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1,100	Family Discovery Stephen M. Omohundro NEC Research Institute 4 Independence Way, Princeton, N J 08540 om@research.nj.nec.com Abstract "Family discovery" is the task of learning the dimension and structure of a parameterized family of stochastic models. It is especially appropriate when the training examples are partitioned into "episodes" of samples drawn from a single parameter value. We present three family discovery algorithms based on surface learning and show that they significantly improve performance over two alternatives on a parameterized classification task. 1 INTRODUCTION Human listeners improve their ability to recognize speech by identifying the accent of the speaker. "Might" in an American accent is similar to "mate" in an Australian accent. By first identifying the accent, discrimination between these two words is improved. We can imagine locating a speaker in a "space of accents" parameterized by features like pitch, vowel formants, "r" -strength, etc. This paper considers the task of learning such parameterized models from data. Most speech recognition systems train hidden Markov models on labelled speech data. Speaker-dependent systems train on speech from a single speaker. Speakerindependent systems are usually similar, but are trained on speech from many different speakers in the hope that they will then recognize them all. This kind of training ignores speaker identity and is likely to result in confusion between pairs of words which are given the same pronunciation by speakers with different accents. Speaker-independent recognition systems could more closely mimic the human approach by using a learning paradigm we call "family discovery". The system would be trained on speech data partitioned into "episodes" for each speaker. From this data, the system would construct a parameterized family of models representing difFamily Discovery 403 Affine Family Affine Patch Family Coupled Map Family Figure 1: The structure of the three family discovery algorithms. ferent accents. The learning algorithms presented in this paper could determine the dimension and structure of the parameterization. Given a sample of new speech, the best-fitting accent model would be used for recognition. The same paradigm applies to many other recognition tasks. For example, an OCR system could learn a parameterized family of font models (Revow, et. al., 1994). Given new text, the system would identify the document's font parameters and use the corresponding character recognizer. In general, we use "family discovery" to refer to the task of learning the dimension and structure of a parameterized family of stochastic models. The methods we present are equally applicable to parameterized density estimation, classification, regression, manifold learning, reinforcement learning, clustering, stochastic grammar learning, and other stochastic settings. Here we only discuss classification and primarily consider training examples which are explicitly partitioned into episodes. This approach fits naturally into the neural network literature on "meta-learning" (Schmidhuber, 1995) and "network transfer" (Pratt, 1994). It may also be considered as a particular case of the "bias learning" framework proposed by Baxter at this conference (Baxter, 1996). There are two primary alternatives to family discovery: 1) try to fit a single model to the data from all episodes or 2) use separate models for each episode. The first approach ignores the information that the different training sets came from distinct models. The second approach eliminates the possibility of inductive generalization from one set to another. In Section 2, we present three algorithms for family discovery based on techniques for "surface learning" (Bregler and Omohundro, 1994 and 1995). As shown in Figure 1, the three alternative representations of the family are: 1) a single affine subspace of the parameter space, 2) a set of local affine patches smoothly blended together, and 3) a pair of coupled maps from the parameter space into the model space and back. In Section 3, we compare these three approaches to the two alternatives on a parameterized classification task. 404 S. M. OMOHUNDRO 2 THE FIVE ALGORITHMS Let the space of all classifiers under consideration be parameterized by 0 and assume that different values of 0 correspond to different classifiers (ie. it is identifiable). For example, 0 might represent the means, covariances, and class priors of a classifier with normal class-conditional densities. O-space will typically have a much higher dimension than the parameterized family we are seeking. We write P9(X) for the total probability that the classifier 0 assigns to a labelled or unlabelled example x. The true models are drawn from a d-dimensional family parameterized by , . Let the training set be partitioned into N episodes where episode i consists of Ni training examples tij, 1 :S j :S Ni drawn from a single underlying model with parameter 0:. A family discovery learning algorithm uses this training data to estimate the underlying parameterized family. From a parameterized family, we may define the projection operator P from O-space to itself which takes each 0 to the closest member of the family. Using this projection operator, we may define a "family prior" on O-space which dies off exponentially with the square distance of a model from the family mp(O) ex e-(9-P(9))2. Each of the family discovery algorithms chooses a family so as to maximize the posterior probability of the training data with respect to this prior. If the data is very sparse, this MAP approximation to a full Bayesian solution can be supplemented by "Occam" terms (MacKay, 1995) or by using a Monte Carlo approximation. The outer loop of each of the algorithms performs the optimization of the fit of the data by re-estimation in a manner similar to the Expectation Maximization (EM) approach (Jordan and Jacobs, 1994). First, the training data in each episode i is independently fit by a model Oi. Then the dimension of the family is determined as described later and the family projection operator P is chosen to maximize the probability that the episode models Oi came from that family ni mp(Oi). The episode models Oi are then re-estimated including the new prior probability mp. These newly re-estimated models are influenced by the other episodes through mp and so exhibit training set "transfer". The re-estimation loop is repeated until nothing changes. The learned family can then be used to classify a set of Ntest unlabelled test examples Xk, 1 :S k :S Ntest drawn from a model O;est in the family. First, the parameter Otest is estimated by selecting the member of the family with the highest likelihood on the test samples. This model is then used to perform the classification. A good approximation to the best-fit family member is often to take the image of the best-fit model in the entire O-space under the projection operator P. In the next five sections, we describe the two alternative approaches and the three family discovery algorithms. They differ only in their choice of family representation as encoded in the projection operator P. 2.1 The Single Model Approach The first alternative approach is to train a single model on all of the training data. It selects 0 to maximize the total likelihood L( 0) = n~l n~l P9 (tij ). New test data is classified by this single selected model. Family Discovery 405 2.2 The Separate Models Approach The second alternative approach fits separate models for each training }£isode. It chooses Bi for 1::; i::; N to maximize the episode likelihood Li(Bi) = TIj~IPIJ(tij). Given new test data, it determines which of the individual models Bi fit best and classifies the data with it. 2.3 The Affine Algorithm The affine model represents the underlying model family as an affine subspace of the model parameter space. The projection operator Pal line projects a parameter vector B orthogonally onto the affine subspace. The subspace is determined by selecting the top principal vectors in a principal components analysis of the bestfit episode model parameters. As described in (Bregler & Omohundro, 1994) the dimension is chosen by looking for a gap in the principal values. 2.4 The Affine Patch Algorithm The second family discovery algorithm is based on the "surface learning" procedure described in (Bregler and Omohundro, 1994). The family is represented by a collection of local affine patches which are blended together using Gaussian influence functions. The projection mapping Ppatch is a smooth convex combination of projections onto the affine patches Ppatch(B) = 2::=1 10: (B)Ao: (B) where Ao: is the projection operator for an affine patch and Io:(B) = E:"J:)(IJ) is a normalized Gaussian blending function. The patches are initialized using k-means clustering on the episode models to choose k patch centers. A local principal components analysis is performed on the episode models which are closest to each center. The family dimension is determined by examining how the principal values scale as successive nearest neighbors are considered. Each patch may be thought of as a "pancake" lying in the surface. Dimensions which belong to the surface grow quickly as more neighbors are considered while dimensions across the surface grow only because of the curvature of the surface. The Gaussian influence functions and the affine patches are then updated by the EM algorithm (Jordan and Jacobs, 1994). With the affine patches held fixed, the Gaussians Go: are refit to the errors each patch makes in approximating the episode models. Then with the Gaussians held fixed, the affine patches Ao: are refit to the epsiode models weighted by the the corresponding Gaussian Go:. Similar patches may be merged together to form a more parsimonious model. 2.5 The Coupled Map Algorithm The affine patch approach has the virtue that it can represent topologically complex families (eg. families representing physical objects might naturally be parameterized by the rotation group which is topologically a projective plane). It cannot, however, provide an explicit parameterization of the family which is useful in some applications (eg. optimization searches). The third family discovery algorithm therefore attempts to directly learn a parameterization of the model family. Recall that the model parameters define B-space, while the family parameters de406 S. M. OMOHUNDRO fine 'Y-space. We represent a family by a mapping G from B-space to 'Y-space together with a mapping F from 'Y-space back to B-space. The projection operation is Pmap(B) = F(G(B)). The map G(O) defines the family parameter l' on the full O-space. This representation is similar to an "auto-associator" network in which we attempt to "encode" the best-fit episode parameters Oi in the lower dimensional 'Y-space by the mapping G in such a way that they can be correctly reconstructed by the function F. Unfortunately, if we try to train F and G using back-propagation on the identity error function, we get no training data away from the family. There is no reason for G to project points away from the family to the closest family member. We can rectify this by training F and G iteratively. First an arbitrary G is chosen and F is trained to send the images 'Yi = G(Oi) back to 0i' G is trained, however, on images under F corrupted by additive spherical Gaussian noise! This provides samples away from the family and on average the training signal sends each point in B space to the closest family member. To avoid iterative training, our experiments used a simpler approach. G was taken to be the affine projection operator defined by a global principal components analysis of the best-fit episode model parameters. Once G is defined, F is chosen to minimize the difference between F(G(Oi)) and Oi for each best-fit episode parameter Oi. Any form of trainable nonlinear mapping could be used for F (eg. backprop neural networks or radial basis function networks). We represent F as a mixture of experts (Jordan and Jacobs, 1994) where each expert is an affine mapping and the mixture coefficients are Gaussians. The mapping is trained by the EM algorithm. 