T-1 T T 1 2 2: L 850 8S0 -~ W " W .. 2 t,.JI< u,.JI< 8W . 8W . . . . t,.JI< u,.JI< t,u=O .JI< T 1 _2 T 1 2 L 2: crSO -~ ~ W .. W .. 2 P t,.JI< u,.JI< 8W .. W . . t,.Jk u,.JI< t,u=O ijl< ;UM ILA IVE Multiplicative Wt,ijl< = W ijl«l + Aijl<)' t = wijl< + 2: Ct,T(Aijl<)T+1 T-O T-1 T T 1 85 85 1 2 2: 2: 850 850 2 .. 8Wt ,ijl< 8Wu ,ijl< t,u=O .JI< -2: ~ " .. " _0_ 0 2 Lt,.JI< u,lmn 8W . . 8W t,.JI< u,lmn t,u=O ijl<,lmn T-1 T 2~~ 8S0 +2~p L-L-"t ijl<--.. ' 8Wt ,ijl< t=O .JI< T-1 _2 T 1 2: L crSO -~ " .. " 2 P t,.JI< u ,lmn 8w .' 8W .. t,.JI< u,lmn t,u=O .JI<,lmn 656 Kam Jim, Bill G. Home, C. Lee Giles These plots illustrate the cases where both convergence and generalization are improved. In figure 311 the curves clearly curl down and to the left for lower noise amplitudes before rising to the right at higher noise amplitudes. These lower regions are important because they represent noise values where generalization and convergence improve simultaneously and do not trade off. 5 CONCLUSIONS We have presented several methods of injecting synaptic noise to recurrent neural networks. We summarized the results of an analysis of these methods and empirically tested them on learning the dual parity automaton from strings encoded as temporal sequences. (For a complete discussion of results, see (Jim, Giles, and Horne, 1994) ). Results show that most of these methods can improve generalization and convergence simultaneously - most other methods previously discussed in literature cast convergence as a cost for improved generalization performance. References [1] Chris M. Bishop. Training with noise is equivalent to Tikhonov Regularization. Neural Computation, 1994. To appear. [2] Robert M. Burton, Jr. and George J. Mpitsos. Event-dependent control of noise enhances learning in neural networks. Neural Networks, 5:627-637, 1992. [3] C.L. Giles, C.B. Miller, D. Chen, H.H. Chen, G.Z. Sun, and Y.C. Lee. Learning and extracting finite state automata with second-order recurrent neural networks. Neural Computation, 4(3):393-405, 1992. [4] Stephen Jose Hanson. A stochastic version ofthe delta rule. Physica D., 42:265272, 1990. [5] Kam Jim, C.L. Giles, and B.G. Horne. Synaptic noise in dynamically-driven recurrent neural networks: Convergence and generalization. Technical Report UMIACS-TR-94-89 and CS-TR-3322, Institute for Advanced Computer Studies, University of Maryland, College Park, MD, 1994. [6] Stephen Judd and Paul W. Munro. Nets with unreliable hidden nodes learn error-correcting codes. In S.J Hanson, J.D. Cowan, and C.L. Giles, editors, Advances in Neural Information Processing Systems 5, pages 89-96, San Mateo, CA, 1993. Morgan Kaufmann Publishers. [7] Anders Krogh and John A. Hertz. A simple weight decay can improve generalization. In J .E. Moody, S.J. Hanson, and R.P. Lippmann, editors, Advances in Neural Information Processing Systems 4, pages 450-957, San Mateo, CA, 1992. Morgan Kaufmann Publishers. [8] Alan F. Murray and Peter J. Edwards. Synaptic weight noise during multilayer perceptron training: Fault tolerance and training improvements. IEEE Trans. on Neural Networks, 4(4):722-725, 1993. [9] Carlo H. Sequin and Reed D. Clay. Fault tolerance in artificial neural networks. In Proc. of IJCNN, volume I, pages 1-703-708, 1990.	1994	59
953	Reinforcement Learning Methods for Continuous-Time Markov Decision Problems Steven J. Bradtke Computer Science Department University of Massachusetts Amherst, MA 01003 bradtkeGcs.umass.edu Michael O. Duff Computer Science Department University of Massachusetts Amherst, MA 01003 duffGcs.umass.edu Abstract Semi-Markov Decision Problems are continuous time generalizations of discrete time Markov Decision Problems. A number of reinforcement learning algorithms have been developed recently for the solution of Markov Decision Problems, based on the ideas of asynchronous dynamic programming and stochastic approximation. Among these are TD(,x), Q-Iearning, and Real-time Dynamic Programming. After reviewing semi-Markov Decision Problems and Bellman's optimality equation in that context, we propose algorithms similar to those named above, adapted to the solution of semi-Markov Decision Problems. We demonstrate these algorithms by applying them to the problem of determining the optimal control for a simple queueing system. We conclude with a discussion of circumstances under which these algorithms may be usefully applied. 1 Introduction A number of reinforcement learning algorithms based on the ideas of asynchronous dynamic programming and stochastic approximation have been developed recently for the solution of Markov Decision Problems. Among these are Sutton's TD(,x) [10], Watkins' Q-Iearning [12], and Real-time Dynamic Programming (RTDP) [1, 394 Steven Bradtke, Michael O. Duff 3]. These learning alogorithms are widely used, but their domain of application has been limited to processes modeled by discrete-time Markov Decision Problems (MDP's). This paper derives analogous algorithms for semi-Markov Decision Problems (SMDP's) extending the domain of applicability to continuous time. This effort was originally motivated by the desire to apply reinforcement learning methods to problems of adaptive control of queueing systems, and to the problem of adaptive routing in computer networks in particular. We apply the new algorithms to the well-known problem of routing to two heterogeneous servers [7]. We conclude with a discussion of circumstances under which these algorithms may be usefully applied. 2 Semi-Markov Decision Problems A semi-Markov process is a continuous time dynamic system consisting of a countable state set, X, and a finite action set, A. Suppose that the system is originally observed to be in state z EX, and that action a E A is applied. A semi-Markov process [9] then evolves as follows: • The next state, y, is chosen according to the transition probabilities Pz,(a) • A reward rate p(z, a) is defined until the next transition occurs • Conditional on the event that the next state is y, the time until the transition from z to y occurs has probability distribution Fz,(·Ja) One form of the SMDP is to find a policy the minimizes the expected infinite horizon discounted cost, the "value" for each state: e {IoOO e-.Bt p(z(t), a(t»dt}, where z(t) and aCt) denote, respectively, the state and action at time t. For a fixed policy 71', the value of a given state z must satisfy v,..(z) Defining L Pz,(7I'(z» (00 r e-.B· p(z, 71'(z»dsdFz,(tJ7I'(z» + X 10 10 ,E L Pz,(7I'(Z» fooo e-.Bt V,..(y)dFz,(tJ7I'(z». ,EX R(z, y, a) = foOO fot e-.B· p(z, 71'(z»dsdFz,(tJ7I'(z», (1) the expected reward that will be received on transition from state z to state y on action a, and Reinforcement Learning Methods for Continuous-Time Markov Decision Problems 395 the expected discount factor to be applied to the value of state y on transition from state z on action a, it is clear that equation (1) is nearly identical to the value-function equation for discrete time Markov reward processes, Vw(z) = R(z, 1I"(z» + "Y I: Pzr( 1I"(z»Vw(Y), rEX (2) where R(z, a) = :ErEx Pzr(a)R(z, y, a). If transition times are identically one for an SMDP, then a standard discrete-time MDP results. Similarly, while the value function associated with an optimal policy for an MDP satisfies the Bellman optimality equation Ve(z) = max {R(Z' a) + "Y I: pzr(a)v(y)} , (3) ilEA X rE the optimal value function for an SMDP satisfies the following version of the Bellman optimality equation: V(z) = max { I: Pzr(a) 1 00 r e-fJa p(z, a)dsdFzr(tJa) + ilEA X 0 10 rE I: Pzr(a) loo e-fJtv*(y)dFzr(tJa)} . (4) rEX 3 Temporal Difference learning for SMDP's Sutton's TD(O) [10] is a stochastic approximation method for finding solutions to the system of equations (2). Having observed a transition from state z to state y with sample reward r(z, y, 1I"(z», TD(O) updates the value function estimate V(A:)(z) in the direction of the sample value r(z, y, 1I"(z»+"YV(A:)(y). The TD(O) update rule for MDP's is V(A:+l)(Z) = V(A:)(z) + QA:[r(z, y, 1I"(z» + "YV(A:)(y) - V(A:)(z)], (5) where QA: is the learning rate. The sequence of value-function estimates generated by the TD(O) proceedure will converge to the true solution, Vw , with probability one [5,8, 11] under the appropriate conditions on the QA: and on the definition of the MDP. The TD(O) learning rule for SMDP's, intended to solve the system of equations (1) given a sequence of sampled state transitions, is: [ 1 -fJT ] V(A:+1)(z) = V(A:)(z) + QA: -; r(z, y, 1I"(z» + e-fJTV(A:)(y) - V(A:)(z) , (6) where the sampled transition time from state z to state y was T time units, I_p-tl .. r(z, y, 1I"(z» is the sample reward received in T time units, and e-fJT is the sample discount on the value of the next state given a transition time of T time units. The TD(>.) learning rule for SMDP's is straightforward to define from here. 396 Steven Bradtke. Michael 0. Duff 4 Q-Iearning for SMDP's Denardo [6] and Watkins [12] define Q.f) the Q-function corresponding to the policy 71", as Q'II"(z, a) = R(z, a) + 'Y 2: PzJ(a)V'II"(Y) (7) YEX Notice that a can be any action. It is not necesarily the action 7I"(z) that would be chosen by policy 71". The function Q. corresponds to the optimal policy. Q'II"(z, a) represents the total discounted return that can be expected if any action is taken from state z, and policy 71" is followed thereafter. Equation (7) can be rewritten as Q'II"(z, a) = R(z, a) + 'Y 2: PZJ(a)Q'II"(Y' 7I"(Y», yEX and Q. satisfies the Bellman-style optimality equation Q·(z, a) = R(z, a) + 'Y 2: Pzy(a) max Q.(y, a'), A'EA JEX (8) (9) Q-Iearning, first described by Watkins [12], uses stochastic approximation to iteratively refine an estimate for the function Q •. The Q-Iearning rule is very similar to TD(O). Upon a sampled transition from state z to state y upon selection of a, with sampled reward r(z, y, a), the Q-function estimate is updated according to Q(A:+l)(Z, a) = Q(J:)(z, a) + etJ: [r(z, y, a) + 'Y ~~ Q(J:)(y, a') - Q(J:)(z, a)]. (10) Q-functions may also be defined for SMDP's. The optimal Q-function for an SMDP satisfies the equation 2: PZJ(a) roo t e-tJ• p(z, a)dsdFzJ(tla) + 'V 10 10 JE"Q·(z, a) 2: Pz1I(a) roo e-tJt max Q.(y, a')dFzJ(tla). (11) 'V 10 A'EA JE"This leads to the following Q-Iearning rule for SMDP's: Q(A:+l)(Z, a) = Q(J:)(z, a)+etJ: [1 -;-tJ'r' r(z, y, a) + e-tJ'r' ~~ Q(J:)(y, a') _ Q(J:)(z, a)] (12) 5 RTDP and Adaptive RTDP for SMDP's The TD(O) and Q-Iearning algorithms are model-free, and rely upon stochastic approximation for asymptotic convergence to the desired function (V'll" and Q., respectively). Convergence is typically rather slow. Real-Time Dynamic Programming (RTDP) and Adaptive RTDP [1,3] use a system model to speed convergence. Reinforcement Learning Methods for Continltolts-Time Markov Decision Problems 397 RTDP assumes that a system model is known a priori; Adaptive RTDP builds a model as it interacts with the system. As discussed by Barto et al. [1], these asynchronous DP algorithms can have computational advantages over traditional DP algorithms even when a system model is given. Inspecting equation (4), we see that the model needed by RTDP in the SMDP domain consists of three parts: 1. the state transition probabilities Pzy(a), 2. the expected reward on transition from state z to state y using action a, R(z, y, a), and 3. the expected discount factor to be applied to the value of the next state on transition from state z to state y using action a, 'Y(z, y, a). If the process dynamics are governed by a continuous time Markov chain, then the model needed by RTDP can be analytically derived through uniJormization [2]. In general, however, the model can be very difficult to analytically derive. In these cases Adaptive RTD P can be used to incrementally build a system model through direct interaction with the system. One version of the Adaptive RTDP algorithm for SMDP's is described in Figure 1. 1 Set k = 0, and set Zo to some start state. 2 Initialize P, R, and ~. 3 repeat forever { 4 For all actions a, compute Q(Ie)(ZIe,a) = L P .. "v(a) [ R(zIe,y,a) +~(zIe,y,a)V(Ie)(y) ] veX 5 Perform the update V(le+l)(ZIe) = minoeA Q(Ie)(zIe,a) 6 Select an action, ale. 7 Perform ale and observe the transition to ZIe+l after T time units. Update P. Use the sample reward 1 __ ;;11'" r(ZIe,Zle+l,ale) and the sample discount factor e- f3T to update R and ~. 8 k=k+l 9 } Figure 1: Adaptive RTDP for SMDP's. P, il, and .y are the estimates maintained by Adaptive RTDP of P, R, and 'Y. Notice that the action selection procedure (line 6) is left unspecified. Unlike RTDP, Adaptive RTDP can not always choose the greedy action. This is because it only has an e8timate of the system model on which to base its decisions, and the estimate could initially be quite inaccurate. Adaptive RTDP needs to explore, to choose actions that do not currently appear to be optimal, in order to ensure that the estimated model converges to the true model over time. 398 Steven Bradtke, Michael O. Duff 6 Experiment: Routing to two heterogeneous servers Consider the queueing system shown in Figure 2. Arrivals are assumed to be Poisson with rate ).. Upon arrival, a customer must be routed to one of the two queues, whose servers have service times that are exponentially distributed with parameters J.l.1 and J.l.2 respectively. The goal is compute a policy that minimizes the objective function: e {foOO e-tJt [c1n1(t) + C2n2(t)]dt}, where C1 and C2 are scalar cost factors, and n1(t) and n2(t) denote the number of customers in the respective queues at time t. The pair (n1(t), n2(t)) is the state of the system at time t; the state space for this problem is countably infinite. There are two actions available at every state: if an arrival occurs, route it to queue 1 or route it to queue 2. -.<-~ ___ -.J~ Figure 2: Routing to two queueing systems. It is known for this problem (and many like it [7]), that the optimal policy is a threshold policy; i.e., the set of states Sl for which it is optimal to route to the first queue is characterized by a monotonically nondecreasing threshold function F via Sl = {(nl,n2)ln1 $ F(n2)}' For the case where C1 = C2 = 1 and J.l.1 = J.l.2, the policy is simply to join the shortest queue, and the theshold function is a line slicing diagnonally through the state space. We applied the SMDP version of Q-Iearning to this problem in an attempt to find the optimal policy for some subset of the state space. The system parameters were set to ). = J.l.1 = J.l.2 = 1, /3 = 0.1, and C1 = C2 = 1. We used a feedforward neural network trained using backpropagation as a function approximator. Q-Iearning must take exploratory actions in order to adequately sample all of the available state transitions. At each decision time k, we selected the action aA: to be applied to state ZA: via the Boltzmann distribution where TA: is the "computational temperature." The temperature is initialized to a relatively high value, resulting in a uniform distribution for prospective actions. TA: is gradually lowered as computation proceeds, raising the probability of selecting actions with lower (and for this application, better) Q-values. In the limit, the action that is greedy with respect to the Q-function estimate is selected. The temperature and the learning rate erA: are decreased over time using a "search then converge" method [4]. Reinforcement Learning Methods for Continuous-Time Markov Decision Problems 399 Figure 3 shows the results obtained by Q-Iearning for this problem. Each square denotes a state visited, with nl(t) running along the z-axis, and n2(t) along the yaxis. The color of each square represents the probability of choosing action 1 (route arrivals to queue 1). Black represents probability 1, white represents probability o. An optimal policy would be black above the diagonal, white below the diagonal, and could have arbitrary colors along the diagonal. == == == == = == = II • II • .. !m il lUll @@ ~d r2 it • •• E M m@ mi· 1 @ w • •• ... moo mllw •••••• ,.m @ll A == ;;= == == = == = • .. III ~2 @ %@Ii •• •• Ell =11111 m • . ' . •• IIIIJIIIII •••••••• @w liliiii w •••••••• @m ••••••••• '. B II • • ;;= ;;;;;; == == = == = l1li 0 @oo mm oow mw m]lw •• mill lUll lM]lm •••••••• lIlIg!W =wjini •••••••••• I'm II ... '.' ........ '. c Figure 3: Results of the Q-Iearning experiment. Panel A represents the policy after 50,000 total updates, Panel B represents the policy after 100,000 total updates, and Panel C represents the policy after 150,000 total updates. One unsatisfactory feature of the algorithm's performance is that convergence is rather slow, though the schedules governing the decrease of Boltzmann temperature TA: and learning rate 0A: involve design parameters whose tweakings may result in faster convergence. If it is known that the optimal policies are of theshold type, or that some other structural property holds, then it may be of extreme practical utility to make use of this fact by constraining the value-functions in some way or perhaps by representing them as a combination of appropriate basis vectors that implicity realize or enforce the given structural property. 7 Discussion In this paper we have proposed extending the applicability of well-known reinforcement learning methods developed for discrete-time MDP's to the continuous time domain. We derived semi-Markov versions of TD(O), Q-Iearning, RTDP, and Adaptive RTDP in a straightforward way from their discrete-time analogues. While we have not given any convergence proofs for these new algorithms, such proofs should not be difficult to obtain if we limit ourselves to problems with finite state spaces. (Proof of convergence for these new algorithms is complicated by the fact that, in general, the state spaces involved are infinite; convergence proofs for traditional reinforcement learning methods assume the state space is finite.) Ongoing work is directed toward applying these techniques to more complicated systems, examining distributed control issues, and investigating methods for incorporating prior 400 Steven Bradtke, Michael 0. Duff knowledge (such as structured function approximators). Acknowledgements Thanks to Professor Andrew Barto, Bob Crites, and to the members of the Adaptive Networks Laboratory. This work was supported by the National Science Foundation under Grant ECS-9214866 to Professor Barto. References [1] A. G. Barto, S. J. Bradtke, and S. P. Singh. Learning to act using real-time dynamic programming. Artificial Intelligence. Accepted. [2] D. P. Bertsekas. Dynamic Programming: Deterministic and Stochastic Models. Prentice Hall, Englewood Cliffs, NJ, 1987. [3] S. J. Bradtke. Incremental Dynamic Programming for On-line Adaptive Optimal Control. PhD thesis, University of Massachusetts, 1994. [4] C. Darken, J. Chang, and J. Moody. Learning rate schedules for faster stochastic gradient search. In Neural Networks for Signal Processing ~ Proceedings of the 199~ IEEE Workshop. IEEE Press, 1992. [5] P. Dayan and T. J. Sejnowski. Td(A): Convergence with probability 1. Machine Learning, 1994. [6] E. V. Denardo. Contraction mappings in the theory underlying dynamic programming. SIAM Review, 9(2):165-177, April 1967. [7] B. Hajek. Optimal control of two interacting service stations. IEEE-TAC, 29:491-499, 1984. [8] T. Jaakkola, M. I. Jordan, and S. P. Singh. On the convergence of stochastic iterative dynamic programming algorithms. Neural Computation, 1994. [9] S. M. Ross. Applied Probability Models with Optimization Applications. HoldenDay, San Francisco, 1970. [10] R. S. Sutton. Learning to predict by the method of temporal differences. Machine Learning, 3:9-44, 1988. [11] J. N. Tsitsiklis. Asynchronous stochastic approximation and Q-Iearning. Technical Report LIDS-P-2172, Laboratory for Information and Decision Systems, MIT, Cambridge, MA, 1993. [12] C. J. C. H. Watkins. Learning from Delayed Rewards. PhD thesis, Cambridge University, Cambridge, England, 1989.	1994	6
954	An Integrated Architecture of Adaptive Neural Network Control for Dynamic Systems Robert L. Tokar2 Brian D.McVey2 'Center for Nonlinear Studies, 2Applied Theoretical Physics Division Los Alamos National Laboratory, Los Alamos, NM, 87545 Abstract In this study, an integrated neural network control architecture for nonlinear dynamic systems is presented. Most of the recent emphasis in the neural network control field has no error feedback as the control input, which rises the lack of adaptation problem. The integrated architecture in this paper combines feed forward control and error feedback adaptive control using neural networks. The paper reveals the different internal functionality of these two kinds of neural network controllers for certain input styles, e.g., state feedback and error feedback. With error feedback, neural network controllers learn the slopes or the gains with respect to the error feedback, producing an error driven adaptive control systems. The results demonstrate that the two kinds of control scheme can be combined to realize their individual advantages. Testing with disturbances added to the plant shows good tracking and adaptation with the integrated neural control architecture. 1 INTRODUCTION Neural networks are used for control systems because of their capability to approximate nonlinear system dynamics. Most neural network control architectures originate from work presented by Narendra[I), Psaltis[2) and Lightbody[3). In these architectures, an identification neural network is trained to function as a model for the plant. Based on the neural network identification model, a neural network controller is trained by backpropagating the error through the identification network. After training, the identification network is replaced by the real plant. As is illustrated in Figure 1, the controller receives external inputs as well as plant state feedback inputs. Training procedures are employed such that the networks approximate feed forward control surfaces that are functions of external inputs and state feedbacks of the plant (or the identification network during training). It is worth noting that in this architecture, the error between the plant output and the desired output of the reference model is not fed back to the controller, after the training phase. In other words, this error information is ignored when the neural network applies its control. It is well known in control theory that the error feedback plays a significant role in adaptation. Therefore, when model uncertainty or noise/disturbances are present, a feed forward neural network controller with only state feedback will not adaptively update the control signal. On line training for the neural controller has been proposed to obtaip adaptive ability[I)[3). However, the stability for the on line training of the neural network controller is unresolved[1][4]. In this study, an additional nonlinear recurrent network is combined with the feed forward neural network controller to form an adaptive controller. This added neural network uses feedback error between the reference model output and the plant output as an input In addition, the system's external 1032 Liu Ke, Robert L. Tokar, Brian D. McVey inputs and the plant states are also input to the feedback network. This architecture is used in the control community, but not with neural network components. The approach differs from a conventional error feedback controller, such as a gain scheduled PID controller, in that the neural network error feedback controller implements a continuous nonlinear gain scheduled hypersurface, and after training, adaptive model reference control for nonlinear dynamic systems is achieved without further parameter computation. The approach is tested on well-known nonlinear control problems in the neural network literature, and good results are obtained. 2 NEURAL NETWORK CONTROL In this section, several different neural network control architectures are presented. In these structures, identification neural networks, viewed as accurate models for real plants, are used. 2.1 NEURAL NETWORK FEED FORWARD CONTROL The neural network controllers are trained by backpropagation of errors through a well trained neural identification network. In this architecture, the state variable yet) of the system is sent back to the neural network, and the external input x(t) also is input to the network. With these inputs, the neural network estabJishes a feed forward mapping from the external input x(t) to the control signal u(t). This control mapping is expressed as a function of the external input x(t) and the plant state yet): u(t)==j(x(t), yet»~ (1) where x(t)=[x(t), x(t-l), .. J, andy(t)=[y(t), y(t-l), . .Y. This neural network control architecture is denoted in this study as feed forward neural control even though it includes state feedback. Neural control with error feedback is denoted as feedback neural control. x(t) -..:...r-----~Ref. Modelf-------, e(t+ 1) Control NN y(t+ 1) Figure I Neural Network Control Architecture. ID NN represents the identification network. Ref. Model means reference model, and NN means neural network. x(t) -~----~Ref. Modell-----.., u(t) ,-------, f----+-.. Figure 2 Neural Network Feedback Control Architecture y(t+ 1) During the training phases, based on the assumption that the neural identification network provides a model for the plant, the gradient information needed for error backpropagation is obtained by calculating the Jacobian of the identification network. The following equation describes this process for the control architecture shown in Figure I. If the cost function is defined as E, then the gradient of the cost function with respect to weight w of the neural controller is An Integrated Architecture of Adaptive Neural Network Control for Dynamic Systems 1033 a E a E a u (a E a u a E J a Yt-l a: = a; a w + a u a Yt-l + a Yt-l --a;(2) where u is tbe control signal and YI-1 is tbe plant feedback state. After tbe training stage, tbe neural network supplies a control law. Because neural networks have the ability to approximate any arbitrary nonlinear functions[5], a feed forward neural network can build a nonlinear controller, which is crucial to tbe use of tbe neural network in control engineering. Also, since all tbe parameters of the neural network identification model and tbe neural network controller are obtained from learning through samples, matbematically untraceable features of tbe plant can be extracted from tbe samples and imbedded into tbe control system. However, because tbe feed forward controller has no error feedback, tbe controller can not adapt to tbe disturbances occurring in tbe plant or tbe reference model. This problem is of substantial importance in tbe context of adaptive control. In tbe next subsection, error feedback between tbe reference models and tbe plant outputs is introduced into neural network controllers for adaptation. 2.2 NEURAL ADAPTIVE CONTROL WITH ERROR FEEDBACK It is known that feedback errors from the system are important for adaptation. Due to the flexibility of the neural network architecture, the error between the reference model and the plant can be sent back to the controller as an extra input. In such an architecture, neural networks become nonlinear gain scheduled controllers with smooth continuous gains. Figure 2 shows the architecture for the feedback neural control. With tbis architecture, tbe neural network control surface is not tbe fixed mapping from tbe x(t) to u(t) for each state y(t), but instead it learns tbe slope or tbe gain referring to tbe feedback error e(t) for control. This gain is a continuous nonlinear function of tbe external input x(t) and tbe state feedback yet). Figure 3 shows tbe recurrent network architecture of tbe feedback neural controller. The output node needs to be recurrent because tbe output witbout tbe recurrent link from tbe neural controller is only a correction to tbe old control signal, and tbe new control signal should be tbe combination of old control signal and tbe correction. The otber nodes of tbe network can be feed forward or recurrent. If we denote tbe weight for tbe output node's recurrent link as w., tben tbe output from tbe recurrent link is w.u(t-l). The following equation describes the feedback network. u(t) = wbu(t-I )+j(X(t), y(t), e(t» (3) where j(.) is a nonlinear function established by tbe network for which tbe recurrent link output is not included and e(t)=[e(t), e(t-I), ... f To compare tbe control gain expression with conventional control theory, consider tbe Taylor series expansion of tbe network forward mappingj(.), equation (3) becomes u(t) = w.u(t-l) + !'(x(t). yet»~ e(t)+ j"(x(t), yet»~ e2(t)+... (4) where f'(x(t), y(t»=[ i1j(x(t), y(t), e(t»/ae(t), aJ!:x(t), y(t), e(t»Ii1e(t-I), ... ]. If high order terms are ignored and gO representsf'O, we get u(t) = wbu(t-I)+ g(x(t), yet»~ e(t) (5) 1034 Liu Ke, Robert L. Tokar, Brian D. McVey which is a gain scheduled controller and the gain is the function of external input x(/) and the plant state y(/). It is clear that when w.=l.O, g(.) is a constant vector and e(/)=[e(t), e(t-l), e(t-2)]T, the feedback neural network controller degenerates to a discrete PID controUer. Because the neural network can approximate arbitrary nonlinear functions through learning, the neural network feedback controller can generate a nonlinear continuous gain hypersurface. Ref. Model Figure 3 Feedback Neural Network Controller Figure 4 Integrated NN Control Architeture. In the training process, error backpropagating through the identification network is used. The process is similar to the training of a feed forward neural controller, but the resulting control surface is completely different due to the different inputs. After training, the neural network is able to provide a nonlinear control law, that is, the desired model following response can be obtained with fixed controller parameters for nonlinear dynamic systems. Traditionally, the control of the nonlinear plant is derived from continuous computing of the controller gains. This feedback controller is error driven. As long as an error exists, the control signal is updated according to the error and the gain. This kind of neural controller is an adaptive controller in principle. 2.3 INTEGRATED NEURAL NETWORK CONTROLLER The characteristics of feed forward and error feedback neural control networks are described in the previous subsections. In this section. the two controllers are combined. Figure 4 shows the architecture. In this architecture, we include both feed forward and feedback neural network controllers. The control signal is the combination from these two networks' outputs. In the training stage, it is our experience that the feed forward network should be trained first. The feedback network is not included while training the feed forward network. After training the feed forward controller, the error feedback network is trained with the feed forward network, but the feed forward networks' weights are unchanged. Backpropagating the error through the identification network is applied for the training of both networks. When training the feedback control network, the feed forward calculation is u(t) = ujt)+u/b(t). y(t+ 1) = P(x(t), y(t), u(t», (6) (7) where uj/) is the output from the feed forward controller network and u,..(t) is the output from the feedback controller network, P(.) is the identification mapping. An Integrated Architecture of Adaptive Neural Network Control for Dynamic Systems 1035 3 CONTROL ON EXAMPLE PROBLEMS In this section, the control architecture described above is applied to a well-known problem from the literature[I). The plants and the reference model of the sample problems are described by difference equations plant: yet) y(t + 1) = 2 + (u(t) -1.O)u(t)(u(t) + 1.0) 1.0+ Y (t) (II) reference model: y(t + 1) = 0.6y(t) + u(t) (12) This is a nonlinear time varying dynamic system with no analytical inverse. 3.1 FEED FORWARD CONTROL A feed forward neural network is trained to control the system to follow the reference model. The plant state yet) and external inputx(t) are fed to the controller. During the training, the x(t) is randomly generated. After training, the controller generates a control signal u(t) such that the plant can follow the reference model output. Figure 5 shows the testing result of the reference model output and the controlled plant output. The input function is x(t)=sin(21ttf25)+sin(21tt/1O). The controller network architecture is (2, 20, 1). 4 OJ 0 c 2 Q) 10OJ ~ 0 '" 1J c 0 -2 '" 20 40 60 80 100 Figure 5 Tracking Result From the Feed Forward NN. Output of reference (solid line) and plant (dash line). 2 2 ? (j ::J ~ 0 0 0 ~ () c ..... 1 0 u -1 ..... 2 Figure 6 Feed Forward Control Surface The output surface of the controller network is shown in Figure 6. By examining the controller output surface, we can see that the neural network builds a feed forward mapping from x(t) to u(t). This feed forward mapping is also a function of the plant state yet). Under each state, the neural network controller accepts input x(t) to produce control signal u(t) such that the plant follows the reference model reasonably well. In Figure 6, the x axis is the external input x(t) and the y axis is the plant feedback output yet). The z axis represents the control surface. The feed forward controller laCks the ability to adapt to plant uncertainty, noise or changes in the reference model. As an example, we apply this feed forward controller to the disturbed plant with a bias 0.5 added to the original plant. The tracking result is shown in Figure 7. With this slight bias, the plant does not follow the reference model. Clearly, the feed forward controller has no adaptive ability to this model bias. 1036 Liu Ke, Robert L. Tokar, Brian D. McVey 3.2 FEEDBACK CONTROL FtrSt, we compare the neural network feedback controller with fixed gain PID controllers. For many nonlinear systems, the fixed gain PID controllers will give poor tracking and continuous adaptation of the controller parameters is needed. The neural network approach offers an alternative control approach for nonlinear systems. Through the training, control gains, imbedded in the neural network, are established as a continuous function of system external inputs x(t) and plant states yet). The sample problem in the above section is now employed to describe how the neural network creates a nonlinear control gain surface with error feedback and additional inputs. First, we show one simple case of neural adaptive feedback controller. This controller can only adapt to the system nonlinearity with a fixed linear input pattern. The reason to show this simple adaptation case first is that its control gain surface can be illustrated graphically. Figure 8 illustrates, for the system in equations (11) and (12) that a fixed gain PI controller fails to track the reference model, for even one fixed linear input pattern x(t)=0.2t-2.5, because the plant nonlinearity. Figure 9 illustrates the result from a recurrent neural network with feedback error e(t) and x(t) as inputs. The neural network is trained by backpropagation error through the identification network. Compared to the flXed gain PI controller, the neural network improves the tracking ability significantly. 4 ., 0 c ., 2 .. ., '!! >. 0 -.::I C 0 -2 >. 20 40 60 80 100 t Figure 7 Tracking Result for Shifted Plant, plant output (dash line) and reference output (solid line). 6 OJ , <J 3 I C ~ I .2 0 ~ >. '0 - 3 c 0 >. o 5 10 15 20 25 30 35 t Figure 8 Reference Model Output (solid line) and PID Controlled Plant Output (dashed line) The control surface of the updating output fl.) is shown in Figure 10, which is the output from the neural network controller without recurrent link (see equation (3». We plot the surface of the updating output from the controller with respect to input x(t) and error feed back input e(t). The gain of the controller is equivalent to the updating output from the network when error=l.O. As shown in the figure, the gain in the neighborhood about x(t)=O changes largely according to the direction of changes in the plant in the corresponding region. The updating surface for a PID controller is a plane. The neural network implements a nonlinear continuous control gain surface. For a more complicated case, we addx(t-I) as another input to the neural network as well as e(t-l), and train by error backpropagation through the identification network. These two inputs, x(t) and x(t-I) add difference information to the network. The network can adapt to not only different operating regions indicated by x(t), but also different input patterns. Figure 11 shows the tracking results with two different input patterns. In Figure II (a), input pattern is x(t)=4.0sin(tI4.0). In Figure 11 (b) input pattern is x(t)=sin(21t1!25)+sin(21t111O). An Integrated Architecture of Adaptive Neural Network Control for Dynamic Systems 1037 6 Q) u 3 c ~ 2 0 ~ , >. , -0. -3 I I c 0 >. - 6 o 5 1 0 1 5 20 25 30 35 t Figure 9 Reference Model Output (solid line) and Neural Network Controled Output (dashed line) Figure 10 Feedback Neural Controller Updating Surface 5 OJ ., 4 u u 3 c c 2 ~ ~ ~ OJ 1 ~ 0 l' 0 >. >. -1 " - 2 " - 2 c: - 4 c - 3 0 0 >. -6 >. - 4 - 8 -5 -10 -6 0 20 40 60 80 10C 0 20 40 60 80 100 (a) (b) Figure 11 Output of the Reference Model (solid line) and the Plant (dash line) 3.3 INTEGRA TED NEURAL CONTROLLER As shown in the above section, when only error feedback neural controller is used, the control result is not very accurate. Now we combine feed forward and feedback control to realize good tracking and adaptation. Figure 12 shows the control result from the integrated controller when the plant is shifted O.S. Compared to only feed forward control (Figure 7), the integrated controller has much better adaptation to the shifted plant. When the plant changes, adding an extra feed back controller can avoid on-line training of feed forward network which may induce potential instability, and the adaptation is achieved. The output from the feedback network controller is driven by the error between the reference model and the plant. 4 DISCUSSIONS We have emphasized in the above sections that a feed forward controller with only state feedback does not adapt when model uncertainties or noise/disturbance are present. The presence of a feed back controller can make the on line training of the feed forward network unnecessary, thus avoiding potential instability. The main reason for the instability of on-line training is the incompleteness of sample sets, which is referred to as a lack of persistent excitation in control theory[6]. First, it leads to an inaccurate identification network. Training with this network can result in an unstable controller. Second, it makes the training of controller away from global representation. With an error feedback adaptive network, the output from the feedback network controller is driven by the error between the reference model and the plant. In the simplest case when all the activity functions are linear and only the feedback errors are inputs, this kind of neural network is equivalent to a PID controller. However, 1038 Liu Ke, Robert L Tokar, Brian D. McVey beyond the scope of PID controllers, the neural networks are capable to approximating nonlinear time variant control gain surfaces corresponding to different operating regions. Also, unlike a PID controller, the coefficients for the neural adaptive controller are obtained through a training procedure. 4 Q) u c 1:' Q) 0; ... >, "0 C 0 >, o 20 40 60 80 100 Figure 12 Integrated Network Controller Tracking Result for Shifted Plant. Plant Output (dash line) and Reference Output (solid line). The error feedback network behaves as a gain scheduling controller. It has rise time, overshoot consideration and delay problem. Feed forward control can compensate for these problems to some degree. For example, the feed forward network can perform a nonlinear mapping with designed time delay. Therefore with the feed forward network, the delay problem maybe overcame significantly. Also the feed forward controller can help to reduce rise time compare to use only feedback controller. With the feed forward network, the feedback network controller can have much smaller gains compared to using a feedback network alone. This increases the noise rejection ability. Also this reduces the overshoot as well as settle time. The neural network control architecture offers an alternative to the conventional approach. It gives a generic model for the broadest class of systems considered in control theory. However this model needs to be configured depending on the details of the control problem. With different inputs, the neural network controllers establish different internal hyperstates. When plant states are fed back to the network, a feed forward mapping is established as a function of the plant states by the neural network controller. When the errors between the reference model and the plant are used as the error feedback inputs to a dynamic neural network controller, the network functions as an associative memory nonlinear gain scheduled controller. The above two kinds of neural controller can be combined and complemented to achieve accurate tracking and adaptation. References [1] Kumpati S. Narendra and Kannan Parthasarathy. "Gradient Methods for the Optimization of DynamiCal Systems Containing Neural Networks," IEEE Trans. Neural Networks. vol. 2. pp252-262 Mar. 1991 [2] Psaltis. D .• Sideris. A. and Yamamura. A., "Neural controllers." Proc. of 1st International Conference on Neural Networks. Vol. 4. pp551-558. San Diego. CA. 1987 [3) G. lightbOdy. Q. H. Wu and G. W. Irwin. "Control applications for feed forward networks." Chapter 4. Neural Networks for Control and Systems. Edited by K.warwich, G. W. Irwin and K. J. Hunt 1992 [4) R. Abikowski and P. 1. Gawthrop. "A survey of neural networks for control" Chapter 3. NeUral Networks for Control and Systems. ISBN 0-86341-279-3. Edited by K.warwich. G. W. Irwin and K. 1. Hunt 1992 [5] John Hertz. Anders Krogh and Richard G. Palmer. "Introduction to the Theory of Neural Computation." [6J Thomas Miller. RiChard S. Sutton and Paul 1. Werbos. "Neural Networks for Control"	1994	60
955	Implementation of Neural Hardware with the Neural VLSI of URAN in Applications with Reduced Representations ll-Song Han Korea Telecom Research Laboratories 17, Woomyun-dong, Suhcho-ku Seoul 137-140, KOREA Hwang-Soo Lee Dept. of Info and Comm KAIST Seoul, 130-012, Korea Abstract Ki-Chul Kim Dept. of Info and Comm KAIST Seoul, 130-012, Korea This paper describes a way of neural hardware implementation with the analog-digital mixed mode neural chip. The full custom neural VLSI of Universally Reconstructible Artificial Neural network (URAN) is used to implement Korean speech recognition system. A multi-layer perceptron with linear neurons is trained successfully under the limited accuracy in computations. The network with a large frame input layer is tested to recognize spoken korean words at a forward retrieval. Multichip hardware module is suggested with eight chips or more for the extended performance and capacity. 812 ll-Song Han, Hwang-Soo Lee, Ki-Chul Kim 1 INTRODUCTION In general, the neural network hardware or VLSI has been preferred in respects of its relatively fast speed, huge network size and effective cost comparing to software simulation. Universally Reconstructible Artificial Neural-network(URAN), the new analog-digital mixed VLSI neural network, can be used for the implementation of the real world neural network applications with digital interface. The basic electronic synapse circuit is based on the electrically controlled MOSFET resistance and is operated with discrete pulses. The URAN's adaptability is tested for the multi-layer perceptron with the reduced precision of connections and states. The linear neuron function is also designed for the real world applications. The multi-layer network with back propagation learning is designed for the speaker independent digit/word recognition. The other case of application is for the servo control, where the neural input and output are extended to 360 levels for the suitable angle control. With the servo control simulation, the flexibility of URAN is proved to extend the accuracy of input and output from external. 2. Analog-Digital Mixed Chip - URAN In the past, there have been improvements in analog or analog-digital mixed VLSI chips. Analog neural chips or analog-digital mixed neural chips are still suffered from the lack of accuracy, speed or flexibility. With the proposed analog-digital mixed neural network circuit of URAN, the accuracy is improved by using the voltage-controlled linear MOSFET resistance for the synapse weight emulation. The speed in neural computation is also improved by using the simple switch controlled by the neural input as described in previous works. The general flexibility is attained by the independent characteristic of each synapse cell and the modular structure of URAN chip. As in Table 1 of URAN chip feature, the chip is operated under the flexible control, that is, the various mode of synaptic connection per neuron or the extendable weight accuracy can be implemented. It is not limited for the asynchronous/direct interchip expansion in size or speed. In fact, 16 fully connected module of URAN is selected from external and independently - it is possible to select either one by one or all at once. Table 1. URAN Chip Features Total Synapses Computation Speed Weight Accuracy Module No. Module Size 135,424 connections 200 Giga Connections Per S 8 Bit 16 92 X 92 Implementation of Neural Hardware with the Neural VLSI of URAN 813 As all circuits over the chip except digital decoder unit are operated in analog transistor level, the computation speed is relatively high and even can be improved substantially. The cell size including interconnection area in conventional short-channel technology is reduced less than 900 1.1 m2. From its expected and measured linear characteristic, URAN has the accuracy more than 256 linear levels. The accuracy extendability and flexible modularity are inherent in electrical wired- OR characteristics as each synapse is an independent bipolar current source with switch. No additional clocking or any limited synchronous operation is required in this case, while it is indispensible in most of conventional digital neural hardware or analog-digital neural chip. Therefore, any size of neural network can be integrated in VLSI or module hardware merely by placing the cell in 2 dimensional array without any timing limitation or loading effect. 3. Neural Hardware with URAN - Module Expansion URAN is the full custom VLSI of analog-digital mixed operation. The prototype of URAN chip is fabricated in 1.0tI digital CMOS technology. The chip contains 135,424 synapses with 8 bit weight accuracy on a 13 X 13 mm2 die size using single poly double metal technology. As summarized in Table 1 of chip features, the chip allows the variety of configuration. In the prototype chip, 16 fully connected module of 92 X 92 can be selected from external and independently selecting independent module either one by one, several or all at a time IS possible. With URAN's synapse circuit of linear Voltage-controlled bipolar current source, the synaptic multiplication with weight value is done with the switching transistor, in a similar way of analog-sampled data type. The accuracy enhancement and flexible modularity of URAN are inherent in its electrical wired-OR interface from each independent bipolar current source. And the neural network hardware module can be realized in any size with the multi URAN chips. 4. Considerations on the Reduced Precision URAN chip is applied for the case of Korean speaker independent speech recognition. By changing numbers of hidden units and input accuracy, the result of simulations has not shown any problems in recognition accuracy. It means that the overall performance is not severely affected from the accuracy of weight, input, and output with URAN. Also, it was possible to train with 2 or 1 decimal accuracy for input and output, which is equivalent to 8 bit or 4 bit precisions. With 20 hidden units for the Korean spoken 10 digit recognition, 2 decimal input accuracy yields 99.2% and l ' decimal input accuracy yields 98.6%, 814 II-Song Han, Hwang-Soo Lee, Ki-Chul Kim while binary I-bit input results 96.6%. The following is the condition for the experimentation. The general result is summarized in Table 2. Conditions for Training and Test • 2,000 samples from 10 women and 10 men ( 10 times X 10 digits X 20 persons) [] Training with 500 spoken samples of 10 digits in Korean from 10 persons (5 times X 10 digits X [ 5 women and 5 men ] ) from 2,000 samples [] Recognition Test with 1,000 spoken samples from the other 10 persons of women and men. Preprocessing of samples [] sampled at 10KHz with 12bit accuracy [] preemphasis with 0.95 [] Hamming window of 20ms [] 17 channel critical-band filter bank [] noise added for the SNR of 3OdB, 2OdB, 10dB, OdB Table 2. Low Accuracy Connection with Linear Neuron Input / Output Accuracy SNR Ratio 2 decimal 1 decimal 1 bit clean 97.5% 97.2% 90.7% 30 dB 96.2% 96.6% 90.5% 20 dB 90.1% 91.3% 86.6% 10 dB 59.8% 59.9% 68.0% o dB 30.8% 29.5% 38.5% In case of servo control, the digital VCR for industrial purpose is modelled for the application. Six inputs are used to minimize the number of hidden units and 20 hidden units are configured for one output. For the adaptation to URAN, the linear neuron function is used during the simulation. The weight accuracy during the learning phase using conventional computer is 4 byte and that in the recall phase using URAN chip is 1 byte. With this limitation, the overall performance is not severely degraded, that is, the reduction of error is attained up to 70% improvement comparing to the conventional method. The nonideal factor of 30% results from the limitation in learning data as well as the limited hardware. Current results are suitable for the digital VCR or compact cam coder in noisy environment Implementation of Neural Hardware with the Neural VLSI of URAN 815 5. Conclusion In this paper, it is proved to be suitable for the application to the multi-layer perceptron with the use of URAN chip, which is fabricated in conventional digital CMOS technology 1.0# single poly double metal. The reduced weight accuracy of 1 byte is proved to be enough to obtain high perfonnance using the linear neuron and URAN. With 8 test chips of 135,424 connections, it is now under development of the practical module of neural hardware with million connections and tera connections per second comparable to the power of biological neuro-system of some insects. The size of the hardware is smaller than A4 size and is designed for more general recognition system. The flexible modularity of URAN makes it possible to realize a 1,000,000 connections neural chip in 0.5# CMOS technology and a general purpose neural hardware of hundreds of tera connections or more. References II Song Han and Ki-Hwan Ahn, "Neural Network VLSI Chip Implementation of Analog-Digital Mixed Operation for more than 100,000 Connections" MicroNeuro'93, pp. 159-162, 1993 M. Brownlow, L. Tara s senko , A. F. Murray, A. Hamilton, I SHan, H. M. Reekie, "Pulse Firing Neural Chips Implementing Hundreds of Neurons," NIPS2, pp. 785-792, 1990	1994	61
956	Estimating Conditional Probability Densities for Periodic Variables Chris M Bishop and Claire Legleye Neural Computing Research Group Department of Computer Science and Applied Mathematics Aston University Birmingham, B4 7ET, U.K. c.m.bishop@aston.ac.uk Abstract Most of the common techniques for estimating conditional probability densities are inappropriate for applications involving periodic variables. In this paper we introduce three novel techniques for tackling such problems, and investigate their performance using synthetic data. We then apply these techniques to the problem of extracting the distribution of wind vector directions from radar scatterometer data gathered by a remote-sensing satellite. 1 INTRODUCTION Many applications of neural networks can be formulated in terms of a multi-variate non-linear mapping from an input vector x to a target vector t. A conventional neural network approach, based on least squares for example, leads to a network mapping which approximates the regression of t on x. A more complete description of the data can be obtained by estimating the conditional probability density of t, conditioned on x, which we write as p(tlx). Various techniques exist for modelling such densities when the target variables live in a Euclidean space. However, a number of potential applications involve angle-like output variables which are periodic on some finite interval (usually chosen to be (0,271")). For example, in Section 3 642 Chris M. Bishop, Claire Legleye we consider the problem of determining the wind direction (a periodic quantity) from radar scatterometer data obtained from remote sensing measurements. Most of the existing techniques for conditional density estimation cannot be applied in such cases. A common technique for unconditional density estimation is based on mixture models of the form m pet) = L Cki¢i(t) (1) i=l where Cki are called mixing coefficients, and the kernel functions ¢i(t) are frequently chosen to be Gaussians. Such models can be used as the basis of techniques for conditional density estimation by allowing the mixing coefficients, and any parameters governing the kernel functions, to be general functions of the input vector x. This can be achieved by relating these quantities to the outputs of a neural network which takes x as input, as shown in Figure 1. Such an approach forms the basis of conditional probability density n p(tlx) 1\ 1\ 1\ parameter (> z vector U mixture model neural network Figure 1: A general framework for conditional density estimation is obtained by using a feed-forward neural network whose outputs determine the parameters in a mixture density model. The mixture model then represents the conditional probability density of the target variables, conditioned on the input vector to the network. the 'mixture of experts' model (Jacobs et al., 1991) and has also been considered by a number of other authors (White, 1992; Bishop, 1994; Lui, 1994). In this paper we introduce three techniques for estimating conditional densities of periodic variables, based on extensions of the above formalism for Euclidean variables. Estimating Conditional Probability Densities for Periodic Variables 643 2 DENSITY ESTIMATION FOR PERIODIC VARIABLES In this section we consider three alternative approaches to estimating the conditional density p(Blx) of a periodic variable B, conditioned on an input vector x. They are based respectively on a transformation to an extended domain representation, the use of adaptive circular normal kernel functions, and the use of fixed circular normal kernels. 2.1 TRANSFORMATION TO AN EXTENDED VARIABLE DOMAIN The first technique which we consider involves finding a transformation from the periodic variable B E (0,27r) to a Euclidean variable X E (-00,00), such that standard techniques for conditional density estimation can be applied in X-space. In particular, we seek a conditional density function p(xlx) which is to be modelled using a conventional Gaussian mixture approach as described in Section 1. Consider the transformation 00 p(Blx) = L p(B + L27rlx) (2) L=-oo Then it is clear by construction that the density model on the left hand side satisfies the periodicity requirement p(B + 27rlx) = p(Blx). Furthermore, if the density function p(xlx) is normalized, then we have {21f Jo p(Blx) dB = (3) and so the corresponding periodic density p(Olx) will also be normalized. We now model the density function p(xlx) using a mixture of Gaussians of the form m p(xlx) = L fri(X)4>i(xlx) (4) i=l where the kernel functions are given by 1 ( {X - Xi(X)P) 4>i(xlx) = (27r)1/2Uj(x) exp 2u;(x) (5) and the parameters fri(X), Uj(x) and Xi(X) are determined by the outputs of a feed-forward network. In particular, the mixing coefficients frj(x) are governed by a 'softmax' activation function to ensure that they lie in the range (0,1) and sum to 644 Chris M. Bishop, Claire Legleye unity; the width parameters O'i(X) are given by the exponentials ofthe corresponding network outputs to ensure their positivity; and the basis function centres Xi(X) are given directly by network output variables. The network is trained by maximizing the likelihood function, evaluated for set of training data, with respect to the weights and biases in the network. For a training set consisting of N input vectors xn and corresponding targets (r, the likelihood is given by N .c = II peon Ixn)p(xn) (6) n=l where p(x) is the unconditional density of the input data. Rather than work with .c directly, it is convenient instead to minimize an error function given by the negative log of the likelihood. Making use of (2) we can write this in the form E = -In.c ~ - 2: In 2:P'(on + L271'1xn) (7) n L where we have dropped the term arising from p(x) since it is independent of the network weights. This expression is very similar to the one which arises if we perform density estimation on the real axis, except for the extra summation over L, which means that the data point on recurs at intervals of 271' along the x-axis. This is not equivalent simply to replicating the data, however, since the summation over L occurs inside the logarithm, rather than outside as with the summation over data points n. In a practical implementation, it is necessary to restrict the summation over L. For the results presented in the next section, this summation was taken over 7 complete periods of 271' spanning the range (-771', 771'). Since the Gaussians have exponentially decaying tails, this represents an extremely good approximation in almost all cases, provided we take care in initializing the network weights so that the Gaussian kernels lie in the central few periods. Derivatives of E with respect to the network weights can be computed using the rules of calculus, to give a modified form of back-propagation. These derivatives can then be used with standard optimization techniques to find a minimum of the error function. (The results presented in the next section were obtained using the BFGS quasi-Newton algorithm). 2.2 MIXTURES OF CIRCULAR NORMAL DENSITIES The second approach which we introduce is also based on a mixture of kernel functions of the form (1), but in this case the kernel functions themselves are periodic, thereby ensuring that the overall density function will be periodic. To motivate this approach, consider the problem of modelling the distribution of a velocity vector v in two dimensions (this arises, for example, in the application considered in Section 3). Since v lives in a Euclidean plane, we can model the density function p(v) using a mixture of conventional spherical Gaussian kernels, where each kernel has Estimating Conditional Probability Densities for Periodic Variables 645 the form ( ) 1 ({ Vx flx F {Vy fly F ) ¢ Vx , Vy = 211"cr2 exp 2cr2 2cr2 (8) where (vx , v y ) are the Cartesian components of v, and (flx, fly) are the components of the center I-' of the kernel. From this we can extract the conditional distribution of the polar angle () of the vector v, given a value for v = I/vll. This is easily done with the transformation Vx = v cos (), Vy = v sin (), and defining ()o to be the polar angle of 1-', so that flx = fl cos ()o and fly = fl sin ()o, where fl = 111-'11· This leads to a distribution which can be written in the form 1 ¢(()) = (A) exp {A cos(() - ()on 211"10 (9) where the normalization coefficient has been expressed in terms of the zeroth order modified Bessel function of the first kind, Io(A) . The distribution (9) is known as a circular normal or von Mises distribution (Mardia, 1972). The parameter A (which depends on v in our derivation) is analogous to the (inverse) variance parameter in a conventional normal distribution. Since (9) is periodic, we can construct a general representation for the conditional density of a periodic variable by considering a mixture of circular normal kernels, with parameters given by the outputs of a neural network. The weights of the network can again be determined by maximizing the likelihood function defined over a set of training data. 2.3 FIXED KERNELS The third approach introduced here is again based on a mixture model in which the kernel functions are periodic, but where the kernel parameters (specifying their width and location) are fixed. The only adaptive parameters are the mixing coefficients, which are again determined by the outputs of a feed-forward network having a softmax final-layer activation function. Here we consider a set of equally-spaced circular normal kernels in which the width parameters are chosen to give a moderate degree of overlap between the kernels so that the resulting representation for the density function will be reasonably smooth. Again, a maximum likelihood formalism is employed to train the network. Clearly a major drawback of fixed-kernel methods is that the number of kernels must grow exponentially with the dimensionality of the output space. For a single output variable, however, they can be regarded as practical techniques. 3 RESULTS In order to test and compare the methods introduced above, we first consider a simple problem involving synthetic data, for which the true underlying distribution function is known. This data set is intended to mimic the central properties of the real data to be discussed in the next section. It has a single input variable x and an output variable () which lies in the range (0,211"). The distribution of () is governed 646 Chris M. Bishop, Claire Leg/eye by a mixture of two triangular functions whose parameters (locations and widths) are functions of x. Here we present preliminary results from the application of the method introduced in section 2.1 (involving the transformation to Euclidean space) to this data. Figure 2 shows a plot of the reconstructed conditional density in both the extended X variable, and in the reconstructed polar variable (), for a particular value of the input variable x . 0.6 r-----r-------,~-_, 0.5,..-----"'T""-----., x=0.5 -network -- - - true p(X) sinO 0.3 0.0 ~-~---+-----~ , 0.0 L..-_---L_"----'-----';.......oooooL..-......... --I -0.5 '--____ ...L..-____ .... -21£ o x 21£ -0.5 0.0 cosO Figure 2: The left hand plot shows the predicted density (solid curve) together with the true density (dashed curve) in the extended X space. The right hand plot shows the corresponding densities in the periodic () space. In both cases the input variable is fixed at x = 0.5. 0.5 One of the original motivations for developing the techniques described in this paper was to provide an effective, principled approach to the analysis of radar scatterometer data from satellites such as the European Remote Sensing Satellite ERS-I. This satellite is equipped with three C-band radar antennae which measure the total backscattered power (called 0"0) along three directions relative to the satellite track, as shown in Figure 3. When the satellite passes over the ocean, the strengths of the backscattered signals are related to the surface ripples of the water (on length-scales of a few cm.) which in turn are determined by the low level winds. Extraction of the wind speed and direction from the radar signals represents an inverse problem which is typically multi-valued. For example, a wind direction of (}1 will give rise to similar radar signals to a wind direction of (}1 + 1r. Often, there are additional such 'aliases' at other angles. A conventional neural network approach to this problem, based on least-squares, would predict wind directions which were given by conditional averages of the target data. Since the average of several valid wind directions is typically not itself a valid direction, such an approach would clearly fail. Here we aim to extract the complete distribution of wind directions (as a function of the three 0"0 values and on the angle of incidence of the radar beam) and hence avoid Estimating Conditional Probability Densities for Periodic Variables satellite 785km 5COkm aft beam mid beam fore beam Figure 3: Schematic illustration of the ERS-l satellite showing the footprints of the three radar scatterometers. 647 such difficulties. This approach also provides the most complete information for the next stage of processing (not considered here) which is to 'de-alias' the wind directions to extract the most probable overall wind field. A large data set of ERS-l measurements, spanning a wide range of meteorological conditions, has been assembled by the European Space Agency in collaboration with the UK Meteorological Office. Labelling of the data set was performed using wind vectors from the Meteorological Office Numerical Weather Prediction code. An example of the results from the fixed-kernel method of Section 2.3 are presented in Figure 4. This clearly shows the existence of a primary alias at an angle of 1r relative to the principal direction, as well as secondary aliases at ±1r /2. Acknowledgements We are grateful to the European Space Agency and the UK Meteorological Office for making available the ERS-l data. We would also like to thank lain Strachan and Ian Kirk of AEA Technology for a number of useful discussions relating to the interpretation of this data. References Bishop C M (1994). Mixture density networks. Neural Computing Research Group Report, NCRG/4288, Department of Computer Science, Aston University, Birmingham, U.K. Jacobs R A, Jordan M I, Nowlan S J and Hinton G E (1991). Adaptive mixtures 648 Chris M. Bishop, Claire Leg/eye 0.5 ~--~---r---~--""'" 0.0 1--------.....;;::~~-__1 -0.5 -1.0 -1.5 ~ __ ~_----II.....-_---I __ --J -1.5 -1.0 -0.5 0.0 0.5 Figure 4: An example of the results obtained with the fixed-kernel method applied to data from the ERS-1 satellite. As well as the primary wind direction, there are aliases at 1r and ±1r /2. of local experts. Neural Computation, 3 79-87. Lui Y (1994) Robust parameter estimation and model selection for neural network regression. Advances in Neural Information Processing Systems 6 Morgan Kaufmann, 192-199 .. Mardia K V (1972). Statistics of Directional Data. Academic Press, London. White H (1992). Parametric statistical estimation with artificial neural networks. University of California, San Diego, Technical Report.	1994	62
957	Analysis of Unstandardized Contributions in Cross Connected Networks Thomas R. Shultz shultz@psych.mcgill.ca Yuriko Oshima-Takane yuriko@psych.mcgill.ca Department of Psychology McGill University Montreal, Quebec, Canada H3A IBI Abstract Yoshio Takane takane@psych.mcgill.ca Understanding knowledge representations in neural nets has been a difficult problem. Principal components analysis (PCA) of contributions (products of sending activations and connection weights) has yielded valuable insights into knowledge representations, but much of this work has focused on the correlation matrix of contributions. The present work shows that analyzing the variance-covariance matrix of contributions yields more valid insights by taking account of weights. 1 INTRODUCTION The knowledge representations learned by neural networks are usually difficult to understand because of the non-linear properties of these nets and the fact that knowledge is often distributed across many units. Standard network analysis techniques, based on a network's connection weights or on its hidden unit activations, have been limited. Weight diagrams are typically complex and weights vary across mUltiple networks trained on the same problem. Analysis of activation patterns on hidden units is limited to nets with a single layer of hidden units without cross connections. Cross connections are direct connections that bypass intervening hidden unit layers. They increase learning speed in static networks by focusing on linear relations (Lang & Witbrock, 1988) and are a standard feature of generative algorithms such as cascadecorrelation (Fahlman & Lebiere, 1990). Because such cross connections do so much of the work, analyses that are restricted to hidden unit activations furnish only a partial picture of the network's knowledge. Contribution analysis has been shown to be a useful technique for multi-layer, cross connected nets. Sanger (1989) defined a contribution as the product of an output weight, the activation of a sending unit, and the sign of the output target for that input. Such contributions are potentially more informative than either weights alone or hidden unit activations alone since they take account of both weight and sending activation. Shultz and Elman (1994) used PCA to reduce the dimensionality of such contributions in several different types of cascade-correlation nets. Shultz and Oshima-Takane (1994) demonstrated that PCA of unscaled contributions produced even better insights into cascade-correlation solutions than did comparable analyses of contributions scaled by the sign of output targets. Sanger (1989) had recommended scaling contributions by the signs of output targets in order to determine whether the contributions helped or hindered the network's solution. But since the signs of output targets are only available to networks during error 602 Thomas R. Shultz, Yuriko Oshima-Takane, Yoshio Takane correction learning, it is more natural to use unscaled contributions in analyzing knowledge representations. There is an issue in PCA about whether to use the correlation matrix or the variancecovariance matrix. The correlation matrix contains Is in the diagonal and Pearson correlation coefficients between contributions off the diagonal. This has the effect of standardizing the variables (contributions) so that each has a mean of 0 and standard deviation of 1. Effectively, this ensures that the PCA of a correlation matrix exploits variation in input activation patterns but ignores variation in connection weights (because variation in connection weights is eliminated as the contributions are standardized). Here, we report on work that investigates whether more useful insights into network knowledge structures can be revealed by PCA of un standardized contributions. To do this, we apply PCA to the variance-covariance matrix of contributions. The variance-covariance matrix has contribution variances along the diagonal and covariances between contributions off the diagonal. Taking explicit account of the variation in connection weights in this way may produce a more valid picture of the network's knowledge. We use some of the same networks and problems employed in our earlier work (Shultz & Elman, 1994; Shultz & Oshima-Takane, 1994) to facilitate comparison of results. The problems include continuous XOR, arithmetic comparisons involving addition and mUltiplication, and distinguishing between two interlocking spirals. All of the nets were generated with the cascade-correlation algorithm (Fahlman & Lebiere, 1990). Cascade-correlation begins as a perceptron and recruits hidden units into the network as it needs them in order to reduce error. The recruited hidden unit is the one whose activations correlate best with the network's current error. Recruited units are installed in a cascade, each on a separate layer and receiving input from the input units and from any previously existing hidden units. We used the default values for all cascade-correlation parameters. The goal of understanding knowledge representations learned by networks ought to be useful in a variety of contexts. One such context is cognitive modeling, where the ability of nets to merely simulate psychological phenomena is not sufficient (McCloskey, 1991). In addition, it is important to determine whether the network representations bear any systematic relation to the representations employed by human subjects . 2 PCA OF CONTRIBUTIONS Sanger's (1989) original contribution analysis began with a three-dimensional array of contributions (output unit x hidden unit x input pattern). In contrast, we start with a twodimensional output weight x input pattern array of contributions. This is more efficient than the slicing technique used by Sanger to focus on particular output or hidden units and still allows identification of the roles of specific contributions (Shultz & Elman, 1994; Shultz & Oshima-Takane, 1994). We subject the variance-covariance matrix of contributions to PCA in order to identify the main dimensions of variation in the contributions (Jolliffe, 1986). A component is a line of best fit to a set of data points in multi-dimensional space. The goal of PCA is to summarize a multivariate data set with a relatively small number of components by capitalizing on covariance among the variables (in this case, contributions). We use the scree test (Cattell, 1966) to determine how many components are useful to include in the analysis. Varimax rotation is applied to improve the interpretability of the solution. Component scores are plotted to identify the function of each component 3 APPLICATION TO CONTINUOUS XOR The classical binary XOR problem does not have enough training patterns to make contribution analysis worthwhile. However, we constructed a continuous version of the XOR problem by dividing the input space into four quadrants. Starting from 0.1, input values were incremented in steps of 0.1, producing 100 x, y input pairs that can be partitioned into four quadrants of the input space. Quadrant a had values of x less than Analysis of Unstandardized Contributions in Cross Connected Networks 603 0.55 combined with values of y above 0.55. Quadrant b had values of x and y greater than 0.55. Quadrant c had values of x and y less than 0.55. Quadrant d had values of x greater than 0.55 combined with values of y below 0.55. Similar to binary XOR, problems from quadrants a and d had a positive output target (0.5) for the net, whereas problems from quadrants band c had a negative output target (-0.5). There was a single output unit with a sigmoid activation. Three cascade-correlation nets were trained on continuous XOR. Each of these nets generated a unique solution, recruiting five or six hidden units and taking from 541 to 765 epochs to learn to correctly classify all of the input patterns. Generalization to test patterns not in the training set was excellent. PCA of unscaled, unstandardized contributions yielded three components. A plot of rotated component scores for the 100 training patterns of net 1 is shown in Figure 1. The component scores are labeled according to their respective quadrant in the input space. Three components are required to account for 96.0% of the variance in the contributions. Figure 1 shows that component 1, with 44.3% of the variance in contributions, has the role of distinguishing those quadrants with a positive output target (a and d) from those with a negative output target (b and c). This is indicated by the fact that the black shapes are at the top of the component space cube in Figure 1 and the white shapes are at the bottom. Components 2 and 3 represent variation along the x and y input dimensions, respectively. Component 2 accounted for 26.1 % of the variance in contributions, and component 3 accounted for 25.6% of the variance in contributions. Input pairs from quadrants b and d (square shapes) are concentrated on the negative end of component 2, whereas input pairs from quadrants a and c (circle shapes) are concentrated on the positive end of component 2. Similarly, input pairs from quadrants a and b cluster on the negative end of component 3, and input pairs from quadrants c and d cluster on the positive end of component 3. Although the network was not explicitly trained to represent the x and y input dimensions, it did so as an incidental feature of its learning the distinction between quadrants a and d vs. quadrants band c. Similar results were obtained from the other two nets learning the continuous XOR problem. In contrast, PCA of the correlation matrix from these nets had yielded a somewhat less clear picture with the third component separating quadrants a and d from quadrants b and c, and the first two components representing variation along the x and y input dimensions (Shultz & Oshima-Takane, 1994). PCA of the correlation matrix of scaled contributions had performed even worse, with plots of component scores indicating interactive separation of the four quadrants, but with no clear roles for the individual components (Shultz & Elman, 1994). Standardized, rotated component loadings for net 1 are plotted in Figure 2. Such plots can be examined to determine the role played by each contribution in the network. For example, hidden units 2, 3, and 4 all playa major role in the job done by component 1, distinguishing positive from negative outputs. 4 APPLICATION TO COMPARATIVE ARITHMETIC Arithmetic comparison requires a net to conclude whether a sum or a product of two integers is greater than, less than, or equal to a comparison integer. Several psychological simulations have used neural nets to make additive and multiplicative comparisons and this has enhanced interest in this type of problem (McClelland, 1989; Shultz, Schmidt, Buckingham, & Mareschal, in press). The first input unit coded the type of arithmetic operation to be performed: 0 for addition and 1 for multiplication. Three additional linear input units encoded the integers. Two of these input units each coded a randomly selected integer in the range of 0 to 9, inclusive; another input unit coded a randomly selected comparison integer. For addition problems, comparison integers ranged from 0 to i9, inclusive; for multiplication, comparison integers ranged from 0 to 82, inclusive. Two sigmoid output units coded the results of the comparison operation. Target outputs of 0.5, -0.5 represented a greater than result, targets of -0.5, 0.5 represented less than, and targets of 0.5,0.5 represented equal to. 604 Thomas R. Shultz, Yuriko Oshima-Takane, Yoshio Takane 2 Component 1 o -1 3 2 Component 2 -2 -2 Component 3 Figure 1. Rotated component scores for a continuous XOR net. Component scores for the x, y input pairs in quadrant a are labeled with black circles, those from quadrant b with white squares, those from quadrant c with white circles, and those from quadrant d with black squares. The network's task is to distinguish pairs from quadrants a and d (the black shapes) from pairs from quadrants b and c (the white shapes). Some of the white shapes appear black because they are so densely packed, but all of the truly black shapes are relatively high in the cube. Hidden6 Hidden5 Hidden4 c: 0 Hidden3 '5 :g C Hidden2 0 () Hidden1 Input2 Input1 -1.0 Component III 3 II 2 • -0.5 0.0 Loading 0.5 1.0 Figure 2. Standardized, rotated component loadings for a continuous XOR net. Rotated loadings were standardized by dividing them by the standard deviation of the respective contribution scores. Analysis of Unstandardized Contributions in Cross Connected Networks 605 The training patterns had 100 addition and 100 multiplication problems, randomly selected, with the restriction that 45 of each had correct answers of greater than, 45 of each had correct answers of less than, and 10 of each had correct answers of equal to. These constraints were designed to reduce the natural skew of comparative values in the high direction on multiplication problems. We ran three nets for 1000 epochs each, at which point they were very close to mastering the training patterns. Either seven or eight hidden units were recruited along the way. Generalization to previously unseen test problems was very accurate. Four components were sufficient to account for most the variance in un standardized contributions, 88.9% in the case of net 1. Figure 3 displays the rotated component scores for the first two components of net 1. Component I, accounting for 51.1 % of the variance, separated problems with greater than answers from problems with less than answers, and located problems with equal to answers in the middle, at least for addition problems. Component 2, with 20.2% of the variance, clearly separated multiplication from addition. Contributions from the first input unit were strongly associated with component 2. Similar results obtained for the other two nets. Components 3 and 4, with 10.6% and 7.0% of the variance, were sensitive to variation in the second and third inputs, respectively. This is supported by an examination of the mean input values of the 20 most extreme component scores on these two components. Recall that the second and third inputs coded the two integers to be added or multiplied. The negative end of component 3 had a mean second input value of 8.25; the positive end of this component had a mean second input value of 0.55. Component 4 had mean third input value of 2.00 on the negative end and 7.55 on the positive end. In contrast, PCA of the correlation matrix for these nets had yielded a far more clouded picture, with the largest components focusing on input variation and lesser components doing bits and pieces of the separation of answer types and operations in an interactive manner (ShUltz & Oshima-Takane, 1994). Problems with equal to answers were not isolated by any of the components. PCA of scaled contributions had produced three components that interactively separated the three answer types and operations, but failed to represent variation in input integers (ShUltz & Elman, 1994). Essentially similar advantages for using the variance-covariance matrix were found for nets learning either addition alone or multiplication alone. 5 APPLICATION TO THE TWO-SPIRALS PROBLEM The two-spirals problem requires a particularly difficult discrimination and a large number of hidden units. The input space is defined by two interlocking spirals that wrap around their origin three times. There are two sets of 97 real-valued x, y pairs, with each set representing one of the spirals, and a single sigmoid output unit coded for the identity of the spiral. Our three nets took between 1313 and 1723 epochs to master the distinction, and recruited from 12 to 16 hidden units. All three nets generalized well to previously unseen input pairs on the paths of the two spirals. PCA of the variance-covariance matrix for net 1 revealed that six components accounted for a total of 97.9% of the variance in contributions. The second and fourth of these components together distinguished one spiral from the other, with 20.7% and 9.8% of the variance respectively. Rotated component scores for these two components are plotted in Figure 4. A diagonal line drawn on Figure 4 from coordinates -2,2 to 2, -2 indicates that 11 points from each spiral were misclassified by components 2 and 4. This is only 11.3% of the data points in the training patterns. The fact that the net learned all of the training patterns implies that these exceptions were picked up by other components. Components 1 and 6, with 40.7% and 6.4% of the variance, were sensitive to variation in the x and y inputs, respectively. Again, this was confirmed by the mean input values of the 20 most extreme component scores on these two components. On component I, the negative end had a mean x value of 3.55 and the positive end had a mean y value of -3.55. 606 Component 1 2 o -1 Thomas R. Shultz. Yuriko Oshima-Takane. Yoshio Takane x> It" #~ • . ·x +< -2 L..-__ ......L ___ ....L... __ --iL..-__ -J -2 -1 0 2 Component 2 Figure 3. Rotated component scores for an arithmetic comparison net. Greater than problems are symbolized by circles, less than problems by squares, addition by white shapes, and multiplication by black shapes. For equal to problems only, addition is represented by + and multiplication by X. Although some densely packed white shapes may appear black, they have no overlap with truly black shapes. All of the black squares are concentrated around coordinates -1, -1. 2 Component 2 o o o o -1 o Spiral 1 -2 1....-__ --' ___ ....... ___ ........ __ _ -2 -1 o 2 Component 4 Figure 4. Rotated component scores for a two-spirals net. Squares represent data points from spiral 1, and circles represent data points from spiral 2. Analysis of Unstandardized Contributions in Cross Connected Networks On component 6, the negative end had a mean x value of 2.75 and the positive end had a mean y value of -2.75. The skew-symmetry of these means is indicative of the perfectly symmetrical representations that cascade-correlation nets achieve on this highly symmetrical problem. Every data point on every component has a mirror image negative with the opposite signed component score on that same component. This -x, -y mirror image point is always on the other spiral. Other components concentrated on particular regions of the spirals. The other two nets yielded essentially similar results. These results can be contrasted with our previous analyses of the two-spirals problem, none of which succeeded in showing a clear separation of the two spirals. PCAs based on scaled (Shultz & Elman, 1994) or unscaled (Shultz & Oshima-Takane, 1994) correlation matrices showed extensive symmetries but never a distinction between one spiral and another.1 Thus, although it was clear that the nets had encoded the problem's inherent symmetries, it was still unclear from previous work how the nets used this or other information to distinguish points on one spiral from points on the other spiral. 6 DISCUSSION 607 On each of these problems, there was considerable variation among network solutions, as revealed, for example, by variation in numbers of hidden units recruited and signs and sizes of connection weights. In spite of such variation, the present technique of applying peA to the variance-covariance matrix of contributions yielded results that are sufficiently abstract to characterize different nets learning the same problem. The knowledge representations produced by this analysis clearly identify the essential information that the net is being trained to utilize as well as more incidental features of the training patterns such as the nature of the input space. This research strengthens earlier conclusions that PCA of network contributions is a useful technique for understanding network performance (Sanger, 1989), including relatively intractable multi-level cross connected nets (Shultz & Elman, 1994; Shultz & Oshima-Takane, 1994). However, the current study underscores the point that there are several ways to prepare a contribution matrix for PCA, not all of which yield equally valid or useful results. Rather than starting with a three dimensional matrix of output unit x hidden unit x input pattern and focusing on either one output unit at a time or one hidden unit at a time (Sanger, 1989), it is preferable to collapse contributions into a two dimensional matrix of output weight x input pattern. The latter is not only more efficient, but yields more valid results that characterize the network as a whole, rather than small parts of the network. Also, rather than scaling contributions by the sign of the output target (Sanger, 1989), it is better to use unsealed contributions. Unsealed contributions are not only more realistic, since the network has no knowledge of output targets during its feed-forward phase, but also produce clearer interpretations of the nefs knowledge representations (Shultz & Oshima-Takane, 1994). The latter claim is particularly true in terms of sensitivity to input dimensions and to operational distinctions between adding and multiplying. Plots of component scores based on unscaled contributions are typically not as dense as those based on sealed contributions but are more revealing of the network's knowledge. Finally, rather than applying peA to the correlation matrix of contributions, it makes more sense to apply it to the variance-covariance matrix. As noted in the introduction, using the correlation matrix effectively standardizes the contributions to have identical means and variances, thus obseuring the role of network connection weights. The present results indicate much clearer knowledge representations when the variance-covariance matrix is used since connection weight information is explicitly retained. Matrix differences were especially marked on the more difficult problems, such as two-spirals, where the only peAs to reveal how nets distinguished the spirals were those based on 1 Results from un scaled contributions on the two-spirals problem were not actually presented in Shultz & Oshima-Takane (1994) since they were not very clear. 608 Thomas R. Shultz, Yuriko Oshima-Takane, Yoshio Takane variance-covariance matrices. But the relative advantages of using the variance-covariance matrix were evident on the easier problems too. There has been recent rapid progress in the study of the knowledge representations leamed by neural nets. Feed-forward nets can be viewed as function approximators for relating inputs to outputs. Analysis of their knowledge representations should reveal how inputs are encoded and transformed to produce the correct outputs. PCA of network contributions sheds light on how these function approximations are done. Components emerging from PCA are orthonormalized ingredients of the transformations of inputs that produce the correct outputs. Thus, PCA helps to identify the nature of the required transformations. Further progress might be expected from combining PCA with other matrix decomposition techniques. Constrained PCA uses external information to decompose multivariate data matrices before applying PCA (Takane & Shibayama, 1991). Analysis techniques emerging from this research will be useful in understanding and applying neural net research. Component loadings, for example, could be used to predict the results of lesioning experiments with neural nets. Once the role of a hidden unit has been identified by virtue of its association with a particular component, then one could predict that lesioning this unit would impair the function served by the component. Acknowledgments This research was supported by the Natural Sciences and Engineering Research Council of Canada. References Cattell, R. B. (1966). The scree test for the number of factors. Multivariate Behavioral Research, 1,245-276. Fahlman, S. E., & Lebiere, C. (1990.) The Cascade-Correlation learning architecture. In D. Touretzky (Ed.), Advances in neural information processing systems 2, (pp. 524532). Mountain View, CA: Morgan Kaufmann. Jolliffe, I. T. (1986). Principal component analysis. Berlin: Springer Verlag. Lang, K. J., & Wi tbrock , M. J. (1988). Learning to tell two spirals apart. In D. Touretzky, G. Hinton, & T. Sejnowski (Eds)., Proceedings of the Connectionist Models Summer School, (pp. 52-59). Mountain View, CA: Morgan Kaufmann. McClelland, J. L. (1989). Parallel distributed processing: Implications for cognition and development. In Morris, R. G. M. (Ed.), Parallel distributed processing: Implications for psychology and neurobiology, pp. 8-45. Oxford University Press. McCloskey, M. (1991). Networks and theories: The place of connectionism in cognitive science. Psychological Science, 2, 387-395. Sanger, D. (1989). Contribution analysis: A technique for assigning responsibilities to hidden units in connectionist networks. Connection Science, 1, 115-138. Shultz, T. R., & Elman, J. L. (1994). Analyzing cross connected networks. In J. D. Cowan, G. Tesauro, & J. Alspector (Eds.), Advances in Neural Information Processing Systems 6. San Francisco, CA: Morgan Kaufmann. ShUltz, T. R., & Oshima-Takane, Y. (1994). Analysis of un scaled contributions in cross connected networks. In Proceedings of the World Congress on Neural Networks (Vol. 3, pp. 690-695). Hillsdale, NJ: Lawrence Erlbaum. Shultz, T. R., Schmidt, W. C., Buckingham, D., & Mareschal, D. (In press). Modeling cognitive development with a generative connectionist algorithm. In G. Halford & T. Simon (Eds.), Developing cognitive competence: New approaches to process modeling. Hillsdale, NJ: Erlbaum. Takane, Y., & Shibayama, T. (1991). Principal component analysis with external information on both subjects and variables. Psychometrika, 56, 97-120.	1994	63
958	A Rigorous Analysis Of Linsker-type Hebbian Learning J. Feng Mathematical Department University of Rome "La Sapienza» P. Ie A. Moro, 00185 Rome, Italy feng~at.uniroma1.it H. Pan V. P. Roychowdhury School of Electrical Engineering Purdue University West Lafayette, IN 47907 hpan~ecn.purdue.edu vwani~drum.ecn.purdue.edu Abstract We propose a novel rigorous approach for the analysis of Linsker's unsupervised Hebbian learning network. The behavior of this model is determined by the underlying nonlinear dynamics which are parameterized by a set of parameters originating from the Hebbian rule and the arbor density of the synapses. These parameters determine the presence or absence of a specific receptive field (also referred to as a 'connection pattern') as a saturated fixed point attractor of the model. In this paper, we perform a qualitative analysis of the underlying nonlinear dynamics over the parameter space, determine the effects of the system parameters on the emergence of various receptive fields, and predict precisely within which parameter regime the network will have the potential to develop a specially designated connection pattern. In particular, this approach exposes, for the first time, the crucial role played by the synaptic density functions, and provides a complete precise picture of the parameter space that defines the relationships among the different receptive fields. Our theoretical predictions are confirmed by numerical simulations. 320 lian/eng Feng, H. Pan, V. P. Roychowdhury 1 Introduction For the purpose of understanding the self-organization mechanism of primary visual system, Linsker has proposed a multilayered unsupervised Hebbian learning network with random un correlated inputs and localized arborization of synapses between adjacent layers (Linsker, 1986 & 1988). His simulations have shown that for appropriate parameter regimes, several structured connection patterns (e.g., centre-surround and oriented afferent receptive fields (aRFs)) occur progressively as the Hebbian evolution of the weights is carried out layer by layer. The behavior of Linsker's model is determined by the underlying nonlinear dynamics which are parameterized by a set of parameters originating from the Hebbian rule and the arbor density of the synapses. For a nonlinear system, usually, there coexist several attractors for the same set of system parameters. That is, for a given set of the parameters, the state space comprises several attractive basins, each corresponding to a steady state respectively. The initial condition determines which attractor will be eventually reached. At the same time, a nonlinear system could have a different group of coexisting attractors for a different set of system parameters. That is, one could make the presence or absence of a specific state as a fixed point attractor by varying the set of the parameters. For a development model like Linsker's network, what is expected to be observed is that the different aRFs could emerge under different sets of parameters but should be relatively not sensitive to the initial conditions. In other words, the dynamics should avoid the coexistence of several attractors in an appropriate way. The purpose of this paper is to gain more insights into the dynamical mechanism of this self-organization model by performing a rigorous analysis on its parameter space without any approximation. That is, our goal is to reveal the effects of the system parameters on the stability of aRFs, and to predict precisely within which parameter regime the network will have the potential to develop a specially designated aRF. The novel rigorous approach presented here applies not only to the Linsker-type Hebbian learning but also to other related self-organization models about neural development. In Linsker's network, each cell in the present layer M receives synaptic inputs from a number of cells in the preceding layer C. The density of these synaptic connections decreases monotonically with distance rC, from the point underlying the M-cell's position. Since the synaptic weights change on a long time scale compared to the variation of random inputs, by averaging the Hebb rule over the ensemble of inputs in layer C, the dynamical equation for the development of the synaptic strength wT(i) between a M-cell and i-th C-cell at time Tis Nc. WT+1 (i) = f{wT(i) + k1 + l:)Qj + k2]r(j)wT(j)} (1) j=l where k1' k2 are system parameters which are particular combinations of the constants of the Hebb rule, r(·) is a non-negative normalized synaptic density function (SDF) 1, and L:iEC, rei) = 1, and 10 is a limiter function defined by I(x) = Wmax , if x > W max ; = x, if I x I ~ W max ; and = -Wmax , if x < -wmax • The covariance IThe SDF is explicitly incorporated into the dynamics (1) which is equivalent to Linsker's formulation. A rigorous explanation for this equivalence is given in MacKay & Miller, 1990. A Rigorous Analysis of Linsker-type Hebbian Learning 321 matrix {Qij} of the layer C describes the correlation of activities of the i-th and the j-th C-cells. Actually, the covariance matrix of each layer is determined by SDFs r(·) of all layers preceding the layer under consideration. The idea of this paper is the following. It is well known that in general it is intractable to characterize the behavior of a nonlinear dynamics, since the nonlinearity is the cause of the coexistence of many attractors. And one has the difficulty in obtaining the complete characteristics of attractive basins in the state space. But usually for some cases, it is relatively easy to derive a necessary and sufficient condition to check whether a given state is a fixed point of the dynamics. In terms of this condition, the whole parameter regime for the emergence of a fixed point of the dynamics may be obtained in the parameter space. If we are further able to prove the stability of the fixed point, which implies that this fixed point is a steady state if the initial condition is in a non empty vicinity in the state space, we can assert the occurrence of this fixed point attractor in that parameter regime. For Linsker's network, fortunately, the above idea can be carried out because of the specific form of the nonlinear function 1(·). Due to space limitations, the rigorous proofs are in (Feng, Pan, & Roychowdhury, 1995). 2 The Set Of Saturated Fixed Point Attractors And The Criterion For The Division Of Parameter Regimes In fact, Linsker's model is a system of first-order nonlinear difference equations, taking the form wr+l(i) = j[wr(i) + hi(Wr' kl' k2)], Wr = {wr(j),j = 1, ... , Nc}, (2) where hi(Wr , kl' k2) = kl + 2:f~dQ~ + k2]r(j)wr(j). And the aRFs observed in Linsker's simulation are the saturated fixed point attractors of this nonlinear system (2). Since the limiter function 1(·) is defined on a hypercube n = [-wmax , wmax]N C in weight state space within which the dynamics is dominated by the linear system wr+l(i) = wr(i) + hi(Wr, kl' k2) , the short-time behaviors of evolution dynamics of connection patterns can be fully characterized in terms of the properties of eigenvectors and their eigenvalues. But this method of stability analysis will not be suitable for the long-time evolution of equation (1) or (2), provided the hypercube constraint is reached as the first largest component of W reaches saturation. However, it is well-known that a fixed point or an equilibrium state of dynamics (2) satisfies (3) Because of the special form of the nonlinear function 1(·), the fixed point equation (3) implies that 3T, such that for T > T, I wr(i) + hi(Wr, kl' k2) I ~ Wmax , if hi (w, kl , k2) i= o. So a saturated fixed point Wr (i) must have the same sign as hi(wr, kl' k 2 ), i.e. wr(i)hi(Wr, kl' k2) > o. By using the above idea, our Theorems 1 & 2 (proven in Feng, Pan, & Roychowdhury, 1995) state that the set of saturated fixed point attractors of the dynamics in 322 lian/eng Feng, H. Pan, V. P. Roychowdhury equation (1) is given by nFP = {w I w(i)hi(Wr ,kl,k2) > 0,1::; i::; N.d, and w E nFP is stable, where the weight vector w belongs to the set of all extreme points of the hypercube n (we assume W max = 1 without loss of generality). We next derive an explicit necessary and sufficient condition for the emergence of structured aRFs, i.e., we derive conditions to determine whether a given w belongs to nFP. Define J+(w) = {i I wei) = 1} as the index set of cells at the preceding layer C with excitatory weight for a connection pattern w, and J-(w) = {i I wei) = -1} as the index set of C-cells with inhibitory weight for w. Note from the property of fixed point attractors that a connection pattern w is an attractor of the dynamics (1) if and only if for i E J+(w), we have w(i){k1 + l)Q~ + k2]r(j)w(j)} = j w(i){k1 + EjEJ+(w)[Qt + k2]r(j)w(j) + EjEJ-(w)[Qt + k2]r(j)w(j)} > O. By the definition of J+(w) and J-(w), we deduce from the above inequality that kl + 2: [Q~ + k2]r(j) - 2: [Q~ + k2]r(j) > 0 jEJ+(w) jEJ-(w) namely kl + k2[ 2: r(j) - 2: r(j)] > 2: Q~r(j) - 2: Q~r(j). jEJ+(w) jEJ-(w) jEJ-(w) jEJ+(w) Inequality above is satisfied for all i in J+(w), and the left hand is independent of i. Hence, kl + k2[ 2: r(j) - 2: r(j)] > . max [ I: Q~r(j) - I: Q~r(j)]. \EJ+(w) jEJ+(w) jEJ-(w) jEJ-(w) jEJ+(w) On the other hand, for i E J-(w), we can similarly deduce that kl + k2[ 2: r(j) - 2: r(j)] < . min [ 2: Q~r(j) - 2: Q~r(j)] . IEJ-(w) jEJ+(w) jEJ-(w) jEJ-(w) jEJ+(w) We introduce the slope function: c( w) ~f 2: r(j) - 2: r(j) jEJ+(w) jEJ-(w) which is the difference of sums of the SDF r(·) over J+(w) and J-(w), and two kl -intercept functions: d1(w) ~ { maxtEJ+(w)(EjEJ-(w) Qf;r(j) - EjEJ+(w) Qtr(j)), if J+(w) # 0 -00, if J+(w) = 0 A Rigorous Analysis of Linsker-type Hehbian Learning (a) (b) o A ~---k , E o 323 Figure 1: The parameter subspace of (kl, k2 ). (a) Parameter regime of (kl' k 2 ) to ensure the emergence of all-excitatory (regime A) and all-inhibitory (regime B) connection patterns. The dark grey regime C is the coexistence regime for both all-excitatory and all-inhibitory connection patterns. And the regime D without texture are the regime that Linsker's simulation results are based on, in which both all-excitatory and all-inhibitory connection patterns are no longer an attractor. (b) The principal parameter regimes. Now from our Theorem 3 in Feng, Pan, & Roychowdhury, 1995, for every layer of Linsker's network, the new rigorous criterion for the division of stable parameter regimes to ensure the development of various structured connection patterns is d2(w) > ki + c(w)k2 > dl(w). That is, for a given SDF r(-), the parameter regime of (kl' k2) to ensure that w is a stable attractor of dynamics (1) is a band between two parallel lines ki + c(w)k2 > dl(w) and ki +c(w)k2 < d2(w) (See regimes E and F in Fig.1(b)). It is noticed that as dl(w) > d2(w), there is no regime of (kl' k2) for the occurrence of that aRF w as an attractor of equation (1). Therefore, the existence of such a structured aRF w as an attractor of equation (1) is determined by k1-intercept functions dIe) and d2 (·), and therefore by the covariance matrix Q.c or SDFs r(·) of all preceding layers. 3 Parameter Regimes For aRFs Between Layers BAnd C Based on our general theorems applicable to all layers, we mainly focus on describing the stabilization process of synaptic development from the 2nd (B) to the 3rd layer (C) by considering the effect of the system parameters on the weight development. For the sake of convenience, we assume that the input at 1st layer (A) is independent normal distribution with mean 0 and variance 1, and the connection strengths from layer A to B are all-excitatory same as in Linsker's simulations. The emergence of various aRFs between layer Band C have been previously studied in the literature, and in this paper we mention only the following new results made possible by our approach: (1) For the cell in layer C, the all-excitatory and the all-inhibitory connection patterns still have the largest stable regimes. Denote both SDFs from layer A to B and from B to C as rAB( .,. ) and rBC (-) respectively. The parameter plane of (kl' k2) 324 lianfeng F eng, H. Pan, V. P. Roychowdhury Table 1: The Principal Parameter Regimes TYPE Regime A Regime B Regime =AnB Regime F d2{w) > kl + C{W)k2 > d1(w) where c(w) < 0 Regime G d2(W1) > kl + c(w1)k2 > d1 (wI) =EnFnAnB is divided into four regimes by for all-excitatory pattern and for all-inhibitory pattern (See Fig.I(a». ATTRACTOR All-excitatory aRF All-inhibitory aRF All-excitatory and all-inhibitory aRFs coexist The structured aRFs may have separate parameter regimes Any connection pattern in which the excitatory connections constitute the majority Any connection pattern in which the inhibitory connections constitute the majority A small coexistence regime of many connection patterns around the origin point of the parameter plane of ( kl , k2) (2) The parameter with large and negative k2 and approximately -1 < -kdk2 < 1 is favorable for the emergence of various structured connection patterns (e.g., ONcenter cells, OFF-center cells, bi-Iobed cells, and oriented cells). This is because this regime (See regime D in Fig.I) is removed from the parameter regime where both all-excitatory and all-inhibitory aRFs are dominant, including the coexistence regime of many kind of at tractors around the origin point (See regime G in Fig.I(b ». The above results provide a precise picture about the principal parameter regimes summarized in Table 1. (3) The relative size of the radiuses of two SDFs r AS(-,.) and r Sc (-) plays a key role in the evolution of various structured aRFs from B to C. A given SDF r.cM (i, j), i E M, j E e will be said to have a range r M if r.cM (i, j) is 'sufficient small' for lIi-jll ~ rM. For a Gaussian SDF r.cM(j,k) '" exp(-lIj-kll/r~), j E e,k EM, the range r M is its standard deviation. We give the analytic prediction about the influence of the SDF's ranges rs, rc on the dynamics by changing rs from the smallest extreme to the largest one with respect to rc. For the smallest extreme of rs (i.e. the A Rigorous Analysis of Linsker-type Hebbian Learning 325 synaptic connections from A to B are concentrated enough, and those from layer B to C are fully feedforward connected), we proved that any kind of connection pattern has a stable parameter regime and emerge under certain parameters, because each synaptic connection within an aRF is developed independently. As rB is changed from the smallest to the largest extreme, the development of synaptic connections between layer Band C will depend on each other stronger and stronger in the sense that most of connections have the same sign as their neighbors in an aRF. So for the largest extreme of rB (i.e. the weights from layer A to B are fully feedforward but there is no constraint on the SDF rBC(.)), any structured aRFs except for the all-excitatory and the all-inhibitory connection patterns will never arise at all, although there exist correlation in input activities (for a proof see Feng, Pan, & Roychowdhury, 1995). Th_erefore, without localized SDF, there would be no structured covariance matrix Q = {[Qij + k2]r(j)} which embodies localized correlation in afferent activities. And without structured covariance matrix Q, no structured aRFs would emerge. (4) As another application of our analyses, we present several numerical results on the parameter regimes of (kl' k2' rB, rc) for the formation of various structured aRFs (Feng & Pan, 1993; Feng, Pan, & Roychowdhury, 1995) (where we assume that rAB(i,j) "" exp(-lli-jll/r~), i E B,j E A, and rBC(i) "" exp(-llill/r2), i E B as in (Linsker, 1986 & 1988)). For example, we show that various aRFs as at tractors have different relative stability. For a fixed rc, the SDF's range rB of the preceding layer as the third system parameter has various critical values for different attractors. That is, an attractor will no longer be stable if rB exceeds its corresponding critical value (See Fig. 2). For circularly symmetric ON-center cells, those aRFs with large ON-center core (which have positive or small negative slope value c(w) ~ -kt/k2 ) always have a stable parameter regime. But for those ON-center cells with large negative slope value c(w), their stable parameter regimes decrease in size with c(w). Similarly, circularly symmetric OFF-center cells with large OFF-center core (which have negative or small positive slope value c( w)) will be more stable than those with large positive average of weights. But for non-circularly-symmetric patterns (e.g., bi-Iobed cells and oriented cells), only those at tractors with zero average synaptic strength might always have a stable parameter regime (See regime H in Fig.1(b)). If the third parameter rB is large enough to exceed its critical values for other aRFs and k2 is large and negative, then ON-center aRFs with positive c(w) and OFFcenter aRFs with negative c(w) will be almost only at tractors in regime DnE and regime DnF respectively. This conclusion makes it clear why we usually obtain ON-center aRFs in regime DnE and OFF-center aRFs in regime DnF much more easily than other patterns. 4 Concluding Remarks One advantage of our rigorous approach to this kind of unsupervised Hebbian learning network is that, without approximation, it unifies the treatment of many diverse problems about dynamical mechanisms. It is important to notice that there is no assumption on the second item hi(wT ) on the right hand side of equation (1), and there is no restriction on the matrix Q. Our Theorems 1 and 2 provide the general framework for the description of the fixed point attractors for any difference equation of the type stated in (2) that uses a limiter function. Depending on the 326 Jianfeng Feng, H. Pan. V. P. Roychowdhury ~ __ -O-N-C-e-"-te-r-a--,R_FS~~ ____ .(r c= 10) ~~ __ -O-rie-"-ted---,a.R.-Fs------._,(r c= 10) .I~~~~~~~~~·~' .I~~~~~~~~~·~ 000@@@@ ••• 000~~~~ ••• ~ __ .O .... FF.-c .... 8n ___ ter ___ a-R-Fs---__ (r c= 10) ~ __ -B-I-I-obed--a-R-F .. s ..--___ (rc= 10) .I~~~~~~~~~·~' ooooo~a ••• Figure 2: The critical values of the SDF's range ra for different connection patterns. structure of the second item, hi(wT ), it is not difficult to adapt our Theorem 3 to obtain the precise relationship among system parameters in other kind of models as long as 10 is a limiter function. Since the functions in the necessary and sufficient condition are computable (like our slope and k1-intercept functions), one is always able to check whether a designated fixed point is stable for a specific set of parameters. Acknowledgements The work ofV. P. Roychowdhury and H. Pan was supported in part by the General Motors Faculty Fellowship and by the NSF Grant No. ECS-9308814. J. Feng was partially supported by Chinese National Key Project of Fundamental Research "Climbing Program" and CNR of Italy. References R. Linsker. (1986) From basic network principle to neural architecture (series). Proc. Natl. Acad. Sci. USA 83: 7508-7512,8390-8394,8779-8783. R. Linsker. (1988) Self-organization in a perceptual network. Computer 21(3): 105-117. D. MacKay, & K. Miller. (1990) Analysis of Linsker's application of Hebbian rules to linear networks. Network 1: 257-297. J. Feng, & H. Pan. (1993) Analysis of Linsker-type Hebbian learning: Rigorous results. Proc. 1993 IEEE Int. Con! on Neural Networks - San Francisco Vol. III, 1516-1521. Piscataway, NJ: IEEE. J. Feng, H. Pan, & V. P. Roy chowdhury. (1995) Linsker-type Hebbian learning: A qualitative analysis on the parameter space. (submitted).	1994	64