3 ALGORITHM COMPARISON To compare these five algorithms, we consider a two-class classification task with unit-variance normal class-conditional distributions on a 5-dimensional feature space. The means of the class distributions are parameterized by a nonlinear twoparameter family: ml = (1'1 + ~cos¢»e~1 + ('Y2 + ~sin¢»e~2 m2 = ('Yl ~ cos ¢> ) e~1 + ('Y2 ~ sin ¢> ) l2 . where 0 ~ 1'1, 1'2 ~ 10 and ¢> = ('Yl + 1'2)/3. The class means are kept at a unit distance apart, ensuring significant class overlap over the whole family. The angle ¢> varies with the parameters so that the correct classification boundary changes orientation over the family. This choice of parameters introduces sufficient nonlinearity in the task to distinguish the non-linear algorithms from the linear one. Figure 1 shows the comparative performance of the 5 algorithms. The x-axis is the total number of training examples. Each set of examples consisted of approximately N = ..;x episodes of approximately Ni = ..;x examples each. The classifier parameters for an episode were drawn uniformly from the classifier family. The episode training examples were then sampled from the chosen classifier according to the classifier's distribution. Each of the 5 algorithms was then trained on these examples. The number of patches in the surface patch algorithm and the number of affine components in the surface map algorithm were both taken to be the square-root of Family Discovery 407 0.52 r---.---.---""T""----r----,-----r---r---~-__, 0.5 0.48 0.46 I!? 0.44 g w '0 0.42 c: 0 :u I!! 0.4 u. 0.38 0.36 0.34 400 600 800 1000 1200 1400 Number of Examples Single model -+Separate models -+-_. Affine family -EJ -Affine Patch family ··x···· Map Mixture family -A-.1600 1800 2000 Figure 2: A comparison of the 5 family discovery algorithms on the classification task. the number of training episodes. The y-axis shows the percentage correct for each algorithm on an independent test set. Each test set consisted of 50 episodes of 50 examples each. The algorithms were presented with unlabelled data and their classification predictions were then compared with the correct classification label. The results show significant improvement through the use of family discovery for this classification task. The single model approach performed significantly worse than any of the other approaches, especially for larger numbers of episodes (where the family discovery becomes possible). The separate model approach improves with the number of episodes, but is nearly always bested by the approaches which take explicit account of the underlying parameterized family. Because of the nonlinearity in this task, the simple affine model performs more poorly than the two nonlinear methods. It is simple to implement, however, and may well be the method of choice when the parameters aren't so nonlinear. From this data, there is not a clear winner between the surface patch and surface map approaches. 4 TRAINING SET DISCOVERY Throughout this paper, we have assumed that the training set was partitioned into episodes by the teacher. Agents interacting with the world may not be given this explicit information. For example, a speech recognition system may not be told when it is conversing with a new speaker. Similarly, a character recognition system 408 s. M. OMOHUNDRO would probably not be given explicit information about font changes. Learners can sometimes use the data itself to detect these changes, however. In many situations there is a strong prior that successive events are likely to have come from a single model with only occasional model changes. The EM algorithm is often used for segmenting unlabelled speech. It may be used in a similar manner to find the training set episode boundaries. First, a clustering algorithm is used to partition the training examples into episodes. A parameterized family is then fit to these episodes. The data is then repartitioned according to the similarity of the induced family parameters and the process is repeated until it converges. A similar approach may be applied when the model parameters vary slowly with time rather than occasionally jumping discontinously. Acknowledgements I'd like to thank Chris Bregler for work on the affine patch approach to surface learning, Alexander Linden for suggesting coupled maps for surface learning, and Peter Blicher for discussions. References Baxter, J. (1995) Learning model bias. This volume. Bregler, C. & Omohundro, S. (1994) Surface learning with applications to lipreading. In J. Cowan, G. Tesauro and J. Alspector (eds.), Advances in Neural Information Processing Systems 6, pp. 43-50. San Francisco, CA: Morgan Kaufmann Publishers. Bregler, C. & Omohundro, S. (1995) Nonlinear image interpolation using manifold learning. In G. Tesauro, D. Touretzky and T. Leen (eds.), Advances in Neural Information Processing Systems 7. Cambridge, MA: MIT Press. Bregler, C. & Omohundro, S. (1995) Nonlinear manifold learning for visual speech recognition. In W. Grimson (ed.), Proceedings of the Fifth International Conference on Computer Vision. Jordan, M. & Jacobs, R. (1994) Hierarchical mixtures of experts and the EM algorithm. Neural Computation, 6:181-214. MacKay, D. (1995) Probable networks and plausible predictions - a review of practical Bayesian methods for supervised neural networks. Network, to appear. Pratt, L. (1994) Experiments on the transfer of knowledge between neural networks. In S. Hanson, G. Drastal, and R. Rivest (eds.), Computational Learning Theory and Natural Learning Systems, Constraints and Prospects, pp. 523-560. Cambridge, MA: MIT Press. Revow, M., Williams, C. and Hinton, G. (1994) Using generative models for handwritten digit recognition. Technical report, University of Toronto. Schmidhuber, J. (1995) On learning how to learn learning strategies. Technical Report FKI-198-94, Fakultat fur Informatik, Technische Universitat Munchen.	1995	55
1,101	Modeling Interactions of the Rat's Place and Head Direction Systems A. David Redish and David S. Touretzky Computer Science Department & Center for the Neural Basis of Cognition Carnegie Mellon University, Pittsburgh PA 15213-3891 Internet: {dredi sh, ds t}@es . emu. edu Abstract We have developed a computational theory of rodent navigation that includes analogs of the place cell system, the head direction system, and path integration. In this paper we present simulation results showing how interactions between the place and head direction systems can account for recent observations about hippocampal place cell responses to doubling and/or rotation of cue cards in a cylindrical arena (Sharp et at., 1990). Rodents have multiple internal representations of their relationship to their environment. They have, for example, a representation of their location (place cells in the hippocampal formation, see Muller et at., 1991), and a location-independent representation of their heading (head direction cells in the postsubiculum and the anterior thalamic nuclei, see Taube et at., 1990; Taube, 1995). If these representations are to be used for navigation, they must be aligned consistently whenever the animal reenters a familiar environment. This process was examined in a set of experiments by Sharp et at. (1990). 1 The Sharp et al., 1990 experiment Rats spent multiple sessions finding food scattered randomly on the floor of a black cylindrical arena with a white cue card along the wall subtending 90° of arc. The animals were not disoriented before entering the arena, and they always entered at the same location: the northwest corner. See Figure 3a. Hippocampal place fields were mapped by single-cell recording. A variety of probe trials were then introduced. When an identical second cue 62 Head r-----~ Direction ~ Path r-----'--~ Integral_.I.o ........ .J (xp,Y,,> Goal Memory Local View (T~ 'I' .;) Place COde A (It) A. D. REDISH, D. S. TOURETZKY Figure 1: Organization of the rodent navigation model. card was added opposite the first (Figure 3c), most place fields did not double. J Instead, the cells continued to fire at their original locations. However, if the rat was introduced into the double-card environment at the southeast corner (Figure 3d), the place fields rotated by 1800 • But rotation did not occur in single-card probe trials with a southeast entry point (Figure 3b). When tested with cue cards rotated by ±30°, Sharp et al. observed that place field locations were controlled by an interaction of the choice of entry point with the cue card positions (Figure 3f.) 2 The CRAWL model In earlier work (Wan et al., 1994a; Wan et al., 1994b; Redish and Touretzky, 1996) we described a model of rodent navigation that includes analogs of both place cells and the head direction system. This model also includes a local view module representing egocentric spatial information about landmarks, and a separate metric representation of location which serves as a substrate for path integration. The existence of a path integration faculty in rodents is strongly supported by behavioral data; see Maurer and Seguinot (1995) for a discussion. Hypotheses about the underyling neural mechanismss are presently being explored by several researchers, including us. The structure of our model is shown in Figure 1. Visual inputs are represented as triples of form (Ti, 'i, (Ji), each denoting the type, distance, and egocentric bearing ofa landmark. The experiments reported here used two point-type landmarks representing the left and right edges of the cue card, and one surface-type landmark representing the arena wall. For the latter, 'i and (Ji define the normal vector between the rat and the surface. In the local view module, egocentric bearings (Ji are converted to allocentric form <Pi by adding the current value represented in the head direction system, denoted as tPh . The visual angle CYij between pairs of landmarks is also part of the local view, and can be used to help localize the animal when its head direction is unknown. See Figure 2. I Five of the 18 cells recorded by Sharp et al. changed their place fields over the various recording sessions. Our model does not reproduce these effects, since it does not address changes in place cell tuning. Such changes could occur due to variations in the animal's mental state from one trial to the next, or as a result of learning across trials. Modeling Interactions of the Rat's Place and Head Direction Systems (T., r., 4>.) } 1 1 63 Figure 2: Spatial variables used in tuning a place cell to two landmarks i and j when the animal is at path integrator coordinates (xl" Yl') . Our simulated place units are radial basis functions tuned to combinations of individual landmark bearings and distances, visual angles between landmark pairs, and path integrator coordinates. Place units can be driven by visual input alone when the animal is trying to localize itself upon initial entry at a random spot in the environment, or by the path integrator alone when navigating in the dark. But normally they are driven by both sources simultaneously. A key role of the place system is to maintain associations between the two representations, so that either can be reconstructed from the other. The place system also maintains a record of allocentric bearings of landmarks when viewed from the current position; this enables the local view module to compare perceived with remembered landmark bearings, so that drift in the head direction system can be detected and corrected. In computer simulations using a single parameter set, the model reproduces a variety of behavioral and neurophysiological results including control of place fields by visual landmarks, persistence of place fields in the dark, and place fields drifting in synchrony with drift in the head direction system. Its predictions for open-field landmark-based navigation behavior match many of the experimental results of Collett et al. (1986) for gerbils. 