959	Associative Decorrelation Dynamics: A Theory of Self-Organization and Optimization in Feedback Networks Dawei W. Dong* Lawrence Berkeley Laboratory University of California Berkeley, CA 94720 Abstract This paper outlines a dynamic theory of development and adaptation in neural networks with feedback connections. Given input ensemble, the connections change in strength according to an associative learning rule and approach a stable state where the neuronal outputs are decorrelated. We apply this theory to primary visual cortex and examine the implications of the dynamical decorrelation of the activities of orientation selective cells by the intracortical connections. The theory gives a unified and quantitative explanation of the psychophysical experiments on orientation contrast and orientation adaptation. Using only one parameter, we achieve good agreements between the theoretical predictions and the experimental data. 1 Introduction The mammalian visual system is very effective in detecting the orientations of lines and most neurons in primary visual cortex selectively respond to oriented lines and form orientation columns [1). Why is the visual system organized as such? We *Present address: Rockefeller University, B272, 1230 York Avenue, NY, NY 10021-6399. 926 Dawei W Dong believe that the visual system is self-organized, in both long term development and short term adaptation, to ensure the optimal information processing. Linsker applied Hebbian learning to model the development of orientation selectivity and later proposed a principle of maximum information preservation in early visual pathways [2]. The focus of his work has been on the feedforward connections and in his model the feedback connections are isotropic and unchanged during the development of orientation columns; but the actual circuitry of visual cortex involves extensive, columnar specified feedback connections which exist even before functional columns appear in cat striate cortex [3]. Our earlier research emphasized the important role of the feedback connections in the development of the columnar structure in visual cortex. We developed a theoretical framework to help understand the dynamics of Hebbian learning in feedback networks and showed how the columnar structure originates from symmetry breaking in the development of the feedback connections (intracortical, or lateral connections within visual cortex) [4]. Figure 1 illustrates our theoretical predictions. The intracortical connections break symmetry and develop strip-like patterns with a characteristic wave length which is comparable to the developed intracortical inhibitory range and the LGN-cortex afferent range (left). The feedforward (LGN-cortex) connections develop under the influence of the symmetry breaking development of the intracortical connections. The developed feedforward connections for each cell form a receptive field which is orientation selective and nearby cells have similar orientation preference (right). Their orientations change in about the same period as the strip-like pattern of the intracortical connections. Figure 1: The results of the development of visual cortex with feedback connections. The simulated cortex consists of 48 X 48 neurons, each of which connects to 5 X 5 other cortical neurons (left) and receives inputs from 7 X 7 LGN neurons (right). In this figure, white inclicates positive connections and black inclicates negative connections. One can see that the change of receptive field's orientation (right) is highly correlated with the strip-like pattern of intracortical connections (left). Many aspects of our theoretical predictions agree qualitatively with neurobiological observations in primary visual cortex. Another way to test the idea of optimal Associative Correlation Dynamics 927 information processing or any self-organization theory is through quantitative psychophysical studies. The idea is to look for changes in perception following changes in input environments. The psychophysical experiments on orientation illusions offer some opportunities to test our theory on orientation selectivity. Orientation illusions are the effects that the perceived orientations of lines are affected by the neighboring (in time or space) oriented stimuli, which have been observed in many psychophysical experiments and were attributed to the inhibitory interactions between channels tuned to different orientations [5]. But there is no unified and quantitative explanation. Neurophysiological evidences support our earlier computational model in which intracortical inhibition plays the role of gain-control in orientation selectivity [6]. But in order for the gain-control mechanism to be effective to signals of different statistics, the system has to develop and adapt in different environments. In this paper we examine the implication of the hypothesis that the intracortical connections dynamically decorrelate the activities of orientation selective cells, i.e., the intracortical connections are actively adapted to the visual environment, such that the output activities of orientation selective cells are decorrelated. The dynamics which ensures such decorrelation through associative learning is outlined in the next section as the theoretical framework for the development and the adaptation of intracortical connections. We only emphasize the feedback connections in the following sections and assume that the feedforward connections developed orientation selectivities based on our earlier works. The quantitative comparisons of the theory and the experiments are presented in section 3. 2 Associative Decorrelation Dynamics There are two different kinds of variables in neural networks. One class of variables represents the activity of the nerve cells, or neurons. The other class of variables describes the synapses, or connections, between the nerve cells. A complete model of an adaptive neural system requires two sets of dynamical equations, one for each class of variables, to specify the evolution and behavior of the neural system. The set of equations describing the change of the state of activity of the neurons is dVi a-I = -Vi + ~T. .. v.. + 1(1) dt I L..J I}} I j in which a is a time constant, Tij is the strength of the synaptic connection from neuron j to neuron i, and Ii is the additional feedforward input to the neuron besides those described by the feedback connection matrix nj . A second set of equations describes the way the synapses change with time due to neuronal activity. The learning rule proposed here is B dnj = (V,. - V.')!, dt I I} (2) in which B is a time constant and Vi' is the feedback learning signal as described in the following. The feedback learning signal Vi' is generated by a Hopfield type associative memory network: Vi' = Lj T/j Vi , in which T/j is the strength of the associative connection 928 Dawei W Dong from neuron j to neuron i, which is the recent correlation between the neuronal activities Vi and Vj determined by Hebbian learning with a decay term [4] ,dTfj , B dt = -Iij + ViVj (3) in which B' is a time constant. The Vi' and T[j are only involved in learning and do not directly affect the network outputs. It is straight forward to show that when the time constants B > > B' > > a, the dynamics reduces to dT B dt = (1- < VVT » < VIT > (4) where bold-faced quantities are matrices and vectors and <> denotes ensemble average. It is not difficult to show that this equation has a Lyapunov or "energy" function L = Tr(1- < VVT »(1- < VVT >f which is lower bounded and satisfies dL < 0 dt and dL =0 -+dt dTij 0 I' 11" dt = lor at,) (5) (6) Thus the dynamics is stable. When it is stable, the output activities are decorrelated, <VVT >= 1 (7) The above equation shows that this dynamics always leads to a stable state where the neuronal activities are decorrelated and their correlation matrix is orthonormal. Yet the connections change in an associative fashion equation (2) and (3) are almost Hebbian. That is why we call it associative decorrelation dynamics. From information processing point of view, a network, self-organized to satisfy equation (7), is optimized for Gaussian input ensembles and white output noises [7]. Linear First Order Analysis In applying our theory of associative decorrelation dynamics to visual cortex to compare with the psychophysical experiments on orientation illusions, the linear first-order approximation is used, which is T = TO + 6T, V = Va +6V, TO = 0, 6T ex - < I IT > Va = I, 6V = TI (8) where it is assumed that the input correlations are small. It is interesting to notice that the linear first-order approximation leads to anti-Hebbian feedback connections: Iij ex - < /i/j > which is guarantteed to be stable around T = 0 [8]. 3 Quantitative Predictions of Orientation Illusions The basic phenomena of orientation illusions are demonstrated in figure 2 (left). On the top, is the effect of orientation contrast (also called tilt illusion): within the two surrounding circles there are tilted lines; the orientation of a center rectangle Associative Correlation Dynamics 929 appears rotated to the opposite side of its surrounding tilt. Both the two rectangles and the one without surround (at the left-center of this figure) are, in fact, exactly same. On the bottom, is the effect of orientation adaptation (also called tilt aftereffect): if one fixates at the small circle in one of the two big circles with tilted lines for 20 seconds or so and then look at the rectangle without surround, the orientation of the lines of the rectangle appears tilted to the opposite side. These two effects of orientation illusions are both in the direction of repulsion: the apparent orientation of a line is changed to increase its difference from the inducing line. Careful experimental measurements also revealed that the angle with the inducing line is <"V 100 for maximum orientation adaptation effect [9] but <"V 20 0 for orientation contrast [10]. 1 Ol..---~-~-""';:"'''''''''---' -90 -45 o 45 90 Stimulus orientation (J (degree) Figure 2: The effects of orientation contrast (upper-left) and orientation adaptation (lowerleft) are attributed to feedback connections between cells tuned to different orientations (upper-right, network; lower-right, tuning curve). Orientation illusions are attributed to the feedback connections between orientation selective cells. This is illustrated in figure 2 (right). On the top is the network of orientation selective cells with feedback connections. Only four cells are shown. From the left, they receive orientation selective feedforward inputs optimal at -45 0 , 00 ,450 , and 90 0 , respectively. The dotted lines represent the feedback connections (only the connections from the second cell are drawn). On the bottom is the orientation tuning curve of the feedforward input for the second cell, optimally tuned to stimulus of 00 (vertical), which is assumed to be Gaussian of width (T = 200 • Because of the feedback connections, the output of the second cell will have different tuning curves from its feedforward input, depending on the activities of other cells. For primary visual cortex, we suppose that there are orientation selective neurons tuned to all orientations. It is more convenient to use the continuous variable e instead of the index i to represent neuron which is optimally tuned to the orientation of angle e. The neuronal activity is represented by V(e) and the feedforward input to each neuron is represented by I(e). The feedforward input itself is orientation 930 Dawei W. Dong selective: given a visual stimulus of orientation eo, the input is J(e) = e-(9-9o)2/ q 2 (9) This kind of the orientation tuning has been measured by experiments (for references, see [6]). Various experiments give a reasonable tuning width around 20° «(7" = 20° is used for all the predictions). Predicted Orientation Adaptation For the orientation adaptation to stimulus of angle eo, substituting equation (9) into equation (8), it is not difficult to derive that the network response to stimulus of angle 0 (vertical) is changed to V(e) = e_92/ q2 _ ae-(9-9o)2/q2 e-9~/2q2 (10) in which (7" is the feedforward tuning width chosen to be 20° and a is the parameter of the strength of decorrelation feedback. The theoretical curve of perceived orientation ¢(eo) is derived by assuming the maximum likelihood of the the neural population, i.e., the perceived angle ¢ is the angle at which Vee) is maximized. It is shown in figure 3 (right). The solid line is the theoretical curve and the experimental data come from [9] (they did not give the errors, the error bars are of our estimation,...., 0.2°). The parameter obtained through X2 fit is the strength of decorrelation feedback: a = 0.42. 2.0 ;--'"T""---,--...,.....-----.------, ~ 1.5 ~ II) CD 1.0 ~ ." ~ 0.5 'il ~ Q., 0.0 f-------------J o 10 20 30 40 50 Surround angle 80 (degree) 4.0 ~ 3.0 ~ II) } 2.0 ." II) 1.0 > 'il i:! II) 0.0 Q., 0 10 20 30 40 50 Adaptation angle 80 (degree) Figure 3: Quantitative comparison of the theoretical predictions with the experimental data of orientation contrast (left) and orientation adaptation (right). It is very interesting that we can derive a relationship which is independent of the parameter of the strength of decorrelation feedback a, (eo - ¢m)(3eo - 2¢m) = (7"2 (11) in which eo is the adaptation angle at which the tilt aftereffect is most significant and ¢m is the perceived angle. Predicted Orientation Contrast For orientation contrast, there is no specific adaptation angle, i.e., the network has developed in an environment of all possible angles. In this case, when the surround is of angle eo, the network response to a stimulus of angle e1 is Vee) = e-(9-91)2/q 2 _ ae-(9-9o)2/3q2 (12) Associative Correlation Dynamics 931 in which fr and a has the same meaning as for orientation adaptation. Again assuming the maximum likelihood, ¢(eo), the stimulus angle e1 at which it is perceived as angle 0, is derived and shown in figure 3 (left). The solid line is the theoretical curve and the experimental data come from [10] and their estimated error is "" 0.20. The parameter obtained through X 2 fit is the strength of decorrelation feedback: a = 0.32. We can derive the peak position eo, i.e., the surrounding angle eo at which the orientation contrast is most significant, ~e~ = fr2 (13) 3 For fr = 200 , one immediately gets eo = 240 • This is in good agreement with experiments, most people experience the maximum effect of orientation contrast around this angle. Our theory predicts that the peak position of the surround angle for orientation contrast should be constant since the orientation tuning width fr is roughly the same for different human observers and is not going to change much for different experimental setups. But the peak value of the perceived angle is not constant since the decorrelation feedback parameter a is not necessarily same, indeed, it could be quite different for different human observers and different experimental setups. 4 Discussion First, we want to emphasis that in all the comparisons, the same tuning width fr is used and the strength of decorrelation feedback a is the only fit parameter. It does not take much imagination to see that the quantitative agreements between the theory and the experiments are good. Further more, we derived the relationships for the maximum effects, which are independent of the parameter a and have been partially confirmed by the experiments. Recent neurophysiological experiments revealed that the surrounding lines did influence the orientation selectivity of cells in primary visual cortex of the cat [11]. Those single cell experiments land further support to our theory. But one should be cautioned that the cells in our theory should be considered as the average over a large population of cells in cortex. The theory not only explains the first order effects which are dominant in angle range of 00 to 500 , as shown here, but also accounts for the second order effects which can be seen in 500 to 90 0 range, where the sign of the effects is reversed. The theory also makes some predictions for which not much experiment has been done yet, for example, the prediction about how orientation contrast depends on the distance of surrounding stimuli from the test stimulus [7]. Finally, this is not merely a theory for the development and the adaptation of orientation selective cells, it can account for effect such as human vision adaptation to colors as well [7]. We can derive the same equation as Atick etal [12] which agrees with the experiment on the appearance of color hue after adaptation. We believe that future psychophysical experiments could give us more quantitative results to further test our theory and help our understanding of neural systems in general. 932 Dawei W. Dong Acknowledgements This work was supported in part by the Director, Office of Energy Research, Division of Nuclear Physics of the Office of High Energy and Nuclear Physics of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098. References [1] Hubel DH, Wiesel TN, 1962 Receptive fields, binocular interactions, and functional architecture in the cat's visual cortex J Physiol (London) 160, 106- 54. 1963 Shape and arrangement of columns in cat's striate cortex J Physiol (London) 165, 559-68. [2] Linsker R, 1986 From basic network principles to neural architecture ... Proc Natl Acad Sci USA 83, 7508 8390 8779. -, 1989 An application of the principle of maximum information preservation to linear systems Advances in Neural Information Processing Systems 1, Touretzky DS, ed, Morgan Kaufman, San Mateo, CA 186-94. [3] Gilbert C, Wiesel T, 1989 Columnar Specificity of intrinsic horizontal and corticocortical connections in cat visual cortex J Neurosci 9(7), 2432-42. Luhmann HJ, Martinez L, Singer W, 1986 Development of horizontal intrinsic connections in cat striate cortex Exp Brain Res 63, 443-8. [4] Dong DW, 1991 Dynamic properties of neural network with adapting synapses Proc International Joint Conference on Neural Networks, Seattle, 2, 255- 260. -, 1991 Dynamic Properties of Neural Networks Ph D thesis, University Microfilms International, Ann Arbor, ML Dong DW, Hopfield JJ, 1992 Dynamic properties of neural networks with adapting synapses Network: Computation in Neural Systems, 3(3), 267- 83. [5] Gibson J J, Radner M, 1937 Adaptation, after-effect and contrast in the perception of tilted lines J of Exp Psy 20, 453-67. Carpenter RHS, Blakemore C, 1973 Interactions between orientations in human vision Exp Brain Res 18, 287-303. Tolhurst DJ, Thompson PG, 1975 Orientation illusions and after-effects: Inhibition between channels Vis Res 15,967-72. Barlow HB, Foldiak P, 1989 Adaptation and decorrelation in the cortex The Computing Neuron, Durbin R, Miall C, Mitchison G, eds, Addison- Wesley, New York, NY. [6] Wehmeier U, Dong DW, Koch C, Van Essen DC, 1989 Modeling the mammalian visual system Methods in Neuronal Modeling: From Synapses to Networks, Koch C, Segev I, eds, MIT Press, Cambridge, MA 335-60. [7] Dong DW, 1993 Associative Decorrelation Dynamics in Visual Cortex Lawrence Berkeley Laboratory Technical Report LBL-34491. [8] Dong DW, 1993 Anti-Hebbian dynamics and total recall of associative memory Proc World Congress on Neural Networks, Portland, 2, 275-9. [9] Campbell FW, Maffei L, 1971 The tilt after-effect: a fresh look Vis Res 11, 833-40. [10] Westheimer G, 1990 Simultaneous orientation contrast for lines in the human fovea Vis Res 30, 1913-21. [11] Gilbert CD, Wiesel TN, 1990 The influence of contextual stimuli on the orientation selectivity of cells in primary visual cortex of the cat Vis Res 30,1689-701. [12] Atick JJ, Li Z, Redlich AN, 1993 What does post-adaptation color appearance reveal about cortical color representation Vis Res 33, 123-9.	1994	65
960	Visual Speech Recognition with Stochastic Networks Javier R. Movellan Department of Cognitive Science University of California San Diego La Jolla, Ca 92093-0515 Abstract This paper presents ongoing work on a speaker independent visual speech recognition system. The work presented here builds on previous research efforts in this area and explores the potential use of simple hidden Markov models for limited vocabulary, speaker independent visual speech recognition. The task at hand is recognition of the first four English digits, a task with possible applications in car-phone dialing. The images were modeled as mixtures of independent Gaussian distributions, and the temporal dependencies were captured with standard left-to-right hidden Markov models. The results indicate that simple hidden Markov models may be used to successfully recognize relatively unprocessed image sequences. The system achieved performance levels equivalent to untrained humans when asked to recognize the fIrst four English digits. 1 INTRODUCTION Visual articulation is an important source of information in face to face speech perception. Laboratory studies have shown that visual information allows subjects to tolerate an extra 4-dB of noise in the acoustic signal. This is particularly important considering that each decibel of signal to noise ratio translates into a 10-15% error reduction in the intelligibility of entire sentences (McCleod and SummerfIeld, 1990). Lip reading alone provides a basis for understanding for a large majority of the hearing impaired and when supplemented by acoustic or electrical signals it allows fluent understanding of speech in highly trained subjects. However visual information plays more than a simple compensatory role in speech perception. From early on humans are predisposed to integrate acoustic and visual information. Sensitivity to correspondences in auditory and visual information for speech events has been shown in 4 month old infants (Spelke, 1976; Kuhl & Meltzoff, 1982). By 6 years of age, humans consistently use audio visual contingencies to understand speech (Massaro, 1987). By adulthood, visual articulation automatically modulates perception of the acoustic signal. Under laboratory conditions it is possible to create powerful illusions in which subjects mistakenly hear sounds which are biased by visual articulations. Subjects in these experiments are typically unaware cf 852 Javier Movellan the discrepancy between the visual and auditory tracks and their experience is that of a unified auditory percept (McGurk & McDonnald, 1976). Recent years have seen a revival of interest in audiovisual speech perception both in psychology and in the pattern recognition literature. There have been isolated efforts to build synthetic models of visual and audio-visual speech recognition (Petahan, 1985; Nishida, 1986; Yuhas, Goldstein, Sejnowski & Jenkins, 1988; Bregler, Manke, Hild & Waibel, 1993; Wolff, Prassad, Stork, & Hennecke, 1994). The main goal of these efforts has been to explore different architectures and visual processing techniques and to illustrate the potential use of visual information to improve the robustness of current speech recognition systems. Cognitive psychologists have also developed high level models of audio-visual speech perception that describe regularities in the way humans integrate visual and acoustic information (Massaro, 1987). In general these studies support the idea that human responses to visual and acoustic stimuli are conditional independent. This regularity has been used in some synthetic systems to simplify the task of integrating visual and acoustic signals (Wolff, Prassad, Stork, & Hennecke, 1994). Overall, multimodal speech perception is still an emerging field in which a lot cf exploration needs to be done. The work presented here builds on the previous research efforts in this area and explores the potential use of simple hidden Markov models 1ir limited vocabulary, speaker independent visual speech recognition. The task at hand is recognition of the first four English digits, a task with possible applications in car-phone dialing. 2 TRAINING SAMPLE The training sample consisted of 96 digitized movies of 12 undergraduate students (9 males,3 females) from the Cognitive Science Department at UCSD. Video capturing was performed in a windowless room at the Center for Research in Language at UCSD. Subjects were asked to talk into a video camera and to say the first four digits in English twice. Subjects could monitor the digitized images in a small display conveniently located in front of them. They were asked to position themselves so that that their lips be roughly centered in the feed-back display. Gray scale video images were digitized at 30 fps, 100x75 pixels, 8 bits per pixel. The video tracks were hand segmented by selecting a few relevant frames before and after the beginning and end of activity in the acoustic track. Statistics of the entire training sample are shown in table 1. Table 1: Frame number statistics. Digit Average S.D. "One" 8.9 2.1 "Two" 9.6 2.1 "Three" 9.7 2.3 "Four" 10.6 2.2 3 IMAGE PREPROCESSING There are two different approaches to visual preprocessing in the visual speech recognition literature (Bregler, Manke, Hild & Waibel, 1993). The first approach, represented by the work of Wolff and colleagues (Wolff, Prassad, Stork, & Hennecke, 1994) favors sophisticated image preprocessing techniques to extract a limited set of hand-crafted features (e.g., height and width of the lips). The advantage of this approach is that it Visual Speech Recognition with Stochastic Networks 853 drastically reduces the number of input dimensions. This translates into lower variability of the signal, potentially improved generalization, and large savings in computing time. The disadvantage is that vital information may be lost when compressing the image into a limited set of hand-crafted features. Variability is reduced at the possible expense cf bias. Moreover, tests have shown that subtle holistic features such as the wrinkling and protrusion of the lips may play an important role in human lip-reading (Montgomery & Jackson, 1983). The second approach to visual preprocessing emphasizes preserving the original images as much as possible and letting the recognition engine discover the relevant features in the images. In most cases, images are low-pass filtered and dimension-reduced by using principal component analysis. The results in this papers indicate that good results can be obtained even without the use of principal components. In this investigation image preprocessing consisted of the following phases: 1. Symmetry enforcement: At each time frame the raw images were symmetrized by averaging pixel by pixel the left and right side of each image, using the vertical midline as the axis of symmetry. For convenience from now on we will refer to the raw images as "rho-images" and the symmetrized images as "sigma-images." The potential benefits cf sigma-images are robustness, and compression, since the number of relevant pixels is reduced by half. 2. Temporal differentiaion: At each time frame we calculated the pixel by pixel differences between present sigma-images and immediately past sigma-images. For convenience we refer to the resulting images as "delta-images." One of the potential advantages of deltaimages in the visual domain is their robustness to changes in illumination and the fuct that they emphasize the dynamic aspects of the visual track. 3. Low pass filtering and subsampling: The sigma and delta images were compressed and subsampled using 20x 15 equidistant Gaussian filters. Different values of the standard deviation of the Gaussian filters were tested. 4. Logistic thresholding and scaling: The sigma and delta images were independently thresholded by feeding the output of the Gaussian filters through a according to the following equation 7r Y = 256 j(K r;; (x - J.L)) -v3a where f is the logistic function, and J.L, a, are respectively the average and standard deviation of the gray level distribution of entire image sequences. The constant K controls the sharpness of the logistic function. Assuming an approximately Gaussian distribution of gray levels when K=1 the thresholding function approximates histogram equalization, a standard technique in visual processing. Three different K values were tried: 0.3, 0.6 and 1.2. 5. Composites of the relevant portions of the blurred sigma and delta images were fed to the recognition network. The number of pixels of each processed image was 300 (150 from the blurred sigma images and 150 from the blurred delta images). Figure 1 shows the effect of the different preprocessing stages. 854 Javier Movellan Figure 1: Image Preprocessing. 1) Rho-Image. 2) Sigma-Image. 3) Delta-Image. 4) Filtered and Sharpened Composite. 4 RECOGNITION NETWORK We used the standard approach in limited vocabulary systems: a bank of hidden Markov models, one per word category, independently trained on the corresponding word categories. The images were modeled as mixtures of continuous probability distributions in pixel space. We tried mixtures of Gaussians and mixtures of Cauchy distributions. The mixtures of Cauchy distributions were very stable numerically but they did perform very poorly when compared to the Gaussian mixtures. We believe the reason for their poor performance is the tendency of Cauchy-based maximum-likelihood estimates to focus on individual exemplars. Gaussian-based estimates are much more prone to blend exemplars that belong to the same cluster. The initial state probabilities, transition probabilities, mixture coefficients, mixture centroids and variance parameters were trained using the E-M algorithm. We initially encountered severe numerical underflow problems when using the E-M algorithm with Gaussian mixtures. These instabilities were due to the fact that the probability densities of images rapidly went to zero due to the large dimensionality of the images. Trimming the outputs of the Gaussian and using very small Gaussian gains did not work well. We solved the numerical problems in the following way: 1) Constraining all the variance parameters for all the states and mixtures to be equal. This allowed pulling out a constant in the likelihood-function of the mixtures, avoiding most numerical problems. 2) Initializing the mixture centroids using linear segmentation followed by the K-means clustering algorithm. For example, ifthere were 4 visual frames and 2 states, the first 2 frames were assigned to state 1 and the last 2 frames to state 2. Kmeans was then used independently on each of the states and their assigned frames. This is a standard initialization method in the acoustic domain (Rabiner & Bing-Hwang, 1993). Since K-means can be trapped in local minima, the algorithm was repeated 20 times with different starting point and the best solution was fed as the starting point for the E-M algorithm. 5 RESULTS The main purpose of this study was to fmd simple image preprocessing techniques that would work well with hidden Markov models. We tested a wide variety of architectures and preprocessing parameters. In all cases the results were evaluated in terms cf generalization to new speakers. Since the training sample is small, generalization performance was estimated using the jackknife procedure. Models were trained with 11 Visual Speech Recognition with Stochastic Networks 855 subjects, leaving one subject out for generalization testing. The entire procedure was repeated 12 times, each time leaving a different subject out for testing. Results are thus based on 96 generalization trials (4 digits x 12 subjects x 2 observations per subject). In all cases we tested several preprocessing techniques using 20 different architectures with different number of states (1,3,5,7,9) and mixtures per state (1,3,5,7). To compare the effect of each processing technique we used the average generalization performance of the best 4 architectures out of the 20 architectures tested. 85 65 45 25 +---L. __ Rho Sigma Delta+Sigma Figure 2: Average performance with the rho, sigma, and delta images. Figure 2 shows the effects of symmetry enforcement and temporal differentiation. Symmetry enforcement had the benefit of reducing the input dimensionality by half and, as the figure show it did not hinder recognition performance. Using delta images had a very positive effect on recognition performance, as the figure shows. Figure 3 shows the effect of varying the thresholding constant and the standard deviation of the Gaussian filters. Best performance was obtained with blurring windows about 4 pixel wide and with thresholding just about histogram equalization. 84 83 82 81 80 3 4 5 Stamdard Deviation of Gaussian Filter. ---'-K=i.2 ---K=O.6 K=O.3 Figure 3: Effect of blurring and sharpening. 856 Javier Movellan Table 2 shows the effects of variations in the number of states (S) and Gaussian mixtures (G) per state. The number within each cell is the percentage of simulations for which a particular combination of states and mixtures performed best out of the 20 architectures tested. Table 2: Effect of varying the the number of states (S) and Gaussian mixtures (G). Gl G3 G5 G7 SI 0.00% 0.00% 0.00% 0.00% S3 0.00% 21.87% 12.5% 6.25% S5 3.12% 9.37% 15.62% 0.00% S7 6.25% 12.5% 0.00% 3.12% S9 6.25% 0.00% 3.12% 0.00% Best overall performance was obtained with about 3 states and 3 mixtures per state. Peak performance was also obtained with a 3-state, 3-mixture per state network, with a generalization rate of 89.58% correct. To compare these results with human performance, 9 subjects were tested on the same sample. Six subjects were normal hearing adults who were not trained in lip-reading. Three were hearing impaired with profound hearing loss and had received training in lip reading at 2 to 8 years of age. The mean correct response for normal subjects was 89.93 % correct, just about the same rate as the best artificial network. The hearing impaired had an average performance of 95.49% correct, significantly better than our network. Table 3: Confusion matrix of the best artificial system. 1 2 3 4 "One" 100.00% 0.00% 0.00% 0.00% "Two" 4.17% 87.50% 4.17% 4.17% "Three" 12.5% 0.00% 83.33% 4.17% "Four" 8.33% 4.17% 0.00% 87.50% Table 4: Average human confusion matrix. 1 2 3 4 "One" 89.36% 0.46% 8.33% 1.85% "Two" 1.39% 98.61% 0.00% 0.00% "Three" 9.25% 3.24% 85.64% 1.87% "Four" 4.17% 0.46% 1.85% 93.52% Tables 3 and 4 show the confusion matrices for the best network and the average confusion matrix with all 9 subjects combined. The correlation between these two matrices was 0.99. This means that 98% of the variance in human confusions can be accounted for by the artificial model. This suggests that the representational space learned by the artificial system may be a reasonable model of the representational space used by humans. Figure 5 shows the representations learned by a network with 6 states and 1 mixture per state. Each column is a different digit, starting with "one." Each row Visual Speech Recognition with Stochastic Networks 857 is a different temporal state. The two pictures within each cell are sigma and delta image centroids. As the figure shows, the identity of individual exemplars is lost but the underlying dynamics of the digits are preserved. The digits can be easily recognized when played as a movie. Figure 4: Dynamic representations learned by a simple network. 6 CONCLUSIONS This paper shows that simple stochastic networks, like hidden Markov models, can be successfully applied for visual speech recognition using relatively unprocessed images. The performance level obtained with these networks roughly matches untrained human performance. Moreover, the representational space learned by these networks may be a reasonable model of the representations used by humans. More research should be done to better understand how humans integrate visual and acoustic information in speech perception and to develop practical models for robust audio-visual speech recognition. References Bregler c., Manke S., Hild H. & Waibel A. (1993) Bimodal Sensor Integration on the Example of "Speech-Reading". Proc ICNN-93, 11,667-677. Kuhl P. & Meltzoff A (1982) The Bimodal Development of Speech in Infancy. Science, 218, 1138-1141. MacLeod A. & Summerfield Q. (1990) A Procedure for Measuring Auditory and Audiovisual Speech-Reception Measuring Thresholds for Sentences in Noise: Rationale, Evaluation and Recommendations for Use. British Journal of Audiology. 24,29-43. Massaro D. (1987) Speech Perception by Ear and Eye. In Dodd B. & Campbell R. (ed.) Hearing by Eye: The Psychology of Lip-Reading. London, LEA, 53-83. Massaro D., Cohen M & Getsi (1993) Long-Term Training, Transfer and Retention in Learning to Lip-read. Perception and Psychophysics, 53,549-562. 858 Javier Movellan McGurk H. & MacDonald J. (1976) Hearing Lips and Seeing Voices. Nature, 264, 126130. Montgomery A. & Jackson P. (1983) Physical Characteristics of the Lips Underlying Vowel Lipreading Performance. Journal of the Acoustical Society of America, 73,21342144. Nishida S. (1986) Speech Recognition Enhancement by Lip Information Proceedings of ACMlCHI86, 198-204. Petajan E. (1985) Automatic Lip Reading to Enhance Speech Recognition. IEEE CVPR 85, 40-47. Rabiner L., Bing-Hwang J. (1993) Fundamentals of Speech Recognition. New Jersey, Prentice Hall. Spelke E. (1976) Infant's Intermodal Perception of Events. Cognitive Psychology, 8, 533-560. Yuhas B., Goldstein T., Sejnowski T., Jenkins R. (1988) Neural Network Models cf Sensory Integration for Improved Vowel Recognition. Proceedings IEEE 78, 16551668. Wolff G., Prassad L., Stork D., Hennecke M. (1994) Lipreading by Neural Networks: Visual Preprocessing, Learning and Sensory Integration. In J. Cowan, G. Tesauro, J. Alspector (ed.), Advances in Neural Information Processing Systems 6, 1027-1035. San Mateo, CA: Morgan Kaufinann.	1994	66
961	Finding Structure in Reinforcement Learning Sebastian Thrun University of Bonn Department of Computer Science nr R6merstr. 164, D-53117 Bonn, Germany E-mail: thrun@carbon.informatik.uni-bonn.de Abstract Anton Schwartz Dept. of Computer Science Stanford University Stanford, CA 94305 Email: schwartz@cs.stanford.edu Reinforcement learning addresses the problem of learning to select actions in order to maximize one's performance in unknown environments. To scale reinforcement learning to complex real-world tasks, such as typically studied in AI, one must ultimately be able to discover the structure in the world, in order to abstract away the myriad of details and to operate in more tractable problem spaces. This paper presents the SKILLS algorithm. SKILLS discovers skills, which are partially defined action policies that arise in the context of multiple, related tasks. Skills collapse whole action sequences into single operators. They are learned by minimizing the compactness of action policies, using a description length argument on their representation. Empirical results in simple grid navigation tasks illustrate the successful discovery of structure in reinforcement learning. 1 Introduction Reinforcement learning comprises a family of incremental planning algorithms that construct reactive controllers through real-world experimentation. A key scaling problem of reinforcement learning, as is generally the case with unstructured planning algorithms, is that in large real-world domains there might be an enormous number of decisions to be made, and pay-off may be sparse and delayed. Hence, instead of learning all single fine-grain actions all at the same time, one could conceivably learn much faster if one abstracted away the myriad of micro-decisions, and focused instead on a small set of important decisions. But this immediately raises the problem of how to recognize the important, and how to distinguish it from the unimportant. This paper presents the SKILLS algorithm. SKILLS finds partially defined action policies, called skills, that occur in more than one task. Skills, once found, constitute parts of solutions to multiple reinforcement learning problems. In order to find maximally useful skills, a description length argument is employed. Skills reduce the number of bytes required to describe action policies. This is because instead of having to describe a complete action policy for each task separately, as is the case in plain reinforcement learning, skills constrain multiple task to pick the same actions, and thus reduce the total number of actions required 386 Sebastian Thrun, Anton Schwartz for representing action policies. However, using skills comes at a price. In general, one cannot constrain actions to be the same in multiple tasks without ultimately suffering a loss in performance. Hence, in order to find maximally useful skills that infer a minimum loss in performance, the SKILLS algorithm minimizes a function of the form E = PERFORMANCE LOSS + 1J' DESCRIPTION LENGTH. (1) This equation summarizes the rationale of the SKILLS approach. The reminder of this paper gives more precise definitions and learning rules for the terms" PERFORMANCE LOSS" and "DESCRIPTION LENGTH," using the vocabulary of reinforcement learning. In addition, experimental results empirically illustrate the successful discovery of skills in simple grid navigation domains. 2 Reinforcement Learning Reinforcement learning addresses the problem of learning, through experimentation, to act so as to maximize one's pay-off in an unknown environment. Throughout this paper we will assume that the environment of the learner is a partially controllable Markov chain [1]. At any instant in time the learner can observe the state of the environment, denoted by s E S, and apply an action, a E A. Actions change the state of the environment, and also produce a scalar pay-off value, denoted by rs,a E ~. Reinforcement learning seeks to identify an action policy, 7f : S --+ A, i.e., a mapping from states s E S to actions a E A that, if actions are selected accordingly, maximizes the expected discounted sum offuture pay-off R = E [f ,t-til r t]. (2) t=tu Here I (with 0 S,S 1) is a discount factor that favors pay-offs reaped sooner in time, and rt refers to the expected pay-off at time t. In general, pay-off might be delayed. Therefore, in order to learn an optimal 7f, one has to solve a temporal credit assignment problem [11]. To date, the single most widely used algorithm for learning from delayed pay-off is QLearning [14]. Q-Learning solves the problem of learning 7f by learning a value function, denoted by Q : S x A --+~. Q maps states s E S and actions a E A to scalar values. After learning, Q(s, a) ranks actions according to their goodness: The larger the expected cumulative pay-offfor picking action a at state s, the larger the value Q(s, a). Hence Q, once learned, allows to maximize R by picking actions greedily with respect to Q: 7f(s) = argmax Q(s, a) aEA The value function Q is learned on-line through experimentation. Initially, all values Q( s, a) are set to zero. Suppose during learning the learner executes action a at state s, which leads to a new state s' and the immediate pay-off rs ,a' Q-Learning uses this state transition to update Q(s, a): Q(s, a) ;(1 - a) . Q(s, a) + a· (rs,a + I' V(s')) (3) with V(s') m~x Q(s', a) a The scalar a (O<aSl) is the learning rate, which is typically set to a small value that is decayed over time. Notice that if Q(s, a) is represented by a lookup-table, as will be the case throughout this paper, the Q-Learning rule (3) has been shown] to converge to a value function Qopt( s, a) which measures the future discounted pay-off one can expect to receive upon applying action a in state s, and acting optimally thereafter [5, 14]. The greedy policy 7f(s) = argmaxa Qopt(s, a) maximizes R. I under certain conditions concerning the exploration scheme, the environment and the learning rate Finding Structure in Reinforcement Learning 387 3 Skills Suppose the learner faces a whole collection of related tasks, denoted by B, with identical states 5 and actions A. Suppose each task b E B is characterized by its individual payoff function, denoted by rb(s, a). Different tasks may also face different state transition probabilities. Consequently, each task requires a task-specific value function, denoted by Qb(S, a), which induces a task-specific action policy, denoted by '!rb. Obviously, plain QLearning, as described in the previous section, can be employed to learn these individual action policies. Such an approach, however, cannot discover the structure which might inherently exist in the tasks. In order to identify commonalities between different tasks, the SKILLS algorithm allows a learner to acquire skills. A skill, denoted by k, represents an action policy, very much like '!rb. There are two crucial differences, however. Firstly, skills are only locally defined, on a subset Sk of all states S. Sk is called the domain of skill k. Secondly, skills are not specific to individual tasks. Instead, they apply to entire sets of tasks, in which they replace the task-specific, local action policies. Let f{ denote the set of all skills. In general, some skills may be appropriate for some tasks, but not for others. Hence, we define a vector of usage values Uk,b (with 0 ~ Uk,b ~ 1 for all kEf{ and all b E B). Policies in the SKILLS algorithm are stochastic, and usages Uk,b determine how frequently skill k is used in task b. At first glance, Uk,b might be interpreted as a probability for using skill k when performing task b, and one might always want to use skill k in task b if Uk ,b = 1, and never use skill k if Uk ,b = 0.2 However, skills might overlap, i.e., there might be states S which occurs in several skill domains, and the usages might add to a value greater than 1. Therefore, usages are normalized, and actions are drawn probabilistical1y according to the normalized distribution: U~,k . mk(s) (with 8 = 0) (4) k'EK Here Pb( kls) denotes the probability for using skill k at state s, if the learner faces task b. The indicator function mk (s) is the membership function for skill domains, which is 1 if s E Sk and 0 otherwise. The probabilistic action selection rule (4) makes it necessary to redefine the value Vb (s) of a state s. If no skill dictates the action to be taken, actions will be drawn according to the Qb-optimal policy '!r;(s) argmax Qb(S, a) , .lEA as is the case in plain Q-Learning. The probability for this to happen is Pb(s) 1 - L Pb(kls) . kEK Hence, the value of a state is the weighted sum Vb(S) P;(s) . vt(s) + L Pb(kls) . Qb(S, '!rk(S)) (5) kEK with vt(s) Qb(S, '!rb(s)) = TEa; Qb(S, a) Why should a learner use skills, and what are the consequences? Skills reduce the freedom to select actions, since mUltiple policies have to commit to identical actions. Obviously, such 2This is exactly the action selection mechanism in the SKILLS algorithm if only one skill is applicable at any given state s. 388 Sebastian Thrun, Anton Schwartz a constraint will generally result in a loss in peiformance. This loss is obtained by comparing the actual value of each state s, Vb(S), and the value ifno skill is used, VtCs): LOSS = ~ ~ Vb(s) - Vb(S) (6) sES bEB '''---v,----' = LOS S(s ) If actions prescribed by the skills are close to optimal, i.e., if Vb(s) ~ Vb(S)('v'S E S), the loss will be small. If skill actions are poor, however, the loss can be large. Counter-balancing this loss is the fact that skills give a more compact representation of the learner's policies. More specifically, assume (without loss of generality) actions can be represented by a single byte, and consider the total number of bytes it takes to represent the policies of all tasks b E B. In the absence of skills, representing all individual policies requires IBI . lSI bytes, one byte for each state in S and each task in B. If skills are used across multiple tasks, the description length is reduced by the amount of overlap between different tasks. More specifically, the total description length required for the specification of all policies is expressed by the following term: DL ~ ~ P;(s) + ~ ISkl (7) sE S bEB kEK ~~-----v~---------j = DL( s ) If all probabilities are binary, i.e. , Pb(kls) and P;(s) E {O, I}, DL measures precisely the number of bytes needed to represent all skill actions, plus the number of bytes needed to represent task-specific policy actions where no skill is used. Eq. (7) generalizes this measure smoothly to stochastic policies. Notice that the number of skills If{ I is assumed to be constant and thus plays no part in the description length DL. Obviously, minimizing LOSS maximizes the pay-off, and minimizing DL maximizes the compactness of the representation of the learner's policies. In the SKILLS approach, one seeks to minimize both (cf Eq. (1» E = LOSS + TJDL = ~ LOSS(s) + TJDL(s) . (8) sES 11 > 0 is a gain parameter that trades off both target functions. E-optimal policies make heavily use of large skills, yet result in a minimum loss in performance. Notice that the state space may be partitioned completely by skills, and solutions to the individual tasks can be uniquely described by the skills and its usages. If such a complete partitioning does not exist, however, tasks may instead rely to some extent on task-specific, local pOlicies. 4 Derivation of the Learning Algorithm Each skill k is characterized by three types of adjustable variables: skill actions 7rk (s), the skill domain Sk, and skill usages Ub,k. one for each task b E B. In this section we will give update rules that perform hill-cli:nbing in E for each of these variables. As in Q-Learning these rules apply only at the currently visited state (henceforth denoted by s). Both learning action policies (cf Eq. (3» and learning skills is fully interleaved. Actions. Determining skill actions is straightforward, since what action is prescribed by a skill exclusively affects the performance loss, but does not play any part in the description length. Hence, the action policy 7rk(S) minimizes LOSS(s) (cf Eqs. (5) and (6»: 7rk(S) argmax ~ Pb(kls) . Qb(S. a) (9) .lEA bEB Finding Structure in Reinforcement Learning 389 Domains. Initially, each skill domain Sk contains only a single state that is chosen at random. Sk is changed incrementally by minimizing E(s) for states s which are visited during learning. More specifically, for each skill k, it is evaluated whether or not to include sin Sk by considering E(s) = LOSS(s) + TJDL(s). s E Sk, ifandonlyif E(s)lsESk < E(s)lsf'Sk (otherwises ~ Sk) (10) If the domain of a skill k vanishes completely, i.e., if Sk = 0, it is re-initialized by a randomly selected state. In addition all usage values {Ub,klb E B} are initialized randomly. This mechanism ensures that skills, once overturned by other skills, will not get lost forever. Usages. Unlike skill domains, which are discrete quantities, usages are real-valued numbers. Initially, they are chosen at random in [0, 1]. Usages are optimized by stochastic gradient descent in E. According to Eq. (8), the derivative of E(s) is the sum of aL~SS(s ) and Ub,k a~uL(s ) . The first term is governed by b,k 8LOSS(s) _ 8Vb(S) _ _ 8P;(s) . Q ( () ) _ " 8Pb(j!S) . Q ( .(» 8 8 8 b 7rb s , s ~ 8 b S,7rJ S Ub,k 'Ub ,k Ub,k jEK Ub,k with 8Pb(j!s) 8Ub,k and 8Pb(s) Here Dkj denotes the Kronecker delta function, which is 1 if k = j and 0 otherwise. second term is given by 8DL(s) 8Pb(s) 8Ubk 8Ukb ' , , (11 ) (12) The (13) which can be further transformed using Eqs. (12) and (11). In order to minimize E, usages are incrementally refined in the opposite direction of the gradients: Uk ,b ;....-.Uk,b {J. (8V(s) + TJ 8DL(S») 8Ukb 8Ukb , , (14) Here {J > 0 is a small learning rate. This completes the derivation of the SKILLS algorithm. After each action execution, Q-Learning is employed to update the Q-function. SKILLS also re-calculates, for any applicable skill, the skill policy according to Eq. (9), and adjusts skill domains and usage values based upon Eqs. (10) and (14). 5 Experimental Results The SKILLS algorithm was applied to discover skills in a simple, discrete grid-navigation domain, depicted in Fig. 1. At each state, the agent can move to one of at most eight adjacent grid cells. With a 10% chance the agent is carried to a random neighboring state, regardless of the commanded action. Each corner defines a starting state for one out of four task, with the corresponding goal state being in the opposite corner. The pay-off (costs) for executing actions is -1, except for the goal state, which is an absorbing state with zero pay-off. In a first experiment, we supplied the agent with two skills f{ = {kJ, k2 }. All four tasks were trained in a time-shared manner, with time slices being 2,000 steps long. We used the following parameter settings: TJ = 1.2, 'Y = 1, a = 0.1, and {J = 0.001. After 30000 training steps for each task, the SKILLS algorithm has successfully discovered the two ski11s shown in Figure 1. One of these skills leads the agent to the right door, and 390 1 1/ I 1'/ // - /// I I //~ ",. -- // / I r, . - - - Sebastian Thrun. Anton Schwartz Figure 1: Simple 3-room environment. Start and goal states are marked by circles. The diagrams also shows two skills (black states), which lead to the doors connecting the rooms. the second to the left. Each skill is employed by two tasks. By forcing two tasks to adopt a single policy in the region of the skill, they both have to sacrifice performance, but the loss in performance is considerably small. Beyond the door, however, optimal actions point into opposite directions. There, forcing both tasks to select actions according to the same policy would result in a significant performance loss, which would clearly outweigh the savings in description length. The solution shown in Fig. 1 is (approximately) the global minimum of E, given that only two skills are available. It is easy to be seen that these skills establish helpful building blocks for many navigation tasks. When using more than two skills, E can be minimized further. We repeated the experiment using six skills, which can partition the state space in a more efficient way. Two of the resulting skills were similar to the skills shown in Fig. 1, but they were defined only between the doors. The other four skills were policies for moving out of a corner, one for each corner. Each of the latter four skills can be used in three tasks (unlike two tasks for passing through the middle room), resulting in an improVed description length when compared to the two-skill solution shown in Fig. 1. We also applied skill learning to a more complex grid world, using 25 skills for a total of 20 tasks. The environment, along with one of the skills, is depicted in Fig. 2. Different tasks were defined by different starting positions, goal positions and door configurations which could be open or closed. The training time was typically an order of magnitude slower than in the previous task, and skills were less stable over time. However, Fig. 2 illustrates that modular skills could be discovered even in such complex a domain. 6 Discussion This paper presents the SKILLS algorithm. SKILLS learns skills, which are partial policies that are defined on a subset of all states. Skills are used in as many tasks as possible, while affecting the performance in these tasks as little as possible. They are discovered by minimizing a combined measure, which takes a task performance and a description length argument into account. While our empirical findings in simple grid world domains are encouraging, there are several open questions that warrant future research. Learning speed. In our experiments we found that the time required for finding useful skills is up to an order of magnitude larger than the time it takes to find close-to-optimal policies. Finding Structure in Reinforcement Learning Figure 2: SkiIl found in a more complex grid navigation task. 391 Similar findings are reported in [9]. This is because discovering skills is much harder than learning control. Initially, nothing is know about the structure of the state space, and unless reasonably accurate Q-tables are available, SKILLS cannot discover meaningful skills. Faster methods for learning skills, which might precede the development of optimal value functions, are clearly desirable. Transfer. We conjecture that skills can be helpful when one wants to learn new, related tasks. This is because if tasks are related, as is the case in many natural learning environments, skills allow to transfer knowledge from previously learned tasks to new tasks. In particular, if the learner faces tasks with increasing complexity, as proposed by Singh [10], learning skills could conceivable reduce the learning time in complex tasks, and hence scale reinforcement learning techniques to more complex tasks. Using function approximators. In this paper, performance loss and description length has been defined based on table look-up representations of Q. Recently, various researchers have applied reinforcement learning in combination with generalizing function approximators, such as nearest neighbor methods or artificial neural networks (e.g., [2, 4, 12, 13]). In order to apply the SKILLS algorithm together with generalizing function approximators, the notions of skill domains and description length have to be modified. For example, the membership function mk, which defines the domain of a skill, could be represented by a function approximator which allows to derive gradients in the description length. Generalization in state space. In the current form, SKILLS exclusively discovers skills that are used across mUltiple tasks. However, skills might be useful under multiple circumstances even in single tasks. For example, the (generalized) skill of climbing a staircase may be useful several times in one and the same task. SKILLS, in its current form, cannot represent such skills. The key to learning such generalized skills is generalization. Currently, skills generalize exclusively over tasks, since they can be applied to entire sets of tasks. However, they cannot generalize over states. One could imagine an extension to the SKILLS algorithm, in which skills are free to pick what to generalize over. For example, they could chose to ignore certain state information (like the color of the staircase). It remains to be seen if effective learning mechanisms can be designed for learning such generalized skills. Abstractions and action hierarchies. In recent years, several researchers have recognized the importance of structuring reinforcement learning in order to build abstractions and action 392 Sebastian Thrun, Anton Schwartz hierarchies. Different approaches differ in the origin of the abstraction, and the way it is incorporated into learning. For example, abstractions have been built upon previously learned, simpler tasks [9, 10], previously learned low-level behaviors [7], subgoals, which are either known in advance [15] or determined at random [6], or based on a pyramid of different levels of perceptual resolution, which produces a whole spectrum of problem solving capabilities [3]. For all these approaches, drastically improved problem solving capabilities have been reported, which are far beyond that of plain, unstructured reinforcement learning. This paper exclusively focuses on how to discover the structure inherent in a family of related tasks. Using skills to form abstractions and learning in the resulting abstract problem spaces is beyond the scope of this paper. The experimental findings indicate, however, that skills are powerful candidates for operators on a more abstract level, because they collapse whole action sequences into single entities. References [I] A. G. Barto, S. J. Bradtke, and S. P. Singh. Learning to act using real-time dynamic programming. Artijiciallntelligence, to appear. [2] J. A. Boyan. Generalization in reinforcement learning: Safely approximating the value function. Same volume. [3] P. Dayan and G. E. Hinton. Feudal reinforcement learning. In J. E. Moody, S. J. Hanson, and R. P. Lippmann, editors, Advances in Neural Information Processing Systems 5,1993. Morgan Kaufmann. [4) V. Gullapalli, 1. A. Franklin, and Hamid B. Acquiring robot skills via reinforcement learning. IEEE Control Systems, 272( 1708), 1994. [5) T. Jaakkola, M. I. Jordan, and S. P. Singh. On the convergence of stochastic iterative dynamic programming algorithms. Technical Report 9307, Department of Brain and Cognitive Sciences, MIT, July 1993. [6] L. P. Kaelbling. Hierarchical learning in stochastic domains: Preliminary results. In Paul E. Utgoff, editor, Proceedings of the Tenth International Conference on Machine Learning, 1993. Morgan Kaufmann. [7] L.-J. Lin. Self-supervised Learning by Reinforcementand Artijicial Neural Networks. PhD thesis, Carnegie Mellon University, School of Computer Science, 1992. [8) M. Ring. Two methods for hierarchy learning in reinforcement environments. In From Animals to Animates 2: Proceedings of the Second International Conference on Simulation of Adaptive Behavior. MIT Press, 1993. [9] S. P. Singh. Reinforcement learning with a hierarchy of abstract models. In Proceeding of the Tenth National Conference on Artijiciallntelligence AAAI-92, 1992. AAAI Pressffhe MIT Press. [10] S. P. Singh. Transfer of learning by composing solutions for elemental sequential tasks. Machine Learning, 8,1992. [II] R. S. Sutton. Temporal Credit Assignment in Reinforcement Learning. PhD thesis, Department of Computer and Information Science, University of Massachusetts, 1984. [12) G. J. Tesauro. Practical issues in temporal difference learning. Machine Learning, 8, 1992. [13] S. Thrun and A. Schwartz. Issues in using function approximation for reinforcement learning. In M. Mozer, Pa. Smolensky, D. Touretzky, J. Elman, and A. Weigend, editors, Proceedings of the J 993 Connectionist Models Summer School, 1993. Erlbaum Associates. [14] C. J. C. H. Watkins. Learningfrom Delayed Rewards. PhD thesis, King's College, Cambridge, England, 1989. [15] S. Whitehead, J. Karlsson, and J. Tenenberg. Learning multiple goal behavior via task decomposition and dynamic policy merging. In J. H. Connell and S. Mahadevan, editors, Robot Learning. Kluwer Academic Publisher, 1993.	1994	67
962	Active Learning with Statistical Models David A. Cohn, Zoubin Ghahramani, and Michael I. Jordan cohnQpsyche.mit.edu. zoubinQpsyche.mit.edu. jordan~syche.mit.edu Department of Brain and Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 Abstract For many types of learners one can compute the statistically "optimal" way to select data. We review how these techniques have been used with feedforward neural networks [MacKay, 1992; Cohn, 1994]. We then show how the same principles may be used to select data for two alternative, statistically-based learning architectures: mixtures of Gaussians and locally weighted regression. While the techniques for neural networks are expensive and approximate, the techniques for mixtures of Gaussians and locally weighted regression are both efficient and accurate. 1 ACTIVE LEARNING - BACKGROUND An active learning problem is one where the learner has the ability or need to influence or select its own training data. Many problems of great practical interest allow active learning, and many even require it. We consider the problem of actively learning a mapping X Y based on a set of training examples {(Xi,Yi)}~l' where Xi E X and Yi E Y. The learner is allowed to iteratively select new inputs x (possibly from a constrained set), observe the resulting output y, and incorporate the new examples (x, y) into its training set. The primary question of active learning is how to choose which x to try next. There are many heuristics for choosing x based on intuition, including choosing places where we don't have data, where we perform poorly [Linden and Weber, 1993], where we have low confidence [Thrun and Moller, 1992], where we expect it 706 David Cohn, Zoubin Ghahramani, Michael I. Jordon to change our model [Cohn et aI, 1990], and where we previously found data that resulted in learning [Schmidhuber and Storck, 1993]. In this paper we consider how one may select x "optimally" from a statistical viewpoint. We first review how the statistical approach can be applied to neural networks, as described in MacKay [1992] and Cohn [1994]. We then consider two alternative, statistically-based learning architectures: mixtures of Gaussians and locally weighted regression. While optimal data selection for a neural network is computationally expensive and approximate, we find that optimal data selection for the two statistical models is efficient and accurate. 2 ACTIVE LEARNING - A STATISTICAL APPROACH We denote the learner's output given input x as y(x). The mean squared error of this output can be expressed as the sum of the learner's bias and variance. The variance 0'3 (x) indicates the learner's uncertainty in its estimate at x. 1 Our goal will be to select a new example x such that when the resulting example (x, y) is added to the training set, the integrated variance IV is minimized: IV = J 0'3P(x)dx. (1) Here, P(x) is the (known) distribution over X. In practice, we will compute a Monte Carlo approximation of this integral, evaluating 0'3 at a number of random points drawn according to P(x). Selecting x so as to minimize IV requires computing 0-3, the new variance at x given (x, y). Until we actually commit to an x, we do not know what corresponding y we will see, so the minimization cannot be performed deterministically.2 Many learning architectures, however, provide an estimate of PWlx) based on current data, so we can use this estimate to compute the expectation of 0-3. Selecting x to minimize the expected integrated variance provides a solid statistical basis for choosing new examples. 2.1 EXAMPLE: ACTIVE LEARNING WITH A NEURAL NETWORK In this section we review the use of techniques from Optimal Experiment Design (OED) to minimize the estimated variance of a neural network [Fedorov, 1972; MacKay, 1992; Cohn, 1994]. We will assume we have been given a learner y = fwO, a training set {(Xi, yd}f;l and a parameter vector til that maximizes a likelihood measure. One such measure is the minimum sum squared residual 52 = ~ f (Yi - Y(Xi))2. m i=l lUnless explicitly denoted, fI and O'~ are functions of x. For simplicity, we present our results in the univariate setting. All results in the paper extend easily to the multivariate case. 2This contrasts with related work by Plutowski and White [1993], which is concerned with filtering an existing data set. Active Learning with Statistical Models 707 The estimated output variance of the network is O'~ ~ S2 (Oy(X ») T (02 S2) -1 (Oy(X») y ow OW2 OW The standard OED approach assumes normality and local linearity. These assumptions allow replacing the distribution P(ylx) by its estimated mean y(x) and variance S2. The expected value of the new variance, iT~, is then: (-2)...... 2 O'~(x, x) O'g ...... O'g - S2 + O'~(x)' [MacKay, 1992]. (2) where we define _( _) = S2 (OY(X»)T (02S2)-1 (Oy(X») 0' y x, x ow ow2 ow· For empirical results on the predictive power of Equation 2, see Cohn [1994]. The advantages of minimizing this criterion are that it is grounded in statistics, and is optimal given the assumptions. Furthermore, the criterion is continuous and differentiable. As such, it is applicable in continuous domains with continuous action spaces, and allows hillclimbing to find the "best" x. For neural networks, however, this approach has many disadvantages. The criterion relies on simplifications and strong assumptions which hold only approximately. Computing the variance estimate requires inversion of a Iwl x Iwl matrix for each new example, and incorporating new examples into the network requires expensive retraining. Paass and Kindermann [1995] discuss an approach which addresses some of these problems. 3 MIXTURES OF GAUSSIANS The mixture of Gaussians model is gaining popularity among machine learning practitioners [Nowlan, 1991; Specht, 1991; Ghahramani and Jordan, 1994]. It assumes that the data is produced by a mixture of N Gaussians gi, for i = 1, ... , N. We can use the EM algorithm [Dempster et aI, 1977] to find the best fit to the data, after which the conditional expectations of the mixture can be used for function approximation. For each Gaussian gi we will denote the estimated input/output means as JLx,i and JLy,i and estimated covariances as O';,i' O';,i and O'xy,i. The conditional variance of y given x may then be written We will denote as ni the (possibly fractional) number of training examples for which gi takes responsibility: 708 David Cohn, Zoubin Ghahramani, Michael I. Jordon For an input x, each 9i has conditional expectation Yi and variance (1'~,i: A 0-xy ,i ( ) Yi = J.Ly,i + -2- X J.Lx,i , 0-x,i 0-2 . (( .)2) o-~ . = ~ 1 + x J.Lx,~ . y,J n' 0- 2 . t XI' These expectations and variances are mixed according to the prior probability that 9i has of being responsible for x: h. = h.( ) _ P(xli) ,_~xN . 2:j=l P(xlj) For input x then, the conditional expectation Y of the resulting mixture and its variance may be written: N Y = L hi Yi, i:::l In contrast to the variance estimate computed for a neural network, here o-~ can be computed efficiently with no approximations. 3.1 ACTIVE LEARNING WITH A MIXTURE OF GAUSSIANS We want to select x to minimize ( Cr~). With a mixture of Gaussians, the model's estimated distribution of ii given x is explicit: N N P(ylx) = L hiP(ylx, i) = L hiN(Yi(X), o-;lx,i(X)), i=l i=l where hi = hi (x). Given this, calculation of ( Cr~) is straightforward: we model the change in each 9i separately, calculating its expected variance given a new point sampled from P(ylx, i) and weight this change by hi. The new expectations combine to form the learner's new expected variance (3) where the expectation can be computed exactly in closed form: Active Learning with Statistical Models 709 4 LOCALLY WEIGHTED REGRESSION We consider here two forms of locally weighted regression (LWR): kernel regression and the LOESS model [Cleveland et aI, 1988]. Kernel regression computes y as an average of the Yi in the data set, weighted by a kernel centered at x. The LOESS model performs a linear regression on points in the data set, weighted by a kernel centered at x. The kernel shape is a design parameter: the original LOESS model uses a "tricubic" kernel; in our experiments we use the more common Gaussian hi(x) == hex - Xi) = exp( -k(x - xd2), where k is a smoothing constant. For brevity, we will drop the argument x for hi(x), and define n = L:i hi. We can then write the estimated means and covariances as: L:ihiXi 2 L:i hi(Xi- x )2 Lihi(Xi-X)(Yi-J.Ly) J.Lx = , U = , U xy = n x n n _ L:i hiYi 2 _ Li hi(Yi - J.Ly)2 2 _ 2 u;y J.Ly , Uy , Uylx - Uy - -2 . n n ~ We use them to express the conditional expectations and their estimated variances: u2 Y = J.Ly, u? = -1!.. y n kernel: LOESS: , _ ~( ),...? __ u;lx (1 + (x - J.Lx)2) Y J.Ly + q2 X - J.Lx, V % Y n u; 4.1 ACTIVE LEARNING WITH LOCALLY WEIGHTED REGRESSION (4) (5) Again we want to select x to minimize (iT~) . With LWR, the model's estimated distribution of y given x is explicit: P(ylx) = N(y(x), u;lxCx)) The estimate of (iT~) is also explicit. Defining h as the weight assigned to x by the kernel, the learner's expected new variance is k 1. (-2) _ (iT~) erne. uy - --n+h (6) where the expectation can be computed exactly in closed form: 710 David Cohn, Zoubin Ghahramani, Michael 1. Jordon 5 EXPERIMENTAL RESULTS Below we describe two sets of experiments demonstrating the predictive power of the query selection criteria in this paper. In the first set, learners were trained on data from a noisy sine wave. The criteria described in this paper were applied to predict how a new training example selected at point x would decrease the learner's variance. These predictions, along with the actual changes in variance when the training points were queried and added, are plotted in Figure 1. o. -0.5 o. - . _. -. - .• predicted change -- actual 9"8rl97". . \ -" i ~. -0.2 0.2 0.4 0.6 0.8 " , . 0.2 0.4 - . - - . _. - predicted change -- actual change .-.- -.-.- predicted change -- actual change 0.6 0.8 Figure 1: The upper portion of each plot indicates each learner's fit to noisy sinusoidal data. The lower portion of each plot indicates predicted and actual changes in the learner's average estimated variance when x is queried and added to the training set, for x E [0,1]. Changes are not plotted to scale with learners' fits. In the second set of experiments, we a:pplied the techniques of this paper to learning the kinematics of a two-joint planar arm (Figure 2; see Cohn [1994] for details). Below, we illustrate the problem using the LOESS algorithm. An example of the correlation between predicted and actual changes in variance on this problem is plotted in Figure 2. Figure 3 demonstrates that this correlation may be exploited to guide sequential query selection. We compared a LOESS learner which selected each new query so as to minimize expected variance Active Learning with Statistical Models 711 with LOESS learners which selected queries according to various heuristics. The variance-minimizing learner significantly outperforms the heuristics in terms of both variance and MSE. 0 .025r--..---...,......-~---.----...---,---.", 0.02 ~ 0.015 c:: .~ ~ 0.01 til ~ 0.005 "iii ::I ~ 0 -0.005 o 0 o o o o -°$.01 -0.005 0 0.005 0,01 0.015 0.02 0.025 predicted delta variance Figure 2: (left) The arm kinematics problem. (right) Predicted vs. actual changes in model variance for LOESS on the arm kinematics problem. 100 candidate points are shown for a model trained with 50 initial random examples. Note that most of the potential queries produce very little improvement, and that the algorithm successfully identifies those few that will help most. 0.2 0.1 VarianceO.04 0.02 0.01 0.004 50 100 150 200 250 300 350 400 450 500 training examples 3 MSE 0.3 0.1 50 100 150200 250 300 350 400 450 500 training examples Figure 3: Variance and MSE for a LOESS learner selecting queries according to the variance-minimizing criterion discussed in this paper and according to several heuristics. "Sensitivity" queries where output is most sensitive to new data, "Bias" queries according to a bias-minimizing criterion, «Support" queries where the model has the least data support. The variance of "Random" and "Sensitivity" are off the scale. Curves are medians over 15 runs with non-Gaussian noise. 712 David Cohn. Zouhin Ghahramani. Michael 1. Jordon 6 SUMMARY Mixtures of Gaussians and locally weighted regression are two statistical models that offer elegant representations and efficient learning algorithms. In this paper we have shown that they also offer the opportunity to perform active learning in an efficient and statistically correct manner. The criteria derived here can be computed cheaply and, for problems tested, demonstrate good predictive power. Acknowledgements This work was funded by NSF grant CDA-9309300, the McDonnell-Pew Foundation, ATR Human Information Processing Laboratories and Siemens Corporate Research. We thank Stefan Schaal for helpful discussions about locally weighted regression. References W. Cleveland, S. Devlin, and E. Grosse. (1988) Regression by local fitting. Journal of Econometrics 37:87-114. D. Cohn, 1. Atlas and R. Ladner. (1990) Training Connectionist Networks with Queries and Selective Sampling. In D. Touretzky, ed., Advances in Neural Information Processing Systems 2, Morgan Kaufmann. D. Cohn. (1994) Neural network exploration using optimal experiment design. In J. Cowan et al., eds., Advances in Neural Information Processing Systems 6. Morgan Kaufmann. A. Dempster, N. Laird and D. Rubin. (1977) Maximum likelihood from incomplete data via the EM algorithm. J. Royal Statistical Society Series B, 39:1-38. V. Fedorov. (1972) Theory of Optimal Experiments. Academic Press, New York. Z. Ghahramani and M. Jordan. (1994) Supervised learning from incomplete data via an EM approach. In J. Cowan et al., eds., Advances in Neural Information Processing Systems 6. Morgan Kaufmann. A. Linden and F. Weber. (1993) Implementing inner drive by competence reflection. In H. Roitblat et al., eds., Proc. 2nd Int. Conf. on Simulation of Adaptive Behavior, MIT Press, Cambridge. D. MacKay. (1992) Information-based objective functions for active data selection, Neural Computation 4( 4): 590-604. S. Nowlan. (1991) Soft Competitive Adaptation: Neural Network Learning Algorithms based on Fitting Statistical Mixtures. CMU-CS-91-126, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA. Paass, G., and Kindermann, J. (1995). Bayesian Query Construction for Neural Network Models. In this volume. M. Plutowski and H. White (1993). Selecting concise training sets from clean data. IEEE Transactions on Neural Networks, 4, 305-318. S. Schaal and C. Atkeson. (1994) Robot Juggling: An Implementation of Memory-based Learning. Control Systems Magazine, 14(1):57-71. J. Schmidhuber and J. Storck. (1993) Reinforcement driven information acquisition in nondeterministic environments. Tech. Report, Fakultiit fiir Informatik, Technische Universitiit Munchen. D. Specht. (1991) A general regression neural network. IEEE Trans. Neural Networks, 2(6):568-576. S. Thrun and K. Moller. (1992) Active exploration in dynamic environments. In J. Moody et aI., editors, Advances in Neural Information Processing Systems 4. Morgan Kaufmann.	1994	68
963	From Data Distributions to Regularization in Invariant Learning Todd K. Leen Department of Computer Science and Engineering Oregon Graduate Institute of Science and Technology 20000 N.W. Walker Rd Beaverton, Oregon 97006 tieen@cse.ogi.edu Abstract Ideally pattern recognition machines provide constant output when the inputs are transformed under a group 9 of desired invariances. These invariances can be achieved by enhancing the training data to include examples of inputs transformed by elements of g, while leaving the corresponding targets unchanged. Alternatively the cost function for training can include a regularization term that penalizes changes in the output when the input is transformed under the group. This paper relates the two approaches, showing precisely the sense in which the regularized cost function approximates the result of adding transformed (or distorted) examples to the training data. The cost function for the enhanced training set is equivalent to the sum of the original cost function plus a regularizer. For unbiased models, the regularizer reduces to the intuitively obvious choice a term that penalizes changes in the output when the inputs are transformed under the group. For infinitesimal transformations, the coefficient of the regularization term reduces to the variance of the distortions introduced into the training data. This correspondence provides a simple bridge between the two approaches. 22 4 Todd Leen 1 A pproaches to Invariant Learning In machine learning one sometimes wants to incorporate invariances into the function learned. Our knowledge of the problem dictates that the machine outputs ought to remain constant when its inputs are transformed under a set of operations gl. In character recognition, for example, we want the outputs to be invariant under shifts and small rotations of the input image. In neural networks, there are several ways to achieve this invariance 1. The invariance can be hard-wired by weight sharing in the case of summation nodes (LeCun et al. 1990) or by constraints similar to weight sharing in higher-order nodes (Giles et al. 1988). 2. One can enhance the training ensemble by adding examples of inputs transformed under the desired inval"iance group, while maintaining the same targets as for the raw data. 3. One can add to the cost function a regularizer that penalizes changes in the output when the input is transformed by elements of the group (Simard et al. 1992). Intuitively one expects the approaches in 3 and 4 to be intimately linked. This paper develops that correspondence in detail. 2 The Distortion-Enhanced Input Ensemble Let the input data x be distributed according to the density function p( x). The conditional distribution for the corresponding targets is denoted p(tlx). For simplicity of notation we take t E R. The extension to vector targets is trivial. Let f(x; w) denote the network function, parameterized by weights w. The training procedure is assumed to minimize the expected squared error £(w) = J J dtdx p(tlx) p(x) [t - f(x; w)]2 . (1) vVe wish to consider the effects of adding new inputs that are related to the old by transformations that correspond to the desired invariances. These transformations, or distortions, of the inputs are carried out by group elements g E g. For Lie groups2, the transformations are analytic functions of parameters a E Rk x -t x' = g(x;a) , with the identity transformation corresponding to parameter value zero g(x;O) = x . (2) (3) In image processing, for example, we may want our machine to exhibit invariance with respect to rotation, scaling, shearing and translations of the plane. These 1 We assume that the set forms a group. 2See for example (Sattillger, 1986). From Data Distributions to Regularization in Invariant Learning 225 transformations form a six-parameter Lie group3. By adding distorted input examples we alter the original density p( x). To describe the new density, we introduce a probability density for the transformation parameters p(a). Using this density, the distribution for the distortion-enhanced input ensemble is p(x') = j j dadx p(x'lx,a) p(a) p(x) = j jdadxt5(x'-g(x;a»p(a)p(x) where t5(.) is the Dirac delta function4 Finally we impose that the targets remain unchanged when the inputs are transformed according to (2) i.e., p(tlx') = p(tlx). Substituting p(x') into (1) and using the invariance of the targets yields the cost function t = j j J dtdxda p(tlx)p(x)p(a) [t - f(g(x;a);w)]2 (4) Equation (4) gives the cost function for the distortion-enhanced input ensemble. 3 Regularization and Hints The remainder of the paper makes precise the connection between adding transformed inputs, as embodied in (4), and various regularization procedures. It is straightforward to show that the cost function for the distortion-enhanced ensemble is equivalent to the cost function for the original data ensemble (1) plus a regularization term. Adding and subtracting f(x; w) to the term in square brackets in (4), and expanding the quadratic leaves t = E + ER , (5) where the regularizer is ER = EH + Ec J da p(a) j dx p(x) [f(x, w) - f(g(x; a); w)]2 - 2 J j J dtdxda p(tlx) p(x) p(a) x[t - f{x;w)] [f(g(x;a);w) - f(x;w)] (6) 3The parameters for rotations, scaling and shearing completely specify elements of G L2, the four parameter group of 2 x 2 invertible matrices. The translations carry an additional two degrees of freedom. 4 In general the density on 0 might vary through the input space, suggesting the conditional density p(o I :1'). This introduces rather minor changes in the discussion that will not be considered here. 226 ToddLeen Training with the original data ensemble using the cost function (5) is equivalent to adding transformed inputs to the data ensemble. The first term of the regularizer £ H penalizes the average squared difference between I(x;w) and I(g(x;a);w). This is exactly the form one would intuitively apply in order to insure that the network output not change under the transformation x -4 g( x, a). Indeed this is the similar to the form of the invariance "hint" proposed by Abu-Mostafa (1993). The difference here is that there is no arbitrary parameter multiplying the term. Instead the strength of the regularizer is governed by the average over the density pea). The term £H measures the error in satisfying the invariance hint. The second term £a measures the correlation between the error in fitting to the data, and the errol' in satisfying the hint. Only when these correlations vanish is the cost function for the enhanced ensemble equal to the original cost function plus the invariance hint penalty. The correlation term vanishes trivially when either 1. The invariance I (g( x; a); w) = I (x; w) is satisfied, or 2. The network function equals the least squares regression on t I(x; w) = J dt p(tlx) t = E[tlx] . (7) The lowest possible £ occurs when I satisfies (7), at which £ becomes the variance in the targets averaged over p( x ). By substituting this into £a and carrying out the integration over dt p( tlx), the correlation term is seen to vanish. If the minimum of t occurs at a weight for which the invariance is satisfied (condition 1 above). then minimizing t ( w) is equivalent to minimizing £ ( w). If the minimum of t occurs at a weight for which the network function is the regression (condition 2), then minimizing t is equivalent to minimizing the cost function with the intuitive regularizer £ H 5. 3.1 Infinitesimal Transformations Above we enumerated the conditions under which the correlation term in £R vanishes exactly for unrestricted transformations. If the transformations are analytic in the paranleters 0', then by restricting ourselves to small transformations (those close to the identity) we can-show how the correlation term approximately vanishes for unbiased models. To implement this, we assume that p( a) is sharply peaked up about the origin so that large transformations are unlikely. 51£ the data is to be fit optimally, with enough freedom left over to satisfy the invariance hint, then there must be several weight values (perhaps a continuum of such values) for which the network function satisfies (7). That is, the problem must be under-specified. If this is the case, then the interesting part weight space is just the subset on which (7) is satisfied. On this subset the correlation term in (6) vanishes and the regularizer assumes the intuitive form. From Data Distributions to Regularization in Invariant Learning 227 We obtain an approximation to the cost function t by expanding the integrands in (6) in power series about 0 = 0 and retaining terms to second order. This leaves t = c + J J dxdo p(x) p(o) (Oi :~ L=o :!" f -2 J J J dt dx do p(tlx)p(x)p(o) [t-f(x;w)] x [ ( Og"l 1 02 gIl I ) ( of) 0°+-0°0° • OOi 0=0 2' J OOi (0) 0=0 ax" (8) where xP and gP denote the pth components of x and g, OJ denotes the ith component of the transformation parameter vector, repeated Greek and Roman indices are summed over, and all derivatives are evaluated at 0 = o. Note that we have used the fact that Lie group transformations are analytic in the parameter vector o to derive the expansion. Finally we introduce two assumptions on the distribution p(o). First 0 is assumed to be zero mean. This corresponds, in the linear approximation, to a distribution of distortions whose mean is the identity transformation. Second, we assume that the components of 0 are uncorrelated so that the covariance matrix is diagonal with elements ul, i = 1 ... k. 6 With these assumptions, the cost function for the distortion-enhanced ensemble simplifies to t = c + ~ (Tr J dx p( x) ( ~g"l : f ) 2 ~ va. a=O vx" .=1 k - L (T~ J J dx dt p(tlx) p(X) { (f(x; w) - t ) .=1 X [ ~:; 10=0 ( :~ ) + :~ L=o :~ L=o (ox~2£x" )]} This last expression provides a simple bridge between the the methods of adding transformed examples to the data, and the alternative of adding a regularizer to the cost function: The coefficient of the regularization term in the latter approach is equal to the variance of the transformation parameters in the former approach. 6Note that the transformed patterns may be correlated in parts of the pattern space. For example the results of applying the shearing and rotation operations to an infinite vertical line are indistinguishable. In general, there may be regions of the pattern space for which the action of several different group elements are indistinguishable; that is x' = g(x; a) = g(x; (3). However this does not imply that a and (3 are statistically correlated. 228 Todd Leen 3.1.1 Unbiased Models For unbiased models the regularizer in E( w) assumes a particularly simple form. Suppose the network function is rich enough to form an unbiased estimate of the least squares regression on t for the un distorted data ensemble. That is, there exists a weight value Wo such that f(x;wo) = J dt tp(tlx) == E[tlx] (10) This is the global minimum for the original error £( w). The arguments of section 3 apply here as well. However we can go further. Even if there is only a single, isolated weight value for which (10) is satisfied, then to O( 0-2 ) the correlation term in the regularizer vanishes. To see this note that by the implicit function theorem the modified cost function (9) has its global minimum at the new weight 7 (11) At this weight, the network function is no longer the regression on t, but rather f(x;wo) = E[tlx] + 0(0-2 ) • (12) Substituting (12) into (9), we find that the minimum of (9) is, to 0(0-2 ), at the same weight as the minimum of t = £ + L.k o-~ JdX p(x) [ oglJ I of (x, w) ] 2 .=1 0 Q'j Q=O oxlJ (13) To 0(0-2 ), minimizing (13) is equivalent to minimizing (9). So we regard t as the effective cost function. The regularization term in (13) is proportional to the average square of the gradient of the network function along the direction in the input space generated by the lineal' part of g. The quantity inside the square brackets is just the linear part of [f (g( X; Q')) - f (x)] from (6). The magnitude of the regularization term is just the variance of the distribution of distortion parameters. This is precisely the form of the regularizer given by Simard et al. in their tangent prop algorithm (Simard et aI, 1992). This derivation shows the equivalence (to 0(0"2)) between the tangent prop regularizer and the alternative of modifying the input distribution. Furthermore, we see that with this equivalence, the constant fixing the strengt.h of the regularization term is simply the variance of the distortions introduced into the original training set. We should stress that the equivalence between the regularizer, and the distortionenhanced ensemble in (13) only holds to 0(0-2). If one allows the variance of the 7We assume that the Hessian of £ is nonsingular at woo From Data Distributions to Regularization in Invariant Learning 229 distortion parameters u 2 to become arbitrarily large in an effort to mock up an arbitrarily large regularization term, then the equivalence expressed in (13) breaks down since terms of order O( ( 4 ) can no longer be neglected. In addition, if the transformations are to be kept small so that the linearization holds (e.g. by restricting the density on a to have support on a small neighborhood of zero), then the variance will bounded above. 3.1.2 Smoothing Regularizers In the previous sections we showed the equivalence between modifying the input distribution and adding a regularizer to the cost function. We derived this equivalence to illuminate mechanisms for obtaining invariant pattern recognition. The technique for dealing with infinitesimal transformations in section §3.1 was used by Bishop (1994) to show the equivalence between added input noise and smoothing regularizers. Bishop's results, though they preceded our own, are a special case of the results presented here. Suppose the group 9 is restricted to translations by random vectors g( X; a) = X + a where a is spherically distributed with variance u!. Then the regularizer in (13) is (14) This regularizer penalizes large magnitude gradients in the network function and is, as pointed out by Bishop, one of the class of generalized Tikhonov regularizers. 4 Summary We have shown that enhancing the input ensemble by adding examples transformed under a group x -? g(x;a), while maintaining the target values, is equivalent to adding a regularizer to the original cost function. For unbiased models the regulatizer reduces to the intuitive form that penalizes the mean squared difference between the network output for transformed and untransformed inputs - i.e. the error in satisfying the desired invariance. In general the regularizer includes a term that measures correlations between the error in fitting the data, and the error in satisfying the desired inva.riance. For infinitesimal transformations, the regularizer is equivalent (up to terms linear in the variance of the transformation parameters) to the tangent prop form given by Simard et a1. (1992), with regularization coefficient equal to the variance of the transformation parameters. In the special case that the group transformations are limited to random translations of the input, the regularizer reduces to a standard smoothing regularizer. \Ve gave conditions under which enhancing the input ensemble and adding the intuitive regularizer £ H are equivalent. However tins equivalence is only with regard to the optimal weight. We have not compared the training dynamics for the two approaches. In particular, it is quite possible that the full regularizer £H + £c exhibits different training dynamics from the intuitive form £ H. For the approach in which data are added to the input ensemble, one can easily construct datasets and distributions p( a) that either increase the condition number of the Hessian, or decrease it. Finally, it may be that the intuitive regularizer can have either detrimental or positive effects on the Hessian as well. 230 ToddLeen Acknowledgments I thank Lodewyk Wessels, Misha Pavel, Eric Wan, Steve Rehfuss, Genevieve Orr and Patrice Simard for stimulating and helpful discussions, and the reviewers for helpful comments. I am grateful to my father for what he gave to me in life, and for the presence of his spirit after his recent passing. This work was supported by EPRI under grant RP8015-2, AFOSR under grant FF4962-93-1-0253, and ONR under grant N00014-91-J-1482. References Yasar S. Abu-Mostafa. A method for learning from hints. In S. Hanson, J. Cowan, and C. Giles, editors, Advances in Neural Information Processing Systems, vol. 5, pages 73-80. Morgan Kaufmann, 1993. Chris M. Bishop. Training with noise is equivalent to Tikhonov regularization. To appear in Neural Computation, 1994. C.L. Giles, R.D. Griffin, and T. Maxwell. Encoding geometric invariances in higherorder neural networks. In D.Z.Anderson, editor, Neural Information Processing Systems, pages 301-309. American Institute of Physics, 1988. Y. Le Cun, B. Boser, J.S. Denker, D. Henderson, R.E. Howard, W. Hubbard, and L.D. Jackel. Handwritten digit recognition with a back-propagation network. In Advances in Neural Information Processing Systems, vol. 2, pages 396-404. Morgan Kaufmann Publishers, 1990. Patrice Simard, Bernard Victorri, Yann Le Cun, and John Denker. Tangent prop a formalism for specifying selected invariances in an adaptive network. In John E. Moody, Steven J. Hanson, and Richard P. Lippmann, editors, Advances in Neural Information Processing Systems 4, pages 895-903. Morgan Kaufmann, 1992. D.H. Sattinger and O.L. Weaver. Lie Groups and Algebras with Applications to Physics, Geometry and Mechanics. Springer-Verlag, 1986.	1994	69
964	Connectionist Speaker Normalization with Generalized Resource Allocating Networks Cesare Furlanello Istituto per La Ricerca Scientifica e Tecnologica Povo (Trento), Italy furlan«lirst. it Diego Giuliani Istituto per La Ricerca Scientifica e Tecnologica Povo (Trento), Italy giuliani«lirst.it Abstract Edmondo Trentin Istituto per La Ricerca Scientifica e Tecnologica Povo (Trento), Italy trentin«lirst.it The paper presents a rapid speaker-normalization technique based on neural network spectral mapping. The neural network is used as a front-end of a continuous speech recognition system (speakerdependent, HMM-based) to normalize the input acoustic data from a new speaker. The spectral difference between speakers can be reduced using a limited amount of new acoustic data (40 phonetically rich sentences). Recognition error of phone units from the acoustic-phonetic continuous speech corpus APASCI is decreased with an adaptability ratio of 25%. We used local basis networks of elliptical Gaussian kernels, with recursive allocation of units and on-line optimization of parameters (GRAN model). For this application, the model included a linear term. The results compare favorably with multivariate linear mapping based on constrained orthonormal transformations. 1 INTRODUCTION Speaker normalization methods are designed to minimize inter-speaker variations, one of the principal error sources in automatic speech recognition. Training a speech recognition system on a particular speaker (speaker-dependent or SD mode) generally gives better performance than using a speaker-independent system, which is 868 Cesare Furlanello. Diego Giuliani. Edmondo Trentin trained to recognize speech from a generic user by averaging over individual differences. On the other hand, performance may be dramatically worse when a SD system "tailored" on the acoustic characteristics of a speaker (the reference speaker) is used by another one (the new or target speaker). Training a SD system for any new speaker may be unfeasible: collecting a large amount of new training data is time consuming for the speaker and unacceptable in some applications. Given a pre-trained SD speech recognition system, the goal of normalization methods is then to reduce to a few sentences the amount of training data required from a new speaker to achieve acceptable recognition performance. The inter-speaker variation of the acoustic data is reduced by estimating a feature vector transformation between the acoustic parameter space of the new speaker and that of the reference speaker (Montacie et al., 1989; Class et al., 1990; Nakamura and Shikano, 1990; Huang, 1992; Matsukoto and Inoue, 1992). This multivariate transformation, also called spectral mapping given the type of features considered in the parameterization of speech data, provides an acoustic front-end to the recognition system. Supervised speaker normalization methods require that the text of the training utterances required from the new speaker is known, while arbitrary utterances can be used by unsupervised methods (Furui and Sondhi, 1991). Good performance have been achieved with spectral mapping techniques based on MSE optimization (Class et al., 1990; Matsukoto and Inoue, 1992). Alternative approaches presented estimation of the spectral normalization mapping with Multi-Layer Perceptron neural networks (Montacie et al., 1989; Nakamura and Shikano, 1990; Huang, 1992; Watrous, 1994). This paper introduces a supervised speaker normalization method based on neural network regression with a generalized local basis model of elliptical kernels (Generalized Resource Allocating Network: GRAN model). Kernels are recursively allocated by introducing the heuristic procedure of (Platt, 1991) within the generalized RBF schema proposed in (Poggio and Girosi, 1989). The model includes a linear term and efficient on-line optimization of parameters is achieved by an automatic differentiation technique. Our results compare favorably with normalization by affine linear transformations based on orthonormal constrained pseudoinverse. In this paper, the normalization module was integrated and tested as an acoustic front-end for speaker-dependent continuous speech recognition systems. Experiments regarded phone units recognition with Hidden Markov Model (HMM) recognition systems. The diagram in Figure 1 outlines the general structure of the experiment with GRAN normalization modules. The architecture is independent from the specific speech recognition system and allows comparisons between different normalization techniques. The GRAN model and a general procedure for data standardization are described in Section 2 and 3. After a discussion of the spectral mapping problem in Section 4, the APASCI corpus used in the experiments and the characteristics of the acoustic data are described in Section 5. The recognition system and the experiment set-up are detailed in Sections 6-8. Results are presented and discussed in Section 9. Connectionist Speaker Normalization with Generalized Resource Allocating Networks 869 DataBase: phraseS reference phrase (Yj } j-I •...• ] Dynamic Time Warping I (Xi(t), Yj(t}} Neural Network Training -' supervised training : Test fx) i-I •...• I '-------------------"1 GRAN normalizati Feature Extraction Speech Signal corresponding to phrase S uttered by a new speaker Figure 1: System overview 2 THE GRAN MODEL Output Feedforward artificial neural networks can be regarded as a convenient realization of general functional superpositions in terms of simpler kernel functions (Barron and Barron, 1988). With one hidden layer we can implement a multivariate superposition f(z) = Ef=o cxjKj(z,wj) where Kj is a function depending on an input vector z and a parameter vector Wj, a general structure which allows to realize flexible models for multivariate regression. We are interested in the schema: y = H K(x) + Ax + b with input vector x E Rd1 and estimated output vector y E R 2 . K = (Kj) is a n-dimensional vector of local kernels, H is the d2 x n real matrix of kernel coefficients, b E R d 2 is an offset term and A is a d2 x d1 linear term. Implemented kernels are Gaussian, Hardy multiquadrics, inverse of Hardy multiquadrics and Epanenchnikov kernels, also in the N adarayaWatson normalized form (HardIe, 1990). The kernel allocation is based on a recursive procedure: if appropriate novelty conditions are satisfied for the example (x', y/), a new kernel Kn+1 is allocated and the new estimate Yn+l becomes Yn+l (x) = Yn(X) + Kn+1 (llx - x'llw)(y' - Yn(X)) (HardIe, 1990). Global properties and rates of convergence for recursive kernel regression estimates are given in (Krzyzak, 1992). The heuristic mechanism suggested by (Platt, 1991) has been extended to include the optimization of the weighted metrics as requested in the generalized versions of RBF networks of (Poggio and Girosi, 1989). Optimization regards kernel coefficients, locations and bandwidths, the offset term, the coefficient matrix A if considered, and the W matrix defining the weighted metrics in the input space: IIxll~ = xtwtWx. Automatic differentiation is used for efficient on-line gradient-descent procedure w.r. t. different error functions (L2, L1, entropy fit), with different learning rates for each type of parameters. 870 Cesare FurLanello, Diego GiuLiani, Edmondo Trentin Ij;-::=<p X -----------+" Y TJx TJy x -----------" Y -1 TJy Figure 2: Commutative diagram for the speaker normalization problem. The spectral mapping <p between original spaces X and Y is estimated by Ij; = TJy 1 . ip . TJx, obtained by composition of the neural GRAN mapping ip between PCA spaces X and Y with the two invertible PCA transformations TJx and TJy. 3 NETWORKS AND PCA TRANSFORMATIONS The normalization module is designed to estimate a spectral mapping between the acoustic spaces of two different speakers. Inter-speaker variability is reflected by significant differences in data distribution in these multidimensional spaces (we considered 8 dimensions); in particular it is important to take into account global data anisotropy. More generally, it is also crucial to decorrelate the features describing the data. A general recipe is to apply the well-known Principal Component Analysis (PCA) to the data, in this case implemented from standard numerical routines based on Singular Value Decomposition of the data covariance matrices. The network was applied to perform a mapping between the new feature spaces obtained from the PCA transformations, mean translation included (Figure 2). 4 THE SPECTRAL MAPPING PROBLEM A sound uttered by a speaker is generally described by a sequence offeature vectors obtained from the speech signal via short-time spectral analysis (Sec. 5). The spectral representations of the same sequence of sounds uttered by two speakers are subject to significant variations (e.g. differences between male and female speakers, regional accents, ... ). To deal with acoustic differences, a suitable transformation (the spectral mapping) is seeked which performs the "best" mapping between the corresponding spectra oftwo speakers. Let Y = (Yl, Y2, ... , YJ) and X = (x 1, X2, ... , X I) be the spectral feature vector sequences of the same sentence uttered by two speakers, called respectively the reference and the new speaker. The desired mapping is performed by a function <pC Xi) such that the transformed vector sequence obtained from X = (Xi) approximates as close as possible the spectral vector sequence Y = (Yi). To eliminate time differences between the two acoustic realizations, a time warping function has to be determined yielding pairs C(k) = (i(k),j(k))k=1.. .K of corresponding indexes of feature vectors in X and Y, respectively. The desired spectral Connectionist Speaker Normalization with Generalized Resource Allocating Networks 87 J mapping r,o(Xi) is the one which minimizes Ef=l d(Yj(k)' r,o(Xi(k»)) where d(·,·) is a distorsion measure in the acoustic feature space. To estImate the transformation, a set of supervised pairs (Xi(k), Yj(k») is considered. In summary, the training material considered in the experiments consisted of a set of vector pairs obtained by applying the Dynamic Time Warping (DTW) algorithm (Sakoe and Chiba, 1978) to a set of phrases uttered by the reference and the new speaker. 5 THE APASCI CORPUS The experiments reported in this paper were performed on a portion of APASCI, an italian acoustic-phonetic continuous speech corpus. For each utterance, text and phonetic transcriptions were automatically generated (Angelini et al., 1994). The corpus consists of two portions. The first part, for the training and validation of speaker independent recognition systems, consists of a training set (2140 utterances), a development set (900 utterances) and a test set (860 utterances). The sets contain, respectively, speech material from 100 speakers (50 males and 50 females), 36 speakers (18 males and 18 females) and 40 speakers (20 males and 20 females). The second portion of the corpus is for training and validation of speaker dependent recognition systems. It consists of speech material from 6 speakers (3 males and 3 females). Each speaker uttered 520 phrases, 400 for training and 120 for test. Speech material in the test set was acquired in different days with respect to the training set. A subset of 40 utterances from the training material forms the adaptation training set, to be used for speaker adaptation/normalization purposes. For this application, each signal in the corpus was processed to obtain its parametric representation. The signal was preemphasized using a filter with transfer function H(z) = 1 - 0.95 X z-l, and a 20 ms Hamming window is then applied every 10 ms. For each frame, the normalized log-energy as well as 8 Mel Scaled Cepstral Coefficients (MSCC) based on a 24-channel filter-bank were computed. Normalization of log-energy was performed by subtracting the maximum log-energy value in the sentence; for each Mel coefficient, normalization was performed by subtracting the mean value of the whole utterance. For both MSCC and the log-energy, the first order derivatives as well as the second order derivatives were computed. For each frame, all the computed acoustic parameters were combined in a single feature vector with 27 components. 6 THE RECOGNITION SYSTEM For each of the 6 speakers, a SD HMM recognition system was trained with the 400 utterances available in the APASCI corpus; the systems were bootstrapped with gender dependent models trained on the gender dependent speech material (1000 utterances for male and 1140 utterances for female). A set of 38 context independent acoustic-phonetic units was considered. Left-to-right HMMs with three and four states were adopted for short (i.e. p,t,k,b,d,g) and long (e.g. a,i,u,Q,e) sounds respectively. Silence, pause and breath were modeled with a single state ergodic model. The output distribution probabilities were modeled with mixtures of 16 gaussian probability densities, diagonal covariance matrixes. Transitions leaving the same state shared the same output distribution probabilities. 872 Cesare Furlanello, Diego Giuliani, Edmondo Trentin Table 1: Phone Recognition Rate (Unit Accuracy %) without normalization 7 TRAINING THE NORMALIZATION MODULES A set of 40 phrases was considered for each pair (new, re f erence) of speakers to train the normalization modules. In order to take into account alternative pronunciation, insertion or deletion of phonemes, pauses between words and other phenomena, the automatic phonetic transcription and segmentation available in APASCI was used for each utterance. Given two utterances corresponding to the same phrase, we considered only their segments having the same phonetic transcription. To determine these segments the DTW algorithm was applied to the phonetic transcription of the two utterances. The DTW algorithm was applied a second time to the obtained segments and the resulting optimal alignment paths gave the desired set of vector pairs. The DTW algorithm was applied only to the 8 MSCC and the other acoustic parameters were left unmodified. We trained networks with 8 inputs and 8 outputs. The model included a linear term: first the linear term was fit to the data, and then the rest of the expansion was estimated by fitting the residuals of the linear regression. The networks grew up to 50 elliptical gaussian kernels using dynamic allocation. Kernel coefficients, locations and bandwidths were optimized using different learning rates for 10 epochs w.r.t the Ll norm, which proved to be more efficient than the usual L2 norm. 8 THE RECOGNITION EXPERIMENTS Experiments concerned continuous phone recognition without any lexical and phonetical constraint (no phone statistic was used). For all the couples (new, reference) of speakers in the database, a recognition experiment was performed using 90 (of the 120 available) test utterances from the new speaker with the SD recognition system previously trained for the reference speaker. On average the test sets consisted of 4770 phone units. The experiments were repeated transforming the test data with different normalization modules and performance compared. Results are expressed in terms of insertions (Ins), deletions (Del) and substitutions (Sub) of phone units made by the recognizer. Unit Accuracy (U A) and Percent Correct (PC) performance indicators are respectively defined w.r.t. the total number of units nunih as U A = 100 (1 - (Ins + Del + Sub)/nunit.) and PC = 100 (1 - (Del + Sub)/nunit.). In Table 1 the baseline speaker dependent performance for the 6 speaker dependent systems is reported. Row labels indicate the speaker reference model while column labels identify whose target acousConnectionist Speaker Normalization with Generalized Resource Allocating Networks 873 Table 2: Phone Recognition Rate (Unit Accuracy %) with NN normalization tic data are used. Thus U A and PC entries in the main diagonal are for the same speaker who trained the system while the remaining entries relate to performance obtained with new speakers. We also considered the adaptability ratios for a = U A and P = PC (Montacie et al., 1989): Pa = (aRT - aRT )/(aRR - aRT) and Pp = (PRT - PRT )/(PRR - PRT) where aRT indicate accuracy for reference speaker R and target T without normalization, aRR is the speaker dependent baseline accuracy and apex n indicates normalization. The same notation applies to the percent correct adaptability ratio pp. 9 RESULTS AND CONCLUSIONS Normalization experiments have been performed with the set-up described in the previous Section. The phone recognition rates obtained with normalization modules based on the GRAN model are reported in Table 2 in terms of Unit Accuracy (dee Table 1 for the baseline performance). In Table 3 the performance of the GRAN model (NN) and constrained orthonormal linear mapping (LIN) are compared with the baseline performance (SD: no adaptation) in terms of both Unit Accuracy and Percent Correct. The network shows an improvement, as evidenced by the variation in the Pa and Pp values. Results are reported averaging performance over all the pairs (new,reference) of speakers (Total column), and considering pairs of speakers of the same gender and of different genders (Female: only female subjects, Male: only males, Dill: different genders). An analysis of the adaptability ratios shows that the effect of the network normalization is higher than with the linear network for all the 3 subgroups of pairs: p~N = 0.20 vs p~IN = 0.16 for the Female couples and liN = 0.16 vs p~IN = 0.15 for the Male couples. The improvement is higher (p~N = 0.28, p~IN = 0.24) for speaker of different genders. Although these preliminary experiments show only a minor improvement of performance achieved by the network with respect to linear mappings, we expect that the selectivity of the network could be exploited using acoustic contexts and code dependent neural networks. Acknowledgements This work has been developed within a grant of the "Programma N azionale di Ricerca per la Bioelettronica" assigned by the Italian Ministry of University and Technologic Research to Elsag Bailey. The authors would like to thank B. Angelini, F. Brugnara, B. Caprile, R. De Mori, D. Falavigna, G. Lazzari and P. Svaizer. 874 Cesare Furlanello, Diego Giuliani, Edmondo Trentin Table 3: Phone Recognition Rate (%) in terms of both Unit Accuracy, Percent Correct, and adaptability ratio p. References Angelini, B., Brugnara, F., Falavigna, D., Giuliani, D., Gretter, R., and Omologo, M. (September 1994). Speaker Independent Continuous Speech Recognition Using an Acoustic-Phonetic Italian Corpus. In Proc. of ICSLP, pages 1391-1394. Barron, A. R. and Barron, R. L. (1988). Statistical learning networks: a unifying view. In Symp. on the Interface: Statistics and Computing Science, Reston, VI. Class, F., Kaltenmeier, A., Regel, P., and Troller, K. (1990). Fast speaker adaptation for speech recognition system. In Proc. of ICASSP 90, pages 1-133-136. Furui, S. and Sondhi, M. M., editors (1991). Advances in Speech Signal Processing. Marcel Dekker and Inc. HardIe, W. (1990). Applied nonparametric regression, volume 19 of Econometric Society Monographs. Cambridge University Press, New York. Huang, X. D. (1992). Speaker normalization for speech recognition. In Proc. of ICASSP 92, pages 1-465-468. Krzyzak, A. (1992). Global convergence of the recursive kernel regression estimates with applications in classification and nonlinear system estimation. IEEE Transactions on Information Theory, 38(4):1323-1338. Matsukoto, H. and Inoue, H. (1992). A piecewise linear spectral mapping for supervised speaker adaptation. In Proc. of ICASSP 92, pages 1-449-452. Montacie, C., Choukri, K., and Chollet, G. (1989). Speech recognition using temporal decomposition and multi-layer feed-forward automata. In Proc. of ICASSP 89, pages 1-409-412. Nakamura, S. and Shikano, K. (1990). A comparative study of spectral mapping for speaker adaptation. In Proc. of ICASSP 90, pages 1-157-160. Platt, J. (1991). A resource-allocating network for function interpolation. Neural Computation, 3(2):213-225. Poggio, T. and Girosi, F. (1989). A theory of networks for approximation and learning. A.1. Memo No. 1140, MIT. Sakoe, H. and Chiba, S. (1978). Dynamic programming algorithm optimization for spoken word recognition. IEEE-A SSP, 26(1):43-49. Watrous, R. (1994). Speaker normalization and adaptation using second-order connectionist networks. IEEE Trans. on Neural Networks, 4(1):21-30.	1994	7