2.1 Entering a familiar environment Upon entering a familiar environment, the model's four spatial representations (local view, head direction, place code, and path integrator coordinates) must be aligned with the current sensory input and with each other. Note that local view information is completely determined given the visual input and head direction, and place cell activity is completely determined given the local view and path integrator representations. Thus, the alignment process manipulates just two variables: head direction and path integrator coordinates. When the animal enters the environment with initial estimates for them, the alignment process can produce four possible outcomes: (1) Retain the initial values of both variables, (2) Reset the head direction, (3) Reset the path integrator, or (4) Reset both head direction and the path integrator. 2.2 Prioritizing the outcomes When the animal was placed at the northwest entry point and there were two cue cards (Figure 3c), we note that the orientation of the wall segment adjacent to the place field is identical with that in the training case. This suggests that the animal's head direction 64 A. D. REDISH, D. S. TOURETZKY did not change. The spatial relationship between the entry point and place field was also unchanged: notice that the distance from the entry point to the center of the field is the same as in Figure 3a. Therefore, we conclude that the initially estimated path integrator coordinates were retained. Alternatively, the animal could have changed both its head direction (by 180°) and its path integrator coordinates (to those of the southeast comer) and produced consistent results, but to the experimenter the place field would appear to have flipped to the other card. Because no flip was observed, the first outcome must have priority over the fourth. In panel d, where the place field has flipped to the northwest comer, the orientation of the segment of wall adjacent to the field has changed, but the spatial relationship between the entry point and field center has not. Resetting the path integrator and not the head direction would also give a solution consistent with this local view, but with the place field unflipped (as in panel b). We conclude that the second outcome (reset head direction) must have priority over the third (reset the path integrator). The third and fourth outcomes are demonstrated in Figures 3b and 3f. In panel b, the orientation of the wall adjacent to the place field is unchanged from panel a, but the spatial relationship between the entry point and the place field center is different, as evidenced by the fact that the distance between them is much reduced. This is outcome 3. In panel f, both variables have changed (outcome 4). Finally, the fact that place fields are stable over an entire session, even when there are multiple cue cards (and therefore multiple consistent pairings of head directions and path integrator coordinates) implies that animals do not reset their head direction or path integrator in visually ambiguous environments as long as the current values are reasonably consistent with the local view. We therefore assume that outcome 1 is preferred over the others. This analysis establishes a partial ordering over the four outcomes: 1 is preferred over 4 by Figure 3c, and over the others by the stability of place fields, and outcome 2 is preferred over 3 by Figure 3d. This leaves open the question of whether outcome 3 or 4 has priority over the other. In this experiment, after resetting the path integrator it's always safe for the animal to attempt to reset its head direction. If the head direction does not change by more than a few degrees, as in panel b, we observe outcome 3; if it does change substantially, as in panel f, we observe outcome 4. 2.3 Consistency The viability of an outcome is a function of the consistency between the local view and path integrator representations. The place system maintains the association between the two representations and mediates the comparison between them. The activity A(u) of a place unit is the product of a local view term LV(u) and a path integrator term C(u). LV(u) is in turn a product of five Gaussians: two tuned to bearings and two to distances (for the same' pair of landmarks), and one tuned to the retinal angle between a pair of landmarks. C(u) is a Gaussian tuned to the path integrator coordinates of the center of the place field. If the two representations agree, then the place units activated by path integrator input will be the same as those activated by the local view module, so the product A(u) computed by those units will be significantly greater than zero. The consistency K, of the association Modeling Interactions of the Rat's Place and Head Direction Systems 65 between path integrator and local view representations is given by: K, = Lu A(u)/ Lu C(u). Because A(u) < C(u) for all place units, K, ranges between 0 and 1. When the current local view is compatible with that predicted by the current path integrator coordinates, K, will be high; when the two are not compatible, K, will be low. Earlier we showed that the navigation system should choose the highest priority viable outcome. If the consistency of an outcome is more than K, * better than all higher-priority outcomes, that outcome is a viable choice and higher-priority ones are not. K,* is an empirically derived constant that we have set equal to 0.04. 3 Discussion Our results match all of the cases already discussed. (See Figure 3, panels a through d as well as f and h.) Sharp et al. (1990) did not actually test the rotated cue cards with a northwest entry point, so our result in panel e is a prediction. When the animals entered from the northwest, but only one cue card was available at 1800 , Sharp et al. report that the place field did not rotate. In our model the place field does rotate, as a result of outcome 4. This discrepancy can be explained by the fact that this particular manipulation was the last one in the sequence done by Sharp et at. McNaughton et al. (1994) and Knierim et al. (1995) have shown that if rats experience the cue card moving over a number of sessions, they eventually come to ignore it and it loses control over place fields. When we tested our model without a cue card (equivalent to a card being present but ignored), the resulting place field was more diffuse than normal but showed no rotation; see Figure 3g. We thus predict that if this experiment had been done before the other manipulations rather than after, the place field would have foIlowed the cue card. In the Sharp et al. experiment, the animals were always placed in the environment at the same location during training. Therefore, they could reliably estimate their initial path integrator coordinates. They also had a reliable head direction estimate because they were not disoriented. We predict that were the rats trained with a variety of entry points instead of just one, using an environment with a single cue card at 00 (the training environment used by Sharp et al.), and then tested with two cue cards at 00 and 1800 , the place field would not rotate no matter what entry point was used. This is because when trained with a variable entry point, the animal would not learn to anticipate its path integrator coordinates upon entry; a path integratorreset would have to be done every time in order to establish the animal's coordinates. The reset mechanism uses allocentric bearing information derived from the head direction estimate, and in this task the resulting path integrator coordinates will be consistent with the initial head direction estimate. Hence, outcome 3 will always prevail. If the animal is disoriented, however, then both the path integrator and the head direction system must be reset upon entry (because consistency will be low with a faulty head direction), and the animal must choose one cue card or the other to match against its memory. So with disorientation and a variable entry point, the place field will be controlled by one or the other cue card with a 50/50 probability. This was found to be true in a related behavioral experiment by Cheng (1986). Our model shows how interactions between the place and head direction systems handle the various combinations of entry point, number of cue cards, and amount of cue card rotation. It predicts that head direction reset will be observed in certain tasks and not in others. In 66 A. D. REDISH, D. S. TOURETZKY experiments such as the single cue card task with an entry in the southeast, it predicts the place code will shift from an initial value corresponding to the northwest entry point to the value for the southeast entry point, but the head direction will not change. This could be tested by recording simultaneously from place cells and head direction cells. References Cheng, K. (1986). A purely geometric module in the rat's spatial representation. Cognition, 23: 149-178. Collett, T., Cartwright, B. A., and Smith, B. A. (1986). Landmark learning and visuospatial memories in gerbils. Journal of Comparative Physiology A, 158:835-851. Knierim, J. J., Kudrimoti, H. 5., and McNaughton, B. L. (1995). Place cells, head direction cells, and the learning of landmark stability. Journal of Neuroscience, 15: 164859. Maurer, R. and Seguinot, V. (1995). What is modelling for? A critical review of the models of path integration. Journal of Theoretical Biology, 175:457-475. McNaughton, B. L., Mizumori, S. J. Y., Barnes, C. A., Leonard, B. 1., Marquis, M., and Green, E. J. (1994). Cortical rpresentation of motion during unrestrained spatial navigation in the rat. Cerebral Cortex, 4(1):27-39. Muller, R. U., Kubie, 1. L., Bostock, E. M., Taube, J. 5., and Quirk, G. 1. (1991). Spatial firing correlates of neurons in the hippocampal formation of freely moving rats. In Paillard, J., editor, Brain and Space, chapter 17, pages 296-333. Oxford University Press, New York. Redish, A. D. and Touretzky, D. s. (1996). Navigating with landmarks: Computing goal locations from place codes. In Ikeuchi, K. and Veloso, M., editors, Symbolic Visual Learning. Oxford University Press. In press. Sharp, P. E., Kubie, J. L., and Muller, R. U. (1990). Firing properties of hippocampal neurons in a visually symmetrical environment: Contributions of multiple sensory cues and mnemonic processes. Journal of Neuroscience, 10(9):3093-3105. Taube, 1. s. (1995). Head direction cells recorded in the anterior thalamic nuclei of freely moving rats. Journal of Neuroscience, 15(1): 1953-1971. Taube, J. 5., Muller, R. I., and Ranck, Jr., J. B. (1990). Head direction cells recorded from the postsubiculum in freely moving rats. I. Description and quantitative analysis. Journal of Neuroscience, 10:420-435. Wan, H. 5., Touretzky, D. 5., and Redish, A. D. (1994a). Computing goal locations from place codes. In Proceedings of the 16th annual conference of the Cognitive Science society, pages 922-927. Lawrence Earlbaum Associates, Hillsdale N1. Wan, H. 5., Touretzky, D. 5., and Redish, A. D. (1994b). Towards a computational theory of rat navigation. In Mozer, M., Smolen sky, P., Touretzky, D., Elman, J., and Weigend, A., editors, Proceedings of the 1993 Connectionist Models Summer School, pages 11-19. Lawrence Earlbaum Associates, Hillsdale NJ. Modeling Interactions of the Rat's Place and Head Direction Systems (a) 1 cue card at 0° (East) entry in Northwest comer angle of rotation (Sharp et al.) = 2.7° precession of HD system = 0 0 (c) 2 cue cards at 00 (East) & 1800 (West) entry in Northwest comer angle of rotation (Sharp et al.) = -2.3° precession of HD system = 0 0 (e) 2 cue cards at 330 0 & 150 0 entry in Northwest comer not done by Sharp et al. precession of HD system = 331 0 (g) I cue card at 1800 (West) entry in Northwest comer angle of rotation (Sharp et al.) ::: -5.5 0 precession of HD system = 00 (b) 1 cue card at 00 entry in Southeast comer angle of rotation (Sharp et al.) = -6.0 0 precession of HD system = 2° (d) 2 cue cards at 00 & 180 0 entry in Southeast comer angle of rotation (Sharp et al.) = 182.5 0 precession of HD system::: 178 0 (f) 2 cue cards at 3300 & 150 0 entry in Southeast comer angle of rotation (Sharp et al.) = 158.3° precession of HD system = 151 ° (h) 1 cue card at 180 0 entry in Southeast comer angle of rotation (Sharp et al.) = 182.2° precession of HD system = 179 0 67 Figure 3: Computer simulations of the Sharp et al. (1990) experiment showing that place fields are controlled by both cue cards (thick arcs) and entry point (arrowhead). "Angle of rotation" is the angle at which the correlation between the probe and training case place fields is maximal. Because head direction and place code are tightly coupled in our model, precession of HD is an equivalent measure in our model.	1995	56