965	An Input Output HMM Architecture Yoshua Bengio* Dept. Informatique et Recherche Operationnelle Universite de Montreal, Qc H3C-3J7 bengioyOIRO.UMontreal.CA Paolo Frasconi Dipartimento di Sistemi e Informatica Universita di Firenze (Italy) paoloOmcculloch.ing.unifi.it Abstract We introduce a recurrent architecture having a modular structure and we formulate a training procedure based on the EM algorithm. The resulting model has similarities to hidden Markov models, but supports recurrent networks processing style and allows to exploit the supervised learning paradigm while using maximum likelihood estimation. 1 INTRODUCTION Learning problems involving sequentially structured data cannot be effectively dealt with static models such as feedforward networks. Recurrent networks allow to model complex dynamical systems and can store and retrieve contextual information in a flexible way. Up until the present time, research efforts of supervised learning for recurrent networks have almost exclusively focused on error minimization by gradient descent methods. Although effective for learning short term memories, practical difficulties have been reported in training recurrent neural networks to perform tasks in which the temporal contingencies present in the input/output sequences span long intervals (Bengio et al., 1994; Mozer, 1992). Previous work on alternative training algorithms (Bengio et al., 1994) could suggest that the root of the problem lies in the essentially discrete nature of the process of storing information for an indefinite amount of time. Thus, a potential solution is to propagate, backward in time, targets in a discrete state space rather than differential error information. Extending previous work (Bengio & Frasconi, 1994a), in this paper we propose a statistical approach to target propagation, based on the EM algorithm. We consider a parametric dynamical system with discrete states and we introduce a modular architecture, with subnetworks associated to discrete states. The architecture can be interpreted as a statistical model and can be trained by the EM or generalized EM (GEM) algorithms (Dempster et al., 1977), considering the internal state trajectories as missing data. In this way learning is decoupled into • also, AT&T Bell Labs, Holmdel, N J 07733 428 Yoshua Bengio, Paolo Frasconi a temporal credit assignment subproblem and a static learning subproblem that consists of fitting parameters to the next-state and output mappings defined by the estimated trajectories. In order to iteratively tune parameters with the EM or GEM algorithms, the system propagates forward and backward a discrete distribution over the n states, resulting in a procedure similar to the Baum-Welch algorithm used to train standard hidden Markov models (HMMs) (Levinson et al., 1983). HMMs however adjust their parameters using unsupervised learning, whereas we use EM in a supervised fashion . Furthermore, the model presented here could be called Input/Output HMM, or IOHMM, because it can be used to learn to map input sequences to output sequences (unlike standard HMMs, which learn the output sequence distribution) . This model can also be seen as a recurrent version of the Mixture of Experts architecture (Jacobs et al. , 1991), related to the model already proposed in (Cacciatore and Nowlan, 1994). Experiments on artificial tasks (Bengio & Frasconi, 1994a) have shown that EM recurrent learning can deal with long term dependencies more effectively than backpropa~ation through time and other alternative algorithms. However, the model used in (Bengio & Frasconi, 1994a) has very limited representational capabilities and can only map an input sequence to a final discrete state. In the present paper we describe an extended architecture that allows to fully exploit both input and output portions of the data, as required by the supervised learning paradigm. In this way, general sequence processing tasks, such as production, classification, or prediction, can be dealt with. 2 THE PROPOSED ARCHITECTURE We consider a discrete state dynamical system based on the following state space description: x f(x U ) t t-l, t (1) Yt = 9(Xt, Ut) where Ut E R m is the input vector at time t, Yt E R r is the output vector, and Xt E {I , 2, .. . , n} is a discrete state. These equations define a generalized Mealy finite state machine, in which inputs and outputs may take on continuous values. In this paper, we consider a probabilistic version of these dynamics, where the current inputs and the current state distribution are used to estimate the state distribution and the output distribution for the next time step. Admissible state transitions will be specified by a directed graph 9 whose vertices correspond to the model's states and the set of successors for state j is Sj . The system defined by equations (1) can be modeled by the recurrent architecture depicted in Figure l(a) . The architecture is composed by a set of state networks N j, j = 1 .. . n and a set of output networks OJ, j = 1 . .. n. Each one of the state and output networks is uniquely associated to one of the states,and all networks share the same input Ut . Each state network M has the task of predicting the next state distribution, based on the current input and given that Xt-l = j . Similarly, each output network OJ predicts the output of the system, given the current state and input. All the subnetworks are assumed to be static and they are defined by means of smooth mappings Nj (Ut ; 9j ) and OJ (Ut; 1J j), where 9 j and 1J j are vectors of adjustable parameters (e.g., connection weights). The ranges of the functions Nj 0 may be constrained in order to account for the underlying transition graph 9. Each output 'Pij ,t of the state subnetwork Nj (at time t) is associated to one of the successors i of state j . Thus the last layer of M has as many units as the cardinality of Sj. For convenience of notation, we suppose that 'Pij,t are defined for each i, j = 1, ... , n and we impose the condition 'Pij,t = 0 for each i not belonging to S j . The softmax function is used in the last layer: 'Pij,t = e a,j,t ILlEsj eal j,t, j = 1, ... , n , i E Sj where aij,t are intermediate variables that can be thought of as the An Input Output HMM Architecture current •• pectod output, given PIlat Input Mquenc. 11 t EIYt lull cu ... nt Input '-'t (a) current atilt. dlatrtbutton Ct= Pl' t I Ul ) Xt-l Xt ... ... 1 1 Yt-l Yt Xt-l X( \.Yt-l \Yt I I Ut-l Ut (b) 429 Xt+l ... .. .... 1 HMM Yt+l Xt+l \.Yt+l I IOHMM Ut+l Figure 1: (a): The proposed IOHMM architecture. (b): Bottom: Bayesian network expressing conditional dependencies for an IOHMM; top: Bayesian network for a standard HMM activations of the output units of subnetwork N j . In this way L:7=1 'Pij ,t = 1 Tij,t. The vector 't E R n represents the internal state of the model and it is computed as a linear combination of the outputs of the state networks, gated by the previously computed internal state: n where IPj,t = ['PIj,t, ... , 'Pnj,t]'. output of the system 1Jt E R r : 't = L(j,t-IIPj,t (2) j=1 Output networks compete to predict the global n 1Jt = L (jt1Jjt (3) j=1 where 1Jjt E R r is the output of subnetwork OJ. At this level, we do not need to further specify the internal architecture of the state and output subnetworks. Depending on the task, the designer may decide whether to include hidden layers and what activation rule to use for the hidden units. This connectionist architecture can be also interpreted as a probability model. Let us assume a multinomial distribution for the state variable Xt and let us consider 't, the main variable of the temporal recurrence (2). If we initialize the vector '0 to positive numbers summing to 1, it can be interpreted as a vector of initial state probabilities. In general, we obtain relation (it = P(Xt = i I un, having denoted with ui the subsequence of inputs from time 1 to t, inclusively. Equation (2) then has the following probabilistic interpretation: n P(Xt = i lui) = L P(Xt = i I Xt-I = j, ut}P(Xt-1 = j lui-I) (4) j=l i.e., the subnetworks N j compute transition probabilities conditioned on the input sequence Ut: P( . I .) (5) 'Pij,t = Xt = ~ Xt-l = ), Ut As in neural networks trained to minimize the output squared error, the output 1Jt of this architecture can be interpreted as an expected "position parameter" for the probability distribution of the output Yt. However, in addition to being conditional on an input Ut, this expectation is also conditional on the state Xt, i.e. 430 Yoshua Bengio, Paolo Frasconi 7]t = E[Yt I Xt,Ut] . The actual form of the output density, denoted !Y(Yt ;7]t), will be chosen according to the task. For example a multinomial distribution is suitable for sequence classification, or for symbolic mutually exclusive outputs. Instead, a Gaussian distribution is adequate for producing continuous outputs. In the first case we use a softmax function at the output of subnetworks OJ; in the second case we use linear output units for the subnetworks OJ. In order to reduce the amount of computation, we introduce an independency model among the variables involved in the probabilistic interpretation of the architecture. We shall use a Bayesian network to characterize the probabilistic dependencies among these variables. Specifically, we suppose that the directed acyclic graph 9 depicted at the bottom of Figure 1 b is a Bayesian network for the dependency model associated to the variables u I, xI, YI. One of the most evident consequences of this independency model is that only the previous state and the current input are relevant to determine the next-state. This one-step memory property is analogue to the Markov assumption in hidden Markov models (HMM). In fact, the Bayesian network for HMMs can be obtained by simply removing the Ut nodes and arcs from them (see top of Figure Ib) . 3 A SUPERVISED LEARNING ALGORITHM The learning algorithm for the proposed architecture is derived from the maximum likelihood principle. The training data are a set of P pairs of input/ output sequences (of length Tp): 1) = {(uip(p),Yip(p));p = 1 .. . P}. Let €J denote the vector of parameters obtained by collecting all the parameters (Jj and iJi of the architecture. The likelihood function is then given by p L(€J; 1)) = II p(Yip(p) I uip(p); €J). (6) p=l The output values (used here as targets) may also be specified intermittently. For example, in sequence classification tasks, one may only be interested in the output YT at the end of each sequence. The modification of the likelihood to account for intermittent targets is straightforward. According to the maximum likelihood principle, the optimal parameters are obtained by maximizing (6). In order to apply EM to our case we begin by noting that the state variables Xt are not observed. Knowledge of the model's state trajectories would allow one to decompose the temporal learning problem into 2n static learning subproblems. Indeed, if Xt were known, the probabilities (it would be either 0 or 1 and it would be possible to train each subnetwork separately, without taking into account any temporal dependency. This observation allows to link EM learning to the target propagation approach discussed in the introduction. Note that if we used a Viterbi-like approximation (i.e., considering only the most likely path) , we would indeed have 2n static learning problems at each epoch. In order to we derive the learning equations, let us define the complete data as 1)c = {(uiP(p),yiP(p),xiP(p));p = 1 ... P}. The corresponding complete data l%-likelihood is """ T T T Ic(€J;1)c) = L...IOgP(YIP(P),ZlP(P) I u1P(p); €J). (7) p=l Since lc( €J; 1)c) depends on the hidden state variables it cannot be maximized directly. The MLE optimization is then solved by introducing the auxiliary function Q(€J; 0) and iterating the following two,steps for k = 1, 2r ... :, Estimation: Compute Q(€J; €J) = E[lc(€J; 1)c) 1), €J] Maximization: Update the parameters as 0 t- arg max€J Q( €J; 0) (8) An Input Output HMM Architecture 431 The expectation of (7) can be expressed as p Tp N N Q(0; 0) = L: L: L: (it!og P(Yt I Xt = i, Ut i 0) + L: hij,tlog<Pij,t (9) p=1 t=1 i=1 j=1 where hij,t = EIZitzj,t-l I uf, yf; 0J, denoting Zit for an indicator variable = 1 if Xt = i and 0 otherwise. The hat in (it and hij,t means that these variables are computed using the "old" parameters 0 . In order to compute hij,t we introduce the forward probabilities Qit = P(YL Xt = i ; uD and the backward probabilities f3it = p(yf I Xt = i , un, that are updated as follows: f3it = fY(Yt;l1it) Lt <Pti(Ut+df3l,t+l Qit = fY(Yt; l1it) Lt <pa(ut}Qt ,t-l . (10) h f3it Qj,t-l<Pij (ut) (11) ij ,t " wi QiT Each iteration of the EM algorithm requires to maximize Q(0; 0). We first consider a simplified case, in which the inputs are quantized (i.e., belonging to a finite alphabet {0"1, "" O"K}) and the subnetworks behave like lookup tables addressed by the input symbols O"t, i.e. we interpret each parameter as Wi'k = P(Xt = i I Xt-l = j , O"t = k). For simplicity, we restrict the analysis to classification tasks and we suppose that targets are specified as desired final states for each sequence. Furthermore, no output subnetworks are used in this particular application of the algorithm. In this case we obtain the reestimation formulae: Wijk = . . " "p" {j,t(j,t-l w i ESj wp=1 wt :Ut=k , + T • "'T ' (12) In general, however, if the subnetworks have hidden sigmoidal units, or use a softmax function to constrain their outputs to sum to one, the maximum of Q cannot be found analytically. In these cases we can resort to a GEM algorithm, that simply produces an increase in Q, for example by gradient ascent. In this case, the derivatives of Q with respect to the parameters can be easily computed as follows. Let Ojlt be a generic weight in the state subnetwork N j . From equation (9): 8Q(0;0) = L:L:L:hij,t_l_8<pij,t 80jk p t i <Pij,t 80jk (13) where the partial derivatives &:e~;t can be computed using backpropagation. Similarly, denoting with t'Jik a generic weight of the output subnetwork Oi, we have: 8Q( 0; 0) '" '" '" . 8 87]a t 8t'J · = L..JL..JL..J(i ,t~logfY(Yy;l1it) 8t'J .' ,k p t t 7],t,t ,k (14) where ~;;:~t are also computed using backpropagation. Intuitively, the parameters are updated as if the estimation step of EM had provided targets for the outputs of the 2n subnetworks, for each time t. Although GEM algorithms are also guaranteed to find a local maximum of the likelihood, their convergence may be significantly slower compared to EM. In several experiments we noticed that convergence can be accelerated with stochastic gradient ascent. 432 Yoshua Bengio, Paolo Frasconi 4 COMPARISONS It appears natural to find similarities between the recurrent architecture described so far and standard HMMs (Levinson et al., 1983). The architecture proposed in this paper differs from standard HMMs in two respects: computing style and learning. With IOHMMs, sequences are processed similarly to recurrent networks, e.g., an input sequence can be synchronously transformed into an output sequence. This computing style is real-time and predictions of the outputs are available as the input sequence is being processed. This architecture thus allows one to implement all three fundamental sequence processing tasks: production, prediction, and classification. Finally, transition probabilities in standard HMMs are fixed, i.e. states form a homogeneous Markov chain. In IOHMMs, transition probabilities are conditional on the input and thus depend on time, resulting in an inhomogeneous Markov chain. Consequently, the dynamics of the system (specified by the transition probabilities) are not fixed but are adapted in time depending on the input sequence. The other fundamental difference is in the learning procedure. While interesting for their capabilities of modeling sequential phenomena, a major weakness of standard HMMs is their poor discrimination power due to unsupervised learning. An approach that has been found useful to improve discrimination in HMMs is based on maximum mutual information (MMI) training. It has been pointed out that supervised learning and discriminant learning criteria like MMI are actually strictly related (Bridle, 1989). Although the parameter adjusting procedure we have defined is based on MLE, yf is used as desired output in response to the input uf, resulting in discriminant supervised learning. Finally, it is worth mentioning that a number of hybrid approaches have been proposed to integrate connectionist approaches into the HMM frame\'Vork. For example in (Bengio et al. , 1992) the observations used by the HMM are generated by a feedforward neural network. In (Bourlard and Wellekens, 1990) a feedforward network is used to estimate state probabilities, conditional to the acoustic sequence. A common feature of these algorithms and the one proposed in this paper is that neural networks are used to extract temporally local information whereas a Markovian system integrates long-term constraints. We can also establish a link between IOHMMs and adaptive mixtures of experts (ME) (Jacobs et al., 1991). Recently, Cacciatore & Nowlan (1994) have proposed a recurrent extension to the ME architecture, called mixture of controllers (MC), in which the gating network has feedback connections, thus allowing to take temporal context into account. Our IOHMM architecture can be interpreted as a special case of the MC architecture, in which the set of state subnetworks play the role of a gating network having a modular structure and second order connections. 5 REGULAR GRAMMAR INFERENCE In this section we describe an application of our architecture to the problem of grammatical inference. In this task the learner is presented a set of labeled strings and is requested to infer a set of rules that define a formal language. It can be considered as a prototype for more complex language processing problems. However, even in the "simplest" case, i.e. regular grammars, the task can be proved to be NP-complete (Angluin and Smith, 1983). We report experimental results on a set of regular grammars introduced by Tomita (1982) and afterwards used by other researchers to measure the accuracy of inference methods based on recurrent networks (Giles et al. , 1992; Pollack, 1991; Watrous and Kuhn, 1992). We used a scalar output with supervision on the final output YT that was modeled as a Bernoulli variable fy (YT ; 7]T) = 7]~T (1 7] ) l-YT, with YT = 0 if the string is rejected and YT = 1 if it is accepted. In tbis application we did not apply An Input Output HMM Architecture 433 Table 1: Summary of experimental results on the seven Tomita's grammars. Grammar Sizes Convergence Accuracies n* FSA min Average Worst Best W&K Best 1 2 2 .600 1.000 1.000 1.000 1.000 2 8 3 .800 .965 .834 1.000 1.000 3 7 5 .150 .867 .775 1.000 .783 4 4 4 .100 1.000 1.000 1.000 .609 5 4 4 .100 1.000 1.000 1.000 .668 6 3 3 .350 1.000 1.000 1.000 .462 7 3 5 .450 .856 .815 1.000 .557 external inputs to the output networks. This corresponds to modeling a Moore finite state machine. Given the absence of prior knowledge about plausible state paths, we used an ergodic transition graph (i.e., fully connected).In the experiments we measured convergence and generalization performance using different sizes for the recurrent architecture. For each setting we ran 20 trials with different seeds for the initial weights. We considered a trial successful if the trained network was able to correctly label all the training strings. The model size was chosen using a cross-validation criterion based on performance on 20 randomly generated strings of length T ::; 12. For comparison, in Table 1 we also report for each grammar the number of states of the minimal recognizing FSA (Tomita, 1982). We tested the trained networks on a corpus of 213 - 1 binary strings of length T ::; 12. The final results are summarized in Table 1. The column "Convergence" reports the fraction of trials that succeeded to separate the training set. The next three columns report averages and order statistics (worst and best trial) of the fraction of correctly classified strings, measured on the successful trials. For each grammar these results refer to the model size n* selected by cross-validation. Generalization was always perfect on grammars 1,4,5 and 6. For each grammar, the best trial also attained perfect generalization. These results compare very favorably to those obtained with second-order networks trained by gradient descent, when using the learning sets proposed by Tomita. For comparison, in the last column of Table 1 we reproduce the results reported by Watrous & Kuhn (1992) in the best of five trials. In most of the successful trials the model learned an actual FSA behavior with transition probabilities asymptotically converging either to 0 or to 1. This renders trivial the extraction of the corresponding FSA. Indeed, for grammars 1,4,5, and 6, we found that the trained networks behave exactly like the minimal recognizing FSA. A potential training problem is the presence of local maxima in the likelihood function. For example, the number of converged trials for grammars 3, 4, and 5 is quite small and the difficulty of discovering the optimal solution might become a serious restriction for tasks involving a large number of states. In other experiments (Bengio & Frasconi, 1994a), we noticed that restricting the connectivity of the transition graph can significantly help to remove problems of convergence. Of course, this approach can be effectively exploited only if some prior knowledge about the state space is available. For example, applications of HMMs to speech recognition always rely on structured topologies. 6 CONCLUSIONS There are still a number of open questions. In particular, the effectiveness of the model on tasks involving large or very large state spaces needs to be carefully evaluated. In (Bengio & Frasconi 1994b) we show that learning long term dependencies in these models becomes more difficult as we increase the connectivity of the state 434 Yoshua Bengio, Paolo Frasconi transition graph. However, because transition probabilities of IOHMMs change at each t, they deal better with this problem of long-term dependencies than standard HMMs. Another interesting aspect to be investigated is the capability of the model to successfully perform tasks of sequence production or prediction. For example, interesting tasks that could also be approached are those related to time series modeling and motor control learning. References Angluin, D. and Smith, C. (1983). Inductive inference: Theory and methods. Computing Surveys, 15(3):237-269. Bengio, Y. and Frasconi, P. (1994a) . Credit assignment through time: Alternatives to backpropagation. In Cowan, J., Tesauro, G., and Alspector, J., editors, Advances in Neural Information Processing Systems 6. Morgan Kaufmann. Bengio, Y. and Frasconi, P. (1994b). An EM Approach to Learning Sequential Behavior. Tech. Rep. RT-DSI/11-94, University of Florence. Bengio, Y., De Mori, R., Flammia, G., and Kompe, R. (1992). Global optimization of a neural network-hidden markov model hybrid. IEEE Transactions on Neural Networks, 3(2):252-259. Bengio, Y., Simard, P., and Frasconi, P. (1994) . Learning long-term dependencies with gradient descent is difficult. IEEE Trans. Neural Networks, 5(2). Bourlard, H. and Wellekens, C. (1990). Links between hidden markov models and multilayer perceptrons. IEEE Trans. Pattern An. Mach. Intell., 12:1167-1178. Bridle, J. S. (1989). Training stochastic model recognition algorithms as networks can lead to maximum mutual information estimation of parameters. In D.S.Touretzky, ed., NIPS2, pages 211-217. Morgan Kaufmann. Cacciatore, T. W. and Nowlan, S. J. (1994). Mixtures of controllers for jump linear and non-linear plants. In Cowan, J. et. al., editors, Advances in Neural Information Processing Systems 6, San Mateo, CA. Morgan Kaufmann. Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum-likelihood from incomplete data via the EM algorithm. J. Royal Stat. Soc. B,39:1-38. Giles, C. L., Miller, C. B., Chen, D., Sun, G. Z., Chen, H. H., and Lee, Y. C. (1992). Learning and extracting finite state automata with second-order recurrent neural networks. Neural Computation, 4(3):393-405. Jacobs, R. A., Jordan, M. I., Nowlan, S. J., and Hinton, G. E. (1991). Adaptive mixture of local experts. Neural Computation, 3:79-87. Levinson, S. E., Rabiner, L. R., and Sondhi, M. M. (1983). An introduction to the application of the theory of probabilistic functIons of a markov process to automatic speech recognition. Bell System Technical Journal, 64(4):1035-1074. Mozer, M. C. (1992). The induction of multiscale temporal structure. In Moody, J. et. al., eds, NIPS 4 pages 275-282. Morgan Kaufmann. Pollack, J. B. (1991) . The induction of dynamical recognizers. Machine Learning, 7(2):196-227. Tomita, M. (1982). Dynamic construction of finite-state automata from examples using hill-climbing. Proc. 4th Cog. Science Con!, pp. 105-108, Ann Arbor MI. Watrous, R. 1. and Kuhn, G. M. (1992). Induction of finite-state languages using second-order recurrent networks. Neural Computation, 4(3):406-414.	1994	70
966	Grouping Components of Three-Dimensional Moving Objects Area MST of Visual Cortex Richard S. Zemel Carnegie Mellon University Department of Psychology Pittsburgh, PA 15213 zemel«lcmu. edu Terrence J. Sejnowski CNL, The Salk Institute P.O. Box 85800 San Diego, CA 92186-5800 terry«lsalk.edu Abstract Many cells in the dorsal part of the medial superior temporal (MST) area of visual cortex respond selectively to spiral flow patterns-specific combinations of expansion/ contraction and rotation motions. Previous investigators have suggested that these cells may represent self-motion. Spiral patterns can also be generated by the relative motion of the observer and a particular object. An MST cell may then account for some portion of the complex flow field, and the set of active cells could encode the entire flow; in this manner, MST effectively segments moving objects. Such a grouping operation is essential in interpreting scenes containing several independent moving objects and observer motion. We describe a model based on the hypothesis that the selective tuning of MST cells reflects the grouping of object components undergoing coherent motion. Inputs to the model were generated from sequences of ray-traced images that simulated realistic motion situations, combining observer motion, eye movements, and independent object motion. The input representation was modeled after response properties of neurons in area MT, which provides the primary input to area MST. After applying an unsupervised learning algorithm, the units became tuned to patterns signaling coherent motion. The results match many of the known properties of MST cells and are consistent with recent studies indicating that these cells process 3-D object motion information. • In 166 Richard S. Zemel, Terrence J. Sejnowski 1 INTRODUCTION A number of studies have described neurons in the dorsal part of the medial superior temporal (MSTd) monkey cortex that respond best to large expanding/contracting, rotating, or shifting patterns (Tanaka et al., 1986; Duffy & Wurtz, 1991a). Recently Graziano et al. (1994) found that MSTd cell responses correspond to a point in a multidimensional space of spiral motions, where the dimensions are these motion types. Combinations of these motions are generated as an animal moves through its environment, which suggests that area MSTd could playa role in optical flow analysis. When an observer moves through a static environment, a singularity in the flow field known as the focus of expansion may be used to determine the direction of heading (Gibson, 1950; Warren & Hannon, 1988). Previous computational models of MSTd (Lappe & Rauschecker, 1993; Perrone & Stone, 1994) have shown how navigational information related to heading may be encoded by these cells. These investigators propose that each MSTd cell represents a potential heading direction and responds to the aspects of the flow that are consistent with that direction. In natural environments, however, MSTd cells are often faced with complex flow patterns produced by the combination of observer motion with other independentlymoving objects. These complex flow fields are not a single spiral pattern, but instead are composed of multiple spiral patterns. This observation that spiral flows are local subpatterns in flow fields suggests that an MSTd cell represents a particular regular subpattern which corresponds to the aspects of the flow field arising from a single cause-the relative motion of the observer and some object or surface in the scene. Adopting this view implies a new goal for MST: the set of MST responses accounts for the flow field based on the ensemble of motion causes. An MST cell that responds to a local subpattern accounts for a portion of the flow field, specifically the portion that arises from a single motion cause. In this manner, MST can be considered to be segmenting motion signals. As in earlier models, the MSTd cell responds to the aspects of flow consistent with its motion hypothesis, but here a cell's motion hypothesis is not a heading direction but instead represents the 3-D motion of a scene element relative to the observer. This encoding may be useful not only in robustly estimating heading detection, but may also facilitate several other tasks thought to occur further down the motion processing stream, such as localizing objects and parsing scenes containing multiple moving objects. In this paper we describe a computational model based on the hypothesis that an MST cell signals those aspects of the flow that arise from a common underlying cause. We demonstrate how such a model can develop response properties from the statistics of natural flow images, such as could be extracted from MT signals, and show that this model is able to capture several known properties of information processing in MST. 2 THE MODEL The input to the system is a movie containing some combination of observer motion, eye movements, and a few objects undergoing independent 3-D motion. An optical Grouping Components of 3-D Moving Objects in Area MST of Visual Cortex 167 flow algorithm is then applied to yield local motion estimates; this flow field is the input to the network, which consists of three layers. The first layer is designed after monkey area MT. The connectivity between this layer and the second layer is based on MST receptive field properties, and the second layer has the same connectivity pattern to the output layer. The weights on all connections are determined by a training algorithm which attempts to force the network to recreate the input pattern on the output units. We discuss the inputs, the network, and the training algorithm in more detail below. 2.1 STIMULI The flow field input to the network is produced from a movie. The various movies are intended to simulate natural motion situations. Sample situations include one where all motion is due to the observer's movement, and the gaze is in the motion direction. Another situation that produces a qualitatively different flow field is when the gaze is not in the motion direction. A third situation includes independent motion of some of the objects in the environment. Each movie is a sequence of images that simulates one of these situations. The images are created using a flexible ray-tracing program, which allows the simulation of many different objects, backgrounds, observer/camera motions, and lighting effects. We currently employ a database of 6 objects (a block of swiss cheese, a snail, a panther, a fish, a ball, and a teapot) and three different backgrounds. A movie is generated by randomly selecting one to four of the objects, and a background. To simulate one of the motion situations, a random selection of motion parameters follows: a). The observer's motion along (x, z) describes her walking; b). The eyes can rotate in (x, y), simulating the tracking of an object during motion; c). Each object can undergo independent 3-D motion. A sequence of 15 images is produced by randomly selecting 3-D initial positions and then updating the pose of the camera and each object in the image based on these motion parameters. Figure 1 shows 3 images selected from a movie generated in this manner. We apply a standard optical flow technique to extract a single flow field from each synthetic image sequence. Nagel's (1987) flow algorithm is a gradient-based scheme which performs spatiotemporal smoothing on the set of images and then uses a multi-resolution second-order derivative technique, in combination with an oriented smoothness relaxation scheme, to produce the flow field. 2.2 MODEL INPUT AND ARCHITECTURE The network input is a population encoding of these optical flow vectors at each location in a 21x31 array by small sets of neurons that share the same receptive field position but are tuned to different directions of motion. The values for each input unit is computed by projecting the local flow vector onto the cell's preferred direction. We are currently using 4 inputs per location, with evenly spaced preferred directions and a tuning half-width of 45°. This popUlation encoding in the input layer is intended to model the response of cells in area MT to a motion sequence. The receptive field (RF) of each model MT unit is determined by the degree of spatial smoothing and subsampling in the flow	1994	71
967	Higher Order Statistical Decorrelation without Information Loss Gustavo Deco SiemensAG Central Research Otto-Hahn-Ring 6 81739 Munich GeIIDany Wilfried Brauer Technische UniversiUit MUnchen Institut fur InfoIIDatik Arcisstr. 21 Abstract 80290 Munich GeIIDany A neural network learning paradigm based on information theory is proposed as a way to perform in an unsupervised fashion, redundancy reduction among the elements of the output layer without loss of information from the sensory input. The model developed performs nonlinear decorrelation up to higher orders of the cumulant tensors and results in probabilistic ally independent components of the output layer. This means that we don't need to assume Gaussian distribution neither at the input nor at the output. The theory presented is related to the unsupervised-learning theory of Barlow, which proposes redundancy reduction as the goal of cognition. When nonlinear units are used nonlinear principal component analysis is obtained. In this case nonlinear manifolds can be reduced to minimum dimension manifolds. If such units are used the network performs a generalized principal component analysis in the sense that non-Gaussian distributions can be linearly decorrelated and higher orders of the correlation tensors are also taken into account. The basic structure of the architecture involves a general transfOlmation that is volume conserving and therefore the entropy, yielding a map without loss of infoIIDation. Minimization of the mutual infoIIDation among the output neurons eliminates the redundancy between the outputs and results in statistical decorrelation of the extracted features. This is known as factorialleaming. 248 Gustavo Deco, Wilfried Brauer 1 INTRODUCTION One of the most important theories of feature extraction is the one proposed by Barlow (1989). Barlow describes the process of cognition as a preprocessing of the sensorial information performed by the nervous system in order to extract the statistically relevant and independent features of the inputs without loosing information. This means that the brain should statistically decorrelate the extracted information. As a learning strategy Barlow (1989) formulated the principle of redundancy reduction. This kind of learning is called factorial learning. Recently Atick and Redlich (1992) and Redlich (1993) concentrate on the original idea of Barlow yielding a very interesting formulation of early visual processing and factorial learning. Redlich (1993) reduces redundancy at the input by using a network structure which is a reversible cellular automaton and therefore guarantees the conservation of information in the transformation between input and output. Some nonlinear extensions of PCA for decorrelation of sensorial input signals were recently introduced. These follow very closely Barlow's original ideas of unsupervised learning. Redlich (1993) use similar information theoretic concepts and reversible cellular automata architectures in order to define how nonlinear decorrelation can be performed. The aim of our work is to formulate a neural network architecture and a novel learning paradigm that performs Barlow's unsupervised learning in the most general fashion. The basic idea is to define an architecture that assures perfect transmission without loss of information. Consequently the nonlinear transformation defined by the neural architecture is always bijective. The architecture performs a volume-conserving transformation (determinant of the Jacobian matrix is equal one). As a particular case we can derive the reversible cellular automata architecture proposed by Redlich (1993). The learning paradigm is defined so that the components of the output signal are statistically decorrelated. Due to the fact that the output distribution is not necessarily Gaussian, even if the input is Gaussian, we perform a cumulant expansion of the output distribution and find the rules that should be satisfied by the higher order correlation tensors in order to be decorrelated. 2 THEORETICAL FORMALISM Let us consider an input vector x of dimensionality d with components distributed according to the probability distribution P (X) , which is not factorial, i.e. the components of x are correlated. The goal of Barlow's unsupervised learning rule is to find a transformation ,. ->,. y = F (x) (2.1) such that the components of the output vector d -dimensional yare statistically decorrelated. This means that the probability distributions of the components Y j are independent and therefore, d pO) = IT P (y) . (2.2) j The objl?,.ctive of factorial learning is to find a neural network, which performs the transformation F ( ) such that the joint probability distribution P 0) of the output signals is factorized as in eq. (2.2). In order to implement factorial learning, the information contained in the input should be transferred to the output neurons without loss but, the probability distribution of the output neurons should be statistically decorrelated. Let us now define Higher Order Statistical Decorrelation without /n/onl1atioll Loss 249 these facts from the information theory perspective. The first aspect is to assure the entropy is conselVed, i.e. H (x) = H (y) (2.3) where the symbol !! (a) denotes the entropy of a and H (a/b) the conditional entropy of a given b. One way to achieve this goal is to construct an architecture that independently of its synaptic parameters satisfies always eq. (2.3). Thus the architecture will conselVe information or entropy. The transmitted entropy satisfies H (J) ~ H (x) + J P (x) In (det (~r» dx (2.4) -" where equality holds only if F is bijective, i.e. reversible. ConselVation of information and bijectivity is assured if the neural transformation conselVes the volume, which mathematically can be expressed by the fact that the Jacobian of the transformation should have determinant unity. In section 3 we fonnulate an architecture that always conselVes the entropy. Let us now concentrate on the main aspect of factorial learning, namely the decorrelation of the output components. Here the problem is to find a volume-conselVing transformation that satisfies eq. (2.2). The major problem is that the distribution of the output signal will not necessarily be Gaussian. Therefore it is impossible to use the technique of minimizing the mutual information between the components of the output as done by Redlich (1993). The only way to decorrelate non-Gaussian distributions is to expand the distribution in higher orders of the correlation matrix and impose the independence condition of eq. (2.2). In order to achieve this we propose to use a cumulant expansion of the output distribution. Let us define the Fourier transform of the output distribution, ~(K) = JdY ei(K·h P(J;~(Ki) = fdYi ei(Ki·Yi) P(y) (2.5) The cumulant expansion of a distribution is (Papoulis, 1991) (2.6) In the Fourier space the independence condition is given by (papoulis, 1991) (2.7) which is equivalent to ...>. In(~(K» = Incrl~(Ki» = Eln(~(Ki» (2.8) I I Putting eq. (2.8) and the cumulant expansions of eq. (2.6) together, we obtain that in the case of independence the following equality is satisfied (2.9) In both expansions we will only consider the first four cumulants. Mter an extra transformation ~ .. -,.y' .. y - (y) (2.10) 250 Gustavo Deco, Wilfried Brauer to remove the bias <y). we can rewrite eq. (2.9) using the cumulants expression derived in the Papoulius (1991): 1 (2) i (3) -21: KjKj {Cjj - Cj 5jj} -"6 E KiKjKd Cijl.: - Cj 5jjk} '.J '.J.k (2.11) 1 (4) (2) 2 + 24 E KjKjKkKd (Cjjkl-3CijCkl) - (Cj -3 (C; » 5jjkl} = 0 i,j, t, I Equation (2.11) should be satisfied for all values of K. The multidimensional correlation tensors C j . .. j and the one-dimensional higher order moments C?) are given by ~j ... j = Jdy' P(y') y'j ... y'j ; C?) = Jdy'j P(y'j) (y'i) (2.12) The 5. . denotes Kroenecker's delta. Due to the fact that eq. (2.11) should be satisfied for all K,'iiU coefficients in each summation of eq. (2.11) must be zero. This means that Cjj = 0, if(i oF j) (2.13) Cjjk 0, if(i oF j v i oF k) (2.l4) Cjjkl = 0, if( {ioFjvioFkvioFl} I\-,L) (2.15) CUjj - CijCjj = 0, if(i oF j) . (2.16) In eq. (2.15) L is the logical expression L {(i .. j 1\ k .. 11\ j oF k) v (i = k 1\ j = I 1\ i oF j) v (i - 11\ j .. k 1\ i oF j) }, (2.17) which excludes the cases considered in eq. (2.16). The conditions of independence given by eqs. (2. 13-2.l6) can be achieved by minimization of the cost function E - aE Cfj+ 13 E Cfjk+Y , Ctkl+ 5E (C;;jj - CjjCjj) 2 (2.18) ;<j i<j~k i<j~';'1 i<j where a, 13, Y, 5 are the inverse of the number of elements in each summation respectively. In conclusion, minimizing the cost given by eq. (2.18) with a volume-conserving network, we achieve nonlinear decorrelation of non-Gaussian distributions. It is very easy to test wether a factorized probability distribution (eq. 2.2) satisfies the eqs. (2.13-2.16). As a particular case if only second order terms are used in the cumulant expansion, the learning rule reduces to eq. (2.13), which expresses nothing more than the diagonalization of the second order covariance matrix. In this case, by anti-transforming the cumulant expansion of the Fourier transform of the distribution,we obtain a Gaussian distribution. Diagonalization of the covariance matrix decorrelates statistically the components of the output only if we assume a Gaussian distribution of the outputs. In general the distribution of the output is not Gaussian and therefore higher orders of the cumulant expansion should be taken into account. yielding the learning rule conditions eqs. (2.13-2.l6) (up to fourth order, generalization to higher orders is straightforward). In the case of Gaussian distribution, minimization of the sum of the variances at each output leads to statistically decorrelation. This fact has a nice information theoretic background namely the minimization of the mutual information between the output components. Statistical independence as expressed in eq. (2.2) is equivalent to (Atick and Redlich. 1992) MH = EH(Yj) -H(~) = 0 (2.19) J Higher Order Statistical Decorrelatioll without Illformation Loss 251 This means that in order to minimize the redundancy at the output we minimize the mutual information between the different components of the output vector. Due to the fact that the volume-conserving structure of the neural network conserves the entropy, the minimization of MH reduces to the minimization of E H (Yj) . J 3 VOLUME-CONSERVING ARCHITECTURE AND LEARNING RULE In this section we define a neural network architecture that is volume-conserving and therefore can be used for the implementation of the learning rules described in the last section. Figure l.a shows the basic architecture of one layer. The dimensionality of input and output layer is the same and equal to d. A similar architecture was proposed by Redlich (1993b) using the theory of reversible cellular automata. ~ x (a) ~ x ~ Y (b) Figure 1: Volume-conserving Neural Architectures. ~ x (c) The analytical definition of the transformation defined by this architecture can be written as, )'i = xi +1, (xo' ... , Xj' Wi)' with j < i (3.1) where Wi represents a set of parameters of the function h' Note that independently of the functions fi the network is always volume-conserving. In particular h can be calculated by another neural network, by a sigmoid neuron, by polynomials (higher order neurons), etc. Due to the asymmetric dependence on the input variables and the direct connections with weights equal to 1 between corresponding components of input and output neurons, the Jacobian matrix of the transformation defined in eq. (3.1) is an upper triangular matrix with diagonal elements all equal to one, yielding a determinant equal to one. A network with inverted asymmetry also can be defined as )'i = Xi + gi (xj' ... , Xd' 8;>, with i <j (3.2) corresponding to a lower triangular Jacobian matrix 'fith diagonal elements all equal to one, being therefore volume-conserving. The vectors 8 j represent the parameters of functions gj. In order to yield a general nonlinear transformation from inputs to outputs (without asymmetric dependences) it is possible to build a multilayer architecture like the one shown in Fig. 1.b, which involves mixed versions of networks described by eq. (3.1) and eq. (3.2), respectively. Due to the fact that successive application of volume-conserving transformation is also volume-conserving, the multilayer architecture is also volume-con252 Gustavo Deco, Wilfried Brauer serving. In the two-layer case (Fig. I.c) the second layer can be interpreted as asymmetric lateral connections between the neurons of the first layer. However, in our case the feedfOlWard connections between input layer and first layer are also asymmetric. As demonstrated in the last section, we minimize a cost function E to decorrelate nonlinearly correlated non-Gaussian inputs. Let us analyze for simplicity a two-layer architecture (Fig. l.c) with the first layer given by eq. (3.l) and the second layer by eq. (3.2). Let us denote the output of the hidden layer by h and use it as input of eq. (3.2) with output y. The extension to multilayer architectures is straightfolWard. The learning rule can be easily expressed by gradient descent method: .>. .>. aE e.=e.-T\I I ..>. ae; In order to calculate the derivative of the cost functions we need (3.3) C i ... j 1 a _ _ _ a _. ay i 1 a == N-~ {",,\"::"(y.-y .) ... (y .-y.) + (y.-y.) ... ""\"::"(y .-y, -a = -~ {,\::"(y .)}(3.4) o i..J 00 I I ] ] I I 00 ] J 0 N i..J 00 I P p where e represents the parameters e j and Wi. The sums in both equations extend over the N training patterns. The gradients of the different outputs are a _ a .j _ a a a ---;:-Yi - - .. -g; ,~Yt (ahgt) (a ..... !;)&i>k+ (a ..... !;) (3.5) ae; ae; [OJ I W; Wi where &i> k is equal to I if i > k and 0 othelWise. In this paper we choose a polynomial fonn for the functions I and g. This model involves higher order neurons. In this case each function Ii or g; is a product of polynomial functions of the inputs. The update equations are given by (3.6) where R is the order of the polynomial used. In this case the two-layer architecture is a higher order network with a general volume-conserving structure. The derivatives involved in the learning rule are given by a _ -a-Yk 0) • . ljr (3.7) Higher Order Statistical Decorrelation without Information Loss 253 4 RESULTS AND SIMULATIONS We will present herein two different experiments using the architecture defined in this paper. The input space in all experiments is two-dimensional in order to show graphically the results and effects of the presented model. The experiments aim at learning noisy nonlinear polynomial and rational curves. Figure 2.a and 2.b plot the input and output space of the second experiment after training is finished, respectively. In this case the noisy logistic map was used to generate the input: (4.1) where'\) introduces I % Gaussian noise. In this case a one-layer polynomial network with R - 2 was used. The learning constant was 11 - 0.01 and 20000 iterations of training were performed. The result of Fig. 2.b is remarkable. The volume-conserving network decorrelated the output space extracting the strong nonlinear correlation that generated the curve in the input space. This means that after training only one coordinate is important to describe the curve. (a) (b) •• o • •• os •• 04 02 ' ·1'=" .2 ---.,....---:.:'-:-.---:.:':-.---,.,':-.-----::':: .. -----7----,,' .. .g7.-~-~.:'-:-2---,O~.--0~.-~ •• ~~-~ Figure 2: Input and Output space distribution after training with a one-layer polynomial volume-conseIVing network of order for the logistic map. (a) input space; (b) output space. The whole information was compressed into the first coordinate of the output. This is the generalization of data compression normally performed by using linear peA (also called Karhunen-Loewe transformation). The next experiment is similar, but in this case a twolayer network of order R .. 4 was used. The input space is given by the rational function 3 Xl (4.2) x2 = O.2xI + 2 + '\l (1 +xI ) where Xl and '\) are as in the last case. The results are shown in Fig. 4.a (input space) and Fig. 4.b (output space). Fig. 4.c shows the evolution of the four summands of eq (2.18) during learning. It is important to remark that at the beginning the tensors of second and third order are equally important. During learning all summands are simultaneously minimized, resulting in a statistically decorrelated output. The training was performed during 20000 iterations and the learning constant was 11 = 0.005 . 254 Gustavo Deco, Wilfried Brauer (a) (b) (c) " " etrIlIt' II. ~st2 " .. " .. " .II • .l •• /' I. I ~. _,,, ...... H"'t . , ....... 1· .. -"'-' ... .. I. t. ,'" .. I ... .. I .. ... t.. cost4b '1. ,'I. .. .. " " " " .~ .. II. M .. Figure 4: Input and Output space distribution after training with a two-layer polynomial volume-conselVing network of order for the noisy CUIVe of~. (4.2). (a) input space; (b) output space (c) Development of the four summands of the cost function (~ 2.18) during learning: (cost2) fiist summand (second order COIrelation tensor); (cost 3) second summand (tliird correlation order tensor); (cost 4a) third summand (fourth order correlation tensor); (cost4b) fourth summand (fourth order correlation tensor). 5 CONCLUSIONS We proposed a unsupervised neural paradigm, which is based on Infonnation Theory. The algorithm perfonns redundancy reduction among the elements of the output layer without loosing infonnation, as the data is sent through the network. The model developed perfonns a generalization of Barlow's unsupervised learning, which consists in nonlinear decorrelation up to higher orders of the cumulant tensors. After training the components of the output layer are statistically independent. Due to the use of higher order cumulant expansion arbitrary non-Gaussian distributions can be rigorously handled. When nonlinear units are used nonlinear principal component analysis is obtained. In this case nonlinear manifolds can be reduced to a minimum dimension manifolds. When linear units are used, the network performs a generalized principal component analysis in the sense that non-Gaussian distribution can be linearly decorrelated.This paper generalizes previous works on factorial learning in two ways: the architecture performs a general nonlinear transformation without loss of information and the decorrelation is perfonned without assuming Gaussian distributions. References: H. Barlow. (1989) Unsupervised Learning. Neural Computation, 1,295-311. A. Papoulis. (1991) Probability, Random Variables, and Stochastic Processes. 3. Edition, McGraw-Hill, New York. A. N. Redlich. (1993) Supervised Factorial Learning. Neural Computation, 5, 750-766.	1994	72
968	Sample Size Requirements For Feedforward Neural Networks Michael J. Turmon Cornell U niv. Electrical Engineering Ithaca, NY 14853 mjt@ee.comell.edu Terrence L. Fine Cornell Univ. Electrical Engineering Ithaca, NY 14853 tlfine@ee.comell.edu Abstract We estimate the number of training samples required to ensure that the performance of a neural network on its training data matches that obtained when fresh data is applied to the network. Existing estimates are higher by orders of magnitude than practice indicates. This work seeks to narrow the gap between theory and practice by transforming the problem into determining the distribution of the supremum of a random field in the space of weight vectors, which in turn is attacked by application of a recent technique called the Poisson clumping heuristic. 1 INTRODUCTION AND KNOWN RESULTS We investigate the tradeofi"s among network complexity, training set size, and statistical performance of feedforward neural networks so as to allow a reasoned choice of network architecture in the face of limited training data. Nets are functions 7](x; w), parameterized by their weight vector w E W ~ Rd , which take as input points x E Rk. For classifiers, network output is restricted to {a, 1} while for forecasting it may be any real number. The architecture of all nets under consideration is N, whose complexity may be gauged by its Vapnik-Chervonenkis (VC) dimension v, the size of the largest set of inputs the architecture can classify in any desired way ('shatter'). Nets 7] EN are chosen on the basis of a training set T = {(Xi, YiHr=l. These n samples are i.i.d. according to an unknown probability law P. Performance of a network is measured by the mean-squared error E(w) E(7](x; w) - y)2 (1) = P(7](x;w);/; y) (for classifiers) (2) 328 Michael Turman, Terrence L. Fine and a good (perhaps not unique) net in the architecture is WO = argmiIlwew £(w). To select a net using the training set we employ the empirical error 1 n VT(W) = - I)11(Xi; w) - Yi)2 (3) n i=l sustained by 11(·; w) on the training set T. A good choice for a classifier is then w· = argmiIlwew VT(W). In these terms, the issue raised in the first sentence ofthe section can be restated as, "How large must n be in order to ensure £(w·)-£(WO) $ i with high probability?" For purposes of analysis we can avoid dealing directly with the stochastically chosen network w· by noting £(w·) - £(WO) $ IVT(W·) - £(w·)1 + IVT(WO) - £(wo)1 $ 2 sup IVT(W) - £(w)1 wEW A bound on the last quantity is also useful in its own right. The best-known result is in (Vapnik, 1982), introduced to the neural network community by (Baum & Haussler, 1989): (2n)V ~ P( sup IVT(W) - £(w)1 ~ i) $ 6-,-e-n ( /2 (4) wEW v. This remarkable bound not only involves no unknown constant factors, but holds independent of the data distribution P . Analysis shows that sample sizes of about nc = (4V/i2) log 3/i (5) are enough to force the bound below unity, after which it drops exponentially to zero. Taking i = .1, v = 50 yields nc = 68000, which disagrees by orders of magnitude with the experience of practitioners who train such simple networks. More recently, Talagrand (1994) has obtained the bound ( K2ni2)v ~ P( sup IVT(W) - £(w)1 ~ i) $ Kl e- 2n ( , wew V (6) yielding a sufficient condition of order V/i2, but the values of Kl and K2 are inaccessible so the result is of no practical use. Formulations with finer resolution near £(w) = 0 are used. Vapnik (1982) bounds P(suPwew IVT(W) - £(w)I/£(w)1/2 ~ i)-note £(w)1/2 ~ Var(vT(w»1/2 when £(w) ~ O-while Blumer et al. (1989) and Anthony and Biggs (1992) work with P(suPWEW IVT(W) - £(w)ll{o}(VT(W» ~ i). The latter obtain the sufficient condition nc = (5.8v/i) log 12/i (7) for nets, if any, having VT( w) = o. If one is guaranteed to do reasonably well on the training set, a smaller order of dependence results. Results (Turmon & Fine, 1993) for perceptrons and P a Gaussian mixture imply that at least v/280i2 samples are needed to force £(w·) - £(WO) < 2i with high probability. (Here w· is the best linear discriminant with weights estimated from the data.) Combining with Talagrand's result, we see that the general (not assuming small VT(W» functional dependence is V/i2. Sample Size Requirements for Feedforward Neural Networks 329 2 APPLYING THE POISSON CLUMPING HEURISTIC We adopt a new approach to the problem. For the moderately large values of n we anticipate, the central limit theorem informs us that Vn[lIT(W) - E(w)] has nearly the distribution of a zero-mean Gaussian random variable. It is therefore reasonablel to suppose that P( sup IlIT(W) - E(w)1 ~ f) ~ P( sup IZ(w)1 ~ fJ1i) ~ 2P( sup Z(w) ~ fVn) wEW wEW wEW where Z( w) is a Gaussian process with mean zero and covariance R(w, v) = EZ(w)Z(v) = Cov(y -1J(x; w»2, (y -1J(x; V»2) The problem about extrema of the original empirical process is equivalent to one about extrema of a corresponding Gaussian process. The Poisson clumping heuristic (PCR), introduced in the remarkable (Aldous, 1989), provides a general tool for estimating such exceedance probabilities. Consider the excursions above level b(= fVn ~ 1) by a stochastic process Z(w). At left below, the set {w : Z( w) ~ b} is seen as a group of "clumps" scattered in weight space W. The PCR says that, provided Z has no long-range dependence and the level b is large, the centers of the clumps fall according to the points of a Poisson process on W, and the clump shapes are independent. The vertical arrows (below right) illustrate two clump centers (points of the Poisson process); the clumps are the bars centered about the arrows. w w In fact, with PheW) = P(Z(w) ~ b), Ch(W) the size of a clump located at w, and Ah (w) the rate of occurrence of clump centers, the fundamental equation is (8) The number of clumps in W is a Poisson random variable Nh with parameter 1, Ah( w) dw. The probability of a clump is P(Nb > 0) = 1-exp( - fwAh( w) dW) =::: fw Ah(W) dw where the approximation holds because our goal is to operate in a regime where this probability is near zero. Letting ~(b) = P(N(0, 1) > b) and (T2(w) = R(w, w), we have PheW) = ~(b/(T(w». The fundamental equation becomes P( sup Z(w) ~ b) ~ r ~(b/(T(w» dw (9) wEW Jw ECh(W) It remains only to find the mean clump size ECh( w) in terms of the network architecture and the statistics of (x, y). lSee ch. 7 of (Pollard, 1984) for treatment of some technical details in this limit. 330 Michael Tunnon, Terrence L. Fine 3 POISSON CLUMPING FOR SMOOTH PROCESSES Assume Z(w) has two mean-square derivatives in w. (If the network activation functions have two derivatives in w, for example, Z( w) will have two almost sure derivatives.) Z then has a parabolic approximation about some Wo via its gradient G = 'VZ(w) and Hessian matrix H = 'V'VZ(w) at woo Provided Zo ~ b, that is that there is a clump at Wo, simple computations reveal (2(Zo - b) - cP'H- 1G)d/2 (10) Cb( wo) ~ Kd IHI I / 2 where Kd is the volume of the unit ball in Rd and I· 1 is the determinant. The mean clump size is the expectation of this conditioned on Z(wo) ~ b. The same argument used to show that Z(w) is approximately normal shows that G and H are approximately normal too. In fact, z E[HIZ(wo) = z] 2( )A(wo) (F Wo A(wo) -EZ(wo)H = -'Vw 'VwR(wo, w)lw=wo so that, since b (and hence z) is large, the second term in the numerator of (10) may be neglected. The expectation is then easily computed, resulting in Lemma 1 (Smooth process clump size) Let the network activation functions be twice continuously differentiable, and let b » (F( w). Then ECb(W) ~ (21r)d/21 ~~~) 1-112 ((F(:») d Substituting into (9) yields P( sup Z(w) ~ b) ~ (21r)-~ ( 1 A(w) 11/2 (_b_) d-~_b~/2q~(W) dw, (11) wEW iw (F2(w) (F(w) where use of the asymptotic expansion ~(z) ~ (zv'21r)-l exp( _Z2 /2) is justified since ('v'w)b » (F( w) is necessary to have the individual P( Z( w) ~ b) low-let alone the supremum. To go farther, we need information about the variance (F2 (w) of (y - 11( x; w»2. In general this must come from the problem at hand, but suppose for example the process has a unique variance maximum 0'2 at w. Then, since the level b is large, we can use Laplace's method to approximate the d-dimensional integral. Laplace's method finds asymptotic expansions for integrals fw g(w) exp( - f(w)2 /2) dw when few) is C2 with a unique positive minimum at Wo in the interior of W ~ Rd , and g( w) is positive and continuous. Suppose I( wo) » 1 so that the exponential factor is decreasing much faster than the slowly varying g. Expanding f to second order about Wo, substituting into the exponential, and performing the integral shows that iw g( w) exp( - f(w)2 /2) dw ~ (21r)d/2If( wo)KI- 1/ 2g( wo) exp( - f( wo)2 /2) Sample Size Requirements for Feedforward Neural Networks 331 where K = V'V'f(w)lwo, the Hessian of f. See (Wong, 1989) for a proof. Applying this to (11) and using the asymptotic expansion for ~ in reverse yields Theorem 1 Let the network activation functions be twice continuously differentiable. Let the variance have a unique maximum u at w in the interior of Wand the level b ~ u. Then the peH estimate of exceedance probability is given by IA(w)1 1/ 2 _ P(:~fv Z(w) ~ b) ~ IA(w) _ r(w)1 1/ 2 ~(b/u) (12) where r(w) = V'wV'tlR(w,v)lw=tI=w. Furthermore, A- r is positive-definite at w; it is -1/2 the Hessian of cr2(w). The leading constant thus strictly e:cceeds unity. The above probability is just P(Z(w) ~ b) multiplied by a factor accounting for the other networks in the supremum. Letting b = f...;n reveals u2 10g(IA(w)I/IA(w) - r(w)!) nc = 2 f. (13) samples force P(supw IlIT(W) - &(w)\ ~ {) below unity. If the variance maximum is not unique but occurs over a d-dimensional set within W, the sample size estimate becomes proportional to u2d/{2. With d playing the role of VC dimension v, this is similar to Vapnik's bound although we retain dependence on P and N. The above probability is determined by behavior near the maximum-variance point, which for example in classification is where &(w) = 1/2. Such nets are uninteresting as classifiers, and certainly it is undesirable for them to dominate the entire probability. This problem is avoided by replacing Z(w) with Z(w)/cr(w), which additionally allows a finer resolution where &(w) nears zero. Indeed, for classification, if n is such that with high probability IlIT(W) - &(w)1 IlIT(W) - &(w)1 (14) sup = sup < { , weW cr(w) wew J&(W)(I- &(w» then lIT(W·) = 0 ::} &(w·) < {2(1 + (2)-1 ~ {2 <t:: {. Near lIT(W·) = 0, condition (14)/ is much more powerful than the corresponding unnormalized one. Sample size estimates using this setup give results having a functional form similar to (7). 4 ANOTHER MEANS OF COMPUTING CLUMP SIZE Conditional on there being a clump center at w, the upper bound Cb(W) ~ Db(W) == iw l[o,oo)(Z(w') - b) dw' (15) is evidently valid: the volume of the clump at w is no larger than the total volume of all clumps. (The right hand side is indeed a function of w because we condition on occurrence of a clump center at w.) The bound is an overestimate when the number Nb of clumps exceeds one, but recall that we are in a regime where b (equivalently n) is large enough so that P( Nb > 1)/ P( Nb = 1) ~ fw ).b (w) dw <t:: 1. Thus error in (15) due to this source is negligible. To compute its mean, we approximate EDb(W) = iw P(Z(~f) ~ blw a clump center)dw' 332 Michael Turmon, Terrence L. Fine (16) The point is that occurrence of a clump center at Wo is a smaller class of events than merely Z( wo) ~ b: the latter can arise from a clump center at a nearby w E W capturing woo Since Z(w) and Z(w') are jointly normal, abbreviate u = u(w), u' = u(w'), p = p(w,w') = R(w,w')/(uu'), and let (=(w,w') (UIU,)I;r;!; (17) 1- p2 (1- p)/(1 + p»)1/2 (constant variance case) (18) Evaluating the conditional probabilities of (16) presents no problem, and we obtain Lemma 2 (Clump size estimate) For b ~ u the mean clump size is ECb(W) ~ EDb(W) ~ /w cf>«blu)() dw' (19) Remark 1. This integral will be used in (9) to find ( cf>(blu) P(s~pZ(w) > b) ~ Jw fw~«blu)() dw,dw (20) Since b is large, the main contribution to the outer integral occurs for w near a variance maximum, i.e. for u' I u ~ 1. If the variance is constant then all w E W contribute. In either case ( is nonnegative. By lemma 1 we expect (19) to be, as a function of b, of the form (const ulb)P for, say, p = d. In particular, we do not anticipate the exponentially small clump sizes resulting if (Vw')( w, w') ~ M ~ O. Therefore ( should approach zero over some range of w', which happens only when p ~ 1, that is, for w' near w. The behavior of pew, w') for w' ~ w is the key to finding the clump size. Remark 2. There is a simple interpretation of the clump size; it represents the volume of w' E W for which Z(w') is highly correlated with Z(w) . The exceedance probability is a sum of the point exceedance probabilities (the numerator of (20», each weighted according to how many other points are correlated with it. In effect, the space W is partitioned into regions that tend to "have exceedances together," with a large clump size ECb( w) indicating a large region. The overall probability can be viewed as a sum over all these regions of the corresponding point exceedance probability. This has a similarity to the Vapnik argument which lumps networks together according to their nV Iv! possible actions on n items in the training set. In this sense the mean clump size is a fundamental quantity expressing the ability of an architecture to generalize. 5 EMPIRICAL ESTIMATES OF CLUMP SIZE The clump size estimate of lemma 2 is useful in its own right if one has information about the covariance of Z. Other known techniques of finding ECb( w) exploit special features of the process at hand (e.g. smoothness or similarity to other wellstudied processes); the above expression is valid for any covariance structure. In Sample Size Requirements for Feedf01ward Neural Networks 333 this section we show how one may estimate the clump size using the training set, and thus obtain probability approximations in the absence of analytical information about the unknown P and the potentially complex network architecture N. Here is a practical way to approximate the integral giving EDb{W). For'Y < 1 define a set of significant w' S-y{W) = {w' E W: (w,w') $ 'Y} V-y{W) = vol{S-y(w)) (21) then monotonicity of ~ yields EDb{W) ~ Is ~((b/(1X) dw' ~ V-y(W) ~((b/uh) . .., This apparently crude lower bound for ~ is accurate enough near the origin to give satisfactory results in the cases we have studied. For example, we can characterize the covariance R( w, w') of the smooth process oflemma 1 and thus find its ( function. The bound above is then easily calculated and differs by only small constant factors from the clump size in the lemma. The lower bound for EDb(W) yields the upper bound 1 ~(b/(1-) P(s~p Z(w) ~ b) $ w V-y(w) ~«b/uh) dw (22) We call V-y(w) the correlation volume, as it represents those weight vectors w' whose errors Z(w') are highly correlated with Z(w); one simple way to estimate the correlation volume is as follows. Select a weight w' and using the training set compute (Yl - 17( Xl; w))2, ... , (Yn - 17( Xn; w)? & (Yl - 17( Xl; w'))2 , ... , (Yn - 17( Xn; w'))2 . It is then easy to estimate u2, u,2, and p, and finally (w ,w'), which is compared to the chosen 'Y to decide if w' E S-y ( w) . The difficulty is that for large d, S-y (w) is far smaller than any approximatelyenclosing set. Simple Monte Carlo sampling and even importance sampling methods fail to estimate the volume of such high-dimensional convex bodies because so few hits occur in probing the space (Lovasz, 1991). The simplest way to concentrate the search is to let w' = w except in one coordinate and probe along each coordinate axis. The correlation volume is approximated as the product of the one-dimensional measurements. Simulation studies of the above approach have been performed for a perceptron architecture in input uniform over [-1, l]d. The integral (22) is computed by Monte Carlo sampling, and based ona training set of size lOOd, V-y (w) is computed at each point via the above method. The result is that an estimated sample size of 5.4d/f2 is enough to ensure (14) with high probability. For nets, if any, having VT(W) = 0, sample sizes larger than 5.4d/f will ensure reliable generalization, which compares favorably with (7). 6 SUMMARY AND CONCLUSIONS To find realistic estimates of sample size we transform the original problem into one of finding the distribution of the supremum of a derived Gaussian random field, which is defined over the weight space of the network architecture. The latter problem is amenable to solution via the Poisson clumping heuristic. In terms of the PCH the question becomes one of estimating the mean clump size, that 334 Michael Turman, Terrence L. Fine is, the typical volume of an excursion above a given level by the random field. In the "smooth" case we directly find the clump volume and obtain estimates of sample size that are (correctly) of order v/€2. The leading constant, while explicit, depends on properties of the architecture and the data-which has the advantage of being tailored to the given problem but the potential disadvantage of our having to compute them. We also obtain a useful estimate for the clump size of a general process in terms of the correlation volume V-y(w). For normalized error, (22) becomes approximately p (sup lIr(w) - £(w) > €) ~ E [vol(W)] e-(1--y2)nf2/2 weW u(w) V-y(w) where the expectation is taken with respect to a uniform distribution on W. The probability of reliable generalization is roughly given by an exponentially decreasing factor (the exceedance probability for a single point) times a number representing degrees of freedom. The latter is the mean size of an equivalence class of "similarlyacting" networks. The parallel with the Vapnik approach, in which a worst-case exceedance probability is multiplied by a growth function bounding the number of classes of networks in N that can act differently on n pieces of data, is striking. In this fashion the correlation volume is an analog of the VC dimension, but one that depends on the interaction of the data and the architecture. Lastly, we have proposed practical methods of estimating the correlation volume empirically from the training data. Initial simulation studies based on a perceptron with input uniform on a region in Rd show that these approximations can indeed yield informative estimates of sample complexity. References Aldous, D. 1989. Probability Approximations via the Poisson Clumping Heuristic. Springer. Anthony, M., & Biggs, N. 1992. Computational Learning Theory. Cambridge Univ. Baum, E., & Haussler, D. 1989. What size net gives valid generalization? Pages 81-90 of' Touretzky, D. S. (ed), NIPS 1. Blumer, A., Ehrenfeucht, A., Haussler, D., & Warmuth, M. K. 1989. Learnability and the Vapnik-Chervonenkis dimension. Jour. Assoc. Compo Mach., 36,929-965. LovMz, L. 1991. Geometric Algorithms and Algorithmic Geometry. In: Proc. Internat. Congr. Mathematicians. The Math. Soc. of Japan. Pollard, D. 1984. Convergence of Stochastic Processes. Springer. Talagrand, M. 1994. Sharper bounds for Gaussian and empirical processes. Ann. Probab., 22, 28-76. Turmon, M. J., & Fine, T. L. 1993. Sample Size Requirements of Feedforward Neural Network Classifiers. In: IEEE 1993 Intern. Sympos. Inform. Theory. Vapnik, V. 1982. Estimation of Dependences Based on Empirical Data. Springer. Wong, R. 1989. Asymptotic Approximations of Integrals. Academic.	1994	73
969	Generalisation in Feedforward Networks Adam Kowalczyk and Herman Ferra Telecom Australia, Research Laboratories 770 Blackburn Road, Clayton, Vic. 3168, Australia (a.kowalczyk@trl.oz.au, h.ferra@trl.oz.au) Abstract We discuss a model of consistent learning with an additional restriction on the probability distribution of training samples, the target concept and hypothesis class. We show that the model provides a significant improvement on the upper bounds of sample complexity, i.e. the minimal number of random training samples allowing a selection of the hypothesis with a predefined accuracy and confidence. Further, we show that the model has the potential for providing a finite sample complexity even in the case of infinite VC-dimension as well as for a sample complexity below VC-dimension. This is achieved by linking sample complexity to an "average" number of implement able dichotomies of a training sample rather than the maximal size of a shattered sample, i.e. VC-dimension. 1 Introduction A number offundamental results in computational learning theory [1, 2, 11] links the generalisation error achievable by a set of hypotheses with its Vapnik-Chervonenkis dimension (VC-dimension, for short) which is a sort of capacity measure. They provide in particular some theoretical bounds on the sample complexity, i.e. a minimal number of training samples assuring the desired accuracy with the desired confidence. However there are a few obvious deficiencies in these results: (i) the sample complexity bounds are unrealistically high (c.f. Section 4.), and (ii) for some networks they do not hold at all since VC-dimension is infinite, e.g. some radial basis networks [7]. 216 Adam Kowalczyk, Hemzan Ferra One may expect that there are at least three main reasons for this state of affairs: (a) that the VC-dimension is too crude a measure of capacity, (b) since the bounds are universal they may be forced too high by some malicious distributions, (c) that particular estimates themselves are too crude, and so might be improved with time. In this paper we will attack the problem along the lines of (a) and (b) since this is most promising. Indeed, even a rough analysis of some proofs of lower bound (e.g. [I)) shows that some of these estimates were determined by clever constructions of discrete, malicious distributions on sets of "shattered samples" (of the size of VC-dimension). Thus this does not necessarily imply that such bounds on the sample complexity are really tight in more realistic cases, e.g. continuous distributions and "non-malicious" target concepts, the point eagerly made by critics of the formalism. The problem is to find such restrictions on target concepts and probability distributions which will produce a significant improvement. The current paper discusses such a proposition which significantly improves the upper bounds on sample complexity. 2 A Restricted Model of Consistent Learning First we introduce a few necessary concepts and some basic notation. We assume we are given a space of samples X with a probability measure J.L, a set H of binary functions X t-+ {O, I} called the hypothesis space and a target concept t E H. For an n-sample i = (Z1, ... , zn) E xn and h : H -+ {O, I} the vector (h(Z1), ... , h(zn)) E {O, l}n will be denoted by h(i). We define two projections 7r:e and 7rt,:e of H onto } n de! (_ de! de! {0,1 as follows 7r:e(h) = h z) = (h(Z1), ... , h(zn)) and 7rt,:e(h) = 7r:e(lt - hi) = It - hl(i) for every h E H . Below we shall use the notation 151 for the cardinality of the set 5 . The average density of the sets of projections 7r:e( H) or 7rt,:e( H) in {O, l}n is defined as PrH(i) d~ 17r:e(H)I/2n = l7rt,:e(H)I/2n (equivalently, this is the probability of a random vector in {O, l}n belonging to the set 7r:e(H)). Now we define two associated quantities: PrH'Il(n) d.:J J PrH(i)J.Ln(di) = 2-n J 17r:e(H) lJ.Ln (di), (1) de! We recall that dH = max{n ; 3:eEX .. I7r:e(H)1 = 2n} is called the VapnikChervonenkis dimension (VC-dimension) of H [1, 11]. If dH ~ 00 then Sauer's lemma implies the estimates (c.f. [1, 2, 10)) PrH'Il(n) ~ PrH,max(n) ~ 2- ncf?(dH, n) ~ 2-n (en/dH )dH , (2) where cf?(d,n) d;J 2::1=0 (~) (we assume C) d;J ° ifi > n). Now we are ready to formulate our main assumption in the model. We say that the space of hypotheses His (J.Ln, C)-uniform around t E 2x iffor every set 5 C {O, l}n J l7rt,:e(H) n 51J.Ln(dx) ~ CI5 IPrH'Il(n). (3) Generalisation in Feedforward Networks 217 The meaning of this condition is obvious: we postulate that on average the number of different projections 7rt,ar(h) of hypothesis h E H falling into S has a bound proportional to the probability PrH,~(n) of random vector in {O, l}n belonging to the set 7rt,ar(H). Another heuristic interpretation of (3) is as follows. Imagine that elements of 7rt,ar(H) are almost uniformly distributed in {O, 1}n, i.e. with average density Par ~ CI7rt,ar(H)1/2n. Thus the "mass" of the volume lSI is l7rt,ar(H) n SI ~ PariSI and so its average f l7rt,ar(H) n SIJ.tn(di) has the estimate ~ lSI f parJ.tn(di) ~ CISiPrH,~(n). Of special interest is the particular case of con8istent learning [1], i.e. when the target concept and the hypothesis fully agree on the training sample. In this case, for any E > 0 we introduce the notation QE(m) d;J {i E Xm ; 3hEH ert,ar(h) = 0 & ert,~(h) ~ E}, de! m de! f where ert,ar(h) = Ei=llt-hl(zi)/mand ert,~(h) = It-hl(z)J.t(dz) denote error rates on the training sample i = (ZlJ ... , zm) and X, respectively. Thus QE(m) is the set of all m-samples for which there exists a hypothesis in H with no error on the sample and the error at least E on X. Theorem 1 If the hypothesis space H is (J.t2m, C)-uniform around t E H then for anyE > 8/m (4) (5) Proof of the theorem is given in the Appendix. Given E,6 > O. The integer mL(6, E) d;J minim > 0 j J.tm(QE) ~ 6} will be called the 8ample complezity following the terminology of computational learning theory (c.f. [1]). Note that in our case the sample complexity depends also (implicitly) on the target concept t, the hypothesis space H and the probability measure J.t. Corollary 2 If the hypothesis space H is (J.tn, C)-uniform around t E H for any n> 0, then mL(6, E) < maz{8/E, minim j 2CPrH,~(2m)(3/2r < 6}} (6) C < maz{8/E, 6.9 dH + 2.4log2 6}' 0 (7) The estimate (7) of Corollary 2 reduces to the estimate mL(6, E) ~ 6.9 dH independent of 6 and E. This predicts that under the assumption of the corollary a transition to perfect generalisation occurs for training samples of size ~ 6.9 dH , which is in agreement with some statistical physics predictions showing such transition occurring below ~ 1.5dH for some simple neural networks (c.f. [5]). Proof outline. Estimate (6) follows from the first estimate (5). Estimate (7) can be derived from the second bound in (5) (virtually by repeating the proof of [1, Theorem 8.4.1] with substitution of 4log2(4/3) for E and 6/C for 6). Q.E.D. 2I8 Adam Kowalczyk, Herman Ferra 10-1 fie 10-2 :; . 10-3 102 103 104 10 10-300 '---'-..J....J. ........... L.U.L_.L-J'--'--<u..LJw......L...L!-O.!.-'-'-u.uJ 10 m m Figure 1: Plots of estimates on PrH,J.I(m) and f(m, £)/C ~ f(m,O)/C = (3/2) 3m PrH,J.I (2m) (Fig_ a) for the analytic threshold neuron (PrH'J.I(m) = <P(dH' 2m)(3/4)3~ and (Fig_ b) for the abstract perceptron according to the estimate (11) on PrH,J.I(m) for m1 = 50, m3 = 1000 and p = 0.015_ The upper bound on the sample complexity, mL(£, 6), corresponds to the abscissa value of the intersection of the curve f(m,O) with the level 6/C (c.f. point A in Fig. b). In this manner for 6/C = 0.01 we obtain estimates mL ~ 602 and mL < 1795 in the case of Fig. a for dH = 100 and dH = 300 and mL ~ 697 in the case of Fig. b (dH = ~ = 1000), respectively. 3 An application to feedforward networks In this section we shall discuss the problem of estimation of PrH'J.I(m) which is crucial for application of the above formalism. First we discuss an example of an analytic threshold neuron on R n [6] when H is the family of all functions R n -+ {O, 1}, x 1-+ 9(~:=1 Wiai(X)), where ai : R n -+ R are fixed real analytic functions and 9 the ordinary hard threshold. In this case dH equals the number of linearly independent functions among ab a3, ... , all. For any continuous probability distribution J.L on R n we have: PrH(i) = <p(dH, m)/2m (Vm and 'Vi E (Rnr with probability 1), (8) and consequently PrH'J.I(m) = PrH,max(m) = <P(dH' m)/2m for every m. (9) Note that this class of neural networks includes as particular cases, the linear threshold neuron (if Ie = n+ 1 and a1, ... , a n +1 are chosen as 1, Xl, ... , xn) and higher order networks (if ai(x) are polynomials); in the former case (8) follows also from the classical result of T. Cover [3]. Now we discuss the more complex case of a linear threshold multilayer perceptron on X = R n , with H defined as the family of all functions R n -+ {0,1} that such an architecture may implement and J.L is any continuous probability measure Generalisation in Feedforward Networks 219 on Rn. In this case there exist two functions, Blow (m) and Bup(m), such that Blow(m) ~ PrH(x) ~ Bup(m)foranyxE (Rn)mwithprobabilityl. In other words, PrH(x) takes values within a "bifurcation region" similar to the shaded region in Fig. l.b. Further, it is known that Blow(m) = 1 for m ~ nhl + 1, Bup(m) = 1 for m ~ dH and, in general, Blow (m) ~ cI.i(nhl + 1, m) and Bup(m) ~ cI.i(dH, m), where hl is the number of neurons in the first hidden layer. Given this we can say that PrH'I'(m) takes values somewhere within the "bifurcation region". It is worth noting that the width of the "bifurcation region" , which approximately equals 2( dH - nhl ) (since it is known that values on the boundaries have a positive probability of being randomly attained) increases with increasing hl since [9] O(nhllog:z(hl )) = maxp(n - p)(hd2 - 2P ) ~ dH • (10) p Estimate (6) is better than (7) in general, and in particular, if PrH,I'(m) "drops" to 0 much quicker than Bup = cI.i(dH , m), we may expect that it will provide an estimate of sample complexity mL even below dHi if PrH'I'(m) is close to Bup , then the difference between both estimates will be negligible. In order to clarify this issue we shall consider now a third, abstract example. We introduce the abstract perceptron defined as the set of hypotheses H on a probabilistic space (X, J.£) with the following property for a random m + 1-tuple (x,z) E xm x X: dvc(x, z) = dvc(x) + 1 with probability = 1 if dvc(x) < ml and with probability pifml ~ dvc(x) < m:z, and dvc(x,z) = dvc(X), otherwise. Here 0 ~ ml ~ m2 ~ 00 are two (integer) constants, 0 ~ p ~ 1 is another constant and dvc(z1J ... , zm) is the maximal n such that 17r(:Z:'l, ... ,:z:,,,)(H)1 = 2n for some 1 ~ i l < ... < in ~ m. It can easily be seen that dH = m:z in this case and that the threshold analytic neuron is a particular example of abstract percept ron (with de! p = 0 and ml = m:z = dH, Blow(m) = Bup = cI.i(mlJ m)J2m ). Note further, that if 0 < p < 1, then for any m ~ ml, dvc(x) = ml with probability> 0, and for any m ~ m:z, dvc(x) = dH = m:z with probability> o. In this regard the abstract percept ron resembles the linear threshold multilayer percept ron (with ml and m:z corresponding to nhl + 1 and dH , respectively). However, the main advantage of this model is that we can derive the following estimate: PrH,.( m) <:; 2-m ~~' (m ~ m, ) pm-m, -'( 1 - p)'<J> ( min( m - i, m,), m) (11) U sing this estimate we find that for sufficiently low p (and sufficiently large m:z) the sample complexity upper bound (6) is determined by ml and can even be lower than m:z = dH (c.f. Figure l.b). In particular, the sample complexity determined by Eqns. (6) and (11) can be finite even if dH = m:z = 00 (c.f. the curve E(m, 0) for p = .05 in Fig. l.b which is the same for m:z = 1000 and m:z = (0). 4 Discussion The paper strongly depends on the postulate (3) of (J.£n, C)-uniformity. We admit that this is an ad hoc assumption here as we do not give examples when it is 220 Adam Kowalczyk, Herman Ferra satisfied nor a method to determine the constant C. From this point of view our results at the current stage have no predictive power, perchaps only explanatory one. The paper should be viewed as an attempt in the direction to explain within VC-formalism some known generalisation properties of neural networks which are out of the reach of the formalism to date, such as the empirically observed peak generalisation for backpropagation network trained with samples of the size well below VC-dimension [8] or the phase transitions to perfect generalisation below 1.5. xVC-dimension [5]. We see the formalism in this paper as one of a number of possible approaches in this direction. There are other possibilities here as well (e.g. [5, 12]) and in particular other, weaker versions of (p.fI., C)-uniformity can be used leading to similar results. For instance in Theorem 1 and Corollary 2 it was enough to assume (p.fI. , C)-uniformity for a special class of sets S (S = s;;:;m, c.f. the Appendix); we intend to discuss other options in this regard on another occasion. Now we relate this research to some previous results (e.g. [2, 4]) which imply the following estimates on sample complexity (c.f. [1, Theorems 8.6.1-2]): ( dH - 1 ).. r4 ( 12 2)1 max 32€ ,-In(<<5)/€ ~mL(<<5,€)~ -; dH log:l-;-+log:l6" ' (12) where the lower bound is proved for all € ~ 1/8 and «5 ~ 1/100; here mL(<<5, €) is the "universal" sample complexity, i.e. for all target concepts t and all probability distributions p.. For € = «5 = 0.01 and dH » 1 this estimate yields 3dH < mi.(.Ol, .01) < 4000dH' These bounds should be compared against estimates of Corollary 2 of which (7) provides a much tighter upper bound, mL(.Ol, .01) ~ 6.9dH, if the assumption on (p.m, C)-uniformity of the hypothesis space around the target concept t is satisfied. 5 Conclusions We have shown that und~r appropriate restriction on the probability distribution and target concept, the upper bound on sample complexity (and "perfect generalisation") can be lowered to ~ 6.9x VC-dimension, and in some cases even below VC-dimension (with a strong possibility that multilayer perceptron could be such). We showed that there are other parameters than VC-dimension potentially impacting on generalisation capabilities of neural networks. In particular we showed by example (abstract perceptron) that a system may have finite sample complexity and infinite VC dimension at the same time. The formalism of this paper predicts transition to perfect generalisation at relatively low training sample sizes but it is too crude to predict scaling laws for learning curves (c.f. [5, 12] and references in there). Acknowledgement. The permission of Managing Director, Research and Information Technology, Telecom Australia, to publish this paper is gratefully acknowledged. References [1] M. Anthony and N. Biggs. Computational Learning Theory. Cambridge UniGeneralisation in Feedfom'ard Networks 221 versity Press, 1992. [2] A. Blumer, A. Ehrenfeucht, D. Haussler, and M.K. Warmuth. Learnability and the Vapnik-Chervonenkis dimensions. Journal of the ACM, 36:929-965, (Oct. 1989). [3] T.M. Cover. Geometrical and statistical properties of linear inequalities with applications to pattern recognition. IEEE Trans. Elec. Comp., EC-14:326334, 1965. [4] A. Ehrenfeucht, D. Haussler, M. Kearns, and L. Valiant. A general lower bound on the number of examples needed for learning. Information and Computation, 82:247-261, 1989. [5] D. Hausler, M. Kearns, H.S. Seung, and N. Tishby. Rigorous learning curve bounds from statistical mechanics. Technical report, 1994. [6] A. Kowalczyk. Separating capacity of analytic neuron. In Proc. ICNN'94, Orlando, 1994. [7] A. Macintyre and E. Sontag. Finiteness results for sigmoidal "neural" networks. In Proc. of the 25th Annual ACM Symp. Theory of Comp., pages 325-334, 1993. [8] G.L. Martin and J .A. Pitman. Recognizing handprinted letters and digits using backpropagation learning. Neural Comput., 3:258-267, 1991. [9] A. Sakurai. Tighter bounds of the VC-dimension of three-layer networks. In Proceedings of the 1999 World Congress on Neural Networks, 1993. [10] N. Sauer. On the density of family of sets. Journal of Combinatorial Theory (Series A), 13:145-147, 1972). [11] V. Vapnik. Estimation of Dependences Based on Empirical Data. SpringerVerlag, 1982. [12] V. Vapnik, E. Levin, and Y. Le Cun. Measuring the vc-dimension of a learning machine. Neural Computation, 6 (5):851-876, 1994). 6 Appendix: Sketch of the proof of Theorem 1 The proof is a modification of the proof of [1, Theorem 8.3.1]. We divide it into three stages. Stage 1. Let . de! 2 'RJ = {(x,y) E Xm X Xm ~ X m ; 3hEHer:ch = 0 & eryh = jim} (13) for j E {O, 1, ... , m}. Using a Chernoff bound on the "tail" of binomial distribution it can be shown [1, Lemma 8.3.2] that for m ~ 8/£ m Jr(Qf(m)) ::; 2 L j.L2m(nj) j~rmf/21 (14) Stage 2. Now we use a combinatorial argument to estimate j.L2m(1~j). We consider the 2m -element commutative group Gm of transformations of xm x xm ~ X 2m generated by all "co-ordinate swaps" of the form (Xb ... , Xm, Y1I ... , Ym) 1-+ (Xl, ... , Xi-1, Yi, Xi+1I ... , Xm, Yb ... , Yi-1, Xi, Yi+b ... , Ym), 222 Adam Kowalczyk, Herman Ferra for 1 ~ i ~ m. We assume also that Gm transforms {O, l}m x {O, l}m ~ {O, 1}2m in a similar fashion. Note that (15) As transformation u E Gm preserve the measure p.2m on xm X xm we obtain 2mp.2m(ni) = IGmlp.2m(ni) = I: J p.2m(d£dY)X'RAu(£, Y» uEG ... J p.2m(didY) I: X'R.i(u(i, Y». (16) uEG ... de! - Let s;;:;m = {h = (hI, h2) E {O,l}m x {O, l}m hI = ° & IIh211 = j} and 2 de! 2 de! 8j m = {h E {O,l} m ; Ilhll = j}, where Ilhll = hI + ... + hm for any h = (h1, ... , hm) E {O, l}m. Then 181ml = (2j), u(s;;:;m) C 81m for any u E Gm and ni = {(i, YJ E xm x xm ; 3h E 7rt,(z,y)(H) n s;;:;m}, (17) Thus from Eqn. (16) we obtain 2mp.2m(ni) ~ J p.2m(didY) I: I: xs;:-;-(uh) uEG ... hEWt,(.,f)(H)nS;J p.2m(didY) I: I: xs;:-;",(uh) hEwt,{.,f)(H)ns;'" uEG ... J p.2m(didY) _ I: _ ... I{u E Gm j uh E 81m}1 hEw t,(.,f)(H)nSo,; = J p.2m (didY) I {7rt,(:ii,y) (H) n 8;m}1 2m- j . Applying now the condition of (p.2m, C)-uniformity (Eqn. 3), Eqn. 17 and dividing by 2m we get Stage 3. On substitution of the above estimate into (14) we obtain estimate (4). To derive (5) let us observe that 2:7=rmE/21 (2j) 2-j ~ (1 + 1/2)2m. Q.E.D.	1994	74
970	The Use of Dynamic Writing Information in a Connectionist On-Line Cursive Handwriting Recognition System Stefan Manke Michael Finke Alex Waibel University of Karlsruhe Computer Science Department D-76128 Karlsruhe, Germany mankeCO)ira. uka.de, finkem@ira.uka.de Carnegie Mellon University School of Computer Science Pittsburgh, PA 15213-3890, U.S.A. w ai bel CO) cs. cm u. ed u Abstract In this paper we present NPen ++, a connectionist system for writer independent, large vocabulary on-line cursive handwriting recognition. This system combines a robust input representation, which preserves the dynamic writing information, with a neural network architecture, a so called Multi-State Time Delay Neural Network (MS-TDNN), which integrates rec.ognition and segmentation in a single framework. Our preprocessing transforms the original coordinate sequence into a (still temporal) sequence offeature vectors, which combine strictly local features, like curvature or writing direction, with a bitmap-like representation of the coordinate's proximity. The MS-TDNN architecture is well suited for handling temporal sequences as provided by this input representation. Our system is tested both on writer dependent and writer independent tasks with vocabulary sizes ranging from 400 up to 20,000 words. For example, on a 20,000 word vocabulary we achieve word recognition rates up to 88.9% (writer dependent) and 84.1 % (writer independent) without using any language models. 1094 Stefan Manke, Michael Finke, Alex Waibel 1 INTRODUCTION Several preprocessing and recognition approaches for on-line handwriting recognition have been developed during the past years. The main advantage of on-line handwriting recognition in comparison to optical character recognition (OCR) is the temporal information of handwriting, which can be recorded and used for recognition. In general this dynamic writing information (i.e. the time-ordered sequence of coordinates) is not available in OCR, where input consists of scanned text. In this paper we present the NPen++ system, which is designed to preserve the dynamic writing information as long as possible in the preprocessing and recognition process. During preprocessing a temporal sequence of N-dimensional feature vectors is computed from the original coordinate sequence, which is recorded on the digitizer. These feature vectors combine strictly local features, like curvature and writing direction [4], with so-called context bitmaps, which are bitmap-like representations of a coordinate's proximity. The recognition component of NPen++ is well suited for handling temporal sequences of patterns, as provided by this kind of input representation. The recognizer, a so-called Multi-State Time Delay Neural Network (MS-TDNN), integrates recognition and segmentation of words into a single network architecture. The MS-TDNN, which was originally proposed for continuous speech recognition tasks [6, 7], combines shift-invariant, high accuracy pattern recognition capabilities of a TDNN [8, 4] with a non-linear alignment procedure for aligning strokes into character sequences. Our system is applied both to different writer dependent and writer independent, large vocabulary handwriting recognition tasks with vocabulary sizes up to 20,000 words. Writer independent word recognition rates range from 92.9% with a 400 word vocabulary to 84.1 % with a 20,000 word vocabulary. For the writer dependent system, word recognition rates for the same tasks range from 98.6% to 88.9% [1]. In the following section we give a description of our preprocessing performed on the raw coordinate sequence, provided by the digitizer. In section 3 the architecture and training of the recognizer is presented. A description of the experiments to evaluate the system and the results we have achieved on different tasks can be found in section 4. Conclusions and future work is described in section 5. 2 PREPROCESSING The dynamic writing information, i.e. the temporal order of the data points, is preserved throughout all preprocessing steps. The original coordinate sequence {(x(t), y(t»hE{O ... T'} recorded on the digitizer is transformed into a new temporal sequence X6 = Xo ... XT, where each frame Xt consists of an N-dimensional realvalued feature vector (h(t), . .. , fN(t» E [-1, l]N. Several normalization methods are applied to remove undesired variability from the original coordinate sequence. To compensate for different sampling rates and varying writing speeds the coordinates originally sampled to be equidistant in time are resampled yielding a new sequence {(x(t),y(t»hE{O ... T} which is equidistant in Dynamic Writing Information in Cursive Handwriting Recognition 1095 normalized coordinate (b) writing direction , , x(.·2),y( •• 2) 'f ,';'1 ~ 1" x(t+2),y(t+2) x(t-l),y('·l) t~, ' • " .... , x('+1),y('+l) x('),y(') curvature , , , , x( •• 2),y('.2) ~ /1 , ' : ' x(.+2),Y(1+2) x(t-11y('·1). ~ " \. .. ...... x(t+l),y(t+l) x(.),y(') Figure I: Feature extraction for the normalized word "able". The final input representation is derived by c,alculating a 15-dimensional feature vector for each data point, which consists of a context bitmap (a) and information about the curvature and writing direction (b). space. This resampled trajectory is smoothed using a moving average window in order to remove sampling noise. In a final normalization step the goal is to find a representation of the trajectory that is reasonably invariant against rotation and scaling of the input. The idea is to determine the words' baseline using an EM approach similar to that described in [5] and rescale the word such that the center region of the word is assigned to a fixed size. From the normalized coordinate sequence {(x(t), y(t))hE{O ... T} the temporal sequence 2::r; of N-dimensional feature vectors ~t = (!l(t), ... .IN(t)) is computed (Figure I). Currently the system uses N = 15 features for each data point. The first two features fl(t) = x(t)-x(t-I) and h(t) = y(t)-b describe the relative X movement and the Y position relative to the baseline b. The features /get) to f6(t) are used to desc,ribe the curvature and writing direction in the trajectory [4] (Figure I(b)). Since all these features are strictly local in the sense that they are local both in time and in space they were shown to be inadequate for modeling temporal long range context dependencies typically observed in pen trajectories [2]. Therefore, nine additional features h(t) to !J5(t) representing :3 x:3 bitmaps were included in each feature vector (Figure I(a». These so-called context bitmaps are basically low resolution, bitmap-like descriptions of the coordinate's proximity, which were originally described in [2]. Thus, the input representation as shown in Figure I combines strictly local features like writing direction and curvature with the c,ontext bitmaps, which are still local 1096 Stefan Manke, Michael Finke, Alex Waibel in space but global in time. That means, each point of the trajectory is visible from each other point of the trajectory in a small neighbourhood. By using these context bitmaps in addition to the local features, important information about other parts of the trajectory, which are in a limited neighbourhood of a coordinate, are encoded. 3 THE NPen++ RECOGNIZER The NPell ++ recognizer integrates recognition and segmentation of words into a single network architecture, the Multi-State Time Delay Neural Network (MSTDNN). The MS-TDNN, which was originally proposed for continuous speech recognition tasks [6, 7], combines the high accuracy single character recognition capabilities of a TDNN [8, 4] with a non-linear time alignment algorithm (dynamic time warping) for finding stroke and character boundaries in isolated handwritten words. 3.1 MODELING ASSUMPTIONS Let W = {WI, .. . WK} be a vocabulary consisting of K words. Each of these words Wi is represented as a sequence of characters Wi == Ci l Ci 2 • •• Cik where each character Cj itself is modelled by a three state hidden markov model Cj == qj q] qJ. The idea of using three states per character is to model explicitly the imtial, middle and final section of the characters. Thus, Wi is modelled by a sequence of states Wi == qioqil · .. qhk . In these word HMMs the self-loop probabilities p(qij/qij) and the transition probabilities p(qij/qij_l) are both defined to be ~ while all other transition probabilities are set to zero. During recognition of an unknown sequence of feature vectors ~'(; = ~o . . . ~T we have to find the word Wi E W in the dictionary that maximizes the a-posteriori probability p( Wi /~a 10) given a fixed set of parameters 0 and the observed coordinate sequence. That means, a written word will be recognized such that Wj = argmaXw,EWp(Wi/Za,O). In our Multi-State Time Delay Neural Network approach the problem of modeling the word posterior probability p( wdz'{; , 0) is simplified by using Bayes' rule which expresses that probability as p(Z'{;/WilO)P(Wi/O) p(z'{;/O) Instead of approximating p( Wi /z'{;, 0) directly we define in the following section a network that is supposed to model the likelihood of the feature vector sequence p(z'{; /Wi, 0). 3.2 THE MS-TDNN ARCHITECTURE In Figure 2 the basic MS-TDNN architecture for handwriting recognition is shown. The first three layers constitute a standard TDNN with sliding input windows in each layer. In the current implementation of the system, a TDNN with 15 input Dynamic Writing InJonnation in Cursive Handwriting Recognition 1097 i::' "E -g :; ] E '" ..'!l '" " 1i 0 0 0 0 '" .D .D ~ .D 0 '" '" '" '" '" N 0 mm.o ~.m .W.J 0 ::::::: :." . • • • • • ... -....... -.... ~ r:'?~ " ""~L-,;~··§i] ~ ",. . ........ ... ~ [~~~::~~~::~ ~~~~~~~~:: :~::~:~~~~:J: ~ __ - _. _____ n ___ ___ . . . __ n n_. n ~. __ nn __ ____ ~ t. ~-- -- n---:011- ~7-n •• --- -- __ .. ~.:~ __ •• -:. ____ n --.~-- "_ ·· ····· ;r?5_ -m~;.·i~::~:~·:·:·:I ; .: I I : 11 ! I I ,I L" ... X ~ Y •. -------------------.-- -~~~: ~ ~~~~~~~:~~~:~~~}:_ ~ z'---______ -l::±-_____ --' --------------. time ------------------Figure 2: The Multi-State TDNN architecture, consisting of a 3-layer TDNN to estimate the a posteriori probabilities of the character states combined with word units, whose scores are derived from the word models by a Viterbi approximation of the likelihoods p(x6'IWi). units, 40 units in the hidden layer, and 78 state output units is used . There are 7 time delays both in the input and hidden layer. The softmax normalized output of the states layer is interpreted as an estimate of the probabilities of the states qj given the input window x!~~ = Xt-d .. . Xt+d for each time frame t, i.e. exp(1/j (t)) 2::k exp(1/k(t)) ( 1 ) where 1lj (t) represents the weighted sum of inputs to state unit j at time t. Based on these estimates, the output of the word units is defined to be a Viterbi approximation of the log likelihoods of the feature vector sequence given the word model 1098 Stefan Manke, Michael Finke, Alex Waibel T logp(zrlwi) ~ m~ L 10gp(z;~~lqt, Wi) + logp(qtlqt-I, Wi) qo t=1 T ~ m~ Llogp(qtlz;~~) -logp(qt) + logp(qtlqt-1, Wi). (2) qo t=1 Here, the maximum is over all possible sequences of states q'{; = qo . .. qT given a word model, p(qtlz!~~) refers to the output of the states layer as defined in (1) and p(qt) is the prior probability of observing a state qt estimated on the training data. 3.3 TRAINING OF THE RECOGNIZER During training the goal is to determine a set of parameters 0 that will maximize the posterior probability p( wlzr, 0) for all training input sequences. But in order to make that maximization computationally feasible even for a large vocabulary system we had to simplify that maximum a posteriori approach to a maximum likelihood training procedure that maximizes p(zrlw, 0) for all words instead. The first step of our maximum likelihood training is to bootstrap the recognizer using a subset of approximately 2,000 words of the training set that were labeled manually with the character boundaries to adjust the paths in the word layer correctly. After training on this hand-labeled data, the recognizer is used to label another larger set of unlabeled training data. Each pattern in this training set is processed by the recognizer. The boundaries determined automatically by the Viterbi alignment in the target word unit serve as new labels for this pattern. Then, in the second phase, the recognizer is retrained on both data sets to achieve the final performance of the recognizer. 4 EXPERIMENTS AND RESULTS We have tested our system both on writer dependent and writer independent tasks with vocabulary sizes ranging from 400 up to 20,000 words. The word recognition results are shown in Table 1. The scaling of the recognition rates with respect to the vocabulary size is plotted in Figure 3b. T bl 1 W' a e rlter d d epen ent an d' d d In epen ent recogmtIOn resu ts Vocabulary Writer Dependent Writer Independent Task Size Test RecognitIon Test KecogmtIOn Patterns Rate Patterns Rate crtAOO 400 800 98.6% 800 92.9% wsj_l,OOO 1,000 800 97.8% wsj_7,000 7,000 2,500 89.3% wsj_l0,000 10,000 1,600 92.1% 2,500 87.7% wsj..20,000 20,000 1,600 88.9% 2,500 84.f% Dynamic Writing Information in Cursive Handwriting Recognition 1099 100r-----.------.----~------., ... ................... writer dependent .... -. writer independent .-+95 ......... . ...... -----... .................................. . 90 .................... .. .................. -.. .. ............... "'-"'-'. 85 -------.. 80 75~----~----~----~------~ 5000 10000 15000 20000 size of vocabulary (b) Figure 3: (a) Different writing styles in the database: cursive (top), hand-printed (middle) and a mixture of both (bottom) (b) Recognition results with respect to the vocabulary size For the writer dependent evaluation, the system was trained on 2,000 patterns from a 400 word vocabulary, written by a single writer, and tested on a disjunct set of patterns from the same writer. In the writer dependent case, the training set consisted of 4,000 patterns from a 7,000 word vocabulary, written by approximately 60 different writers. The test was performed on data from an independent set of 40 writers. All data used in these experiments was collected at the University of Karlsruhe, Germany. Only minimal instructions were given to the writers. The writers were asked to write as natural as they would normally do on paper, without any restrictions in writing style. The consequence is, that the database is characterized by a high variety of different writing styles, ranging from hand-printed to strictly cursive patterns or a mixture of both writing styles (for example see Figure 3a). Additionally the native language of the writers was german, but the language of the dictionary is english. Therefore, frequent hesitations and corrections can be observed in the patterns of the database. But since this sort of input is typical for real world applications, a robust recognizer should be able to process these distorted patterns, too. From each of the writers a set of 50-100 isolated words, choosen randomly from the 7,000 word vocabulary, was collected. The used vocabularies CRT (Conference Registration Task) and WSJ (ARPA Wall Street Journal Task) were originally defined for speech recognition evaluations. These vocabularies were chosen to take advantage of the synergy effects between handwriting recognition and speech recognition, since in our case the final goal is to integrate our speech recognizer JANUS [10] and the proposed NPen++ system into a multi-modal system. 1100 Stefan Manke, Michael Finke, Alex Waibel 5 CONCLUSIONS In this paper we have presented the NPen++ system, a neural recognizer for writer dependent and writer independent on-line cursive handwriting recognition. This system combines a robust input representation, which preserves the dynamic writing information, with a neural network integrating recognition and segmentation in a single framework. This architecture has been shown to be well suited for handling temporal sequences as provided by this kind of input. Evaluation of the system on different tasks with vocabulary sizes ranging from 400 to 20,000 words has shown recognition rates from 92.9% to 84.1 % in the writer independent case and from 98.6% to 88.9% in the writer dependent case. These results are especially promising because they were achieved with a small training set compared to other systems (e.g. [3]). As can be seen in Table 1, the system has proved to be virtually independent of the vocabulary. Though the system was trained on rather small vocabularies (e.g. 400 words in the writer dependent system), it generalizes well to completely different and much larger vocabularies. References [1] S. Manke and U. Bodenhausen, "A Connectionist Recognizer for Cursive Handwriting Recognition", Proceedings of the ICASSP-94, Adelaide, April 1994. [2] S. Manke, M. Finke, and A. Waibel, "Combining Bitmaps with Dynamic Writing Information for On-Line Handwriting Recognition", Proceedings of the ICPR94, Jerusalem, October 1994. [3] M. Schenkel, I. Guyon, and D. Henderson, "On-Line Cursive Script Recognition Using Time Delay Neural Networks and Hidden Markov Models", Proceedings of the ICASSP-94, Adelaide, April 1994. [4] I. Guyon, P. Albrecht, Y. Le Cun, W. Denker, and W. Hubbard, "Design of a Neural Network Character Recognizer for a Touch Terminal", Pattern Recognition, 24(2), 1991. [5] Y. Bengio and Y. LeCun. "Word Normalization for On-Line Handwritten Word Recognition", Proceedings of the ICPR-94, Jerusalem, October 1994. [6] P. Haffner and A. Waibel, "Multi-State Time Delay Neural Networks for Continuous Speech Recognition", Advances in Neural Information Processing Systems (NIPS-4) , Morgan Kaufman, 1992. [7] C. Bregler, H. Hild, S. Manke, and A. Waibel, "Improving Connected Letter Recognition by Lipreading", Proceedings of the ICASSP-93, Minneapolis, April 1993. [8] A. Waibel, T. Hanazawa, G. Hinton, K. Shiano, and K. Lang, "Phoneme Recognition using Time-Delay Neural Networks", IEEE Transactions on Acoustics, Speech and Signal Processing, March 1989. [9] W. Guerfali and R. Plamondon, "Normalizing and Restoring On-Line Handwriting", Pattern Recognition, 16(5), 1993. [10] M. Woszczyna et aI., "Janus 94: Towards Spontaneous Speech Translation", Proceedings of the ICASSP-94, Adelaide, April 1994.	1994	75
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972	Capacity and Information Efficiency of a Brain-like Associative Net Bruce Graham and David Willshaw Centre for Cognitive Science, University of Edinburgh 2 Buccleuch Place, Edinburgh, EH8 9LW, UK Email: bruce@cns.ed.ac.uk&david@cns.ed.ac.uk Abstract We have determined the capacity and information efficiency of an associative net configured in a brain-like way with partial connectivity and noisy input cues. Recall theory was used to calculate the capacity when pattern recall is achieved using a winners-takeall strategy. Transforming the dendritic sum according to input activity and unit usage can greatly increase the capacity of the associative net under these conditions. For moderately sparse patterns, maximum information efficiency is achieved with very low connectivity levels (~ 10%). This corresponds to the level of connectivity commonly seen in the brain and invites speculation that the brain is connected in the most information efficient way. 1 INTRODUCTION Standard network associative memories become more plausible as models of associative memory in the brain if they incorporate (1) partial connectivity, (2) sparse activity and (3) recall from noisy cues. In this paper we consider the capacity of a binary associative net (Willshaw, Buneman, & Longuet-Higgins, 1969; Willshaw, 1971; Buckingham, 1991) containing these features. While the associative net is a very simple model of associative memory, its behaviour as a storage device is not trivial and yet it is tractable to theoretical analysis. We are able to calculate 514 Bruce Graham, David Willshaw the capacity of the net in different configurations and with different pattern recall strategies. Here we consider the capacity as a function of connectivity level when winners-take-all recall is used. The associative net is an heteroassociative memory in which pairs of binary patterns are stored by altering the connection weights between input and output units via a Hebbian learning rule. After pattern storage, an output pattern is recalled by presenting a previously stored input pattern on the input units. Which output units become active during recall is determined by applying a threshold of activation to measurements that each output unit makes of the input cue pattern. The most commonly used measurement is the weighted sum of the inputs, or dendritic sum. Amongst the simpler thresholding strategies is the winners-take-all (WTA) approach, which chooses the required number of output units with the highest dendritic sums to be active. This works well when the net is fully connected (each input unit is connected to every output unit), and input cues are noise-free. However, recall performance deteriorates rapidly if the net is partially connected (each input unit is connected to only some of the output units) and cues are noisy. Marr (1971) recognised that when an associative net is only partially connected, another useful measurement for threshold setting is the total input activity (sum of the inputs, regardless of the connection weights). The ratio of the dendritic sum to the input activity can be a better discriminator of which output units should be active than the dendritic sum alone. Buckingham and Willshaw (1993) showed that differences in unit usage (the number of patterns in which an output unit is active during storage) causes variations in the dendritic sums that makes accurate recall difficult when the input cues are noisy. They incorporated both input activity and unit usage measurements into a recall strategy that minimised the number of errors in the output pattern by setting the activity threshold on a unit by unit basis. This is a rather more complex threshold setting strategy than a simple winners-take-all. We have previously demonstrated via computer simulations (Graham & Wills haw , 1994) that the WTA threshold strategy can achieve the same recall performance as this minimisation approach if the dendritic sums are transformed by certain functions of the input activity and unit usage before a threshold is applied. Here we calculate the capacity of the associative net when WTA recall is used with three different functions of the dendritic sums: (1) pure dendritic sums, (2) modified by input activity and (3) modified by input activity and unit usage. The results show that up to four times the capacity can be obtained by transforming the dendritic sums by a function of both input activity and unit usage. This increase in capacity was obtained without a loss of information efficiency. For the moderately sparse patterns used, WTA recall is most information efficient at low levels of connectivity (~ 10%), as is the minimisation approach to threshold setting (Buckingham, 1991). This connectivity range is similar to that commonly seen in the brain. Capacity and Infonnation Efficiency of a Brain-Like Associative Net 515 2 NOTATION AND OPERATION The associative net consists of N B binary output units each connected to a proportion Z of the N A binary input units. Pairs of binary patterns are stored in the net. Input and output patterns contain MA and MB active units, respectively (activity level a = M / N « 1). All connection weights start at zero. On presentation to the net of a pattern pair for storage, the connection weight between an active input unit and an active output unit is set to 1. During recall an input cue pattern is presented on the input units. The input cue is a noisy version of a previously stored input pattern in which a fraction, s, of the MA active units do not come from the stored pattern. A thresholding strategy is applied to the output units to determine which of them should be active. Those that should be active in response to the input cue will be called high units, and those that should be inactive will be called low units. We consider winners-take-all (WTA) thresholding strategies that choose to be active the MB output units with the highest values of three functions of the dendritic sum, d, the input activity, a, and the unit usage, r. These functions are listed in Table 1. The normalised strategy deals with partial connectivity. The transformed strategy reduces variations in the dendritic sums due to differences in unit usage. This function minimises the variance of the low unit dendritic sums with respect to the unit usage (Graham & Willshaw, 1994). Table 1: WTA Strategies WTA Strategy FUnction Basic d Normalised d' = d/a Transformed d* = 1 - (1 - d/a)l/r 3 RECALL THEORY The capacity of the associative net is defined to be the number of pattern pairs that can be stored before there is one bit in error in a recalled output pattern. This cannot be calculated analytically for the net configuration under study. However, it can be determined numerically for the WTA recall strategy by calculating the recall response for different numbers of stored patterns, R, until the minimum value of R is found for which a recall error occurs. The WTA recall response can be calculated theoretically using expressions for the distributions of the dendritic sums of low and high output units. The probability that the dendritic sum of a low or high output unit should have a particular value x is, respectively (Buckingham & Willshaw, 1993; Buckingham, 1991) P(d, = x) = t. ( ~ ) "'8(1-"B)R-, ( "!,A ) (Zp[rJ)"(I- Zp[rJ)MA-" (I) 516 Bruce Graham. David Willshaw P(dh = x) = ~ ( ~ ) a;'(l-aB)R-" ( ~A ) (ZI'[r +1])"(1- ZI'[r+1])MA-" (2) where p[r] and /L[r] are the probabilities that an arbitrarily selected active input is on a connection with weight 1. For a low unit, p[r] = 1- (1- OA)r. For a high unit a good approximation for /L is /L[r + 1] ~ g + sp[r] = 1 - s(l - OAY where g and s are the probabilities that a particular active input in the cue pattern is genuine (belongs to the stored pattern) or spurious, respectively (g + s = 1) (Buckingham & Willshaw, 1993). The basic WTA response is calculated using these distributions by finding the threshold, T, that gives (3) The number of false positive and false negative errors of the response is given by (4) The actual distributions of the normalised dendritic sums are the distributions of dja. For the purposes of calculating the normalised WTA response, it is possible to use the basic distributions for the situation where every unit has the mean input activity, am = MAZ. In this case the low and high unit distributions are approximately P(til = x) = t. ( ~ ) ,,;'(1- "B)R-" ( "; ) (p[r])"(l- p[r])"m-" (5) P(d'h = x) = ~ ( ~ ) aB(1-"B )R-" ( a; ) (I'[r + 1])" (l-l'[r +1 ])"m -" (6) Due to the nonlinear transformation used, it is not possible to calculate the transformed distributions as simple sums of binomials, so the following approach is used to generate the transformed WTA response. For a given transformed threshold, T, and for each possible value of unit usage, r, an equivalent normalised threshold is calculated via T'[r] = am (l - (1 - TY) (7) The transformed cumulative probabilities can then be calculated from the normalised distributions: P( dj :2: TO) = t. ( ~ ) aB(1 - "B )R-" P( til :2: T'[r]) (8) P( di. :2: T') = ~ ( ~ ) ,,;'(1 - "B )R-" P(d' :2: T'[r + 1]) (9) The normalised and transformed WTA responses are calculated in the same manner as the basic response, using the appropriate probability distributions. Capacity and Information Efficiency of a Brain-Like Associative Net 5000 4000 g ~ 3000 ·u ca Co ca 2000 U 1000 4 RESULTS (a) 0% noise --B -- N T ~ ... .-~ , ............ ......... ./" ...-., 3000 2500 2000 1500 1000 500 (b) 40% noise --B ·-N ---T O~---...L ___ --'---'-.....L-....o...-..I.-..........J 20 40 60 80 100 0 20 40 60 80 100 Connectivity (%) Connectivity (%) Figure 1: Capacity Versus Connectivity 517 Extensive simulations were previously carried out of WTA recall from a large associative net with the following specifications (Graham & Willshaw, 1994): NA = 48000, MA = 1440, NB = 6144, MB = 180. Agreement between the simulations and the theoretical recall described above is extremely good, indicating that the approximations used in the theory are valid. Here we use the theoretical recall to calculate capacity results for this large associative net that are not easily obtained via simulations. All the results shown have been generated using the theory described in the previous section. Figure 1 shows the capacity as a function of connectivity for the different WTA strategies when there is no noise in the input cue, or 40% noise in the cue (legend: B = basic WTA, N = normalised WTA, T = transformed WTA; for clarity, individual data points are omitted). With no noise in the cue the normalised and transformed methods perform identically, so only the normalised results are shown. Figure l(a) highlights the effectiveness of normalising the dendritic sums against input activity when the net is partially connected. Figure 1 (b) shows the effect of noise on capacity. The capacity of each recall strategy at a given connectivity level is much reduced compared to the noise-free case. However, for connectivities greater than 10% the capacity of the transformed WTA is now much greater than either the normalised or basic WTA. The relative capacities of the different strategies are shown in Figure 2 (legend: NIB = ratio of normalised to basic capacity, T I B = ratio of transformed to basic, TIN = ratio of transformed to normalised). In the noise-free case (Figure 2(a)), at low levels of connectivity the relative capacity is distorted because the basic capacity 518 Bruce Graham, David Willshaw drops to near zero, so that even low normalised capacities are relatively very large. For most connectivity levels (10-90%) the normalised WTA provides 2-4 times the capacity of the basic WTA. In the noisy case (Figure 2(b)), the normalised capacity is only up to 1.5 times the basic capacity over this range of connectivities. The transformed WTA, however, provides 3 to nearly 4 times the basic capacity and 2.5 to nearly 3 times the normalised capacity for connectivities greater than 10%. The capacities can be interpreted in information theoretic terms by considering the information efficiency of the net. This is the ratio of the amount of information that can be retrieved from the net to the number of bits of storage available and is given by "'0 = Ro10jZNANB' where Ro is the capacity, 10 is the amount of information contained in an output pattern and Z NAN B is the number of weights, or bits of storage required (Willshaw et al., 1969; Buckingham & Willshaw, 1992). Information efficiency as a function of connectivity is shown in Figure 3. There is a distinct peak in information efficiency for each of the recall strategies at some low level of connectivity. The peak information efficiencies and the efficiencies at full connectivity are summarised in Table 2. The greatest contrast between full and partial connectivity is seen with the normalised WTA and noise-free cues. At 1 % connectivity the normalised WTA is nearly 14 times more efficient than at full connectivity. In absolute terms, however, the normalised capacity is only 694 at 1 % connectivity, compared with 5122 at full connectivity. The peak efficiency of 53% obtained by the normalised WTA is approaching the theoretically approximate maximum of 69% for a fully connected net (Willshaw et al., 1969). 5 DISCUSSION Previous simulations (Graham & Willshaw, 1994) have shown that, when the input cues are noisy, the recall performance of the winners-take-all thresholding strategy applied to the partially connected associative net is greatly improved if the dendritic sums of the output units are transformed by functions of input activity and unit usage. We have confirmed and extended these results here by calculating the theoretical capacity of the associative net as a function of connectivity. For the moderately sparse patterns used, all of the recall strategies are most information efficient at very low levels of connectivity (~ 10%). However, the optimum connectivity level is dependent on the pattern coding rate. Extending the analysis of Willshaw et al. (1969) to a partially connected net using normalised WTA recall yields that maximum information efficiency is obtained when ZMA = log2(NB). So for input coding rates higher than log2(NB), a partially connected net is most information efficient. For the input coding rate used here, this relationship gives an optimum connectivity level of 0.87%, very close to the 1% obtained from the recall theory. Comparing the peak efficiencies across the different strategies for the noisy cue case, the normalised WTA is about twice as efficient as the basic WTA, while the transformed WTA is three times as efficient. This comparison does not include the Capacity and Information Efficiency of a Brain-Like Associative Net 519 (a) 0% noise (b) 40% noise 12 4 " ...... .............. -- NIB, TIB " ..... , 10 I , I , .~ 3 ( , -~ 0 8 , ...... -------III I , ,..a. I III \1 ' u 6 ~ 2 }{ ~ ;; ' ... III 4 1;·--___ Q) a: 1 I, -- NIB -_ -----II 2 ., ' --- T/B \ I TIN 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Connectivity (%) Connectivity (%) Figure 2: Relative Capacity Versus Connectivity (a) 0% noise (b) 40% noise 60 6 ", --B ~ 50 5 ' '\ --N --B , '\ ---T >. , '\ g 40 ._._- N, T 4 , '\ \ , Q) f.l '" .(3 ffi 30 3 .\ " ;'\ " c: ~ . , ............ . Q I - ... ia 20 2 ., ' ... E .... '-0 - 10 --------1: r-----"::7 o 0 L-'--' ........ ---L_--L_---L.--'---I o 20 40 60 80 100 0 20 40 60 80 100 Connectivity (%) Connectivity (%) Figure 3: Information Efficiency Versus Connectivity Table 2: Information Efficiency 0% Noise 40% Noise WTA TJo at Z at TJo at T]o at Z at TJo at Strategy Peak Peak Z=1 Peak Peak Z=1 (%) (%) (%) (%) (%) (%) Basic 6.1 4 3.9 2.0 7 0.8 Normalised 53.3 1 3.9 3.6 5 0.8 Transformed 53.3 1 3.9 5.7 10 2.3 520 Bruce Graham, David Willshaw cost of storing input activity and unit usage information. If one bit of storage per connection is required for the input activity, and another bit for the unit usage, then the information efficiency of the normalised WTA is halved, and the information efficiency of the transformed WTA is reduced by two thirds. This results in all the strategies having about the same peak efficiency. However, the absolute capacities of the different strategies at their peak efficiencies are 183,237 and 741 for the basic, normalised and transformed WTA, respectively. So, at the same level of efficiency, the transformed WTA delivers four times the capacity of the basic WTA. In conclusion, numerical calculations of the capacity of the associative net show that it is most information efficient at a very low level of connectivity when moderately sparse patterns are stored. Including input activity and unit usage information into the recall calculations results in a four-fold increase in storage capacity without loss of efficiency. Acknowledgements To the Medical Research Council for financial support under Programme Grant PG 9119632 References Buckingham, J., & Willshaw, D. (1992). Performance characteristics of the associative net. Network, 8, 407-414. Buckingham, J., & Willshaw, D. (1993). On setting unit thresholds in an incompletely connected associative net. Network, 4, 441-459. Buckingham, J. (1991). Delicate nets, faint recollections: a study of partially connected associative network memories. Ph.D. thesis, University of Edinburgh. Graham, B., & Willshaw, D. (1994). Improving recall from an associative memory. Bioi. Cybem., in press. Marr, D. (1971). Simple memory: a theory for archicortex. Phil. Trans. Roy. Soc. Lond. B, 262, 23-81. Shepherd, G. (Ed.). (1990). The Synaptic Organization of the Brain (Third edition). Oxford University Press, New York, Oxford. Willshaw, D. (1971). Models of distributed associative memory. Ph.D. thesis, University of Edinburgh. Willshaw, D., Buneman, 0., & Longuet-Higgins, H. (1969). Non-holographic associative memory. Nature, 222, 960-962.	1994	77
973	SARDNET: A Self-Organizing Feature Map for Sequences Daniel L. James and Risto Miikkulainen Department of Computer Sciences The University of Texas at Austin Austin, TX 78712 dljames,risto~cs.utexas.edu Abstract A self-organizing neural network for sequence classification called SARDNET is described and analyzed experimentally. SARDNET extends the Kohonen Feature Map architecture with activation retention and decay in order to create unique distributed response patterns for different sequences. SARDNET yields extremely dense yet descriptive representations of sequential input in very few training iterations. The network has proven successful on mapping arbitrary sequences of binary and real numbers, as well as phonemic representations of English words. Potential applications include isolated spoken word recognition and cognitive science models of sequence processing. 1 INTRODUCTION While neural networks have proved a good tool for processing static patterns, classifying sequential information has remained a challenging task. The problem involves recognizing patterns in a time series of vectors, which requires forming a good internal representation for the sequences. Several researchers have proposed extending the self-organizing feature map (Kohonen 1989, 1990), a highly successful static pattern classification method, to sequential information (Kangas 1991; Samarabandu and Jakubowicz 1990; Scholtes 1991). Below, three of the most recent of these networks are briefly described. The remainder of the paper focuses on a new architecture designed to overcome the shortcomings of these approaches. 578 Daniel L. James, Risto Miikkulainen Recently, Chappel and Taylor (1993) proposed the Temporal Kohonen Map (TKM) architecture for classifying sequences. The TKM keeps track of the activation history of each node by updating a value called leaky integrator potential, inspired by the membrane potential in biological neural systems. The activity of a node depends both on the current input vector and the previous input vectors, represented by the node's potential. A given sequence is processed by mapping one vector at a time, and the last winning node serves to represent the entire sequence. This way, there needs to be a separate node for every possible sequence, which is a disadvantage when the number of sequences to be classified is large. The TKM also suffers from loss of context. Which node wins depends almost entirely upon the most recent input vectors. For example, the string baaaa would most likely map to the same node as aaaaa, making the approach applicable only to short sequences. The SOFM-S network proposed by van Harmelen (1993) extends TKM such that the activity of each map node depends on the current input vector and the past activation of all map nodes. The SOFM-S is an improvement of TKM in that contextual information is not lost as quickly, but it still uses a single node to represent a sequence. The TRACE feature map (Zandhuis 1992) has two feature map layers. The first layer is a topological map of the individual input vectors, and is used to generate a trace (i.e. path) of the input sequence on the map. The second layer then maps the trace pattern to a single node. In TRACE, the sequences are represented by distributed patterns on the first layer, potentially allowing for larger capacity, but it is difficult to encode sequences where the same vectors repeat, such as baaaa. All a-vectors would be mapped on the same unit in the first layer, and any number of a-vectors would be indistinguishable. The architecture described in this paper, SARDNET (Sequential Activation Retention and Decay NETwork), also uses a subset of map nodes to represent the sequence of vectors. Such a distributed approach allows a large number of representations be "packed" into a small map-like sardines. In the following sections, we will examine how SARDNET differs from conventional self-organizing maps and how it can be used to represent and classify a large number of complex sequences. 2 THE SARDNET ARCHITECTURE Input to SARDNET consists of a sequence of n-dimensional vectors S = V I, V 2 , V 3 , ... , VI (figure 1) . The components of each vector are real values in the interval [0,1]. For example, each vector might represent a sample of a speech signal in n different frequencies, and the entire sequence might constitute a spoken word. The SARDNET input layer consists of n nodes, one for each component in the input vector, and their values are denoted as A = (aI, a2, a3, ... , an). The map consists of m x m nodes with activation Ojk , 1 ~ j, k ~ m. Each node has an n-dimensional input weight vector Wjk, which determines the node's response to the input activation. In a conventional feature map network as well as in SARDNET, each input vector is mapped on a particular unit on the map, called the winner or the maximally responding unit. In SARDNET, however, once a node wins an input, it is made SARDNET: A Self-Organizing Feature Map for Sequences 579 Sequence of Input vectors S V1 V2 V3 V4 V, ---II Previous winners Input weight vector wJk.l Winning unit jlc Figure 1: The SARDNET architecture. A sequence of input vectors activates units on the map one at a time. The past winners are excluded from further competition, and their activation is decayed gradually to indicate position in the sequence. INITIALIZATION: Clear all map nodes to zero. MAIN LOOP: While not end of seihence 1. Find unactivated weight vector t at best matches the input. 2. Assign 1.0 activation to that unit. 3. Adjust weight vectors of the nodes in the neighborhood. 4. Exclude the winning unit from subse~ent competition. S. Decrement activation values for all ot er active nodes. RESULT: Sequence representation = activated nodes ordered by activation values Table 1: The SARDNET training algorithm. uneligible to respond to the subsequent inputs in the sequence. This way a different map node is allocated for every vector in the sequence. As more vectors come in, the activation of the previous winners decays. In other words, each sequence of length 1 is represented by 1 active nodes on the map, with their activity indicating the order in which they were activated. The algorithm is summarized in table 1. Assume the maximum length ofthe sequences we wish to classify is I, and each input vector component can take on p possible values. Since there are pn possible input vectors, Ipn map nodes are needed to represent all possible vectors in all possible positions in the sequence, and a distributed pattern over the Ipn nodes can be used to represent all pnl different sequences. This approach offers a significant advantage over methods in which pnl nodes would be required for pnl sequences. The specific computations of the SARDNET algorithm are as follows: The winning node (j, k) in each iteration is determined by the Euclidean distance Djk of the 580 Daniel L. James, Risto Miikkulainen input vector A and the node's weight vector W j k: n Djk = 1) Wjk ,i a;)2. i=O (1) The unit with the smallest distance is selected as the winner and activated with 1.0. The weights of this node and all nodes in its neighborhood are changed according to the standard feature map adaptation rule: (2) where a denotes the learning rate. As usual, the neighborhood starts out large and is gradually decreased as the map becomes more ordered. As the last step in processing an input vector, the activation 7]jk of all active units in the map are decayed proportional to the decay parameter d: O<d<1. (3) As in the standard feature map, as the weight vectors adapt, input vectors gradually become encoded in the weight vectors of the winning units. Because weights are changed in local neighborhoods, neighboring weight vectors are forced to become as similar as possible, and eventually the network forms a topological layout of the input vector space. In SARDNET, however, if an input vector occurs multiple times in the same input sequence, it will be represented multiple times on the map as well. In other words, the map representation expands those areas of the input space that are visited most often during an input sequence. 3 EXPERIMENTS SARDNET has proven successful in learning and recognizing arbitrary sequences of binary and real numbers, as well as sequences of phonemic representations for English words. This section presents experiments on mapping three-syllable words. This data was selected because it shows how SARDNET can be applied to complex input derived from a real-world task. 3.1 INPUT DATA The phonemic word representations were obtained from the CELEX database of the Max Planck Institute for Psycholinguistics and converted into International Phonetic Alphabet (IPA)-compliant representation, which better describes similarities among the phonemes. The words vary from five to twelve phonemes in length. Each phoneme is represented by five values: place, manner, sound, chromacity and sonority. For example, the consonant p is represented by a single vector (bilabial, stop, unvoiced, nil, nil) , or in terms of real numbers, (.125, .167, .750,0, 0). The diphthong sound ai as in "buy" , is represented by the two vectors (nil, vowel, voiced, front, low) and (nil, vowel , voiced, front-center, hi-mid), or in real numbers, (0, 1, .25, .2, 1) and (0, 1, .25, .4, .286). There are a total of 43 phonemes in this data set, including 23 consonants and 20 vowels. To represent all phonemic sequences of length 12, TKM and SOFM-S would SARDNET: A Self-Organizing Feature Map for Sequences 581 o.ee 0.118 0.87 0.118 0.85 ....... -Figure 2: Accuracy of SARDNET for different map and data set sizes. The accuracy is measured as a percentage of unique representations out of all word sequences. need to have 4512 ~ 6.919 map nodes, whereas SARDNET would need only 45 x 12 = 540 nodes. Of course, only a very small subset of the possible sequences actually occur in the data. Three data sets consisting of 713, 988, and 1628 words were used in the experiments. If the maximum number of occurrences of phoneme i in any single sequence is Cj I then the number of nodes SARDNET needs is C = L:~o Cj I where N is the number of phonemes. This number of nodes will allow SARDNET to map each phoneme in each sequence to a unit with an exact representation of that phoneme in its weights. Calculated this way, SARDNET should scale up very well with the number of words: it would need 81 nodes for representing the 713 word set, 84 for the 988 set and 88 for the 1628 set. 3.2 DENSENESS AND ACCURACY A series of experiments with the above three data sets and maps of 16 to 81 nodes were run to see how accurately SARDNET can represent the sequences. Self-organization was quite fast: each simulation took only about 10 epochs, with a = 0.45 and the neighborhood radius decreasing gradually from 5-1 to zero. Figure 2 shows the percentage of unique representations for each data set and map SIze. SARDNET shows remarkable representational power: accuracy for all sets is better than 97.7%, and SARDNET manages to pack 1592 unique representations even on the smallest 16-node map. Even when there are not enough units to represent each phoneme in each sequence exactly, the map is sometimes able to "reuse" units to represent multiple similar phonemes. For example, assume units with exact representations for the phonemes a and b exist somewhere on the map, and the input data does not contain pairs of sequences such as aba-abb, in which it is crucial to distinguished the second a from the second b. In this case, the second occurrence of both phonemes could be represented by the same unit with a weight vector that is the average of a and b. This is exactly what the map is doing: it is finding the most descriptive representation of the data, given the available resources. 582 Daniel L. James, Risto Miikkulainen Note that it would be possible to determine the needed C = L:f:o Cj phoneme representation vectors directly from the input data set, and without any learning or a map structure at all, establish distributed representations on these vectors with the SARDNET algorithm. However, feature map learning is necessary ifthe number of available representation vectors is less than C. The topological organization of the map allows finding a good set of reusable vectors that can stand for different phonemes in different sequences, making the representation more efficient. 3.3 REPRESENTING SIMILARITY Not only are the representations densely packed on the map, they are also descriptive in the sense that similar sequences have similar representations. Figure 3 shows the final activation patterns on the 36-unit, 713-word map for six example words. The first two words, "misplacement" and "displacement," sound very similar, and are represented by very similar patterns on the map. Because there is only one m in "displacement" , it is mapped on the same unit as the initial m of "misplacement." Note that the two IDS are mapped next to each other, indicating that the map is indeed topological, and small changes in the input cause only small changes in the map representation. Note also how the units in this small map are reused to represent several different phonemes in different contexts. The other examples in figure 3 display different types of similarities with "misplacement". The third word, "miscarried", also begins with "mis", and shares that subpart of the representation exactly. Similarly, "repayment" shares a similar tail and "pessimist" the subsequence "mis" in a different part or the word. Because they appear in a different context, these subsequences are mapped on slightly different units, but still very close to their positions with "misplacement." The last word, "burundi" sounds very different, as its representation on the map indicates. Such descriptive representations are important when the map has to represent information that is incomplete or corrupted with noise. Small changes in the input sequence cause small changes in the pattern, and the sequence can still be recognized. This property should turn out extremely important in real-world applications of SARDNET, as well as in cognitive science models where confusing similar patterns with each other is often plausible behavior. 4 DISCUSSION AND FUTURE RESEARCH Because the sequence representations on the map are distributed, the number of possible sequences that can be represented in m units is exponential in m, instead of linear as in most previous sequential feature map architectures. This denseness together with the tendency to map similar sequences to similar representations should turn out useful in real-world applications, which often require scale-up to large and noisy data sets. For example, SARDNET could form the core of an isolated word recognition system. The word input would be encoded in durationnormalized sequences of sound samples such as a string of phonemes, or perhaps representations of salient transitions in the speech signal. It might also be possible to modify SARDNET to form a more continuous trajectory on the map so that SARDNET itself would take care of variability in word duration. For example, a SARDNEf: A Self-Organizing Feature Map for Sequences 583 (a) (b) (c) ~~t • ript.ItDt (d) (e) (f) Figure 3: Example map representations. sequence of redundant inputs could be reduced to a single node if all these inputs fall within the same neighborhood. Even though the sequence representations are dense, they are also descriptive. Category memberships are measured not by labels of the maximally responding units, but by the differences in the response patterns themselves. This sort of distributed representation should be useful in cognitive systems where sequential input must be mapped to an internal static representation for later retrieval and manipulation. Similarity-based reasoning on sequences should be easy to implement, and the sequence can be easily recreated from the activity pattern on the map. Given part of a sequence, SARDNET may also be modified to predict the rest of the sequence. This can be done by adding lateral connections between the nodes in the map layer. The lateral connections between successive winners would be strengthened during training. Thus, given part of a sequence, one could follow the strongest lateral connections to complete the sequence. 584 Daniel L. James, Risto Miikkulainen 5 CONCLUSION SARDNET is a novel feature map architecture for classifying sequences of input vectors. Each sequence is mapped on a distributed representation on the map, making it possible to pack a remarkable large number of category representations on a small feature map. The representations are not only dense, they also represent the similarities of the sequences, which should turn out useful in cognitive science as well as real-world applications of the architecture. Acknowledgments Thanks to Jon Hilbert for converting CELEX data into the International Phonetic Alphabet format used in the experiments. This research was supported in part by the National Science Foundation under grant #IRI-9309273. References Chappel, G. J., and Taylor, J. G. (1993). The temporal Kohonen map. Neural Networks, 6:441-445. Kangas, J. (1991). Time-dependent self-organizing maps for speech recognition. In Proceedings of the International Conference on Artificial Neural Networks (Espoo, Finland), 1591-1594. Amsterdam; New York: North-Holland. Kohonen, T. (1989). Self-Organization and Associative Memory. Berlin; Heidelberg; New York: Springer. Third edition. Kohonen, T . (1990) . The self-organizing map. Proceedings of the IEEE, 78:14641480. Samarabandu, J. K., and Jakubowicz, O. G. (1990). Principles of sequential feature maps in multi-level problems. In Proceedings of the International Joint Conference on Neural Networks (Washington, DC), vol. II, 683-686. Hillsdale, NJ: Erlbaum. Scholtes, J. C. (1991). Recurrent Kohonen self-organization in natural language processing. In Proceedings of the International Conference on Artificial Neural Networks (Espoo, Finland), 1751-1754. Amsterdam; New York: NorthHolland. van Harmelen, H. (1993). Time dependent self-organizing feature map for speech recognition. Master's thesis, University of Twente, Enschede, the Netherlands. Zandhuis, J. A. (1992). Storing sequential data in self-organizing feature maps. Internal Report MPI-NL-TG-4/92, Max-Planck-Institute fur Psycholinguistik, Nijmegen, the Netherlands.	1994	78
974	Deterministic Annealing Variant of the EM Algorithm N aonori U eda Ryohei N alcano ueda@cslab.kecl.ntt.jp nakano@cslab.kecl.ntt.jp NTT Communication Science Laboratories Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02 Japan Abstract We present a deterministic annealing variant of the EM algorithm for maximum likelihood parameter estimation problems. In our approach, the EM process is reformulated as the problem of minimizing the thermodynamic free energy by using the principle of maximum entropy and statistical mechanics analogy. Unlike simulated annealing approaches, this minimization is deterministically performed. Moreover, the derived algorithm, unlike the conventional EM algorithm, can obtain better estimates free of the initial parameter values. 1 INTRODUCTION The Expectation-Maximization (EM) algorithm (Dempster, Laird & Rubin, 1977) is an iterative statistical technique for computing maximum likelihood parameter estimates from incomplete data. It has generally been employed to a wide variety of parameter estimation problems. Recently, the EM algorithm has also been successfully employed as the learning algorithm of the hierarchical mixture of experts (Jordan & Jacobs, 1993). In addition, it has been found to have some relationship to the learning of the Boltzmann machines (Byrne, 1992). This algorithm has attractive features such as reliable global convergence, low cost per iteration, economy of storage, and ease of programming, but it is not free from problems in practice. The serious practical problem associated with the algorithm 546 Naonori Veda, Ryohei Nakano is the local maxima problem. This problem makes the performance dependent on the initial parameter value. Indeed, the EM algorithm should be performed from as wide a choice of starting values as possible according to some ad hoc criterion. To overcome this problem, we adopt the principle of statistical mechanics. Namely, by using the principle of maximum entropy, the thermodynamic free energy is defined as an effective cost function that depends on the temperature. The maximization of log-likelihood is done by minimizing the cost function. Unlike simulated annealing (Geman & Geman, 1984) where stochastic search is performed on the given energy surface, this cost function is deterministically optimized at each temperature. Such deterministic annealing (DA) approach has been successfully adopted for vector quantization or clustering problems (Rose et al., 1992; Buhmann et al., 1993; Wong, 1993). Recently, Yuile et al.(Yuile, Stolorz, & Utans, 1994) have shown that the EM algorithm can be used in conjunction with the DA. In our previous paper, independent ofYuile's work, we presented a new EM algorithm with DA for mixture density estimation problems (Ueda & Nakano, 1994). The aim of this paper is to generalize our earlier work and to derive a DA variant of the general EM algorithm. Since the EM algorithm can be used not only for the mixture estimation problems but also for other parameter estimation problems, this generalization is expected to be of value in practice. 2 GENERAL THEORY OF THE EM ALGORITHM Suppose that a measure space Y of "unobservable data" exists corresponding to a measure space X of "observable data". An observable data sample ~(E X) with density p(~; @) is called incomplete and (~, y) with joint density p(~, y;@) is called complete, where y is an unobservable data sample 1 corresponding to~. Note that ~ E'R,n and y E 'R,m. @ is parameter of the density distribution to be estimated. Given incomplete data samples X = {~klk = 1"", N}, the goal of the EM algorithm is to compute the maximum-likelihood estimate of @ that maximizes the following log-likelihood function: N L(@;X) = Llogp(~k;@)' (1) 1:=1 by using the following complete data log-likelihood function: N Le(@;X) = I: logp(zk' Yk;@) ' (2) k=l In the EM algorithm, the parameter @ is iteratively estimated. Suppose that @(t) denotes the current estimate of @ after the tth iteration of the algorithm. Then @(t+1) at the next iteration is determined by the following two steps: 1 In such unsupervised learning as mixture problems, 1/ reduces to an integer value (1/ E {I, 2, ... , C}, where C is the number of components), indicating the component from which a data sample x originates. Deterministic Annealing Variant of the EM Algorithm 547 E-step: Compute the Q-function defined by the conditional expectation of the complete data log-likelihood given X and @(t): Q(@; @(t» ~ E{Lc(@;X)IX, @(t)}. (3) M-step: Set @(t+l) equal to @ which maximizes Q(@, @(t». It has theoretically been shown that an iterative procedure for maximizing Q over @ will cause the likelihood L to monotonically increase, e.g., L(@(t+l» 2: L(@(t». Eventually, L(@(t» converges to a local maximum. The EM algorithm is especially useful when the maximization of the Q-function can be more easily performed than that of L. By substituting Eq. 2 into Eq. 3, we have N N Q(@;@{t» = ~ J ... J{logp(~j:'Yj:;@)} IIp(Yjl~j;@(t))dYl . .. dYN k=l j=l N ~ J {logp(~k' Yk; @)}p(Ykl~k; @(t»dYk' k=l (4) @ that maximizes Q(@; @(t» should satisfy aQla@ = 0, or equivalently, N J a ~ {a@ logp( ~k, Yk; @) }P(Yk I~k; @(t»dYk = o. k=l (5) Here, p(Ykl~k' @(t» denotes the posterior and can be computed by the following Bayes rule: (6) It can be interpreted that the missing information is estimated by using the posterior. However, because the reliability of the posterior highly depends on the parameter @(t), the performance of the EM algorithm is sensitive to an initial parameter value @(O). This has often caused the algorithm to become trapped by some local maxima. In the next section, we will derive a new variant of the EM algorithm as an attempt at global maximization of the Q-function in the EM process. 3 DETERMINISTIC ANNEALING APPROACH 3.1 DERIVATION OF PARAMETERIZED POSTERIOR Instead of the posterior given in Eq. 6, we introduce another posterior f(Ykl~k)' The function form of f will be specified later. Using f(Ykl~k)' we consider a new function instead of Q, say E, defined as: N E ~r ~ J {-logp(~k' Yk; @)}f(Yk l~k)dYk' k=l (7) 548 Naonori Ueda. Ryohei Nakano (N ote: E is always nonnegative.) One can easily see that (-E) is also the conditional expectation of the complete data log-likelihood but it differs from Q in that the expectation is taken with respect to !(lIk I:l:k) instead of the posterior given by Eq. 6. In other words, if !(Ykl:l:k) = P(lIkl:l:k; B(t», then E = -Q. Since we do not have a priori knowledge about !(lIk I:l:k), we apply the principle of maximum entropy to specify it. That is, by maximizing the entropy given by: N H = - L J {log !(1I k I:l:k)} !(Yk I:l:k )dllb (8) k=1 with respect to !, under the constraints of Eq. 7 and J !dYk = 1, we can obtain the following Gibbs distribution: 1 !(lIk I:l:k) = exp{ -{J( -logp(:l:k.lIk; B»)}, (9) Z:l:" where Z:I:" = J exp{ -{J( -logp(:l:k, 11k; B»)}dllb and is called the partition function. The parameter {J is the Lagrange multiplier determined by the value E. From an analogy of the annealing, 1 j {J corresponds to the "temperature". By simplifying Eq. 9, we obtain a new posterior parameterized by {J, !( I) p(:l:k, 11k; B)fJ (10) lIk:l:k = J p(:l:k, 11k; B)fJdllk . Clearly, when {J = 1, !(lIkl:l:k) reduces to the original posterior given in Eq. 6. The effect of {J will be explained later. Since :1:1, ..• ,:l:N are identically and independently distributed observations, the partition function Zp(B) for X becomes Ok Z:l:". Therefore, N Zp(B) = II J P(lIbllk; B)Pdllk · (11) k=l Once the partition function is obtained explicitly, using statistical mechanics analogy, we can define the free energy as an effective cost function that depends on the temperature: (12) At equilibrium, it is well known that a thermodynamic system settles into a configuration that minimizes its free energy. Hence, B should satisfy oFp(B)joB = O. It follows that (13) Interestingly, We have arrived at the same equation as the result ofthe maximization of the Q-function, except that the posterior P(lIk I:l:k; B(t» in Eq. 5 is replaced by !(lIk I:l:k). Deterministic Annealing Variant of the EM Algorithm 549 3.2 ANNEALING VARIANT OF THE EM ALGORITHM Let Qf3(@; @(I» be the expectation of the complete data log-likelihood by the parameterized posterior f(y" I~,,). Then, the following deterministic annealing variant of the EM algorithm can be naturally derived to maximize -Ff3(@). [Annealing EM (AEM) algorithm] 1. Set {3 +- {3min(O < {3min <t:: 1). 2. Arbitrarily choose an initial estimate @(O). Set t +- O. 3. Iterate the following two steps until convergence2 : E-step: Compute N J p(~" y . @(t»f3 QfJ(@;@(t» = ~ {logp(~", y,,; @)} , '" dy". "=1 I p(~", y";@(t» f3dy,, (14) M-step: Set @(t+l) equal to @ which maximizes Qf3(@; @(t». 4. Increase {3. 5. If {3 < {3max, set t +- t + 1, and repeat from step 3; otherwise stop. One can see that in the proposed algorithm, an outer loop is added to the original EM algorithm for the annealing process. An important distinction to keep in mind is that unlike simulated annealing, the optimization in step 3 is deterministically performed at each {3. Now let's consider the effect of the posterior parameterization of Eq. 10. The annealing process begins at small {3 (high temperature). Clearly, at this time, since f(y"I~,,) becomes uniform, -Ff3(@) has only one global maximum. Hence, the maximum can be easily found. Then by gradually increasing {3, the influence of each ~" is gradually localized. At {3 > 0, function Qf3 will have several local maxima. However, at each step, it can be assumed that the new global maximum is close to the previous one. Hence, by this assumption, the algorithm can track the global maximum at each {3 while increasing {3. Clearly, when {3 = 1 the parameterized posterior coincides with the original one. Moreover, noting that -Fl(@) = L(@), {3max ought be one. 4 Demonstration To visualize how the proposed algorithm works, we consider a simple onedimensional, two-component normal mixture problem. The mixture is given by p(x;m},m2) = 0.3kexp{-~(x - ml?} + 0.7*exp{-~(x - m2)2}. In this case, @ = (mb m2), y" E {I, 2}, and therefore, the joint density p(x, 1; ml, m2) = 0.3$ exp{ -~(x - ml)2}, while p(x, 2; ml, m2) = 0.7 $ exp{ -~(x - m2)2}. One hundred samples in total were generated from this mixture with ml = -2 and m2 = 2. Figure 1 shows contour plots of the -FfJ(ml,m2)/N surface. It is interesting to see how FfJ(m}, m2) varies with {3. One can see that a finer and truer structure emerges by increasing {3. Note that as explained before, when {3 = 1, 2When the sequence converges to a saddle point (e.g., when the Hessian matrix of -Fp(9) has at least one positive eigen value), a local line search in the direction of the eigen vector corresponding to the largest eigen value should be performed to escape the solution from the saddle point. 550 Naonori Ueda, Ryohei Nakano -4 -2 0 2 -2 0 2 -4 -2 0 2 (a) ~ .. 0.1 (b) ~ - 0.25 (c) ~ '" 0.3 -4 -2 0 2 4 ml -4 -2 0 2 4 ml -4 -2 0 2 (d) ~ = 0.5 (e) ~ .. 0.7 (f) ~ = 1 Figure 1: Contour plots of -Fp(mhm2)fN surface. ("+" denotes a global maximum at each JJ.) -4 -2 0 2 4 ml -4 -2 0 2 4 ml (a) (~),m~~ = (4, -1) (b) (m(~),m~~ =(-2,-4) Figure 2: Trajectories for EM and AEM procedures. Deterministic Annealing Variant of the EM Algorithm 551 -Fl(ml, m2) == L(mb m2). The maximum of -Fl (or L) occurs at m1 = -2.0 and m2 = 1.9. As initial points, (m~O),m~O» = (4,-1),(-2,-4) were tested. Note that these points are close to a local maximum. Figure 2 shows how both algorithms converge from these starting points3. The original EM algorithm converges to (mp2), m~12» = (2.002, -1.687) (Figure 2( a» and to (m~19), m~19» = (1.999, -1.696) (Figure 2(b». In contrast, the proposed AEM algorithm converges to (m~43), m~43» = (-2.022,1.879) (Figure 2(a» and (m~43),m~43» = (-2.022,1.879) (Figure 2(b». Since these starting points are close to the local maximum point, the original EM algorithm becomes trapped by the local maximum point, while the proposed algorithm successfully converges to the global maximum in both cases. 5 Discussion A new view of the EM algorithm has recently been presented (Hathaway, 1986; Neal & Hinton, 1993). This view states that the EM steps can be regarded as a grouped version of the method of coordinate ascent of the following objective function: N N J ~f I: E p(lIlo) {logp(:l:l:, 111:; @)} - I: E p(lIlo) {logp(lIl:)} (15) I: I: over P(lIl:) and @. That is, the E-step corresponds to the maximization of J with respect to P(lIl:) with fixed @, while the M-step corresponds to that of J with respect to @ with fixed P(lIl:)' Neal & Hinton mentioned that apart from the sign change, J is analogous to the "free energy" well known in statistical physics. It is worth mentioning another interpretation of the derived parameterized posterior; i.e., it plays a central role in the AEM algorithm. Taking the logarithm on both sides of Eq. 10, we have (16) Moreover, taking the conditional expectation4 given :1:1:, summing over k, and using Eq. 12, we have N 1 N FfJ(@) = I: EJ(lIlol:l:lo){ -logp(:l:I:, 111:; @)}+ p I: EJ(lIlo 1:1:10) {log 1(111: 1:l:1:)}· (17) 1:=1 1:=1 From Eqs. 7 and 8, Eq. 17 can be rewritten as FfJ(@) = E - !H. Noting that l/f3 corresponds to the "temperature", one can see that this expression exactly agrees with the free energy, while Eq. 15 is completely without the "temperature". In other words, FfJ(@) can be interpreted as an annealing variant of -J. Clearly, when f3 = 1, FfJ(@) == -J. 3 Although the AEM procedure was actually performed for successive fJ (fJnew fJ x 1.4), the results a.re superimposed on -Fl(ml, m2)/N surfa.ce for convenience. 4Since the LHS of Eq. 16 is independent of 111" it does not cha.nge a.fter expecta.tion. 552 Naonori Veda. Ryohei Nakano The proposed algorithm is also applicable to the learning of the (Generalized) Radial Basis Function (RBF, GRBF) networks. Indeed, Nowlan (Nowlan, 1990), for instance, proposes a maximum likelihood competitive learning algorithm for the RBF networks. In his study, "soft competition" and "hard competition" are eXperimentally compared and it is shown that the soft competition can give better performance. In our algorithm, on the other hand, the soft model exactly corresponds to the case f3 = 1, while the hard model corresponds to the case f3 -+ 00. Consequently, both models can be regarded as special cases in our algorithm. Acknowledgements We would like to thank Dr. Tsukasa Kawaoka, NTT CS labs, for his encouragement. References J. Buhmann & H. Kuhnel. (1993) Complexity optimized data clustering by competitive neural networks. Neural Computation, 5:75-88. W. Byrne. (1992) Alternating minimization and Boltzmann machine learning. IEEE Trans. Neural Networks, 3:612-620. A. P. Dempster, N. M. Laird & D. B. Rubin. (1977) Maximum-likelihood from incomplete data via the EM algorithm. J. Royal Statist. Soc. Ser. B (methodological),39:1-38. S. Geman & D. Geman. (1984) Stochastic relaxation, Gibbs distribution and the Baysian restortion in images. IEEE Trans. Pattern Anal. Machine Intel/" 6,6:721741. R. J. Hathaway. (1986) Another interpretation of the EM algorithm for mixture distributions. Statistics fj Probability Leiters, 4: 53-56. M. 1. Jordan & R. A. Jacob. (1993) Hierarchical mixtures of experts and the EM algorithm. MIT Dept. of Brain and Cognitive Science preprint. R. M. Neal & G. E. Hinton. (1993) A new view of the EM algorithm that justifies incremental and other variants. submitted to Biometrika. S. J. Nowlan. (1990) Maximum likelihood competitve learning. in D. S. Touretzky et al. eds., Advances in Neural Information Systems 2, Morgan Kaufmann. 574-582. K. Rose, E. Gurewitz & G. C. Fox. (1992) Vector quantization by deterministic annealing. IEEE Trans. Information Theory, 38,4:1249-1257. N. Ueda & R. Nakano. (1994) Mixture density estimation via EM algorithm with deterministic annealing. in proc. IEEE Neural Networks for Signal Processing, 69-77. Y. Wong. (1993) Clustering data by melting. Neural Computation, 5:89-104. A. 1. Yuille, P. Stolorz & J. Utans. (1994) Statistical physics, mixtures of distributions, and the EM algorithm. Neural Computation, 6:334-340.	1994	79
975	A Novel Reinforcement Model of Birdsong Vocalization Learning Kenji Doya ATR Human Infonnation Processing Research Laboratories 2-2 Hikaridai, Seika, Kyoto 619-02, Japan Abstract Terrence J. Sejnowski Howard Hughes Medical Institute UCSD and Salk Institute, San Diego, CA 92186-5800, USA Songbirds learn to imitate a tutor song through auditory and motor learning. We have developed a theoretical framework for song learning that accounts for response properties of neurons that have been observed in many of the nuclei that are involved in song learning. Specifically, we suggest that the anteriorforebrain pathway, which is not needed for song production in the adult but is essential for song acquisition, provides synaptic perturbations and adaptive evaluations for syllable vocalization learning. A computer model based on reinforcement learning was constructed that could replicate a real zebra finch song with 90% accuracy based on a spectrographic measure. The second generation of the birdsong model replicated the tutor song with 96% accuracy. 1 INTRODUCTION Studies of motor pattern generation have generally focussed on innate motor behaviors that are genetically preprogrammed and fine-tuned by adaptive mechanisms (Harris-Warrick et al., 1992). Birdsong learning provides a favorable opportunity for investigating the neuronal mechanisms for the acquisition of complex motor patterns. Much is known about the neuroethology of bird song and its neuroanatomical substrate (see Nottebohm, 1991 and Doupe, 1993 for reviews), but relatively little is known about the overall system from a computational viewpoint. We propose a set of hypotheses for the functions of the brain nuclei in the song system and explore their computational strength in a model based on biological constraints. The model could reproduce real and artificial birdsongs in a few hundred learning trials. 102 Kenji Doya, Terrence 1. Sejnowski ----+ Direct Motor Pathway ----+ Anterior Forebrain Pathway Figure 1: Major songbird brain nuclei involved in song control. The dark arrows show the direct motor control pathway and the gray arrows show the anterior forebrain pathway. Abbreviations: Uva, nucleus uvaeforrnis of the thalamus; NIf, nucleus interface of the neostriatum; L, field L (primary auditory are of the forebrain); HV c, higher vocal center; RA, robust nucleus of the archistriatum; DM, dorso-medial part of the nucleus intercollicularis; nXllts, tracheosyringeal part of the hypoglossal nucleus; AVT, ventral area of Tsai of the midbrain; X, area X of lobus parolfactorius; DLM, medial part of the dorsolateral nucleus of the thalamus; LMAN, lateral magnocellular nucleus of the anterior neostriatum. 2 NEUROETHOLOGY OF BIRDSONG Although songs from individual birds of the same species may sound quite similar, a young male songbird learns to sing by imitating the song of a tutor, which is usually the father or another adult male in the colony. If a young bird does not hear a tutor song during a critical period, it will sing short, poorly structured songs, and if a bird is deafened in the period when it practices vocalization, it develops highly abnormal songs. These observations indicate that there are two phases in song learning: the sensory learning phase when a young bird memorizes song templates and the motor learning phase in which the bird establishes the motor programs using auditory feedback (Konishi, 1965). These two phases can be separated by several months in some species, implying that birds have remarkable capability for memorizing complex temporal sequences. Once a song is crystallized, its pattern is very stable. Even deafening the bird has little immediate effect. The brain nuclei involved in song learning are shown in Figure 1. The primary motor control pathway is composed ofUva, NIf, HVc, RA, DM, and nXllts. If any of these nuclei is lesioned, a bird cannot sing normally. Experimental studies suggest that HVc is involved in generating syllable sequences and that RA produces motor commands for each syllable (Vu et aI., 1994). Interestingly, neurons in HVc, RA and nXllts show vigorous auditory responses, suggesting that the motor control system is closely coupled with the auditory system (Nottebohm, 1991). There is also a "bypass" from HV c to RA which consists of area X, DLM, and LMAN called the anterior forebrain pathway (Doupe, 1993). This pathway is not directly involved A Novel Reinforcement of Birdsong Vocalization Learning auditory syllable encoding preprocessing sequence generation motor pattern generation reinforcement attention memory of tutor song normalized evaluation synaptic perturbation gradient estimate 103 Figure 2: Schematic of primary song control nuclei and their proposed functions in the present model of bird song learning. in vocalization because lesions in these nuclei in adult birds do not impair their crystallized songs. However, lesions in area X and LMAN during the motor learning phase result in contrasting deficits. The songs of LMAN-Iesioned birds crystallize prematurely, whereas the songs of area X-Iesioned birds remain variable (Scharff and Nottebohm, 1991). It has been suggested that this pathway is responsible for the storage of song templates (Doupe and Konishi, 1991) or guidance of the synaptic connection from HVc to RA (Mooney, 1992). 3 FUNCTIONAL NEUROANATOMY OF BIRDSONG The song learning process can be decomposed into three stages. In the first stage, suitable internal acoustic representations of syllables and syllable combinations are constructed. This "auditory template" can be assembled by unsupervised learning schemes like clustering and principal components analysis. The second stage involves the encoding of phonetic sequences using the internal representation. If the representation is sparse or nearly orthogonal, sequential transition can be easily encoded by Hebbian learning. The third stage is an inverse mapping from the internal auditory representation into spatio-temporal patterns of motor commands. This can be accomplished by exploration in the space of motor commands using reinforcement learning. The responses of the units that encode the acoustic primitives can be used to the evaluate the resulting auditory signal and direct the exploration. How are these three computational stages organized within the brain areas and pathways of the songbird? Figure 2 gives an overview of our current working hypothesis. Auditory inputs are pre-processed in field L. Some higher-order representations, such as syllables and syllable combinations, are established in HVc depending on the bird's auditory experience. Moreover, transitions between syllables are encoded in the HVc network. The sequential activation of syllable coding units in HVc are transformed into the time course of motor commands in RA. DM and nXIIts control breathing and the muscles in syrinx, bird's vocal organ. 104 Kenji Doya, Terrence 1. Sejnowski a bronchus Figure 3: (a) The syrinx of songbirds. (b) The model syrinx. The consequences of selective lesions of areas in the anterior forebrain pathway (Scharff and Nottebohm, 1991) are consistent with the failures expected for a reinforcement learning system. In particular, we suggest that this pathway serves the function of an adaptive critic with stochastic search elements (Barto et al., 1983). We propose that LMAN perturbs the synaptic connections from HV c to RA and area X regulates LMAN by the song evaluation. Modulation of HVc to RA connection by LMAN is biologically plausible since LMAN input to RA is mediated mainly by NMDA type synapses, which can modulate the amplitude of mainly non-NMDA type synaptic input from HVc (Mooney, 1992). The assumption that area X provides evaluation is supported by the fact that it receives catecholaminergic projection (dopamine of norepinephrine) from a midbrain nucleus AVT (Lewis et al., 1981). These neurotransmitters are used in many species for reinforcement or attention signals. It is known that auditory learning is enhanced when associated with visual or social interaction with the tutor. Area X is a candidate region where auditory inputs from HVc are associated with reinforcing input from AVT during auditory learning. 4 CONSTRUCTION OF SONG LEARNING MODEL In order to test the above hypothesis, we constructed a computer model of the bird song learning system. The specific aim was to simulate the process of explorative motor learning, in which the time course of motor command for each syllable is determined by auditory template matching. We assumed that orthogonal representations for syllables and their sequential activation were already established in HVc and that an auditory template matching mechanism exists in area X. 4.1 The syrinx The bird's syrinx is located near the junction of the trachea and the bronchi (Vicario, 1991). Its sound source is the tympani form membrane which faces to the bronchus on one side and the air sac on the other (Figure 3a). When some of the syringeal muscles contract, the lumen of the bronchus is throttled and produces vibration in the membrane. When stretched along one dimension, the membrane produces harmonic sounds, but when stretched along two dimensions, the sound contains non-harmonic components (Casey and Gaunt, 1985). Accordingly, we provided two sound sources for the model syrinx (Figure 3b). The fundamental frequency of the harmonic component was controlled by the membrane tension in one direction (Tl). The amplitu6e of the noisy component was proportional to the A Novel Reinforcement of Birdsong Vocalization Learning 105 Ex ! ~Th ! • ! ~T' • ~ T2 HVc RA OM nXlIts Figure 4: RA units with different spatio-temporal output profiles are driven by locallycoded RVc units. LMAN perturbs the weights W between RVc and RA. The output units in DM and nXIIts drive the model syrinx. membrane tension in an orthogonal direction (T2). Mixture of these sounds went through a bandpass filter whose resonance frequency was controlled by the throttling of the bronchus (Th). The overall sound amplitude was determined by the strength of expiration (Ex). By controlling the time course of these four variables (Ex,Th,Tl,T2), the model could produce wide variety of "bird-like" chirps and warbles. 4.2 Motor pattern generation in RA RA is topographically organized, each part projecting to different motoneuron pools in nXIIts (Vicario, 1991). Also, RA neurons have complex temporal responses to the inputs from RVc (Mooney, 1992). Therefore, we assumed that RA consists of groups of neurons with specific spatial and temporal output properties, as shown in Figure 4. For each of the four motor command variables, we provided several units with different temporal response kernels. The sequential activation of syllable coding units in RVc drove the RA units through the weights W. Their responses were linearly combined and squashed between 0 and 1 to make the final motor commands. 4.3 Weight space search by LMAN and area X With the above model of the motor output, the task is to find a connection matrix W that maximizes a template matching measure. One way for doing this is to perturb the output of the units and correlate it with the input and the evaluation (Barto et aI., 1983). An alternative way, adopted here, is to perturb the weights and correlate them with the evaluation. We used the following stochastic gradient ascent algorithm. The weight matrix W is modulated by ~ W, given by the sum of the evaluation gradient estimate G and a random component. The modulated weight persists if the resulting vocalization is better than the recent average evaluation E[r]. The evaluation gradient estimate G is updated by the sum 106 Kenji Doya, Terrence J. Sejnowski of the perturbations ~ W weighted with the normalized evaluation. ~ W := G + random perturbation r := evaluation of the song generated with W + ~ W W:= W+~W if r > E[r] r - E[r] G := 0: JVR" ~W + (1- o:)G, V[r] where 0 < 0: < 1 provides a form of "momentum" in weight space similar to that used in supervised learning. The average and the variance of evaluation are also estimated on-line as follows. E[r] := o:r + (1 0: )E[r] V[r] := o:(r - E[r]? + (1 - o:)V[r] . 4.4 Spectrographic template matching We assumed that evaluation for vocalization is given separately for each of the syllables in a song. The sound signal was analyzed by an 80 channel spectrogram. Each output channel was sent to an analog delay line similar to the gamma filter (de Vries and Principe, 1992). The snapshot image of this (80 channels) x (12 steps) delay line at the end of each syllable was stored as the template. The same delay line image of the syllable generated by the model was compared with the template. This allowed some compensation for variable syllable duration. The direction cosine between the two delay line images was used for the reinforcement signal, r. 5 SIMULATION RESULTS One phrase of a recorded zebra finch song (Figure 5a) was the target. Templates were stored for the five syllables in the phrase. Five HVc units coded the five different syllables and 16 RA units represented the four output variables and four different temporal kernels. Learning was started with small random weights. After 200 to 300 trials, the syllable evaluation by direction cosine reached 0.9 (Figure 5d, solid line). The syllables of the learned song resembled the overall frequency profiles of the original syllables. The complex spectrographic structure of the original syllables were, however, not accurate (Figure 5b). One reason for this imperfect replication could be the difference between the vocal organs of the real zebra finch syrinx and our model syrinx. In order to check the significance of this difference, we took syllable templates from the model song (Figure 5b) and trained another model with random initial weights. In this case, the direction cosine went up to 0.96 (Figure 5d, dotted line) and they sounded quite similar to human ears (Figure 5c). We also checked the importance of the gradient estimate G in our algorithm. The dashed line in Figure 5d shows the performance of the model with G = 0: a simple random walk learning. The learning was hopelessly slow and resembles the deficit seen after lesion of area X (Figure 2). A Novel Reinforcement of Birdsong Vocalization Learning 107 1.0 ~-~--~--~-~---, d trilla zebra finch JOng model song - - - - random walk Figure 5: Spectrograms of (a) the original zebra finch song, (b) the learned song based on the tutor in (a), and (c) the second generation learned song based on the tutor in (b). (d) Learning curves for two tutors: a zebra finch song (solid line) and a model song (dotted line) compared with an undirected search in weight space (dashed line). Weight perturbation was given by a Gaussian distribution with (J = 0.1. The averaging parameter was a = 0.2. Simulating 500 trials took 30 minutes on Sparc Station 10. 6 Discussion We have assumed that each vocalized syllable was separately evaluated. If the evaluation is given only at the end of one song or a phrase, learning can be much more difficult because of the temporal credit assignment problem. If we assume that birds take the easiest strategy available, there should be syllable specific evaluation and separate perturbation mechanisms. In some songbirds, individual syllables are practiced out of order at an early stage, and only later is the sequence matched to the auditory template. Selectivity of auditory responses in both HVc and area X develop during motor learning (Volman, 1993; Doupe, 1993). We can expect such change in response tuning in area X if the evaluations of syllables or syllable sequences are normalized with respect to recent average performance, as we assumed in our model. Many simplifying assumptions were made in the present model: syllables were unary coded in HVc; simple spectrographic template matching was used; the number of motor output variables and temporal kernels were fairly small; and the sound synthesizer was much simpler than a real syrinx. However, it is not difficult to replace these idealizations with more biologically accurate models. Since the number of learning trials needed in the present model was much less than in the real birdsong learning (tens of thousands of trials), there is margin for further elaboration. 108 Kenji Doya, Terrence 1. Sejnowski Acknowledgments We thank M. Lewicki for the zebra finch song data and M. Konishi, A. Doupe, M. Lewicki, E. Vu, D. Perkel and G. Striedter for their helpful discussions. References Barto, A. G., Sutton, R. S., and Anderson, C. W. (1983). Neuronlike adaptive elements that can solve difficult learning control problems. IEEE Transactions on System, Man, and Cybernetics, SMC-13:834-846. Casey, R. M. and Gaunt, A. S. (1985). Theoretical models of the avian syrinx. Journal of Theoretical Biology, 116:45-64. de Vries, B. and Principe, J. C. (1992). The gamma model-A new neural model for temporal processing. Neural Networks, 5:565-576. Doupe, A. J. (1993). A neural circuit specialized for vocal learning. Current Opinion in Neurobiology, 3:104-111. Doupe, A. J. and Konishi, M. (1991). Song-selective auditory circuits in the vocal control system of the zebra finch. Proceedings of the National Academy of Sciences, USA, 88: 11339-11343. Harris-Warrick, R. M., Marder, E., Selverston, A. I., and Moulins, M. (1992). Dynamic Biological Networks-The Stomatogastric Nervous System. MIT Press, Cambridge, MA. Konishi, M. (1965). fhe role of auditory feedback in the control of vocalization in the white-crowned sparrow. Zeitschrift fur Tierpsychologie, 22:770-783. Lewis, J. W., Ryan, S. M., Arnold, A. P., and Butcher, L. L. (1981). Evidence for a catecholarninergic projection to area x in the zebra finch. Journal of Comparative Neurology, 196:347-354. Mooney, R. (1992). Synaptic basis of developmental plasticity in a birdsong nucleus. Journal of Neuroscience, 12:2464-2477. Nottebohm, F. (1991). Reassessing the mechanisms and origins of vocal learning in birds. Trends in Neurosciences, 14:206-211. Scharff, C. and Nottebohm, F. (1991). A comparative study of the behavioral deficits following lesions of various parts of the zebra finch song systems: Implications for vocal learning. Journal of Neuroscience, 11 :2896-2913. Vicario, D. S. (1991). Neural mechanisms of vocal production in songbirds. Current Opinion in Neurobiology, 1 :595-600. Volman, S. F. (1993). Development of neural selectivity for birdsong during vocal learning. Journal of Neuroscience, 13:4737--4747. Vu, E. T., Mazurek, M. E., and Kuo, Y.-C. (1994). Identification of a forebrain motor programming network for the learned song of zebra finches. Journal of Neuroscience, 14:6924-6934.	1994	8
976	A Non-linear Information Maximisation Algorithm that Performs Blind Separation. Anthony J. Bell tonylOsalk.edu Terrence J. Sejnowski terrylOsalk.edu Computational Neurobiology Laboratory The Salk Institute 10010 N. Torrey Pines Road La Jolla, California 92037-1099 and Department of Biology University of California at San Diego La Jolla CA 92093 Abstract A new learning algorithm is derived which performs online stochastic gradient ascent in the mutual information between outputs and inputs of a network. In the absence of a priori knowledge about the 'signal' and 'noise' components of the input, propagation of information depends on calibrating network non-linearities to the detailed higher-order moments of the input density functions. By incidentally minimising mutual information between outputs, as well as maximising their individual entropies, the network 'factorises' the input into independent components. As an example application, we have achieved near-perfect separation of ten digitally mixed speech signals. Our simulations lead us to believe that our network performs better at blind separation than the HeraultJ utten network, reflecting the fact that it is derived rigorously from the mutual information objective. 468 Anthony J. Bell, Terrence J. Sejnowski 1 Introduction Unsupervised learning algorithms based on information theoretic principles have tended to focus on linear decorrelation (Barlow & Foldiak 1989) or maximisation of signal-to-noise ratios assuming Gaussian sources (Linsker 1992). With the exception of (Becker 1992), there has been little attempt to use non-linearity in networks to achieve something a linear network could not. Non-linear networks, however, are capable of computing more general statistics than those second-order ones involved in decorrelation, and as a consequence they are capable of dealing with signals (and noises) which have detailed higher-order structure. The success of the 'H-J' networks at blind separation (Jutten & Herault 1991) suggests that it should be possible to separate statistically independent components, by using learning rules which make use of moments of all orders. This paper takes a principled approach to this problem, by starting with the question of how to maximise the information passed on in non-linear feed-forward network. Starting with an analysis of a single unit, the approach is extended to a network mapping N inputs to N outputs. In the process, it will be shown that, under certain fairly weak conditions, the N ---. N network forms a minimally redundant encoding ofthe inputs, and that it therefore performs Independent Component Analysis (ICA). 2 Information maximisation The information that output Y contains about input X is defined as: I(Y, X) = H(Y) - H(YIX) (1) where H(Y) is the entropy (information) in the output, while H(YIX) is whatever information the output has which didn't come from the input. In the case that we have no noise (or rather, we don't know what is noise and what is signal in the input), the mapping between X and Y is deterministic and H(YIX) has its lowest possible value of -00 . Despite this, we may still differentiate eq.l as follows (see [5]): 8 8 8w I(Y, X) = 8w H(Y) (2) Thus in the noiseless case, the mutual information can be maximised by maximising the entropy alone. 2.1 One input, one output. Consider an input variable, x, passed through a transforming function, g( x), to produce an output variable, y, as in Fig.2.1(a). In the case that g(x) is monotonically increasing or decreasing (ie: has a unique inverse), the probability density function (pdf) of the output fy(y) can be written as a function of the pdfofthe input fx(x), (Papoulis, eq. 5-5): f (y) = fx(x) (3) y 8y/8x A Non-Linear Information Maximization Algorithm That Performs Blind Separation 469 The entropy of the output, H(y), is given by: H(y) = -E[lnfy(y)] = -1: fy(y)lnfy(y)dy (4) where E[.] denotes expected value. Substituting eq.3 into eq.4 gives H(y) = E [In ~:] - E[lnfx(x)] (5) The second term on the right may be considered to be unaffected by alterations in a parameter, w, determining g(x). Therefore in order to maximise the entropy of y by changing w, we need only concentrate on maximising the first term, which is the average log of how the input affects the output. This can be done by considering the 'training set' of x's to approximate the density fx(x), and deriving an 'online', stochastic gradient descent learning rule: ~W ex oH = ~ (In OY) = (oy) -1 ~ (oy) OW OW OX OX OW ox (6) In the case of the logistic transfer function y = (1 + e- U )-l , u = wx + Wo in which the input is multiplied by a weight wand added to a bias-weight wo, the terms above evaluate as: wY(l - y) (7) y(l - y)(l + wX(l - 2y)) (8) Dividing eq.8 by eq.7 gives the learning rule for the logistic function, as calculated from the general rule of eq.6: 1 ~w ex - + x(l - 2y) W Similar reasoning leads to the rule for the bias-weight: ~wo ex 1- 2y (9) (10) The effect of these two rules can be seen in Fig. 1a. For example, if the input pdf fx(x) was gaussian, then the ~wo-rule would centre the steepest part of the sigmoid curve on the peak of fx(x), matching input density to output slope, in a manner suggested intuitively by eq.3. The ~w-rule would then scale the slope of the sigmoid curve to match the variance of fx (x). For example, narrow pdfs would lead to sharply-sloping sigmoids. The ~w-rule is basically anti-Hebbian1 , with an anti-decay term. The anti-Hebbian term keeps y away from one uninformative situation: that of y being saturated to 0 or 1. But anti-Hebbian rules alone make weights go to zero, so the anti-decay term (l/w) keeps y away from the other uninformative situation: when w is so small that y stays around 0.5. The effect of these two balanced forces is to produce an output pdf fy(y) which is close to the flat unit distribution, which is the maximum entropy distribution for a variable 11£ y = tanh(wx + wo) then ~w ex !;; - 2xy 470 Anthony 1. Bell, Terrence 1. Sejnowski 4 ~---' 0 --~---4"""",""::~---- X y (a) ----~~~--~----X ox (b) Figure 1: (a) Optimal information flow in sigmoidal neurons (Schraudolph et al 1992). Input x having density function fx(x), in this case a gaussian, is passed through a non-linear function g(x). The information in the resulting density, fy(Y) depends on matching the mean and variance of x to the threshold and slope of g( x). In (b) fy(y) is plotted for different values of the weight w. The optimal weight, Wopt transmits most information. bounded between 0 and 1. Fig. 1b illustrates a family of these distributions, with the highest entropy one occuring at Wopt . A rule which maximises information for one input and one output may be suggestive for structures such as synapses and photoreceptors which must position the gain of their non-linearity at a level appropriate to the average value and size of the input fluctuations. However, to see the advantages of this approach in artificial neural networks, we now analyse the case of multi-dimensional inputs and outputs. 2.2 N inputs, N outputs. Consider a network with an input vector x, a weight matrix Wand a monotonically transformed output vector y = g(Wx + wo). Analogously to eq.3, the multivariate probability density function of y can be written (Papoulis, eq. 6-63): f ( ) = fx(x) y y IJI (11) where IJI is the absolute value ofthe Jacobian of the transformation. The Jacobian is the determinant of the matrix of partial derivatives: [ ~ ~] 8Xl 8x n J = det: : ~ ~ 8Xl 8xn (12) The derivation proceeds as in the previous section except instead of maximising In(8y/8x), now we maximise In IJI. For sigmoidal units, y = (1 + e-U)-l, U = A Non-Linear Information Maximization Algorithm That Performs Blind Separation 471 Wx + Wo, the resulting learning rules are familiar in form: D. W ex: [WTrl + x(1 - 2y? D.wo ex: 1- 2y (13) (14) except that now x, y, Wo and 1 are vectors (1 is a vector of ones), W is a matrix, and the anti-Hebbian term has become an outer product. The anti-decay term has generalised to the inverse of the transpose of the weight matrix. For an individual weight, Wij, this rule amounts to: cof Wij D.wij ex: det W + xj(l - 2Yi) (15) where cof Wij, the cofactor of Wij, is (-1 )i+j times the determinant of the matrix obtained by removing the ith row and the jth column from W. This rule is the same as the one for the single unit mapping, except that instead of W = 0 being an unstable point of the dynamics, now any degenerate weight matrix is unstable, since det W = 0 if W is degenerate. This fact enables different output units, Yi, to learn to represent different components in the input. When the weight vectors entering two output units become too similar, det W becomes small and the natural dynamic of learning causes these weight vectors to diverge. This effect is mediated by the numerator, cof Wij. When this cofactor becomes small, it indicates that there is a degeneracy in the weight matrix of the rest of the layer (ie: those weights not associated with input Xj or output Yi). In this case, any degeneracy in W has less to do with the specific weight Wij that we are adjusting. 3 Finding independent components blind separation Maximising the information contained in a layer of units involves maximising the entropy of the individual units while minimising the mutual information (the redundancy) between them. Considering two units: (16) For I(Yl, Y2) to be zero, Yl and Y2 must be statistically independent of each other, so that fY1Y2(Yl, Y2) = fYl (ydfY2(Y2). Achieving such a representation is variously called factorial code learning, redundancy reduction (Barlow 1961, Atick 1992), or independent component analysis (ICA), and in the general case of continuously valued variables of arbitrary distributions, no learning algorithm has been shown to converge to such a representation. Our method will converge to a minimum redundancy, factorial representation as long as the individual entropy terms in eq.16 do not override the redundancy term, making an I(Yl, Y2) = 0 solution sub-optimal. One way to ensure this cannot occur is if we have a priori knowledge of the general form of the pdfs of the independent components. Then we can tailor our choice of node-function to be optimal for transmitting information about these components. For example, unit distributions require piecewise linear node-functions for highest H(y), while the more common gaussian forms require roughly sigmoidal curves. Once we have chosen our nodefunctions appropriately, we can be sure that an output node Y cannot have higher 472 slt--_---<x\. __ ---4!'.~ o o o )<----~ 1'-----..:·0 unknown mixing process BUND SEPARATION (learnt weights) Anthony 1. Bell, Terrence 1. SejnowskiWA after learning: '" 1-4.091 0.13 0.09 -0.07 -0.01 0.07 1-2.921 0.00 0.02 -0.06 0.02 -0.02 -0.06 -0.08 1-2.2ij 0.02 0.03 0.00 11.971 0.02 -0.07 0.14 1-3.501 -0.01 0.04 Figure 2: (a) In blind separation, sources, s, have been linearly scrambled by a matrix, A, to form the inputs to the network, x. We must recover the sources at our output y, by somehow inverting the mapping A with our weight matrix W. The problem: we know nothing about A or the sources. (b) A successful 'unscrambling' occurs when WA is a 'permutation' matrix. This one resulted from separating five speech signals with our algorithm. entropy by representing some combination of independent components than by representing just one. When this condition is satisfied for all output units, the residual goal, of minimising the mutual information between the outputs, will dominate the learning. See [5] for further discussion of this. With this caveat in mind, we turn to the problem of blind separation, (Jutten & Herault 1991), illustrated in Fig.2. A set of sources, Sl, ... , SN, (different people speaking, music, noise etc) are presumed to be mixed approximately linearly so that all we receive is N superpositions of them, Xl, ... , X N, which are input to our single-layer information maximisation network. Providing the mixing matrix, A, is non-singular then the original sources can be recovered if our weight matrix, W, is such that W A is a 'permutation' matrix containing only one high value in each row and column. Unfortunately we know nothing about the sources or the mixing matrix. However, if the sources are statistically independent and non-gaussian, then the information in the output nodes will be maximised when each output transmits one independent component only. This problem cannot be solved in general by linear decorrelation techniques such as those of (Barlow & Foldicik 1989) since second-order statistics can only produce symmetrical decorrelation matrices. We have tested the algorithm in eq.13 and eq.14 on digitally mixed speech signals, and it reliably converges to separate the individual sources. In one example, five separately-recorded speech signals from three individuals were sampled at 8kHz. Three-second segments from each were linearly mixed using a matrix of random values between 0.2 and 4. Each resulting mixture formed an incomprehensible babble. Time points were generated at random, and for each, the corresponding five mixed values were entered into the network, and weights were altered according to eq.13 and eq.14. After on the order of 500,000 points were presented, the network A Non-Linear Information Maximization Algorithm That Performs Blind Separation 473 had converged so that WA was the matrix shown in Fig.2b. As can be seen from the permutation structure of this matrix, on average, 95% of each output unit is dedicated to one source only, with each unit carrying a different source. Any residual interference from the four other sources was inaudible. We have not yet performed any systematic studies on rates of convergence or existence of local minima. However the algorithm has converged to separate N independent components in all our tests (for 2 ~ N ~ 10). In contrast, we have not been able to obtain convergence of the H-J network on our data set for N > 2. Finally, the kind of linear static mixing we have been using is not equivalent to what would be picked up by N microphones positioned around a room. However, (Platt & Faggin 1992) in their work on the H-J net, discuss extensions for dealing with time-delays and non-static filtering, which may also be applicable to our methods. 4 Discussion The problem of Independent Component Analysis (ICA) has become popular recently in the signal processing community, partly as a result ofthe success ofthe H-J network. The H-J network is identical to the linear decorrelation network of (Barlow & Foldicik 1989) except for non-linearities in the anti-Hebb rule which normally performs only decorrelation. These non-linearities are chosen somewhat arbitrarily in the hope that their polynomial expansions will yield higher-order cross-cumulants useful in converging to independence (Comon et aI, 1991). The H-J algorithm lacks an objective function, but these insights have led (Comon 1994) to propose minimising mutual information between outputs (see also Becker 1992). Since mutual information cannot be expressed as a simple function of the variables concerned, Comon expands it as a function of cumulants of increasing order. In this paper, we have shown that mutual information, and through it, ICA, can be tackled directly (in the sense of eq.16) through a stochastic gradient approach. Sigmoidal units, being bounded, are limited in their 'channel capacity'. Weights transmitting information try, by following eq.13, to project their inputs to where they can make a lot of difference to the output, as measured by the log of the Jacobian of the transformation. In the process, each set of statistically 'dependent' information is channelled to the same output unit. The non-linearity is crucial. If the network were just linear, the weight matrix would grow without bound since the learning rule would be: (17) This reflects the fact that the information in the outputs grows with their variance. The non-linearity also supplies the higher-order cross-moments necessary to maximise the infinite-order expansion of the information. For example, when y = tanh( u), the learning rule has the form D.. W ex: [wT] -1 - 2yxT , from which we can write that the weights stabilise (or (D..W) = 0) when 1= 2(tanh(u)uT ). Since tanh is an odd function, its series expansion is of the form tanh( u) = L:j bj u2P+1 , the bj being coefficients, and thus this convergence criterion amounts to the condition L:i,j bijp(U;P+1Uj) = 0 for all output unit pairs i 1= j, for p = 0,1,2,3 ... , 474 Anthony J. Bell, Terrence J. Sejnowski and for the coefficients bijp coming from the Taylor series expansion of the tanh function. These and other issues are covered more completely in a forthcoming paper (Bell & Sejnowski 1995). Applications to blind deconvolution (removing the effects of unknown causal filtering) are also described, and the limitations of the approach are discussed. Acknowledgements We are much indebted to Nici Schraudolph, who not only supplied the original idea in Fig.l and shared his unpublished calculations [13], but also provided detailed criticism at every stage of the work. Much constructive advice also came from Paul Viola and Alexandre Pouget. References [1] Atick J.J. 1992. Could information theory provide an ecological theory of sensory processing, Network 3, 213-251 [2] Barlow H.B. 1961. Possible principles underlying the transformation of sensory messages, in Sensory Communication, Rosenblith W.A. (ed), MIT press [3] Barlow H.B. & Foldicik P. 1989. Adaptation and decorrelation in the cortex, in Durbin R. et al (eds) The Computing Neuron, Addison-Wesley [4] Becker S. 1992. An information-theoretic unsupervised learning algorithm for neural networks, Ph.D. thesis, Dept. of Compo Sci., Univ. of Toronto [5] Bell A.J. & Sejnowski T.J. 1995. An information-maximisation approach to blind separation and blind deconvolution, Neural Computation, in press [6] Comon P., Jutten C. & Herault J. 1991. Blind separation of sources, part II: problems statement, Signal processing, 24, 11-21 [7] Comon P. 1994. Independent component analysis, a new concept? Signal processing, 36, 287-314 [8] Hopfield J.J. 1991. Olfactory computation and object perception, Proc. N atl. Acad. Sci. USA, vol. 88, pp.6462-6466 [9] Jutten C. & Herault J. 1991. Blind separation of sources, part I: an adaptive algorithm based on neuromimetic architecture, Signal processing, 24, 1-10 [10] Linsker R. 1992. Local synaptic learning rules suffice to maximise mutual information in a linear network, Neural Computation, 4, 691-702 [11] Papoulis A. 1984. Probability, random variables and stochastic processes, 2nd edition, McGraw-Hill, New York [12] Platt J .C. & Faggin F. 1992. Networks for the separation of sources that are superimposed and delayed, in Moody J.E et al (eds) Adv. Neur. Inf. Proc. Sys. 4, Morgan-Kaufmann [13] Schraudolph N.N., Hart W.E. & Belew R.K. 1992. Optimal information flow in sigmoidal neurons, unpublished manuscript	1994	80