1,102	Empirical Entropy Manipulation for Real-World Problems Paul Viola: Nicol N. Schraudolph, Terrence J. Sejnowski Computational Neurobiology Laboratory The Salk Institute for Biological Studies 10010 North Torrey Pines Road La Jolla, CA 92037-1099 viola@salk.edu Abstract No finite sample is sufficient to determine the density, and therefore the entropy, of a signal directly. Some assumption about either the functional form of the density or about its smoothness is necessary. Both amount to a prior over the space of possible density functions. By far the most common approach is to assume that the density has a parametric form. By contrast we derive a differential learning rule called EMMA that optimizes entropy by way of kernel density estimation. Entropy and its derivative can then be calculated by sampling from this density estimate. The resulting parameter update rule is surprisingly simple and efficient. We will show how EMMA can be used to detect and correct corruption in magnetic resonance images (MRI). This application is beyond the scope of existing parametric entropy models. 1 Introduction Information theory is playing an increasing role in unsupervised learning and visual processing. For example, Linsker has used the concept of information maximization to produce theories of development in the visual cortex (Linsker, 1988). Becker and Hinton have used information theory to motivate algorithms for visual processing (Becker and Hinton, 1992). Bell and Sejnowski have used information maximization • Author to whom correspondence should be addressed. Current address: M.LT., 545 Technology Square, Cambridge, MA 02139. 852 P. VIOLA, N. N. SCHRAUDOLPH, T. J. SEJNOWSKI to solve the "cocktail party" or signal separation problem (Bell and Sejnowski, 1995). In order to simplify analysis and implementation, each of these techniques makes specific assumptions about the nature of the signals used, typically that the signals are drawn from some parametric density. In practice, such assumptions are very inflexible. In this paper we will derive a procedure that can effectively estimate and manipulate the entropy of a wide variety of signals using non-parametric densities. Our technique is distinguished by is simplicity, flexibility and efficiency. We will begin with a discussion of principal components analysis (PCA) as an example of a simple parametric entropy manipulation technique. After pointing out some of PCA's limitation, we will then derive a more powerful non-parametric entropy manipulation procedure. Finally, we will show that the same entropy estimation procedure can be used to tackle a difficult visual processing problem. 1.1 Parametric Entropy Estimation Typically parametric entropy estimation is a two step process. We are given a parametric model for the density of a signal and a sample. First, from the space of possible density functions the most probable is selected. This often requires a search through parameter space. Second, the entropy of the most likely density function is evaluated. Parametric techniques can work well when the assumed form of the density matches the actual data. Conversely, when the parametric assumption is violated the resulting algorithms are incorrect. The most common assumption, that the data follow the Gaussian density, is especially restrictive. An entropy maximization technique that assumes that data is Gaussian, but operates on data drawn from a non-Gaussian density, may in fact end up minimizing entropy. 1.2 Example: Principal Components Analysis There are a number of signal processing and learning problems that can be formulated as entropy maximization problems. One prominent example is principal component analYllill (PCA). Given a random variable X, a vector v can be used to define a new random variable, Y" = X . v with variance Var(Y,,) = E[(X . v - E[X . v])2]. The principal component v is the unit vector for which Var(Yv) is maximized. In practice neither the density of X nor Y" is known. The projection variance is computed from a finite sample, A, of points from X, Var(Y,,) ~ Var(Y,,) == EA[(X . v - EA[X . v])2] , (1) A where VarA(Y,,) and E A [·] are shorthand for the empirical variance and mean evaluated over A. Oja has derived an elegant on-line rule for learning v when presented with a sample of X (Oja, 1982). Under the assumption that X is Gaussian is is easily proven that Yv has maximum entropy. Moreover, in the absence of noise, Yij, contains maximal information about X. However, when X is not Gaussian Yij is generally not the most informative projection. 2 Estimating Entropy with Parzen Densities We will now derive a general procedure for manipulating and estimating the entropy of a random variable from a sample. Given a sample of a random variable X, we can Empirical Entropy Manipulation for Real-world Problems 853 construct another random variable Y = F(X,l1). The entropy, heY), is a function of v and can be manipulated by changing 11. The following derivation assumes that Y is a vector random variable. The joint entropy of a two random variables, h(Wl' W2), can be evaluated by constructing the vector random variable, Y = [Wl' w2jT and evaluating heY). Rather than assume that the density has a parametric form, whose parameters are selected using maximum likelihood estimation, we will instead use Parzen window density estimation (Duda and Hart, 1973). In the context of entropy estimation, the Parzen density estimate has three significant advantages over maximum likelihood parametric density estimates: (1) it can model the density of any signal provided the density function is smooth; (2) since the Parzen estimate is computed directly from the sample, there is no search for parameters; (3) the derivative of the entropy of the Parzen estimate is simple to compute. The form of the Parzen estimate constructed from a sample A is p.(y, A) = ~A I: R(y - YA) = EA[R(y - YA)] , YAEA (2) where the Parzen estimator is constructed with the window function R(·) which integrates to 1. We will assume that the Parzen window function is a Gaussian density function. This will simplify some analysis, but it is not necessary. Any differentiable function could be used. Another good choice is the Cauchy density. Unfortunately evaluating the entropy integral hey) ~ -E[log p.(~, A)] = -i: log p.(y, A)dy is inordinately difficult. This integral can however be approximated as a sample mean: (3) where EB{ ] is the sample mean taken over the sample B. The sample mean converges toward the true expectation at a rate proportional to 1/ v' N B (N B is the size of B). To reiterate, two samples can be used to estimate the entropy of a density: the first is used to estimate the density, the second is used to estimate the entropyl. We call h· (Y) the EMMA estimate of entropy2. One way to extremize entropy is to use the derivative of entropy with respect to v. This may be expressed as ~h(Y) ~ ~h·(Y) = __ 1_ '" LYAEA f;gt/J(YB - YA) (4) dl1 dv N B L....iB Ly EA gt/J(YB - YA) YBE A 1 d 1 = NB I: I: Wy (YB , YA) dl1 "2 Dt/J(YB - YA), (5) YBEB YAEA _ gt/J(Yl - Y2) where WY(Yl' Y2) = L ( ) , (6) YAEA gt/J Yl - YA Dt/J(Y) == yT.,p-ly, and gt/J(Y) is a multi-dimensional Gaussian with covariance .,p. Wy(Yl' Y2) is an indicator of the degree of match between its arguments, in a "soft" lUsing a procedure akin to leave-one-out cross-validation a single sample can be used for both purposes. 2EMMA is a random but pronounceable subset of the letters in the words "Empirical entropy Manipulation and Analysis". 854 P. VIOLA, N. N. SCHRAUDOLPH, T. J. SEJNOWSKl sense. It will approach one if Yl is significantly closer to Y2 than any element of A. To reduce entropy the parameters v are adjusted such that there is a reduction in the average squared distance between points which Wy indicates are nearby. 2.1 Stochastic Maximization Algorithm Both the calculation of the EMMA entropy estimate and its derivative involve a double summation. As a result the cost of evaluation is quadratic in sample size: O(NANB). While an accurate estimate of empirical entropy could be obtained by using all of the available data (at great cost), a stochastic estimate of the entropy can be obtained by using a random subset of the available data (at quadratically lower cost). This is especially critical in entropy manipulation problems, where the derivative of entropy is evaluated many hundreds or thousands of times. Without the quadratic savings that arise from using smaller samples entropy manipulation would be impossible (see (Viola, 1995) for a discussion of these issues). 2.2 Estimating the Covariance In addition to the learning rate .A, the covariance matrices of the Parzen window functions, g,p, are important parameters of EMMA. These parameters may be chosen so that they are optimal in the maximum likelihood sense. For simplicity, we assume that the covariance matrices are diagonal,.,p = DIAG(O"~,O"~, ... ). Following a derivation almost identical to the one described in Section 2 we can derive an equation analogous to (4), d. 1"" "" ( 1 ) ([y]~ ) -h (Y) = L...J L...J WY(YB' YA) -- - 1 dO"k N B b O"k O"~ YsE YAEa (7) where [Y]k is the kth component of the vector y. The optimal, or most likely, .,p minimizes h· (Y). In practice both v and .,p are adjusted simultaneously; for example, while v is adjusted to maximize h· (YlI ), .,p is adjusted to minimize h· (y,,). 3 Principal Components Analysis and Information As a demonstration, we can derive a parameter estimation rule akin to principal components analysis that truly maximizes information. This new EMMA based component analysis (ECA) manipulates the entropy of the random variable Y" = X·v under the constraint that Ivl = 1. For any given value of v the entropy of Yv can be estimated from two samples of X as: h·(Yv ) = -EB[logEA[g,p(xB·v - XA· v)]], where .,p is the variance of the Parzen window function. Moreover we can estimate the derivative of entropy: d~ h·(YlI ) = ; L L Wy(YB, YA) .,p-l(YB - YA)(XB - XA) , B B A where YA = XA . v and YB = XB . v. The derivative may be decomposed into parts which can be understood more easily. Ignoring the weighting function Wy.,p-l we are left with the derivative of some unknown function f(y"): d 1 dvf(Yv ) = N N L L(YB - YA)(XB - XA) (8) B A B A What then is f(y")? The derivative of the squared difference between samples is: d~ (YB - YA)2 = 2(YB - YA)(XB - XA) . So we can see that f(Y,,) = 2N IN L L(YB - YA)2 B A B A Empirical Entropy Manipulation for Real-world Problems 3 2 o -I -2 •• t -3 -4 -2 • I o : . ECA-MIN ECA-MAX BCM BINGO PCA 2 4 Figure 1: See text for description. 