977	Real-Time Control of a Tokamak Plasma Using Neural Networks Chris M Bishop Neural Computing Research Group Department of Computer Science Aston University Birmingham, B4 7ET, U.K. c.m.bishop@aston.ac.uk Paul S Haynes, Mike E U Smith, Tom N Todd, David L Trotman and Colin G Windsor AEA Technology, Culham Laboratory, Oxfordshire OX14 3DB (Euratom/UKAEA Fusion Association) Abstract This paper presents results from the first use of neural networks for the real-time feedback control of high temperature plasmas in a tokamak fusion experiment. The tokamak is currently the principal experimental device for research into the magnetic confinement approach to controlled fusion. In the tokamak, hydrogen plasmas, at temperatures of up to 100 Million K, are confined by strong magnetic fields. Accurate control of the position and shape of the plasma boundary requires real-time feedback control of the magnetic field structure on a time-scale of a few tens of microseconds. Software simulations have demonstrated that a neural network approach can give significantly better performance than the linear technique currently used on most tokamak experiments. The practical application of the neural network approach requires high-speed hardware, for which a fully parallel implementation of the multilayer perceptron, using a hybrid of digital and analogue technology, has been developed. 1008 C. Bishop, P. Haynes, M. Smith, T. Todd, D. Trotman, C. Windsor 1 INTRODUCTION Fusion of the nuclei of hydrogen provides the energy source which powers the sun. It also offers the possibility of a practically limitless terrestrial source of energy. However, the harnessing of this power has proved to be a highly challenging problem. One of the most promising approaches is based on magnetic confinement of a high temperature (107 - 108 Kelvin) plasma in a device called a tokamak (from the Russian for 'toroidal magnetic chamber') as illustrated schematically in Figure 1. At these temperatures the highly ionized plasma is an excellent electrical conductor, and can be confined and shaped by strong magnetic fields. Early tokamaks had plasmas with circular cross-sections, for which feedback control of the plasma position and shape is relatively straightforward. However, recent tokamaks, such as the COMPASS experiment at Culham Laboratory, as well as most next-generation tokamaks, are designed to produce plasmas whose cross-sections are strongly noncircular. Figure 2 illustrates some of the plasma shapes which COMPASS is designed to explore. These novel cross-sections provide substantially improved energy confinement properties and thereby significantly enhance the performance of the tokamak. z R Figure 1: Schematic cross-section of a tokamak experiment showing the toroidal vacuum vessel (outer D-shaped curve) and plasma (shown shaded). Also shown are the radial (R) and vertical (Z) coordinates. To a good approximation, the tokamak can be regarded as axisymmetric about the Z-axis, and so the plasma boundary can be described by its cross-sectional shape at one particular toroidal location. Unlike circular cross-section plasmas, highly non-circular shapes are more difficult to produce and to control accurately, since currents through several control coils must be adjusted simultaneously. Furthermore, during a typical plasma pulse, the shape must evolve, usually from some initial near-circular shape. Due to uncertainties in the current and pressure distributions within the plasma, the desired accuracy for plasma control can only be achieved by making real-time measurements of the position and shape of the boundary, and using error feedback to adjust the currents in the control coils. The physics of the plasma equilibrium is determined by force balance between the Real-Time Control of Tokamak Plasma Using Neural Networks circle ellipse O-shape bean Figure 2: Cross-sections of the COMPASS vacuum vessel showing some examples of potential plasma shapes. The solid curve is the boundary of the vacuum vessel, and the plasma is shown by the shaded regions. 1009 thermal pressure of the plasma and the pressure of the magnetic field, and is relatively well understood. Particular plasma configurations are described in terms of solutions of a non-linear partial differential equation called the Grad-Shafranov (GS) equation. Due to the non-linear nature of this equation, a general analytic solution is not possible. However, the GS equation can be solved by iterative numerical methods, with boundary conditions determined by currents flowing in the external control coils which surround the vacuum vessel. On the tokamak itself it is changes in these currents which are used to alter the position and cross-sectional shape of the plasma. Numerical solution of the GS equation represents the standard technique for post-shot analysis of the plasma, and is also the method used to generate the training dataset for the neural network, as described in the next section. However, this approach is computationally very intensive and is therefore unsuitable for feedback control purposes. For real-time control it is necessary to have a fast (typically:::; 50J.lsec.) determination of the plasma boundary shape. This information can be extracted from a variety of diagnostic systems, the most important being local magnetic measurements taken at a number of points around the perimeter of the vacuum vessel. Most tokamaks have several tens or hundreds of small pick up coils located at carefully optimized points around the torus for this purpose. We shall represent these magnetic signals collectively as a vector m. For a large class of equilibria, the plasma boundary can be reasonably well represented in terms of a simple parameterization, governed by an angle-like variable B, given by R(B) Z(B) Ro + a cos(B + 8 sinB) Zo + a/\,sinB where we have defined the following parameters (1) 1010 C. Bishop, P. Haynes, M. Smith, T. Todd, D. Trotman, C. Windsor Ro radial distance of the plasma center from the major axis of the torus, Zo vertical distance of the plasma center from the torus midplane, a minor radius measured in the plane Z = Zo, K elongation, 6 triangularity. We denote these parameters collectively by Yk. The basic problem which has to be addressed, therefore, is to find a representation for the (non-linear) mapping from the magnetic signals m to the values of the geometrical parameters Yk, which can be implemented in suitable hardware for real-time control. The conventional approach presently in use on many tokamaks involves approximating the mapping between the measured magnetic signals and the geometrical parameters by a single linear transformation. However, the intrinsic non-linearity of the mappings suggests that a representation in terms of feedforward neural networks should give significantly improved results (Lister and Schnurrenberger, 1991; Bishop et a/., 1992; Lagin et at., 1993). Figure 3 shows a block diagram of the control loop for the neural network approach to tokamak equilibrium control. Neural Network Figure 3: Block diagram of the control loop used for real-time feedback control of plasma position and shape. 2 SOFTWARE SIMULATION RESULTS The dataset for training and testing the network was generated by numerical solution of the GS equation using a free-boundary equilibrium code. The data base currently consists of over 2,000 equilibria spanning the wide range of plasma positions and shapes available in COMPASS. Each equilibrium configuration takes several minutes to generate on a fast workstation. The boundary of each configuration is then fitted using the form in equation 1, so that the equilibria are labelled with the appropriate values of the shape parameters. Of the 120 magnetic signals available on COMPASS which could be used to provide inputs to the network, a Real-Time Control o/Tokamak PLasma Using Neural Networks 1011 subset of 16 has been chosen using sequential forward selection based on a linear representation for the mapping (discussed below). It is important to note that the transformation from magnetic signals to flux surface parameters involves an exact linear invariance. This follows from the fact that, if all of the currents are scaled by a constant factor, then the magnetic fields will be scaled by this factor, and the geometry of the plasma boundary will be unchanged. It is important to take advantage of this prior knowledge and to build it into the network structure, rather than force the network to learn it by example. We therefore normalize the vector m of input signals to the network by dividing by a quantity proportional to the total plasma current. Note that this normalization has to be incorporated into the hardware implementation of the network, as will be discussed in Section 3. 4 01 2 c .5. 0- ° CIS :E 1iI ~ -2 ::J -4 4 ~ 2 ~ ° CD Z ~ -2 :::I CD z -4 Database Database 01 c .5. 2 ~ :E ° 1iI CD c ::J -2 Database Database 1.2 go.8 .5. 0CIS :E 1iI0.4 CD c ::J ° Database 1.2 ~O.8 ~ CD z ~O.4 :::I CD Z ° Database Figure 4: Plots of the values from the test set versus the values predicted by the linear mapping for the 3 equilibrium parameters, together with the corresponding plots for a neural network with 4 hidden units. .2 .2 The results presented in this paper are based on a multilayer perceptron architecture having a single layer of hidden units with 'tanh' activation functions, and linear output units. Networks are trained by minimization of a sum-of-squares error using a standard conjugate gradients optimization algorithm, and the number of hidden J012 C. Bishop, P. Haynes, M. Smith, T. Todd, D. Trotman, C. Windsor units is optimized by measuring performance with respect to an independent test set. Results from the neural network mapping are compared with those from the optimal linear mapping, that is the single linear transformation which minimizes the same sum-of-squares error as is used in the neural network training algorithm, as this represents the method currently used on a number of present day tokamaks. Initial results were obtained on networks having 3 output units, corresponding to the values of vertical position ZQ, major radius RQ, and elongation K; these being parameters which are of interest for real-time feedback control. The smallest normalized test set error of 11.7 is obtained from the network having 16 hidden units. By comparison, the optimal linear mapping gave a normalized test set error of 18.3. This represents a reduction in error of about 30% in going from the linear mapping to the neural network. Such an improvement, in the context of this application, is very significant. For the experiments on real-time feedback control described in Section 4 the currently available hardware only permitted networks having 4 hidden units, and so we consider the results from this network in more detail. Figure 4 shows plots of the network predictions for various parameters versus the corresponding values from the test set portion of the database. Analogous plots for the optimal linear map predictions versus the database values are also shown. Comparison of the corresponding figures shows the improved predictive capability of the neural network, even for this sub-optimal network topology. 3 HARDWARE IMPLEMENTATION The hardware implementation of the neural network must have a bandwidth of 2: 20 kHz in order to cope with the fast timescales of the plasma evolution. It must also have an output precision of at least (the the analogue equivalent of) 8 bits in order to ensure that the final accuracy which is attainable will not be limited by the hardware system. We have chosen to develop a fully parallel custom implementation of the multilayer perceptron, based on analogue signal paths with digitally stored synaptic weights (Bishop et al., 1993). A VME-based modular construction has been chosen as this allows flexibility in changing the network architecture, ease of loading network weights, and simplicity of data acquisition. Three separate types of card have been developed as follows: • Combined 16-input buffer and signal normalizer. This provides an analogue hardware implementation of the input normalization described earlier. • 16 x 4 matrix multiplier The synaptic weights are produced using 12 bit frequency-compensated multiplying DACs (digital to analogue converters) which can be configured to allow 4-quadrant multiplication of analogue signals by a digitally stored number. • 4-channel sigmoid module There are many ways to produce a sigmoidal non-linearity, and we have opted for a solution using two transistors configured as along-tailed-pair, Real-Time Control of Tokamak Plasma Using Neural Networks 1013 to generate a 'tanh' sigmoidal transfer characteristic. The principal drawback of such an approach is the strong temperature sensitivity due to the appearance of temperature in the denominator of the exponential transistor transfer characteristic. An elegant solution to this problem has been found by exploiting a chip containing 5 transistors in close thermal contact. Two of the transistors form the long-tailed pair, one of the transistors is used as a heat source, and the remaining two transistors are used to measure temperature. External circuitry provides active thermal feedback control, and stability to changes in ambient temperature over the range O°C to 50°C is found to be well within the acceptable range. The complete network is constructed by mounting the appropriate combination of cards in a VME rack and configuring the network topology using front panel interconnections. The system includes extensive diagnostics, allowing voltages at all key points within the network to be monitored as a function of time via a series of multiplexed output channels. 4 RESULTS FROM REAL-TIME FEEDBACK CONTROL Figure 5 shows the first results obtained from real-time control of the plasma in the COMPASS tokamak using neural networks. The evolution of the plasma elongation, under the control of the neural network, is plotted as a function of time during a plasma pulse. Here the desired elongation has been preprogrammed to follow a series of steps as a function of time. The remaining 2 network outputs (radial position Ro and vertical position Zo) were digitized for post-shot diagnosis, but were not used for real-time control. The solid curve shows the value of elongation given by the corresponding network output, and the dashed curve shows the post-shot reconstruction of the elongation obtained from a simple 'filament' code, which gives relatively rapid post-shot plasma shape reconstruction but with limited accuracy. The circles denote the elongation values given by the much more accurate reconstructions obtained from the full equilibrium code. The graph clearly shows the network generating the required elongation signal in close agreement with the reconstructed values. The typical residual error is of order 0.07 on elongation values up to around 1.5. Part of this error is attributable to residual offset in the integrators used to extract magnetic field information from the pick-up coils, and this is currently being corrected through modifications to the integrator design. An additional contribution to the error arises from the restricted number of hidden units available with the initial hardware configuration. While these results represent the first obtained using closed loop control, it is clear from earlier software modelling of larger network architectures (such as 32- 16-4) that residual errors of order a few % should be attainable. The implementation of such larger networks is being persued, following the successes with the smaller system. Acknowledgements We would like to thank Peter Cox, Jo Lister and Colin Roach for many useful discussions and technical contributions. This work was partially supported by the UK Department of Trade and Industry. 1014 c: o 1.8 ~ 14 C) • c: o as 1.0 C. Bishop, P. Haynes, M. Smith, T. Todd, D. Trotman, C. Windsor shot 9576 0.0 0.1 0.2 time (sec.) Figure 5: Plot of the plasma elongation K. as a function of time during shot no. 9576 on the COMPASS tokamak, during which the elongation was being controlled in real-time by the neural network. References Bishop C M, Cox P, Haynes P S, Roach C M, Smith M E U, Todd T N and Trotman D L, 1992. A neural network approach to tokamak equilibrium control. In Neural Network Applications, Ed. J G Taylor, Springer Verlag, 114-128. Bishop C M, Haynes P S, Roach C M, Smith ME U, Todd T N, and Trotman D L. 1993. Hardware implementation of a neural network for plasma position control in COMPASS-D. In Proceedings of the 17th. Symposium on Fusion Technology, Rome, Italy. 2 997-1001. Lagin L, Bell R, Davis S, Eck T, Jardin S, Kessel C, Mcenerney J, Okabayashi M, Popyack J and Sauthoff N. 1993. Application of neural networks for real-time calculations of plasma equilibrium parameters for PBX-M, In Proceedings of the 17th. Symposium on Fusion Technology, Rome, Italy. 21057-106l. Lister J Band Schnurrenberger H. 1991. Fast non-linear extraction of plasma parameters using a neural network mapping. Nuclear Fusion. 31, 1291-1300. Pulsestream Synapses with Non-Volatile Analogue Amorphous-Silicon Memories. A.J. Holmes, A.F. Murray, S. Churcher and J. Hajto Department of Electrical Engineering University of Edinburgh Edinburgh, EH9 3JL M. J. Rose Dept. of Applied Physics and Electronics, Dundee University Dundee DD14HN Abstract A novel two-terminal device, consisting of a thin lOooA layer of p+ a-Si:H sandwiched between Vanadium and Chromium electrodes, exhibits a non-volatile, analogue memory action. This device stores synaptic weights in an ANN chip, replacing the capacitor previously used for dynamic weight storage. Two different synapse designs are discussed and results are presented. 1 INTRODUCTION Analogue hardware implementations of neural networks have hitherto been hampered by the lack of a straightforward (local) analogue memory capability. The ideal storage mechanism would be compact, non-volatile, easily reprogrammable, and would not interfere with the normal silicon chip fabrication process. Techniques which have been used to date include resistors (these are not generally reprogrammable, and suffer from being large and difficult to fabricate with any accuracy), dynamic capacitive storage [4] (this is compact, reprogrammable and simple, but implies an increase in system complexity, arising from off-chip refresh circuitry), 764 A. J. Holmes, A. F. Murray, S. Churcher, J. Hajto, M. J. Rose EEPROM ("floating gate") memory [5] (which is compact, reprogrammable, and non-volatile, but is slow, and cannot be reprogrammed in situ), and local digital storage (which is non-volatile, easily programmable and simple, but consumes area horribly). Amorphous silicon has been used for synaptic weight storage [1, 2], but only as either a high-resistance fixed weight medium or a binary memory. In this paper, we demonstrate that novel amorphous silicon memory devices can be incorporated into standard CMOS synapse circuits, to provide an analogue weight storage mechanism which is compact, non-volatile, easily reprogrammable, and simple to implement. 2 a-Si:H MEMORY DEVICES The a-Si:H analogue memory device [3] comprises a lOooA thick layer of amorphous silicon (p+ a-Si:H) sandwiched between Vanadium and Chromium electrodes. The a-Si device takes the form of a two-terminal, programmable resistor. It is an "add-on" to a conventional CMOS process, and does not demand that the normal CMOS fabrication cycle be disrupted. The a-Si device sits on top of the completed chip circuitry, making contact with the CMOS arithmetic elements via holes cut in the protective passivation layer, as shown in Figure 1. CMOS Passivation Figure 1: The construction of a-Si:H Devices on a CMOS chip After fabrication a number of electronic procedures must be performed in order to program the device to a given resistance state. Programming, and Pre-Programming Procedures Before the a-Si device is usable, the following steps must be carried out: • Forming: This is a once-only process, applied to the a-Si device in its "virgin" state, where it has a resistance of several MO. A series of 300ns pulses, increasing in amplitude from 5v to 14v, is applied to the device electrodes. This creates a vertical conducting channel or filament whose approximate resistance is 1KO. This filament can then be programmed to a value in the range lKO to 1 MO. The details of the physical mechanisms are not yet fully established, but it is clear that conduction occurs through a narrow (sub-micron) conducting channel. Pulsestream Synapses with Non-Volatile Analogue Amorphous-Silicon Memories 765 • Write: To decrease the device's resistance, negative "Write", pulses are applied. • Erase: To increase the device's resistance, positive" Erase" , pulses are applied. • Usage: Pulses below O.5v do not change the device resistance. The resistance can therefore be utilised as a weight storage medium using a voltage of less than O.5v without causing reprogramming. Programming pulses, which range between 2v and 5v, are typically 120ns in duration. Programming is therefore much faster than for other EEPROM (floating gate) devices used in the same context, which use a series of 100jls pulses to set the threshold voltage [5]. The following sections describe synapse circuits using the a-Si:H devices. These synapses use the reprogrammable a-Si:H resistor in the place of a storage capacitor or EEPROM cell. These new synapses were implemented on a chip referred to as ASiTEST2, consisting of five main test blocks, each comprising of four synapses connected to a single neuron. 3 The EPSILON based synapse The first synapse to be designed used the a-Si:H resistor as a direct replacement for the storage capacitor used in the EPSILON [4] synapse. +Sv V .. t. Storage 1 I Original ~:l: Capacitor O.5v ><!: ~ E 30 ... <0-----------.,... __ Mirror Set a-Si => Vw EPSILON Synapse Neuron Circuitry Figure 2: The EPSILON Synapse with a-Si:H weight storage In the original EPSILON chip the weight voltage was stored as a voltage on a capacitor. In this new synapse design, shown in Figure 2, the a-Si:H resistance is set such that the voltage drop produced by Iset is equivalent to the original weight voltage, Vw, that was stored dynamically on the capacitor. A new, simpler, synapse, which can be operated from a single +5v supply, was also be included on the ASiTEST2 chip. 766 A. J. Holmes, A. F. Murray, S. Churcher, J. Hajto, M. J. Rose 4 The MkII synapse The circuit is shown in Figure 3. The a-Si:H memory is used to store a current, Iasi. This current is subtracted from a zero current, Isy...:z" to give a weight current , +/-Iw, which adds or subtracts charge from the activity capacitor, Cact, thus implementing excitation or inhibition respectively. For the circuit to function correctly we must limit the voltage on the activity capacitor to the range [1.5v,3.5v], to ensure that the transistors mirroring Isy_z and Iasi remain in saturation. As Figure 3 shows, there are few reference signals and the circuit operates from a single +5v power supply rail, in sharp contrast to many earlier analogue neural circuits, including our own. +5v PWm 1 v"" II . .r--\. Vsel ~ 881 ....L -.L Comparator * Cact Vramp ~ Ov E ;. "'E~-----------:~~ E ;. Mirror Set Synapse Neuron Power Supplies References Tail Currents Vrstv·2.5v V5_0=5.Ov Isy_z=5uA Ineu=4uA Figure 3: The MkII synapse PWout ..rL On first inspection the main drawback of this design would appear to be a reliance on the accuracy with which the zero current Isy...:z, is mirrored across an entire chip. The variation in this current means that two cells with the same synapse resistance could produce widely differing values of Iw. However, during programming we do not use the resistance of the a-Si:H device as a target value. We monitor the voltage on Cact for a given PWin signal, increasing or decreasing the resistance of the a-Si:H device until the desired voltage level is achieved. Example: To set a weight to be the maximum positive value, we adjust the a-Si resistance until a PWin signal of 5us, the maximum input signal, gives a voltage of 3.5v on the integration capacitor. We are able to set the synapse weight using the whole integration range of [1.5v,3.5v] by only closing V sel for the desired synapse during programming. In normal operating mode all four Vsel switches will be closed so that the integration charge is summed over all four local capacitors. Pulsestream Synapses with Non-Volatile Analogue Amorphous-Silicon Memories 767 4.1 Example - Stability Test As an example of the use of integration voltage as means of monitoring the resistance of a particular synapse we have included a stability test. This was carried out on one of the test chips which contained the MkII synapse. The four synapses on the test chip were programmed to give different levels of activation. The chip was then powered up for 30mins each day during a 7-day period, and the activation levels for each synapse were measured three times. Stability Test - PWin = 3us 3.5 ,----~-_._--;--__..___r...:....,.-_._,.__-,.--.,..,.-__..__:___., testl test2 test3 test4 testS test6 test7 3 · . I , I I t:'" • • ~. • • • • ~ - ~ -:- - ~ -:- -:- - -. of - ~-:- - {,o-:- - ~-s4 -~ , . ,.... 25 . . ' . . . . •1:1 • • • . • • • • 't - - ~ - - ~ - - --:- - - ~ - - -~- - -w. ----r s2 -- - .;. '" - -: -011>- - :.. ~ - ~ -oGii - -:- - - i-- ~ - ..; -sl · . . . . -. . 2 ~ - - ~ - - -~- - - ~ - - L ---L---~ --~ -s3 • I • , • , • I , • , • • • I I • , • • • • , • , I , • • I • I • • • • • I • I • • I • : : • • • 10 20 30 40 50 60 70 80 90 Measurement Index Figure 4: ASiTEST2- Stability Test As figure 4 shows, the memories remain in the same resistance state (i.e retain their programmed weight value) over the whole 7-day period. Separate experiments on isolated devices indicate much longer hold times - of the order of months at least. 5 ASiTEST3 Recently we have received our latest, overtly neural, a-Si:H based test chip. This contains an 8x8 array of the MkII synapses. The circuit board for this device has been constructed and partially tested while the ASiTEST3 chips are awaiting the deposition of the a-Si:H layers. We have been able to use an ASiTEST2 chip containing two of the MkII synapse test blocks i.e. 8 synapses and 2 neurons to exercise much of the board's functionality. The test board contains a simple state machine which has four different states: • State 0: Load Input Pulsewidths into SRAM from PC. • State 1: Apply Input Pulsewidth signals to chipl. • State 2: Use Vramp to generate threshold function for chipl. The resulting Pulsewidth outputs are used as the inputs to chip2, as well as being stored 768 A. J. Holmes, A. F. Murray, S. Churcher, J. Hajto, M. J. Rose in SRAM . • State 3: Use Vramp to generate threshold function for chip2. Read resulting Pulsewidth Outputs into SRAM . • State 0: Read Output Pulsewidths from SRAM into PC. The results obtained during a typical test cycle are shown in Figure 5. IE-- Statel --;,,;,,*1 EE-- State2 -----;>~i~ State3 ---;!>~I ~r-IF~~~--r-------~---------l 4v 3v 2v PWin_O Iv Ov~ ......... 3.5v r;;;;;""-~,"",,,or;:::::;;:;::;;;;:;;;::;;t---~----t--------, ~:~ ....... ~;a~;"""'" ~'Sig~;,id""""" ... ~"Li~"""" ...... 2.Ov ....................... ,. ....... . .... ..... .......... .. .. l.~ e·_e _____ ...... ___ • _____ • ____________ ........ _____ •••• ____ .......... . 5v 4v 3v 2v :;; ~,...,. ............... -..--t; •. ~ . ..:... ~ __ --! .......... ~~.n-.-~~~~~~......J 10. IS. Figure 5: ASiTEST3 Board Scope Waveforms As this figure shows different ramp signals, corresponding to different threshold functions, can be applied to chipl and chip2 neurons. Single Buffer PulscWidth Sweeps 10.0 .----.,..----r------.----r----.------, 9.0 8.0 ! 7.0 ~ i6.o~ ~ 5'O~~~~~~~-~~-+++~~~~N~~~ J:: ~---2D 1.0 o.oL---~--~- --~-- o 0.5 1.0 1.5 2.0 2.5 Pulscwldlh Input [WI) Neal-Syal N~~JIII N~~ya2 3.0 Figure 6: ASiTEST3 Board - MkII Synapse Characteristic While the signals shown in Figure 5 appear noisy the multiplier characteristic that the chip produces is still admirably linear, as shown in Figure 6. In this experiment all eight synapses on a test chip were programmed into different resistance states and PWin was swept from 0 to 3us. Pulsestream Synapses with Non- Volatile Analogue Amorphous-Silicon Memories 769 6 Conclusions We have demonstrated the use of novel a-Si:H analogue memory devices as a means of storing synaptic weights in a Pulsewidth ANN. We have also demonstrated the operation of an interface board which allows two 8x8 ANN chips, operating as a two layer network, to be controlled by a simple PC interface card. This technology is most suitable for small networks in, for example, remote control and other embedded-system applications where cost and power considerations favour a single all-inclusive ANN chip with non-volatile, but programmable weights. Another possible application of this technology is in large networks constructed using Thin Film Technology(TFT). If TFT's were used in place of the CMOS transistors then the area constraint imposed by crystalline silicon would be removed, allowing truly massively parallel networks to be integrated. In summary - the a-Si:H analogue memory devices described in this paper provide a route to an analogue, non-volatile and fast synaptic weight storage medium. At the present time neither the programming nor storage mechanisms are fully understood making it difficult to compare this new device with more established technologies such as the ubiquitous Floating-Gate EEPROM technique. Current research is focused on firstly, improving the yield on the a-Si:H device which is unacceptably low at present, a demerit that we attribute to imperfections in the a-Si fabrication process and secondly, improving understanding of the device physics and hence the programming and storage mechanisms. Acknowledgements This research has been jointly funded by BT, and EPSRC (formerly SERC), the Engineering and Physical Sciences Research Council. References [1] W. Hubbard et al.(1986) Electronic Neural Networks AlP Conference Proceedings - Snowbird 1986 :227-234 [2] H.P. Graf (1986) VLSI Implementation of a NN memory with several hundreds of neurons AlP Conference Proceedings - Snowbird 1986 :182-187. [3] M.J. Rose et al (1989) Amorphous Silicon Analogue Memory Devices Journal of Non-Crystalline Solids 1(115):168-170 [4] A.Hamilton et al. (1992) Integrated Pulse-Stream Neural Networks - Results, Issues and Pointers IEEE Transactions on N.N.s 3(3):385-393 [5] M.Holler, S.Tam, H.Castro and R.Benson (1989) An Electrically Trainable ANN with 10240 Floating Gate Synapses. Int Conf on N.N.s Proc :191-196 [6] A.F.Murray and A.V.W.Smith.(1987) Asynchronous Arithmetic for VLSI Neural Systems. Electronics Letters 23(12):642-643 [7] A.J. Holmes et al. (1993) Use of a-Si:H Memory Devices for Non-volatile Weight Storage in ANNs. Proc lCAS 15 :817-820	1994	81
978	Dynamic Cell Structures Jorg Bruske and Gerald Sommer Department of Cognitive Systems Christian Albrechts University at Kiel 24105 Kiel- Germany Abstract Dynamic Cell Structures (DCS) represent a family of artificial neural architectures suited both for unsupervised and supervised learning. They belong to the recently [Martinetz94] introduced class of Topology Representing Networks (TRN) which build perlectly topology preserving feature maps. DCS empI'oy a modified Kohonen learning rule in conjunction with competitive Hebbian learning. The Kohonen type learning rule serves to adjust the synaptic weight vectors while Hebbian learning establishes a dynamic lateral connection structure between the units reflecting the topology of the feature manifold. In case of supervised learning, i.e. function approximation, each neural unit implements a Radial Basis Function, and an additional layer of linear output units adjusts according to a delta-rule. DCS is the first RBF-based approximation scheme attempting to concurrently learn and utilize a perfectly topology preserving map for improved performance. Simulations on a selection of CMU-Benchmarks indicate that the DCS idea applied to the Growing Cell Structure algorithm [Fritzke93] leads to an efficient and elegant algorithm that can beat conventional models on similar tasks. 1 Introduction The quest for smallest topology preserving maps motivated the introduction of growing feature maps like Fritzke's Growing Cell Structures (GCS). In GCS, see [Fritzke93] for details, one starts with a k-dimensional simplex of N = k+ 1 neural units and (k + 1) . kl2 lateral connections (edges). Growing of the network is performed such that after insertion 498 Jorg Bruske, Gerald Sommer of a new unit the network consists solely of k dimensional simplices again. Thus, like Kohonen's SOM, GCS can only learn a perfectly topology preserving feature mapl if k meets the actual dimension of the feature manifold. Assuming that the lateral connections do reflect the actual topology the connections serve to define a neighborhood for a Kohonen like adaptation of the synaptic vectors Wj and guide the insertion of new units. Insertion happens incrementally and does not necessitate a retraining of the network. The principle is to insert new neurons in such a way that the expected value of a certain local error measure, which Fritzke ca11s the resource, becomes equal for all neurons. For instance, the number of times a neuron wins the competition, the sum of distances to stimuli for which the neuron wins or the sum of errors in the neuron's output can all serve as a resource and dramatically change the behavior of GCS. Using different error measures and guiding insertion by the lateral connections contributes much to the success of GCS. The principle of DCS is to avoid any restriction of the topology of the network (lateral connection scheme between the neural units) but to concurrently learn and utilize a perfectly topology preserving map. This is achieved by adapting the lateral connection structure according to a competitive Hebbian learning rule2: { max{Yj'YpCij(t)} : Yj'Yj~Yk'YI V'(1S,k,IS,N) CIj(t+ 1) = 0 : Cjj(t) <9 fIfI aCjj (t) : otherwise, till (1) where a, 0 < a < 1 is a forgettin¥. constant, 9, 0 < 9 < 1 serves as a threshold for deleting lateral connections, and Yj = R (j/v - Wj//) is the activation of the i-th unit with Wj as the centre of its receptive field on presentatIOn of stimulus v. R(.) can be any positive continuously monotonically decreasing function. For batch learning with a training set T of fixed size 111 , a = 17JJ9 is a good choice. Since the isomorphic representation of the topology of the feature manifold M in the lateral connection structure is central to performance, in many situations a DCS algorithm may be the right choice. These situations are characterized by missing a priori knowledge of the topology of the feature manifold M or a topology of M which cannot be readily mapped to the existing models. Of course, if such a priori knowledge is available then models like GCS or Kohonen's SOM allowing to incorporate such knowledge have an advantage, especially if training data are sparse. Note that DCS algorithms can also aid in cluster analysis: In a perfectly topology preserving map clusters which are bounded by regions of P(v) = 0 can be identified simply by a connected component analysis. However, without prior knowledge about the feature manifold M it is in principal impossible to check for perfect topology preservation of S. Noise in the input data may render perfect topology learning even more difficult. So what can perfect topology learning be used for? The answer is simply that for every s~t S of reference vectors perfect topology learning yields maximum topology preservation with respect to this set. And connected components with respect to the lateral connection structure C may well serve as an initialization for postprocessing by hierarchical cluster algorithms. 1. We use the term "perfectly topology preserving feature map" in accordance with its rigorous definition in [Martinetz93]. 2. In his very recent and recommendable article [Martinetz94] the term Topology Representing Network (TRN) is coined for any network employing competitive Hebbian learning for topology learning. 3. if topology preservation is measured by the topographic function as defined in [Villmann94]. Dynamic Cell Structures 499 The first neural algorithm attempting to learn perfectly topology preserving feature maps is the Neural Gas algorithm ofT. Martinetz [Martinetz92]. However, unlike DCS the Neu· ral Gas does not further exploit this information: In every step the Neural Gas computes the k nearest neighbors to a given stimulus and, in the supervised case, employs a11 of them for function approximation. DCS avoids this computational burden by utilizing the lateral connection structure (topology) learned so far, and it restricts interpolation between activated units to the submanifold of the current stimulus. Applying the principle ofDCS to Fritzke's GCS yields our DCS·GCS algorithm. This algorithm sticks very closely to the basic structure of its ancestor GCS except the predefined k-dimensional simplex connection structure being replaced by perfect topology learning. Besides the conceptual advantage of perfect topology learning, DCS·GCS does decrease overhead (Fritzke has to handle quite sophisticated data structures in order to maintain the k-dimensional simplex structure after insertion! deletion of units) and can be readily implemented on any serial computer. 2 Unsupervised DCS-GCS The unsupervised DCS-GCS algorithm starts with initializing the network (graph) to two neural units (vertices) n, and n2. Their weight vectors wI' WI (centres of receptive fields) are set to points v I' v2 E M which are drawn from M according to P(v). They are connected by a lateral connection of weight Cl2 = C21 = I. Note that lateral connections in DCS are always bidirectional and have symmetric weights. Now the algorithm enters its outer loop which is repeated until some stopping criterium is fulfi11ed. This stopping criterium could for instance be a test whether the quantization error has already dropped below a predefined accuracy. The inner loop is repeated for A. times. In off-line learning A. can be set to the number examples in the training set T. In this case, the inner loop just represents an epoch of training. Within the inner loop, the algorithm first draws an input stimulus v E M from M according to P(v) and then proceeds to calculate the two neural units which weight vectors are first and second closest to v. In the next step, the lateral connections between the neural units are modified according to eq. (1), the competitive Hebbian learning rule. As has already been mentioned, in off-line learning it is a good idea to set a = va . Now the weight vectors Wj of the best matching unit and its neighbors are adjusted in a Kohonen like fashion: .1.wbmu = £8 (v - wbmu> and .1.Wj = £Nh (v - wj) , (2) where the neighborhood N h U) of a unit j is defined by NhU) = {il(Cjj*O,l~i~N)} . The inner loop ends with updating the resource value of the best matching unit. The resource of a neuron is a local error measure attached to each neural unit. As has been pointed out, one can choose alternative update functions corresponding to different error measures. For our ~xperim~nts (section 2.1 and sectpn 3.1) we used the accumulated squared distance to the stImulus, I.e . .1.'tbmu = Ilv - wbmull . The outer loop now proceeds by adding a new neural unit r to the network. This unit is located in-between the unit I with largest resource value and its neighbor n with second largest resource value:4 500 Jorg Bruske, Gerald Sommer The exact location of its centre of receptive field w r is calculated according to the ratio of the resource values 'tl, 'tn' and the resource values of units n and I are redistributed among r, n and I: W r = wI + yew n - wI) , 'tr = ~'tn + ~'tr' 'tl = 't[- ~'tr and'tn = 'tn ~'tn· (4) This gives an estimate of the resource values if the new unit had been in the network right from the start. Finally the lateral connections are changed, Cr=C[ =l,C =C =IandCr=Cr =0, r r rn rn n n (5) connecting unit r to unit I and disconnecting n and I. This heuristic guided by the lateral connection structure and the resource values promises insertion of new units at good initial positions. It is responsible for the better performance of DCS-GCS and GCS compared to algorithms which do not exploit the neighborhood relation between existing units. The outer loop closes by decrementing the resource values of all units, 'ti (t + 1) = ~'ti (t) , 1:::; i:::; N, where ° < ~ < 1 is a constant. This last step just avoids overflow of the resource variables. For off-line learning, ~ = ° is the natural choice. 2.1 Unsupervised DCS simulation results Let us first turn to our simulation on artificial data. The training set T contains 2000 examples randomly drawn from a feature manifold M consisting of three squares, two of them connected by a line. The development of our unsupervised DCS-GCS network is depicted in Figure 1, with the initial situation of only two units shown in the upper left. Examples are represented by small dots, the centres of receptive fields by small circles and the lateral connections by lines connecting the circles. From left to right the network is examined after 0, 9 and 31 epochs of training (i.e. after insertion of 2, 11 and 33 neural units). After 31 epochs the network has built a perfectly topology preserving map of M, the lateral connection structure nicely reflecting the shape of M: Where Mis 2-dimensional the lateral connection structure is 2-dimensional, and it is I-dimensional where M is I-dimensional. Note, that a connected component analysis could recognize that the upper right square is separated from the rest of M. The accumulated squared distance to stimuli served as the resource. Th.e quantization error Eq = ! L Ilv - wbmu (v) 112 dropped from 100% (3 units) to 3% (33 umts). n VE T The second simulation deals with the two-spirals benchmark. Data were obtained by running the program "two-spirals" (provided by eMU) with parameters 5 (density) and 6.5 (spiral radius) resulting in a training set T of 962 examples. The data represent two distinct spirals in the x-y-plane. Unsupervised DCS-GCS at work is shown in Figure 2, after insertion of 80, 154 and, finally, 196 units. With 196 units a perfectly topology preserving map of M has emerged, and the two spirals are clearly separated. Note that the algorithm has learned the separation in a totally unsupervised manner, i.e. not using the labels of the data 4. Fritzke inserts new units at a slightly different location, using not the neighbor with second largest resource but the most distant neighbor. Dynamic Cell Structures 501 Figure 1: Unsupervised DCS-GCS on artificial data points (which are provided by CMU for supervised learning). Again, the accumulated squared distance to stimuli served as the resource. -;..--------,--- -- - . '-::---.:-:r~~ Figure 2: Unsupervised learning of two spirals 3 Supervised DCS-GCS , ~.:~:.~.';{';';'; ~.:.;.! ~~~!~I In supervised DCS-GCS examples consist not only of an input vector v but also include an additional teaching output vector u. The supervised algorithm actually does work very similar to its unsupervised version except • when a neural unit nj is inserted an output vector OJ will be attached to it with OJ = u. • the output y of the network is calculated as a weighted sum of the best matching unit's output vector 0hmu and the output vectors of its neighbors OJ' i E Nh (bmu) , Y = (~ a.o.) , ~jE {bmuuNh(hmu)} I I (6) 502 Jorg Brnske, Gerald Sommer where a· = I/(crllv-wiI12+ 1) is the activation of neuron i on stimulus v, cr, cr> (1, representing the size of the receptive fields. In our simulations, the size of receptive fields have been equal for all units. • adaption of output vectors by the delta-rule: A simple delta-rule is employed to adjust the output vectors of the best matching unit and its neighbors. Most important, the approximation (classification) error can be used for resource updating. This leads to insertion of new units in regions where the approximation error is worst, thus promising to outperform dynamic algorithms which do not employ such a criterion for insertion. In our simulations we used the accumulated squared distance of calculated and teaching output, ~tbmu = Ily - u11 2 . 3.1 Supervised DCS-GCS simulation results We applied our supervised DCS-GCS algorithm to three CMU benchmarks, the supervised two-spiral problem, the speaker independent vowel recognition problem and the sonar mine! rock separation problem.5 The t~o spirals benchmark contains 194 examples, each consisting of an input vector v E ~ and a binary label indicating to which spiral the point belongs. The spirals can not be linearly separated. The task is to train the examples until the learning system can produce the correct output for all of them and to record the time. The decision regions learned by supervised DCS-GCS are depicted in Figure 3 after 110 and 135 epochs of training, where the classification error on the training set has dropped to 0%. Black indicates assignment to the fist, white assignment to the second spiral. The network and the examples are overlaid. Figure 3: Supervised learning of two spirals Results reported by others are 20000 epochs of Backprop for a MLP by Lang and Witbrok [Lang89], 10000 epochs of Cross Entropy Backprop and 1700 epochs of Cascade-Correlation by Fahlman and Lebiere [Fahlman90] and 180 epochs of GCS training by Fritzke [Fritzke93]. 5. For details of simulation, parameters and additional statistics for all of the reported experiments the reader is refered to [Bruske94] which is also available viaftp.infomuztik.uni-kiel.de in directory publkiellpublicationsffechnicalReportslPs.ZI as 1994tr03.ps.Z Dynamic Cell Structures 503 The data for the speaker independent recognition of 11 vowels comprises a training set of 582 examples and a test set of 462 examples, see [Robinson89]. We obtained 65% correctly classified test samples with only 108 neural units in the DCS· GCS network. This is superior to conventional models (including single and multi layer perceptron, Kanerva Model, Radial Basis Functions, Gaussian Node Network, Square Node Network and Nearest Neighbor) for which figures well below 57% have been reported by Robinson. It also qualitatively compares to GCS Gumps above the 60% margin), for which Fritzke reports best classification results of 61 %(158 units) up to 67% (154 units) for a 3-dim GCS. On the other hand, our best DCS·GCS used much fewer units. Note that DCS·GCS did not rely on a pre-specified connection structure (but learned it!). Our last simulation concerns a data set used by Gorman and Sejnowski in their study of classification of sonar data, [Gorman88]. The training and the test set contain 104 examples each. Gorman and Sejnowski report their best results of 90.4% correctly classified test examples for a standard BP network with 12 hidden units and 82.7% for a nearest neighbor classifier. Supervised DCS·GCS reached a peak classification rate of 95% after only 88 epochs of training. 4 Conclusion We have introduced the idea ofRBF networks which concurrently learn and utilize perfectly topology preserving feature maps for adaptation and interpolation. This family of ANNs, which we termed Dynamic Cell Structures, offers conceptual advantage compared to classical Kohonen type SOMs since the emerging lateral connection structure maximally preserves topology. We have discussed the DCS-GCS algorithm as an instance of DCS. Compared to its ancestor GCS of Fritzke, this algorithm elegantly avoids computational overhead for handling sophisticated data structures. If connection updates (eq.(I)) are restrJcted to the best matching unit and its neighbors, DCS has linear (serial) time complexity and thus may also be considersd as an improvement of Martinetz's Neural Gas idea7. Space complexity of DCS is 0 (N) in general and can be shown to become linear if the feature manifold M is two dimensional. The simulations on CMU-Benchmarks indicate that DCS indeed has practical relevance for classification and approximation. Thus encouraged, we look forward to apply DCS at various sites in our active computer vision project, including image compression by dynamic vector quantization, sensorimotor maps for the oculomotor system and hand-eye coordination, cartography and associative memories. A recent application can be found in [Bruske95] where a DCS network attempts to learn a continous approximation of the Q-function in a reinforcement learning problem. 6. Here we refer to the serial time a DeS algorithm needs to process a single stimulus (including response calculation and adaptation). 7. The serial time complexity of the Neural Gas is Q (N) ,approaching 0 (NlogN) for k ~ N , k the number of nearest neighbors. 504 larg Bruske, Gerald Sommer References [Bruske94] J. Bruske and G. Sommer. Dynamic Cell Structures: Radial Basis Function Networks with Perfect Topology Preservation. Inst. f. Inf. u. Prakt. Math. CAU zu Kiel. Technical Report 9403. [Bruske9S] J. Bruske. I. Ahms and G. Sommer. Heuristic Q-Learning. submitted to ECML 95. [Fahlman90] S.E. Fahlman. C.Lebiere. The Cascade-Correlation Learning Architecture. Advances in Neural Information processing systems 2, Morgan Kaufman, San Mateo, pp.524-534. [Fahlman93] S.E. Fahlman, CMU Benchmark Collection for Neural Net Learning Algorithms, Carnegie Mellon Univ., School of Computer Science, machine-readable data repository. Pittsburgh. [Fritzke92] B. Fritzke. Growing Cell Structures - a self organizing network in k dimensions, Artificial Neural Networks 2. I.Aleksander & J .Taylor eds .• North-Holland, Amsterdam, 1992. [Fritzke93] B. Fritzke, Growing Cell Structures - a self organizing network for unsupervised and supervised training, ICSI Berkeley, Technical Report, tr-93-026. [Gorman88] R.B. Gorman and TJ. Sejnowski, Analysis of Hidden Units in a Layered Network Trained to Classify Sonar Targets, Neural Networks, VoU. pp. 75-89 [Lang89] K.J. Lang & M.J. Witbrock, Learning to tell two spirals apart, Proc. of the 1988 Connectionist Models Summer School. Morgan Kaufmann, pp.52-59. [Martinetz92] Thomas Martinetz, Selbstorganisierende neuronale Netzwerke zur Bewegungssteuerung, Dissertation. DIFKI-Verlag, 1992. [Martinetz93] Thomas Martinetz, Competitive Hebbian Learning Rule Forms Perfectly Topology Preserving Maps, Proc. of the ICANN 93. p.426-438, 1993. [Martinetz94] Thomas Martinetz and Klaus Schulten, Topology Representing Networks, Neural Networks. No.7, Vol. 3. pp. 505-522, 1994. [Moody89] J.Moody, c.J. Darken. Fast Learning in Networks of Locally-Tuned Processing Units, Neural Computation VoLl Num.2, Summer 1989. [Robinson89] AJ. Robinson, Dynamic Error Propagation Networks, Cambridge Univ., Ph.D. thesis, Cambridge. [Villmann94] T. Villmann and R. Der and T. Martinetz, A Novel Approach to Measure the Topology Preservation of Feature Maps, Proc. of the ICANN 94, 1994.	1994	82
979	Single Transistor Learning Synapses Paul Hasler, Chris Diorio, Bradley A. Minch, Carver Mead California Institute of Technology Pasadena, CA 91125 (818) 395 - 2812 paul@hobiecat.pcmp.caltech.edu Abstract We describe single-transistor silicon synapses that compute, learn, and provide non-volatile memory retention. The single transistor synapses simultaneously perform long term weight storage, compute the product of the input and the weight value, and update the weight value according to a Hebbian or a backpropagation learning rule. Memory is accomplished via charge storage on polysilicon floating gates, providing long-term retention without refresh. The synapses efficiently use the physics of silicon to perform weight updates; the weight value is increased using tunneling and the weight value decreases using hot electron injection. The small size and low power operation of single transistor synapses allows the development of dense synaptic arrays. We describe the design, fabrication, characterization, and modeling of an array of single transistor synapses. When the steady state source current is used as the representation of the weight value, both the incrementing and decrementing functions are proportional to a power of the source current. The synaptic array was fabricated in the standard 21'm double - poly, analog process available from MOSIS. 1 INTRODUCTION The past few years have produced a number of efforts to design VLSI chips which "learn from experience." The first step toward this goal is developing a silicon analog for a synapse. We have successfully developed such a synapse using only 818 Paul Hasler, Chris Diorio, Bradley A. Minch, Carver Mead Aoating Gate ... I._High ,oltage Drain Gate ! Source --~~--'-~------r-~--~~--~ Figure 1: Cross section of the single transistor synapse. Our single transistor synapse uses a separate tunneling voltage terminal The pbase implant results in a larger threshold voltage, which results in all the electrons reaching the top of the Si02 barrier to be swept into the floating gate. a single transistor. A synapse has two functional requirements. First, it must compute the product of the input multiplied by the strength (the weight) of the synapse. Second, the synapse must compute the weight update rule. For a Hebbian synapse, the change in the weight is the time average of the product of the input and output activity. In many supervised algorithms like backpropagation, this weight change is the time average of the product of the input and some fed back error signal. Both of these computations are similar in function. We have developed single transistor synapses which simultaneously perform long term weight storage, compute the product of the input and the weight value, and update the weight value according to a Hebbian or a backpropagation learning rule. The combination of functions has not previously been achieved with floating gate synapses. There are five requirements for a learning synapse. First, the weight should be stored permanently in the absence of learning. Second, the synapse must compute as an output the product of the input signal with the synaptic weight. Third, each synapse should require minimal area, resulting in the maximum array size for a given area. Fourth, each synapse should operate with low power dissipation so that the synaptic array is not power constrained. And finally, the array should be capable of implementing either Hebbian or Backpropagation learning rule for modifying the weight on the floating gate. We have designed, fabricated, characterized, and modeled an array of single transistor synapses which satisfy these five criteria. We believe this is the first instance of a single transistor learning synapse fabricated in a standard process. 2 OVERVIEW Figure 1 shows the cross section for the single transistor synapse. Since the floating gate is surrounded by Si02 , an excellent insulator, charge leakage is negligible resulting in nearly permanent storage of the weight value. An advantage of using floating gate devices for learning rules is the timescales required to add and remove charge from the floating gate are well matched to the learning rates of visual and auditory signals. In addition, these learning rates can be electronically controlled. Typical resolution of charge on the floating gate after ten years is four bits (Holler 89). The FETs are in a moderately doped (1 x 1017 em-3 ) substrate, to achieve a Single Transistor Learning Synapses 819 Vg1 t Vg2 (1,1) (1,2) Vtun1 t Vd1 ---+----~--------~----~----VS1~ ---+-..-------------+_e_--------- Vtun2 _ Vd2 (2,1) (2,2) ---+--~~---1---~----VS2~ Figure 2: Circuit diagram of the single - transistor synapse array. Each transistor has a floating gate capacitively coupled to an input column line. A tunneling connection (arrow) allows weight increase. Weight decreased is achieved by hot electron injection in the transistor itself. Each synapse is capable of simultaneous feedforward computations and weight updates. A 2 x 2 section of the array allows us to characterize how modifying a single floating gate (such as synapse (1,1)) effects the neighboring floating gate values. The synapse currents are a measure of the synaptic weights, and are summed along each row by the source (Vs) or drain (Vd) lines into some soma circuit. high threshold voltage. The moderately doped substrate is formed in the 2pm MOSIS process by the pbase implant. npn transistor. The implant has the additional benefit of increasing the efficiency of the hot electron injection process by increasing the electric field in the channel. Each synapse has an additional tunneling junction for modifying the charge on the floating gate. The tunneling junction is formed with high quality gate oxide separating a well region from the floating gate. Each synapse in our synaptic array is a single transistor with its weight stored as a charge on a floating silicon gate. Figure 2 shows the circuit diagram of a 2 x 2 array of synapses. The column 'gate' inputs (V g) are connected to second level polysilicon which capacitively couples to the floating gate. The inputs are shared along a column. The source (Vs), drain (lid), and tunneling (Viun) terminals are shared along a row. These terminals are involved with computing the output current and feeding back 'error' signal voltages. Many other synapses use floating gates to store the weight value, as in (Holler 89), but none of the earlier approaches update the charge on the floating gate during the multiplication of the input and floating gate value. In these previous approaches one must drive the floating gate over large a voltage range to tunnel electrons onto the floating gate. Synaptic computation must stop for this type of weight update. The synapse computes as an output current a product of weight and input signal, 820 15 S j 10.7 10" 10" 10.10 10.11 10-12 Paul Hasler. Chris Diorio. Bradley A. Minch. Carver Mead Iojec:tion Opel1lliolll synapse (2,1), synapse (2.2) ..... Tunneling Opel1llions synapse (1.2) ... . . . '. "."" synapse (1.1) row 2 background alln:nl row I IJeckarouod alrlent 10-13 "-_--'-__ -'-__ -'--__ '"--_----' __ --'-__ ...J o SO I 00 I SO 200 2S0 300 3SO Step iteration Number Figure 3: Output currents from a 2 x 2 section of the synapse array, showing 180 injection operations followed by 160 tunneling operations. For the injection operations, the drain (V dl) is pulsed from 2.0 V upto 3.3 V for 0.5s with Vg1 at 8V and Vg2 at OV. For the tunneling operations, the tunneling line (Vtunl) is pulsed from 20 V up to 33.5 V with Vg2 at OV and "91 at 8V. Because our measurements from the 2 x 2 section come from a larger array, we also display the 'background' current from all other synapses on the row. This background current is several orders of magnitude smaller than the selected synapse current, and therefore negligible. and can simultaneously increment or decrement the weight as a function of its input and error voltages. The particular learning algorithm depends on the circuitry at the boundaries of the array; in particular the circuitry connected to each of the source, drain, and tunneling lines in a row. With charge Qlg on the floating gate and Vs equal to 0 the subthreshold source current is described by (1) where Qo is a device dependent parameter, and UT is the thermal voltage k;. The coupling coefficient, 6, of the gate input to the transistor surface potential is typically less than 0.1. From ( 1) We can consider the weight as a current I, defined by ( ~~) 6gVp 6gVp ISllnapse = Ioe---qo- e T e T = Ie T (2) where Vgo is the input voltage bias, and ~ Vg is Vg - "90. The synaptic current is thus the product of the weight, I, and a weak exponential function of the input voltage. The single transistor learning synapses use a combination of electron tunneling and hot electron injection to adapt the charge on the floating gate, and thereby the Single Transistor Learning Synapses 821 weight of the synapse. Hot electron injection adds electrons to the floating gate, thereby decreasing the weight. Injection occurs for large drain voltages; therefore the floating gate charge can be reduced during normal feedforward operation by raising the drain voltage. Electron tunneling removes electrons from the floating gate, thereby increasing the weight. The tunneling line controls the tunneling current; thus the floating gate charge can be increased during normal feedforward operation by raising the tunneling line voltage. The tunneling rate is modulated by both the input voltage and the charge on the floating gate. Figure 3 shows an example the nature of the weight update process. The source current is used as a measure of the synapse weight. The experiment starts with all four synapses set to the same weight current. Then, synapse (1,1) is injected for 180 cycles to preferentially decrease its weight. Finally, synapse (1,1) is tunneled for 160 cycles to preferentially increase its weight. This experiment shows that a synapse can be incremented by applying a high voltage on tunneling terminals and a low voltage on the input, and can be decremented by applying a high voltage on drain terminals and a high voltage on the input. In the next two sections, we consider the nature of these update functions. In section three we examine the dependence of hot electron injection on the source current of our synapses. In section four we examine the dependence of electron tunneling on the source current of our synapses. 3 Hot Electron Injection Hot electron injection gives us a method to add electrons to the floating gate. The underlying physics of the injection process is to give some electrons enough energy and direction in the channel to drain depletion region to surmount the Si02 energy barrier. A device must satisfy two requirements to inject an electron on a floating gate. First, we need a region where the potential drops more than 3.1 volts in a distance of less than 0.2pm to allow electrons to gain enough energy to surmount the oxide barrier. Second, we need a field in the oxide in the proper direction to collect electrons after they cross the barrier. The moderate substrate doping level allows us to easily achieve both effects in subthreshold operation. First, the higher substrate doping results in a much higher threshold voltage (6.1 V), which guarantees that the field in the oxide at the drain edge of the channel will be in the proper direction for collecting electrons over the useful range of drain voltages. Second, the higher substrate doping results in higher electric fields which yield higher injection efficiencies. The higher injection efficiencies allow the device to have a wide range of drain voltages substantially below the threshold voltage. Figure 4 shows measured data on the change in source current during injection vs. source current for several values of drain voltage. Because the source current, I, is related to the floating gate charge, Q,g as shown in ( 1) and the charge on the floating gate is related to the tunneling or injection current (I, g) by dQ,g _ I dt Ig an approximate model for the change of the weight current value is dl I - = -1'9 dt Qo (3) (4) 822 ! 10" i 10.10 i!, ~ 10'" J 10.12 .S .. g 10'" 3.6 Paul Hasler, Chris Diorio, Bradley A. Minch, Carver Mead synapse (1,1) 3.3 synapse (1,2) Sour.-e Curren! (Weiaht Value) Figure 4: Source Current Decrement during injection vs. Source Current for several values of drain voltage. The injection operation decreases the synaptic weight. V 92 was held at OV, and V 91 was at 8V during the 0.5s injecting pulses. The change in source current is approximately proportional to the source current to the f3 power, where of f3 is between 1.7 and 1.85 for the range of drain voltages shown. The change in source current in synapse (1,2) is much less than the corresponding change in synapse (1,1) and is nearly independent of drain voltage. The effect of this injection on synapses (2,1) and (2,2) is negligible. The injection current can be approximated over the range of drain voltages shown in Fig. 4 by (Hasler 95) (5) where Vd-c is the voltage from the drain to the drain edge of the channel, Vd is the drain voltage, /0 is a slowly varying function defined in (Hasler 95), and Vini is in the range of 60m V to 100m V. A is device dependent parameter, Since hot electron injection adds electrons to the floating gate, the current into the floating gate (If 9) is negative, which results in dI IfJ ~ - = -A-e .nJ (6) dt Qo The model agrees well with the data in Fig. 4, with f3 in the range of 1. 7 - 1.9. Injection is very selective along a row with a selectivity coefficient between 102 and 107 depending upon drain voltage and weight. The injection operations resulted in negligible changes in source current for synapses (2,1) and (2,2). 4 ELECTRON TUNNELING Electron tunneling gives us a method for removing electrons from the floating gate. Tunneling arises from the fact that an electron wavefunction has finite extent. For a Single Transistor Learning Synapses i 10" ~ I 'Ii i! 10.9 I a )10.10 .5 110.11 3S.S 33.0 VlUn = 36.0·' 3S.0 ..-34.0 34.S 33.S Source Cumnt (Weig/ll Value) 823 Figure 5: Synapse (1,1) source current increment vs. Source Current for several values of tunneling voltage. The tunneling operation increases the synaptic weight. V 91 was held at OV and V 92 was 8V while the tunneling line was pulsed for 0.5s from 20V to the voltage shown. The change in source current is approximately proportional to the a power of the svurce current where a is between 0.7 and 0.9 for the range of tunneling voltages shown. The effect of this tunneling procedure on synapse (2,1) and (2,2) are negligible. The selectivity ratio of synapses on the same row is typically between 3-7 for our devices. thin enough barrier, this extent is sufficient for an electron to penetrate the barrier. An electric field across the oxide will result in a thinner barrier to the electrons on the floating gate. For a high enough electric field, the electrons can tunnel through the oxide. When traveling through the oxide, some electrons get trapped in the oxide, which changes the barrier profile. To reduce this trapping effect we tunnel through high quality gate oxide, which has far less trapping than interpoly oxide. Both injection and tunneling have very stable and repeatable characteristics. When tunneling at a fixed oxide voltage, the tunneling current decreases only 50 percent after lOnG of charge has passed through the oxide. This quantity of charge is orders of magnitude more than we would expect a synapse to experience over a lifetime of operation. Figure 5 shows measured data on the change in source current during tunneling as a function of source current for several values of tunneling voltage. The functional form of tunneling current is of the form (Lenzlinger 69) (7) where Yo, f Olun are model parameters which roughly correspond with theory. Tunneling removes electrons from the floating gate; therefore the floating gate current is positive. By expanding Vfg for fixed Viun as VfgO + ~ Vfg and inserting ( 1), the 824 Paul Hasler, Chris Diorio, Bradley A. Minch, Carver Mead II Parameter I Typical Values II Parameter I Typical Values II {3 1.7 - 1.9 Q' 0.7 - 0.9 Qo .2 pC 6 0.02 - 0.1 A 8.6 x 10 ·~u Vinj 78mV Table 1: Typical measured values of the parameters in the modeling of the single transistor synapse array. resulting current change is dI I Vg ( I )Q dt = Q:n e - Viu v/ gO Iso Iso (8) where IsO is the bias current corresponding to VfgO. The model qualitatively agrees with the data in Fig. 5, with Q' in the range of 0.7- 0.9. The tunneling selectivity between synapses on different rows is very good, but tunneling selectivity along along a row is poor. We typically measure tunneling selectivity ratios along a row between 3 - 7 for our devices. 5 Model of the Array of Single Transistor Synapses Finally, we present an approximate model of our array of these single transistor synapses. The learning increment of the synapse at position (i,i) can be modeled as 61l.V ISllnapse'j = Iije ~ == Iso WijXj dW. . 1 _ Vq _ ;d; _ (9) =..:.:...!.L =.:.£t.II..A.e VI"n v/ gO W~x~ 1 _....d....e inj WP.xl? 1 dt Qo I) ) Qo I) ) for the synapse at position (i,i), where Wi,j can be considered the weight value, and x j are the effective inputs network. Typical values for the parameters in ( 9 ) are given in Table 1. Acknowledgments The work was supported by the office of Naval Research, the Advanced Research Projects Agency, and the Beckman Foundation. References P. Hasler, C. Diorio, B. Minch, and C. Mead (1995) "An Analytic model of Hot Electron Injection from Boltzman Transport", Tech. Report 123456 M. Holler, S. Tam, H. Castro, and R. Benson (1989), "An electrically trainable artificial neural network with 10240 'floating gate' synapses", International Joint Conference on Neural Networks, Wasllington, D.C., June 1989, pp. 11-191 - 11-196. M. Lenzlinger and E. H. Snow (1969), "Fowler-Nordheim tunneling into thermally grown Si02 ," J. Appl. Phys., vol. 40, pp. 278-283, 1969. PARTVll SPEECH AND SIGNAL PROCESSING	1994	83
980	Comparing the prediction accuracy of artificial neural networks and other statistical models for breast cancer survival Harry B. Burke Department of Medicine New York Medical College Valhalla, NY 10595 David B. Rosen Department of Medicine New York Medical College Valhalla, NY 10595 Philip H. Goodman Department of Medicine University of Nevada School of Medicine Reno, Nevada 89520 Abstract The TNM staging system has been used since the early 1960's to predict breast cancer patient outcome. In an attempt to increase prognostic accuracy, many putative prognostic factors have been identified. Because the TNM stage model can not accommodate these new factors, the proliferation of factors in breast cancer has lead to clinical confusion. What is required is a new computerized prognostic system that can test putative prognostic factors and integrate the predictive factors with the TNM variables in order to increase prognostic accuracy. Using the area under the curve of the receiver operating characteristic, we compare the accuracy of the following predictive models in terms of five year breast cancer-specific survival: pTNM staging system, principal component analysis, classification and regression trees, logistic regression, cascade correlation neural network, conjugate gradient descent neural, probabilistic neural network, and backpropagation neural network. Several statistical models are significantly more ac1064 Harry B. Burke, David B. Rosen, Philip H. Goodman curate than the TNM staging system. Logistic regression and the backpropagation neural network are the most accurate prediction models for predicting five year breast cancer-specific survival 1 INTRODUCTION For over thirty years measuring cancer outcome has been based on the TNM staging system (tumor size, number of lymph nodes with metastatic disease, and distant metastases) (Beahr et. al., 1992). There are several problems with this model (Burke and Henson, 1993). First, it is not very accurate, for breast cancer it is 44% accurate. Second its accuracy can not be improved because predictive variables can not be added to the model. Third, it does not apply to all cancers. In this paper we compare computerized prediction models to determine if they can improve prognostic accuracy. Artificial neural networks (ANN) are a class of nonlinear regression and discrimination models. ANNs are being used in many areas of medicine, with several hundred articles published in the last year. Representative areas of research include anesthesiology (Westenskow et. al., 1992), radiology (Tourassi et. al., 1992), cardiology (Leong and Jabri, 1982), psychiatry (Palombo, 1992), and neurology (Gabor and Seyal, 1992). ANNs are being used in cancer research including image processing (Goldberg et. al., 1992) , analysis of laboratory data for breast cancer diagnosis (0 Leary et. al., 1992), and the discovery of chemotherapeutic agents (Weinstein et. al., 1992). It should be pointed out that the analyses in this paper rely upon previously collected prognostic factors. These factors were selected for collection because they were significant in a generalized linear model such as the linear or logistic models. There is no predictive model that can improve upon linear or logistic prediction models when the predictor variables meet the assumptions of these models and there are no interactions. Therefore he objective of this paper is not to outperform linear or logistic models on these data. Rather, our objective is to show that, with variables selected by generalized linear models, artificial neural networks can perform as well as the best traditional models . There is no a priori reason to believe that future prognostic factors will be binary or linear, and that there will not be complex interactions between prognostic factors. A further objective of this paper is to demonstrate that artificial neural networks are likely to outperform the conventional models when there are unanticipated nonmonotonic factors or complex interactions. 2 METHODS 2.1 DATA The Patient Care Evaluation (PCE) data set is collected by the Commission on Cancer of the American College of Surgeons (ACS). The ACS, in October 1992, requested cancer information from hospital tumor registries in the United States. The ACS asked for the first 25 cases of breast cancer seen at that institution in 1983, and it asked for follow up information on each of these 25 patients through the date of the request. These are only cases of first breast cancer. Follow-up information included known deaths. The PCE data set contains, at best, eight year follow-up. Prediction Accuracy of Models for Breast Cancer Survival 1065 We chose to use a five year survival end-point. This analysis is for death due to breast cancer, not all cause mortality. For this analysis cases with missing data, and cases censored before five years, are not included so that the prediction models can be compared without putting any prediction model at a disadvantage. We randomly divided the data set into training, hold-out, and testing subsets of 3,100, 2,069, and 3,102 cases, respectively. 2.2 MODELS The TMN stage model used in this analysis is the pathologic model (pTNM) based on the 1992 American Joint Committee on Cancer's Manual for the Staging of Cancer (Beahr et. al., 1992). The pathologic model relies upon pathologically determined tumor size and lymph nodes, this contrasts with clinical staging which relies upon the clinical examination to provide tumor size and lymph node information. To determine the overall accuracy of the TNM stage model we compared the model's prediction for each patient, where the individual patient's prediction is the fraction of all the patients in that stage who survive, to each patient's true outcome. Principal components analysis, is a data reduction technique based on the linear combinations of predictor variables that minimizes the variance across patients (Jollie, 1982). The logistic regression analysis is performed in a stepwise manner, without interaction terms, using the statistical language S-PLUS (S-PLUS, 1992), with the continuous variable age modeled with a restricted cubic spline to avoid assuming linearity (Harrell et. al., 1988). Two types of Classification and Regression Tree (CART) (Breiman et. al., 1984) analyses are performed using S-PLUS. The first was a 9-node pruned tree (with 10-fold cross validation on the deviance), and the second was a shrunk tree with 13.7 effective nodes. The multilayer perceptron neural network training in this paper is based on the maximum likelihood function unless otherwise stated, and backpropagation refers to gradient descent. Two neural networks that are not multilayer perceptrons are tested. They are the Fuzzy ARTMAP neural network (Carpenter et. al., 1991) and the probabilistic neural network (Specht, 1990). 2.3 ACCURACY The measure of comparative accuracy is the area under the curve of the receiver operating characteristic (Az). Generally, the Az is a nonparametric measure of discrimination. Square error summarizes how close each patient's predicted value is to its true outcome. The Az measures the relative goodness of the set of predictions as a whole by comparing the predicted probability of each patient with that of all other patients. The computational approach to the Az that employs the trapezoidal approximation to the area under the receiver operating characteristic curve for binary outcomes was first reported by Bamber (Bamber, 1975), and later in the medical literature by Hanley (Hanley and McNeil, 1982). This was extended by Harrell (Harrell et. al., 1988) to continuous outcomes. 1066 Harry B. Burke, David B. Rosen, Philip H. Goodman Table 1: PCE 1983 Breast Cancer Data: 5 Year Survival Prediction, 54 Variables. PREDICTION MODEL ACCURACY· SPECIFICATIONS pTNM Stages Principal Components Analysis CART, pruned CART, shrunk Stepwise Logistic regression Fuzzy ARTMAP ANN Cascade correlation ANN Conjugate gradient descent ANN Probabilistic ANN Backpropagation ANN .720 .714 .753 .762 .776 .738 .761 .774 .777 .784 O,I,I1A,I1B,IIIA,I1IB,IV one scaling iteration 9 nodes 13.7 nodes with cubic splines 54-F2a, 128-1 54-21-1 54-30-1 bandwidth = 16s 54-5-1 * The area under the curve of the receiver operating characteristic. 3 RESULTS All results are based on the independent variable sample not used for training (i.e., the testing data set), and all analyses employ the same testing data set. Using the PCE breast cancer data set, we can assess the accuracy of several prediction models using the most powerful of the predictor variables available in the data set (See Table 1). Principal components analysis is not expected to be a very accurate model; with one scaling iteration, its accuracy is .714. Two types of classification and regression trees (CART), pruned and shrunk, demonstrate accuracies of .753 and .762, respectively. Logistic regression with cubic splines for age has an accuracy of .776. In addition to the backpropagation neural network and the probabilistic neural network, three types of neural networks are tested. Fuzzy ARTMAP's accuracy is the poorest at .738. It was too computationally intensive to be a practical model. Cascade-correlation and conjugate gradient descent have the potential to do as well as backpropagation. The PNN accuracy is .777. The PNN has many interesting features, but it also has several drawbacks including its storage requirements. The backpropagation neural network's accuracy is .784.4. 4 DISCUSSION For predicting five year breast cancer-specific survival, several computerized prediction models are more accurate than the TNM stage system, and artificial neural networks are as good as the best traditional statistical models. References Bamber D (1975). The area above the ordinal dominance graph and the area below the receiver operating characteristic. J Math Psych 12:387-415. Beahrs OH, Henson DE, Hutter RVP, Kennedy BJ (1992). Manual for staging of Prediction Accuracy of Models for Breast Cancer Survival 1067 cancer, 4th ed. Philadelphia: JB Lippincott. Burke HB, Henson DE (1993). Criteria for prognostic factors and for an enhanced prognostic system. Cancer 72:3131-5. Breiman L, Friedman JH, Olshen RA (1984). Classification and Regression Trees. Pacific Grove, CA: Wadsworth and Brooks/Cole. Carpenter GA, Grossberg S, Rosen DB (1991). Fuzzy ART: Fast stable learning and categorization of analog patterns by an adaptive resonance system. Neural Networks 4:759-77l. Gabor AJ, M. Seyal M (1992) . Automated interictal EEG spike detection using artificial neural networks. Electroencephalogr Clin Neurophysiology 83:271-80. Goldberg V, Manduca A, Ewert DL (1992). Improvement in specificity of ultrasonography for diagnosis of breast tumors by means of artificial intelligence. Med Phys 19:1275-8l. Hanley J A, McNeil BJ (1982). The meaning of the use of the area under the receiver operating characteristic (ROC) curve. Radiology 143:29-36. Harrell FE, Lee KL, Pollock BG (1988). Regression models in clinical studies: determining relationships between predictors and response. J Natl Cancer Instit 80:1198-1202. Jollife IT (1986). Principal Component Analysis. New York: Springer-Verlag, 1986. Leong PH, J abri MA (1982). MATIC - an intracardiac tachycardia classification system. PACE 15:1317-31,1982. O'Leary TJ, Mikel UV, Becker RL (1992). Computer-assisted image interpretation: use of a neural network to differentiate tubular carcinoma from sclerosing adenosis. Modern Pathol 5:402-5. Palombo SR (1992). Connectivity and condensation in dreaming. JAm Psychoanal Assoc 40:1139-59. S-PLUS (1991), v 3.0. Seattle, WA; Statistical Sciences, Inc. Specht DF (1990). Probabilistic neural networks. Neural Networks 3:109-18. Tourassi GD, Floyd CE, Sostman HD, Coleman RE (1993). Acute pulmonary embolism: artificial neural network approach for diagnosis. Radiology 189:555-58. Weinstein IN, Kohn KW, Grever MR et. al. (1992) Neural computing in cancer drug development: predicting mechanism of action. Science 258:447-51. Westenskow DR, Orr JA, Simon FH (1992). Intelligent alarms reduce anesthesiologist's response time to critical faults. Anesthesiology 77:1074-9, 1992.	1994	84
981	Learning direction in global motion: two classes of psychophysically-motivated models V. Sundareswaran Lucia M. Vaina* Intelligent Systems Laboratory, College of Engineering, Boston University 44 Cummington Street, Boston, MA 02215 Abstract Perceptual learning is defined as fast improvement in performance and retention of the learned ability over a period of time. In a set of psychophysical experiments we demonstrated that perceptual learning occurs for the discrimination of direction in stochastic motion stimuli. Here we model this learning using two approaches: a clustering model that learns to accommodate the motion noise, and an averaging model that learns to ignore the noise. Simulations of the models show performance similar to the psychophysical results. 1 Introduction Global motion perception is critical to many visual tasks: to perceive self-motion, to identify objects in motion, to determine the structure of the environment, and to make judgements for safe navigation. In the presence of noise, as in random dot kinematograms, efficient extraction of global motion involves considerable spatial integration. Newsome and Colleagues (1989) showed that neurons in the macaque middle temporal area (MT) are motion direction-selective, and perform global integration of motion in their large receptive fields. Psychophysical studies in humans have characterized the limits of spatial and temporal integration in motion (Watamaniuk et. aI, 1984) and the nature of the underlying motion computations (Vaina et. al 1990). ·Please address all correspondence to Lucia Vaina 918 V. Sundareswaran, Lucia M. Vaina Since the psychophysical and neural substrate of global motion are fairly well understood, we were interested to see whether the perception of direction in such global motion stimuli can improve with practice. Studies specifically addressing this question for other early perceptual tasks have shown that improvements of performance obtained in the first experimental session are preserved in a subsequent session and retained over weeks. This is considered as perceptual learning (Gibson, 1953). Psychophysical studies of perceptual learning show that the beneficial effects of practice are lost if some stimulus parameters are changed significantly, such as orientation, spatial frequency or location in the visual field. Based on the time scale necessary for the improvement to occur, two major learning paradigms have been used in perceptual learning : slow, progressive learning (several thousand trials are required to reach stable performance) and fast !earning (improvement occurs and stabilizes in the first 100-200 trials). The idea of fast learning and the nature of its limits is attractive from a computational point of view because it encourages the exploration of practice-dependent plasticity found in the adult early visual system (Fregnac et. al., 1988, Gilbert and Wiesel 1992). A recent line of research in biologically motivated learning models originated by Poggio (Poggio, 1990) takes perceptual learning as evidence that"the brain may be able to synthesize-possibly in the cortex-appropriate task-specific modules that receive input from retinotopic cells and learn to solve the task after a short training phase in which they are exposed to examples of the task". Poggio and colleagues (1992) have illustrated this approach in learning vernier hyperacuity. Here, we adopted this general framework to study learning of direction in global motion. In contrast to Poggio et. aI's supervised learning paradigm, we used unsupervised learning both in the psychophysical experiments and modeling of learning. We designed a set of psychophysical tasks to study whether fast learning occurs in discrimination of opposite directions of global motion and to explore the limits of this learning. To model the learning, we studied two models that differ in the way they deal with noise. 2 Psychophysics Ball and Sekuler (1982, 1987) showed that discriminability of the direction of motion of two random dot patterns improved with training. In this learning paradigm, more than 2000 trials are required for reaching a stable performance. Such a "slow" learning time scale has been reported for the learning of other perceptual tasks, such as vernier acuity (McKee and Westheimer 1978, Fahle 1994), stereoacuity (Fendick and Westheimer, 1983; Ramachandran and Braddick 1973) and discrimination of line orientation (Vogels and Orban 1985). In contrast to this "slow learning," Fiorentini and Berardi (1981) showed that for learning the discrimination of complex gratings with two harmonics of different spatial phase, 100-200 trials suffice. Similarly Poggio et. al. (1992) show that a small number of trials suffice for significantly improving performance on a vernier hyperacuity task. Both studies discussed the specificity of learning to the stimulus attributes. In our study, we used a two-alternative, forced-choice psychophysical procedure to measure the subject's ability to discriminate between two opposite directions of motion in dynamic random dot patterns in which 25% of the dots provide a correlated Learning Direction in Global Motion 919 Figure 1: (a) Stimulus: a fraction of the dots (filled circles) move coherently in the signal direction; the rest (open circles) move randomly, (b) Fast improvement is observed on the first day of testing, and retained over a period of time. Each block consisted of 40 trials; data points averaged over 4 subjects, and errorbars show standard error. motion signal spatially dispersed in a masking motion noise due to random motion of the reminder 75% of dots (Fig. 1(a)). Each trial lasted 90 msec during which two frames were presented (with inter-frame interval equal to zero). A session consisted of 4-6 blocks of 40 trials each. Feedback was not provided during the experimental sessions. Observers were required to maintain fixation on a fixation mark placed at 2° from the imaginary circumference of the stimulus. To investigate the effects of practice and their retention, the discrimination of leftward vs rightward direction of motion in the display was tested first on each experimental session for 3 consecutive days and repeated 10 days later. The results are presented in Fig. 1(b). For most observers, a fast and dramatic improvement was seen in the first day. Fig. 1(b) shows that learning was maintained in subsequent days, and even ten days later without any training in between. Examination of the individual observers' data revealed that improvement of performance occurred only if they started above chance level. This suggests that this fast learning might imply the improvement of an existing representation of the stimulus. In additional experiments, we did not find transfer to another direction of motion (up/down), indicating that the learning is selective to specific characteristics of the stimulus. Details of experiments testing the limits of the learning appear in Vaina et. al (1995). 3 Modeling We propose two paradigms to model the learning found in psychophysics. Both use directionally-tuned units with properties similar to those of neurons found in MT (Maunsell and Van Essen, 1983). Schematic MT neurons used in our modeling integrate global information by summing over localized responses: ~ -(~(oi-od) Xt = ~e 2<1" • (1) i=l where (Jh is the standard deviation of the tuning, and information from n local responses is taken into account. The responses of a collection of such units (each 920 V. Sundareswaran, Lucia M. Vaina (a) (b) Figure 2: Architectures of the models: (a) Learning to Accommodate: Cluster Gaussians !I and h operate on input vector X to decide global motion direction, (b) Learning to Ignore: Global motion direction is computed as a weighted combination of units' preferred directions. tuned to a different direction) form the input vector to the models. 3.1 Learning to accommodate This model is based on grouping similar data. In this paradigm, after learning, the network has an estimate of the contribution of noise, and takes it into account when performing the task of discriminating between two different directions of motion; so, we say that the model "learns to accommodate" the noise. Fig 2(a) contains a schematic of this model. The representation vector consists of responses Xl.X2, ..• , Xn of the directionally-tuned units .. Clustering is done in the space of the representation vectors. The model is a combination of HyperBF-like functions and clustering (Poggio et. aI, 1992, Moody and Darken 1989). We use gaussians with mean at the cluster centers, a.nd "move" the cluster centers by a learning algorithm. A cluster gaussian computes a gaussian function of the representation vector for the current stimulus from the current center of the corresponding cluster. We say "current" because the center is moved as the learning proceeds. At any given point, a center is at the current best estimate of the center of the corresponding data cluster. More precisely, the learning rule to modify the jth coordinate of the center is given by: c(t-+:l) = c(t) . + '11 * (x~t) _ c(t) .) W,) W,) '/ ) W,) , (2) where w is the index of "winning" cluster and Cw,j is the jth coordinate of the center for the wth cluster. This moves the center towards the new data vector X that has been judged to belong to the wth cluster. The parameter TJ controls the learning rate. Learning Direction in Global Motion 921 3.2 Learning to ignore The learning involved in this approach is Hebbian, and is termed "learning to ignore," because in this weighted averaging scheme, as learning occurs, the weights for the noise response are progressively reduced to zero, leaving only the contribution from the signal. In other words, the network learns to ignore the noise (Vaina et. al 1995). The model's output is a global motion direction. If the weights associated with the responses Xl, X2,· . · Xn are WI, W2,'" W n , the global motion direction is calculated as II -1 (L: wiXisinOi ) Uo = tan '" ' w WiXiCOSOj where ti is the tuned direction (angle) of the ith unit. A schematic of the model is shown in Fig. 2(b). The global direction is judged to be rightward if +Ot > 80 > -Ot. We have examined two different learning rules: exposure-based learning, and selfsupervised learning. Exposure-based learning: The weight corresponding to a unit is incremented by an amount proportional to the current weight. Only units whose response values are above a certain threshold are allowed to increase their weights. This learning rule favors units that fire consistently: (3) where rt is a threshold, and 1] is a small fraction that controls the learning rate. Self-supervised learning: The weight corresponding to a unit is increased by an amount proportional to the product of the current weight and a decreasing function (exponential) of the angular difference between the calculated global motion direction and the direction of tuning of the unit: Wi ~ Wi + 1]Wie(- (O;-OO)2/2u e) , if Xi> rt . In this case, the model uses its own estimate of the global direction as an internal feedback to determine the learning. An approach similar to this model for learning vernier hyperacuity was proposed by Weiss et. al (1993). 4 Experiments For the simulations, the input (motion) vectors were represented by their angles relative to the positive horizontal axis (Le., the magnitude is ignored). The responses of the directionally-tuned units can be directly computed from the angles (see Eqn. 1; alternatively, cosine tuning functions were used, and similar results were obtained). For each trial, the coherent motion direction was randomly decided. In the experiments, O'h was set to ~ , where nd is the number of unit preferred directions; the n d directions Ot are chosen by uniformly dividing nd in to 211'. In all the experiments, eight preferred directions were used; each trial contained 40 random dots moving with a correlation of25% (the same correlation as in the tests with human subjects); 922 V. Sundareswaran, Lucia M. Vaina 1.00 0.9 I 0.90 0.8 u 0.80 -G- Espoeure 8 0.70 0.7 --.-- Self·npervised 'f ~ 0.60 0.6 Do O.S 14 0 s 10 1S 20 Block No Block No Figure 3: Results from typical simulation runs. On the left is the curve for the learning to accommodate model (1} = 0.005; cluster (j = 0.5) and on the right are the curves for the learning to ignore model. each block consisted of 50 trials. The performance is measured by fraction correct, corresponding to the fraction of the inputs that were correctly classified. For both models, we did simulations to justify the architecture of the models by disabling learning, and studying the performance of the models for increasing correlation. The performance was qUalitatively similar to that described in perceptual studies of these stimuli in humans and monkeys. Learning curves from simulations of both models are shown in Fig. 3. These are results from averaging over the performance in ten simulation runs. As can be seen, the models learn to do the discrimination. While both models successfully learned to do the direction discrimination, there are quantitative differences in the learning curves. The learning to accommodate paradigm improved very rapidly (somewhat faster than the human subjects). The exposure-based learning rule for the learning to ignore paradigm learned at a rate comparable with the human observers. However, the final performance in this case was not very stable, and oscillated (not shown), consistent with the observations of Weiss et. al (1993) in learning vernier hyperacuity. The self-supervised rule, on the other hand, reached a stable level of performance, but the learning was slower; the reason is that this learning rule exhibits instability if a high value of learning rate (1}) is used. If either model trained on inputs containing horizontal correlated motion was later presented with inputs containing vertical correlated motion, the performance dropped to pre-training levels. This non-transfer is consistent with psychophysical results mentioned in Section 2. 5 Discussion In this study, we focused on learning of global motion direction, and not on motion perception. Motion perception is well-understood both physiologically and psyLearning Direction in Global Motion 923 chophysically. In a series of studies of neuronal correlates of the perceptual decision of direction of motion in stochastic motion signals like the stimuli used here, together with studies of performance on these stimuli on monkeys with MT lesions, Newsome and his collaborators (for a review, Newsome et. aI, 1989) provide strong support for the hypothesis that perceptual judgements of motion direction are by and large based on the directional signals carried by MT neurons. Salzman and Newsome (1994) reported that in trained monkeys, the perception of motion in stochastic noise is more likely mediated by a winner-take-all mechanism than by a weighted averaging mechanism. Interestingly, our clustering model, after learning, is behaviorally similar to the winner-take-all mechanism: the dominating component of the representation results in the assignment to the closest cluster. Our psychophysical studies clearly demonstrate that perceptual learning of global motion direction occurs. While significant progress has been made to understand the mechanisms and underlying circuitry of learning (Zohary et. al 1994) as yet there is no satisfactory biologically explanation of how and where this learning may occur (Zohary and Newsome 1994). Our focus in this paper was on computational models of learning direction in global motion. For this, we proposed two approaches, which differ in the way they deal with noise: one learns to accommodates noise, and the other learns to ignore it. We do not advocate that one or the other of the models we proposed here provides the biologically correct choice for this task. However, together with the psychophysics described here these models suggest new experiments which we are now exploring both psychophysically and computationally. We hope that by closely connecting models and psychophysics while keeping in mind the aim of neuronal compatibility, we will make progress in understanding how the cortex learns so fast to discriminate direction of motion in extremely noisy situations. Acknowledgments This research was conducted at the Intelligent Systems Laboratory of Boston University, College of Engineering. LMV and VS were supported in part by grants from the Office of Naval Research (# N00014-93-1-0381) and the National Institute of Health (EY R01- 07861) to LMV. VS was in part supported by the Boston University College of Engineering Dean's postdoctoral fellowship. We gratefully acknowledge additional financial support for VS from the Dean's special research fund. References [1] K. Ball and R. Sekuler. Direction-specific improvement in motion discrimination. Vision Research, 27(6):953-965, 1987. [2] M. Fahle. Human pattern recognition: parallel processing and perceptual learning. Perception, 23:411-427, 1994. [3] M. Fendick and G. Westheimer. Effects of practice and the separation of test targets on foveal and perifoveal hyperacuity. Vision Research, 23: 145-150, 1983. 924 V. Sundareswaran, Lucia M. Vaina [4] A. Fiorentini and N. Berardi. Perceptual learning specific for orientation and spatial frequency. Nature, 287(5777):43-44, September 1980. [5] Y. Fregnac, D. Shulz, S. Thorpe, and E. Bienstock. A cellular analogue of visual cortical plasticity. Nature, 333:367-370, 1988. [6] E. J. Gibson. Improvements in perceptual judgement as a function of controlled practice of training. Psychology bulletin, 50:402-431, 1953. [7] C. D. Gilbert and T. N. Wiesel. Receptive field dynamics in adult primary visual cortex. Nature, 356:150-152, 1992. [8] A. Karni and D. Sagi. Where practice makes perfect in texture discrimination: Evidence for primary visual cortex plasticity. Proc. Natl. Acad. Sci. USA, 88:4966- 4970, June 1991. [9] J. H. R. Maunsell and D. C. Van Essen. Functional properties of neurons in the middle temporal visual area of the macaque monkey i: selectivity for stimulus direction, speed, and orientation. J. Neurophysiology, 49:1127-1147, 1983. [10] S. P. McKee and G. Westheimer. Improvement in venier acuity with practice. Perception €3 Psychophysics, 24:258-262, 1978. [11] W. T. Newsome, K. H. Britten, and J. A. Movshon. Neuronal correlates of a perceptual decision. Nature, 341:52-54, 1989. [12] T. Poggio. A theory of how the brain might work. In Cold Spring Harbor Symposia on Quantitative Biology, pages 899-910. Cold Spring Harbor Laboratory Press, 1990. [13] T. Poggio, M. Fahle, and S. Edelman. Fast perceptual learning in visual hyperacuity. Science, 256:1018-1021, 1992. [14] T. Poggio and F. Girosi. A theory of networks for approximation and learning. AI Memo 1140, M.LT, July 1989. [15] V. S. Ramachandran and O. Braddick. Orientation-specific learning in stereopsis. Perception, 2:371-376, 1973. [16] C. D. Salzman and W. T. Newsome. Neural mechanisms for forming a perceptual decision. Science, 264:231-237, 1994. [17] L. M. Vaina, V. Sundareswaran, and J. Harris. Computational learning and natural learning. Cognitive Brain Research, 1995. (in press). [18] L. M. Vaina, N. M. Grzywacz, and M. LeMay. Structure from motion with impaired local-speed and global motion-field computations. Neuml Computation, 2:420-435, 1990. [19] R. Vogels and G. A. Orban. The effect of practice on the oblique effect in line orientation judgements. Vision Research, 25:1679-1687, 1985. [20] S. Watamaniuk, R. Sekuler, and D. Williams. Direction perception in complex dynamic displays: the integration of direction information. Vision Research, 24:55- 62, 1984. [21] Y. Weiss, S. Edelman, and M. Fahle. Models of perceptual learning in vernier hyperacuity. Neural Computation, 5:695-718, 1993. [22] E. Zohary, S. Celebrini, K. H. Britten, and W. T. Newsome. Neuronal plasticity that underlies improvement in perceptual performance. Science, 263:1289-1292, March 1994. [23] E. Zohary and W. T. Newsome. Perceptual learning in a direction discrimination task is not based upon enhanced neuronal sensitivity in the sts. Investigative Ophthalmology and Visual Science supplement, page 1663, 1994.	1994	85
982	On-line Learning of Dichotomies N. Barkai Racah Institute of Physics The Hebrew University Jerusalem, Israel 91904 naamaCfiz.huji.ac.il H. S. Seung AT&T Bell Laboratories Murray Hill, NJ 07974 seungCphysics.att.com H. Sompolinsky Racah Institute of Physics The Hebrew University Jerusalem, Israel 91904 and AT&T Bell Laboratories haimCfiz.huji.ac.il Abstract The performance of on-line algorithms for learning dichotomies is studied. In on-line learning, the number of examples P is equivalent to the learning time, since each example is presented only once. The learning curve, or generalization error as a function of P, depends on the schedule at which the learning rate is lowered. For a target that is a perceptron rule, the learning curve of the perceptron algorithm can decrease as fast as p- 1 , if the schedule is optimized. If the target is not realizable by a perceptron, the perceptron algorithm does not generally converge to the solution with lowest generalization error. For the case of unrealizability due to a simple output noise, we propose a new on-line algorithm for a perceptron yielding a learning curve that can approach the optimal generalization error as fast as p-l/2. We then generalize the perceptron algorithm to any class of thresholded smooth functions learning a target from that class. For "well-behaved" input distributions, if this algorithm converges to the optimal solution, its learning curve can decrease as fast as p-l. 1 Introduction Much work on the theory of learning from examples has focused on batch learning, in which the learner is given all examples simultaneously, or is allowed to cycle through them repeatedly. In many situations, it is more natural to consider on-line learning paradigms, in which at each time step a new example is chosen. The examples are never recycled, and the learner is not allowed to simply store them (see e.g, Heskes, 1991; Hansen, 1993; Radons, 1993). Stochastic approximation theory (Kushner, 1978) provides a framework for understanding of the local convergence properties of on-line learning of smooth functions. This paper addresses the problem of on-line learning of dichotomies, for which no similarly complete theory yet exists. 304 N. Barkai, H. S. Seung, H. Sompolinsky We begin with on-line learning of perceptron rules. Since its introduction in the early 60's, the perceptron algorithm has been used as a simple model of learning a binary classification rule. The algorithm has been proven to converge in finite time and to yield a half plane separating any set of linearly separable examples. The perceptron algorithm, however, is not efficient in the sense of distribution-free PAC learning (Valiant, 1984), for one can construct input distributions that require an arbitrarily long convergence time. In a recent paper (Baum, 1990) Baum proved that the perceptron algorithm applied in an on-line mode, converges as p-I/3 when learning a half space under a uniform input distribution, where P is the number of presented examples drawn at random. For on-line learning P is also the number of time steps. Baum also generalized his result to any "non-malicious" distribution. Kabashima has found the same power law for learning a two-layer parity machine with non-overlapping inputs, using an on-line least action algorithm (Kabashima, 1994). If efficiency is measured only by the number of examples used (disregarding time), these particular on-line algorithms are much worse than batch algorithms. Any batch algorithm which is able to correctly classify a given set of P examples will converge as p- I (Vapnik, 1982; Amari, 1992; Seung, 1992). In this paper, we construct on-line algorithms that can actually achieve the same power law as batch algorithms, demonstrating that the results of Baum and Kabashima do not reflect a fundamental limitation of on-line learning. In Section 3, we study on-line algorithms for perceptron learning of a target rule that is not realizable by a perceptron. Here it is nontrivial to construct an algorithm that even converges to the optimal one, let alone to optimize the rate of convergence. For the special case of a target rule that is a percept ron corrupted by output noise this can be done. In Section 4, our results are generalized to dichotomies generated by thresholding smooth functions. In Section 5 we summarize the results. 2 On-line learning of a perceptron rule We consider a half space rule generated by a normalized teacher perceptron Wo ERN, Wo' Wo = 1 such that any vector 5 E RN is given a label uo(5) = sgn(Wo ·5). We study the case of a Gaussian input distribution centered at zero with a unit variance in each direction in space: N 1 2 P(5) = II _e-s, /2 ;=1 v'21r (1) Averages over this input distribution will be written with angle brackets (). A student percept ron W is trained by an on-line perceptron algorithm. At each time step, an input 5 E RN is drawn at random, according to distribution Eq. (1) and the student's output u(5) = sgn(W . 5) is calculated. The student is then updated according to the perceptron rule: W' = W + ~f(5;W)uo(5)5 (2) and is then normalized so that W . W = 1 at all times. The factor f(5; W) denotes the error of the student perceptron on the input 5: f = 1 if u(5)uo(5) = 1, and 0 otherwise. The learning rate 1/ is the magnitude of change of the weights at each time step. It is scaled by N to ensure that the change in the overlap R = W· Wo is of order lIN. Thus, a change of 0(1) occurs only after presentation of P = O(N) examples. The performance of the student is measured by the generalization error, defined as the probability of disagreement between the student and the teacher on an arbitrary input fg = (f(5; W»). In the present case, fg is cos- I R fg = --7r-' (3) Although for simplicity we analyze below the performance of the perceptron rule (2) only for large N, our results apply to finite N as well. Multiplying Eq. (2) by Wo after incorporation of the normalization operation and averaging with respect to the input distribution (1), yields the following differential equation for R(a) where a = PIN, (4) On-line Learning of Dichotomies 305 Here terms of order J17/N have been neglected. The evolution of the overlap R, and thus of the generalization error, depends on the schedule at which the learning rate 17 decreases. We consider two cases, a constant 17 and a time-dependent 17. Constant learning rate: When 17 is held fixed, Eq. (4) has a stable fixed point at R < 1, and hence eg converges to an '7-dependent nonzero value eoo ('7). For '7 «: 1, 1- Roo('7) oc '72 and eg oc VI - R is therefore proportional to '7, eoo('7) = '7/..f2i3 . (5) The convergence to this value is exponential in 0, eg(o) - eoo('7) '" exp(-'7o/$). Time-dependent learning rate: Convergence to eg = 0 can be achieved if '7 decreases slowly enough with o. We study the limiting behaviour of the system for '7 which is decreasing with time as '7 = (7]0$) o-z. z > 1. In this case the rate is reduced too fast before a sufficient number of examples have been seen. This results in R which does not converge to 1 but instead to a smaller value that depends on its initial value. z < 1. The system follows the change in '7 adiabatically. Hence, to first order in 0-1, eg(o) = eoo('7(o». Thus, eg converges to zero with an asymptotic rate eg(o) '" o-z. z = 1. The behaviour of the system depends on the prefactor '70: eg (1 '7~)1 71"'70- 10 y'logo a A 0'1<> '70> 1 7]0 = 1 7]0 < 1 (6) where A depends on the initial condition. Thus the optimal asymptotic change of '7 is 2..;21i / 0, in which case the error will behave asymptotically as eg(o) '" 1.27/0. This is not far from the batch asymptotic (Seung, 1992) eg(o) '" 0.625/0. We have confirmed these results by numerical simulation of the algorithm Eq. (2). Figure 1 presents the results of the optimalleaming schedule, i.e., '7 = 2..;21i/o. The numerical results are in excellent agreement with the prediction eg(o) = 1.27/0 for the asymptotic behavior. Finally, we note that our analysis of the time-dependent case is similar to that of Kabashima and Shinomoto for a different on-line learning problem (Kabashima, 1993). 3 On-line learning of a percept ron with output noise In the case discussed above, the task can be fully realized by a perceptron, i.e., there is a perceptron W such that eg = O. In more realistic situations a percept ron will only provide an approximation of the target function, so that the minimal value of eg is greater than zero. These cases are called unrealizable tasks. A drawback of the above on-line algorithm is that, for a general unrealizable task, it does not converge to the optimal perceptron, i.e., it does not approach the minimum of ego To illustrate this fact we consider a perceptron rule corrupted by output noise. The label of an input S is O"o(S), where O"o(S) = sgn(Wo . S) with probability 1 - p, and - sgn(Wo . S) with probability p. We assume 0 $ p $ 1/2. For reasons which will become clear later, the input distribution is taken as a Gaussian centered at U N P(S) = n _1_e-(S,-u.)'/2 (7) ;=1 ..;21i In this case eg is given by eg =p+ (lq DYH(~)+l°O DYH(~»). -00 1 - R _q 1 - R (8) where qo = U . W 0 denotes the overlap between the center of the distribution and the teacher perceptron, and q = U . W is the overlap between the center of the distribution and W. The integrals in Eq. (8) are 306 N. Barkai, H. S. Seung, H. Sompolinsky with respect to a Gaussian measure Dy = exp(-y2/2)/.J21i and H(x) = J.,oo Dy. Note that the optimal perceptron is the teacher W = Wo i.e., R = 1, q = 110, which yields the minimal error fmin = p. First, we consider training with the normalized perceptron rule (2). In this case, we obtain differential equations for two variables: R and q. Solving these equations we find that in general, W converges to a vector with a direction which is in the plane of Wo and U and is does not point in the direction of Wo even in the limit of,., -+ O. Here we present the result for the limit of,., -+ 0 and small noise level, i.e., p ¢: 1. In this case, we obtain for foo('" = 0) foo(O) = P + p (1 - 2H(qo»~ + O(p2) 1+(u2-q~) (9) where u = lUI is the magnitude of the center of the input distribution. For p = 0, the only solution is R = 1 and q = qo, in agreement with the previous results. For p > 0 the optimal solution is retrieved only in the following special cases: (i) the input distribution is isotropic, i.e., qo = u = OJ (ii) when U is parallel to W o, i.e., u = qoj and (iii) when U is orthogonal to W o, i.e., qo = O. This holds also for large value of p. In these special cases, the symmetry of the input distribution relative to the teacher vector, guarantees that the deviations from W = Wo incurred by the inputs that come with the wrong label cancel each other on average. According to Eq. (9), for other directions of U, fg is above the optimal value. Note that the additional term in fg is of the same order of magnitude (O(P» as the minimal error. In the following we suggest a modified on-line algorithm for learning a perceptron rule with output noise. The student weights are changed according to W' = W + ~f(S; W)ITo(S)(S - T(S» (10) followed by a normalization of W. This algorithm differs from the perceptron algorithm in that the change in W is not proportional to the present input, but to a shifted vector. The shifting vector T(S), is determined by the requirement that the teacher Wo will be a fixed point of the algorithm in the limit of,., -+ O. This is equivalent to the condition (fo(S)lTo(S)(S - T(S») = 0 (11) where foeS) is the error function for S when W = Woo This condition does not determine T uniquely. A simple choice is one for which T is independent of S. This leads to T = (sgn(Wo' S)S) = (IToS) (sgn(Wo . S» (ITo) (12) where we used the fact that for any S, fo(S)lTo(S) equals - sgn(Wo . S) with probability p, and zero with probability (1 - p). This uniform shift is possible only when (O'o) ~ 0, namely when the average frequencies of +1 and -1 labels are not equal. If this is not the case, one has to choose nonuniform forms of T(S). Note that in general T has to be learned so that Eq. (10) has to be supplemented by appropriate equations for changing T. In the case of Eq. (12), one can easily learn separately the numerator and denominator by running averages of O'oS and 0'0, respectively. We have studied analytically the above algorithm for the case of the Gaussian input distribution Eq. (7), in the limit of large N. The shifting vector is given by T = U + Wo ~ exp( -q~/2) V 1T 1 - 2H(qo) (13) The differential equations for the overlaps R and q in the neighborhood of the point R = 1 and q = qo are, daR da doq da (14) where oR = 1 - R and oq = qo - q. In the limit,., -+ 0, R = 1 and q = qo is indeed a stable fixed point of the algorithm, so that the student converges to the optimal perceptron W o, and hence the feneralization error converges to its minimal value fmin = p. Since, unlike Eq. (4), the coefficient of the,., term in Eq. On-line Learning of Dichotomies 307 (14) is constant, c5Roo(1]) <X 1], for small fixed 1], and not to 1]2. Thus, in this case, the generalization error approaches, in the limit a -t 00, the value exp(-qa/4) t:oo (1]) :;;: p +.,fiiP (271"3)1/4 (15) For a time-dependent 1], the convergence to the optimal weights depends on the choice of 1](n), as in the case of the noiseless perceptron rule. For 1] :;;: (1]0 J1i72 exp( q5 12) ) a-' , with z ~ 1, the error converges to p. For z < 1, to first order in 1/a, t:g(a) :;;: t:oo(1](a)) , yielding t:g(a) - p ~ a-·/2 • When z :;;: 1, the rate of convergence depends on the value of 1]0. ( ) { a-l/2 1] > 1 t:g a - p ~ a-'1oj:~, ~ < 1 and logarithmic corrections to a- 1/ 2 for 1]0 :;;: 1. Thus, the optimal rate of convergence is t: (a) _ p ~ f2P 9 V~ which is achieved for 1]0 = 2. (16) (17) (18) We have tested successfully this algorithm by simulations of learning a perceptron rule with output noise with several input distributions, including the Gaussian, of Eq. (7). Figure 2 presents the generalization error as a function of a for the Gaussian distribution, with p = 0.2, and we have chosen 1]0 = 2. The error converges to the optimal value 0.2 as a- I / 2 in agreement with the theory. For comparison the result of the usual perceptron algorithm is also presented. This algorithm converges to t:g ~ 0.32, clearly larger than the optimal value. 4 On-line learning of thresholded smooth functions Our results for the realizable perceptron can be extended to a more general class of dichotomies, namely thresholded smooth functions. They are defined as dichotomies of the form O"(S; W) = sgn(f(Sj W)) (19) where 1 is a differentiable function of a set of parameters, denoted by W, and S is the input vector. We consider here the case of a realizable task, where the examples are given with labels 0"0 corresponding to a target machine W 0 which is in the W space. For this task we propose the following generalization of the perceptron rule (2) W' = W + 1]t:(Sj W)O"o(S)V I(S; W) (20) where V denotes a gradient w.r.t. W. Then, as we argue below, the vector Wo is a stable fixed point in the limit of 1] -t O. Furthermore, for constant small 1] the residual error scales as t:oo <X 1]. For 1] ~ a-' , z < 1, t:g(a) ~ t:oo(1](a)) '""' a-·. To show this, let us consider for simplicity the one-dimensional case, w' = w + 1]g(w, s), where 81 g(w,s) = 9(-I(w,s)l(wo,s))sgn(f(wo,s))8w . (21) This equation can be converted into a Markov equation for the probability distribution, P(w,n) (Van Kampen, 1981) P(w,n + 1):;;: J dw'W(w'lw)P(w' ,n) (22) where W(wlw') =< c5(w' - w -1]g(w, s)) > is the transition rate from w to w'. In the limit of small fixed 1], the equilibrium distribution, Poo, can be shown to have the following scaling form, 1 Poo(w;1]):;;: -F(c5wl1]) (23) 1] 308 N. Barkai, H. S. Seung, H. Sompolinsky where ow = w - Wo and F(x) obeys the following difference equation LF(x) == L 9«(f~ + O"x)J~)I(f~ + O"x)IF(x + o"/~) -lxIF(x) = 0 (24) C1=±1 where J6 is the value of the gradient aJ(wo, s)law at the decision boundary of J(wo, s), namely at the point s obeying J(wo,s) = O. Note that since we are interested in normalizable solutions of Eq. (24), F(x) has to vanish for for all x > 1161. This result is valid provided the input distribution is smooth and nonvanishing near the decision boundary. Furthermore, aJ law at Wo may not vanish on the decision boundary. Under the same conditions, it can be shown that the error is homogeneous in ow with degree 1, hence it should scale linearly with "I, i.e., 1':00 oc "I. It should be noted that, unlike other on-line learning problems (Heskes, 1991; Hansen, 1993; Radons, 1993), the equilibrium distribution is our case is not Gaussian. For a time-dependent "I of the form "I = TJon- z , z < I, pew, n) at long times is of the form (25) where F is the stationary distribution, given by Eq. (24) and the coefficient of the correction, G, solves the inhomogeneous equation dF • zx dx + zF(x) = TJoLG(x) (26) where the linear operator L is defined in Eq. (24). Thus, to leading order in inverse time, the system follows adiabatically the finite-TJ stationary distribution, yielding I':g(n) which vanishes asymptotically as I':g(n) oc TJ(n) ~ n-z . The optimal schedule is obtained for z = 1. In this case, P(w,n) = "1-1 (n)F(ow/TJ(n)) where F(x) solves the homogeneous equation dF • zx dx + zF(x) = TJoLF(x) (27) For sufficiently large "10, this equation has a solution, implying that I':g oc n-1 Similarly, the results of Section 3 can also be extended to the case of thresholded- smooth functions with a probability p of an error due to isotropic output noise. In this case, the optimal choice is again "I oc n-I yielding I':g -p Rl ,;ri. It should be noted that for this case, the probability distribution for small "I does reduce to a Gaussian distribution in owl,;ri. Using a multidimensional Markov equation, it is straightforward to extend these results to higher dimensions. The small "I limit yields equations similar to Eqs. (24-26), that involve integration over the decision boundary of J(W, S). 5 Summary and Discussion We have found that the perceptron rule (2) with normalization can lead to a variety of learning curves, depending on the schedule at which the learning rate is decreased. The optimal schedule leads to an inverse power law learning curve, I':g ~ 0-1 . Baum's results (Baum, 1990) of a non-normalized perceptron with a constant learning rate can be viewed as a special case of the above analysis. In the non-normalized perceptron algorithm, the magnitude of the student's weights grow with 0 as IWI ~ 0 1/ 3 . The time evolution of the overlap R, and thus of the generalization error is governed by the effective learning rate TJe/f = T//IWlleading via Eq. (6) to the result I':g ~ 0-1/ 3 . Similar results apply to the two-layer parity machine studied in (Kabashima, 1994). Our analysis, leading to the equations of motion (4) and (14), was based on the limit of large N and P, such that 0 = PIN remains finite. We would like to stress however, that this limit is only necessary in deriving the full form of the learning curve, i.e., R(o) for all o. On the other hand, our results for the large P asymptote of the learning curve for smailT/ are valid for finite N as well, as implied by the general treatment of the previous section. Unrealizable percept ron rules present a more complicated problem. We have presented here a modified perceptron algorithm that converges to the optimal solution in the special case of an isotropic output noise. On-line Learning of Dichotomies 309 In this case, the convergence to the optimal error is as 01-1/ 2 . This is the same power law as obtained in the standard sample complexity upper bounds (Vapnik, 1982) and in the approximate replica symmetric calculations (Seung, 1992) for batch learning of unrealizable rules. It should be stressed however, that the success of the modified algorithm in the case of an output noise depends on the fact that the errors made by the optimal solution are uncorrelated with the input. Thus, finding an on-line algorithm that can cope with other types of unrealizability remains an important problem. The learning algorithms for the perceptron rule, without and with output noise, can be generalized to learning thresholded smooth functions, assuming certain reasonable properties of the input distribution are present, as shown in Section 4. The dependence of the learning curve on the learning rate schedule remains roughly the same as in the percept ron case. This implies that on-line learning of realizable dichotomies, with possible output noise, can achieve the same power laws in the number of examples that is typical of batch learning of such rules. Furthermore, the on-line formulation possesses the theoretical virtues of addressing time as well as sample complexity, so that the same power laws imply the polynomial relationship between the time and the achieved error level. The above conclusions assume that the equilibrium state at small learning rates is unique, which in general is not the case. The issue of overcoming local minima in on-line learning is a difficult problem (Heskes, 1992) Finally, the theoretical results for on-line learning has the important advantage of not requiring the use of the often problematic replica formalism. Acknowledgements We are grateful for helpful discussions with Y. Freund, M. Kearns, R. Schapire, and E. Shamir, and thank Y. Kabashima for bringing his paper to our attention. HS is partially supported by the Fund for Basic Research of the Israeli Academy of Arts and Sciences. References S. Amari, N. Fujita, and S. Shinomoto. Four types of learning curves. Neural Comput., 4:605-618, 1992. E. B. Baum. The perceptron algorithm is fast for nonmalicious distributions. Neural Comput., 2:248-260, 1990. H. J. Kushner and D. S. Clark. Stochastic approximation methods for constrained and unconstrained systems. Springer, Berlin, 1978. L. K. Hansen, R. Pathria, and P. Salamon. Stochastic dynamics of supervised learning. J. Phys., A26:63-71, 1993. T. Heskes and B. Kappen. Learning processes in neural networks. Phys. Rev., A44:2718-2762, 1991. T. Heskes, E. T. P. Slijpen, and B. Kappen. Learning in neural networks with local minima. Phys. Rev., A46:5221-5231, 1992. Y. Kabashima. Perfect loss of generalization due to noise in k = 2 parity machines. J. Phys., A27:1917-1927, 1994. Y. Kabashima and S. Shinomoto. Incremental learning with and without queries in binary choice problems. In Proc. of IJCNN, 1993. G. Radons. On stochastic dynamics of supervised learning. J. Phys., A26:3455-3461, 1993. H. S. Seung, H. Sompolinsky, and N. Tishby. Statistical mechanics of learning from examples. Phys. Rev., A45:6056-6091, 1992. L. G. Valiant. A theory of the learnable. Commun. ACM, 27:1134-1142,1984. N. G. Van Kampen. Stochastic processes in physics and chemistry. North holland 1981. V. N. Vapnik. Estimation of Dependences based on Empirical Data. Springer-Verlag, New York, 1982. 310 N. Barkai, H. S. Seung, H. Sompolinsky 0.03 0.025 0.02 f9 0.015 0.01 0.005 0 0 0.005 0.01 0.015 0.02 1/a Figure 1: Asymptotic performance of a realizable perceptron. Simulation results for 110 :;;: 2 and N :;;: 50 (solid curve) are compared with the theoretical prediction f g :;;: 1.271a (dashed curve). 0.35 0.3 0.25 0.2 fg - P 0.15 0.1 0.05 0 0 0.1 ..... .... .. ' .. . .' . .' o • • 'c" .. ' '.:.... 0.2 0.3 0.4 lIfo .' . ' .. ~ . 0.5 0.6 0.7 Figure 2: Simulation results for on-line learning of a perceptron with output noise. Here 1Jo :;;: 2, P :;;: 0.2, N :;;: 250, U = 4, and qo :;;: -1.95. The regular percept ron learning (dashed curve) is compared with the modified algorithm (solid curve). The dashed line shows the theoretical prediction Eq. (18)	1994	86