855 is one half the expectation of the squared difference between pairs of trials of Yv • Recall that PCA searches for the projection, Yv , that has the largest sample variance. Interestingly, f(Yv ) is precisely the sample variance. Without the weighting term Wll,p-l, ECA would find exactly the same vector that PCA does: the maximum variance projection vector. However because of Wll , the derivative of ECA does not act on all points of A and B equally. Pairs of points that are far apart are forced no further apart. Another way of interpreting ECA is as a type of robust variance maximization. Points that might best be interpreted as outliers, because they are very far from the body of other points, playa very small role in the minimization. This robust nature stands in contrast to PCA which is very sensitive to outliers. For densities that are Gaussian, the maximum entropy projection is the first principal component. In simulations ECA effectively finds the same projection as PCA, and it does so with speeds that are comparable to Oja's rule. ECA can be used both to find the entropy maximizing (ECA-MAX) and minimizing (ECA-MIN) axes. For more complex densities the PCA axis is very different from the entropy maximizing axis. To provide some intuition regarding the behavior of ECA we have run ECAMAX, ECA-MIN, Oja's rule, and two related procedures, BCM and BINGO, on the same density. BCM is a learning rule that was originally proposed to explain development of receptive fields patterns in visual cortex (Bienenstock, Cooper and Munro, 1982). More recently it has been argued that the rule finds projections that are far from Gaussian (Intrator and Cooper, 1992). Under a limited set of conditions this is equivalent to finding the minimum entropy projection. BINGO was proposed to find axes along which there is a bimodal distribution (Schraudolph and Sejnowski, 1993). Figure 1 displays a 400 point sample and the projection axes discussed above. The density is a mixture of two clusters. Each cluster has high kurtosis in the horizontal direction. The oblique axis projects the data so that it is most uniform and hence has the highest entropy; ECA-MAX finds this axis. Along the vertical axis the data is clustered and has low entropy; ECA-MIN finds this axis. The vertical axis also has the highest variance. Contrary to published accounts, the first principal component can in fact correspond to the minimum entropy projection. BCM, while it may find minimum entropy projections for some densities, is attracted to the kurtosis along the horizontal axis. For this distribution BCM neither minimizes nor maximizes entropy. Finally, BINGO successfully discovers that the vertical axis is very bimodal. 856 P. VIOLA, N. N. SCHRAUOOLPH, T. J. SEJNOWSKI 1200 1000 800 600 400 200 ~ .1 0 0.1 0.2 0.3 0.4 \ Corrupted:. Corrected .• . ' .. . : '. 0.7 0.8 0.9 Figure 2: At left: A slice from an MRI scan of a head. Center: The scan after correction. Right: The density of pixel values in the MRI scan before and after correction. 4 Applications EMMA has proven useful in a number of applications. In object recognition EMMA has been used align 3D shape models with video images (Viola and Wells III, 1995). In the area of medical imaging EMMA has been used to register data that arises from differing medical modalities such as magnetic resonance images, computed tomography images, and positron emission tomography (Wells, Viola and Kikinis, 1995). 4.1 MRI Processing In addition, EMMA can be used to process magnetic resonance images (MRI). An MRI is a 2 or 3 dimensional image that records the density of tissues inside the body. In the head, as in other parts of the body, there are a number of distinct tissue classes including: bone, water, white matter, grey matter, and fat. ~n principle the density of pixel values in an MRI should be clustered, with one cluster for each tissue class. In reality MRI signals are corrupted by a bias field, a multiplicative offset that varies slowly in space. The bias field results from unavoidable variations in magnetic field (see (Wells III et al., 1994) for an overview of this problem). Because the densities of each tissue type cluster together tightly, an uncorrupted MRI should have relatively low entropy. Corruption from the bias field perturbs the MRI image, increasing the values of some pixels and decreasing others. The bias field acts like noise, adding entropy to the pixel density. We use EMMA to find a low-frequency correction field that when applied to the image, makes the pixel density have a lower entropy. The resulting corrected image will have a tighter clustering than the original density. Call the uncorrupted scan s(z); it is a function of a spatial random variable z. The corrupted scan, c( x) = s( z) + b( z) is a sum of the true scan and the bias field. There are physical reasons to believe b( x) is a low order polynomial in the components of z. EMMA is used to minimize the entropy of the corrected signal, h( c( x) - b( z, v», where b( z, v), a third order polynomial with coefficients v, is an estimate for the bias corruption. Figure 2 shows an MRI scan and a histogram of pixel intensity before and after correction. The difference between the two scans is quite subtle: the uncorrected scan is brighter at top right and dimmer at bottom left. This non-homogeneity Empirical Entropy Manipulation for Real-world Problems 857 makes constructing automatic tissue classifiers difficult. In the histogram of the original scan white and grey matter tissue classes are confounded into a single peak ranging from about 0.4 to 0.6. The histogram of the corrected scan shows much better separation between these two classes. For images like this the correction field takes between 20 and 200 seconds to compute on a Sparc 10. 5 Conclusion We have demonstrated a novel entropy manipulation technique working on problems of significant complexity and practical importance. Because it is based on nonparametric density estimation it is quite flexible, requiring no strong assumptions about the nature of signals. The technique is widely applicable to problems in signal processing, vision and unsupervised learning. The resulting algorithms are computationally efficient. Acknowledgements This research was support by the Howard Hughes Medical Institute. References Becker, S. and Hinton, G. E. (1992). A self-organizing neural network that discovers surfaces in random-dot stereograms. Nature, 355:161-163. Bell, A. J. and Sejnowski, T. J. (1995). An information-maximisation approach to blind separation. In Tesauro, G., Touretzky, D. S., and Leen, T. K., editors, Advance8 in Neural Information Proce88ing, volume 7, Denver 1994. MIT Press, Cambridge. Bienenstock, E., Cooper, L., and Munro, P. (1982). Theory for the development of neuron selectivity: Orientation specificity and binocular interaction in visual cortex. Journal of Neur08cience, 2. Duda, R. and Hart, P. (1973). Pattern Cla88ification and Scene AnalY8i8. Wiley, New York. Intrator, N. and Cooper, L. N. (1992). Objective function formulation of the bcm theory of visual cortical plasticity: Statistical connections, stability conditions. Neural Network., 5:3-17. Linsker, R. (1988). Self-organization in a perceptual network. IEEE Computer, pages 105-117. Oja, E. (1982). A simplified neuron model as a principal component analyzer. Journal of Mathematical Biology, 15:267-273. 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, Advance. in Neural Information Proce88ing, volume 5, pages 499-506, Denver 1992. Morgan Kaufmann, San Mateo. Viola, P. A. (1995). Alignment by Ma:cimization of Mutual Information. PhD thesis, Massachusetts Institute of Technology. MIT AI Laboratory TR 1548. Viola, P. A. and Wells III, W. M. (1995). Alignment by maximization of mutual information. In Fifth Inti. Conf. on Computer Vi8ion, pages 16-23, Cambridge, MA. IEEE. Wells, W., Viola, P., and Kikinis, R. (1995). Multi-modal volume registration by maximization of mutual information. In Proceeding. of the Second International Sympo8ium on Medical Robotic. and Computer A88i8ted Surgery, pages 55 - 62. Wiley. Wells III, W., Grimson, W., Kikinis, R., and Jolesz, F. (1994). Statistical Gain Correction and Segmentation of MRI Data. In Proceeding. of the Computer Society Conference on Computer Vi.ion and Pattern Recognition, Seattle, Wash. IEEE, Submitted.	1995	57
1,103	A Neural Network Model of 3-D Lightness Perception Luiz Pessoa Federal Univ. of Rio de Janeiro Rio de Janeiro, RJ, Brazil pessoa@cos.ufrj.br Abstract William D. Ross Boston University Boston, MA 02215 bill@cns.bu.edu A neural network model of 3-D lightness perception is presented which builds upon the FACADE Theory Boundary Contour System/Feature Contour System of Grossberg and colleagues. Early ratio encoding by retinal ganglion neurons as well as psychophysical results on constancy across different backgrounds (background constancy) are used to provide functional constraints to the theory and suggest a contrast negation hypothesis which states that ratio measures between coplanar regions are given more weight in the determination of lightness of the respective regions. Simulations of the model address data on lightness perception, including the coplanar ratio hypothesis, the Benary cross, and White's illusion. 1 INTRODUCTION Our everyday visual experience includes surface color constancy. That is, despite 1) variations in scene lighting and 2) movement or displacement across visual contexts, the color of an object appears to a large extent to be the same. Color constancy refers, then, to the fact that surface color remains largely constant despite changes in the intensity and composition of the light reflected to the eyes from both the object itself and from surrounding objects. This paper discusses a neural network model of 3D lightness perception i.e., only the achromatic or black to white dimension of surface color perception is addressed. More specifically, the problem of background constancy (see 2 above) is addressed and mechanisms to accomplish it in a system exhibiting illumination constancy (see 1 above) are proposed. A landmark result in the study of lightness was an experiment reported by Wallach (1948) who showed that for a disk-annulus pattern, lightness is given by the ratio of disk and annulus luminances (i.e., independent of overall illumination); the A Neural Network Model of 3-D Lightness Perception 845 so-called ratio principle. In another study, Whittle and Challands (1969) had subjects perform brightness matches in a haploscopic display paradigm. A striking result was that subjects always matched decrements to decrements, or increments to increments, but never increments to decrements. Whittle and Challands' (1969) results provide psychophysical support to the notion that the early visual system codes luminance ratios and not absolute luminance. These psychophysical results are in line with results from neurophysiology indicating that cells at early stages of the visual system encode local luminance contrast (Shapley and Enroth-Cugell, 1984). Note that lateral inhibition mechanisms are sensitive to local ratios and can be used as part of the explanation of illumination constancy. Despite the explanatory power of the ratio principle, and the fact that the early stages of the visual system likely code contrast, several experiments have shown that, in general, ratios are insufficient to account for surface color perception. Studies of background constancy (Whittle and Challands, 1969; Land and McCann, 1971; Arend and Spehar, 1993), of the role of 3-D spatial layout and illumination arrangement on lightness perception (e.g. , Gilchrist, 1977) as well as many other effects, argue against the sufficiency of local contrast measures (e.g., Benary cross, White's, 1979 illusion). The neural network model presented here addresses these data using several fields of neurally plausible mechanisms of lateral inhibition and excitation. 2 FROM LUMINANCE RATIOS TO LIGHTNESS The coplanar ratio hypothesis (Gilchrist, 1977) states that the lightness of a given region is determined predominantly in relation to other coplanar surfaces, and not by equally weighted relations to all retinally adjacent regions. We propose that in the determination of lightness, contrast measures between non-coplanar adjacent surfaces are partially negated in order to preserve background constancy. Consider the Benary Cross pattern (input stimulus in Fig. 2). If the gray patch on the cross is considered to be at the same depth as the cross, while the other gray patch is taken to be at the same depth as the background (which is below the cross), the gray patch on the cross should look lighter (since its lightness is determined in relation to the black cross), and the other patch darker (since its lightness is determined in relation to the white background) . White's (1979) illusion can be discussed in similar terms (see the input stimulus in Fig. 3). The mechanisms presented below implement a process of partial contrast negation in which the initial retinal contrast code is modulated by depth information such that the retinal contrast consistent with the depth interpretation is maintained while the retinal contrast not supported by depth is negated or attenuated. 3 A FILLING-IN MODEL OF 3-D LIGHTNESS Contrast/Filling-in models propose that initial measures of boundary contrast followed by spreading of neural activity within filling-in compartments produce a response profile isomorphic with the percept (Gerrits & Vendrik, 1970; Cohen & Grossberg, 1984; Grossberg & Todorovic, 1988; Pessoa, Mingolla, & Neumann, 1995). In this paper we develop a neural network model of lightness perception in the tradition of contrast/filling-in theories. The neural network developed here is an extension of the Boundary Contour System/Feature Contour System (BCS/FCS) proposed by Cohen and Grossberg (1984) and Grossberg and Mingolla (1985) to explain 3-D lightness data. 846 L. PESSOA. W. D. ROSS A fundamental idea of the BCS/FCS theory is that lateral inhibition achieves illumination constancy but requires the recovery of lightness by the filling-in, or diffusion, of featural quality ("lightness" in our case). The final diffused activities correspond to lightness, which is the outcome of interactions between boundaries and featural quality, whereby boundaries control the process of filling-in by forming gates of variable resistance to diffusion. H ow can the visual system construct 3-D lightness percepts from contrast measures obtained by retinotopic lateral inhibition? A mechanism that is easily instantiated in a neural model and provides a straightforward modification to the contrast/fillingin proposal of Grossberg and Todorovic (1988) is the use of depth-gated filling-in. This can be accomplished through a pathway that modulates boundary strength for boundaries between surfaces or objects across depth. The use of permeable or "leaky" boundaries was also used by Grossberg and Todorovic (1988) for 2-D stimuli. In the current usage, permeability is actively increased at depth boundaries to partially negate the contrast effect since filling-in proceeds more freely and thus preserve lightness constancy across backgrounds. Figure 1 describes the four computational stages of the system. I BOUNDARIES ,...---------, ~ ~ ON/OFF ~FILTERING j ~ I RLLlNG-IN I Figure 1: Model components. I '" I DEPTH I MAP Stage 1: Contrast Measurement. At this stage both ON and OFF neural fields with lateral inhibitory connectivity measure the strength of contrast at image regions in uniform regions a contrast measurement of zero results. Formally, the ON field is given by dyi; _ + + ) + (+ ) + dt - -aYij + ((3 Yij Cij Yij + 'Y Eij (1) where a , (3 and 'Yare constants; ct is the total excitatory input to yi; and Et; is the total inhibitory input to yi;. These terms denote discrete convolutions of the input Iij with Gaussian weighting functions, or kernels. An analogous equation specifies Yi; for the OFF field. Figure 2 shows the ON-contrast minus the OFF-contrast. Stage 2: 2-D Boundary Detection. At Stage 2, oriented odd-symmetric boundary detection cells are excited by the oriented sampling of the ON and OFF Stage 1 cells. Responses are maximal when ON activation is strong on one side of a cell's receptive field and OFF activation is strong on the opposite side. In other words, the cells are tuned to ON/OFF contrast co-occurrence, or juxtaposition (see Pessoa et aI., 1995). The output at this stage is the sum of the activations of such cells at each location for all orientations. The output responses are sharpened and localized through lateral inhibition across space; an equation similar to Equation 1 is used. The final output of Stage 2 is given by the signals Zij (see Fig. 2, Boundaries). Stage 3: Depth Map. In the current implementation a simple scheme was employed for the determination of the depth configuration. Initially, four types of A Neural Network Model of 3-D Lightness Perception 847 T-junction cells detect such configurations in the image. For example, Iij = Zi-d,j x Zi+d,j x Zi ,j+d, (2) where d is a constant, detects T-junctions, where left, right, and top positions of the boundary stage are active; similar cells detect T-junctions of different orientations. The activities of the T-junction cells are then used in conjunction with boundary signals to define complete boundaries. Filling-in within these depth boundaries results in a depth map (see Fig. 2, Depth Map). Stage 4: Depth-modulated Filling-in. In Stage 4, the ON and OFF contrast measures are allowed to diffuse across space within respective filling-in regions. Diffusion is blocked by boundary activations from Stage 2 (see Grossberg & Todorovic, 1988, for details). The diffusion process is further modulated by depth information. The depth map provides this information; different activities code different depths. In a full blown implementation of the model, depth information would be obtained by the depth segmentation of the image supported by both binocular disparity and monocular depth cues. Depth-modulated filling-in is such that boundaries across depths are reduced in strength. This allows a small percentage of the contrast on either side ofthe boundary to leak across it resulting in partial contrast negation, or reduction, at these boundaries. ON and OFF filling-in domains are used which receive the corresponding ON and OFF contrast activities from Stage 1 as inputs (see Fig. 2, Filled-in). 4 SIMULATIONS The present model can account for several important phenomena, including 2 - D effects of lightness constancy and contrast (see Grossberg and Todorovic, 1988). The simulations that follow address 3 -D lightness effects. 4.1 Benary Cross Figure 2 shows the simulation for the Benary Cross. The plotted gray level values for filling-in reflect the activities of the ON filling-in domain minus the OFF domain. The model correctly predicts that the patch on the cross appears lighter than the patch on the background. This result is a direct consequence of contrast negation. The depth relationships are such that the patch on the cross is at the same depth as the cross and the patch on the background is at the same depth as the background (see Fig. 2, Depth Map) . Therefore, the ratio of the background to the patch on the cross (across a depth boundary) and the ratio of the cross to the patch on the background (also across a depth boundary), are given a smaller weight in the lightness computation. Thus, the background will have a stronger effect on the appearance of the patch on the background, which will appear darker. At the same time, the cross will have a greater effect on the appearance of the patch on the cross, which will appear lighter. 4.2 White's lllusion White's (1979) illusion (Fig. 3) is such that the gray patches on the black stripes appear lighter than the gray patches on the white stripes. This effect is considered a puzzling violation of simultaneous contrast since the contour length of the gray patches is larger for the stripes they do not lie on. Simultaneous contrast would predict that the gray patches on the black stripes appear lighter than the ones on white. 848 Stimulus I I I L~ - ---I Boundaries ON-OFF Contrast L. PESSOA, W. D. ROSS Depth Map Filled-in Figure 2: Benary Cross. The filled-in values of the gray patch on the cross are higher than the ones for the gray patch on the background. Gray levels code intensity; darker grays code lower values, lighter grays code higher values. Figure 3 shows the result of the model for White's effect. The T-junction information in the stimulus determines that the gray patches are coplanar with the patches they lie on. Therefore, their appearance will be determined in relation to the contrast of their respective backgrounds. This is obtained, again, through contrast modulation, where the contrast of, say, the gray patch on a black stripe is preserved, while the contrast of the same patch with the white is partially negated (due to the depth arrangement). 4.3 Coplanar Hypothesis Gilchrist (1977) showed that the perception of lightness is not determined by retinal adjacency, and that depth configuration and spatial layout help specify lightness. More specifically, it was proposed that the ratio of coplanar surfaces, not necessarily retinally adjacent, determines lightness, the so-called coplanar ratio hypothesis. Gilchrist was able to convincingly demonstrate this by comparing the perception of lightness in two equivalent displays (in terms of luminance values), aside from the perceived depth relationships in the displays. Figure 4 shows computer simulations of the coplanar ratio effect. The same stimulus is given as input in two simulations with different depth specifications. In one (Depth Map 1), the depth map specifies that the rightmost patch is at a different depth than the two leftmost patches which are coplanar. In the other (Depth Map 2), the two rightmost patches are coplanar and at a different depth than the leftmost patch. In all, the depth organization alters the lightness of the central region, which should appear darker in the configuration of Depth Map 1 than the one for Depth Map 2. For Depth Map 1, since the middle patch is coplanar with a white patch, this patch is darkened by simultaneous contrast. For Depth Map 2, the middle patch will be lightened by contrast since it is coplanar with a black patch. It should be noted that the depth maps for the simulations shown in Fig. 4 were given as input. A Neural Network Model of 3-D Lightness Perception 849 -, - --- 1 1 Boundaries Stimulus ON-OFF Contrast Filled-in Figure 3: White's effect. The filled-in values of the gray patches on the black stripes are higher than the ones for the gray patches on white stripes. The current implementation cannot recover depth trough binocular disparity and only employs monocular cues as in the previous simulations. 5 CONCLUSIONS In this paper, data from experiments on lightness perception were used to extend the BCSjFCS theory of Grossberg and colleagues to account for several challenging phenomena. The model is an initial step towards providing an account that can take into consideration the complex factors involved in 3-D vision see Grossberg (1994) for a comprehensive account of 3-D vision. Acknowledgements The authors would like to than Alan Gilchrist and Fred Bonato for their suggestions concerning this work. L. P. was supported in part by Air Force Office of Scientific Research (AFOSR F49620-92-J-0334) and Office of Naval Research (ONR N0001491-J-4100); W. R. was supported in part by HNC SC-94-001. Reference Arend, L., & Spehar, B. (1993) Lightness, brightness, and brightness contrast: 2. Reflectance variation. Perception {3 Psychophysics 54:4576-468. Cohen, M., & Grossberg, S. (1984) Neural dynamics of brightness perception: Features, boundaries, diffusion, and resonance. Perception {3 Psychophysics 36:428-456. Gerrits, H. & Vendrik, A. (1970) Simultaneous contrast, filling-in process and information processing in man's visual system. Experimental Brain Research 11:411-430. 850 L. PESSOA, W. D. ROSS Filled-in 2 Stimulus Depth Map 1 Filled-in 1 Figure 4: Gilchrist's coplanarity. The Filled-in values for the middle patch on top are higher than on bottom. Gilchrist, A. (1977) Perceived lightness depends on perceived spatial arrangement. Science 195:185-187. Grossberg, S. (1994) 3-D vision and figure-ground separation by visual cortex. Perception & Psychophysics 55:48-120. Grossberg, S., & Mingolla, E. (1985) Neural dynamics of form perception: Boundary completion, illusory figures, and neon color spreading. Psychological Review 92:173-211. Grossberg, S., & Todorovic. D. (1988). Neural dynamics of 1-D and 2-D brightness perception: A unified model of classical and recent phenomena. Perception & Psychophysics 43:241-277. Land, E., & McCann, J. (1971). Lightness and retinex theory. Journal of the Optical Society of America 61:1-11. Pessoa, L., Mingolla, E., & Neumann, H. (1995) A contrast- and luminance-driven multiscale network model of brightness perception. Vision Research 35:22012223. Shapley, R., & Enroth-Cugell, C. (1984) Visual adaptation and retinal gain controls. In N. Osborne and G. Chader (eds.), Progress in Retinal Research, pp. 263346. Oxford: Pergamon Press. Wallach, H. (1948) Brightness constancy and the nature of achromatic colors. Journal of Experimental Psychology 38: 310-324. White, M. (1979) A new effect of pattern on perceived lightness. Perception 8:413416. Whittle, P., & Challands, P. (1969) The effect of background luminance on the brightness of flashes. Vision Research 9:1095-1110.	1995	58