983	Asymptotics of Gradient-based Neural Network 'fraining Algorithms Sayandev Mukherjee saymukh~ee.comell.edu School of Electrical Engineering Cornell University Ithaca, NY 14853 Terrence L. Fine tlfine~ee.comell.edu School of Electrical Engineering Cornell University Ithaca, NY 14853 Abstract We study the asymptotic properties of the sequence of iterates of weight-vector estimates obtained by training a multilayer feed forward neural network with a basic gradient-descent method using a fixed learning constant and no batch-processing. In the onedimensional case, an exact analysis establishes the existence of a limiting distribution that is not Gaussian in general. For the general case and small learning constant, a linearization approximation permits the application of results from the theory of random matrices to again establish the existence of a limiting distribution. We study the first few moments of this distribution to compare and contrast the results of our analysis with those of techniques of stochastic approximation. 1 INTRODUCTION The wide applicability of neural networks to problems in pattern classification and signal processing has been due to the development of efficient gradient-descent algorithms for the supervised training of multilayer feedforward neural networks with differentiable node functions. A basic version uses a fixed learning constant and updates all weights after each training input is presented (on-line mode) rather than after the entire training set has been presented (batch mode). The properties of this algorithm as exhibited by the sequence of iterates are not yet well-understood. There are at present two major approaches. 336 Sayandev Mukherjee, Terrence L. Fine Stochastic approximation techniques (Bucklew,Kurtz,Sethares, 1993; Finnoff, 1993; Kuan,Hornik, 1991; White, 1989) study the limiting behavior of the stochastic process that is the piecewise-constant or piecewise-linear interpolation of the sequence of weight-vector iterates (assuming infinitely many U.d. training inputs) as the learning constant approaches zero. It can be shown (Bucklew,Kurtz,Sethares, 1993; Finnoff, 1993) that as the learning constant tends to zero, the fluctuation between the paths and their limit, suitably normalized, tends to a Gaussian diffusion process. Leen and Moody (1993) and Orr and Leen (1993) have considered the Markov process formed by the sequence of iterates (again, assuming infinitely many Li.d. training inputs) for a fixed nonzero learning constant. This approach has the merit of dealing with the nonzero learning constant case anq of linking the study of the training algorithm with the well-developed literature on Markov processes. In particular, it is possible to solve (Leen,Moody, 1993) for the asymptotic distribution of the sequence of weight-vector iterates from the Chapman-Kolmogorov equation after certain assumptions have been used to simplify it considerably. However, the assumptions are unrealistic: in particular, the assumption of detailed balance does not hold in more than one dimension. This approach also fails to establish the existence of a limiting distribution in the general case. This paper follows the method of considering the sequence of weight-vector iterates as a discrete-time continuous state-space Markov process, when the learning constant is fixed and nonzero. We shall first seek to establish the existence of an asymptotic distribution, and then examine this distribution through its first few moments. It can be proved (Mukherjee, 1994), using Foster's criteria (Tweedie, 1976) for the positive-recurrence of a Markov process, that when a single sigmoidal node with one parameter is trained using the iterative form of the basic gradient-descent training algorithm (without batch-processing), the sequence of iterates of the parameter has a limiting distribution which is in general non-Gaussian, thereby qualifying the oftstated claims in the literature (see, for example, (Bucklew,Kurtz,Sethares, 1993; Finnoff, 1993; White, 1989)). However, this method proves to be intractable in the multiple parameter case. 2 THE GENERAL CASE AND LINEARIZATION IN W n The general version of this problem for a neural network", with scalar output involves training ", with the i. i.d. training sequence {( X n' Yn )}, loss function £ (;f, y, w) = ~ [y ",(~, w W (~ E lRd, Y E lR, w E lR m) and the gradient-descent updating equation for the estimates of the weight vector given by Wn+1 = Wn -/LV'1££(;f,y,w)I(Xn+1 ,Yn+l,Wn ) = W n + /L[Yn+1 - ",(W n' X n+1)]V' J£."'(w, ;f)lw",x n +1 • As is customary in this kind of analysis, the training set is assumed infinite, so that {W n}~=O forms a homogeneous Markov process in discrete time. In our analysis, the training data is assumed to come from the model Y = ",(WO,X) + Z, Asymptomatics of Gradient-Based Neural Network Training Algorithms 337 where Z and X are independent, and Z has zero mean and variance (J'2. Hence, the unrestricted Bayes estimator of Y given X, E(YIX) = "l(wO,X), is in the class of neural network estimators, and WO is the goal of training. For convenience, we define W = w - woo Assuming that J.L is small and that after a while, successive iterates, with high probability, jitter about in a close neighborhood of the optimal value wO, we make the important assumption that (1) for some 0 < k < 1 (see Section 4) 1. Applying Taylor series expansions to "l and \1 w"l and neglecting all terms Op(J.L1+2k) and higher, we obtain the following linearized form of the updating equation: where Bn+1 = J.LZn+l \1 1£"l( W, ;£)11£0 ,x n+l ' An+1 = 1m J.L(\11£"l(w,;£)I~o , Xn+l)(\11£"l(w,;£)I~o ,Xn+lf + J.LZn+1 \1 w \1 w"l( w,;£) Iwo X '-n+l 1m - J.L(Gn+1 - Zn+1Jn+1), Gn+1 = (\11!!."l(w,~)I~o ,Xn+l)(\11!!."l(w,~)I~o,Xn+l f I n +1 = (\1w\1w"l(w,;£)lwo X ) '-n+l (2) (3) (4) do not depend on W n. The matrices {(An+1, B n+1)} form an Li.d. sequence, but An+! and Bn+1 are dependent for each n. Hence the linearized W n again forms a homogeneous Markov process in discrete time. In what follows we analyze this process in the hope that its asymptotics agree with those of the original Markov process. 3 EXISTENCE OF A LIMITING DISTRIBUTION Let A, B, G, J denote random matrices with the common distributions of the i.i.d. sequences {An}, {Bn}, {Gn }, and {In} respectively, and let T : lRm ---+ lRm be the random affine transformation w~Aw+B . The following result establishes the existence of a limiting distribution of W n. Lemma 1 (Berger Thm. V, p.162) Suppose E[1og+ IIAII + log+ IIBlll < ElogIIAn A n - 1 · · ·AtII < where 00· , o for some n log+ X = log x V o. Then the following conclusions hold: ll.e., (VE > O)(3M.)(Vn)lP(J.L- k IlW nil $ M.) ~ 1 - E. (5) (6) 338 Sayandev Mukherjee, Terrence L. Fine 1. Unique stationary distribution: There exists a unique mndom variable WEIR m, upto distribution, that is stationary with respect to T (i. e., W is independent of T, and TW has the same distribution as W). 2. Asymptotic stationarity: We have convergence in distribution: 'D Wn~W. Our choice of norm is the operator norm for the matrix A, /lAII = max IA(A)I, where {A(A)} are the eigenvalues of A, and the Euclidean norm for the vector B, m IIBII = 2: IBiI2 . i=1 We first verify (5). From the inequality "Ix E IR, log+ X ::; x 2 , it is easily seen that if r] is a feedforward net where all activation functions are twice-continuously differentiable in the weights, all hidden-layer activation functions are bounded and have bounded derivatives up to order 2, and if the training sequence (Xn' Yn) is Li.d. with finite fourth moments, then (5) holds for the Euclidean norm for Band the Frobenius norm for A, IIAII2 = 2::1 2:7=1IAijI2. Since m m (max IA(A)1)2 :::; 2: IA(AW :::; 2: 2: IAij12, (7) i=1 j=1 we see that (5) also holds for the operator norm of A. Assumption (6) forces the product An'" A1 to tend to Omxm almost surely (Berger, 1993, p.146) and therefore removes the dependence of the asymptotic distribution of {W n} on that of the initial value ll:=o. A sufficient condition for (6) is given by the following lemma. Lemma 2 Suppose lEG is positive definite (note that it is positive semidefinite by definition), and for all n, lEAn < 00. Then (6) holds for sufficiently small, positive 11· Proof: By assumption, minA(lEG) = 8 > 0 for some 8. Let Hn = ~ 2:~=1 (Gi - ZiJi)' By the Strong Law of Large Numbers applied to the LLd. random matrices (Gi - ZiJi) , we have Hn --+ lEG a.s., so minA(Hn) --+ minA(lEG) a.s. (8) Applying (7) to min A(Hn), it is easily shown that the same conditions on r] and the training sequence that are sufficient for (5) also give sUPn lE[min A(Hn )J2 < 00, which in turn implies that {min A(Hn)} are uniformly integrable. Together with (8), this implies (Loeve, 1977, p.165) that min A(Hn) --+ min A(lEG) in L1. Hence there Asymptomatics of Gradient-Based Neural Network Training Algorithms 339 exists some (nonrandom) N, say, such that ElminA(HN) - minA(EG)1 ::; 8/2. Since IEminA(HN) - minA(EG)1 ::; EI minA(HN) - minA(EG)1 ::; 8/2, we therefore have E [min'\ (~ t,(Gi - ZiJi») 1 ? rnin'\(EG) - 8/2 = 8 - 8/2 = 8/2> O. (9) We shall prove that (6) holds for this N (;:::: m) by showing that Elog IIANAN- I ··· Ad2 = 2lEiog IIANAN-I'" Alii < O. For our choice of norm, we therefore want E[log(max IA(AN .. . Al)I)2] < O. From Jensen's inequality, it is sufficient to have log E[max IA(AN . . . Ad 1]2 < 0, or equivalently, Now, since N is fixed, we can choose /-l small enough that N AN'" Al = 1m - /-l2:)Gi - ZiJi) + Op(/-l2). i=l (10) Hence, A(AN ... AI) = 1 /-lA(l:~1 (Gi - ZiJi» + Op(/-l2), and N is fixed, so I'\(AN ... AI)I' = 1 - 21''\ (t, (G, - Z,J ,») + Op (1"), giving max I'\(AN' .. Adl' <; 1 - 21' min'\ (t,(Gi - Z,J ,») + Op (1"), EmaxIA(AN,,·AIW::; I-N8/-l+o(/-l), (11) where we use (9) and the observation that the structure of the last Op(/-l2) term is such that its expectation (guaranteed finite by the hypothesis EAN < 00) is o (/-l2) , or o(/-l), and we also restrict /-l < 1/ N 8 so that 1- 2p.Ernin'\ (t,(G, - Z,J,») > O. From (11), it is clear that (10) holds for all sufficiently small, positive /-l (<< 1/N8). Therefore (6) holds for n = N. We can combine these two lemmas into the following theorem. Theorem 1 Let 'fJ be a feed forward net where all activation functions are twicecontinuously differentiable in the weights, all hidden-layer activation junctions are bounded and have bounded derivatives up to order 2, and let the training sequence (Xn' Yn) be i.i.d. with finite moments. Further, assume that lEG is positive definite. Then, for all sufficiently small, positive /-l the sequence of random vectors {W n}~l obtained from the updating equation (2) has a unique limiting distribution. 340 Sayandev Mukherjee, Terrence L. Fine We circumvent the generally intractable problem of finding the limiting distribution by calculating and investigating the behavior of its moments. 4 MOMENTS OF THE LIMITING DISTRIBUTION Let us assume that the mean and variance of the limiting distribution exist, and that Z '" N(O, (1'2). From (2) and the form of An+! and Bn+!, it is easy to show that EW = 0, or EW = wo, so the optimal value Wo is the mean of the limiting distribution of the sequence of iterates {W n}' It can also be shown (Mukherjee, 1994) that EWWT = (f.L(1'2/2) 1m, yielding W = Op(yTi). This is consistent with our assumption (1) with k = 1/2. In the one-dimensional case (d = m = 1), we have EW = 0 and EW2 = ~f.L(1'2 if lE[Xn+!'1}'(WO Xn+!W i= O. Using these results, the fact that Z '" N(O, (1'2), EW = 0, the independence of Z and X, and assuming that EXB < 00, it is not difficult to compute the expressions and where 3 _ f.L2(1'4E[X3'1}"'1}'(1 - f.LX2'1}'2)] lEW , E[X2'1}'2 - f.LX4( '1}'4 + (1'2'1}"2) + f.L2 X6'1}'2( '1}'4 /3 + (1'2'1}"2)] E[X2'1}'2(1 - f.LX2'1}'2)2 + f.LX4'1}'4 + 3f.L2 X6'1}"2'1}'2] K(f.L) 18f.L2(1'2E[X3'1}"'1}'(1 - f.LX2'1}'2)2 + f.L4 (1' 4 x 7 '1}"3'1}'] K(f.L) E[X2'1}'2 ~f.LX4('1}14 + (1'2'1}"2(1 - f.LX2'1}'2)2) 2 + f.L2 X6'1}'6 ~f.L3 XB('1}'B + 3(1'4'1}"4)], and '1}' and '1}" are evaluated at the argument wO X for'1}. From the above expressions, it is seen that if'1}O = 1/[1 + e-O] and X has a symmetric distribution (say N(O, 1)), then EW3 i= 0 and lEW4 i= 3(EW2)2, implying that W is non-Gaussian in general. This result is consistent with that obtained by direct application of Foster's criterion (Mukherjee, 1994). 5 RECONCILING LINEARIZATION AND STOCHASTIC APPROXIMATION METHODS The results of stochastic approximation analysis give a Gaussian distribution for W in the limit as f.L ---. 0 (Bucklew,Kurtz,Sethares, 1993; Finnoff, 1993). However, our results establish that the Gaussian distribution result is not valid for small nonzero f.L in general. To reconcile these results, recall W = Op (yTi). Hence, if we consider Asymptomatics of Gradient-Based Neural Network Training Algorithms 341 only moments of the normalized quantity W / fo (and neglect higher-order terms in Op(fo)), we obtain E(W / fo)3 = 0 and E(W / fo)4 = 3[E(W / fo)2j2, which suggests that the normalized quantity W / fo is Gaussian in the limit of vanishing {L, a conclusion also reached from stochastic approximation analysis. In support of this theoretical indication that the conclusions of our analysis (based on linearization for small {L) might tally with those of stochastic approximation techniques for small values of {L, simulations were done on the simple onedimensional training case of the previous section for 8 cases: {L = 0.1,0.2,0.3,0.5, and (72 = 0.1,0.5 for each value of {L, with wO fixed at 3. For each of the 8 cases, either 5 or 10 runs were made, with lengths (for the given values of {L) of 810000, 500000, 300000, and 200000 respectively. Each run gave a pair of sequences {Wn } obtained by starting off at Wo = 0 and training the network independently twice. Each resulting sequence {Wn} was then downs amp led at a large enough rate that the true autocorrelation of the downsampled sequence was less than 0.05, followed by deleting the first 10% of the samples of this downsampled sequence, so as to remove any dependence on initial conditions that might persist. (Autocorrelation at lag unity for this Markov Chain was so high that when {L = 0.1, a decimation rate of 9000 was required.) This was done to ensure that the elements of the resulting downsampled sequences could be assumed independent for the various hypothesis tests that were to follow. (a) For each run of each case, the empirical distribution functions of the two downsampled sequences thus generated were compared by means of the Kolmogorov-Smirnov test (Bickel,Doksum, 1977) at level 0.95, with the null hypothesis being that both sequences had the same actual cumulative distribution function (assumed continuous). This test was passed with ease on all trials, thereby showing that a limiting distribution existed and was attained by such a training algorithm. (b) For each run of each case, a skewness test and a kurtosis test (Bickel,Doksum, 1977) for normality were done at level 0.95 to test for normality. The sequences generated failed both tests for the ({L, (7) pair (0.1,0.1) and passed them both for the pairs (0.1,0.5), (0.3,0.1), (0.5,0.1), and (0.5,0.5). For the pair (0.2,0.5), the skewness test was passed and the kurtosis test failed, and for the pairs (0.2,0.1) and (0.3,0.5), the skewness test was failed and the kurtosis test passed. (c) All trials cleared the Kolmogorov tests (Bickel,Doksum, 1977) for normality at level 0.95, both when the normal distribution was taken to have the sample mean and variance (c.)mputed on the downsampled sequence), and when the normal distribution function had the asymptotic values of mean (zero) and variance ({L(72 /2). Hence we may conclude: 1. The limiting distribution of {Wn } exists. 2. For small values of {L, the deviation from Gaussianness is so small that the Gaussian distribution may be taken as a good approximation to the limiting distribution. 342 Sayandev Mukherjee, Terrence L. Fine In other words, though stochastic approximation analysis states that W / yfji is Gaussian only in the limit of vanishing f.J" our simulation shows that this is a good approximation for small values of f.J, as well. Acknowledgements The research reported here was partially supported by NSF Grant SBR-9413001. References Berger, Marc A. An Introduction to Probability and Stochastic Processes. SpringerVerlag, New York, 1993. Bickel, Peter, and Doksum, Kjell. Mathematical Statistics: Basic Ideas and Selected Topics. Holden-Day, San Francisco, 1977. Bucklew, J.A., Kurtz, T.G., and Sethares, W.A. "Weak Convergence and Local Stability Properties of Fixed Step Size Recursive Algorithms," IEEE Trans. Inform. Theory, vol. 39, pp. 966-978, 1993. Finnoff, W. "Diffusion Approximations for the Constant Learning Rate Backpropagation Algorithm and Resistence to Local Minima." In Giles, C.L., Hanson, S.J., and Cowan, J.D., editors, Advances in Neural Information Processing Systems 5. Morgan Kaufmann Publishers, San Mateo CA, 1993, p.459 ff. Kuan, C-M, and Hornik, K. "Convergence of Learning Algorithms with Constant Learning Rates," IEEE Trans. Neural Networks, vol. 2, pp. 484-488, 1991. Leen, T.K., and Moody, J .E. "Weight Space Probability Densities in Stochastic Learning: 1. Dynamics and Equilibria," Adv. in NIPS 5, Morgan Kaufmann Publishers, San Mateo CA, 1993, p.451 ff. Loeve, M. Probability Theory I, 4th ed. Springer-Verlag, New York, 1977. Mukherjee, Sayandev. Asymptotics of Gradient-based Neural Network Training Algorithms. M.S. thesis, Cornell University, Ithaca, NY, 1994. Orr, G.B., and Leen, T.K. "Probability densities in stochastic learning: II. Transients and Basin Hopping Times," Adv. in NIPS 5, Morgan Kaufmann Publishers, San Mateo CA, 1993, p.507 ff. Rumelhart, D.E., Hinton, G.E., and Williams, R.J. "Learning interval representations by error propagation." In D.E. Rumelhart and J.L. McClelland, editors, Parallel Distributed Processing, Ch. 8, MIT Press, Cambridge MA, 1985. Tweedie, R.L. "Criteria for Classifying General Markov Chains," Adv. Appl. Prob., vol. 8,737-771, 1976. White, H. "Some Asymptotic Results for Learning in Single Hidden-Layer Feedforward Network Models," J. Am. Stat. Assn., vol. 84, 1003-1013, 1989. PART IV REThWORCEMENTLEARMNG	1994	87
984	Convergence Properties of the K-Means Algorithms Leon Bottou Neuristique, 28 rue des Petites Ecuries, 75010 Paris, France leonCneuristique.fr Yoshua Bengio" Dept. LR.O. Universite de Montreal Montreal, Qc H3C-3J7, Canada bengioyCiro.umontreal.ca Abstract This paper studies the convergence properties of the well known K-Means clustering algorithm. The K-Means algorithm can be described either as a gradient descent algorithm or by slightly extending the mathematics of the EM algorithm to this hard threshold case. We show that the K-Means algorithm actually minimizes the quantization error using the very fast Newton algorithm. 1 INTRODUCTION K-Means is a popular clustering algorithm used in many applications, including the initialization of more computationally expensive algorithms (Gaussian mixtures, Radial Basis Functions, Learning Vector Quantization and some Hidden Markov Models). The practice of this initialization procedure often gives the frustrating feeling that K-Means performs most of the task in a small fraction of the overall time. This motivated us to better understand this convergence speed. A second reason lies in the traditional debate between hard threshold (e.g. KMeans, Viterbi Training) and soft threshold (e.g. Gaussian Mixtures, Baum Welch) algorithms (Nowlan, 1991). Soft threshold algorithms are often preferred because they have an elegant probabilistic framework and a general optimization algorithm named EM (expectation-maximization) (Dempster, Laird and Rubin, 1977). In the case of a gaussian mixture, the EM algorithm has recently been shown to approximate the Newton optimization algorithm (Xu and Jordan, 1994). We prove in this "also, AT&T Bell Labs, Holmdel, NJ 07733 586 Leon Bottou, Yoshua Bengio paper that the corresponding hard threshold algorithm) K-Means) minimizes the quantization error using exactly the Newton algorithm. In the next section) we derive the K-Means algorithm as a gradient descent procedure. Section 3 extends the mathematics of the EM algorithm to the case of K-Means. This second derivation of K-Means provides us with proper values for the learning rates. In section 4 we show that this choice of learning rates optimally rescales the parameter space using Newton)s method. Finally) in section 5 we present and discuss a few experimental results comparing various versions of the K-Means algorithm. The 5 clustering algorithms presented here were chosen for a good coverage of the algorithms related to K-Means) but this paper does not have the ambition of presenting a literature survey on the subject. 2 K-MEANS AS A GRADIENT DESCENT Given a set of P examples (Xi) the K-Means algorithm computes k prototypes W = (Wk) which minimize the quantization error, i.e.) the average distance between each pattern and the closest prototype: ( ) def ""' ( ) def ""' 1 . ( 2 E W = L....J L Xi) W = L....J - mln Xi Wk) . . 2 (1) t t Writing Si (w) for the subscript of the closest prototype to example Xi, we have (2) 2.1 GRADIENT DESCENT ALGORITHM We can then derive a gradient descent algorithm for the quantization error: ~W = -(t 8~;:). This leads to the following batch update equation (updating prototypes after presenting all the examples) : (3) We can also derive a corresponding online algorithm which updates the prototypes after the presentation of each pattern Xi: A _ OL(Xi) w) u.W -(t OW ) l.e.) if k = Si(W) otherwise. (4) The proper value of the learning rate (t remain to be specified in both batch and online algorithms. Convergence proofs for both algorithms (Bottou) 1991) exist for decreasing values of the learning rates satisfying the conditions E (t = 00 and E (; < 00. Following (Kohonen) 1989), we could choose (t = (o/t . We prove however in this paper that there exist a much better choice of learning rates. Convergence Properties of the K-Means Algorithms 3 K-MEANS AS AN EM STYLE ALGORITHM 3.1 EM STYLE ALGORITHM 587 The following derivation of K-Means is similar to the derivation of (MacQueen, 1967). We insist however on the identity between this derivation and the mathematics of EM (Liporace, 1976) (Dempster, Laird and Rubin, 1977). Although [{-Means does not fit in a probabilistic framework, this similarity holds for a very deep reason: The semi-ring of probabilies (!R+, +, x) and the idempotent semi-ring of hard-threshold scores (!R, Min, +) share the most significant algebraic properties (Bacceli, Cohen and Olsder, 1992). This assertion completely describes the similarities and the potential differences between soft-threshold and hard-threshold algorithms. A complete discussion however stands outside the scope of this paper. The principle of EM is to introduce additional "hidden" variables whose knowledge would make the optimization problem easier. Since these hidden variables are unknown, we maximize an auxiliary function which averages over the values of the hidden variables given the values of the parameters at the previous iteration. In our case, the hidden variables are . the assignments Si(W) of the patterns to the prototypes. Instead of considering the expected value over the distribution on these hidden variables, we just consider the values of the hidden variables that minimize the cost, given the previous values of the parameters: Q( I ) def '"' 1 ( I )2 W ,W = L...J '2 Xi - W &.(w) i The next step consists then in finding a new set of prototypes Wi which mllllmizes Q( Wi, w) where w is the previous set of prototypes. We can analytically compute the explicit solution of this minimization problem. Solving the equation 8Q(w' , W)/8W'k = 0 yields: I 1 '"' wk = 111 L...J Xi k i:k=&i(w) (5) where Nk is the number of examples assigned to prototype Wk . The algorithm consists in repeatedly replacing w by Wi using update equation (6) until convergence. Since si(w' ) is by definition the best assignment of patterns Xi to the prototypes w~, we have the following inequality: E(w' ) - Q(w' , w) = ~ L(Xi - Wi &.(w,))2 - (Xi - Wi Si(W))2 ::; 0 i Using this result, the identity E(w) = Q(w, w) and the definition of Wi, we can derive the following inequality: E(w' ) - E(w) E(w' ) - Q(w' , w) + Q(w', w) - Q(w, w) < Q(w' , w) - Q(w, w) ::; 0 Each iteration of the algorithm thus decreases the otherwise positive quantization error E (equation 1) until the error reaches a fixed point where condition W' = w is verified (unicity of the minimum of Q(., w)). Since the assignment functions Si (w) are discrete, there is an open neighborhood of w on which the assignments are constant. According to their definition, functions E(.) and Q(., w) are equal on this neighborhood. Being the minimum of function Q(.,w), the fixed point w* of this algorithm is also a local minimum of the quantization error E. 0 588 Leon Bottou, Yoshua Bengio 3.2 BATCH K-MEANS The above algorithm (5) can be rewritten in a form similar to that of the batch gradient descent algorithm (3). if k = S(Xi' w) otherwise. (6) This algorithm is thus equivalent to a batch gradient descent with a specific, prototype dependent, learning rate Jk • 3.3 ONLINE K-MEANS The online version of our EM style update equation (5) is based on the computation of the mean flt of the examples Xl, ... ,Xt with the following recursive formula: flt+l = t~l (t flt + Xt+l) = flt + t~l (Xt+l fld Let us introduce new variables nk which count the number of examples so far assigned to prototype Wk. We can then rewrite (5) as an online update applied after the presentation of each pattern Xi: D..nk { 1 if k = S(Xi' w) 0 otherwise. D..wk { n\ (Xi Wk) if k = S(Xi'W) (7) 0 otherwise. This algorithm is equivalent to an online gradient descent (4) with a specific, prototype dependent, learning rate nlk. Unlike in the batch case, the pattern assignments S(Xi , w) are thus changing after each pattern presentation. Before applying this algorithm, we must of course set nk to zero and Wk to some initial value. Various methods have been proposed including initializing Wk with the first k patterns. 3.4 CONVERGENCE General convergence proofs for the batch and online gradient descent (Bottou, 1991; Driancourt, 1994) directly apply for all four algorithms. Although the derivatives are undefined on a few points, these theorems prove that the algorithms almost surely converge to a local minimum because the local variations of the loss function are conveniently bounded (semi-differentiability). Unlike previous results, the above convergence proofs allow for non-linearity, non-differentiability (on a few points) (Bottou, 1991), and replacing learning rates by a positive definite matrix (Driancourt, 1994). 4 K-MEANS AS A NEWTON OPTIMIZATION We prove in this section that Batch K-Means (6) applies the Newton algorithm. 4.1 THE HESSIAN OF K-MEANS Let us compute the Hessian H of the K-Means cost function (2). This matrix contains the second derivatives of the cost E(w) with respect to each pair of parameters. Since E(w) is a sum of terms L(Xi'W), we can decompose H as the sum Convergence Properties of the K-Means Algorithms 589 of matrices Hi for each term of the cost function: L(Xi'W) = ~in~(xi - Wk)2. Furthermore, the Hi can be decomposed in blocks corresponding to each pair of prototypes. Since L( Xi, w) depends only on the closest prototype to pattern Xi, all these blocks are zero except block (Si (w), Si (w)) which is the identity matrix. Summing the partial Hessian matrices Hi thus gives a diagonal matrix whose diagonal elements are the counts of examples Nk assigned to each prototype. H=(T o We can thus write the Newton update of the parameters as follows: ~w = _H- 1 oE(w) ow which can be exactly rewritten as the batch EM style algorithm (6) presented earlier: ~w =" { ~k (Xi - Wk) if k = ~(Xi' w) (8) k L..J 0 otherwlse. i 4.2 CONVERGENCE SPEED When optimizing a quadratic function, Newton's algorithm requires only one step. In the case of a non quadratic function, Newton's algorithm is superlinear if we can bound the variations of the second derivatives. Standard theorems that bound this variation using the third derivative are not useful for K-Means because the gradient of the cost function is discontinuous. We could notice that the variations of the second derivatives are however nicely bounded and derive similar proofs for K-Means. For the sake of brevity however, we are just giving here an intuitive argument: we can make the cost function indefinitely differentiable by rounding up the angles around the non differentiable points. We can even restrict this cost function change to an arbitrary small region of the space. The iterations of K-Means will avoid this region with a probability arbitrarily close to 1. In practice, we obtain thus a superlinear convergence. Batch K-Means thus searches for the optimal prototypes at Newton speed. Once it comes close enough to the optimal prototypes (i.e. the pattern assignment is optimal and the cost function becomes quadratic), K-Means jumps to the optimum and terminates. Online K-Means benefits of these optimal learning rates because they remove the usual conditioning problems of the optimization. However, the stochastic noise induced by the online procedure limits the final convergence of the algorithm. Final convergence speed is thus essentially determined by the schedule of the learning rates. Online K-Means also benefits from the redundancies of the training set. It converges significantly faster than batch K-Means during the first training epochs (Darken 590 IaI Co.t 2500 2'00 2300 2200 2100 2000 1900 1800 1700 1600 1500 1600 1300 1200 1100 1000 900 800 700 600 500 '00 300 200 100 Leon Bottou, Yoshua Bengio BII Co.t 1 2 3 , 5 6 7 8 9 10 11 12 13 16 15 16 17 18 19 20 Figure 1: Et - Eoo versus t. black circles: online K-Means; black squares: batch K-Means; empty circles: online gradient; empty squares: batch gradient; no mark: EM +Gaussian mixture and Moody, 1991). After going through the first few patterns (depending of the amount of redundancy), online K-Means indeed improves the prototypes as much as a complete batch K-Means epoch. Other researchers have compared batch and online algorithms for neural networks, with similar conclusions (Bengio, 1991). 5 EXPERIMENTS Experiments have been carried out with Fisher's iris data set, which is composed of 150 points in a four dimensional space representing physical measurements on various species of iris flowers. Codebooks of six prototypes have been computed using both batch and online K-Means with the proper learning rates (6) and (7). These results are compared with those obtained using both gradient descent algorithms (3) and (4) using learning rate ft = 0.03/t that we have found optimal. Results are also compared with likelihood maximization with the EM algorithm, applied to a mixture of six Gaussians, with fixed and uniform mixture weights, and fixed unit variance. Inputs were scaled down empirically so that the average cluster variance was around unity. Thus only the cluster positions were learned, as for the K-Means algorithms. Each run of an algorithm consists in (a) selecting a random initial set of prototypes, (b) running the algorithm during 20 epochs and recording the error measure Et after each epoch, (c) running the batch K-Means algorithm! during 40 more epochs in order to locate the local minimum Eoo corresponding to the current initialization of the algorithm. For the four K-Means algorithms, Et is the quantization error (equation 1). For the Gaussian mixture trained with EM, the cost Et is the negative 1 except for the case of the mixture of Gaussians, in which the EM algorithm was applied Convergence Properties of the K-Means Algorithms 591 1 2 3 • 5 6 7 8 9 10 11 12 13 U 15 16 17 18 19 Figure 2: E~-;~~~oo versus t. black circles: online K-Means; black squares: batch K-Means; empty circles: online gradient; empty squares: batch gradient; no mark: EM+Gaussian mixture logarithm of the likelihood of the data given the model. Twenty trials were run for each algorithm. Using more than twenty runs did not improve the standard deviation of the averaged measures because various initializations lead to very different local minima. The value Eoo of the quantization error on the local minima ranges between 3300 and 5800. This variability is caused by the different initializations and not by the different algorithms. The average values of Eoo for each algorithm indeed fall in a very small range (4050 to 4080). Figure 1 shows the average value of the residual error Et - Eoo during the first 20 epochs. Online K-Means (black circles) outperforms all other algorithms during the first five epochs and stabilizes on a level related to the stochastic noise of the online procedure. Batch K-Means (black squares) initially converges more slowly but outperforms all other methods after 5 epochs. All 20 runs converged before the 15th epoch. Both gradients algorithms display poor convergence because they do not benefit of the Newton effect. Again, the online version (white circles) starts faster then the batch version (white square) but is outperformed in the long run. The negative logarithm of the Gaussian mixture is shown on the curve with no point marks, and the scale is displayed on the right of Figure 1. Figure 2 show the final convergence properties of all five algorithms. The evolutions of the ratio (Et+l - Eoo)/(Et - Eoo) characterize the relative improvement of the residual error after each iteration. All algorithms exhibit the same behavior after a few epochs except batch K-Means (black squares). The fast convergence ofthis ratio to zero demonstrates the final convergence of batch K-Means. The EM algorithm displays a better behavior than all the other algorithms except batch K-Means. Clearly, however, its relative improvement ratio doesn't display the fast convergence behavior of batch K-Means. 592 Uon Bottou, Yoshua Bengio The online K-Means curve crosses the batch K-Means curve during the second epoch, suggesting that it is better to run the online algorithm (7) during one epoch and then switch to the batch algorithm (6). 6 CONCLUSION We have shown with theoretical arguments and simple experiments that a well implemented K-Means algorithm minimizes the quantization error using Newton's algorithm. The EM style derivation of K-Means shows that the mathematics of EM are valid well outside the framework of probabilistic models. Moreover the provable convergence properties of the hard threshold K-Means algorithm are superior to those of the EM algorithm for an equivalent soft threshold mixture of Gaussians. Extending these results to other hard threshold algorithms (e.g. Viterbi Training) is an interesting open question. References Bacceli, F., Cohen, G., and Olsder, G. J. (1992). Synchronization and Linearity. Wiley. Bengio, Y. (1991). Artificial Neural Networks and their Application to Sequence Recognition. PhD thesis, McGill University, (Computer Science), Montreal, Qc., Canada. Bottou, 1. (1991). Une approche theorique de l'apprentissage connexioniste; applications a la reconnaissance de la parole. PhD thesis, Universite de Paris XI. Darken, C. and Moody, J. (1991). Note on learning rate schedules for stochastic optimization. In Lippman, R. P., Moody, R., and Touretzky, D. S., editors, Advances in Neural Information Processing Systems 3, pages 832-838, Denver, CO. Morgan Kaufmann, Palo Alto. Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum-likelihood from incomplete data via the EM algorithm. Journal of Royal Statistical Society B, 39:1-38. Driancourt, X. (1994). Optimisation par descente de gradient stochastique ... , PhD thesis, Universite de Paris XI, 91405 Orsay cedex, France. Kohonen, T. (1989). Self-Organization and Associative Memory. Springer-Verlag, Berlin, 3 edition. Liporace, L. A. (1976). PTAH on continuous multivariate functions of Markov chains. Technical Report 80193, Institute for Defense Analysis, Communication Research Department. MacQueen, J. (1967). Some methods for classification and analysis of multivariate observations. In Proceedings of the Fifth Berkeley Symposium on Mathematics, Statistics and Probability, Vol. 1, pages 281-296. Nowlan, S. J. (1991). Soft Competitive Adaptation: Neural Network Learning Algorithms based on Fitting Statistical Mixtures. CMU-CS-91-126, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA. Xu, L. and Jordan, M. (1994). Theoretical and experimental studies of convergence properties of the em algorithm for unsupervised learning based on finite mixtures. Presented at the Neural Networks for Computing Conference.	1994	88
985	Using Voice Transformations to Create Additional Training Talkers for Word Spotting Eric I. Chang and Richard P. Lippmann MIT Lincoln Laboratory Lexington, MA 02173-0073, USA eichang@sst.ll.mit.edu and rpl@sst.ll.mit.edu Abstract Speech recognizers provide good performance for most users but the error rate often increases dramatically for a small percentage of talkers who are "different" from those talkers used for training. One expensive solution to this problem is to gather more training data in an attempt to sample these outlier users. A second solution, explored in this paper, is to artificially enlarge the number of training talkers by transforming the speech of existing training talkers. This approach is similar to enlarging the training set for OCR digit recognition by warping the training digit images, but is more difficult because continuous speech has a much larger number of dimensions (e.g. linguistic, phonetic, style, temporal, spectral) that differ across talkers. We explored the use of simple linear spectral warping to enlarge a 48-talker training data base used for word spotting. The average detection rate overall was increased by 2.9 percentage points (from 68.3% to 71.2%) for male speakers and 2.5 percentage points (from 64.8% to 67.3%) for female speakers. This increase is small but similar to that obtained by doubling the amount of training data. 1 INTRODUCTION Speech recognizers, optical character recognizers, and other types of pattern classifiers used for human interface applications often provide good performance for most users. Performance is often, however, low and unacceptable for a small percentage of "outlier" users who are presumably not represented in the training data. One expensive solution to this problem is to obtain more training data in the hope of including users from these outlier 876 Eric I. Chang, Richard P. Lippmann classes. Other approaches already used for speech recognition are to use input features and distance metrics that are relatively invariant to linguistically unimportant differences between talkers and to adapt a recognizer for individual talkers. Talker adaptation is difficult for word spotting and with poor outlier users because the recognition error rate is high and talkers often can not be prompted to recite standard phrases that can be used for adaptation. An alternative approach, that has not been fully explored for speech recognition, is to artificially expand the number of training talkers using voice transformations. Transforming the speech of one talker to make it sound like that of another is difficult because speech varies across many difficult-to-measure dimensions including linguistic, phonetic, duration, spectra, style, and accent. The transformation task is thus more difficult than in optical character recognition where a small set of warping functions can be successfully applied to character images to enlarge the number of training images (Drucker, 1993). This paper demonstrates how a transformation accomplished by warping the spectra of training talkers to create more training data can improve the performance of a whole-word word spotter on a large spontaneous-speech data base. 2 BASELINE WORD SPOTTER A hybrid radial basis function (RBF) - hidden Markov model (HMM) keyword spotter has been developed over the past few years that provides state-of-the-art performance for a whole-word word spotter on the large spontaneous-speech credit-card speech corpus. This system spots 20 target keywords, includes one general filler class, and uses a Viterbi decoding backtrace as described in (Chang, 1994) to backpropagate errors over a sequence of input speech frames. This neural network word spotter is trained on target and background classes, normalizes target outputs using the background output, and thresholds the resulting score to generate putative hits, as shown in Figure 1. Putative hits in this figure are input patterns which generate normalized scores above a threshold. The performance of this, and other spotting systems, is analyzed by plotting a detection versus false alarm rate curve. This curve is generated by adjusting the classifier output threshold to allow few or many putative hits. The figure of merit (FOM) is defined as the average keyword detection rate when the false alarm rate ranges from 1 to 10 false alarms per keyword per hour. The previous best FOM for this word spotter is 67.8% when trained using 24 male talkers and tested on 11 male talkers, and 65.9% when trained using 24 female talkers and tested on 11 female talkers. The overall FOM for all talkers is 66.3%. CONTINUOUS SPEECH INPUT NEURAL NETWORK WORDSPOTTER Figure 1: Block diagram of neural network word spotter. ATIVE HITS Using Voice Transfonnations to Create Additional Training Talkers for Word Spotting 877 3 TALKER VARIABILITY FOM scores of test talkers vary over a wide range. When training on 48 talkers and then performing testing on 22 talkers from the 70 conversations in the NIST Switchboard credit card database, the FOM ofthe test talkers varies from 16.7% to 100%. Most talkers perform well above 50%, but there are two female talkers with FOM's of 16.7% and 21.4%. The low FOM for individual speakers indicates a lack of training data with voice qualities that are similar to these test speakers. 4 CREATING MORE TRAINING DATA USING VOICE TRANSFORMATIONS Talker adaptation is difficult for word spotting because error rates are high and talkers often can not be prompted to verify adaptation phrases. Our approach to increasing performance across talkers uses voice transformation techniques to generate more varied training examples of keywords as shown in Figure 2. Other researchers have used talker transformation techniques to produce more natural synthesized speech (lwahashi, 1994, Mizuno, 1994), but using talker transformation techniques to generate more training data is novel. We have implemented a new voice transformation technique which utilizes the Sinusoidal Transform Analysis/Synthesis System (STS) described in (Quatieri, 1992). This technique attempts to transform one talker's speech pattern to that of a different talker. The STS generates a 512 point spectral envelope of the input speech 100 times a second and also separates pitch and voicing information. Separation of vocal tract characteristic and pitch information has allowed the implementation of pitch and time transformations in previous work (Quatieri, 1992). The system has been modified to generate and accept a spectral enORIGINAL SPEECH VOICE TRANSFORMATION SYSTEM TRANSFORMED SPEECH Figure 2: Generating more training data by artificially transforming original speech training data. 878 Eric I. Chang, Richard P. Lippmann velo.pe file fro.m an input speech sample. We info.nnally explo.red different techniques to' transfo.nn the spectral envelo.pe to' generate mo.re varied training examples by listening to' transfo.nned speech. This resulted in the fo.llo.wing algo.rithm that transfo.nns a talker's vo.ice by scaling the spectral envelo.pe o.f training talkers. 1. Training co.nversatio.ns are upsampled fro.m 8000 Hz to' 10,000 Hz to' be co.mpatible with existing STS co.ding so.ftware. 2. The STS system pro.cesses the upsampled files and generates a 512 point spectral envelo.pe o.f the input speech wavefo.rm at a frame rate o.f 100 frames a seco.nd and with a windo.w length o.f appro.ximately 2.5 times the length o.f each pitch perio.d. 3. A new spectral envelo.pe is generated by linearly expanding o.r co.mpressing the spectral axis. Each spectral po.int is identified by its index, ranging fro.m 0 to' 511. To. transfo.rm a spectral pro.file by 2, the new spectral value at frequency f is generated by averaging the spectral values aro.und the o.riginal spectral pro.file at frequency o.f 0.5 f The transfo.nnatio.n process is illustrated in Figure 3. In this figure, an o.riginal spectral envelo.pe is being expanded by two.. The spectral value at index 150 is thus transfo.rmed to. spectral index 300 in the new envelo.pe and the o.riginal spectral info.rmatio.n at high frequencies is lo.st. 4. The transfo.rmed spectral value is used to. resynthesize a speech wavefo.rm using the vocal tract excitatio.n info.nnatio.n extracted fro.m the o.riginal file. Vo.ice transfo.rmatio.n with the STS co.der allo.ws listening to. transfo.rmed speech but requires lo.ng co.mputatio.n. We simplified o.ur appro.ach to' o.ne o.f mo.difying the spectral scale in the spectral do.main directly within a mel-scale filterbank analysis pro.gram. The inco.ming speech sample is pro.cessed with an FFT to' calculate spectral magnitudes. Then spectral magnitudes are linearly transfo.rmed. Lastly mel-scale filtering is perfo.rmed with 10 linearly spaced filters up to' 1000 Hz and lo.garithmically spaced filters fro.m 1000 Hz up. A co.sine transfo.rm is then used to' generate mel-scaled cepstral values that are used by the wo.rdspo.tter. Much faster pro.cessing can be achieved by applying the spectral transfo.rmatio.n as part o.f the filterbank analysis. Fo.r example, while perfo.rming spectral transfo.nnatio.n using the STS algo.rithm takes up to' approximately 10 times real time, spectral transfo.rmatio.n within the mel-scale filterbank pro.gram can be acco.mplished within 1110 real time o.n a Sparc 10 wo.rkstatio.n. The rapid pro.cessing rate allo.ws o.n-line spectral transfo.rmatio.n. 5 WORD SPOTTING EXPERIMENTS Linear warping in the spectral do.main, which is used in the above algo.rithm, is co.rrect when the vo.cal tract is mo.delled as a series o.f lo.ssless aco.ustic tubes and the excitatio.n so.urce is at o.ne end o.f the vo.cal tract (Wakita, 1977). Wakita sho.wed that if the vo.cal tract is mo.delled as a series o.f equal length, lo.ssless, and co.ncatenated aco.ustic tubes, then the ratio. o.f the areas between the tubes determines the relative reso.nant frequencies o.f the vo.cal tract, while the o.verall length o.fthe vo.cal tract linearly scales fo.rmant frequencies. Preliminary research was co.nducted using linear scaling with spectral ratio.s ranging fro.m 0.6 to. 1.8 to. alter test utterances. After listening to. the STS transfo.rmed speech and also. o.bserving Using Voice Transformations to Create Additional Training Talkers for Word Spotting 879 Original Spectral Envelope Transformed Spectral Envelope o " o III I I 150 511 300 511 Spectral Value Index Figure 3: An example of the spectral transformation algorithm where the original spectral envelope frequency scale is expanded by 2. spectrograms of the transformed speech, it was found that speech transformed using ratios between 0.9 and 1.1 are reasonably natural and can represent speech without introducing artifacts. Using discriminative training techniques such as FOM training carries the risk of overtraining the wordspotter on the training set and obtaining results that are poor on the testing set. To delay the onset of overtraining, we artificially transformed each training set conversation during each epoch using a different random linear transformation ratio. The transformation ratio used for each conversation is calculated using the following formula: ratio == a + N (0,0.06), where a is the transformation ratio that matches each training speaker to the average of the training set speakers, and N is a normally distributed random variable with a mean of 0.0 and standard deviation of 0.06. For each training conversation, the long term averages of formant frequencies for formant 1, 2, and 3 are calculated. A least square estimation is then performed to match the formant frequencies of each training set conversation to the group average formant frequencies. The transformation equation is described below: 880 100 90 ~ 80 o 50 o Figure 4: Eric I. Chang, Richard P. Lippmann • • • • t • :--! a•• .,. • • NORMAL 234 5 678 9 NUMBER OF EPOCHS OF FOM TRAINING 10 Average detection accuracy (FOM) for the male training and testing set versus the number of epochs of FOM training. The transform ratio for each individual conversation is calculated to improve the naturalness of the transformed speech. In preliminary experiments, each conversation was transformed with fixed ratios of 0.9, 0.95, 1.05, and 1.1. However, for a speaker with already high formant frequencies, pushing the formant frequencies higher may make the transformed speech sound unnatural. By incorporating the individual formant matching ratio into the transformation ratio, speakers with high formant frequencies are not transformed to very high frequencies and speakers with low formant frequencies are not transformed to even lower formant frequency ranges. Male and female conversations from the NIST credit card database were used separately to train separate word spotters. Both the male and the female partition of data used 24 conversations for training and 11 conversations for testing. Keyword occurrences were extracted from each training conversation and used as the data for initialization of the neural network word spotter. Also, each training conversation was broken up into sentence length segments to be used for embedded reestimation in which the keyword models are joined with the filler models and the parameters of all the models are jointly estimated. After embedded reestimation, Figure of Merit training as described in (Lippmann. 1994) was performed for up to 10 epochs. During each epoch. each training conversation is transformed using a transform ratio randomly generated as described above. The performance of the word spotter after each iteration of training is evaluated on both the training set and the testing set. 6 WORD SPOTTING RESULTS Training and testing set FOM scores for the male speakers and the female speakers are shown in Figure 4 and Figure 5 respectively. The x axis plots the number of epochs of FOM training where each epoch represents presenting all 24 training conversations once. The FOM for word spotters trained with the normal training conversations and word spotters Using Voice Transformations to Create Additional Training Talkers for Word Spotting 881 100 90 ?j 80 60 50 .. - ra- ". au • ~ - = I - -. " -• .TRAIN ~= • \ NORMAL o 23456789 NUMBER OF EPOCHS OF FOM TRAINING • 10 Figure 5: Average detection accuracy (FOM) for the female training and testing set versus the number of iterations of FOM training. trained with artificially expanded training conversations are shown in each plot. After the first epoch, the FOM improves significantly. With only the original training conversations (normal), the testing set FOM rapidly levels off while the training set FOM keeps on improving. When the training conversations are artificially expanded, the training set FOM is below the training set FOM from the normal training set due to more difficult training data. However, the testing set FOM continues to improve as more epochs of FOM training are performed. When comparing the FOM of wordspotters trained on the two sets of data after ten epochs of training, the FOM for the expanded set was 2.9 percentage points above the normal FOM for male speakers and 2.5 percentage points above the normal FOM for female speakers. For comparison, Carlson has reported that for a high performance word spotter on this database, doubling the amount of training data typically increases the FOM by 2 to 4 percentage points (Carlson, 1994). 7 SUMMARY Lack of training data has always been a constraint in training speech recognizers. This research presents a voice transformation technique which increases the variety among training talkers. The resulting more varied training set provided up to 2.9 percentage points of improvement in the figure of merit (average detection rate) of a high performance word spotter. This improvement is similar to the increase in performance provided by doubling the amount of training data (Carlson, 1994). This technique can also be applied to other speech recognition systems such as continuous speech recognition, talker identification, and isolated speech recognition. 882 Eric I. Chang. Richard P. Lippmcmn ACKNOWLEDGEMENT This work was sponsored by the Advanced Research Projects Agency. The views expressed are those of the authors and do not reflect the official policy or position of the U.S. Government. We wish to thank Tom Quatieri for providing his sinusoidal transform analysis/synthesis system. BIBLIOGRAPHY B. Carlson and D. Seward. (1994) Diagnostic Evaluation and Analysis of Insufficient and Task-Independent Training Data on Speech Recognition. In Proceedings Speech Research Symposium XlV, Johns Hopkins University. E. Chang and R. Lippmann. (1994) Figure of Merit Training for Detection and Spotting. In Neural Information Processing Systems 6, G. Tesauro, J. Cohen, and J. Alspector, (Eds.), Morgan Kaufmann: San Mateo, CA. H. Drucker, R. Schapire, and P. Simard. (1993) Improving Performance in Neural Networks Using a Boosting Algorithm. In Neural Information Processing Systems 5, S. Hanson, J. Cowan, and C. L. Giles, (Eds.), Morgan Kaufmann: San Mateo, California. N. Iwahashi and Y. Sagisaka. (1994) Speech Spectrum Transformation by Speaker Interpolation. In Proceedings International Conference on Acoustics Speech and Signal Processing, Vol. 1,461-464. R. Lippmann, E. Chang & C. Jankowski. (1994) Wordspotter Training Using Figure-ofMerit Back Propagation. In Proceedings of International Conference on Acoustics Speech and Signal Processing, Vol. 1,389-392. H. Mizuno and M. Abe. (1994) Voice Conversion Based on Piecewise Linear Conversion Rules of Formant Frequency and Spectrum Tilt. In Proceedings International Conference Oil Acoustics Speech and Signal Processing, Vol. 1,469-472. T. Quatieri and R. McAulay. (1992) Shape Invariant Time-Scale and Pitch Modification of Speech. In IEEE Trans. Signal Processing, vol 40, no 3. pp. 497-510. Hisashi Wakita. (1977) Normalization of Vowels by Vocal-Tract Length and Its Application to Vowel Identification. In IEEE Trans. Acoustics, Speech, and Signal Processing, vol. ASSP-25, No.2., pp. 183-192.	1994	89
986	Bias, Variance and the Combination of Least Squares Estimators Ronny Meir Faculty of Electrical Engineering Technion, Haifa 32000 Israel rmeirGee.technion.ac.il Abstract We consider the effect of combining several least squares estimators on the expected performance of a regression problem. Computing the exact bias and variance curves as a function of the sample size we are able to quantitatively compare the effect of the combination on the bias and variance separately, and thus on the expected error which is the sum of the two. Our exact calculations, demonstrate that the combination of estimators is particularly useful in the case where the data set is small and noisy and the function to be learned is unrealizable. For large data sets the single estimator produces superior results. Finally, we show that by splitting the data set into several independent parts and training each estimator on a different subset, the performance can in some cases be significantly improved. Key words: Bias, Variance, Least Squares, Combination. 1 INTRODUCTION Many of the problems related to supervised learning can be boiled down to the question of balancing bias and variance. While reducing bias can usually be accomplished quite easily by simply increasing the complexity of the class of models studied, this usually comes at the expense of increasing the variance in such a way that the overall expected error (which is the sum of the two) is often increased. 296 Ronny Meir Thus, many efforts have been devoted to the issue of decreasing variance, while attempting to keep the concomitant increase in bias as small as possible. One of the methods which has become popular recently in the neural network community is variance reduction by combining estimators, although the idea has been around in the statistics and econometrics literature at least since the late sixties (see Granger 1989 for a review). Nevertheless, it seems that not much analytic work has been devoted to a detailed study of the effect of noise and an effectively finite sample size on the bias/variance balance. It is the explicit goal of this paper to study in detail a simple problem of linear regression, where the full bias/variance curve can be computed exactly for any effectively finite sample size and noise level. We believe that this simple and exactly solvable model can afford us insight into more complex non-linear problems, which are at the heart of much of the recent work in neural networks. A further aspect of our work is related to the question of the partitioning of the data set between the various estimators. Thus, while most studies assume the each estimator is trained on the complete data set, it is possible to envisage a situation where the data set is broken up into several subsets, using each subset of data to form a different estimator. While such a scheme seems wasteful from the bias point of view, we will see that in fact it produces superior foreca..,ts in some situations. This, perhaps suprising, result is due to a large decrease in variance resulting from the independence of the estimators, in the case where the data subsets are independent. 2 ON THE COMBINATION OF ESTIMATORS The basic objective of regression is the following: given a finite training set, D, composed of n input/output pairs, D = {(XIJ' YIJ)}~=l' drawn according to an unkown distribution P(x, y), find a function (,estimator'), f(x; D), which 'best' approximates y. Using the popular mean-squared error criterion and taking expectations with respect to the data distribution one finds the well-known separation of the error into a bias and variance terms respectively (Geman et al. 1992) £(x) = (EDf(x; D) - E[ylx])2 + ED (f(x; D) - EDf(x; D)]2 . (1) We consider a data source of the form y = g(x) + 7], where the 'target' function g(x) is an unknown (and potentially non-linear) function and 7] is a Gaussian random variable with zero mean and variance (72. Clearly then E[Ylx] = g(x). In the usual scenario for parameter estimation one uses the complete data set, D, to form an estimator f(x; D). In this paper we consider the case where the data set D is broken up into I< subsets (not necessarily disjoint), such that D = nf= 1 DC k) , and a separate estimator is found for each subset. The full estimator is then given by the linear combination (Granger 1989) K f(x; D) = L bk/k(x; D(k) (2) k=l The optimal values of the parameters bk can be easily obtained if the data distribution, P(x, y), is known, by simply minimizing the mean-squared error (Granger Bias. Variance and the Combination of Least Squares Estimators 297 1989). In the more typical case where this distribution is unkown, one may resort to other schemes such as least-squares fitting for the parameter vector b = {bl , ... , bK}. The bias and variance of the combined estimator can be simply expressed in this case, and are given by B(x; g) = (t hh:(x) - g(x)) 2 ; V(x; g) = L bkh' {lkfk'(X) - fk(X)fk'(X)} k=l k,k' (3) where the overbars denote an average with respect to the data. It is immediately apparent that the variance term is composed of two contributions. The first term, corresponding to k = k', simply computes a weighted average of the single estimator variances, while the second term measures the average covariance between the different estimators. While the first term in the variance can be seen to decay as 1/ J{ in the case where all the weights h are of the same order of magnitude, the second term is finite unless the covariances between estimators are very small. It would thus seem beneficial to attempt to make the estimators as weakly correlated as possible in order to decrease the variance. Observe that in the extreme case where the data sets are independent of each other, the second term in the variance vanishes identically. Note that the bias term depends only on single estimator properties and can thus be computed from the theory of the single estimator. As mentioned above, however, the second term in the variance expression explicitly depends on correlations between the different estimators, and thus requires the computation of quantities beyond those of single estimators. 3 THE SINGLE LINEAR ESTIMATOR Before considering the case of a combination of estimators, we first review the case of a single linear estimator, given by f(x; D) = wT . x, where w is estimated from the data set D. Following Bos et al. (1993) we further assume that the data arises through an equation of the form y = g(x) + 1] with 9 = g(w~ . x). Looking back at equations (3) it is clear that the bias and variance are explicit functions of x and the weight vector woo In order to remove the explicit dependence we compute in what follows expectations with respect to the probability distribution of x and Wo, denoted respectively by Ep ['] and Eo[']' Thus, we define the averaged bias and variance by B = EoEp[B(x; wo)] and V = EoEp[V(x; wo)] and the expected error is then £ = B + V. In this work we consider least-squares estimation which corresponds to minimizing the empirical error, £emp(w, D) = IIXw - Y112, where X is the n x d data matrix, Y is the n x 1 output vector and w is a d x 1 weight vector. The components of the 'target' vector Yare given by YP = g(w~ . xl-') + 1]p where 1Jp are i.i.d normal random variables with zero mean and variance (J'2 . Note that while we take the estimator itself to be linear we allow the target function g(.) to be non-linear. This is meant to model the common situation where the model we are trying to fit is inadequate, since the correct model (even it exists) is usually unkown. Thus, the least squares estimator is given by wE argminw£emp(w,D). Since in this case the error-function is quadratic it possesses either a unique global minimum 298 Ronny Meir or a degenerate manifold of minima, in the case where the Hessian matrix, XT X, is singular. The solution to the least squares problem is well known (see for example Scharf 1991), and will be briefly summarized. When the number of examples, n, is smaller than the input dimension, d, the problem is underdetermined and there are many solutions with zero empirical error. The solutions can be written out explicitly in the form (n < d), (4) where V is an arbitrary d-dimensional vector. It should be noted that any vector w satisfying this equation (and thus any least-squares estimator) becomes singular as n approaches d from below, since the matrix X XT becomes singular. The minimal norm solution, often referred to as the Moore-Penrose solution, is given in this case by the choice V = O. It is common ill the literature to neglect the study of the underdetermined regime since the solution is not unique in this case. We however will pay specific attention to this case, corresponding to the often prevalent situation where the amount of data is small, attempting to show that the combination of estimators approach can significantly improve the quality of predictors in this regime. Moreover, many important inverse problems in signal processing fall into this category (Scharf 1991). In the overdetermined case, n > d (assuming the matrix X to be of full rank), a zero error solution is possible only if the function g(.) is linear and there is no noise, namely El1]2] = O. In any other case, the problem becomes unrealizable and the minimum error is non-zero. In any event, in this regime the unique solution minimizing the empirical error is given by W = (XT X)-l XTy (n > d). (5) It is eay to see that this estimator is unbiased for linear g(.). In order to compute the bias and variance for this model we use Eqs. (3) with K = 1 and bTl: = 1. In order to actually compute the expectations with respect to x and the weight vector Wo we assume in what follows that the random vector x is distributed according to a multi-dimensional normal distributions of zero mean and covariance matrix (lid)!. The vector Wo is similarly distributed with unit covariance matrix. The reason for the particular scaling chosen for the covariance matrices will become clear below. In the remainder of the paper we will be concerned with exact calculations in the so called thermodynamic limit: n, d -+ 00 and a = nl d finite. This limit is particularly useful in that the central limit theorem allows one to make precise statements about the behavior of the system, for an effectively finite sample size, a. We note in passing that in the thermodynamic limit, d -+ 00, we have Ei x; -+ 1 with probability 1 and similarly for (lid) Ei W5i' Using these simple distributions we can, after some algerbra, directly compute the bias and variance. Denoting R = Eo[wT . wo], r = Eollwll2, Q = Eollwl1 2 , one can show that the bias and variance are given by B = 7' - 2ugR + g2 v = Q -r. (6) In the above equations we have used g2 = f Dug2(u) and ug = f Du ug(u) where the Gaussian measure Du is defined by Du = (e-u~/2 1...j2-i)du. We note in passing Bias, Variance and the Combination of Least Squares Estimators 299 that the same result is obtained for any i.i.d variables, Xi, with zerO mean and variance lid. We thus note that a complete calculation of the expected bias and variance requires the explicit computation of the variables R, rand Q defined above. In principle, with the explicit expressions (4) and (5) at hand one may proceed to compute all the quantities relevant to the evaluation of the bias and variance. Unfortunately, it turns out that a direct computation of r, Rand Q using these expressions is a rather difficult task in the theory of random matrices, keeping in mind the potential non-linearity of the function g{.). A way to solve the problem can be undertaken via a slightly indirect route, using tools from statistical physics. The variables Rand Q above have been recently computed by Bos et al. (1993) using replicas and by Opper and Kinzel (1994) by a direct calculation. The variable r can be computed along similar lines resulting in the following expressions for the bias and variance (given for brevity for the Moore-Penrose solution): 0'< 1 : B a> 1 : v = _0'_ [g2 + u2 _ 0'(2 _ a )ttg2] 1-0' (7) We see from this solution that for Q > 1 the bias is constant, while the variance is monotonically decreasing with the sample size Q. For Q < 1, there are of course multiple solutions corresponding to different normalizations Q. It is easy to see, however, that the Moore-Penrose solution, gives rise to the smallest variance of all least-squares estimators (the bias is unaffected by the normalization of the solution) . The expected (or generalization) error is given simply by £ = B + V, and is thus smallest for the Moore-Penrose solution. Note that this latter result is independent of whether the function g(.) is linear or not. We note in passing that in the simple case where the target function g(-) is linear and the data is noise-free (u 2 = 0) one obtains the particularly simple result £ = 1 - a for a < 1 and £ = 0 above Q = 1. Note however that in any other case the expected error is a non-linear function of the normalized sample size a. 4 COMBINING LINEAR ESTIMATORS Having summarized the situation in the case of a single estimator, we proceed to the case of K linear estimators. In this case we assume that the complete data set, D, is broken up into K subsets D(k) of size nk = Qkd each. In particular we consider two extreme situations: (i) The data sets are independent of each other, namely D(k) n D(k') = 0 for all k, and (ii) D(k) = D for all k. We refer to the first situation as the non-overlapping case, while to the second case where all estimators are trained on the same data set as the fully-overlappin~ case. Denoting by w(k) the k'th least-squares estimator, based on training set D( ), we define the following quantities: R(k) = Eo [w(k)T . w o] , r(k,k') = Eo [w(k)T . w(k')] , p(k,k') = Eo [w(k)T . w(k')] . (8) Making use of Eqs. (3) and the probability distribution of Wo one then straighfor300 Ronny Meir wardly finds for the mixture estimator L blebk,r(k,Ie') - 2ug L bkR(k) + g2, 1e,Ie' Ie L b~ [p<k,le) r(k,Ie)] + L: bleble, [p(k,k') _ r(k,k')] . (9) Ie k#Ie' Now, the computation of the parameters p(le,k) and R(k) is identical to the case of a single estimator and can be directly read from the results of Bos et al. (1993) with the single modification of rescaling the sample size a to ak. The only interaction between estimators arises from the terms p(Ie,k') and r(k ,k') which couple two different estimators. Note however that r(k,k') is independent of the degree of overlap between the data sets D(Ie) since each estimator is averaged independently. In fact, a straighforward, if lengthy, calculation leads to the conclusion that in general r(Ie,Ie') = R(Ie) R(1e'). For the case where the data sets are independent of each other, namely D(k) n D(k') = 0 it is obvious that p(k ,k') = r(k,k'). One then finds p(Ie,Ie') = r(Ie,Ie') = R(Ie) R(k') for independent data sets. In the case where the data sets are identical, D(k) = D for all k, the variable p(k,k') has been computed by BOs et al. (1993). At this point we proceed by discussing individually the two data generation scenarios described in section 1. 4.1 NON·OVERLAPPING DATA SETS As shown above, since the data sets are independent of each other one has in this case p(le,k') = r(k,k'), implying that the second term in the expression for the variance in Eq. (9) vanishes. Specializing now to the case where the sizes of the data sets are equal, namelyak = nk/d = a/I{ for all k, we expect the variables R(k) and O(k) to be independent of the specific index k. In the particular case of a uniform mixture (bk = 1/1< for all k) we obtain (10) where the subscript 1< indicates that the values of the corresponding variables must be taken with respect to aK = a/1<. The specific values of the parameters RK and OK can be found directly from the solution of Bos et al. (1993) by replacing a by a/I{ and dividing the variance term by a further ]{ factor. Since the bias is a non-increasing function a it is always increased (or at best unchanged) by combining estimators from non-overlapping data subsets. It can also be seen from the above equation that as 1< increases the variance can be made arbitrarily small, albeit at the expense of increasing the bias. It can thus be expected that for any sample size a there is an optimal number of estimators minimizing the expected error, £ = B + V. In fact, for small a one may expand the equations for the bias and variance obtaining the expected error from which it is easy to see that the optimal value of 1< is given by 1<* = (g2 + (72)/ug2. As could be expected we see that the optimal value of I{ scales with the noise level (72, since the effect of combining multiple independent estimators is to reduce the effective noise variance by a factor of I{. As the sample size a increases and outweighs the effect of the noise, we find that the bias increase due to the data partitioning dominates the Bias, Variance and the Combination of Least Squares Estimators 301 decrease in variance and the combined estimator performs more and more poorly relative to the single estimator. Finally, for a > K we find the choice 1< = 1 always yields a lower expected error. Thus, while we have shown that for small sample sizes the effect of splitting the data set into independent subsets is helpful, this is no longer the case if the sample size is sufficiently large, in which case a single estimator based on the complete data set is superior. For a -+ 00, however, one finds that all uniform mixtures converge (to leading order) at the same rate, namely £K(a) ::::::: £00 + (£00 + (!2)ja , where £00 = g2 - ug2. For finite values of cr, however, the value of 1< has a strong effect on the quality of the stimator. 4.2 FULLY OVERLAPPING DATA SETS We focus now on the case where all estimators are formed from the same data set D, namely DUe) = D for all k. Since there is a unique solution to the least-squares estimation problem for a > 1, all least-squares estimators coincide in this regime. Thus, we focus here on the case a < 1, where multiple least-squares estimators coexist. We further assume that only mixtures of estimators of the same norm Q are allowed. We obtain for the uniform mixture Clearly the expression for the bias in this case is identical to that obtained for the single estimator, since all estimators are based On the same data set and the bias term depends only on single estimator properties. The variance term, however , is modified due to the correlation between the estimators expressed through the variable p(k,k'). Since the variance for the case of a single estimator is Q - R2 and since q $ Q it is clear in this case that the variance is reduced while the bias remains unchanged. Thus we conclude that the mixture of estimators in this case indeed produces superior performance to that of the single estimator. However, it can be seen that in the case of the Moore-Penrose solution, corresponding to choosing the smallest possible norm Q, the expected error is minimal. We thus conclude that for a < 1 the Moore-Penrose pseudo-inverse solution yields the lowest expected error, and this cannot be improved on by combining least-squares estimators obtained from the full data set D . Recall that we have shown in the previous section that (for small and noisy data sets) combining estimators formed using non-overlapping data subsets produced results superior to those of any single estimator trained on the complete data set. An interesting conclusion of these results is that splitting the data set into nonoverlapping subsets is a better strategy than training each estimator with the full data. As mentioned previously, the basic reason for this is the independence of the estimators formed in this fashion, which helps to reduce the variance term more drastically than in the case where the estimators are dependent (having been exposed to overlapping data sets). 5 CONCLUSIONS In this paper we have studied the effect of combining different estimators on the performance of linear regression. In particular we have focused on the case of linear 302 Ronny Meir least-squares estimation, computing exactly the full bias and variance curves for the case where the input dimension is very large (the so called thermodynamic limit). While we have focused specifically on the case of linear estimators, it should not be hard to extend these results to simple non-linear functions ofthe form f(wT ·x) (see section 2). The case of a combination of more complex estimators (such as multilayered neural networks) is much more demanding, as even the case of a single such network is rather difficult to analyzes. Several positive conclusions we can draw from our study are the following. First, the general claim that combining experts is always helpful is clearly fallacious. While we have shown that combining estimators is beneficial in some cases (such as small noisy data sets), this is not the case in general. Second, we have shown that in some situations (specifically unrealizable rules and small sample size) it is advantageous to split the data into several non-overlapping subsets. It turns out that in this case the decrease in variance resulting from the independence of the different estimators, is larger than the concomitant increase in bias. It would be interesting to try to generalize our results to the case where the data is split in a more efficient manner. Third, our results agree with the general notion that when attempting to learn an unrealizable function (whether due to noise or to a mismatch with the target function) the best option is to learn with errors. Ultimately one would like to have a general theory for combining empirical estimators. Our work has shown that the effect of noise and finite sample size is expected to produce non-trivial effects which are impossible to observe when considering only the asymptotic limit. Acknowledgements The author thanks Manfred Opper for a very helpful conversation and the Ollendorff center of the Electrical Engineering department at the Technion for financial support. References S. Bos., W. Kinzel and M. Opper 1993, The generalization ability of perceptrons with continuous outputs, Phys. Rev. A 47:1384-1391. S. Geman, E. Bienenstock and R. Dorsat 1992, Neural networks and the bias/variance dilemma, Neural Computation 4:1-58. C.W.J. Granger 1989, Combining forecasts - twenty years later, J. of Forecast. 8:167-173. M. Opper and W. Kinzel 1994, Statistical mechanics of generalization, in Physics of Neural networks, van Hemmen, J.S., E. Domany and K. Schulten eds., SpringerVerlag, Berlin. L.1. Scharf, 1991 Statistical Signal Processing: Detection, Estimation and Time Series Analysis, Addison-Wesley, Massachusetts.	1994	9
987	Forward dynamic models in human motor control: Psychophysical evidence Daniel M. Wolpert wolpert@psyche.mit.edu Zouhin Ghahramani zoubin@psyche.mit.edu Michael I. Jordan jordan@psyche.mit.edu Department of Brain & Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 Abstract Based on computational principles, with as yet no direct experimental validation, it has been proposed that the central nervous system (CNS) uses an internal model to simulate the dynamic behavior of the motor system in planning, control and learning (Sutton and Barto, 1981; Ito, 1984; Kawato et aI., 1987; Jordan and Rumelhart, 1992; Miall et aI., 1993). We present experimental results and simulations based on a novel approach that investigates the temporal propagation of errors in the sensorimotor integration process. Our results provide direct support for the existence of an internal model. 1 Introduction The notion of an internal model, a system which mimics the behavior of a natural process, has emerged as an important theoretical concept in motor control (Jordan, 1995). There are two varieties of internal models-"forward models," which mimic the causal flow of a process by predicting its next state given the current state and the motor command, and "inverse models," which are anticausal, estimating the motor command that causes a particular state transition. Forward modelsthe focus of this article-have been been shown to be of potential use for solving four fundamental problems in computational motor control. First, the delays in most sensorimotor loops are large, making feedback control infeasible for rapid 44 Daniel M. Wolpert, Zoubin Ghahramani, Michaell. Jordan movements. By using a forward model for internal feedback the outcome of an action can be estimated and used before sensory feedback is available (Ito, 1984; Miall et al., 1993). Second, a forward model is a key ingredient in a system that uses motor outflow ("efference copy") to anticipate and cancel the reafferent sensory effects of self-movement (Gallistel, 1980; Robinson et al., 1986). Third, a forward model can be used to transform errors between the desired and actual sensory outcome of a movement into the corresponding errors in the motor command, thereby providing appropriate signals for motor learning (Jordan and Rumelhart, 1992). Similarly by predicting the sensory outcome of the action, without actually performing it, a forward model can be used in mental practice to learn to select between possible actions (Sutton and Barto, 1981). Finally, a forward model can be used for state estimation in which the model's prediction of the next state is combined with a reafferent sensory correction (Goodwin and Sin, 1984). Although shown to be of theoretical importance, the existence and use of an internal forward model in the CNS is still a major topic of debate. When a subject moves his arm in the dark, he is able to estimate the visual location of his hand with some degree of accuracy. Observer models from engineering formalize the sources of information which the CNS could use to construct this estimate (Figure 1). This framework consists of a state estimation process (the observer) which is able to monitor both the inputs and outputs of the system. In particular, for the arm, the inputs are motor commands and the output is sensory feedback (e.g. vision and proprioception). There are three basic methods whereby the observer can estimate the current state (e.g. position and velocity) of the hand form these sources: It can make use of sensory inflow, it can make use of integrated motor outflow (dead reckoning), or it can combine these two sources of information via the use of a forward model. u(t) Input r Moto Comma nd System X(t) Observer Output y(t) S ensory edback Fe State estimate x(t) Figure 1. Observer model of state estimation. Forward Dynamic Models in Human Motor Control 45 2 State Estimation Experiment To test between these possibilities, we carried out an experiment in which subjects made arm movements in the dark. The full details of the experiment are described in the Appendix. Three experimental conditions were studied, involving the use of null, assistive and resistive force fields. The subjects' internal estimate of hand location was assessed by asking them to localize visually the position of their hand at the end of the movement. The bias of this location estimate, plotted as a function of movement duration shows a consistent overestimation of the distance moved (Figure 2). This bias shows two distinct phases as a function of movement duration, an initial increase reaching a peak of 0.9 cm after one second followed by a sharp transition to a region of gradual decline. The variance of the estimate also shows an initial increase during the first second of movement after which it plateaus at about 2 cm2 . External forces had distinct effects on the bias and variance propagation. Whereas the bias was increased by the assistive force and decreased by the resistive force (p < 0.05), the variance was unaffected. a c 1.0 1.5 E 1.0 E 0 0.5 0 0.5 (J) 0.0 - - ~ . -- .:... :~::- . -' (J) <tI <tI i:i5 -0.5 CD <] -1 .0 0.0 -1.5 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 b Time (5) d Time (5) 2.5 2 (\I (\I E E 0 2.0 0 1 ..... . Q) 1.5 Q) 0 0 0 c:: c:: 1.0 <tI <tI L... L... -1 <tI 0.5 <tI > > 0.0 <] -2 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 Time (5) Time (5) Figure 2. The propagation of the (a) bias and (b) variance of the state estimate is shown, with standard error lines, against movement duration. The differential effects on (c) bias and (d) variance of the external force, assistive (dotted lines) and resistive (solid lines), are also shown relative to zero (dashed line). A positive bias represents an overestimation of the distance moved. 46 Daniel M. Wolpert, Zoubin Ghahramani, Michael I. Jordan 3 Observer Model Simulation These experimental results can be fully accounted for only if we assume that the motor control system integrates the efferent outflow and the reafferent sensory inflow. To establish this conclusion we have developed an explicit model of the sensorimotor integration process which contains as special cases all three of the methods referred to above. The model-a Kalman filter (Kalman and Bucy, 1961)-is a linear dynamical system that produces an estimate of the location of the hand by monitoring both the motor outflow and the feedback as sensed, in the absence of vision, solely by proprioception. Based on these sources of information the model estimates the arm's state, integrating sensory and motor signals to reduce the overall uncertainty in its estimate. Representing the state of the hand at time t as x(t) (a 2 x 1 vector of position and velocity) acted on by a force u = [Uint, Uext]T, combining both internal motor commands and external forces, the system dynamic equations can be written in the general form of x(t) = Ax(t) + Bu(t) + w(t), (1) where A and B are matrices with appropriate dimension. The vector w(t) represents the process of white noise with an associated covariance matrix given by Q = E[w(t)w(t)T]. The system has an observable output y(t) which is linked to the actual hidden state x(t) by y(t) = Cx(t) + v(t), (2) where C is a matrix with appropriate dimension and the vector v(t) represents the output white noise which has the associated covariance matrix R = E[v(t)v(t)T]. In our paradigm, y(t) represents the proprioceptive signals (e.g. from muscle spindles and joint receptors). In particular, for the hand we approximate the system dynamics by a damped point mass moving in one dimension acted on by a force u(t). Equation 1 becomes (3) where the hand has mass m and damping coefficient {3. We assume that this system is fully observable and choose C to be the identity matrix. The parameters in the simulation, {3 = 3.9 N ,s/m, m = 4 kg and Uint = 1.5 N were chosen based on the mass of the arm and the observed relationship between time and distance traveled. The external force Uext was set to -0.3, 0 and 0.3 N for the resistive, null and assistive conditions respectively. To end the movement the sign of the motor command Uint was reversed until the arm was stationary. Noise covariance matrices of Q = 9.5 X 10-5 [ and R = 3.3 x 1O-4[ were used representing a standard deviation of 1.0 cm for the position output noise and 1.8 cm s-l for the position component of the state noise. At time t = 0 the subject is given full view of his arm and, therefore, starts with an estimate X(O) = x(O) with zero bias and variance-we assume that vision calibrates the system. At this time the light is extinguished and the subject must rely on the inputs and outputs to estimate the system's state. The Kalman filter, using a Forward Dynamic Models in Human Motor Control 47 model of the system A, Band C, provides an optimal linear estimator of the state given by i(t) = Ax(t) + Bu(t) + K(t)[y(t) - Cx(t)] , .I , I 'V' V forward model sensory correction where K(t) is the recursively updated gain matrix. The model is, therefore, a combination of two processes which together contribute to the state estimate. The first process uses the current state estimate and motor command to predict the next state by simulating the movement dynamics with a forward model. The second process uses the difference between actual and predicted reafferent sensory feedback to correct the state estimate resulting from the forward model. The relative contributions of the internal simulation and sensory correction processes to the final estimate are modulated by the Kalman gain matrix K(t) so as to provide optimal state estimates. We used this state update equation to model the bias and variance propagation and the effects of the external force. By making particular choices for the parameters of the Kalman filter, we are able to simulate dead reckoning, sensory inflow-based estimation, and forward modelbased sensorimotor integration. Moreover, to accommodate the observation that subjects generally tend to overestimate the distance that their arm has moved, we set the gain that couples force to state estimates to a value that is larger than its veridical value; B = ~ [1~4 1~6] while both A and C accurately reflected the true system. This is consistent with the independent data that subjects tend to under-reach in pointing tasks suggesting an overestimation of distance traveled (Soechting and Flanders, 1989). Simulations of the Kalman filter demonstrate the two distinct phases of bias propagation observed (Figure 3). By overestimating the force acting on the arm the forward model overestimates the distance traveled, an integrative process eventually balanced by the sensory correction. The model also captures the differential effects on bias of the externally imposed forces. By overestimating an increased force under the assistive condition, the bias in the forward model accrues more rapidly and is balanced by the sensory feedback at a higher level. The converse applies to the resistive force. In accord with the experimental results the model predicts no change in variance under the two force conditions. 4 Discussion We have shd-ttn that the Kalman filter is able to reproduce the propagation of the bias and variance of estimated position of the hand as a function of both movement duration and external forces. The Kalman filter also simulates the interesting and novel empirical result that while the variance asymptotes, the bias peaks after about one second and then gradually declines. This behavior is a consequence pf a trade off between the inaccuracies accumulating in the internal simulation of the arm's dynamics and the feedback of actual sensory information. Simple models which do not trade off the contributions of a forward model with sensory feedback, such as those based purely on sensory inflow or on motor outflow, are unable to reproduce the observed pattern of bias and variance propagation. The ability of the Kalman filter to parsimoniously model our data suggests that the processes embodied in the 48 Daniel M. Wolpert, Zoubin Ghahramani, Michaell. Jordan a c 1.0 1.5 E 1.0 E 0 0.5 0 - 0.5 en 0.0 ~-------:.:...:.:...:,;.;.:.;.. en n1 n1 [Ii -0.5 en <l -1.0 -1.5 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 b Time (s) d Time (s) - 2.5 2 C\I C\I E E 0 2.0 0 Q) 1.5 Q) 0 0 0 c:: 1.0 c:: n1 n1 .... .... -1 n1 0.5 n1 > > 0.0 <l -2 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 Time (s) Time (s) Figure 3. Simulated bias and variance propagation, in the same representation and scale as Figure 2, from a Kalman filter model of the sensorimotor integration process. filter, namely internal simulation through a forward model together with sensory correction, are likely to be embodied in the sensorimotor integration process. We feel that the results of this state estimation study provide strong evidence that a forward model is used by the CNS in maintaining its estimate of the hand location. Furthermore, the state estimation paradigm provides a framework to study the sensorimotor integration process in both normal and patient populations. For example, the specific predictions of the sensorimotor integration model can be tested in both patients with sensory neuropathies, who lack proprioceptive reafference, and in patients with damage to the cerebellum, a proposed site for the forward model (Miall et al., 1993). Acknowledgements We thank Peter Dayan for suggestions about the manuscript. This project was supported by grants from the McDonnell-Pew Foundation, ATR Human Information Processing Research Laboratories, Siemens Corporation, and by grant N00014-94-10777 from the Office of Naval Research. Daniel M. Wolpert and Zoubin Ghahramani are McDonnell-Pew Fellows in Cognitive Neuroscience. Michael!. Jordan is a NSF Presidential Young Investigator. Forward Dynamic Models in Human Motor Control 49 Appendix: Experimental Paradigm To investigate the way in which errors in the state estimate change over time and with external forces we used a setup (Figure 4) consisting of a combination of planar virtual visual feedback with a two degree of freedom torque motor driven manipulandum (Faye, 1986). The subject held a planar manipulandum on which his thumb was mounted. The manipulandum was used both to accurately measure the position of the subject's thumb and also, using the torque motors, to constrain the hand to move along a line across the subject's body. A projector was used to create virtual images in the plane of the movement by projecting a computer VGA screen onto a horizontal rear projection screen suspended above the manipulandum. A horizontal semi-silvered mirror was placed midway between the screen and manipulandum. The subject viewed the reflected image of the rear projection screen by looking down at the mirror; all projected images, therefore, appeared to be in the plane of the thumb, independent of head position. Projector ,', \ Cursor '0' \ Image I : I , I' \ I' \ " 1. \ Screen Finger .,,> '> It; Semi·silvered mirror .~ Bulb ~orque motors Manlpulandum Figure 4. Experimental Setup Eight subjects participated and performed 300 trials each. Each trial started with the subject visually placing his thumb at a target square projected randomly on the movement line. The arm was then illuminated for two seconds, thereby allowing the subject to perceive visually his initial arm configuration. The light was then extinguished leaving just the initial target. The subject was then required to move his hand either to the left or right, as indicated by an arrow in the initial starting square. This movement was made in the absence of visual feedback of arm configuration. The subject was instructed to move until he heard a tone at which point he stopped. The timing of the tone was controlled to produce a uniform distribution of path lengths from 0-30 cm. During this movement the subject either moved in a randomly selected null or constant assistive or resistive 0.3N force field generated by the torque motors. Although it is not possible to directly probe a subject's internal representation of the state of his arm, we can examine a function of this state-the estimated visual location of the thumb. (The relationship between the state of the arm and the visual coordinates of the hand is known as 50 Daniel M. Wolpert, Zoubin Ghahramani, Michaell. Jordan the kinematic transformation; Craig, 1986.) Therefore, once at rest the subject indicated the visual estimate of his unseen thumb position using a trackball, held in his other hand, to move a cursor projected in the plane of the thumb along the movement line. The discrepancy between the actual and visual estimate of thumb location was recorded as a measure of the state estimation error. The bias and variance propagation of the state estimate was analyzed as a function of movement duration and external forces. A generalized additive model (Hastie and Tibshirani, 1990) with smoothing splines (five effective degrees of freedom) was fit to the bias and variance as a function of final position, movement duration and the interaction of the two forces with movement duration, simultaneously for main effects and for each subject. This procedure factors out the additive effects specific to each subject and, through the final position factor, the position-dependent inaccuracies in the kinematic transformation. References Craig, J. (1986). Introduction to robotics. Addison-Wesley, Reading, MA. Faye, I. (1986). An impedence controlled manipulandum for human movement studies. MS Thesis, MIT Dept. Mechanical Engineering, Cambridge, MA. Gallistel, C. (1980). The organization of action: A new synthesis. Erlbaum, Hilladale, NJ. Goodwin, G. and Sin, K. (1984). Adaptive filtering prediction and control. PrenticeHall, Englewood Cliffs, NJ. Hastie, T. and Tibshirani, R. (1990). Generalized Additive Models. Chapman and Hall, London. Ito, M. (1984). The cerebellum and neural control. Raven Press, New York. Jordan, M. and Rumelhart, D. (1992). Forward models: Supervised learning with a distal teacher. Cognitive Science, 16:307-354. Jordan, M. I. (1995). Computational aspects of motor control and motor learning. In Heuer, H. and Keele, S., editors, Handbook of Perception and Action: Motor Skills. Academic Press, New York. Kalman, R. and Bucy, R. S. (1961). New results in linear filtering and prediction. Journal of Basic Engineering (ASME), 83D:95-108. Kawato, M., Furawaka, K., and Suzuki, R. (1987). A hierarchical neural network model for the control and learning of voluntary movements. Biol.Cybern., 56:117. Miall, R., Weir, D., Wolpert, D., and Stein, J. (1993). Is the cerebellum a Smith Predictor? Journal of Motor Behavior, 25(3):203-216. Robinson, D., Gordon, J., and Gordon, S. (1986). A model of the smooth pursuit eye movement system. Biol.Cybern., 55:43-57. Soechting, J. and Flanders, M. (1989). Sensorimotor representations for pointing to targets in three- dimensional space. J.Neurophysiol., 62:582-594. Sutton, R. and Barto, A. (1981). Toward a modern theory of adaptive networks: expettation and prediction. Psychol.Rev., 88:135-170.	1994	90
988	Direction Selectivity In Primary Visual Cortex Using Massive Intracortical Connections Humbert Suarez CNS Program 216-76 Caltech Pasadena, CA 91125 Christof Koch CNS Program 216-76 Caltech Pasadena, CA 91125 Rodney Douglas MRC Anatomical Neuropharmacology Unit University of Oxford Oxford UK Abstract Almost all models of orientation and direction selectivity in visual cortex are based on feedforward connection schemes, where geniculate input provides all excitation to both pyramidal and inhibitory neurons. The latter neurons then suppress the response of the former for non-optimal stimuli. However, anatomical studies show that up to 90 % of the excitatory synaptic input onto any cortical cell is provided by other cortical cells. The massive excitatory feedback nature of cortical circuits is embedded in the canonical microcircuit of Douglas &. Martin (1991). We here investigate analytically and through biologically realistic simulations the functioning of a detailed model of this circuitry, operating in a hysteretic mode. In the model, weak geniculate input is dramatically amplified by intracortical excitation, while inhibition has a dual role: (i) to prevent the early geniculate-induced excitation in the null direction and (ii) to restrain excitation and ensure that the neurons fire only when the stimulus is in their receptive-field. Among the 4 Humbert Suarez, Christo! Koch, Rodney Douglas insights gained are the possibility that hysteresis underlies visual cortical function, paralleling proposals for short-term memory, and strong limitations on linearity tests that use gratings. Properties of visual cortical neurons are compared in detail to this model and to a classical model of direction selectivity that does not include excitatory corti co-cortical connections. The model explain a number of puzzling features of direction-selective simple cells, including the small somatic input conductance changes that have been measured experimentally during stimulation in the null direction. The model also allows us to understand why the velocity-response curve of area 17 neurons is different from that of their LG N afferents, and the origin of expansive and compressive nonlinearities in the contrast-response curve of striate cortical neurons. 1 INTRODUCTION Direction selectivity is the property of neurons that fire more strongly for one direction of motion of a bar (the preferred direction) than the other (nUll direction). It is one of the fundamental properties of neurons in visual cort~x and is intimately related to the processing of motion by the visual system. LGN neurons that provide input to visual cortex respond approximately symmetrically to motion in different directions; so cortical neurons must generate that direction specificity. Models of direction selectivity in primary visual. cortex generally overlook two important constraints. 1. at least 80% of excitatory synapses on cortical pyramidal cells originate from other pyramidal cells, and less than 10 % are thalamic afferents (Peters & Payne, 1993). 2. Intracellular in vivo recordings in cat simple cells by Douglas, et al. (1988) failed to detect significant changes in somatic input conductance during stimulation in the null direction, indicating that there is very little synaptic input to direction-selective neurons in that condition, including no massive "shunting" inhibition. One very attractive solution incorporating these two constraints was proposed by Douglas & Martin (1991) in the form of their canonical microcircuit: for motion of a visual stimulus in the preferred direction, weak geniculate excitation excites cortical pyramidal cells to respond moderately. This relatively small amount of cortical excitation is amplified via excitatory cortico-cortical connections. For motion in the null direction, the weak geniculate excitation is vetoed by weak inhibition (mediated via an interneuron) and the cortical loop is never activated. In order to test quantitatively this circuit against the large body of experimental data, it is imperative to model its operation through mathematical analysis and detailed simulations. 2 MODEL DESCRIPTION AND ANALYSIS The model consists of a retino-geniculate and a cortical stage (Fig. 1). The former includes a center-surround receptive field and bandpass temporal filtering (Victor, 1987). We simulate a I-D array of ON LGN neurons, with 208 LGN neurons at each of 6 spatial positions in this array. The output ofthe LGN-action potentials-feeds Direction Selectivity in Primary Visual Cortex 5 into the cortical module consisting of 640 pyramidal (excitatory) and 160 inhibitory neurons. Each neuron is modeled using 3-4 compartments whose parameters reflect, in a simplified way, the biophysics and morphology of cortical neurons; the somatic compartment produces action potentials in response to current injection. There are excitatory connections among all pyramidal neurons, and inhibitory connections from the inhibitory population to itself and to the pyramidal population. The receptive field of the geniculate input to the inhibitory cortical neurons is displaced in space from the geniculate input to the pyramidal neurons, so that in the null direction inhibition overlaps with geniculate excitation in the pyramidal neurons, resulting in direction selectivity in the pyramidal neurons. The time courses of the post-synaptic potentials (PSP's) are consistent with physiologically recorded PSP's. The EPSP's in our model arise exclusively from non-NMDA synapses, and the IPSP's originate from both GABAA and GABAB synapses. There are two operating modes of this cortical amplifier circuit, depending on parameter values. In the first mode, the pyramidal neurons' response increases proportionally to the stimulus strength over a substantial range of input values, before saturating. In the second mode, the response increases much faster over a narrow range of stimulus strengths, then saturates. Analytically, one can define a steadystate transfer function for the network of pyramidal neurons; then, in the first mode, the transfer function has a slope that is less than 1, so that the network's firing rate at equilibrium increases proportionally to the input strength and the network dynamics are rather slow. In the second mode the initial slope is larger than 1, so that the network can discharge briskly at equilibrium even without any input, show hysteresis, and has rather fast dynamics. The network does not fire without any input, because of the neuron's threshold. In this paper, we will show responses in that second, hysteretic mode of operation. We will compare the cortical amplifier model's response properties to a pure feedforward model, that has no excitatory connections between pyramidal neurons. In order to maintain strong responses in that model, the LGN was connected more strongly to the pyramidal neurons, and in order to maintain direction selectivity, the weights of the connections of the inhibitory neurons to the pyramids were increased as well. 3 AMPLIFICATION AND CONDUCTANCE CHANGE IN THE NULL DIRECTION The input conductance of pyramidal neurons, a measure of total synaptic input, changes by only 50 % in the cortical amplifier model versus 400 % in the feedforward model, and so is more consistent with physiology (Fig. 2). Indeed, most of the current causing firing in the preferred direction originates from other pyramidal neurons, so the connection weight from the LGN is small. Consequently, the inhibitory weight is also small, since it needs only be large enough to balance out the LG N current in the null direction. Since there is little firing in other pyramidal neurons in the null direction, there is also little total synaptic input to the fiducial cell. The cortical amplifier circuit provides substantial amplification of the LGN input. In the preferred direction, excitatory intracortical connections amplify the LGN input, providing a feedback current that is about 2.2 times larger than the 6 Humbert Suarez, Christo! Koch, Rodney Douglas AREA 17 LGN RETINA 640 160 smooth cells 111I11111111111111 ==-1111111111111111111 208 neurons at each position 125 pixels Figure 1: Wiring diagram of the direction selectivity model. Input to LGN neurons comes from a one-dimensional array of retinal pixels. There are 208 LGN neurons at each of six spatial positions. The LG N neurons connect slightly differently with the two populations of cortical neurons (pyramidal and inhibitory) so that as a group the LGN inputs to the pyramids are displaced spatially by 5' from those to the inhibitory neurons. The open triangle symbols denote excitatory connections, the filled triangles inhibitory GABAA-mediated synapses, and the filled circles inhibitory GABAB-mediated synapses. The capital sigma symbol indicates convergence of inputs from many LGN neurons onto cortical neurons. Direction Selectivity in Primary Visual Cortex 0.5 0 4 ~ .s 0.3 E ~ 0.2 :; 0 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 4()0 c 300 c j 200 ~: .... 100 -: ... ..,...,..-..,. •• . .... ; .. :"-:~ Time (seconds) b r25 1 tOO ~:~I ~:':-~_.'~~::'>:~ :: ~"~:'~:':'.~:..~ .,)0.1:' 75 -2sec Cumal 0 Tesl · 2iec 150 d 125 ..... '. 100 o\V_,MJ _ ... : ---..,;" ·o\...-ohl 75 501----------' o 7 Figure 2: (a) Total synaptic current in a simulated pyramidal cell from the LGN (with symbols) and from other pyramidal cells (continuous without symbols), during stimulation by a bar moving in the preferred direction. For the same stimulus moving in the null direction, somatic input conductance as a function of time, (b) for a direction-selective pyramidal neuron (data from Douglas et ai., 1988); (c) feedforward model; (d) cortical amplifier model; LGN current at 60 % contrast (Fig. 2). 4 CONTRAST-RESPONSE CURVES Contrast-response curves plot the peak firing rate to a grating moving in the preferred direction as function of its contrast, or stimulus amplitude (Albrecht & Hamilton, 1982). The cortical amplifier's contrast-response curve is very different from the LGN inputs' and is similar to those that have been described experimentally in cortex (Albrecht & Geisler 1991), having a steep power-function portion followed by abrupt saturation (Fig. 3). The network firing saturates at the fixed point of the transfer function mentioned above (see section 2) and the steep portion results from the fast rise to that fixed point when the stimulus has exceeded the neurons' threshold. In contrast, the feedforward model's contrast-response curve is similar to the LGN inputs' and does not match physiology. The response is very small in the null direction, resulting in very good direction selectivity at all contrasts (i.e., the average direction index is above 0.9). 5 VELOCITY-RESPONSE CURVES 8 Humbert Suarez, Christo! Koch, Rodney Douglas KlO 100 a u b S! '" <II L .: a. ~ 10 :< .. .. ~ 2' :,.; u. 1 1 10 100 Contrast ('X.) 100 100 UUS! c ~ S! d <II <II .. .. ~ 1 10 ! 10 .. ~ ~ go 2' :~ J; u. IL 1 1 1 10 100 1 10 100 Contrast ('X.) Contrast('X.) Figure 3: Peak firing rate versus contrast during stimulation by a moving grating. (a) Visual cortical neuron. Data from Albrecht &, Hamilton, 1982. (b) LGN model. ( c) Feedforward model. (d) Cortical amplifier model. Velocity-response curves plot the peak response to a bar moving in the preferred direction as a function of its velocity. Again, the cortical amplifier's velocityresponse curve is very different from the LGN inputs' and is similar to physiology (Orban, 1984; Fig. 4). The LGN model is firing strongly at high velocities but the model pyramidal neurons are totally silent, due to a combination of GABAA inhibition, neuron threshold, and membrane low-pass filtering. At low velocities, the LGN model does not fire much, while the cortical neurons respond strongly. Indeed, the network firing has enough time to reach the fixed point of the transfer function and will reach it as long as the input is suprathreshold. In contrast, the feedforward model's velocity-response curve is again similar to the LGN input's and does not match physiology. Also shown in Fig. 4 is the response in the null direction for the cortical amplifier model. There is very good direction selectivity at all velocities, consistent with physiological data (Orban, 1984). The persistence of direction selectivity down to low velocities depends critically on the time constant of GABAB and the presence of a very small displacement between the LGN inputs to the pyramidal and inhibitory neurons. 6 OTHER PROPERTIES Recently, direction-selective cortical neurons have been tested for linearity using an intracellular grating superposition test and found to be quite linear (Jagadeesh et al., 1993). Despite that amplification in the present model is so nonlinear, the model is also linear according to that superposition test. An analysis of the test in the context of the model shows that such a test has limited usefulness and suggests improvements. Given that the network transfer function has a fixed point at high firing without any Direction Selectivity in Primary Visual Cortex 9 U 400 ~ ~ lit LGN X ~ .~'OO a .~ !! .. 200 D .. GO 1n0 .E .. iL 0 0.1 10 100 1000 Stimulus velocity (e/sec) 120 100 'S 'S 51100 c 51 80 d CiJ CiJ Q) 80 Q) '! '! 60 CIl 60 ~ P -Q) Q) 40 Cii 40 GN-::J Cii ... ... CI ell c FF c: 20 :~ 20 :~ u.. u.. NP 0 0 0.1 1 10 100 0.1 1 10 100 Velocity (deg/sec) Velocity (deg/sec) Figure 4: Peak firing rate versus velocity during stimulation by a bar. (a) LGN neuron in cat (Frishman et aJ., 1983). (b) area 17 visual cortical neuron in cat (Orban, 1984). (c) LGN model neuron (curve labelled "LGN") and feedforward model (FF). (d) Cortical amplifier model in the preferred (curve labelled "P") and null directions (NP). 10 Humbert Suarez, Christo! Koch, Rodney Douglas input (see section 2), hysteresis may occur, whereby the network's discharge persists after the initial stimulus is withdrawn. Because of hysteresis, it is imperative to reset the network by presenting a negative, or inhibitory, input. A parallel can be drawn to several recent proposals for the mechanisms underlying short-term memory. 7 CONCLUSIONS From early on, neurophysiologists have proposed that the LG N provides most of the input to visual cortical neurons and shapes their receptive properties (Hubel and Wiesel, 1962). However, direction selectivity and several other stimulus selectivities are not present in LGN neurons, and other important discrepancies have appeared between the receptive field properties of cortical neurons and those of their LG N afferents. Anatomically, synaptic inputs from the LGN account for less than 10 % of synapses to pyramidal neurons in visual cortex; the remaining 90 % could clearly provide a substrate for these receptive field transformations. Although intracortical inhibition has often been invoked to explain various cortical properties, excitation is usually not mentioned, despite that most corti co-cortical synapses are excitatory. In this paper, we show that intracortical excitation can better account for several key properties of cortical neurons than a purely feedforward model, including the magnitude of the conductance change in the null direction, contrast-response curves, and velocity-response curves. Furthermore, surprinsingly, other key cell properties that are appear to point to feedforward models, such as linearity measured by superposition tests, are also properties of a model based on intracortical excitation. Acknowledgements This research was supported by the Office of Naval Research, the National Science Foundation, the National Eye Institute, and the McDonnell Foundation. References Albrecht, D.G., and Geisler, W.S. (1991) Visual Neuroscience 7, 531-546. Albrecht, D.B., and Hamilton, D.B. (1982) J. Neurophysiol. 48,217-237. Douglas, R.J., and Martin, K.A.C. (1991) J. Physiol. 440,735-769. Douglas, R.J ., Martin, K.A.C., and Whitteridge, D. (1988) Nature 332,642-644. Frishman, L.J., Schweitzer-Tong, D.E., and Goldstein, E.B. (1983) J. Neurophysiol. 50, 1393-1414. Hubel, D.H., and Wiesel, T.N. (1962) J. Physiol. 165,559-568. Jagadeesh, B., Wheat, H.S. and Ferster, D. (1993) Science 262, 1901-1904 Orban, G.A. (1984) Neuronal operations in the visual cortex. Springer, Berlin. Peters, A. and Payne, B. R. (1993) Cerebral Cortex 3, 69-78. Victor, J.D. (1987) J.Physiol. 386, 219-246.	1994	91