1,104	Adaptive Retina with Center-Surround Receptive Field Shih-Chii Lin and Kwabena Boahen Computation and Neural Systems 139-74 California Institute of Technology Pasadena, CA 91125 shih@pcmp.caltech.edu, buster@pcmp.caltech.edu Abstract Both vertebrate and invertebrate retinas are highly efficient in extracting contrast independent of the background intensity over five or more decades. This efficiency has been rendered possible by the adaptation of the DC operating point to the background intensity while maintaining high gain transient responses. The centersurround properties of the retina allows the system to extract information at the edges in the image. This silicon retina models the adaptation properties of the receptors and the antagonistic centersurround properties of the laminar cells of the invertebrate retina and the outer-plexiform layer of the vertebrate retina. We also illustrate the spatio-temporal responses of the silicon retina on moving bars. The chip has 59x64 pixels on a 6.9x6.8mm2 die and it is fabricated in 2 J-tm n-well technology. 1 Introduction It has been observed previously that the initial layers of the vertebrate and invertebrate retina systems perform very similar processing functions on the incoming input signal[1]. The response versus log intensity curves of the receptors in invertebrate and vertebrate retinas look similar. The curves show that the receptors have a larger gain for changes in illumination than to steady illumination, i.e, the receptors adapt. This adaptation property allows the receptor to respond over a large input range without saturating. Anatomically, the eyes of invertebrates differ greatly from that of vertebrates. VerAdaptive Retina with Center-Surround Receptive Field 679 tebrates normally have two simple eyes while insects have compound eyes. Each compound eye in the fly consists of 3000-4000 ommatidia and each ommatidium consists of 8 photoreceptors. Six of these receptors (which are also called RI-R6) are in a single spectral class. The other two receptors, R7 and R8 provide channels for wavelength discrimination and polarization. The vertebrate eye is divided into the outer-plexiform layer and the inner-plexiform layer. The outer-plexiform layer consists of the rods and cones, horizontal cells and bipolar cells. Invertebrate receptors depolarise in response to an increase in light, in contrast to vertebrate receptors, which hyperpolarise to an increase in light intensity. Both vertebrate and invertebrate receptors show light adaptation over at least five decades of background illumination. This adaptation property allows the retina to maintain a high transient gain to contrast over a wide range of background intensities. The invertebrate receptors project to the next layer which is called the lamina layer. This layer consists primarily of monopolar cells which show a similar response versus log intensity curve to that of vertebrate bipolar cells in the outer-plexiform layer. Both cells respond with graded potentials to changes in illumination. These cells also show a high transient gain to changes in illumination while ignoring the background intensity and they possess center-surround receptive fields. In vertebrates, the cones which are excited by the incoming light, activate the horizontal cells which in tum inhibit the cones. The horizontal cells thus mediate the lateral inhibition which produces the center-surround properties. In insects, a possible process of this lateral inhibition is done by current flow from the photoreceptors through the epithelial glial cells surrounding an ommatidium or the modulation of the local field potential in the lamina to influence the transmembrane potential of the photoreceptor[2]. The center-surround receptive fields allow contrasts to be accentuated since the surround computes a local mean and subtracts that from the center signal. Mahowald[3] previously described a silicon retina with adaptive photoreceptors and Boahen et al.[4] recently described a compact current-mode analog model of the outer-plexiform layer of the vertebrate retina and analysed the spatio-temporal processing properties of this retina[5]. A recent array of photoreceptors from Delbriick[6] uses an adaptive photoreceptor circuit that adapts its operating point to the background intensity so that the pixel shows a high transient gain over 5 decades of background illumination. However this retina does not have spatial coupling between pixels. The pixels in the silicon retina described here has a compact circuit that incorporates both spatial and temporal filtering with light adaptation over 5 decades of background intensity. The network exhibits center-surround behavior. Boahen et al.[4] in their current-mode diffusor retina, draw an analogy between parts of the diffusor circuit and the different cells in the outer-plexiform layer. While the same analogy cannot be drawn from this silicon retina to the invertebrate retina since the function of the cells are not completely understood, the output responses of the retina circuit are similar to the output responses of the photoreceptor and monopolar cells in invertebrates. The circuit details are described in Section 2 and the spatio-temporal processing performed by the retina on stimulus moving at different speeds is shown in Section 680 S.-C. LIU, K. BOAHEN 3. 2 Circuit -----VI Vb VI VI 1 p1 1 M4 Vh VI·I Vh VI+I .1. .1. .bel --------Vr MI (a) im.l iia iI ... I rrr rrr rrr 'II 'II (b) Figure 1: (a) One-dimensional version of the retina. (b) Small-signal equivalent of circuit in (a). A one-dimensional version of the retina is shown in Figure l(a). The retina consists of an adaptive photoreceptor circuit at each pixel coupled together with diffusors, controlled by voltages, Vg and Vh. The output of this network can either be obtained at the voltage output, V, or at the current output, 10 but the outputs have different properties. Phototransduction is obtained by using a reverse-biased photodiode which produces current that is proportional to the incident light. The logarithmic properties are obtained by operating the feedback transistor shown in Figure l(a) in the subthreshold region. The voltage change at the output photoreceptor, Vr , is proportional to a small contrast since UT UTdI UT i Vr = -d(logl) = -- = -K, K, 1 K, h g where UT is the thermal voltage, K, = CO:rCd ' Coz is the oxide capacitance and Cd is the depletion capacitance of a transistor. The circuit works as follows: If the photocurrent through the photodiode increases, Vr will be pulled low and the output voltage at V, increases by VI = AVr where A is the amplifier gain of the output stage. This output change in V, is coupled into Vel through a capacitor Adaptive Retina with Center-Surround Receptive Field 681 divider ratio, Cl~2C2. The feedback transistor, M4, operates in the subthreshold region and supplies the current necessary to offset the photocurrent. The increase in Vel (i.e. the gate voltage of M4) causes the current supplied by M3 to increase which pulls the node voltage, Vr , back to the voltage level needed by Ml to sink the bias current from transistor, M2. 3.5 3.45 3.4 ... -= 0 ~ • 3.35 • -2 c 0 Q. • • a: 3.3 -1 3.25 0 3.2 0 5 10 15 20 25 Time (Sec) Figure 2: This figure shows the output response of the receptor to a variation of about 40% p-p in the intensity of a flickering LED light incident on the chip. The response shows that the high sensitivity of the receptor to the LED is maintained over 5 decades of differing background intensities. The numbers on the section of the curve indicate the log intensity of the mean value. 0 log is the absolute intensity from the LED. The adaptive element, M3, has an I-V curve which looks like a hyperbolic sine. The small slope of the I-V curve in the middle means that for small changes of voltages across M3, the element looks like an open-circuit. With large changes of voltage across M3, the current through M3 becomes exponential and Vel is charged or discharged almost instantaneously. Figure 2 shows the output response of the photoreceptor to a square-wave variation of about 40% p-p in the intensity of a red LED (635 nm). The results show that the circuit is able to discern the small contrast over five decades of background intensity while the steady-state voltage of the photoreceptor output varies only about 15mV. Further details of the photoreceptor circuit and its adaptation properties are described in Delbriick[6]. 3 Spatio-Temporal Response The spatio-temporal response of the network to different moving stimuli is explored in this section. The circuit shown in Figure l(a) can be transferred to an equivalent network of resistors and capacitors as shown in Figure l(b) to obtain the transfer function of the circuit. The capacitors at each node are necessary to model the 682 8.5 i ~ 7.5 :; ... :; o i I ~ r. ... (a) 0.4 0.6 0.8 Time (Sec) 1.2 1.4 3.8 ~_---:-":--_--::'= __ ':"':-_--::':-_--::,'::-_--.J 0.3 0.4 0.5 0.6 0.7 0.8 (b) Time (Sec) S.-C. LIU, K. BOAHEN 1 lJ; Figure 3: (a) Response of a pixel to a grey strip 2 pixels wide of gray-level "0.4" on a dark background of level "0" moving past the pixel at different speeds. (b) Response of a pixel to a dark strip of gray-level "0.6" on a white background of level "1" moving past the pixel at different speeds. The voltage shown on these curves is not the direct measurement of the voltage at V, but rather V, drives a current-sensing transistor and this current is then sensed by an offchip current sense-amplifier. Adaptive Retina with Center-Surround Receptive Field 683 temporal responses of the circuit. The chip results from the experiments below illustrate the center-surround properties of the network and the difference in time-constants between the surround and center. 