989	Bayesian Query Construction for Neural Network Models Gerhard Paass Jorg Kindermann German National Research Center for Computer Science (GMD) D-53757 Sankt Augustin, Germany paass@gmd.de kindermann@gmd.de Abstract If data collection is costly, there is much to be gained by actively selecting particularly informative data points in a sequential way. In a Bayesian decision-theoretic framework we develop a query selection criterion which explicitly takes into account the intended use of the model predictions. By Markov Chain Monte Carlo methods the necessary quantities can be approximated to a desired precision. As the number of data points grows, the model complexity is modified by a Bayesian model selection strategy. The properties of two versions of the criterion ate demonstrated in numerical experiments. 1 INTRODUCTION In this paper we consider the situation where data collection is costly, as when for example, real measurements or technical experiments have to be performed. In this situation the approach of query learning ('active data selection', 'sequential experimental design', etc.) has a potential benefit. Depending on the previously seen examples, a new input value ('query') is selected in a systematic way and the corresponding output is obtained. The motivation for query learning is that random examples often contain redundant information, and the concentration on non-redundant examples must necessarily improve generalization performance. We use a Bayesian decision-theoretic framework to derive a criterion for query construction. The criterion reflects the intended use of the predictions by an appropriate 444 Gerhard Paass. Jorg Kindermann loss function. We limit our analysis to the selection of the next data point, given a set of data already sampled. The proposed procedure derives the expected loss for candidate inputs and selects a query with minimal expected loss. There are several published surveys of query construction methods [Ford et al. 89, Plutowski White 93, Sollich 94]. Most current approaches, e.g. [Cohn 94], rely on the information matrix of parameters. Then however, all parameters receive equal attention regardless of their influence on the intended use of the model [Pronzato Walter 92]. In addition, the estimates are valid only asymptotically. Bayesian approaches have been advocated by [Berger 80], and applied to neural networks [MacKay 92]. In [Sollich Saad 95] their relation to maximum information gain is discussed. In this paper we show that by using Markov Chain Monte Carlo methods it is possible to determine all quantities necessary for the selection of a query. This approach is valid in small sample situations, and the procedure's precision can be increased with additional computational effort. With the square loss function, the criterion is reduced to a variant of the familiar integrated mean square error [Plutowski White 93]. In the next section we develop the query selection criterion from a decision-theoretic point of view. In the third section we show how the criterion can be calculated using Markov Chain Monte Carlo methods and we discuss a strategy for model selection. In the last section, the results of two experiments with MLPs are described. 2 A DECISION-THEORETIC FRAMEWORK Assume we have an input vector x and a scalar output y distributed as y "" p(y I x, w) where w is a vector of parameters. The conditional expected value is a deterministic function !(x, w) := E(y I x, w) where y = !(x, w)+£ and £ is a zero mean error term. Suppose we have iteratively collected observations D(n) := ((Xl, iii), .. . , (Xn, Yn)). We get the Bayesian posterior p(w I D(n)) = p(D(n) I w) p(w)/ J p(D(n) I w) p(w) dw and the predictive distribution p(y I x, D(n)) = J p(y I x, w)p(w I D(n)) dw if p(w) is the prior distribution. We consider the situation where, based on some data x, we have to perform an action a whose result depends on the unknown output y. Some decisions may have more severe effects than others. The loss function L(y, a) E [0,00) measures the loss if y is the true value and we have taken the action a E A. In this paper we consider real-valued actions, e.g. setting the temperature a in a chemical process. We have to select an a E A only knowing the input x. According to the Bayes Principle [Berger 80, p.14] we should follow a decision rule d : x --t a such that the average risk J R(w, d) p(w I D(n)) dw is minimal, where the risk is defined as R(w, d) := J L(y, d(x)) p(y I x, w) p(x) dydx. Here p(x) is the distribution of future inputs, which is assumed to be known. For the square loss function L(y, a) = (y - a)2, the conditional expectation d(x) := E(y I x, D(n)) is the optimal decision rule. In a control problem the loss may be larger at specific critical points. This can be addressed with a weighted square loss function L(y, a) := h(y)(y - a)2, where h(y) 2: a [Berger 80, p.1U]. The expected loss for an action is J(y - a)2h(y) p(y I x, D(n)) dy. Replacing the predictive density p(y I x, D(n)) with the weighted predictive density Bayesian Query Construction for Neural Network Models 445 p(y I x, Den) := h(y) p(y I x, Den)/G(x), where G(x) := I h(y) p(y I x, Den) dy, we get the optimal decision rule d(x) := I yp(y I x, Den) dy and the average loss G(x) I(y - E(y I x, D(n))2 p(y I x, Den) dy for a given input x. With these modifications, all later derIvations for the square loss function may be applied to the weighted square loss. The aim of query sampling is the selection of a new observation x in such a way that the average risk will be maximally reduced. Together with its still unknown y-value, x defines a new observation (x, y) and new data Den) U (x, y). To determine this risk for some given x we have to perform the following conceptual steps for a candidate query x: 1. Future Data: Construct the possible sets of 'future' observations Den) U (x, y), where y ""' p(y I x, Den). 2. Future posterior: Determine a 'future' posterior distribution of parameters p(w I Den) U (x, y» that depends on y in the same way as though it had actually been observed. 3. Future Loss: Assuming d~,x(x) is the optimal decision rule for given values of x, y, and x, compute the resulting loss as 1';,x(x):= J L(y,d;,x(x»p(ylx,w)p(wIDen)U(x,y»dydw (1) 4. Averaging: Integrate this quantity over the future trial inputs x distributed as p(x) and the different possible future outputs y, yielding 1';:= Ir;,x(x)p(x)p(ylx,Den)dxdy. This procedure is repeated until an x with minimal average risk is found. Since local optima are typical, a global optimization method is required. Subsequently we then try to determine whether the current model is still adequate or whether we have to increase its complexity (e.g. by adding more hidden units). 3 COMPUTATIONAL PROCEDURE Let us assume that the real data Den) was generated according to a regression model y = !(x, w)+{ with i.i.d. Gaussian noise {""' N(O, (T2(w». For example !(x, w) may be a multilayer perceptron or a radial basis function network. Since the error terms are independent, the posterior density is p( w I Den) ex: p( w) rr~=l P(Yi I Xi, w) even in the case of query sampling [Ford et al. 89]. As the analytic derivation of the posterior is infeasible except in trivial cases, we have to use approximations. One approach is to employ a normal approximation [MacKay 92], but this is unreliable if the number of observations is small compared to the number of parameters. We use Markov Chain Monte Carlo procedures [PaaB 91, Neal 93] to generate a sample WeB) := {WI, .. . WB} of parameters distributed according to p( w I Den). If the number of sampling steps approaches infinity, the distribution of the simulated Wb approximates the posterior arbitrarily well. To take into account the range of future y-values, we create a set of them by simulation. For each Wb E WeB) a number of y ""' p(y I x, Wb) is generated. Let 446 Gerhard Paass. JiJrg Kindermann y(x.R) := {YI, ... , YR} be the resulting set. Instead of performing a new Markov Monte Carlo run to generate a new sample according to p(w I DCn) U (x, y)), we use the old set WCB) of parameters and reweight them (importance sampling). In this way we may approximate integrals of some function g( w) with respect to p(w I DCn) U (x, y)) [Kalos Whitlock 86, p.92]: j ( ) ( ID U( - -))d __ L~-lg(Wb)P(ylx,Wb) 9 w P W Cn) X, Y W -B Lb=l p(Y I x, Wb) (2) The approximation error approaches zero as the size of WCB) increases. 3.1 APPROXIMATION OF FUTURE LOSS Consider the future loss f;,x(x) given new observation (x, y) and trial input Xt. In the case of the square loss function, (1) can be transformed to f~,.t(Xt) = j[!(Xt,w)-E(yIXt,Dcn)U(X,y)Wp(wIDcn)U(x,y))dw (3) + j £T2(w) p(w I DCn) U (x, y)) dw where £T2(w) := Var(y I x, w) is independent of x. Assume a set XT = {Xl, ... , XT} is given, which is representative of trial inputs for the distribution p(x). Define S(x, y) := L~=i p(Y I x, Wb) for y E YCx,R) . Then from equations (2) and (3) we get E(ylxt,DCn)U(x,y)):= 1/S(x,Y)L~=1!(Xt,Wb)P(Ylx,Wb) and 1 B S(x -) L£T2(Wb)P(Ylx,Wb) (4) ,y b=l 1 B + S(x -) I)!(Xt, Wb) - E(y I Xt, DCn) U (x, y))]2 p(Y I x, Wb) ,y b=l The final value of f; is obtained by averaging over the different y E YCx,R) and different trial inputs Xt E XT. To reduce the variance, the trial inputs Xt should be selected by importance sampling (2) to concentrate them on regions with high current loss (see (5) below). To facilitate the search for an x with minimal f; we reduce the extent of random fluctuations of the y values. Let (Vi, ... , VR) be a vector of random numbers Vr -- N(O,1), and let jr be randomly selected from {1, ... , B}. Then for each x the possible observations Yr E YCx,R) are defined as Yr := !(x, wir) + Vr£T2(wir). In this way the difference between neighboring inputs is not affected by noise, and search procedures can exploit gradients. 3.2 CURRENT LOSS As a proxy for the future loss, we may use the current loss at x, rcurr(x) = p(x) j L(y, d*(x)) p(y I x, DCn)) dy (5) Bayesian Query Construction for Neural Network Models 447 where p(x) weights the inputs according to their relevance. For the square loss function the average loss at x is the conditional variance Var(y I x, DCn». We get Tcurr(X) = p(x) jU(x,w)-E(YIX,DCn»)2p(wIDcn»dw (6) + p(x) j 0"2(w) p(w I D(n» dw If E(y I x,DCn» fr~~=lf(x,wb) and the sample WCB):= {Wl, ... ,WB} is representative of p(w I DCn» we can approximate the current loss with p( x) ~ A 2 p( x) ~ 2 Tcurr(X) ~ 13 L..tU(x, Wb) - E(y I x, DCn») + 13 L..t 0" (Wb) b=l b=l (7) If the input distribution p( x) is uniform, the second term is independent of x. 3.3 COMPLEXITY REGULARIZATION Neural network models can represent arbitrary mappings between finite-dimensional spaces if the number of hidden units is sufficiently large [Hornik Stinchcombe 89]. As the number of observations grows, more and more hidden units are necessary to catch the details of the mapping. Therefore we use a sequential procedure to increase the capacity of our networks during query learning. White and Wooldridge call this approach the "method of sieves" and provide some asymptotic results on its consistency [White Wooldridge 91]. Gelfand and Dey compare Bayesian approaches for model selection and prove that, in the case of nested models Ml and M2, model choice by the ratio of popular Bayes factors p(DCn) I Mi) := J p(DCn) I W, Mi) p(w I Mi) dw will always choose the full model regardless of the data as n --t 00 [Gelfand Dey 94]. They show that the pseudoBayes factor, a Bayesian variant of crossvalidation, is not affected by this paradox n n A(Ml' M2) := II p(y; I x;, DCn,j), Mt}j II p(Y; I x;, DCn,j), M2) (8) ;=1 j=1 Here DCn,;) := D(n) \ (x;, y;). As the difference between p(w I DCn» and p( wi D(n,j» is usually small, we use the full posterior as the importance function (2) and get p(Y; I x;, DCn,j),Mi) = j p(Y; IXj,w,Mi)p(wIDCn,j),Mi)dw '" B/(t,l/P (Y;li;,W"M,)) (9) 4 NUMERICAL DEMONSTRATION In a first experiment we tested the approach for a small a 1-2-1 MLP target function with Gaussian noise N(0,0.05 2). We assumed the square loss function and a uniform input distribution p(x) over [-5,5]. Using the "true" architecture for the approximating model we started with a single randomly generated observation. We 448 ::::.:::::.::::\.... .... =~!¥~ --- ~tuo:io_ d \~. '\ ------ ------------\., 1 \l .......... _ .. _-_._ ........... __ .................... _ .... _ ....... _ .. \ ! ~ \! .. -2 ~ ~ 1'1 0 on :; ~ a: 0 :; , \ ~ :.,. '" 0 Gerhard Paass, JiJrg Kindermann I --'~ ' =~ I .. .' . " . . . . . \, ' " 10 15 20 25 30 No.d_ Figure 1: Future loss exploration: predicted posterior mean, future loss and current loss for 12 observations (left), and root mean square error of prediction (right). estimated the future loss by (4) for 100 different inputs and selected the input with smallest future loss as the next query. B = 50 parameter vectors were generated requiring 200,000 Metropolis steps. Simultaneously we approximated the current loss criterion by (7). The left side of figure 1 shows the typical relation of both measures. In most situations the future loss is low in the same regions where the current loss (posterior standard deviation of mean prediction) is high. The queries are concentrated in areas of high variation and the estimated posterior mean approximates the target function quite well. In the right part of figure 1 the RMSE of prediction averaged over 12 independent experiments is shown. After a few observations the RMSE drops sharply. In our example there is no marked difference between the prediction errors resulting from the future loss and the current loss criterion (also averaged over 12 experiments). Considering the substantial computing effort this favors the current loss criterion. The dots indicate the RMSE for randomly generated data (averaged over 8 experiments) using the same Bayesian prediction procedure. Because only few data points were located in the critical region of high variation the RMSE is much larger. In the second experiment, a 2-3-1 MLP defined the target function I(x, wo), to which Gaussian noise of standard deviation 0.05 was added. I( x, wo) is shown in the left part of figure 2. We used five MLPs with 2-6 hidden units as candidate models Ml, .. . , M5 and generated B = 45 samples WeB) of the posterior pew I D(n)' M.), where D(n) is the current data. We started with 30,000 Metropolis steps for small values of n and increased this to 90,000 Metropolis steps for larger values of n. For a network with 6 hidden units and n = 50 observations, 10,000 Metropolis steps took about 30 seconds on a Sparc10 workstation. Next, we used equation (9) to compare the different models, and then used the optimal model to calculate the current loss (7) on a regular grid of 41 x 41 = 1681 query points x. Here we assumed the square loss function and a uniform input distribution p(x) over [-5,5] x [-5,5]. We selected the query point with maximal current loss and determined the final query point with a hillclimbing algorithm. In this way we were rather sure to get close to the true global optimum. The main result of the experiment is summarized in the right part of figure 2. It Bayesian Query Construct.ion for Neural Network Models ". :2 \ "': <:> \ 449 • eXDlorati~n o .m random a ~\·{l·· .. o .. o .. o ............. __ (). ... \ ............ 0 ............. --0 ., ." ~. .. 20 40 60 80 100 No. of Observations Figure 2: Current loss exploration: MLP target function and root mean square error. shows - averaged over 3 experiments - the root mean square error between the true mean value and the posterior mean E(y I x) on the grid of 1681 inputs in relation to the sample size. Three phases of the exploration can be distinguished (see figure 3). In the beginning a search is performed with many queries on the border of the input area. After about 20 observations the algorithm knows enough detail about the true function to concentrate on the relevant parts of the input space. This leads to a marked reduction ofthe mean square error. After 40 observations the systematic part of the true function has been captured nearly perfectly. In the last phase of the experiment the algorithm merely reduces the uncertainty caused by the random noise. In contrast, the data generated randomly does not have sufficient information on the details of f(x , w), and therefore the error only gradually decreases. Because of space constraints we cannot report experiments with radial basis functions which led to similar results. Acknowledgements This work is part of the joint project 'REFLEX' of the German Fed. Department of Science and Technology (BMFT), grant number 01 IN 111Aj4. We would like to thank Alexander Linden, Mark Ring, and Frank Weber for many fruitful discussions. References [Berger 80] Berger, J. (1980): Statistical Decision Theory, Foundations, Concepts, and Methods. Springer Verlag, New York. [Cohn 94] Cohn, D. (1994): Neural Network Exploration Using Optimal Experimental Design. In J. Cowan et al. (eds.): NIPS 5. Morgan Kaufmann, San Mateo. [Ford et al. 89] Ford, I., Titterington, D.M., Kitsos, C.P. (1989): Recent Advances in Nonlinear Design. Technometrics, 31, p.49-60. [Gelfand Dey 94] Gelfand, A.E., Dey, D.K. (1994): Bayesian Model Choice: Asymptotics and Exact Calculations. J. Royal Statistical Society B, 56, pp.501-514. 450 Gerhard Paass, Jorg Kindermann Figure 3: Squareroot of current loss (upper row) and absolute deviation from true function (lower row) for 10,25, and 40 observations (which are indicated by dots) . [Hornik Stinchcombe 89] Hornik, K., Stinchcombe, M. (1989): Multilayer Feedforward Networks are Universal Approximators. Neural Networks 2, p.359-366. [Kalos Whitlock 86] Kalos, M.H., Whitlock, P.A. (1986): Monte Carlo Methods, Wiley, New York. [MacKay 92] MacKay, D. (1992): Information-Based Objective Functions for Active Data Selection. Neural Computation 4, p.590-604. [Neal 93] Neal, R.M. (1993): Probabilistic Inference using Markov Chain Monte Carlo Methods. Tech. Report CRG-TR-93-1, Dep. of Computer Science, Univ. of Toronto. [PaaB 91] PaaB, G. (1991): Second Order Probabilities for Uncertain and Conflicting Evidence. In: P.P. Bonissone et al. (eds.) Uncertainty in Artificial Intelligence 6. Elsevier, Amsterdam, pp. 447-456. [Plutowski White 93] Plutowski, M., White, H. (1993): Selecting Concise Training Sets from Clean Data. IEEE Tr. on Neural Networks, 4, p.305-318. [Pronzato Walter 92] Pronzato, L., Walter, E. (1992): Nonsequential Bayesian Experimental Design for Response Optimization. In V. Fedorov, W.G. Miiller, I.N. Vuchkov (eds.): Model Oriented Data-Analysis. Physica Verlag, Heidelberg, p. 89-102. [Sollich 94] Sollich, P. (1994): Query Construction, Entropy and Generalization in Neural Network Models. To appear in Physical Review E. [Sollich Saad 95] Sollich, P., Saad, D. (1995): Learning from Queries for Maximum Information Gain in Unlearnable Problems. This volume. [White Wooldridge 91] White, H., Wooldridge, J. (1991): Some Results for Sieve Estimation with Dependent Observations. In W. Barnett et al. (eds.) : Nonparametric and Semiparametric Methods in Econometrics and Statistics, New York, Cambridge Univ. Press.	1994	92
990	A Silicon Axon Bradley A. Minch, Paul Hasler, Chris Diorio, Carver Mead Physics of Computation Laboratory California Institute of Technology Pasadena, CA 91125 bminch, paul, chris, carver@pcmp.caltech.edu Abstract We present a silicon model of an axon which shows promise as a building block for pulse-based neural computations involving correlations of pulses across both space and time. The circuit shares a number of features with its biological counterpart including an excitation threshold, a brief refractory period after pulse completion, pulse amplitude restoration, and pulse width restoration. We provide a simple explanation of circuit operation and present data from a chip fabricated in a standard 2Jlm CMOS process through the MOS Implementation Service (MOSIS). We emphasize the necessity of the restoration of the width of the pulse in time for stable propagation in axons. 1 INTRODUCTION It is well known that axons are neural processes specialized for transmitting information over relatively long distances in the nervous system. Impulsive electrical disturbances known as action potentials are normally initiated near the cell body of a neuron when the voltage across the cell membrane crosses a threshold. These pulses are then propagated with a fairly stereotypical shape at a more or less constant velocity down the length of the axon. Consequently, axons excel at precisely preserving the relative timing of threshold crossing events but do not preserve any of the initial signal shape. Information, then, is presumably encoded in the relative timing of action potentials. 740 Bradley A. Minch, Paul Hasler, Chris Diorio, Carver Mead The biophysical mechanisms underlying the initiation and propagation of action potentials in axons have been well studied since the seminal work of Hodgkin and Huxley on the giant axon of Loligo. (Hodgkin & Huxley, 1952) Briefly, when the voltage across a small patch of the cell membrane increases to a certain level, a population of ion channels permeable to sodium opens, allowing an influx of sodium ions, which, in turn, causes the membrane voltage to increase further and a pulse to be initiated. This population of channels rapidly inactivates, preventing the passage of additional ions. Another population of channels permeable to potassium opens after a brief delay causing an efflux of potassium ions, restoring the membrane to a more negative potential and terminating the pulse. This cycle of ion migration is coupled to neighboring sections of the axon, causing the action potential to propagate. The sodium channels remain inactivated for a brief interval of time during which the affected patch of membrane will not be able to support another action potential. This period of time is known as the refractory period. The axon circuit which we present in this paper does not attempt to model the detailed dynamics of the various populations of ion channels, although such detailed neuromimes are both possible (Lewis, 1968; Mahowald & Douglas, 1991) and useful for learning about natural neural systems. Nonetheless, it shares a number of important features with its biological counterpart including having a threshold for excitation and a refractory period. It is well accepted that the amplitude of the action potential must be restored as it propagates. It is not as universally understood is that the width of the action potential must be restored in time if it is to propagate over any appreciable distance. Otherwise, the pulse would smear out in time resulting in a loss of precise timing information, or it would shrink down to nothing and cease to propagate altogether. In biological axons, restoration of the pulse width is accomplished through the dynamics of sodium channel inactivation and potassium channel activation. In our silicon model, the pulse width is restored through feedback from the successive stage. This feedback provides an inactivation which is similar to that of the sodium channels in biological axons and is also the underlying cause of refractoriness in our circuit. In the following section we provide a simple description of how the circuit behaves. Following this, data from a chip fabricated in a standard 2p.m CMOS process through MOSIS are presented and discussed. 2 THE SILICON AXON CmCUIT An axon circuit which is to be used as a building block in large-scale computational systems should be made as simple and low-power as possible, since it would be replicated many times in any such system. Each stage of the axon circuit described below consists of five transistors and two small capacitors, making the axon circuit very compact. The axon circuit uses the delay through a stage to time the signal which is fed back to restore the pulse width, thus avoiding the need for an additional delay circuit for each section. Additonally, the circuit operates with low power; during typical operation (a pulse of width 2ms travelling at 103stages/s), pulse propagation costs about 4pJ / stage of energy. Under these circumstances, the circuit consumes about 2nW/stage of static power. A Silicon Axon 741 Figure 1: Three sections of the axon circuit. Three stages of the axon circuit are depicted in Figure 1. A single stage consists of two capacitors and what would be considered a pseudo-nMOS NAND gate and a pseudo-nMOS inverter in digital logic design. These simple circuits are characterized by a threshold voltage for switching and a slew rate for recharging. Consider the inverter circuit. If the input is held low for a sufficiently long time, the pull-up transistor will have charged the output voltage almost completely to the positive rail. If the input voltage is ramped up toward the positive rail, the current in the pull-down transistor will increase rapidly. At some input voltage level, the current in the pull-down transistor will equal the saturation current of the pull-up transistor; this voltage is known as the threshold. The output voltage will begin to discharge at a rapidly increasing rate as the input voltage is increased further. After a very short time, the output will have discharged almost all the way to the negative rail. Now, if the input were decreased rapidly, the output voltage would ramp linearly in time (slew) up toward the positive rail at a rate set by the saturation current in the pull-up transistor and the capacitor on the output node. The NAND gate is similar except both inputs must be (roughly speaking) above the threshold in order for the output to go low. If either input goes low, the output will charge toward the positive rail. Note also that if one input of the NAND gate is held high, the circuit behaves exactly as an inverter. The axon circuit is formed by cascading multiple copies of this simple five transistor circuit in series. Let the voltage on the first capacitor of the nth stage be denoted by Un and the voltage on the second capacitor by vn . Note that there is feedback from Un +1 to the lower input of the NAND gate of the nth stage. Under quiescent conditions, the input to the first stage is low (at the negative rail) , the U nodes of all stages are high (at the positive rail), and all of the v nodes are held low (at the negative rail). The feedback signal to the final stage in the line would be tied to the positive rail. The level of the bias voltages 71 and 72 determine whether or not a narrow pulse fed into the input of the first stage will propagate and, if so, the width and velocity with which it does. In order to obtain a semi-quantitative understanding of how the axon circuit behaves, we will first consider the dynamics of a cascade of simple inverters (three sections of which are depicted in Figure 2) and then consider the addition of feedback. Under most circumstances, discharges will occur on a much faster time scale 742 Bradley A. Minch, Paul Hasler, Chris Diorio, Carver Mead than the recharges, so we make the simplifying assumption that when the input of an inverter reaches the threshold voltage, the output discharges instantaneously. Additionally, we assume that saturated transistors behave as ideal current sources (i.e., we neglect the Early effect) so that the recharges are linear ramps in time. TTT TTT TTT TTT TTT TTT Figure 2: A cascade of pseudo-nMOS inverters. Let hand h be the saturation currents in the pull-up transistors with bias voltages 1"1 and 1"2, respectively. Let 6 1 and 62 be the threshold voltages of the first and second inverters in a single stage, respectively. Also, let u = IdC and v = I2/C be the slew rates for the U and v nodes, respectively. Let a 1 = 6dv and a 2 = 6 2/u be the time required for Un to charge from the negative rail up to E>1 and for Vn to charge from the negative rail up to 6 2 , respectively. Finally, let v~ denote the peak value attained by the v signal of the nth stage. IE ~IE ~I 671 a 2 (a) II > h v· 71 vn V· IE ~IE ~I 671 a 2 (b) h < 12 Figure 3: Geometry of the idealized Un and Vn signals under the bias conditions (a) h > 12 and (b) II < 12, Consider what would happen if Vn -l exceeded 6 1 for a time 671 , In this case, Un would be held low for 671 and then released. Meanwhile, Vn would ramp up to v6n . Then, Un would begin to charge toward the positive rail while Vn continues to charge. This continues for a time a2 at which point Un will have reached 6 2 causing A Silicon Axon 743 Vn to be discharged to the negative rail. Now, Un+l is held low while Vn exceeds 8 1 this interval of time is precisely On+!. Figures 3a and 3b depict the geometry of the Un and Vn signals in this scenario under the bias conditions h > 12 and h < 12, respectively. Simple Euclidean geometry implies that the evolution of On will be governed by the first-order difference equation which is trivially solved by Thus, the quantity a2-al determines what happens to the width of the pulse as it propagates. In the event that a 2 < aI, the pulse will shrink down to nothing from its initial width. If a2 > aI, the pulse width will grow without bound from its initial width. The pulse width is preserved only if a 2 = a l . This last case, however, is unrealistic. There will always be component mismatches (with both systematic and random parts), which will cause the width of the pulse to grow and shrink as it propagates down the line, perhaps cancelling on average. Any systematic offsets will cause the pulse to shrink to nothing or to grow without bound as it propagates. In any event, information about the detailed timing of the initial pulse will have been completely lost. Now, consider the action of the feedback in the axon circuit (Figure 1). If Un were to be held low for a time longer than a l (i.e., the time it takes Vn to charge up to 8d, Un+1 would come back and release Un, regardless of the state of the input. Thus, the feedback enforces the condition On ::; a l . If 11 > h (i.e., a2 < ad, a pulse whose initial width is larger than a 1 will be clipped to a 1 and then shrink down to nothing and disappear. In the event that II < h (i.e., a2 > ad, a pulse whose initial duration is too small will grow up until its width is limited by the feedback. The axon circuit normally operates under the latter bias condition. The dynamics of the simple inverter chain cause a pulse which is to narrow to grow and the feedback loop serves to limit the pulse width; thus, the width of the pulse is restored in time. The feedback is also the source of the refractoriness in the axon; that is, until Un +l charges up to (roughly) 8 1 , Vn -l can have no effect on Un. 3 EXPERIMENTAL DATA In this section, data from a twenty-five stage axon will be shown. The chip was fabricated in a standard 2f.lm p-well (Orbit) CMOS process through MOSIS. Uniform Axon A full space-time picture of pulse (taken at the v nodes of the circuit) propagation down a uniform axon is depicted on the left in Figure 4. The graph on the right in Figure 4 shows the same data from a different perspective. The lower sloped curve represents the time of the initial rapid discharge of the U node at each successive stage-this time marks the leading edge of the pulse taken at the v node of that stage. 744 Bradley A. Minch. Paul Hasler. Chris Diorio. Carver Mead The upper sloped curve marks the time of the final rapid discharge of the v node of each stage-this time is the end of the pulse taken at the v node of that stage. The propagation velocity of the pulse is given (in units of stages/s) by the reciprocal of the slope of the lower inclined curve. The third curve is the difference of the other two and represents the pulse width as a function of position along the axon. The graph on the left of Figure 5 shows propagation velocity as a function of the 72 bias voltage- so long as the pulse propagates, the velocity is nearly independent of 71. Two orders of magnitude of velocity are shown in the plot; these are especially well matched to the time scales of motion in auditory and visual sensory data. The circuit is tunable over a much wider range of velocities (from about one stage per second to well in excess of 104stages/ s). The graph on the right of Figure 5 shows pulse width as a function of 71 for various values of 72-the pulse width is mainly determined by 71 with 72 setting a lower limit. Tapered Axon In biological axons, the propagation velocity of an action potential is related to the diameter of the axon- the bigger the diameter, the greater the velocity. If the axon were tapered, the velocity of the action potential would change as it propagated. If the bias transistors in the axon circuit are operated in their subthreshold region, the effect of an exponentially tapered axon can be simulated by applying a small voltage difference to the ends of each ofthe 71 and 72 bias lines. (Lyon & Mead, 1989) These narrow wires are made with a relatively resistive layer (polysilicon); hence, putting a voltage difference across the ends will linearly interpolate the bias voltages for each stage along the line. In subthreshold, the bias currents are exponentially related to the bias voltages. Since the pulse width and velocity are related to the bias currents, we expect that a pulse will either speed up and get narrower or slow down and get wider (depending on the sign of the applied voltage) exponentially as a function of position along the line. The graph on the left of Figure 6 depicts the boundaries of a pulse as it propagates along of the axon circuit for a positive ('s) and negative (x's) voltage difference applied to the 7 lines. The graph on the right of Figure 6 shows the corresponding pulse width for each applied voltage difference. Note that in each case, the width changes by more than an order of magnitude, but the pulse maintains its integrity. That is, the pulse does not disappear nor does it split into multiple pulses-this behavior would not be possible if the pulse width were not restored in time. 4 CONCLUSIONS In this paper we have presented a low-power, compact axon circuit, explained its operation, and presented data from a working chip fabricated through MOSIS. The circuit shares a number of features with its biological counterpart including an excitation threshold, a brief refractory period after pulse completion, pulse amplitude restoration, and pulse width restoration. It is tunable over orders of magnitude in pulse propagation velocity-including those well matched to the time scales of auditory and visual signals- and shows promise for use in synthetic neural systems which perform computations by correlating events which occur over both space and time such as those presented in (Horiuchi et ai, 1991) and (Lazzaro & Mead, 1989). A Silicon Axon 745 Acknowledgements This material is based upon work supported in part under a National Science Foundation Graduate Research Fellowship, the Office of Naval Research, DARPA, and the Beckman Foundation. References A. 1. Hodgkin and A. K. Huxley, (1952). A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve. Journal of Physiology, 117:6, 500-544. T. Horiuchi, J. Lazzaro, A. Moore, and C. Koch, (1991) . A Delay-Line Based Motion Detection Chip. Advances in Neural Information Processing Systems 3. San Mateo, CA: Morgan Kaufmann Publishers, Inc. 406-412. J. Lazzaro and C. Mead, (1989). A Silicon Model of Auditory Localization. Neural Computation, 1:1, 47-57. R. Lyon and C. Mead, (1989). Electronic Cochlea. Analog VLSI and Neural Systems. Reading, MA: Addison-Wesley Publishing Company, Inc. 279-302. E. R. Lewis, (1968). Using Electronic Circuits to Model Simple Neuroelectric Interactions. Proceedings of the IEEE, 56:6, 931-949. M. Mahowald and R. Douglas, (1991). A Silicon Neuron. Nature, 354:19, 515-518. o.o35r-----.,-----.,--------, 0.03 2 1.5 0.025 ~ 0 .02 ., .& 0 .5 '0 > ~ f= 0.015 0 -0.5 30 0.06 Stage o 0 TilTIe (s) °0~--~1~0---~2~0---~30 Stage Figure 4: Pulse propagation along a uniform axon. (Left) Perspective view. (Right) Overhead view. : pulse boundaries, x: pulse width. T1 = 0.720V, T2 = 0.780V. Velocity = 1, 100stages/s, Width = 3.8ms 746 Bradley A. Minch, Paul Hasler, Chris Diorio, Carver Mead II< "" "" "" "" "" "" II< 'ttl" II< II< "" "" "" ... II< "" II< ~ "" "" 10~~ . 6-------0~.~7-------0~ . 8-------0~. 9 Tau2 (V) 3: ~ ~ l!l £ 10" r----~--~--~--____, 10.2 10-3 o o 00 Tau2 = 0 .660 V ~ 110 11<... Tau2 = 0.700 V ~ Xx Tau2 = 0 .740 ~ + Tau2 = 0.7 0 V +d> ~o T a u2 = .820 V ~ , Ta\lp = 0 .860 V 10~~.5-----0~.6-----0~.~7-----0~.8~---0~.9 Taul (V) Figure 5: Uniform axon. (Left) Pulse velocity as a function of r2. (Right) Pulse width as a function of rl for various values of r2. 0 . 14r-------~-------~----~ 0 .12 0 .1 il:! 0 .08 E::: 0.06 x x x x II< )0( )0( )0( )O()O( )0( )0( )0( )0( )0( )0( .... )0()0( 0.04# .... II< 0 . 020~------~1~0~------2~0~------370 Stage 1 0" r------~--------~------___, )0( )O()O( )0( )0( x x ><"XxX X X X 1 O·3~------,'---------_'_------__=' o 10 20 30 Stage Figure 6: (Left) Pulse propagation along a tapered axon. (Right) Pulse width as a function of position along a tapered axon. *: rieft = 0.770V, r;ight = 0.600V, r~eft = 0.B20V, r;ight = 0.650. x : rieft = 0.600V, r;ight = 0.770V, r~eft = 0.650V, r;ight = 0.B20V.	1994	93
991	Plasticity-Mediated Competitive Learning Nicol N. Schraudolph nici@salk.edu Terrence J. Sejnowski terry@salk.edu Computational Neurobiology Laboratory The Salk Institute for Biological Studies San Diego, CA 92186-5800 and Computer Science & Engineering Department University of California, San Diego La Jolla, CA 92093-0114 Abstract Differentiation between the nodes of a competitive learning network is conventionally achieved through competition on the basis of neural activity. Simple inhibitory mechanisms are limited to sparse representations, while decorrelation and factorization schemes that support distributed representations are computationally unattractive. By letting neural plasticity mediate the competitive interaction instead, we obtain diffuse, nonadaptive alternatives for fully distributed representations. We use this technique to Simplify and improve our binary information gain optimization algorithm for feature extraction (Schraudolph and Sejnowski, 1993); the same approach could be used to improve other learning algorithms. 1 INTRODUCTION Unsupervised neural networks frequently employ sets of nodes or subnetworks with identical architecture and objective function. Some form of competitive interaction is then needed for these nodes to differentiate and efficiently complement each other in their task. 476 1.00 0.50 0.00 Nicol Schraudolph, Terrence 1. Sejnowski f(y) ·4r(y)' .... j ................................. '.' .......... // .... ··1 '.:! ' .... "··,,··.,, ... ' .. 1 ••••••••••••• = ... = ... :::. ... :::: ... :j:. ... :..... ... -.. -~ y -4.00 -2.00 0.00 2.00 4.00 Figure 1: Activity f and plasticity f' of a logistic node as a function of its net input y. Vertical lines indicate those values of y whose pre-images in input space are depicted in Figure 2. Inhibition is the simplest competitive mechanism: the most active nodes suppress the ability of their peers to learn, either directly or by depressing their activity. Since inhibition can be implemented by diffuse, nonadaptive mechanisms, it is an attractive solution from both neurobiological and computational points of view. However, inhibition can only form either localized (unary) or sparse distributed representations, in which each output has only one state with significant information content. For fully distributed representations, schemes to decorrelate (Barlow and Foldiak, 1989; Leen, 1991) and even factorize (Schmidhuber, 1992; Bell and Sejnowski, 1995) node activities do exist. Unfortunately these require specific, weighted lateral connections whose adaptation is computationally expensive and may interfere with feedforward learning. While they certainly have their place as competitive learning algorithms, the capability of biological neurons to implement them seems questionable. In this paper, we suggest an alternative approach: we extend the advantages of simple inhibition to distributed representations by decoupling the competition from the activation vector. In particular, we use neural plasticity the derivative of a logistic activation function as a medium for competition. Plasticity is low for both high and low activation values but high for intermediate ones (Figure 1); distributed patterns of activity may therefore have localized plasticity. If competition is controlled by plasticity then, simple competitive mechanisms will constrain us to localized plasticity but allow representations with distributed activity. The next section reintroduces the binary information gain optimization (BINGO) algorithm for a single node; we then discuss how plasticity-mediated competition improves upon the decorrelation mechanism used in our original extension to multiple nodes. Finally, we establish a close relationship between the plasticity and the entropy of a logistiC node that provides an intuitive interpretation of plasticity-mediated competitive learning in this context. Plasticity-Med;ated Competitive Learning 477 2 BINARY INFORMATION GAIN OPTIMIZATION In (Schraudolph and Sejnowski, 1993), we proposed an unsupervised learning rule that uses logistic nodes to seek out binary features in its input. The output 1 z = f(y), where f(y) = 1 + e- Y and y = tV · x (1) of each node is interpreted stochastically as the probability that a given feature is present. We then search for informative directions in weight space by maximizing the information gained about an unknown binary feature through observation of z. This binary infonnation gain is given by D.H(z) = H(Z) - H(z) , (2) where H(z) is the entropy of a binary random variable with probability z, and z is a prediction of z based on prior knowledge. Gradient ascent in this objective results in the learning rule D.w <X J'(y) . (y - fI) . x, (3) where fI is a prediction of y. In the simplest case, fI is an empirical average (y) of past activity, computed either over batches of input data or by means of an exponential trace; this amounts to a nonlinear version of the covariance rule (Sejnowski, 1977). Using just the average as prediction introduces a strong preference for splitting the data into two equal-sized clusters. While such a bias is appropriate in the initial phase of learning, it fails to take the nonlinear nature of f into account. In order to discount data in the saturated regions of the logistic function appropriately, we weigh the average by the node's plasticity J'(y): (y . f'(y)) fI = --'-'---'--'-'--'-'-(f'(y)) + C , (4) where c is a very small positive constant introduced to ensure numerical stability for large values of y. Now the bias for splitting the data evenly is gradually relaxed as the network's weights grow and data begins to fall into saturated regions of f. 3 PLASTICITY-MEDIATED COMPETITION For multiple nodes the original BINGO algorithm used a decorrelating predictor as the competitive mechanism: g = y + (Qg - 2I)(y - (y)) , (5) where Qg is the autocorrelation matrix of y, and I the identity matrix. Note that Qg is computationally expensive to maintain; in connectionist implementations it 478 Nicol Schraudolph, Terrence J. Sejnowski j ! i .: f ,. . .... .. . . . ~' i.. " .. ~.', " . . .. . .. . , .. : ..... . :. ·,"f.e: 1',. , , :~X ~." . . ·'IJ"~~ .~~~ .. .. . ~~ .. . .j I . . :: ':" ! Figure 2: The "three cigars" problem. Each plot shows the pre-image of zero net . input, superimposed on a scatter plot of the data set, in input space. The two flanking lines delineate the "plastic region" where the logistic is not saturated, providing an indication of weight vector size. Left, two-node BINGO network using decorrelation (Equations 3 & 5) fails to separate the three data clusters. Right, same network using plasticity-mediated competition (Equations 4 & 6) succeeds. is often approximated by lateral anti-Hebbian connections whose adaptation must occur on a faster time scale than that of the feedforward weights (Equation 3) for reasons of stability (Leen, 1991). In practice this means that learning is slowed significantly. In addition, decorrelation can be inappropriate when nonlinear objectives are optimized in our case, two prominent binary features may well be correlated. Consider the "three cigars" problem illustrated in Figure 2: the decorrelating predictor (left) forces the two nodes into a near-orthogonal arrangement, interfering with their ability to detect the parallel gaps separating the data clusters. For our purposes, decorrelation is thus too strong a constraint on the discriminants: all we require is that the discovered features be distinct. We achieve this by reverting to the simple predictor of Equation 4 while adding a global, plasticity-mediated excitationl factor to the weight update: ~Wi ex: f'(Yi) . (Yi - 1li) . X· L f'(Yj) j (6) As Figure 2 (right) illustrates, this arrangement solves the "three cigars" problem. In the high-dimensional environment of hand-written digit recognition, this algorithm discovers a set of distributed binary features that preserve most of the information needed to classify the digits, even though the network was never given any class labels (Figure 3). 1 The interaction is excitatory rather than inhibitory since a node's plasticity is inversely correlated with the magnitude of its net input. Plasticity-Mediated Competitive Learning 479 ................... .. .... . .... , •••• ................... . . .. " .... ....... ... . .... , ..... .. .. . ... ' ..... • . .... • ........ · ... « ...... .. ...... ...... • ••••••• · ... " I. • a .... .. ..... ........ . ......... « I ............ t ••••• ........ .., .. -..... , ...... .. ....... .. .... .. . .. .. " I ...... a ....... .. •••••• . .......... " ..... . ..... • " , .... • •••••••• ~ ..... '" •••• ' . • I ... .............. . _ .... • ••• . .. , .. • ••••••••• • •••• . .. ' , ...... I l •• ... ~ ••• . ....... , •••• . .. '" ...... to. .... ....... " ....... ••••• • ••• • ••• .. ... a .. ' . . . . ..... ..... . .. ' ........ , ........ •••••••• .... • •• . ........... Figure 3: Weights found by a four-node network running the improved BINGO algorithm (Equations 4 & 6) on a set of 1200 handwritten digits due to (Guyon et aI., 1989). Although the network is unsupervised, its four-bit output conveys most of the information necessary to classify the digits. 4 PLASTICITY AND BINARY ENTROPY It is possible to establish a relationship between the plasticity /' of a logistiC node and its entropy that provides an intuitive account of plasticity-mediated competition as applied to BINGO. Consider the binary entropy H(z) = - z logz - (1 - z) log(l - z) (7) A well-known quadratic approximation is H(z) = 8e- 1 z (1 - z) ~ H(z) (8) Now observe that the plasticity of a logistic node !'(Y)=:Y l+le_y =, .. =z(l-z) (9) is in fact proportional to H(z) that is, a logistic node's plasticity is in effect a convenient quadratic approximation to its binary output entropy. The overall entropy in a layer of such nodes equals the sum of individual entropies less their redundancy: H(z) = L H(zj) - R(Z) j The plasticity-mediated excitation factor in Equation 6 j j (10) (11) is thus proportional to an approximate upper bound on the entropy of the layer, which in turn indicates how much more information remains to be gained by learning from a particular input. In the context of BINGO, plasticity-mediated 480 Nicol SchraudoLph. Terrence J. Sejnowski competition thus scales weight changes according to a measure of the network's ignorance: the less it is able to identify a given input in terms of its set of binary features, the more it tries to learn doing so. 5 CONCLUSION By using the derivative of a logistic activation function as a medium for competitive interaction, we were able to obtain differentiated, fully distributed representations without resorting to computationally expensive decorrelation schemes. We have demonstrated this plasticity-mediated competition approach on the BINGO feature extraction algorithm, which is significantly improved by it. A close relationship between the plasticity of a logistic node and its binary output entropy provides an intuitive interpretation of this unusual form of competition. Our general approach of using a nonmonotonic function of activity rather than activity itself to control competitive interactions may prove valuable in other learning schemes, in particular those that seek distributed rather than local representations. Acknowledgements We thank Rich Zemel and Paul Viola for stimulating discussions, and the McDonnell-Pew Center for Cognitive Neuroscience in San Diego for financial support. References Barlow, H. B. and Foldiak, P. (1989). Adaptation and decorrelation in the cortex. In Durbin, R. M., Miall, c., and Mitchison, G. J., editors, The Computing Neuron, chapter 4, pages 54-72. Addison-Wesley, Wokingham. Bell, A. J. and Sejnowski, T. J. (1995). A non-linear information maximisation algorithm that performs blind separation. In Advances in Neural Information Processing Systems, volume 7, Denver 1994. Guyon,!., Poujaud, 1., Personnaz, L., Dreyfus, G., Denker, J., and Le Cun, Y. (1989). Comparing different neural network architectures for classifying handwritten digits. In Proceedings of the International Joint Conference on Neural Networks, volume II, pages 127-132. IEEE. Leen, T. K. (1991). Dynamics of learning in linear feature-discovery networks. Network, 2:85-105. Schmidhuber, J. (1992). Learning factorial codes by predictability minimization. Neural Computation, 4(6):863-879. Schraudolph, N. N. and Sejnowski, T. J. (1993). Unsupervised discrimination of clustered data via optimization of binary information gain. In Hanson, S. J., Cowan, J. D., and Giles, C. L., editors, Advances in Neural Information Processing Systems, volume 5, pages 499-506, Denver 1992. Morgan Kaufmann, San Mateo. Sejnowski, T. J. (1977). Storing covariance with nonlinearly interacting neurons. Journal of Mathematical Biology, 4:303-321.	1994	94
992	Active Learning for Function Approximation Kah Kay Sung (sung@ai.mit.edu) Massachusetts Institute of Technology Artificial Intelligence Laboratory 545 Technology Square Cambridge, MA 02139 Partha Niyogi (pn@ai.mit.edu) Massachusetts Institute of Technology Artificial Intelligence Laboratory 545 Technology Square Cambridge, MA 02139 Abstract We develop a principled strategy to sample a function optimally for function approximation tasks within a Bayesian framework. Using ideas from optimal experiment design, we introduce an objective function (incorporating both bias and variance) to measure the degree of approximation, and the potential utility of the data points towards optimizing this objective. We show how the general strategy can be used to derive precise algorithms to select data for two cases: learning unit step functions and polynomial functions. In particular, we investigate whether such active algorithms can learn the target with fewer examples. We obtain theoretical and empirical results to suggest that this is the case. 1 INTRODUCTION AND MOTIVATION Learning from examples is a common supervised learning paradigm that hypothesizes a target concept given a stream of training examples that describes the concept. In function approximation, example-based learning can be formulated as synthesizing an approximation function for data sampled from an unknown target function (Poggio and Girosi, 1990). Active learning describes a class of example-based learning paradigms that seeks out new training examples from specific regions of the input space, instead of passively accepting examples from some data generating source. By judiciously selecting ex594 Kah Kay Sung, Parlha Niyogi amples instead of allowing for possible random sampling, active learning techniques can conceivably have faster learning rates and better approximation results than passive learning methods. This paper presents a Bayesian formulation for active learning within the function approximation framework. Specifically, here is the problem we want to address: Let Dn = {(Xi, Yi)li = 1, ... , n} be a set of n data points sampled from an unknown target function g, possibly in the presence of noise. Given an approximation function concept class, :F, where each f E :F has prior probability P;:-[J], one can use regularization techniques to approximate 9 from Dn (in the Bayes optimal sense) by means of a function 9 E:F. We want a strategy to determine at what input location one should sample the next data point, (XN+l, YN+d, in order to obtain the "best" possible Bayes optimal approximation of the unknown target function 9 with our concept class :F. The data sampling problem consists of two parts: 1) Defining what we mean by the "best" possible Bayes optimal approximation of an unknown target function. In this paper, we propose an optimality criterion for evaluating the "goodness" of a solution with respect to an unknown target function. 2) Formalizing precisely the task of determining where in input space to sample the next data point. We express the above mentioned optimality criterion as a cost function to be minimized, and the task of choosing the next sample as one of minimizing the cost function with respect to the input space location of the next sample point. Earlier work (Cohn, 1991; MacKay, 1992) have tried to use similar optimal experiment design (Fedorov, 1972) techniques to collect data that would provide maximum information about the target function. Our work differs from theirs in several respects. First, we use a different, and perhaps more general, optimality criterion for evaluating solutions to an unknown target function, based on a measure of function uncertainty that incorporates both bias and variance components of the total output generalization error. In contrast, MacKay and Cohn use only variance components in model parameter space. Second, we address the important sample complexity question, i.e., does the active strategy require fewer examples to learn the target to the same degree of uncertainty? Our results are stated in PAC-style (Valiant, 1984). After completion of this work, we learnt that Sollich (1994) had also recently developed a similar formulation to ours. His analysis is conducted in a statistical physics framework. The rest of the paper is organized as follows: Section 2, develops our active sampling paradigm. In Sections 3 and 4, we consider two classes offunctions for which active strategies are obtained, and investigate their performance both theoretically and empirically. 2 THE MATHEMATICAL FRAMEWORK In order to optimally select examples for a learning task, one should first have a clear notion of what an "ideal" learning goal is for the task. We can then measure an example's utility in terms of how well the example helps the learner achieve the Active Learning for Function Approximation 595 goal, and devise an active sampling strategy that selects examples with maximum potential utility. In this section, we propose one such learning goal to find an approximation function g E :F that "best" estimates the unknown target function g. We then derive an example utility cost function for the goal and finally present a general procedure for selecting examples. 2.1 EVALUATING A SOLUTION TO AN UNKNOWN TARGET THE EXPECTED INTEGRATED SQUARED DIFFERENCE Let 9 be the target function that we want to estimate by means of an approximation function 9 E :F. If the target function 9 were known, then one natural measure of how well (or badly) g approximates 9 would be the Integrated Squared Difference (ISD) of the two functions: 1x", 8(g,g) = (g(x) - g(x))2dx. (1) Xlo In most function approximation tasks, the target 9 is unknown, so we clearly cannot express the quality of a learning result, g, in terms of g. We can, however, obtain an expected integrated squared difference (EISD) between the unknown target, g, and its estimate, g, by treating the unknown target 9 as a random variable from the approximation function concept class:F. Taking into account the n data points, Dn , seen so far, we have the following a-posteriori likelihood for g: P(gIDn) ex PF[g]P(Dnlg). The expected integrated squared difference (EISD) between an unknown target, g, and its estimate, g, given Dn , is thus: EF[8(g,g)IDn] = f P(gIDn)8(g,g)dg = f PF[g]P(Dnlg)8(y,g)dg. (2) 19EF 19EF 2.2 SELECTING THE NEXT SAMPLE LOCATION We can now express our learning goal as minimizing the expected integrated squared difference (EISD) between the unknown target 9 and its estimate g. A reasonable sampling strategy would be to choose the next example from the input location that minimizes the EISD between g and the new estimate gn+l. How does one predict the new EISD that results from sampling the next data point at location Xn+l ? Suppose we also know the target output value (possibly noisy), Yn+l, at Xn+l. The EISD between 9 and its new estimate gn+l would then be EF[8(gn+l, g)IDn U (xn+l,Yn+d]' where gn+l can be recovered from Dn U (Xn+l,Yn+l) via regularization. In reality, we do not know Yn+l, but we can derive for it the following conditional probability distribution: P(Yn+llxn+l,Dn) ex f P(Dn U(Xn+l,Yn+dlf)PF[J]df. (3) liEF This leads to the following expected value for the new EISD, if we sample our next data point at X n+l: U(Yn+lIDn, xn+1) = 1: P(Yn+lIXn+1' Dn)EF[8(gn+1' g)IDn U (Xn+b Yn+l)]dYn+1' Clearly, the optimal input location to sample next is the location that minimizes ~l cost function in Equation 4 (henceforth referred to as the total output uncertainty), l.e., (5) 596 Kah Kay Sung, Partha Niyogi 2.3 SUMMARY OF ACTIVE LEARNING PROCEDURE We summarize the key steps involved in finding the optimal next sample location: 1) Compute P(gIDn). This is the a-posteriori likelihood of the different functions 9 given Dn , the n data points seen so far. 2) Fix a new point Xn+l to sample. 3) Assume a value Yn+l for this Xn+l . One can compute P(glDn U (Xn+l' Yn+d) and hence the expected integrated squared difference between the target and its new estimate. This is given by EF[8(Un+l, g)IDn U (Xn+l' Yn+t>]. See also Equation 2. 4) At the given Xn+l, Yn+l has a probability distribution given by Equation 3. Averaging over all Yn+l 's, we obtain the total output uncertainty for Xn+l, given by U(Un+lIDn, xn+t> in Equation 4. 5) Sample at the input location that minimizes the total output uncertainty cost function. 3 EXAMPLE 1: UNIT STEP FUNCTIONS To demonstrate the usefulness of the above procedure, let us first consider the following simple class of indicator functions parameterized by a single parameter a which takes values in [0,1]. Thus :F = {I[a,l]IO ~ a ~ I} We obtain a prior peg = l[a,l]» by assuming that a has an a-priori uniform distribution on [0,1]. Assume that data, Dn = {(Xi;Yi); i = 1, .. n} consistent with some unknown target function l[at,l] (which the learner is to approximate) has been obtained. We are interested in choosing a point x E [0,1] to sample which will provide us with maximal information. Following the general procedure outlined above we go through the following steps. For ease of notation, let x R be the right most point belonging to Dn whose Y value is 0, i.e., XR = maXi=l, .. n{xdYi = OJ. Similarly let XL = millj=l, .. n{Xs!Yi = I} and let XL - XR = W. 1) We first need to get P(gIDn). It is easy to show that P(g = l[a,l]IDn) = ~ if a E [XR' XL]; 0 otherwise 2) Suppose we sample next at a particular X E [0,1]' we would obtain Y with the distribution P(y = OIDn, x) = (XL - x) = (XL - x) if x E [XR' XL]; 1 if x ~ XR; 0 otherwise XL - XR W For a particular y, the new data set would be Dn+l = Dn U (x, y) and the corresponding EISD can be easily obtained using the distribution P(gIDn+d. Averaging this over P(YIDn, x) as in step 4 of the general procedure, we obtain if x ~ x R or x ~ XL otherwise Active Learning for Function Approximation 597 Clearly the point which minimizes the expected total output uncertainty is the midpoint of XL and XR. . Xn+1 = arg mm U(gIDn, x) = (XL + xR)/2 XE[O,l] Thus applying the general procedure to this special case reduces to a binary search learning algorithm which queries the midpoint of XR and XL. An interesting question at this stage is whether such a strategy provably reduces the sample complexity; and if so, by how much. It is possible to. prove the following theorem which shows that for a certain pre-decided total output uncertainty value, the active learning algorithm takes fewer examples to learn the target to the same degree of total output uncertainty than a random drawing of examples according to a uniform distribution. Theorem 1 Suppose we want to collect examples so that we are guaranteed with high probability (i. e. probability> 1 - 6) that the total output uncertainty is less than f. Then a passive learner would require at least ~ In(I/6) examples while y(48f) the active strategy described earlier would require at most (1/2) In(1 / 12f) examples. 4 . EXAMPLE 2: THE CASE OF POLYNOMIALS In this section we turn our attention to a class of univariate polynomials (from [-5,5] to ~) of maximum degree K, i.e., K :F = {g(ao, ... , aK) = L: aixi} i:=O As before, the prior on :F is obtained here by assuming a prior on the parameters; in particular we assume that a = (ao, aI, ... , aK) has a multivariate normal distribution N(O, S). For simplicity, it is assumed that the parameters are independent, i.e., S is a diagonal matrix with Si,i = (7;' In this example we also incorporate noise (distributed normally according to N(O, (72)). As before, there is a target gt E G which the learner is to approximate on the basis of data. Suppose the learner is in possession of a data set Dn = {(Xi, Yi = gt(Xi) + 1]); i = 1 ... n} and is to receive another data point. The two options are 1) to sample the function at a point X according to a uniform distribution on the domain [-5,5] (passive learning) and 2) follow our principled active learning strategy to select the next point to be sampled. 4.1 ACTIVE STRATEGY Here we derive an exact expression for Xn+1 (the next query point) by applying the general procedure described earlier. Going through the steps as before, 1)It is possible to show that p(g(a)IDn) = P(aIDn) is again a multivariate normal distribution N(J1., 1:n ) where J1. = L~l Yixi, Xi = (1, Xi, xl, ... , xf)T and 1 n 1:-1 = S-l + _" (x.x. T ) n 2(72 L..J 1 1 i:=l 2)Computation of the total output uncertainty U(gn+1IDn, x) requires several steps. Taking advantage of the Gaussian distribution on both the parameters a and the noise, we obtain (see Niyogi and Sung, 1995 for details): U(gIDn, x) = l1:n+1AI 598 Kah Kay Sung. Partha Niyogi 20 20 J i\ Noise=0.1 \ Noise=1.0 15 to/ I~J \ A 15 0\ '\ l I \J\ "\ r \ \ 10 \.j\ 10 10". '\ 'v .J\ "'5 5 0 50 0 50 Figure 1: Comparing active and passive learning average error rates at different sample noise levels. The two graphs above plot log error rates against number of samples. See text for detailed explanation. The solid and dashed curves are the active and passive learning error rates respectively. where A is a matrix of numbers whose i,j element is J~5 t(i+i- 2)dt. En+l has the same form as En and depends on the previous data, the priors, noise and Xn+l. When minimized over Xn+l, we get xn+! as the maximum utility location where the active learner should next sample the unknown target function. 4.2 SIMULATIONS We have performed several simulations to compare the performance of the active strategy developed in the previous section to that of a passive learner (who receives examples according to a uniform random distribution on the domain [-5,5]). The following issues have been investigated. 1) Average error rate as a function of the number of examples: Is it indeed the case that the active strategy has superior error performance for the same number of examples? To investigate this we generated 1000 test target polynomial functions (of maximum degree 9) according ~o the following Gaussian prior on the parameters: for each ai,P(ai) = N(0,0.9'). For each target polynomial, we collected data according to the active strategy as well as the passive (random) strategy for varying number of data points. Figure 1 shows the average error rate (i.e., the integrated squared difference between the actual target function and its estimate, averaged over the 1000 different target polynomials) as a function of the number of data points. Notice that the active strategy has a lower error rate than the passive for the same number of examples and is particularly true for small number of data. The active strategy uses the same priors that generate the test target functions. We show results of the same simulation performed at two noise levels (noise standard deviation 0.1 and 1.0). In both cases the active strategy outperforms the passive learner indicating robustness in the face of noise. 2) Incorrect priors: How sensitive is the active learner to possible differences between its prior assumptions on the class :F and the true priors? We repeated the function learning task of the earlier case with the test targets generated in the same way as before. The active learner assumes a slightly different Gaussian prior and polynomial degree from the target (Std(ai) = 0.7i and K = 7 for the active learner versus Std(ad = O.Si and K = S for the target). Despite its inaccurate priors, the Active Learning for Function Approximation 8.5 r--------r----r---~--...---___, OJ 8 1tS ~7.5 o ~ ~ w C) o 7 -6.5 Y' '- Different Gauss Prior & Deg. \ \ "-'\ " "' ....... ........ --... ....... ----6~--~~--~----~----~----~ o 10 20 30 40 50 599 Figure 2: Active lea.rning results with different Gaussian priors for coefficients, and a lower a priori polynomial degree K. See text for detailed explanation. The solid and dashed curves a.re the active a.nd passive learning error rates respectively. active learner outperforms the passive case. 3) The distribution of points: How does the active learner choose to sample the domain for maximally reducing uncertainty? There are a few sampling trends which are noteworthy here. First, the learner does not simply sample the domain on a uniform grid. Instead it chooses to cluster its samples typically around K + 1 locations for concept classes with maximum degree K as borne out by simulations where K varies from 5 to 9. One possible explanation for this is it takes only K + 1 points to determine the target in the absence of noise. Second, as the noise increases, although the number of clusters remains fixed, they tend to be distributed away from the origin. It seems that for higher noise levels, there is less pressure to fit the data closely; consequently the prior assumption of lower order polynomials dominates. For such lower order polynomials, it is profitable to sample away from the origin as it reduces the variance of the resulting fit. (Note the case of linear regression) . Remarks 1) Notice that because the class of polynomials is linear in its model parameters, a, the new sample location (xn+d does not depend on the y values actually observed but only on the x values sampled. Thus if the learner is to collect n data points, it can pre-compute the n points at which to sample from the start. In this sense the active algorithm is not really adaptive. This behavior has also been observed by MacKay (1992) and Sollich (1994). 2) Needless to say, the general framework from optimal design can be used for any function class within a Bayesian framework. We are currently investigating the possibility of developing active strategies for Radial Basis Function networks. While it is possible to compute exact expressions for Xn+l for such RBF networks with fixed centers, for the case of moving centers, one has to resort to numerical minimization. For lack of space we do not include those results in this paper. 600 Kah Kay Sung, Partha Niyogi 10 (DO 0 00 (D CJX:» 0 2 o o 8 00 (I) 0 0 00 0 1 6 0 0 0 0 00 0 o~~~~----~~~~ 4 -5 o 5 -5 0 5 Figure 3: Distribution of active learning sample points as a function of (i) noise strength and (ii) a-priori polynomial degree. The horizontal axis of both graphs represents the input space [-5,5]. Each circle indicates a sample location. The Left graph shows the distribution of sample locations (on x axis) for different noise level (indicated on y-axis). The Right graph shows the distribution of sample locations (on x-axis) for different assumptions on the maximum polynomial degree K (indicated on y-axis). 5 CONCLUSIONS We have developed a Bayesian framework for active learning using ideas from optimal experiment design. Our focus has been to investigate the possibility of improved sample complexity using such active learning schemes. For a simple case of unit step functions, we are able to derive a binary search algorithm from a completely different standpoint. Such an algorithm then provably requires fewer examples for the same error rate. We then show how to derive specific algorithms for the case of polynomials and carry out extensive simulations to compare their performance against the benchmark of a passive learner with encouraging results. This is an application of the optimal design paradigm to function learning and seems to bear promise for the design of more efficient learning algorithms. References D. Cohn. (1991) A Local Approach to Optimal Queries. In D. Touretzky (ed.), Proc. of 1990 Connectionist Summer School, San Mateo, CA, 1991. Morgan Kaufmann Publishers. V. Fedorov. (1972) Theory of Optimal Experiments. Academic Press, New York, 1972. D. MacKay. (1992) Bayesian Methods for Adaptive Models. PhD thesis, CalTech, 1992. P. Niyogi and K. Sung. (1995) Active Learning for Function Approximation: Paradigms from Optimal Experiment Design. Tech Report AIM-1483, AI Lab., MIT, In Preparation. M. Plutowski and H. White. (1991) Active Selection of Training Examples for Network Learning in Noiseless Environments. Tech Report CS91-180, Dept. of Computer Science and Engineering, University of California, San Diego, 1991. T. Poggio and F. Girosi. (1990) Regularization Algorithms for Learning that are Equivalent to Multilayer Networks. Science, 247:978-982, 1990. P. Sollich. (1994) Query Construction, Entropy, Generalization in Neural Network Models. Physical Review E, 49:4637-4651, 1994. 1. Valiant. (1984) A Theory of Learnable. Proc. of the 1984 STOC, p436-445, 1984.	1994	95
993	Patterns of damage in neural networks: The effects of lesion area, shape and number Eytan Ruppin and James A. Reggia • Department of Computer Science A.V. Williams Bldg. University of Maryland College Park, MD 20742 ruppin@cs.umd.edu reggia@cs.umd.edu Abstract Current understanding of the effects of damage on neural networks is rudimentary, even though such understanding could lead to important insights concerning neurological and psychiatric disorders. Motivated by this consideration, we present a simple analytical framework for estimating the functional damage resulting from focal structural lesions to a neural network. The effects of focal lesions of varying area, shape and number on the retrieval capacities of a spatially-organized associative memory. Although our analytical results are based on some approximations, they correspond well with simulation results. This study sheds light on some important features characterizing the clinical manifestations of multi-infarct dementia, including the strong association between the number of infarcts and the prevalence of dementia after stroke, and the 'multiplicative' interaction that has been postulated to occur between Alzheimer's disease and multi-infarct dementia. *Dr. Reggia is also with the Department of Neurology and the Institute of Advanced Computer Studies at the University of Maryland. 36 Eytan Ruppin, James A. Reggia 1 Introduction Understanding the response of neural nets to structural/functional damage is important for a variety of reasons, e.g., in assessing the performance of neural network hardware, and in gaining understanding of the mechanisms underlying neurological and psychiatric disorders. Recently, there has been a growing interest in constructing neural models to study how specific pathological neuroanatomical and neurophysiological changes can result in various clinical manifestations, and to investigate the functional organization of the symptoms that result from specific brain pathologies (reviewed in [1, 2]). In the area of associative memory models specifically, early studies found an increase in memory impairment with increasing lesion severity (in accordance with Lashley's classical 'mass action' principle), and showed that slowly developing lesions have less pronounced effects than equivalent acute lesions [3]. Recently, it was shown that the gradual pattern of clinical deterioration manifested in the majority of Alzheimer's patients can be accounted for, and that different synaptic compensation rates can account for the observed variation in the severity and progression rate of this disease [4]. However, this past work is limited in that model elements have no spatial relationships to one another (all elements are conceptually equidistant). Thus, as there is no way to represent focal (localized) damage in such networks, it has not been possible to study the functional effect of focal lesions and to compare them with that caused by diffuse lesions. The limitations of past work led us to use spatially-organized neural network for studying the effects of different types of lesions (we use the term lesion to mean any type of structural and functional damage inflicted on an initially intact neural network). The elements in our model, which can be thought of as representing neurons, or micro-columnar units, form a 2-dimensional array (whose edges are connected, forming a torus to eliminate edge effects), and each unit is connected primarily to its nearby neighbors, as is the case in the cortex [5]. It has recently been shown that such spatially-organized attractor networks can function reasonably well as associative memory devices [6]. This paper presents the first detailed analysis of the effects of lesions of various size, form and number on the memory performance of spatially-organized attractor neural networks. Assuming that these networks are a plausible model of some frontal and associative cortical areas (see, e.g., [7]), our results shed light on the clinical progress of disorders such as stroke and dementia. In the next section, we derive a theoretical framework that characterizes the effects of focal lesions on an associative network's performance. This framework, which is formulated in very general terms, is then examined via simulations in Section 3, which show a remarkable quantitative fit with the theoretical predictions, and are compared with simulations examining performance with diffuse damage. Finally, the clinical significance of our results is discussed in Section 4. 2 Analyzing the effects of focal lesions Our analysis pertains to the case where in the pre-damaged network, all units have an approximately similar average level of activity 1. A focal structural lesion IThis is true in general for associative memory networks, when the activity of each unit is averaged over a sufficiently long time span. Patterns of Damage ill NeuraL Networks 37 (anatomical lesion), denoting the area of damage and neuronal death, is modeled by clamping the activity of the lesioned units to zero. As a result of this primary lesion, the activity of surrounding units may be decreased, resulting in a secondary area of functional lesion, as illustrated in Figure 1. We are primarily interested in large focal lesions, where the area s of the lesion is significantly greater than the local neighborhood region from which each unit receives its inputs. Throughout our analysis we shall hold the working assumption that, traversing from the border of the lesion outwards, the activity of units gradually rises from zero until it reaches its normal, predamaged levels, at some distance d from the lesion's border (see Figure 1). As s is large and d is determined by local interactions on the borders of the structural lesion, we may reasonably assume that the value of d is independent of the lesion size, and depends primarily on the specific network characteristics, such as it architecture, dynamics, and memory load. Figure 1: A sketch of a structural (dark shading) and surrounding functional (light shading) rectangular lesion. Let the intact baseline performance level of the network be denoted as P(O), and let the network size be A. The network's performance denotes how accurately it retrieves the correct memorized patterns given a set of input cues, and is defined formally below. A structural lesion of area s (dark shading in Figure 1), causing a functional lesion of area ~s (light shading in Figure 1), will then result In a performance level of approximately P(s) = P(O) [A - (s + ~3)l + Pt:..~3 = P(O) _ (~P~3)/(A _ s) , (1) A-s where Pt:.. denotes the average level of performance over ~3 and ~P = P(O) - Pt:.. . P( s) hence reflects the performance level over the remaining viable parts of the network, discarding the structurally damaged region. Bearing these definitions in mind, a simple analysis shows that the effect of focal lesions is governed by the following rules. Consider a symmetric, circular structural lesion of size s = 71-r2 . ~3' the area of functional damage following such a lesion is then (assuming large lesions and hence VS> d) Rule 1: (2) 38 Eytan Ruppin, James A. Reggia 1.0 .--~----r-------r------.--, 0.8 -k.1 --- k=5 k=25 0.2 0.0 '---~----'"---~-~---'--' 0.0 500.0 1000.0 1500.0 Lesion size Figure 2: Theoretically predicted network performance as a function of a single focal structural lesion's size (area): analytic curves obtained for different k values; A = 1600. and pes) ~ P(O) Ak..JS , -s (3) for some constant k = yi4;db..P. Thus, the area of functional damage surrounding a single focal structural lesion is proportional to the square root of the structural lesion's area. Some analytic performancejlesioning curves (for various k values) are illustrated in Figure 2. Note the different qualitative shape of these curves as a function of k. Letting x = sjA be the fraction of structural damage, we have P(x) ~ P(O) _ k.JX _1_ 1-x.,jA , (4) that is, the same fraction x of damage results in less performance decrease in larger networks. This surprising result testifies to the possible protective value of having functional 'modular' cortical networks of large size. Expressions 3 and 4 are valid also when the structural lesion has a square shape. To study the effect of the structural lesion's shape, we consider the area b.. 8 [n] of a functional lesion resulting from a rectangular focal lesion of size s = a . b (see Figure 1), where, without loss of generality, n = ajb ~ 1. Then, for large n, we find that the functional damage of a rectangular structural lesion of fixed size increases as its shape is more elongated, following Rule 2: and pes) ~ P(O) _ k.,foS 2(A - s) (5) (6) Patterns of Damage in Neural Networks 39 To study the effect of the number of lesions, consider the area .6. 3 m of a functional lesion composed of m focal rectangular structural lesions (with sides a = n· b), each of area s/m. We find that the functional damage increases with the number offocal sub-lesions (while total structural lesion area is held constant), according to Rule 3: and pes) ~ P(O) _ k.;mns 2(A - s) (7) (8) While Rule 3 presents a lower bound on the functional damage which may actually be significantly larger and involves no approximations, Rule 2 presents an upper bound on the actual functional damage. As we shall show in the next section, the number of lesions actually affects the network performance significantly more than its precise shape. 3 Numerical Simulation Results We now turn to examine the effect of lesions on the performance of an associative memory network via simulations. The goal of these simulations is twofold. First, to examine how accurately the general but approximate theoretical results presented above describe the actual performance degradation in a specific associative network. Second, to compare the effects of focal lesions to those of diffuse ones, as the effect of diffuse damage cannot be described as a limiting case within the framework of our analysis. Our simulations were performed using a standard Tsodyks-Feigelman attractor neural network [8]. This is a Hopfield-like network which has several features which make it more biologically plausible [4], such as low activity and non-zero positive thresholds. In all the experiments, 20 sparse random {O, I} memory patterns (with a fraction of p ~ 1 of l's) were stored in a network of N = 1600 units, placed on a 2-dimensionallattice. The network has spatially organized connectivity, where each unit has 60 incoming connections determined randomly with a Gaussian probability <J>(z) = J1/27rexp( _z2 /2(1'2), where z is the distance between two units in the array. When (1' is small, each unit is connected primarily to its nearby neighbors. As in [4], the cue input patterns are presented via an external field of magnitude e = 0.035, and the noise level is T = 0.005. The performance of the network is measured (over the viable, non-lesioned units) by the standard overlap measure which denotes the similarity between the final state S the network converges to and the memory pattern ~!l that was cued in that trial, defined by N m!l(t) = (1 ~ )N I:(~r - p)Si(t) . P P i=l (9) In all simulations we report the average overlap achieved over 100 trials. We first studied the network's performance at various (1' values. Figure 3a displays how the performance of the network degrades when diffuse structural lesions of increasing size are inflicted upon it (i.e., randomly selected units are clamped to zero), while Figure 3b plots the performance as a function of the size of a single square-shaped focal lesion. As is evident, spatially-organized connectivity enables 40 Eytan Ruppin, James A. Reggia the network to maintain its memory retrieval capacities in face of focal lesions of considerable size. Diffuse lesions are always more detrimental than single focal lesions of identical size. Also plotted in Figure 3b is the analytical curve calculated via expression (3) (with k = 5), which shows a nice fit with the actual performance of the spatially-connected network parametrized by (J' = 1. Concentrating on the study of focal lesions in a spatially-connected network, we adhere to the values (J' = 1 and k = 5 hereafter, and compare the analytical and numerical results. With these values, the analytical curves describing the performance of the network as a function of the fraction of the network lesioned (obtained using expression 4) are similar to the corresponding numerical results. (a) 1.0 r-----,----~-------, 0.8 0.6 0.4 0.2 - -- ",gna_1 signa =3 "'gna= 10 --- "'gnam30 0.0 L-_~ _ _'___~_~ ___ ---' 0.0 500.0 1000.0 1500.0 Lesion sIZe (b) 1.0,-----,-----r------, 0.8 0.6 0.4 0.2 .... ~ .... ~...: .... ~.: .. -~, ',~''', ~-... ~. ; - .. -\ , \ \ ".\', .\ \ \ \ \ \ \ \ \ \ \ , \ \ \ --- sigma =1 \ ... , sigma _3 \ sigma -10 \ --- sigma =30 " Analytical .osuns, k = 5 " , '. . \ '. \ '. \ , \ '.\ ....................... '0.0 ':--~__;:::'::_:_-~-::-:':c_:__---= 0.0 500.0 1000.0 1500.0 Lesion size Figure 3: Network performance as a function of lesion size: simulation results obtained in four different networks, each characterized by a distinct distribution of spatially-organized connectivity. (a) Diffuse lesions. (b) Focal lesions. To examine Rule 2, a rectangular structural lesion of area s = 300 was induced in the network. As shown in Figure 4a, as the ratio n between the sides is increased, the network's performance further decreases, but this effect is relatively mild. The markedly stronger effect of varying the lesion number (described by Rule 3) is demonstrated in figure 4b, which shows the effect of multiple lesions composed of 2,4,8 and 16 separate focal lesions. For comparison, the performance achieved with a diffuse lesion of similar size is plotted on the 20'th x-ordinate. It is interesting to note that a sufficiently large multiple focal lesion (s = 512) can cause a larger performance decrease than a diffuse lesion of similar size. That is, at some point, when the size of each individual focal lesion becomes small in relation to the spread of each unit's connectivity, our analysis looses its validity, and Rule 3 ceases to hold. Patterns of Damage in Neural Networks (a) li-i ~ 0.930 0.910 " 0.890 0.870 <>--<> SlrnuIeIion "'1U~s - . Analytical reou~ " , " " 0.850 '---~-'---'----:"-~~-~-----' 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Rectangular ratio n (b) 0.90 ,---'-~-"-----'----r-----, 0.80 0.80 <>--tI Sinulation (s D 256) ...... " Sinulation (s • 512) - - - - Analytic (s =256) - . Analytic (s • 512) 0.50 '----'----"------'-~--'-----' 0.0 5.0 10.0 15.0 20.0 25.0 No. 01 sub-lesions m 41 Figure 4: Network performance as a function of focal lesion shape (a) and number (b). Both numerical and analytical results are displayed. In Figure 4b, the x-ordinate denotes the number of separate sub-lesions (1,2,4,8,16) , and, for comparison, the performance achieved with a diffuse lesion of similar size is plotted on the 20'th x-ordinate. 4 Discussion We have presented a simple analytical framework for studying the effects of focal lesions on the functioning of spatially organized neural networks. The analysis presented is quite general and a similar approach could be adopted to investigate the effect offocallesions in other neural models. Using this analysis, specific scaling rules have been formulated describing the functional effects of structural focal lesions on memory retrieval performance in associative attractor networks. The functional lesion scales as the square root of the size of a single structural lesion, and the form of the resulting performance curve depends on the impairment span d. Surprisingly, the same fraction of damage results in significantly less performance decrease in larger networks, pointing to their relative robustness. As to the effects of shape and number, elongated structural lesions cause more damage than more symmetrical ones. However, the number of sub-lesions is the most critical factor determining the functional damage and performance decrease in the model. Numerical studies show that in some conditions multiple lesions can damage performance more than diffuse damage, even though the amount of lost innervation is always less in a multiple focal lesion than with diffuse damage. Beyond its computational interest, the study of the effects of focal damage on the performance of neural network models can lead to a better understanding of functional impairments accompanying focal brain lesions. In particular, we are interested in multi-infarct dementia, a frequent cause of dementia (chronic deterioration of cognitive and memory capacities) characterized by a series of multiple, aggregat42 Eytan Ruppin, James A. Reggia ing focal lesions. Our results indicate a significant role for the number of infarcts in determining the extent of functional damage and dementia in multi-infarct disease. In our model, multiple focal lesions cause a much larger deficit than their simple 'sum', i.e., a single lesion of equivalent total size. This is consistent with clinical studies that have suggested the main factors related to the prevalence of dementia after stroke to be the infarct number and site, and not the overall infarct size, which is related to the prevalence of dementia in a significantly weaker manner [9, 10]. Our model also offers a possible explanation to the 'multiplicative' interaction that has been postulated to occur between co-existing Alzheimer and multi-infarct dementia [10], and to the role of cortical atrophy in increasing the prevalence of dementia after stroke; in accordance with our model, it is hypothesized that atrophic degenerative changes will lead to an increase in the value of d (and hence of k) and increase the functional damage caused by a lesion of given structural size. This hypothesis, together with a detailed study of the effects of the various network parameters on the value of d, are currently under further investigation. Acknowledgements This research has been supported by a Rothschild Fellowship to Dr. Ruppin and by Awards NS29414 and NS16332 from NINDS. References [1] J. Reggia, R. Berndt, and L. D'Autrechy. Connectionist models in neuropsychology. In Handbook of Neuropsychology, volume 9. 1994, in press. [2] E. Ruppin. Neural modeling of psychiatric disorders. Network: Computation in Neural Systems, 1995. Invited review paper, to appear. [3] J .A. Anderson. Cognitive and psychological computation with neural models. IEEE Trans. on Systems,Man, and Cybernetics, SMC-13(5):799-815, 1983. [4] D. Horn, E. Ruppin, M. Usher, and M. Herrmann. Neural network modeling of memory deterioration in alzheimer's disease. Neural Computation, 5:736-749, 1993. [5] A. M. Thomson and J. Deuchars. Temporal and spatial properties of local circuits in the neocortex. Trends in neuroscience, 17(3):119-126, 1994. [6] J .M. Karlholm. Associative memories with short-range higher order couplings. Neural Networks, 6:409-421, 1993. [7] F.A.W. Wilson, S.P.O Scalaidhe, and P.S. Goldman-Rakic. Dissociation of object and spatial processing domains in primate prefrontal cortex. Science, 260:1955-1958,1993. [8] M.V. Tsodyks and M.V. Feigel'man. The enhanced storage capacity in neural networks with low activity level. Europhys. Lett., 6:101 - 105, 1988. [9] T.K. Tatemichi, M.A. Foulkes, J.P. Mohr, J.R. Hewitt, D. B. Hier, T .R. Price, and P.A. Wolf. Dementia in stroke survivors in the stroke data bank cohort. Stroke, 21:858-866, 1990. [10] T. K. Tatemichi. How acute brain failure becomes chronic: a view of the mechanisms of dementia related to stroke. Neurology, 40:1652-1659, 1990.	1994	96
994	A Study of Parallel Perturbative Gradient Descent D. Lippe· J. Alspector Bellcore Morristown, NJ 07960 Abstract We have continued our study of a parallel perturbative learning method [Alspector et al., 1993] and implications for its implementation in analog VLSI. Our new results indicate that, in most cases, a single parallel perturbation (per pattern presentation) of the function parameters (weights in a neural network) is theoretically the best course. This is not true, however, for certain problems and may not generally be true when faced with issues of implementation such as limited precision. In these cases, multiple parallel perturbations may be best as indicated in our previous results. 1 INTRODUCTION Motivated by difficulties in analog VLSI implementation of back-propagation [Rumelhart et al., 1986] and related algorithms that calculate gradients based on detailed knowledge of the neural network model, there were several similar recent papers proposing to use a parallel [Alspector et al., 1993, Cauwenberghs, 1993, Kirk et al., 1993] or a semi-parallel [Flower and Jabri, 1993] perturbative technique which has the property that it measures (with the physical neural network) rather than calculates the gradient. This technique is closely related to methods of stochastic approximation [Kushner and Clark, 1978] which have been investigated recently by workers in fields other than neural networks. [Spall, 1992] showed that averaging multiple parallel perturbations for each pattern presentation may be asymptotically preferable in the presence of noise. Our own results [Alspector et al., 1993] indicated ·Present address: Dept. of EECSj MITj Cambridge, MA 02139; dalippe@mit.edu 804 D. Lippe, 1. Alspector that multiple parallel perturbations are also preferable when only limited precision is available in the learning rate which is realistic for a physical implementation. In this work we have investigated whether multiple parallel perturbations for each pattern are non-asymptotically preferable theoretically (without noise). We have also studied this empirically, to the limited degree that simulations allow, by removing the precision constraints of our previous work. 2 GRADIENT ESTIMATION BY PARALLEL WEIGHT PERTURBATION Following our previous work, one can estimate the gradient of the error, E( w), with respect to any weight, Wi, by perturbing Wi by 6w1 and measuring the change in the output error, 6E, as the entire weight vector, W, except for component Wi is held constant. 6E E(w + 6;1) - E(w) 6w1 6Wi We now consider perturbing all weights simultaneously. However, we wish to have the perturbation vector, 6w, chosen uniformly on a hypercube. Note that this requires only a random sign multiplying a fixed perturbation and is natural for VLSI using a parallel noise generator [Alspector et al., 1991J. This leads to the approximation (ignoring higher order terms) w 6E 8E 2:(8E) (6Wi) - -+ 6w· 8w· 8w· 6w·· , 'i¢1 ] , (1 ) The last term has expectation value zero for random and independently distributed 6w1• The weight change rule where 1] is a learning rate, will follow the gradient on the average but with considerable noise. For each pattern, one can reduce the variance of the noise term in (1) by repeating the random parallel perturbation many times to improve the statistical estimate. If we average over P perturbations, we have where p indexes the perturbation number. A Study of Parallel Perturbative Gradient Descellf 805 3 THEORETICAL RELATIVE EFFICIENCY 3.1 BACKGROUND Spall [Spall, 1992] shows in an asymptotic sense that multiple perturbations may be faster if only a noisy measurement of E( tV) is available, and that one perturbation is superior otherwise. His results are asymptotic in that they compare the rate of convergence to the local minimum if the algorithms run for infinite time. Thus, his results may only indicate that 1 perturbation is superior close to a local minimum. Furthermore, his result implicitly assumes that P perturbations per weight update takes P times as long as 1 perturbation per weight update. Experience shows that the time required to present patterns to the hardware is often the bottleneck in VLSI implementations of neural networks [Brown et al., 1992]. In a hardware implementation of a perturbative learning algorithm, a few perturbations might be performed with no time penalty while waiting for the next pattern presentation. The remainder of this section sketches an argument that multiple perturbations may be desirable for some problems in a non-asymptotic sense, even in a noise free environment and under the assumption of a multiplicative time penalty for performing multiple perturbations. On the other hand, the argument also shows that there is little reason to believe in practice that any given problem will be learned more quickly by multiple perturbations. Space limitations prevent us from reproducing the full argument and discussion of its relevance which can be found in [Lippe, 1994]. The argument fixes a point in weight space and considers the expectation value of the change in the error induced by one weight update under both the 1 perturbation case and the multiple perturbation case. [Cauwenberghs, 1994] contains a somewhat related analysis of the relative speed of one parallel perturbation and weight perturbation as described in [Jabri and Flower, 1991]. The analysis is only truly relevant far from a local minimum because close to a local minimum the variance of the change of the error is as important as the mean of the change of the error. 3.2 Calculations If P is the number of perturbations, then our learning rule is -'TJ ~ 6E(p) ~Wi = P L.J~. P=1 6wi If W is the number of weights, then ~E, calculated to second order in 'TJ, is w 8E 1 W W 82E ~E = '" -8 ~Wi + - '" '" 8 8 ~Wi~Wj. L.J W· 2 L.J L.J W· W · i=l' i=l j=l • J Expanding 6E(p) to second order in (j (where 6Wi = ±(j), we obtain W 8E W W 8 2 6E(p) = '" -6w~P) + ! "'" E 6w~P)6w(P). L.J 8w' J 2 L.J L.J 8w' 8wk J k j=l J j=l k=l J (2) (3) (4) 806 D. Lippe, 1. Alspector [Lippe, 1994] shows that combining (2)-(4), retaining only first and second order terms, and taking expectation values gives 2 < l:1E >= -TJX + ~ (Y + PZ) (5) where x w (8E)2 L 8w' ' i=l ' Z Y Note that first term in (5) is strictly less than or equal to 0 since X is a sum of squaresl . The second term, on the other hand, can be either positive or negative. Clearly then a sufficient condition for learning is that the first term dominates the second term. By making TJ small enough, we can guarantee that learning occurs. Strictly speaking, this is not a necessary condition for learning. However, it is important to keep in mind that we are only focusing on one point in weight space. If, at this point in weight space, < l:1E > is negative but the second term's magnitude is close to the first term's magnitude, it is not unlikely that at some other point in weight space < l:1E > will be positive. Thus, we will assume that for efficient learning to occur, it is necessary that TJ be small enough to make the first term dominate the second term. Assume that some problem can be successfully learned with one perturbation, at learning rate TJ(I). Then the first order term in (5) dominates the second order terms. Specifically, at any point in weight space we have, for some large constant J1., TJ(I)X ~ J1.TJ(I)2IY + ZI In order to learn with P perturbations, we apparently need TJ( P)X ~ J1. TJ(~)2 IY + P ZI (6) The assumption that the first order term of (5) dominates the second order terms implies that convergence time is proportional to ,.lp), Thus, learning is more efficient in the multiple perturbation case if J1.TJ(P) > J1.TJ(I) P (7) It turns out, as shown in [Lippe, 1994] that the conditions (6) and (7) can be met simultaneously with multiple perturbations if =f ~ 2. lIf we are at a stationary point then the first term in (5) is O. A Study of Parallel Perturbative Gradient Descent 807 It is shown in [Lippe, 1994], by using the fact that the Hessian of a quadratic function with a minimum is positive semi-definite, that if E is guadratic and has a minimum, then Y and Z have the same sign (and hence =f < 2). Any well behaved function acts quadratically sufficiently close to a stationary point. Thus, we can not get < flE > more than a factor of P larger by using P perturbations near local minima of well behaved functions. Although, as mentioned earlier, we are entirely ignoring the issue of the variance of flE, this may be some indication of the asymptotic superiority of 1 perturbation. 3.3 Discussion of Results The result that multiple perturbations are superior when -i ~ 2 may seem somewhat mysterious. It sheds some light on our answer to rewrite (5) as Y < flE >= -"IX + "I2(p + Z). For strict gradient descent, the corresponding equation is < flE >= flE = -"IX + "I2Z. The difference between strict gradient descent and perturbative gradient descent, on average, is the second order term "I2~. This is the term which results from not following the gradient exactly, and it Obviously goes down as P goes up and the gradient measurement becomes more accurate. Thus, if Z and Y have different signs, P can be used to make the second order term disappear. There is no way to know whether this situation will occur frequently. Furthermore, it is important to keep in mind that if Y is negative and Z is positive, then raising P may make the magnitude of the second order term smaller, but it makes the term itself larger. Thus, in general, there is little reason to believe that multiple perturbations will help with a randomly chosen problem. An example where multiple perturbations help is when we are at a point where the error surface is convex along the gradient direction, and concave in most other directions. Curvature due to second derivative terms in Y and Z help when the gradient direction is followed, but can hurt when we stray from the gradient. In this case, Z < 0 and possibly Y > 0, so multiple perturbations might be preferable in order to follow the gradient direction very closely. 4 SIMULATIONS OF SINGLE AND MULTIPLE PARALLEL PERTURBATION 4.1 CONSTANT LEARNING RATES The second order terms in (5) can be reduced either by using a small learning rate, or by using more perturbations, as discussed briefly in [Cauwenberghs, 1993]. Thus, if "I is kept constant, we expect a minimum necessary number of perturbations in order to learn. This in itself might be of importance in a limited precision implementation. If there is a non-trivial lower bound on "I, then it might be necessary to use multiple perturbations in order to learn. This is the effect that was noticed in [Alspector et al., 1993]. At that time we thought that we had found empirically 808 D. Lippe, J. Alspector Table 1: Running times for the first initial weight vector P TJ Time for < .1 Time for < .5 1 .0005 1,121,459 32,179 1 .001 831, 684 18,534 1 .002 784, 768 11,008 1 .003 4 94,029 9,933 1 .004 1,695,974 9,728 7 .00625 707,840 23,834 7 .008 583,654 16,845 7 .0125 922,880 13,261 7 .025 1,010,355 12,006 7 .035 Not tested 17,024 that multiple perturbations were necessary for learning. The problem was that we failed to decrease the learning rate with the number of perturbations. 4.2 EMPIRICAL RELATIVE EFFICIENCY OF SINGLE AND MULTIPLE PERTURBATION ALGORITHMS Section 3 showed that, in theory, multiple perturbations might be faster than 1 perturbation. We investigated whether or not this is the case for the 7 input hamming error correction problem as described in [Biggs, 1989]. This is basically a nearest neighbor problem. There exist 16 distinct 7 bit binary code words. When presented with an arbitrary 7 bit binary word, the network is to output the code word with the least hamming distance from the input. After preliminary tests with 50, 25, 7, and 1 perturbation, it seemed that 7 perturbations provided the fastest learning, so we concentrated on running simulations for both the 1 perturbation and the 7 perturbation case. Specifically, we chose two different (randomly generated) initial weight vectors, and five different seeds for the pseudo-random function used to generate the bWi. For each of these ten cases, we tested both 1 perturbation and 7 perturbations with various learning rates in order to obtain the fastest possible learning. The 128 possible input patterns were repeatedly presented in order. We investigated how many pattern presentations were necessary to drive the MSE below .1 and how many presentations were necessary to drive it below .5. Recalling the theory developed in section 3, we know that multiple perturbations can be helpful only far away from a stationary point. Thus, we expected that 7 perturbations might be quicker reaching .5 but would be slower reaching .1. The results are summarized in tables 1 and 2. Each table summarizes information for a different initial weight vector. All of the data presented are averaged over 5 runs, one with each of the different random seeds. The two columns labeled "Time for < .5" and "Time for < .1" are adjusted according to the assumption that one weight update at 7 perturbations takes 7 times as long as one weight update at 1 perturbation. In each table, the following four numbers appear in italics: the shortest time to reach .1 with 1 perturbation, the shortest time to reach .1 with 7 perturbations, the shortest time to reach .5 with 1 perturbation, and the shortest time to reach .5 with 7 perturbations. 7 perturbations were a loss in three out of four of the experiments. Surprisingly, A Study of Parallel Perturbative Gradient Descent 809 Table 2: Running times for the second initial weight vector l' 'T/ TIme for < .1 TIme for < .5 1 .001 928,236 22,133 1 .002 719 , 078 12,817 1 .003 154,139 10,675 1 .004 1,603,354 11,150 1 .00625 629, 530 21,059 1 .008 611,610 19,112 1 .0125 912,333 15,949 1 .025 1,580,442 14,515 1 .035 Not tested 11,141 the one time that multiple perturbations helped was in reaching .1 from the second initial weight vector. There are several possible explanations for this. To begin with, these learning times are averages over only five simulations each, which makes their statistical significance somewhat dubious. Unfortunately, it was impractical to perform too many experiments as the data obtained required 180 computer simulations, each of which sometimes took more than a day to complete. Another possible explanation is that .1 may not be "asymptotic enough." The numbers .5 and .1 were chosen somewhat arbitrarily to represent non-asymptotic and asymptotic results. However, there is no way of predicting from the theory how close the error must be to its minimum before asymptotic results become relevant. The fact that 1 perturbation outperformed 7 perturbations in three out of four cases is not surprising. As explained in section 3, there is in general no reason to believe that multiple perturbations will help on a randomly chosen problem. 5 CONCLUSION Our results show that, under ideal computational conditions, where the learning rate can be adjusted to proper size, that a single parallel perturbation is, except for unusual problems, superior to multiple parallel perturbations. However, under the precision constraints imposed by analog VLSI implementation, where learning rates may not be adjustable and presenting a pattern takes longer than performing a perturbation, multiple parallel perturbations are likely to be the best choice. Acknowledgment We thank Gert Cauwenberghs and James Spall for valuable and insightful discussIons. References [Alspector et al., 1991] Alspector, J., Gannett, J. W., Haber, S., Parker, M. B., and Chu, R. (1991). A VLSI-efficient technique for generating multiple uncorrelated noise sources and its application to stochastic neural networks. IEEE Transactions on Circuits and Systems, 38:109-123. [Alspector et al., 1993] Alspector, J., Meir, R., Yuhas, B., Jayakumar, A., and Lippe, D. (1993). A parallel gradient descent method for learning in analog 810 D. Lippe, J. A/spector VLSI neural networks. In Hanson, S. J., Cowan, J. D., and Giles, C. L., editors, Advances in Neural Information Processing Systems 5, pages 836-844, San Mateo, California. Morgan Kaufmann Publishers. [Biggs, 1989] Biggs, N. L. (1989). Discrete Math. Oxford University Press. [Brown et al., 1992] Brown, T. X., Tran, M. D., Duong, T., and Thakoor, A. P. (1992). Cascaded VLSI neural network chips: Hardware learning for pattern recognition and classification. Simulation, 58(5):340-347. [Cauwenberghs, 1993] Cauwenberghs, G. (1993). A fast stochastic error-descent algorithm for supervised learning and optimization. In Hanson, S. J., Cowan, J. D., and Giles, C. L., editors, Advances in Neural Information Processing Systems 5, pages 244-251, San Mateo, California. Morgan Kaufmann Publishers. [Cauwenberghs, 1994] Cauwenberghs, G. (1994). Analog VLSI Autonomous Systems for Learning and Optimization. PhD thesis, California Institute of Technology. [Flower and Jabri, 1993J Flower, B. and Jabri, M. (1993). Summed weight neuron perturbation: An o(n) improvement over weight perturbation. In Hanson, S. J., Cowan, J. D., and Giles, C. L., editors, Advances in Neural Information Processing Systems 5, pages 212-219, San Mateo, California. Morgan Kaufmann Publishers. [Jabri and Flower, 1991] Jabri, M. and Flower, B. (1991). Weight perturbation: An optimal architecture and learning technique for analog VLSI feedforward and recurrent multilayer networks. In Neural Computation 3, pages 546-565. [Kirk et al., 1993] Kirk, D., Kerns, D., Fleischer, K., and Barr, A. (1993). Analog VLSI implementation of gradient descent. In Hanson, S. J., Cowan, J. D., and Giles, C. L., editors, Advances in Neural Information Processing Systems 5, pages 789-796, San Mateo, California. Morgan Kaufmann Publishers. [Kushner and Clark, 1978] Kushner, H. and Clark, D. (1978). Stochastic Approzimation Methods for Constrained and Unconstrained Systems. Springer-Verlag, New York. [Lippe, 1994] Lippe, D. A. (1994). Parallel, perturbative gradient descent methods for learning in analog VLSI neural networks. Master's thesis, Massachusetts Institute of Technology. [Rumelhart et al., 1986] Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986). Learning internal representations by error propogation. In Rumelhart, D. E. and McClelland, J. L., editors, Parallel Distributed Processing: Ezplorations in the Microstructure of Cognition, page 318. MIT Press, Cambridge, MA. [Spall, 1992] Spall, J. C. (1992). Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Transactions on A utomatic Control, 37(3):332-341.	1994	97
995	A Neural Model of Delusions and Hallucinations in Schizophrenia Eytan Ruppin and James A. Reggia Department of Computer Science University of Maryland, College Park, MD 20742 ruppin@cs.umd.edu reggia@cs.umd.edu David Horn School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel horn@vm.tau.ac.il Abstract We implement and study a computational model of Stevens' [19921 theory of the pathogenesis of schizophrenia. This theory hypothesizes that the onset of schizophrenia is associated with reactive synaptic regeneration occurring in brain regions receiving degenerating temporal lobe projections. Concentrating on one such area, the frontal cortex, we model a frontal module as an associative memory neural network whose input synapses represent incoming temporal projections. We analyze how, in the face of weakened external input projections, compensatory strengthening of internal synaptic connections and increased noise levels can maintain memory capacities (which are generally preserved in schizophrenia). However, These compensatory changes adversely lead to spontaneous, biased retrieval of stored memories, which corresponds to the occurrence of schizophrenic delusions and hallucinations without any apparent external trigger, and for their tendency to concentrate on just few central themes. Our results explain why these symptoms tend to wane as schizophrenia progresses, and why delayed therapeutical intervention leads to a much slower response. 150 Eytan Ruppin, James A. Reggia, David Hom 1 Introduction There has been a growing interest in recent years in the use of neural models to investigate various brain pathologies and their cognitive and behavioral effects. Recent published examples of such studies include models of cortical plasticity following stroke, Alzheimer's disease and schizophrenia, and cognitive and behavioral explorations of aphasia, acquired dyslexia and affective disorders (reviewed in [1, 2]). Continuing this line of study, we present a computational account linking specific pathological synaptic changes that are postulated to occur in schizophrenia, and the emergence of schizophrenic delusions and hallucinations. The latter symptoms denote persistent, unrealistic, psychotic thoughts (delusions) or percepts (hallucinations) that may at times flood the patient in an overwhelming, stressful manner. The wealth of data gathered concerning the pathophysiology of schizophrenia supports the involvement of both the frontal and the temporal lobes. On the one hand, there are atrophic changes in the hippocampus and parahippocampal areas including neuronal loss and gliosis. On the other hand, neurochemical and morphometric studies testify to an expansion of various receptor binding sites and increased dendritic branching in the frontal cortex of schizophrenics. Stevens has recently presented a theory linking these temporal and frontal findings, claiming that the onset of schizophrenia is associated with reactive anomalous sprouting and synaptic reorganization taking place in the projection sites of degenerating temporal neurons, including (among various cortical and subcortical structures) the frontal lobes [3]. This paper presents a computational study of Stevens' theory. Within the framework of a memory model of hippocampal-frontal interaction, we show that the introduction of the 'microscopic' synaptic changes that underlie Stevens' hypothesis can help preserve memory function but results in specific 'pathological' changes in the 'macroscopic' behavior of the network. A small subset of the patterns stored in the network are now spontaneously retrieved at times, without being cued by any specific input pattern. This emergent behavior shares some of the important characteristics of schizophrenic delusions and hallucinations, which frequently appear in the absence of any apparent external trigger, and tend to concentrate on a limited set of recurrent themes [4]. Memory capacities are fairly preserved in schizophrenics, until late stages of the disease [5]. In Section 2 we present our model. The analytical and numerical results obtained are described in Section 3, followed by our conclusions in Section 4. 2 The Model As illustrated in Figure 1, we model a frontal module as an associative memory attractor neural network, receiving its input memory cues from decaying external input fibers (representing the degenerating temporal projections). The network's internal connections, which store the memorized patterns, undergo synaptic strengthening changes that model the reactive synaptic regeneration within the frontal module. The effect of other diffuse external projections is modeled as background noise. A frontal module represents a macro-columnar unit that has been suggested as a basic functional building block of the neocortex [6]. The assumption that memory retrieval from the frontal cortex is invoked by the firing of incoming A Neural Model of Delusions and Hallucinations in Schizophrenia 151 temporal projections is based on the notion that temporal structures have an important role in establishing long-term memory in the neocortex and in the retrieval of facts and events (e.g., [7]). --,-, " Other cortical : modules I I I ....... T : (Influence nxxJeled as,' .... .. ! noise) ," ......... .. \ ,I" .... .. Figure 1: A schematic illustration of the model. A frontal module is modeled as an attractor neural network whose neurons receive inputs via three kinds of connections: internal connections from other frontal neurons, external connections from temporal lobe neurons, and diffuse external connections from other cortical modules, modeled as noise. The attractor network we use is a biologically-motivated variant of Hopfield's ANN model, proposed by Tsodyks & Feigel'man [8]. Each neuron i is described by a binary variable Si = {1,0} denoting an active (firing) or passive (quiescent) state, respectively. M = aN distributed memory patterns eIJ are stored in the network. The elements of each memory pattern are chosen to be 1 (0) with probability p (1- p) respectively, with p ~ 1. All N neurons in the network have a fixed uniform threshold O. In its initial, undamaged state, the weights of the internal synaptic connections are M CO" Wij = N L....t(eIJi - p)(ej - p) , IJ=I (1) where Co = 1. The post-synaptic potential (input field) hi of neuron i is the sum of internal contributions from other neurons and external projections Fi e hi(t) = L WijSj(t - 1) + Fie. j The updating rule for neuron i at time t is given by S.(t) = {I, with p~ob . G(hi(t) - 0) 1 0, otherwIse (2) (3) 152 Eytan Ruppin, James A. Reggia, David Hom where G is the sigmoid function G(x) = 1/(1+exp( -x/T)), and T denotes the noise level. The activation level of the stored memories is measured by their overlaps mil with the current state of the network, defined by 1 N mll(t) = (1 _ )N L)er - p)Si(t) . P P i=l (4) Stimulus-dependent retrieval is modeled by orienting the field Fe with one of the memorized patterns (the cued pattern, say e), such that F/ = e . e\ , (e > 0) . (5) Following the presentation of an external input cue, the network state evolves until it converges to a stable state. The network parameters are tuned such that in its initial, undamaged state it correctly retrieves the cued patterns (eo = 0.035, Co = 1, T = 0.005). We also examine the network's behavior in the absence of any specific stimulus. The network may either continue to wander around in a state of random low baseline activity, or it may converge onto a stored memory state. We refer to the latter process as spontaneous retrieval. Our investigation of Stevens' work proceeds in two stages. First we examine and analyze the behavior of the network when it undergoes uniform synaptic changes that represent the pathological changes occurring in accordance with Stevens' theory. These include the weakening of external input projections (e !) and the increase in the internal projections (c i) and noise levels (T i). In the second stage, we add the assumption that the internal synaptic compensatory changes have an additional Hebbian activity-dependent component, and examine the effect of the rule (6) where Sk is 1 (0) only if neuron k has been consecutively firing (quiescent) for the last r iterations, and 'Y is a constant. 3 Results We now show some simulation and analytic results, examining the effects of the 'microscopic' pathological changes, taking place in accordance with Stevens' theory, on the 'macroscopic' behavior of the network. The analytical results presented have been derived by calculating the magnitude of randomly formed initial 'biases', and comparing their effect on the network's dynamics versus the effect of externally presented input cues. This comparison is performed by formulating a corresponding overlap master equation, whose fixed point dynamics are investigated via phaseplane analysis, as described in [9]. First, we study whether the reactive synaptic changes (occurring in both internal and external, diffuse synapses) are really compensatory, i.e., to what extent can they help maintaining memory capacities in the face of degenerating external input synapses. As illustrated in Figure 2, we find that increased noise levels can (up to some degree) preserve memory retrieval in the face of decreased external input strength. Increased synaptic strengthening preserves A Neural Model of Delusions and Hallucinations in Schizophrenia (a) 1.0 O.B 0.8 ! . 6 0.( 0.2 0.0 0.000 ,/ i' : -._0.035 I . ~_.__ • _ 0.025 - --- •• 0.015 --- •• 0.005 - :... ---0.010 0.020 T (b) 1.0 O.B 0.6 t 6 0.( '--- 0.2 0·8. , -._ ,!O.O35 ------ •• iO.025 ---- •• '0.015 - •• 10.014 G---o •• ;0.013 <)--0._ 10.012 --- •• ; 0.005 153 0.020 T Figure 2: Stimulus-dependent retrieval performance, measured by the average final overlap m, as a function of the noise level T. Each curve displays this relation at a different magnitude of external input projections e. (a) Simulation results. (b) Analytic approximation. memory retrieval in a similar manner, and the combined effect of these synaptic compensatory measures is synergistic. Second, although the compensatory synaptic changes help maintain memory retrieval capacities, they necessarily have adverse effects, leading eventually to the emergence of spontaneous activation of non-cued memory patterns; the network converges to some of its memory patterns in a pathological, autonomous manner, in the absence of any external input stimuli. This emergence of pathological spontaneous retrieval, when either the noise level or the internal synaptic strength (or both) are increased beyond some point, is demonstrated in Figure 3. Third, when the compensatory regeneration of internal synapses has an additional Hebbian component (representing a period of increased activity-dependent plasticity due to the regenerative synaptic changes), a biased spontaneous retrieval distribution is obtained. That is, as time evolves (measured in time units of 'trials'), the distribution of patterns spontaneously retrieved by the network in a pathological manner tends to concentrate only on one or two of all the memory patterns stored in the network, as is shown in Figure 4a. This highly peaked distribution is maintained for a few hundred additional trials until memory retrieval sharply collapses to zero as a global mixed-state attractor is formed. Such a mixed attractor state does not have very high overlap with any memorized pattern, and thus does not represent any well-defined cognitive or perceptual item. It is an end state of the Hebbian, activity-dependent evolution of the network. Yet, even after activitydependent changes ensue, if spontaneous activity does not emerge the distribution of retrieved memories remains homogeneous (see Figure 4b). Eventually, a global 154 Eytan Ruppin. James A. Reggia. David Hom (a) (b) 1.0 , ,---.. 1.0 , ........... . ~ .. .. .. ,1'" O.B , , O.B , 0.6 , 0.6 t , ! f 0.4 AnalytIc _Imollon - - - - Slmulallon 0.4 0.2 0.2 ' , , , 0.0 0.005 0.010 0.015 0.020 0.0 ':--~-'-:-'-::--~~--~----:' 2.0 2.5 3.0 3.5 4.0 T Figure 3: (a) Spontaneous retrieval, measured as the highest final overlap m achieved with any of the stored memory patterns, displayed as a function of the noise level T. c = 1. (b) Spontaneous retrieval as a function of internal synaptic compensation factor c. T = 0.009. mixed-state attractor is formed, and the network looses its retrieval capacities, but during this process no memory pattern gets to dominate the retrieval output. Our results remain qualitatively similar even when bounds are placed on the absolute magnitude of the synaptic weights. 4 Conclusions Our results suggest that the formation of biased spontaneous retrieval requires the concomitant occurrence of both degenerative changes in the external input fibers, and regenerative Hebbian changes in the intra-modular synaptic connections. They add support to the plausibility of Stevens' theory by showing that it may be realized within a neural model, and account for a few characteristics of schizophrenic symptoms: • The emergence of spontaneous, non-homogeneous retrieval is a self-limiting phenomenon (as eventually a cognitively meaningless global attractor is formed) - this parallels the clinical finding that as schizophrenia progresses both delusions and hallucinations tend to wane, while negative symptoms are enhanced [10]. • Once converged to, the network has a much larger tendency to remain in a biased memory state than in a non biased one - this is in accordance with the persistent characteristic of schizophrenic florid symptoms. • As more spontaneous retrieval trials occur the frequency of spontaneous retrieval increases - indeed, while early treatment in young psychotic adults A Neural Model of Delusions and Hallucinations in Schizophrenia (a) 1.0 ~-~-~~-:----~-----. 0.8 ~ 0.6 t 1 .~ ~ 0.4 0.2 G---f] Aft.r 200 tria_ {3 - -(:~Aft.r500trials ~ - v Aft ... 800triala Q .,. >' ., ~ ~ ., . ; , ~ 1 0.100 ~----"-~~--~-----. 0.080 ~ 0.000 t 1 l 0.040 0.020 ~AftOf200trlals ? ( · Aft ... 400tr1a1a 155 (b) Figure 4: (a) The distribution of memory patterns spontaneously retrieved. The xaxis enumerates the memories stored, and the y-axis denotes the retrieval frequency of each memory. 'Y = 0.0025. (b) The distribution of stimulus-dependent retrieval of memories. 'Y = 0.0025. leads to early response within days, late, delayed intervention leads to a much slower response during one or more months [11]. The current model generates some testable predictions: • On the neuroanatomical level, the model can be tested quantitatively by searching for a positive correlation between a recent history of florid psychotic symptoms and postmortem neuropathological findings of synaptic compensation. (For example, this kind of correlation, between indices of synaptic area and cognitive functioning was found in Alzheimer patients [12]). • On the physiological level, the increased compensatory noise should manifest itself in increased spontaneous neural activity. While this prediction is obviously difficult to examine directly, EEG studies in schizophrenics show significant increase in slow-wave delta activity which may reflect increased spontaneous activity [13]. • On the clinical level, due to the formation of a large and deep basin of attraction around the memory pattern which is at the focus of spontaneous retrieval, the proposed model predicts that its retrieval (and the elucidation of the corresponding delusions or hallucinations) may be frequently triggered by various environmental cues. A recent study points in this direction [14]. 156 Eytan Ruppin, James A. Reggia, David Hom Acknowledgements This research has been supported by a Rothschild Fellowship to Dr. Ruppin. References [1] J. Reggia, R. Berndt, and L. D' Autrechy. Connectionist models in neuropsychology. In Handbook of Neuropsychology, volume 9. 1994, in press. [2] E. Ruppin. Neural modeling of psychiatric disorders. Network: Computation in Neural Systems, 1995. Invited review paper, to appear. [3] J .R. Stevens. Abnormal reinnervation as a basis for schizophrenia: A hypothesis. Arch. Gen. Psychiatry, 49:238-243, 1992. [4] S.K. Chaturvedi and V.D. Sinha. Recurrence of hallucinations in consecutive episodes of schizophrenia and affective disorder. Schizophrenia Research, 3: 103106, 1990. [5] M. Marsel Mesulam. Schizophrenia and the brain. New England Journal of Medicine, 322(12):842-845, 1990. [6] P.S. Goldman and W.J .H. Nauta. Columnar distribution of cortica-cortical fibers in the frontal, association, limbic and motor cortex of the developing rhesus monkey. Brain Res., 122:393-413,1977. [7] L. R. Squire. Memory and the hippocampus: A synthesis from findings with rats, monkeys, and humans. Psychological Review, 99:195-231, 1992. [8] M.V. Tsodyks and M.V. Feigel'man. The enhanced storage capacity in neural networks with low activity level. Europhys. Lett., 6:101 - 105, 1988. [9] D. Horn and E. Ruppin. Synaptic compensation in attractor neural networks: Modeling neuropathological findings in schizophrenia. Neural Computation, page To appear, 1994. [10] W.T. Carpenter and R.W. Buchanan. Schizophrenia. New England Journal of Medicine, 330:10, 1994. [11] P. Seeman. Schizophrenia as a brain disease: The dopamine receptor story. Arch. Neurol., 50:1093-1095, 1993. [12] S. T. DeKosky and S.W. Scheff. Synapse loss in frontal cortex biopsies in alzheimer's disease: Correlation with cognitive severity. Ann. Neurology, 27(5):457-464, 1990. [13] Y. Jin, S.G. Potkin, D. Rice, and J. Sramek et. al. Abnormal EEG responses to photic stimulation in schizophrenic patients. Schizophrenia Bulletin, 16(4):627634, 1990. [14] R.E. Hoffman and J .A. Rapaport. A psycholoinguistic study of auditory /verbal hallucinations: Preliminary findings. In David A. and Cutting J. , editors, The Neuropsychology of Schizophrenia. Erlbaum, 1993.	1994	98
996	Correlation and Interpolation Networks for Real-time Expression Analysis/Synthesis. Trevor Darrell, Irfan Essa, Alex Pentland Perceptual Computing Group MIT Media Lab Abstract We describe a framework for real-time tracking of facial expressions that uses neurally-inspired correlation and interpolation methods. A distributed view-based representation is used to characterize facial state, and is computed using a replicated correlation network. The ensemble response of the set of view correlation scores is input to a network based interpolation method, which maps perceptual state to motor control states for a simulated 3-D face model. Activation levels of the motor state correspond to muscle activations in an anatomically derived model. By integrating fast and robust 2-D processing with 3-D models, we obtain a system that is able to quickly track and interpret complex facial motions in real-time. 1 INTRODUCTION An important task for natural and artificial vision systems is the analysis and interpretation of faces. To be useful in interactive systems and in other settings where the information conveyed is of a time critical nature, analysis of facial expressions must occur quickly, or be oflittle value. However, many of the traditional computer vision methods for estimating and modeling facial state have proved difficult to perform fast enough for interactive settings. We have therefore investigated neurally inspired mechanisms for the analysis of facial expressions. We use neurally plausible distributed pattern recognition mechanisms to make fast and robust assessments of facial state, and multi-dimensional interpolation networks to connect these measurements to a facial model. There are many potential applications of a system for facial expression analysis. Person910 Trevor Darrell. Irfan Essa. Alex Pentland alized interfaces which sense a users emotional state, ultra-low bitrate video conferencing which sends only facial muscle activations, as well as the enhanced recognition systems mentioned above. We have focused on a application in computer graphics which stresses both the analysis and synthesis components of our system: interactive facial animation. In the next sections we develop a computational framework for neurally plausible expression analysis, and the connection to a physically-based face model using a radial basis function method. Finally we will show the results of these methods applied to the interactive animation task, in which an computer graphics model of a face is rendered in real time, and matches the state of the users face as sensed through a conventional video camera. 2 EXPRESSION MODELINGffRACKING The modeling and tracking of expressions and faces has been a topic of increasing interest recently. In the neural network field, several successful models of character expression modeling have been developed by Poggio and colleagues. These models apply multidimensional interpolation techniques, using the radial basis function method, to the task of interpolating 2D images of different facial expression. Librande [4] and Poggio and Brunelli [9] applied the Radial Basis Function (RBF) method to facial expression modeling, using a line drawing representation of cartoon faces. In this model a small set of canonical expressions is defined, and intermediate expressions constructed via the interpolation technique. The representation used is a generic "feature vector", which in the case of cartoon faces consists of the contour endpoints. Recently, Beymer et al. [1] extended this approach to use real images, relying on optical flow and image warping techniques to solve the correspondence and prediction problems, respectively. RBF-based techniques have the advantage of allowing for the efficient and fast computation of intermediate states in a representation. Since the representation is simple and the interpolation computation straight-forward, real-time implementations are practical on conventional systems. These methods interpolate between a set of 2D views, so the need for an explicit 3-D representation is sidestepped. For many applications, this is not a problem, and may even be desirable since it allows the extrapolation to "impossible" figures or expressions, which may be of creative value. However, for realistic rendering and recognition tasks, the use of a 3-D model may be desirable since it can detect such impossible states. In the field of computer graphics, much work has been done on on the 3-D modeling of faces and facial expression. These models focus on the geometric and physical qualities of facial structure. Platt and B adler [7], Pieper [6], Waters [11] and others have developed models of facial structure, skin dynamics, and muscle connections, respectively, based on available anatomical data. These models provide strong constraints for the tracking of feature locations on a face. Williams et. al. [12] developed a method in which explicit feature marks are tracked on a 3-D face by use of two cameras. Terzopoulos and Waters [10] developed a similar method to track linear facial features, estimate corresponding parameters of a three dimensional wireframe face model, and reproduce facial expression. A significant limitation of these systems is that successful tracking requires facial markings. Essa and Pentland [3] applied optical flow methods (see also Mase [5]) for the passive tracking of facial motion, and integrated the flow measurement method into a dynamic system model. Their method allowed for completely passive estimation of facial expressions, using all the constraints provided by a full 3-D model of facial expression. Both the view based method of Beymer et. al. and the 3-D model of Essa and Pentland rely Correlation and Interpolation Networks for Real-Time Expression Analysis/Synthesis 911 ( (b) Figure 1: (a) Frame of video being processed to extract view model. Outlined rectangle indicates area of image used for model. (b) View models found via clustering method on training sequence consisting of neutral, smile, and surprise expressions. on estimates of optic flow, which are difficult to compute reliably, especially in real-time. Our approach here is to combine interpolated view-based measurements with physically based models, to take advantage of the fast interpolation capability of the RBF and the powerful constraints imposed by physically based models. We construct a framework in which perceptual states are estimated from real video sequences and are interpolated to control the motor control states of a physically based face model. 3 VIEW-BASED FACE PERCEPTION To make reliable real-time measurements of a complex dynamic object, we use a distributed representation corresponding to distinct views of that object. Previously, we demonstrated the use of this type of representation for the tracking and recognition of hand gestures [2]. Like faces, hands are complex objects with both non-rigid and rigid dynamics. Direct use of a 3-D model for recognition has proved difficult for such objects, so we developed a view-based method for representation. Here we apply this technique to the problem of facial representation, but extend the scheme to connect to a 3-D model for high-level modeling and generation/animation. With this, we gain the representational power and constraints implied by the 3-D model as a high-level representation; however the 3-D model is only indirectly involved in the perception stage, so we can still have the same speed and reliability afforded by the view-based representation. In our method each view characterizes a particular aspect or pose of the object being represented. The view is stored iconically, that is, it is a literal image or template (but with some point-wise statistics) of the appearance of the object in that aspect or pose. A match criteria is defined between views and input images; usually a normalized correlation function is used, but other criteria are possible. An input image is represented by the ensemble of match scores from that image to the stored views. To achieve invariance across a range of transformations, for example translation, rotation and/or scale, units which compute the match score for each view are replicated at different values of each transformation. I The unit which has maximal response across all values of the transformation is selected, and the ensemble response of the view units which share the 1 In a computer implementation this exhaustive sampling may be impractical due to the number of units needed, in which case this stage may be approximated by methods which are hybrid sampling/search methods. 912 Trevor Darrell, Irfan Essa, Alex Pentland same transformation values as the selected unit is stored as the representation for the input image. We set the perceptual state X to be a vector containing this ensemble response. If the object to be represented is fully known a priori, then methods to generate views can be constructed by analysis of the aspect graph if the object is polyhedral, or in general by rendering images of the object at evenly spaced rotations. However, in practice good 3-D models that are useful for describing image intensity values are rare2, so we look to data-driven methods of acquiring object views. As described in [2] a simple clustering algorithm can find a set of views that "span" a training sequence of images, in the sense that for each image in the sequence at least one view is within some threshold similarity to that image. The algorithm is as follows. Let V be the current set of views for an object (initially one view is specified manually). For each frame I of a training sequence, if at least one v E V has a match value M (v, 1) that is greater than a threshold (J, then no action is performed and the next frame is processed. If no view is close, then I is used to construct a new view which is added to the view set. A view v' is created using a window of I centered at the location in the previous image where the closest view was located. (All views usually share the same window size, determined by the initial view.) The view set is then augmented to include the new view: V = V u v'. This algorithm will find a set of views which well-characterizes an object across the range of poses or expressions contained in the training sequence. For example, in the domain of hand gestures, inputing a training sequence consisting of a waving hand will yield views which contain images of the hand at several different rotations. In the domain of faces, when input a training sequence consisting of a user performing 3 different expressions, neutral, smile, and surprise, this algorithm (with normalized correlation and (J = 0.7) found three views corresponding to these expressions to represent the face, as shown in Figure l(b). These 3 views serve as a good representation for the face of this user as long as his expression is similar to one in the training set. The major advantage of this type of distributed view-based representation lies in the reduction of the dimensionality of the processing that needs to occur for recognition, tracking, or control tasks. In the gesture recognition domain, this dimensionality reduction allowed for conventional recognition strategies to be applied successfully and in real-time, on examples where it would have been infeasible to evaluate the recognition criteria on the full signal. In the domain explored in this paper it makes the interpolation problem of much lower order: rather than interpolate from thousands of input dimensions as would be required when the input is the image domain, the view domain for expression modeling tasks typically has on the order of a dozen dimensions. 4 3-D MODELINGIMOTOR CONTROL To model the structure of the face and the dynamics of expression performance, we use the physically based model of Essa et. al. This model captures how expressions are generated by muscle actuations and the resulting skin and tissue deformations. The model is capable of controlled nonrigid deformations of various facial regions, in a fashion similar to how humans generate facial expressions by muscle actuations attached to facial tissue. Finite Element methods are used to model the dynamics of the system. 2 As opposed to modeling forces and shape deformations, for which 3-D models are useful and indeed are used in the method presented here. Correlation and Interpolation Networks for Real-Time Expression Analysis/Synthesis 9 J 3 (a) 1 (b) 900. 8 0<5'> core 700 l\I1odel ParalTleters (e) 0.2 . o ActuatIons -0.2 (d) Figure 2: (a) Face images used as input, (b) normalized correlation scores X(t) for each view model, (c) resulting muscle control parameters Y(t), (d) rendered images of facial model corresponding to muscle parameters. 914 Trevor Darrell, Irfan Essa, Alex Pentland This model is based on the mesh developed by Platt and Badler [7], extended into a topologically invariant physics-based model through the addition of a dynamic skin and muscle model [6, 11]. These methods give the facial model an anatomically-based facial structure by modeling facial tissue/skin, and muscle actuators, with a geometric model to describe force-based deformations and control parameters. The muscle model provides us with a set of control knobs to drive the facial state, defined to be a vector Y. These serve to define the motor state of the animated face. Our task now is to connect the perceptual states of the observed face to these motor states. 5 CONNECTING PERCEPTION WITH ACTION We need to establish a mapping from the perceptual view scores to the appropriate muscle activations on the 3-D face model. To do this, we use multidimensional interpolation strategies implemented in network form. Interpolation requires a set of control points or exemplars from which to derive the desired mapping. Example pairs of real faces and model faces for different expressions are presented to the interpolation method during a training phase. This can be done in one of two ways, with either a user-driven or model-driven paradigm. In the model-driven case the muscle states are set to generate a particular expression by an animator/programmer and then the user is asked to make the equivalent expression. The resulting perceptual (view-model) scores are then recorded and paired with the muscle activation levels. In the user-driven case, the user makes an expression of hislher own choosing, and the optic flow method of Essa et. al. is used to derive the corresponding muscle activation levels. The modeldriven paradigm is simpler and faster, but the user-driven paradigm yields more detailed and authentic facial expressions. We use the Radial Basis Function (RBF) method presented in [8], and define the interpolated motor controls to be a weighted sum of radial functions centered at each example: n Y = I:ci9(X - Xi) (1) i=1 where Y are the muscle states, X are the observed view-model scores, Xi are the example scores, 9 is an RBF (and in our case was simply a linear ramp 9(§) = II§II), and the weights Ci are computed from the example motor values Yi using the pseudo-inverse method [8]. 6 INTERACTIVE ANIMATION SYSTEM The correlation network, RBF interpolator, and facial model described above have been combined into a single system for interactive animation. The entire system can be updated at over 5 Hz, using a dedicated single board accelerator to compute the correlation network, and an SGI workstation to render the facial mesh. · Here we present two examples of the processing performed by the system, using different strategies for coupling perceptual and motor state. Figure 2 illustrates one example of real-time facial expression tracking using this system, using a full-coupling paradigm. Across the top, labeled (a), are five frames of a video sequence of a user making a smile expression. This was one of the expressions used in the training sequence for the view models shown in Figure 1 (b), so they were applicable to be Correlation and Interpolation Networks for Real-Time Expression Analysis/Synthesis 915 (a) (b) Figure 3: (a) Processing of video frame with independent view model regions for eyes, eye-brows, and mouth region. (b) Overview shot of full system. User is on left, vision system and camera is on right, and animated face is in the center of the scene. The animated face matches the state of the users face in real-time, including eye-blinks (as is the case in this shot.) used here. Figure 2(b) shows the correlation scores computed for each of the 3 view models for each frame of the sequence. This constituted the perceptual state representation, X(t). In this example the full face is coupled with the full suite of motor control parameters. An RBF interpolator was trained using perceptual/motor state pairs for three example full-face expressions (neutral, smile, surprise); the resulting (interpolated) motor control values, yet), for the entire sequence are shown in Figure 2(c). Finally, the rendered facial mesh for five frames of these motor control values is shown in Figure 2( d). When there are only a few canonical expressions that need be tracked/matched, this full-face template approach is robust and simple. However if the user wishes to exercise independent control of the various regions of the face, then the full coupling paradigm will be overly restrictive. For example, if the user trains two expressions, eyes closed and eyes open, and then runs the system and attempts to blink only one eye, the rendered face will be unable to match it. (In fact closing one eye leads to the rendered face half-closing both eyes.) A solution to this is to decouple the regions of the face which are independent geometrically (and to some degree, in terms of muscle effect.) Under this paradigm, separate correlation networks are computed for each facial regions, and multiple RBF interpolations are performed for each system. Each interpolator drives a distinct subset of the motor state vector. Figure 3(a) shows the regions used for decoupled local templates. In these examples independent regions were used for each eye, eyebrow, and the mouth region. Finally, figure 3 (b) shows a picture of the set-up of the system as it is being run in an interactive setting. The animated face mimics the facial state of the user, matching in real time the position of the eyes, eyelids, eyebrows and mouth of the user. In the example shown in this picture, the users eyes are closed, so the animated face's eyes are similarly closed. Realistic performance of animated facial expressions and gestures are are possible 916 Trevor Darrell, Irfan Essa, Alex Pentland through this method, since the timing and levels of the muscle activations react immediately to changes in the users face. 7 CONCLUSION We have explored the use of correlation networks and Radial Basis Function techniques for the tracking of real faces in video sequences. A distributed view-based representation is computed using a network of replicated normalized correlation units, and offers a fast and robust assesment of perceptual state. 3-D constraints on facial shape are achieved through the use of a an anatomically derived facial model, whose muscle activations are controled via interolated perceptual states using the RBF method. With this framework we have been able to acheive the fast and robust analysis and synthesis of facial expressions. A modeled face mimics the expression of a user in real-time, using only a conventional video camera sensor and no special marking on the face of the user. This system has promise as a new approach in the interactive animation, video tele-conferencing, and personalized interface domains. References [1] D. Beymer, A. Shashua, and T. Poggio, Example Based Image Analysis and Synthesis, MIT AI Lab TR-1431, 1993. [2] T. Darrell and A. Pentland. Classification of Hand Gestures using a View-Based Distributed Representation In NIPS-6, 1993. [3] I. Essa and A. Pentland. A vision system for observing and extracting facial action parameters. In Proc. IEEE Conf. Computer Vision and Pattern Recognition, 1994. [4] S. Librande. Example-based Character Drawing. S.M. Thesis, Media Arts and SciencelMedia Lab, MIT. 1992 [5] K. Mase. Recognition of facial expressions for optical flow. IEICE Transactions, Special I ssue on Computer Vision and its Applications, E 74( 1 0), 1991. [6] S. Pieper, J. Rosen, and D. Zeltzer. Interactive graphics for plastic surgery: A task level analysis and implementation. Proc. Siggraph-92, pages 127-134, 1992. [7] S. M. Platt and N. I. Badler. Animating facial expression. ACM SIGGRAPH Conference Proceedings, 15(3):245-252,1981. [8] T. Poggio and F. Girosi. A theory of networks for approximation and learning. MIT AI Lab TR-1140, 1989. [9] T. Poggio and R. Brunelli, A Novel Approach to Graphics, MIT AI Lab TR- 1354. 1992. [10] D. Terzopoulus and K. Waters. Analysis and synthesis offacial image sequences using physical and anatomical models. IEEE Trans. PAMI, 15(6):569-579, June 1993. [11] K. Waters and D. Terzopoulos. Modeling and animating faces using scanned data. The Journal of Visualization and Computer Animation, 2: 123-128, 1991. [12] L. Williams. Performance-driven facial animation. ACM SIGGRAPH Conference Proceedings, 24(4):235-242, 1990.	1994	99
997	Learning to Predict Visibility and Invisibility from Occlusion Events Jonathan A. Marshall Richard K. Alley Robert S. Hubbard Department of Computer Science, CB 3175, Sitterson Hall University of North Carolina, Chapel Hill, NC 27599-3175, U.S.A. marshall@cs.unc.edu, 919-962-1887, fax 919-962-1799 Abstract Visual occlusion events constitute a major source of depth information. This paper presents a self-organizing neural network that learns to detect, represent, and predict the visibility and invisibility relationships that arise during occlusion events, after a period of exposure to motion sequences containing occlusion and disocclusion events. The network develops two parallel opponent channels or "chains" of lateral excitatory connections for every resolvable motion trajectory. One channel, the "On" chain or "visible" chain, is activated when a moving stimulus is visible. The other channel, the "Off" chain or "invisible" chain, carries a persistent, amodal representation that predicts the motion of a formerly visible stimulus that becomes invisible due to occlusion. The learning rule uses disinhibition from the On chain to trigger learning in the Off chain. The On and Off chain neurons can learn separate associations with object depth ordering. The results are closely related to the recent discovery (Assad & Maunsell, 1995) of neurons in macaque monkey posterior parietal cortex that respond selectively to inferred motion of invisible stimuli. 1 INTRODUCTION: LEARNING ABOUT OCCLUSION EVENTS Visual occlusion events constitute a major source of depth information. Yet little is known about the neural mechanisms by which visual systems use occlusion events to infer the depth relations among visual objects. What is the structure of such mechanisms? Some possible answers to this question are revealed through an analysis of learning rules that can cause such mechanisms to self-organize. Evidence from psychophysics (Kaplan, 1969; Nakayama & Shimojo, 1992; Nakayama, Shimojo, & Silverman, 1989; Shimojo, Silverman, & Nakayama, 1988, 1989; Yonas, Craton, & Thompson, 1987) and neurophysiology (Assad & Maunsell, 1995; Frost, 1993) suggests that the process of determining relative depth from occlusion events operates at an early stage of visual processing. Marshall (1991) describes evidence that suggests that the same early processing mechanisms maintain a representation of temporarily occluded objects for some amount Learning to Predict Visibility and Invisibility from Occlusion Events 817 of time after they have disappeared behind an occluder, and that these representations of invisible objects interact with other object representations, in much the same manner as do representations of visible objects. The evidence includes the phenomena of kinetic subjective contours (Kellman & Cohen, 1984), motion viewed through a slit (Parks' Camel) (Parks, 1965), illusory occlusion (Ramachandran, Inada, & Kiama, 1986), and interocular occlusion sequencing (Shimojo, Silverman, & Nakayama, 1988). 2 PERCEPTION OF OCCLUSION AND DISOCCLUSION EVENTS: AN ANALYSIS The neural network model exploits the visual changes that occur at occlusion boundaries to form a mechanism for detecting and representing object visibility/invisibility information. The set of learning rules used in this model is an extended version of one that has been used before to describe the formation of neural mechanisms for a variety of other visual processing functions (Hubbard & Marshall, 1994; Marshall, 1989, 1990ac, 1991, 1992; Martin & Marshall, 1993). Our analysis is derived from the following visual predictivity principle, which may be postulated as a fundamental principle of neural organization in visual systems: Visual systems represent the world in terms of predictions of its appearance, and they reorganize themselves to generate better predictions. To maximize the correctness and completeness of its predictions, a visual system would need to predict the motions and visibility/invisibility of all objects in a scene. Among other things, it would need to predict the disappearance of an object moving behind an occluder and the reappearance of an object emerging from behind an occluder. A consequence of this postulate is that occluded objects must, at some level, continue to be represented even though they are invisible. Moreover, the representation of an object must distinguish whether the object is visible or invisible; otherwise, the visual system could not determine whether its representations predict visibility or invisibility, which would contravene the predictivity principle. Thus, simple single-channel prediction schemes like the one described by Marshall (1989, 1990a) are inadequate to represent occlusion and disocclusion events. 3 A MODEL FOR GROUNDED LEARNING TO PREDICT VISIBILITY AND INVISIBILITY The initial structure of the Visible/Invisible network model is given in Figure 1A. The network self-organizes in response to a training regime containing many input sequences representing motion with and without occlusion and disocclusion events. After a period of self-organization, the specific connections that a neuron receives (Figure 1B) determine whether it responds to visible or invisible objects. A neuron that responds to visible objects would have strong bottom-up input connections, and it would also have strong time-delayed lateral excitatory input connections. A neuron that responds selectively to invisible objects would not have strong bottomup connections, but it would have strong lateral excitatory input connections. These lateral inputs would transmit to the neuron evidence that a previously visible object existed. The neurons that respond to invisible objects must operate in a way that allows lateral input excitation alone to activate the neurons supraliminally, in the absence of bottom-up input excitation from actual visible objects. 4 SIMULATION OF A SIMPLIFIED NETWORK 4.1 INITIAL NETWORK STRUCTURE The simulated network, shown in Figure 2, describes a simplified onedimensional subnetwork (Marshall & Alley, 1993) of the more general twodimensional network. Layer 1 is restricted to a set of motion-sensitive neurons corresponding to one rightward motion trajectory. The L+ connections in the simulation have a signal transmission latency of one time unit. Restricting the lateral connections to a single time delay and to a single direction limits the simulation to representing a single speed and direction of motion; these results are therefore preliminary. This restriction reduced the number of connections and made the simulation much faster. 818 J. A. MARSHALL, R. K. ALLEY, R. S. HUBBARD (A) 0 0 ,'9 (B) . Figure 1: Model of a self-organized occlusion-event detector network. (A) Network is initially organized nonspecifically, so that each neuron receives roughly homogeneous input connections: feedforward, bottom-up excitatory ("B+") connections from a preprocessing stage of motion-tuned neurons (bottom-up solid arrows), lateral inhibitory ("L-") connections (dotted arrows), and timedelayed lateral excitatory ("L+") connections (lateral solid arrows). (B) After exposure during a developmental period to many motion sequences containing occlusion and disocclusion events, the network learns a highly specific connection structure. The previously homogeneous network bifurcates into two parallel opponent channels for every resolvable motion trajectory: some neurons keep their bottom-up connections and others lose them. The channels for one trajectory are shown. Neurons from the two opponent channels are strongly linked by lateral inhibitory connections (dotted arrows). Time-delayed lateral excitatory connections cause stimulus information (priming excitation, or "prediction signals") to propagate along the channels. Layer 2 Layer 1 Figure 2: Simula.tion results. (Left) Simulated network structure before training. Neurons are wired homogeneously from the input la.yer. (Right) After training, some of the neurons lose their bottom-up input connections. 4.2 USING DISINHIBITION TO CONTROL THE LEARNING OF OCCLUSION RELATIONS This paper describes one method for learning occlusion relations. Other methods may also work. The method involves extending the EXIN (excitatory+inhibitory) learning scheme described by Marshall (1992, 1995). The EXIN scheme uses a variant of a Hebb rule to govern learning in the bottom-up and timedelayed lateral excitatory connections, plus an anti-Hebb rule to govern learning in the lateral inhibitory connections. The EX IN system was extended by letting inhibitory connections exert a disinhibitory effect under certain regulated conditions. The disinhibition rule was chosen because it constitutes a simple way that the unexpected failure of a neuron to become activated (e.g., when an object disappears behind an occluder) can cause some other neuron to become activated. That other neuron can then learn, becoming selective for invisible object motion. Thus, the representations of visible objects are protected from losing their bottom-up input connections during occlusion events. In this way, the network can learn separate representations for visible and invisible stimuli. The representations of invisible objects are allowed to develop only to the extent that the neurons representing visible objects explicitly disclaim the "right" to represent the objects. These properties prevent the network from losing complete grounded contact with actual bottom-up visual input, while at the same time allowing some neurons to lose their direct bottom-up input connections. The disinhibition produces an excitatory response at the target neurons. Disinhibition is generated according to the following rule: When a neuron has strong, Learning to Predict Visibility and Invisibility from Occlusion Events 819 active lateral excitatory input connections and strong but inactive bottom-up input connections, then it tends to disinhibit the neurons to which it projects inhibitory connections. This implements a type of differencing operation between lateral and bottom-up excitation. Because the disinhibition tends to excite the recipient neurons, it causes one (or possibly more) of the recipient neurons to become active and thereby enables that neuron to learn. The lateral excitation that a neuron receives can be viewed as a prediction of the neuron's activation. If that prediction is not matched by actual bottom-up excitation, then a shortfall (prediction failure) has occurred, probably indicating an occlusion event. Each neuron's disinhibition input was combined with its bottom-up excitatory input and its lateral excitatory input to form a total excitatory input signal. Either bottom-up excitation or disinhibition alone could contribute toward a neuron's excitation. However, lateral excitation could merely amplify the other signals and could not alone excite the neuron. This prevented neurons from learning in response to lateral excitation alone. 4.3 DISINH.IBITION LETS THE NETWORK LEARN TO RESPOND TO INVISIBLE OBJECTS During continuous motion sequences, without occlusion or disocclusion, the system operates similarly to a system with the standard EXIN learning rules (Marshall, 1990b, 1995): lateral excitatory "chains" of connections are learned across sequences of neurons along a motion trajectory. Marshall (1990a) showed that such chains form in 2-D networks with multiple speeds and multiple directions of motion. During occlusion events, some predictive lateral excitatory signals reach neurons that have strong but inactive bottom-up excitatory connections. The neurons reached by this excitation pattern disinhibit, rather than inhibit, their competitor neurons. Over the course of many occlusion events, such neurons become increasingly selective for the inferred motion of an invisible object: their bottom-up input connections weaken, and their lateral inhibitory input connections strengthen. More than one neuron receives L+ signals after every neuron activation; the recipients of each neuron's L+ output connections represent the (learned) possible sequents of the neuron's activation. But at most one of those sequents actually receives both B+ and L+ signals: the one that corresponds to the actual stimulus. This winner neuron receives the disinhibition from the other neurons receiving L+ excitation; its competitive advantage over the other neurons is thus reinforced. 4.4 SIMULATION TRAINING The sequences of input training data consisted of a single visual feature moving with constant velocity across the I-D visual field. When this stimulus was visible, its presence was indicated by strong activation of an input neuron in Layer 1. While occluded, the stimulus would produce no activation in Layer 1. The stimulus occasionally disappeared "behind" an occluder and reappeared at a later time and spatial position farther along the same trajectory. After some duration, the stimulus was removed and replaced by a new stimulus. The starting positions and lifetimes of the stimuli and occluders were varied randomly within a fixed range. The network was trained for 25,000 input pattern presentations. The stability of the connection weights was verified by additional training for 50,000 presentations. 4.5 SIMULATION RESULTS: ARCHITECTURE The second stage of neurons gradually underwent a self-organized bifurcation into two distinct pools of neurons, as shown in Figure 2B. These pools consist of two parallel opponent channels or "chains" of lateral excitatory connections for every resolvable motion trajectory. One channel, the "On" chain or "visible" chain, was active when a moving stimulus became visible. The other channel, the "Off" chain or "invisible" chain, was active when a formerly visible stimulus became invisible. The model is thus named the Visible/Invisible model. The bifurcation may be analogous to the activity-dependent stratification of cat retinal gan~lion cells into separate On and Off layers, described by Bodnarenko and Chalupa (1993). 4.6 SIMULATION RESULTS: OPERATION The On chain carries a predictive modal representation of the visible stimulus. The Off chain carries a persistent, amodal representation that predicts the motion 820 J. A. MARSHALL, R. K. ALLEY, R. S. HUBBARD of the invisible stimulus. The shading of the neurons in Figure 3 shows the neuron acti vat ions of the final, trained network simulation during an occlusion-disocclusion sequence. The following noteworthy behaviors were observed in the test. • When the stimulus was visible, it was represented by activation in the On channel. • When the stimulus became invisible, its representation was carried in the Off channel. The Off channel did not become active until the visible stimulus disappeared. • The activations representing the visible stimulus became stronger (toward an asymptote) at successive spatial positions, because of the propagation of accumulating evidence for the presence of the stimulus (Martin & Marshall, 1993). • The activation representing the invisible stimulus decayed at successive spatial positions. Thus, representations of invisible stimuli did not remain active indefinitely. • When the stimulus reappeared (after a sufficiently brief occlusion), its activation in the On channel was greater than its initial activation in the On channel. Thus, the representation carried across the Off channel helps maintain the perceptual stability of the stimulus despite its being temporarily occluded along parts of its trajectory. fflIfff f'ffffTff Lay.,' ~ •. ( '.L ..... ; •. .. ; .. ::::.·.·.·.· •. i .. '.;.; .. ': .. , .• ' .. >:: ~:: : . ~', .". '." . -, .. Layer 1 :<:P}::;";;:;':'r Figure 3: Simulated network operation after learning. The learning pr()c~d~'re cahses the representation of each trajectory to split into two parallel opponent channels. The Visible and Invisible channel pair for a single trajectory are shown. The display has been arranged so that all the Visible channel neurons are on the same row (Layer 2, lower row); likewise the Invisible channel neurons (Layer 2, upper row). Solid arrows indicate excitatory connections. Gray arrows indicate lateral inhibitory connections. (Left) The network's responses to an unbroken rightward motion of the stimulus are shown. The activities of the network at successive moments in time have been combined into a single network display; each horizontal position in the figure represents a different moment in time as well as a different position in the network. The stimulus successively activates motion detectors (solid circles) in Layer 1. The activation of the responding neuron in the second layer builds toward an asymptote, reaching full activation by the fourth frame. (Right) The network's responses to a broken (occluded) rightward motion sequence are shown. When the stimulus reaches the region indicated by gray shading, it disappears behind a simulated occluder. The network responds by successively activating neurons in the Invisible channel. When the stimulus emerges from behind the occluder (end of gray shading), it is again represented by activation in the Visible channel. 5 DISCUSSION 5.1 PSYCHOPHYSICAL ISSUES AND PREDICTIONS Several visual phenomena (Burr, 1980; Piaget, 1954; Shimojo, Silverman, & Nakayama, 1988) support the notion that early processing mechanisms maintain a dynamic representation of temporarily occluded objects for some amount of time after they disappear (Marshall, 1991). In general, the duration of such representations should vary as a function of many factors, including top-down cognitive expectations, stimulus complexity, and Gestalt grouping. 5.2 ALTERNATIVE MECHANISMS Another model besides the Visible/Invisible model was studied extensively: a Visible/Virtual system, which would develop some neurons that respond to visible objects and others that respond to both visible and invisible objects (Le., to "virtual" objects). There is a functional equivalence between such a Visible/.Virtual system and a Visible/Invisible system: the same information about visibllity and invisibility can be determined by examining the activations of the neurons. Activity in a Virtual channel neuron, paired with inactivity in a corresponding Visible channel neuron, would indicate the presence of an invisible stimulus. Learning to Predict Visibility and Invisibility from Occlusion Events 821 5.3 NEUROPHYSIOLOGICAL CORRELATES Assad and Maunsell (1995) recently described their remarkable discovery ofneurons in macaque monkey posterior parietal cortex that respond selectively to the inferred motion of invisible stimuli. This type of neuron responded more strongly to the disappearance and reappearance of a stimulus in a task where the stimulus' "inferred" trajectory would pass through the neuron's receptive field than in a task where the stimulus would disappear and reappear in the same position. Most of these neurons also had a strong off-response, which in the present models is closely correlated with inferred motion. Thus, the results of Assad and Maunsell (1995) are more directly consistent with the Visible/Virtual model than with the Visible/Invisible model. Although this paper describes only one of these models, both models merit investigation. 5.4 LEARNING ASSOCIATIONS BETWEEN VISIBILITY AND RELATIVE DEPTH The activation of neurons in the Off channels is highly correlated with the activation of other neurons elsewhere in the visual system, specifically neurons whose activation indicates the presence of other objects acting as occluders. Simple associative Hebb-type learning lets such occluder-indicator neurons and the Off channel neurons gradually establish reciprocal excitatory connections to each other. After such reciprocal excitatory connections have been learned, activation of occluder-indicator neurons at a given spatial position causes the network to favor the Off channel in its predictions - i.e., to predict that a moving object will be invisible at that position. Thus, the network learns to use occlusion information to generate better predictions of the visibility/invisibility of objects. Conversely, the activation of Off channel neurons causes the occluder-indicator neurons to receive excitation. The disappearance of an object excites the representation of an occluder at that location. If the representation of the occluder was not previously activated, then the excitation from the Off channel may even be strong enough to activate it alone. Thus, disappearance of moving visual objects constitutes evidence for the presence of an inferred occluder. These results will be described in a later paper. 5.5 LIMITATIONS AND FUTURE WORK The Visible/Invisible model presented in this paper describes SOme of the processes that may be involved in detecting and representing depth from occlusion events. There are other major issues that have not been addressed in this paper. For example, how can the system handle real 2-D or 3-D objects, composed of many visual features grouped together across space, instead of mere point stimuli? How can it handle partial occlusion of objects? How can it handle nonlinear trajectories? How exactly can the associative links between occluding and occluded objects be formed? How can it handle transparency? 6 CONCLUSIONS Perception of relative depth from occlusion events is a powerful, useful, but poorlyunderstood capability of human and animal visual systems. We have presented an analysis based on predictivity: a visual system that can predict the visibility /invisibility of objects during occlusion events possesses (ipso facto) a good representation of relative depth. The analysis implies that the representations for visible and invisible objects must be distinguishable. We have implemented a model system in which distinct representations for visible and invisible features self-organize in response to exposure to motion sequences containing simulated occlusion and disocclusion events. When a moving feature fails to appear approximately where and when it is predicted to appear, the mismatch between prediction and the actual image triggers an unsupervised learning rule. Over many motions, the learning leads to a bifurcation of a network layer into two parallel opponent channels of neurons. Prediction signals in the network are carried along motion trajectories by specific chains of lateral excitatory connections. These chains also cause the representation of invisible features to propagate for a limited time along the features' trajectories. The network uses shortfall (differencing) and disinhibition to maintain grounding of the representations of invisible features. 822 J. A. MARSHALL, R. K. ALLEY, R. S. HUBBARD Acknowledgements Supported in part by ONR (NOOOl4-93-1-0208), NEI (EY09669), a UNC-CH Junior Faculty DeveloP!llent Award, an ORAU Junior Faculty Enhancement Award from Oak Ridge Associated Universities, the Univ. of Minnesota Center for Research in Learning, Perception, and Cognition, NICHHD (HD-07151), and the Minnesota Supercomputer Institute. We thank Kevin Martin, Stephen Aylward, Eliza Graves, Albert Nigrin, Vinay Gupta, Geor~e Kalarickal", Charles SchmItt, Viswanath Srikanth, David Van Essen, Christof Koch, and Ennio Mingolla for valuable discussions. References Assad JA, Maunsell JHR (1995) Neuronal correlates of inferred motion in macaque posterior parietal cortex. Nature 373:518-521. Bodnarenko SR, Chalupa LM (1993) Stratification of On and Off ganglion cell dendrites depends on glutamate-mediated afferent activity in the developing retina. Nature 364:144-146. Burr D (1980) Motion smear. Nature 284:164-165. Frost BJ (1993) Subcortical analysis of visual motion: Relative motion, figure-ground discrimination and induced optic flow. Visual Motion and Its Role in the Stabilization of Gaze, Miles FA, Wallman J (Eds). Amsterdam: Elsevier Science, 159-175. Hubbard RS, Marshall JA (1994) Self-organizing neural network model of the visual inertia phenomenon in motion perception. Technical Report 94-001, Department of Computer Science, University of North Carolina at Chapel Hill. 26 pp. Kaplan GA (1969) Kinetic disruption of optical texture: The perception of depth at an edge. Perception fj Psychophysics 6:193-198. Kellman PJ, Cohen MH (1984) Kinetic subjective contours. Perception fj Psychophysics 35:237-244. Marshall JA (1989) Self-organizing neural network architectures for computing visual depth from motion parallax. Proceedings of the International Joint Conference on Neural Networks, Washington DC, 11:227-234. Marshall JA (1990a) Self-organizing neural networks for perception of visual motion. Neural Networks 3:45-74. Marshall JA (1990b) A self-organizing scale-sensitive neural network. Proceedings of the International Joint Conference on Neural Networks, San Diego, CA, 111:649-654. Marshall JA (1990c) Adaptive neural methods for multiplexing oriented edges. Intelligent Robots and Computer Vision IX: Neural, Biological, and 3-D Methods, Casasent DP (Ed), Proceedings of the SPIE 1382, Boston, MA, 282- 291. Marshall JA (1991) Challenges of vision theory: Self-organization of neural mechanisms for stable steering of obiect-g!ouping data in visual motion perception. Stochastic and Neural Methods in Signal Processing, Image Processing, and Computer Vision, Chen SS (Ed), Proceedings of the SPIE 1569, San Diego, CA, 200-215. Marshall JA (1992) Unsupervised learning of contextual constraints in neural networks for simultaneous visual processing of multiple objects. Neural and Stochastic Methods in Image and Signal Processing, Chen SS (Ed), Proceedings of the SPIE 1766, San Diego, CA, 84-93. Marshall JA (1995) Adaptive perceptual pattern recognition by self-organizing neural networks: Context, uncertainty, multiplicity, and scale. Neural Networks 8:335-362. Marshall JA, Alley RK (1993) A self-organizing neural network that learns to detect and represent visual depth from occlusion events. Proceedings of the AAAI Fall Symposium on Machine Learning and Computer Vision, Bowyer K, Hall L (Eds), 70-74. Martin KE, Marshall JA (1993) Unsmearing visual motion: Development of long-range horizontal intrinsic connections. Advances in Neural Information Processing Systems, 5, Hanson SJ, Cowan JD, Giles CL (Eds). San Mateo, CA: Morgan Kaufmann Publishers, 417-424. Nakayama K, Shimojo S (1992) Experiencing and perceiving visual surfaces. Science 257:1357-1363. Nakayama K, Shimojo S, Silverman GH (1989) Stereoscopic depth: Its relation to image segmentation, grouping, and the recognition of occluded objects. Perception 18:55-68. Parks T (1965) Post-retinal visual storage American Journal of Psychology 78:145-147. Piaget J (1954) The Construction of Reality in the Child. New York: Basic Books. Ramachandran VS, Inada V, Kiama G (1986) Perception of illusory occlusion in apparent motion. Vision Research 26:1741-1749. Shimojo S, Silverman GH, Nakayama K (1989) Occlusion and the solution to the aperture problem for motion. Vision Research 29:619-626. Yonas A, Craton LG, Thompson WB (1987) Relative motion: Kinetic information for the order of depth at an edge. Perception fj Psychophysics 41:53-59.	1995	1
998	Model Matching and SFMD Computation Steve Rehfuss and Dan Hammerstrom Department of Computer Science and Engineering Oregon Graduate Institute of Science and Technology P.O.Box 91000, Portland, OR 97291-1000 USA stever@cse.ogi.edu, strom@asi.com Abstract In systems that process sensory data there is frequently a model matching stage where class hypotheses are combined to recognize a complex entity. We introduce a new model of parallelism, the Single Function Multiple Data (SFMD) model, appropriate to this stage. SFMD functionality can be added with small hardware expense to certain existing SIMD architectures, and as an incremental addition to the programming model. Adding SFMD to an SIMD machine will not only allow faster model matching, but also increase its flexibility as a general purpose machine and its scope in performing the initial stages of sensory processing. 1 INTRODUCTION In systems that process sensory data there is frequently a post-classification stage where several independent class hypotheses are combined into the recognition of a more complex entity. Examples include matching word models with a string of observation probabilities, and matching visual object models with collections of edges or other features. Current parallel computer architectures for processing sensory data focus on the classification and pre-classification stages (Hammerstrom 1990).This is reasonable, as those stages likely have the largest potential for speedup through parallel execution. Nonetheless, the model-matching stage is also suitable for parallelism, as each model may be matched independently of the others. We introduce a new style of parallelism, Single Function Multiple Data (SFMD), that is suitable for the model-matching stage. The handling of interprocessor synchronization distinguishes the SFMD model from the SIMD and MIMD models: SIMD synchronizes implicitly at each instruction, SFMD synchronizes implicitly at conditional expression or loop boundaries, and MIMD synchronizes explicitly at 714 S. REHFUSS, D. HAMMERSTROM arbitrary inter-processor communication points. Compared to MIMD, the use of implicit synchronization makes SFMD easier to program and cheaper to implement. Compared to SIMD, the larger granularity of synchronization gives SFMD increased flexibility and power. SFMD functionality can be added with small hardware expense to SIMD architectures already having a high degree of processor autonomy. It can be presented as an incremental addition to programmer's picture of the machine, and applied as a compiler optimization to existing code written in an SIMD version of 'C'. Adding SFMD to an SIMD machine will not only allow faster model matching, but also increase its flexibility as a general purpose machine, and increase its scope in performing the initial stages of sensory processing. 2 SIMD ARCHITECTURE AND PROGRAMMING As background, we first review SIMD parallelism. In SIMD, multiple processing elements, or PEs, simultaneously execute identical instruction sequences, each processing different data. The instruction stream is produced by a controller, or sequencer. Generally, each PE has a certain amount of local memory, which only it can access directly. All PEs execute a given -instruction in the stream at the same time, so are synchronized at each instruction. Thus synchronization is implicit, the hardware need not support it, and the programmer need (can) not manage it. SIMD architectures differ in the functionality of their PEs. If PEs can independently address local memory at differing locations, rather than all having to access the same address at a given step, the architecture is said to have local addressing. If PEs can independently determine whether to execute a given instruction, rather than having this determined by the sequencer, the architecture has local conditional execution. Note that all PEs see the same instruction stream, yet a given PE executes only one branch of any if-then-else, and so must idle while other PEs execute the other branch. This is the cost of synchronizing at each instruction. 3 MODEL MATCHING We view models as pieces of a priori knowledge, interrelating their components. Models are matched against some hypothesis set of possible features. Matching produces a correspondence between components of the model and elements of the hypothesis set, and also aligns the model and the set ("pose estimation" in vision, and "time-alignment" in speech). An essential fact is that, because models are known a priori, in cases where there are many models it is usually possible and profitable to construct an index into the set of models. Use of the index at runtime restricts the set of models that need actually be matched to a few, high-probability ones. Model-matching is a common stage in sensory data processing. Phoneme, character and word HMMs are models, where the hypothesis set is a string of observations and the matching process is either of the usual Viterbi or trellis procedures. For phonemes and characters, the HMMs used typically all have the same graph structure, so control flow in the matching process is not model-dependent and may be encoded in the instruction stream. Word models have differing structure, and control flow is model-dependent. In vision, model-matching has been used in a variety of complicated ways (cf. (Suetens, FUa & Hanson 1992)), for example, graph models may have constraints between node attribute values, to be resolved during matching. Model Matching and SFMD Computation 715 4 DATA AND KNOWLEDGE PARALLELISM SIMD is a type of computer architecture. At the algorithm level, it corresponds to data parallelism. Data parallelism, applying the same procedure in parallel to multiple pieces of data, is the most common explicit parallelization technique. and is the essence of the Single Program Multiple Data (SPMD) programming model. On a distributed memory machine, SPMD can be stylized as "given a limited amount of (algorithmic) knowledge to be applied to a large piece of data, distribute the data and broadcast the knowledge" . In sensory processing systems, conversely, one may have a large amount of knowledge (many models) that need to be applied to a (smallish) piece of data, for example, a speech signal frame or segment, or a restricted region of an image. In this case, it makes sense to "distribute the knowledge and broadcast the data". Modelmatching often works well on an SIMD architecture, e.g. for identical phoneme models. However, when matching requires differing control flow between models, an SIMD implementation can be inefficient. Data and knowledge parallelism are asymmetrical, however, in two ways. First, all data must normally be processed, while there are usually indexing techniques that greatly restrict the number of models that actually must be matched. Second, processing an array element frequently requires information about neighboring elements; when the data is partitioned among multiple processors, this may require inter-processor communication and synchronization. Conversely, models on different processors can be matched to data in their local memories without any inter-processor communication. The latter observation leads to the SFMD model. 5 PROGRAMMING MODEL We view support for SFMD as functionality to be added to an existing SIMD machine to increase its flexibility, scope, and power. As such, the SFMD programming model should be an extension of the SIMD one. Given an SIMD architecture with the local addressing and local conditional execution, SFMD programming is made available at the assembly language level by adding three constructs: distribute n tells the sequencer and PEs that the next n instructions are to be distributed for independent execution on the PEs. We call the next n instructions an SFMD block. sync tells the individual PEs to suspend execution and signal the controller (barrier synchronization). This is a no-op if not within an SFMD block. branch-local one or more local branch instruction(s), including a loop construct; the branch target must lie within the enclosing SFMD block. This is a no-op if not within an SFMD block. We further require that code within an SFMD block contain only references to PElocal memory; none to global (sequencer) variables, to external memory or to the local memory of another PE. It must also contain no inter-PE communication .. When the PEs are independently executing an SFMD block, we say that the system is in SFMD mode, and refer to normal execution as SIMD mode. When programming in a data-parallel 'C'-like language for an SIMD machine, use of SFMD functionality can be an optimization performed by the compiler, completely hidden from the user. Variable type and usage analysis can determine for any given block of code whether the constraints on non-local references are met, and emit 716 S. REHFUSS, D. HAMMERSTROM code for SFMD execution if so. No new problems are introduced for debugging, as SFMD execution is semantically equivalent to executing on each PE sequentially, and can be executed this way during debugging. To the programmer, SFMD ameliorates two inefficiencies o~SIMD programming: (i) in conditionals, a PE need not be idle while other PEs execute the branch it didn't take, and (ii) loops and recursions may execute a processor-dependent number of times. 6 HARDWARE MODEL AND COST We are interested in embedded, "delivery system" applications. Such systems must have few chips; scalability to 100's or 1000's of chips is not an issue. Parallelism is thus achieved with multiple PEs per chip. As off-chip I/O is always expensive compared to computation!, such chips can contain only a relatively small number of processors. Thus, as feature size decreases, area will go to local memory and processor complexity, rather than more processors. Adding SFMD functionality to an architecture whose PEs have local addressing and local conditional execution is straightforward. Here we outline an example implementation. Hardware for branch tests and decoding sequencer instructions in the instruction register (IR) already exists. Local memory is suitable for local addressing. A very simple "micro-sequencer" must be added, consisting essentially of a program counter (PC) and instruction buffer (1M), and some simple decode logic. The existing PE output path can be used for the barrier synchronization. A 1-bit path from the sequencer to each PE is added for interrupting local execution. Execution of a distribute n instruction on a PE causes the next n instructions to be stored sequentially in 1M, starting at the current address In the PC. The (n+ 1) 'st instruction is executed in SPMD mode, it is typically either a branch-local to start execution, or possibly a sync if the instructions are just being cached2 . Almost the entire cost of providing SFMD functionality is silicon area used by the 1M. The 1M contains inner loop code, or model-driven conditional code, which is likely to be small. For a 256 4-byte instruction buffer on the current ASI CNAPS 1064, having 64 PEs with 4KB memory each, this is about 11% of the chip area; for a hypothetical 16 PE, 16K per PE chip of the same size, it is 3%. These numbers are large, but as feature size decreases, the incremental cost of adding SFMD functionality to an SIMD architecture quickly becomes small. 7 PERFORMANCE What performance improvement may be expected by adding SFMD to SIMD? There are two basic components, improvement on branches, and improvement on nested loops, where the inner loop count varies locally. U nnested (equiprobable) branches speed up most when the branch bodies have the same size, with a factor of 2 improvement. For nested branches of depth d, the factor is 2d, but these are probably unusual. An exception would be applying a decision tree classifier in a data-parallel way. To examine improvement on nested loops, suppose we have a set of N models (or any independent tasks) to be evaluated en an architecture with P processors. On IE.g., due to limited pin count, pad area, and slower clock off-chip. 2For example, if the distributed code is a subroutine that will be encountered again. Model Matching and SFMD Computation 717 an SFMD architecture, we partition the set into P groups, assign each group to a processor, and have each processor evaluate all the models in its group. If evaluating the j'th model of the i'th group takes time t~;tmd), then the total time is N; Tsfmd = rriax "t(~fmd) i=l ~ 1J j=l (1) where Ni is the size of the i'th group, 2::: 1 Ni = N. On an SIMD architecture, we partition the set into r N! Pl groups of size P and sequentially evaluate each group in parallel. Each group has a model that takes the most time to evaluate; SIMD execution forces the whole group to have this time complexity. So, evaluating a single group, Gi , takes time maxj t~;imd), where j indexes over the elements of the group, 1 ::; j ::; P. The total time for SIMD execution is then rNIPl T L P (simd) simd= maxt· · '-1 1J i=l J(2) Ignoring data-dependent branching and taking t~;imd) = t~;tmd) == tij, we see that optimal (i, j)-indexing of the N models for either case is a bin packing problem. As such, (i, j)-indexing will be heuristic, and we examine Tsimd!Tsfmd by simulation. It should be clear that the expected improvement due to SFMD cannot be large unless the outer loop count is large. So, for model matching, improvement on nested loops is likely not an important factor, as usually only a few models are matched at once. To examine the possible magnitude of the effect in general, we look instead at multiplication of an input vector by a large sparse matrix. Rows are partitioned among the PEs, and each PE computes all the row-vector inner products for its set of rows3 . Tsfmd is given by equation (1), with {tijl1 ::; j ::; Nd the set of all rows for processor i. Tsimd is given by equation (2), with {tijl1 ::; j ::; P} the set of rows executed by all processors at time i. Here tij is the time to perform a row-vector inner product. Under a variety of choices of matrix size (256 x 256 to 2048 x 2048), number of processors (16,32,64), distribution of elements (uniform, clustered around the diagonal), and sparsity (fraction of nonzero elements from 0.001 to 0.4) we get that the ratio Tsimd!Tsfmd decreases from around 2.2-2.7 for sparsities near 0.001, to 1.2 for sparsities near 0.06, and to 1.1 or less for more dense matrices (Figure 1). The effect is thus not dramatic. As an example of the potential utility of SFMD functionality for model matching, we consider interpretation tree search (ITS), a technique used in vision4 . ITS is a technique for establishing a correspondence between image and model features. It consists essentially of depth-first search (DFS) , where a node on level d of the tree corresponds to a pairing of image features with the first d model features. The search is limited by a variety of unary and binary geometric constraints on the allowed pairings. Search complexity implies small models are matched to small 3We assume the assignment of rows to PEs is independent of the number of nonzero elements in the rows. If not, then for N » P, simply sorting rows by number of elements and then assigning row i to processor i mod P is a good enough packing heuristic to make Ts;tTld ~ T s/ md . 4See (Grimson 1990) for a complete description of ITS and for the complexity results alluded to here. 718 ' .0 ... , .. .., ! " t ! 1 . ' ~ 1.6 1 . ' 1.2 1 0.0001 spar .. Matzic •• · . . · .' " • •• + :. . , ., . •• t-. +. :,-.+. : .. ~ ~ • • par •• . af_ . gnu · • ..... :.n .... • f :... • . q •• Figure 1: Sparse matrices: speedup vs. sparsity numbers of data features, so distributing models and data to local memories is practical. To examine the effect of SFMD on this form of model matching, we performed some simple simulations. To match a model with D features to a set of B data points, we attempt to match the first model feature with each data point in order, with some probability of success, Pmatch. If we succeed, we attempt to match the second model feature with one of the remaining B-1 data points, and so on. If we match all D features, we then check for global consistency of the correspondence, with some probability of success, Pcheck. This procedure is equivalent to DFS in a tree with branching factor B -d at level d of the tree, 1 ~ d ~ D, where the probability of expanding any given node is Pmatch, and the probability of stopping the search at any given leaf is 1 - Pcheck. By writing the search as an iteration managing an explicit stack, one obtains a loop with some common code and some code conditional on whether the current node has any child nodes left to be expanded. The bulk of the "no-child" code deals with leaf nodes, consisting of testing for global consistency and recording solutions. The relative performance of SIMD and SFMD thus depends mainly on the probability, Pleat, that the node being traversed is a leaf. If, for each iteration, the time for the leaf code is taken to be 1, that for common code is t, and that for the non-leaf code is k, then t+k+1 TsimdlTstmd = t + (1- p)k + p' (3) Panel 1 of figure 2 shows values of P from a variety of simulations of ITS, with B,D E {8, 10, 12, 14, 16}, Pmatch E {0.1,0.2, liB}, Pcheck E {0,1}. Grimson (1990) reports searches on realistic data of around 5000-10000 expansions; this corresponds to P ~ 0.2 - 0.4. Panel 2 of figure 2 shows how equation 3 behaves for P in this regime and for realistic values of k. We see speedups in the range 2-4 unless the leaf code is very small. In fact, the code for global consistency checking is typically larger than that for local consistency, corresponding to log2 k < 0. 8 OTHER USES There are a number of uses for SFMD, other than model matching. First, common "subroutines" involving branching may be kept in the 1M. Analysis of code for IEEE floating point emulation on an SIMD machine shows an expected 2x improvement by Model Matching and SFMD Computation 0.7 0.5 o 4 0.' o l 0.1 protwlbl1it.y o f t.raver.lng a l .. t i n ITS . ' .' .. . ' .. . ... ". ~. . . 719 .~ ... Figure 2: DFS speedup. Panel 1 shows the probability, p, of traversing a leaf. Panel 2 plots equation 3 for realistic values of p and k, with t = 0.1. using SFMD. Second, simple PE-local searches and sorts should show a significant, sub-2x, improvement in expected time. Third, more speculatively, different PEs can execute entirely different tasks by having the SFMD block consist of a single (nested) if-then-else. This would allow a form of (highly synchronized) pipeline parallelism by communicating results in SIMD mode after the end of the SFMD block. 9 CONCLUSION We have introduced the SFMD computation model as a natural way of implementing the common task of model matching, and have shown how it extends SIMD computing, giving it greater flexibility and power. SFMD functionality can easily, and relatively cheaply, be added to existing SIMD designs already having a high degree of processor autonomy. The addition can be made without altering the user's programming model or environment. We have argued that technology trends will force multiple-processor-per-chip systems to increase processor complexity and memory, rather than increase the number of processors model per chip, and believe that the SFMD model is a natural step in that evolution. Acknowledgements The first author gratefully acknowledges support under ARPA/ONR grants N0001494-C-0130, N00014-92-J-4062, and N00014-94-1-0071. References Grimson, W. E. L. (1990), Object Recognition by Computer: The Role of Geometric Constraints, MIT Press. Hammerstrom, D. (1990), A VLSI architecture for high-performance, low-cost, on-chip learning, in 'The Proceedings of the IJCNN'. Suetens, P., Fua, P. & Hanson, A. J. (1992), 'Computational strategies for object recognition', Computing Surveys 24(1), 5 - 61.	1995	10
999	Using Feedforward Neural Networks to Monitor Alertness from Changes in EEG Correlation and Coherence Scott Makeig Naval Health Research Center, P.O. Box 85122 San Diego, CA 92186-5122 Tzyy-Ping Jung Naval Health Research Center and Computational Neurobiology Lab The Salk Institute, P.O. Box 85800 San Diego, CA 92186-5800 Terrence J. Sejnowski Howard Hughes Medical Institute and Computational Neurobiology Lab The Salk Institute, P.O. Box 85800 San Diego, CA 92186-5800 Abstract We report here that changes in the normalized electroencephalographic (EEG) cross-spectrum can be used in conjunction with feedforward neural networks to monitor changes in alertness of operators continuously and in near-real time. Previously, we have shown that EEG spectral amplitudes covary with changes in alertness as indexed by changes in behavioral error rate on an auditory detection task [6,4]. Here, we report for the first time that increases in the frequency of detection errors in this task are also accompanied by patterns of increased and decreased spectral coherence in several frequency bands and EEG channel pairs. Relationships between EEG coherence and performance vary between subjects, but within subjects, their topographic and spectral profiles appear stable from session to session. Changes in alertness also covary with changes in correlations among EEG waveforms recorded at different scalp sites, and neural networks can also estimate alertness from correlation changes in spontaneous and unobtrusivelyrecorded EEG signals. 1 Introduction When humans become drowsy, EEG scalp recordings of potential oscillations change dramatically in frequency, amplitude, and topographic distribution [3]. These changes are complex and differ between subjects [10]. Recently, we have shown 932 S. MAKEIG, T.-P. JUNG, T. J. SEJNOWSKl that using principal components analysis in conjunction with feedforward neural networks, minute-scale changes in performance on a sustained auditory detection task can be estimated in near real-time from changes in the EEG spectrum at one or more scalp channels [4, 6]. Here, we report, first, that loss of alertness during auditory detection task performance is also accompanied by changes in spectral coherence of EEG signals recorded at different scalp sites. The extent, topography, and frequency content of coherence changes linked to changes in alertness differ between subjects, but within subjects they appear stable from session to session. Second, since most coherence changes linked to alertness are not associated with significant phase differences, moving correlation measures applied to wideband or bandlimited EEG waveforms also covary with changes in alertness. Incorporating coherence and/or correlation information into neural network algorithms for estimating alertness from the EEG spectrum should enhance their accuracy and robustness and contribute to the design of practical neural human-system interfaces performing real-time monitoring of changes in operator alertness. 2 Methods Concurrent EEG and behavioral data were collected for the purpose of developing a method of objectively monitoring the alertness of operators of complex systems [6] . Ten adult volunteers participated in three or more half-hour sessions during which they pushed one button whenever they detected an above-threshold auditory target stimulus (a brief increase in the level of the continuously-present background noise). To maximize the chance of observing alertness decrements, sessions were conducted in a small, warm, and dimly-lit experimental chamber, and subjects were instructed to keep their eyes closed. Targets were 350 ms increases in the intensity of a 62 dB white noise background, 6 dB above their threshold of detectability, presented at random time intervals at a mean rate of 10/min. Short, and task-irrelevant probe tones oftwo frequencies (568 and 1098 Hz) were interspersed between the target noise bursts at 2-4 s intervals. EEG was collected from thirteen electrodes located at sites of the Internation 10-20 System, referred to the right mastoid, at a sampling rate of 312.5 Hz. A bipolar diagonal electrooculogram (EOG) channel was also recorded for use in eye movement artifact correction and rejection. Two sessions each from three of the subjects were chosen for analysis on the basis of their including more than 50 detection lapses. A continuous performance measure, local error rate, was computed by convolving an irregularly-spaced performance index (hit=0/lapse=1) with a 95 s smoothing window advanced through the performance data in 1.64 s steps. Target hits were defined as targets responded to within a 100-3000 ms window; other targets were called lapses. After eye movement artifacts were removed from the data using a selective regression procedure [5], and data containing other large artifacts were rejected from analysis, complex EEG spectra were computed by advancing a 512point (1.64 s) data window through the data in 0.41 s steps, multiplying by a Hanning window, and converting to frequency domain using an FFT. Complex coherence was then computed for each channel pair in 1.64 s spectral epochs. In the coherence studies, error rate was smoothed with a bell-shaped Papoulis window; a 36 s rectangular window was used to smooth the coherence estimates. Finally, complex coherence was converted to coherence amplitude and phase and results were correlated with local error rate. A moving correlation measure between (1-20 Hz) bandlimited EEG waveforms was computed for each channel pair in a moving 1.64 s smoothing window, and then smoothed using a causal 95-s exponential window. The same window was used to smooth the error rate time series for the correlation studies. Using Feedforward Neural Networks to Monitor Alertness from EEG 933 3 Results u 1 40 ~ ~ 0.8 30 < 0.6 B 20 ~ \9Chan"I~~ e 0.4 .. l! Frequency: 9.1 Hz ~ 10 0 u ~ 0 8 0.8 g ·10 0.6 ~ ·20 B : 0.4 ~ 0.2 0 0 30 Time on Task (min) Time On Task (min) (a) (b) Figure 1: (a) Changes in coherence amplitude at 9.1 Hz (upper traces) are correlated with simultaneous changes in error rate during a half auditory detection task (lower trace) in nine indicated central-frontal channel pairs. (b) Concurrent changes in coherence phase at 15.25 Hz (upper traces) and local error rate (lowertrace) for the same session and channel-pairs. 3.1 Relation of Coherence Changes to Detection Performance. During the first 2-3 minutes of the session shown in Fig. la, the subject detected all targets presented, and coherence amplitudes remained high (0.9). In minutes 8-10, however, when the subject failed to make a single detection response(lower trace), coherence amplitude fell to as low as 0.6. Overall correlations for this session between the coherence and error rate time series in these channel pairs ranged from -0.590 to -0.776. In the same session, coherence phase at 15 Hz also covaried with performance (Fig. 1 b). During low-error portions of the session, there was no detectable coherence phase lag at 15 Hz within the same nine channel pairs, whereas while the subject performed poorly, a 20 degree phase lag appeared during which 15 Hz activity at frontal sites lead activity at frontal sites by 3 ms. Overall correlations for this session between coherence phase and error rate for these channel pairs ranged from 0.416 to 0.689. Correlations between coherence amplitude and error rate at 80 EEG frequencies (Fig. 2a, upper traces) included two broad bands of strong negative correlations (3-12 Hz and 15-20 Hz), while appreciable correlations between coherence phase and performance were confined to much narrower frequency bands (lower traces). To estimate the significance ofthese coherence correlations, surrogate moving coherence records were collected 10 times using randomly-selected, asynchronous blocks of contiguous EEG data for each channeL Correlations between the resulting surrogate moving coherence time series and error rate were computed, and the 99.936th percentile of the distribution of (absolute) correlations was determined. For the subject whose data is shown here, this value was 0.485. Under conservative assumptions of complete independence of adjacent frequencies, this should give the (p=0.05) significance level for the maximum absolute correlation in each 80-bin correlation spectrum. (The heuristic estimate of this significance level from the surrogate data was 0.435). In the two sessions from this subject, however, more than 20% of all the 78 channel-pair coherence correlations were larger in absolute value than 0.485, implying that coherence amplitude changes at many scalp sites and frequencies are significantly related to changes in alertness in this subject. 934 S. MAKEIG, T.-P. JUNG, T. J. SEJNOWSKI 3.2 Spectral and Topographic Stability c 0 ~ t .'" Subject p37 .... 1-~ir@t;B i ~ 9 C:annel pairs ] 25essions (30.5 0 ." ..... c. -S ®2 e 0 ..... .. .••. ..... . ....... .. .. ... .~ < ]-0.5 t ~: 0 (,) § ~ c 0 'i o. 1l ~ ~: 0 o . " (,) ·i ~ -0. il 0 ...: (,) 1! 0 5 10 15 20 25 0 (,) Frequency (Hz) Frequency (Hz) (a) (b) Figure 2: (a) Correlation spectra showing correlations between moving-average coherence and error rate for the same session and channel-pairs. Small letters 'a,b,c' indicate the frequencies analyzed in Figs. 1 and 3. (b) Cluster analysis of correlations between coherence amplitude and error rate at 41 frequencies (0.6 Hz to 25 Hz). Means of six sets of channel pairs derived from cluster analysis of 78 similar coherence correlation spectra from all pairs of 13 scalp channels; superimposed on the same means for a second session from the same subject. The sign, size, and spectral and topographic structure of correlations between coherence amplitude and error rate at each frequency were stable across two sessions for most channel pairs and frequency bands. Fig. 2b shows mean spectral correlations in both sessions from the same subject for six clusters of similar channel-pair correlation spectra identified by cluster analysis on results of the first session. Except near 5 Hz, the size and structure of the correlation spectra for the second session replicate results of the first session. The spectral stability of monotonic relationships between EEG coherence and auditory detection performance suggests that coherence may be used to predict changes in performance level from spontaneous EEG data collected continuously and unobtrusively from two or more scalp channels. 3.3 EEG Waveform Correlations and Performance In most cases, coherence phase lags in these data are small, and correlations between changes in phase lag and performance were insignificant. We therefore investigated whether moving-average correlations between band-limited EEG signals in different scalp channels might also be used to predict changes in alertness, possibly at a lower computational cost, by studying the relationship between error rate and changes in moving-average correlations of time-domain EEG waveforms (1-20 Hz bandpass) in the same 6 sessions. Again, we found that the strength and topographic structure of significant relationships between moving-correlation and performance measures are stable within, and variable between subjects. For each subject, we selected 8 EEG channel pairs whose moving-correlation time series correlated most highly with error rate, and used these to train a multilinear regression network and three feedforward three-layer perceptrons to estimate error rate from moving-average correlations. The feedforward neural networks had 3, 4, and 5 hidden units, respectively. Weights and biases of the network were adjusted using the error backpropagation algorithm [9]. Conjugate gradient descent was used to minimize the mean-squared error between network output and the actual error rate 30 Using Feedforward Neural Networks to Monitor Alertness from EEG 935 time series. Cross-validation [7] was used to prevent the network from overfitting the training data. For each of the 6 training-testing session pairs and each neural network architecture, the time course of error rate was estimated five times using different random initial weights between -0.3 and 0.3. We tested the generalization ability of the models on second sessions from the same subjects. The procedure simulated potential real-world alertness monitoring applications in which pilot data for each operator would be used to train a network to estimate his or her alertness in subsequent sessions from unobtrusively-recorded EEG data. Accuracy of error rate estimation in the test sessions was almost identical for neural networks with 3, 4, and 5 hidden units. Each was more accurate than multivariate linear regression. Figure 3 shows the time courses of actual and estimated error rate in one pair of training (top paneQ and test sessions. Results for two other subjects were equivalent. Table 1 shows the average correlations and root-mean-squared estimation error between actual and estimated error rate time series for 6 sessions, 2 each on 3 subjects using a feedforward neural network with 3 hidden units. Results using 4 or 5 hidden units are equivalent. Diagonal cells show results for training sessions, off-diagonal cells for test sessions. The nonlinear adaptability of threelayer perceptrons give improved estimation performance over multivariate linear regression, reducing the RMS estimation error in the test sessions from 0.255 to 0.225 (F(l, 5) = 1234.29;p ~ 0.0001), and increasing the mean correlation between actual and estimated error rate time series from 0.63 to 0.67 (F(l, 5) = 549.5;p ~ 0.0001). 4 Discussion Spectral coherence of EEG waveforms at different scalp sites has been measured for nearly 30 years [11]. and is the subject of a steadily increasing number of clinical, behavioral, and developmental EEG studies. Coherence values are known to be higher in sleep than in waking [8], and wake-sleep transitions have been noted to be preceded by increased coherence at some frequencies [2]. Our results, from data on three subjects performing a sustained auditory detection task under soporific conditions, suggest that during drowsiness, coherence may either increase or decrease, depending on the subject, analysis frequency, and electrode sites analyzed. However, in individual subjects the spectral and topographic structure of alertness-related coherence changes appears stable from session to session. EEG correlation and coherence are intimately related: changes in moving-average correlations of EEG waveforms reflect changes in broad-band, zero-lag coherence of activity at the same sites. The possibility of using moving-average correlation measures of electrophysiological activity to monitor state changes in animals was discussed by Arduini [1], but to our knowledge this approach has not previously been applied to human EEG. The origin and function of nonstationarity in EEG synchrony are not yet understood. Decreased EEG coherence during drowsiness might result from inactivation of subcortical brain systems coordinating activity in separate cortical EEG generators during wakefulness, or from emergence of drowsiness-related EEG activity projecting preferentially to one part of the scalp surface. Similarly, increases in coherence in drowsiness might either result from increased synchrony between cortical generators, or from volume conduction of enhanced activity generated at a single cortical or subcortical site. Measuring changes in EEG coherence and correlation during other cognitive tasks give clues to the possible role of variable EEG synchrony in brain and cognitive dynamics. We are now investigating to what extent moving EEG coherence and/or correlation 936 S. MAKEIG, T.-P. JUNG, T. J. SEJNOWSKl measures, in combination with spectral amplitude measures [4], will allow practical, robust, continuous, and near-real time estimation of alertness level in auditory detection and other task environments. Estimated and Actual Error Rates Tralnln~ : 3674 ?i Testse : 3674 RMS : 0.1706 8 0.8 Corr : 0.8246 E 0.6 G> OJ 0.4 a: g 0.2 I .-.-.- .~ .~ .W 0 ," .. . :..,.,,.,,,,. -0.2 0 5 10 15 20 25 30 Time on Task (min) Tralnl~: 3674 ~ Test se : 3648 RMS : 0.2418 8 0.8 • .rCorr : 0.7159 ~ p , 0.6 ~ 0.4 a: g 0.2 w 0 -0.2 0 5 10 15 20 25 30 Time on Task (min) Figure 3: Changes in detection rate (95-s exponential window) and their estimate using a feedforward three-layer perceptron on moving correlations between (1-20 Hz) band passed EEG signals for 8 selected pairs of 7 scalp channels. The top panel shows the training session, the bottom panel the testing session. Solid lines show the actual error rate time course; dashed lines, the estimate. Correlation and RMS error between the two are indicated. Table 1: The results of alertness monitoring using moving EEG pairwise correlation. Subject A Subject B Test Training Set Test Training Set set 3648 3674 set 3654 3656 3648 rms: 0.17 rms : 0.26 3654 rms: 0.17 rms : 0.22 corr : 0.87 corr : 0.68 corr : 0.83 corr : 0.73 3674 rms: 0.21 rms: 0.17 3656 rms: 0.25 rms : 0.14 corr : 0.73 corr: 0.83 corr : 0.54 corr: 0.76 Subject C Test Training Set set 3665 3673 3665 rms : 0.19 rms: 0.23 corr : 0.76 corr : 0.65 3673 rms : 0.18 rms: 0.17 corr : 0.67 corr: 0.70 Using Feedforward Neural Networks to Monitor Alertness from EEG 937 Acknowledgments This work was supported by a grant (ONR.Reimb.30020.6429) to the Naval Health Research Center by the Office of Naval Research. The views expressed in this article are those of the authors and do not reflect the official policy or position of the Department of the Navy, Department of Defense, or the U.S. Government. We acknowledge the contributions of Keith Jolley, F.scot Elliott, and Mark Postal in collecting and processing the data, and thank Tony Bell for suggestions. References [1] Arduini A .. 1979. In-phase brain activity and sleep. Electroencephalog clin NeurophysioI47,441-9 [2] Borodkin SM, GrindeI' OM, Boldyreva GN, Zaitsev VA & Luk'ianov V.1.1987. Dynamics of the spectral-coherent characteristics of the human EEG in healthy subjects and brain pathology. Zh Vyssh Nerv Deiat 37, 22-30 [3] Davis H., Davis P.A., Loomis A.L., Harvey E.N., & Hobart G. 1938. Human brain potentials during the onset of sleep. J Neurophysiol1, 24-38 [4] Jung T-P, Makeig S., Stensmo M., & Sejnowski T. Estimating alertness from the EEG power spectrum, submitted for publication. [5] Kenemans J.L., Molenaar P.C.M., Verbaten M.N. & Slangen J.L. 1991. Removal of the ocular artifact from the EEG: a comparison of time and frequency domain methods with simulated and real data. Psychophysiolog 28, 114-121 [6] Makeig S. & Inlow M. 1993. Lapses in alertness: Coherence of fluctuations in performance and EEG spectrum. Electroencephalog clin Neurophysiol86, 23-35 [7] Morgan N., & Bourlard H. 1990. Generalization and parameter estimation in feedforward nets: some experiments. Neural Information Processing Systems, 2, 630-637. [8] Nielsen T, Abel A, Lorrain D, & Montplaisir J . 1990. Interhemispheric EEG coherence during sleep and wakefulness in left- and right-handed subjects. Brain and Cognition 14, 113-25 [9] Rumelhart D, Hinton G, & Williams R, 1986. Learning internal representation by error propagation, Parallel distributed processing, Chap. 8. [10] Santamaria J. & Chiappa K.H. 1987. The EEG of drowsiness in normal adults. J Clin Neurophysiol4, 327-82 [11] Walter D.O. 1968. Coherence as a measure of relationship between EEG records. Electroencephalog clin N europhysiol 24, 282	1995	100

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