3.1 Chip Results Data from the 2D chip is shown in the next few figures. In these experiments, we are only looking at one pixel of the 2D array. A rotating circular fly-wheel stimulus with strips of alternating contrasts is mounted above the chip. The stimulus was created using Mathematica. Figure 3a shows the spati~temporal impulse response of one pixel measured at V, with a small strip at level "0.4" on a dark background of level "0" moving past the pixels on the row. At slow speeds, the impulse response shows a center-surround behavior where the pixel first receives inhibition from the preceding pixels which are excited by the stimulus. When the stimulus moves by the pixel of interest, it is excited and then it is inhibited by the subsequent pixels seeing the stimulus. Tim. (Sec) I o f I i Figure 4: Response of a pixel to a strip of varying contrasts on a dark background moving past the pixel at a constant speed. At faster speeds, the initial inhibition in the response grows smaller until at some even faster speed, the initial inhibition is no longer observed. This response comes about because the inhibition from the surround has a longer-time constant than the center. When the stimulus moves past the pixel of interest, the inhibition from the preceding pixels excited by the stimulus does not have time to inhibit the pixel of interest. Hence the excitation is seen first and then the inhibition comes into place when the stimulus passes by. Note that in these figures (Figures 3-4), the curves have been displaced to show the pixel response at different speeds of the moving stimulus. The voltage shown on these curves is not the direct measurement of the voltage at V, but rather V, drives a current-sensing transistor and this current is then sensed by an off-chip current sense-amplifier. Figure 3b shows the spati~temporal impulse response of one pixel with a similar 684 s.-c. LlU, K. BOAHEN size strip of level "0.6" on a light background of level "1" moving past the row of pixels. The same inhibition behavior is seen for increasing stimulus speeds. Figure 4 shows the output response at V, for the same stimulus of gray-levels varying from "0.2" to "0.8" on a dark background of level "0" moving at one speed. The peak excitation response is plotted against the contrast in Figure 5. A level of "0.2" corresponds to a irradiance of 15mW/m2 while a level of "0.8" corresponds to a irradiance of 37.4mW/m2. These measurements are done with a photometer mounted about 1.5in above a piece of paper with the contrast which is being measured. The irradiance varies exponentially with increasing level. 4 Conclusion In this paper, we described an adaptive retina with a center-surround receptive field. The system properties of this retina allows it to model functionally either the responses of the laminar cells in the invertebrate retina or the outer-plexiform layer of vertebrate retina. We show that the circuit shows adaptation to changes over 5 decades of background intensities. The center-surround property of the network can be seen from its spatio-temporal response to different stimulus speeds. This property serves to remove redundancy in space and time of the input signal. Acknowledgements We thank Carver Mead for his support and encouragement. SC Liu is supported by an NIMH fellowship and K Boahen is supported by a Sloan fellowship. We thank Tobias Delbriick for the inspiration and help in testing the design. We also thank Rahul Sarpeshkar and Bradley Minch for comments. Fabrication was provided by MOSIS. References [1] S. B. Laughlin, "Coding efficiency and design in retinal processing", In: Facets of Vision (D. G. Stavenga and R. C. Hardie, eds) pp. 213-234. Springer, Berlin, 1989. [2] S. R. Shaw, "Retinal resistance barriers and electrica1lateral inhibition", Nature, Lond.255,: 480-483, 1975. [3] M. A. Mahowald, "Silicon Retina with Adaptive Photoreceptors" in SPIE/SPSE Symposium on Electronic Science and Technology: From Neurons to Chips. Orlando, FL, April 1991. [4] K. A. Boahen and A. G. Andreou, "A Contrast Sensitive Silicon Retina with Reciprocal Synapses", In D. S. Touretzky (ed.), Advances in Neural Information Processing Systems 4, 764-772. San Mateo, CA: Morgan Kaufmann, 1992. [5] K. A. Boahen, "Spatiotemporal sensitivity of the retina: A physical model", CNS Memo CNS-TR-91-06, California Institute of Technology, Pasadena, CA 91125, June 1991. [6] T. Delbriick, "Analog VLSI Phototransduction by continous-time, adaptive, logarithmic photoreceptor circuits", CNS Memo No.30, California Institute of Technology, Pasadena, CA 91125, 1994.	1995	59
1,105	Laterally Interconnected Self-Organizing Maps in Hand-Written Digit Recognition Yoonsuck Choe, Joseph Sirosh, and Risto Miikkulainen Department of Computer Sciences The University of Texas at Austin Austin, TX 78712 yschoe,sirosh,risto@cs. u texas .ed u Abstract An application of laterally interconnected self-organizing maps (LISSOM) to handwritten digit recognition is presented. The lateral connections learn the correlations of activity between units on the map. The resulting excitatory connections focus the activity into local patches and the inhibitory connections decorrelate redundant activity on the map. The map thus forms internal representations that are easy to recognize with e.g. a perceptron network. The recognition rate on a subset of NIST database 3 is 4.0% higher with LISSOM than with a regular Self-Organizing Map (SOM) as the front end, and 15.8% higher than recognition of raw input bitmaps directly. These results form a promising starting point for building pattern recognition systems with a LISSOM map as a front end. 1 Introduction Hand-written digit recognition has become one of the touchstone problems in neural networks recently. Large databases of training examples such as the NIST (National Institute of Standards and Technology) Special Database 3 have become available, and real-world applications with clear practical value, such as recognizing zip codes in letters, have emerged. Diverse architectures with varying learning rules have been proposed, including feed-forward networks (Denker et al. 1989; Ie Cun et al. 1990; Martin and Pittman 1990), self-organizing maps (Allinson et al. 1994), and dedicated approaches such as the neocognitron (Fukushima and Wake 1990). The problem is difficult because handwriting varies a lot, some digits are easily confusable, and recognition must be based on small but crucial differences. For example, the digits 3 and 8, 4 and 9, and 1 and 7 have several overlapping segments, and the differences are often lost in the noise. Thus, hand-written digit recognition can be seen as a process of identifying the distinct features and producing an internal representation where the significant differences are magnified, making the recognition easier. Laterally Interconnected Self-organizing Maps in Handwritten Digit Recognition 737 In this paper, the Laterally Interconnected Synergetically Self-Organizing Map architecture (LISSOM; Sirosh and Miikkulainen 1994, 1995, 1996) was employed to form such a separable representation. The lateral inhibitory connections of the LISSOM map decorrelate features in the input, retaining only those differences that are the most significant. Using LISSOM as a front end, the actual recognition can be performed by any standard neural network architecture, such as the perceptron. The experiments showed that while direct recognition of the digit bitmaps with a simple percept ron network is successful 72.3% of the time, and recognizing them using a standard self-organizing map (SOM) as the front end 84.1% of the time, the recognition rate is 88.1 % based on the LISSOM network. These results suggest that LISSOM can serve as an effective front end for real-world handwritten character recognition systems. 2 The Recognition System 2.1 Overall architecture The system consists of two networks: a 20 x 20 LISSOM map performs the feature analysis and decorrelation of the input, and a single layer of 10 perceptrons the final recognition (Figure 1 (a)) . The input digit is represented as a bitmap on the 32 x 32 input layer. Each LISSOM unit is fully connected to the input layer through the afferent connections, and to the other units in the map through lateral excitatory and inhibitory connections (Figure 1 (b)). The excitatory connections are short range, connecting only to the closest neighbors of the unit, but the inhibitory connections cover the whole map. The percept ron layer consists of 10 units, corresponding to digits 0 to 9. The perceptrons are fully connected to the LISSOM map, receiving the full activation pattern on the map as their input. The perceptron weights are learned through the delta rule, and the LISSOM afferent and lateral weights through Hebbian learning. 2.2 LISSOM Activity Generation and Weight Adaptation The afferent and lateral weights in LISSOM are learned through Hebbian adaptation. A bitmap image is presented to the input layer, and the initial activity of the map is calculated as the weighted sum of the input. For unit (i, j), the initial response TJij IS TJij = (7 ('2: eabllij,ab) , a,b (1) where eab is the activation of input unit (a, b), Ilij ,ab is the afferent weight connecting input unit ( a, b) to map unit (i, j), and (7 is a piecewise linear approximation of the sigmoid activation function. The activity is then settled through the lateral connections. Each new activity TJij (t) at step t depends on the afferent activation and the lateral excitation and inhibition: TJiAt) = (7 ('2: eabllij,ab + Ie '2: Eij,kITJkl(t - 1) - Ii '2: Iij,kITJkl(t - 1)), (2) a,b k,l k,l where Eij,kl and Iij,kl are the excitatory and inhibitory connection weights from map unit (k, l) to (i, j) and TJkl(t - 1) is the activation of unit (k , I) during the previous time step. The constants I e and Ii control the relative strength of the lateral excitation and inhibition. After the activity has settled, the afferent and lateral weights are modified according to the Hebb rule. Afferent weights are normalized so that the length of the weight 738 Y. CHOE, J. SIROSH, R. MIIKKULAINEN Output Layer (10) .Lq?'Li7.L:17.La7'LV.87..,.Li7.LWLp' : ...... LISSOM Map LaY~/~~~X20) L::7.L7.L7""'-.L::7LI7 ~ ..... ..,. .... ..c:7\ . ..c:7L:7.&l§7'..,. ..... L7 \. ; .L7.L7.£7.L7LSJ7L7 'L7.AlFL7.L7..c:7L7 .L7.A11P".AIIP"L7.L7.o " :mput L~yer (32x32) ". L7L7~~~~~~~L7L7 L7.L7L:7.L7.L7..c:7L7 L7L7~~~~~~~L7L7 . .L7.L7 .......... ..,..L7..c:7 L7L7L7L7L7L7L7L7L7L7L7. ' .L7..,..L7L::7.L7.L7..c:7 L7L7L7L7L7L7L7L7L7L7L70 ' ..c:7..,..L7 ..... ~..c:7..c:7 0 20 . L7..,...,..L7.L?..,.L7 .... ..c:7..,..L7L7.L7..,..L7 .... L7.L:7..,...,...,.L/.L:7 :L:7.L7..c:7.L7.L7..c:7L7 (a) • Unit OJ) tII'd Units with excitatory lateral connections to (iJ) • Units with inhibitory lateral connections to (iJ) (b) Figure 1: The system architecture. (a) The input layer is activated according to the bitmap image of digit 6. The activation propagates through the afferent connections to the LISSOM map, and settles through its lateral connections into a stable pattern. This pattern is the internal representation of the input that is then recognized by the perceptron layer. Through ,the connections from LISSOM to the perceptrons, the unit representing 6 is strongly activated, with weak activations on other units such as 3 and 8. (b) The lateral connections to unit (i, j), indicated by the dark square, are shown. The neighborhood of excitatory connections (lightly shaded) is elevated from the map for a clearer view. The units in the excitatory region also have inhibitory lateral connections (indicated by medium shading) to the center unit. The excitatory radius is 1 and the inhibitory radius 3 in this case. vector remains the same; lateral weights are normalized to keep the sum of weights constant (Sirosh and Miikkulainen 1994): .. (t + 1) Ilij,mn(t) + crinp1]ij~mn IllJ,mn VLmn[llij,mn(t) + crinp1]ij~mnF' (3) .. (t + 1) _ Wij,kl(t) + cr1]ij1]kl W1J,kl "'" [ ( ) ] , wkl Wij ,kl t + cr1]ij1]kl (4) where Ilij,mn is the afferent weight from input unit (m, n) to map unit (i, j), and crinp is the input learning rate; Wij ,kl is the lateral weight (either excitatory Eij ,kl or inhibitory Iij ,kl) from map unit (k, I) to (i, j), and cr is the lateral learning rate (either crexc or crinh). 2.3 Percept ron Output Generation and Weight Adaptation The perceptrons at the output of the system receive the activation pattern on the LISSOM map as their input. The perceptrons are trained after the LISSOM map has been organized. The activation for the perceptron unit Om is Om = CL1]ij Vij,m, i,j (5) where C is a scaling constant, 1]ij is the LISSOM map unit (i,j), and Vij,m is the connection weight between LISSOM map unit (i,j) and output layer unit m. The delta rule is used to train the perceptrons: the weight adaptation is proportional to the map activity and the difference between the output and the target: Vij,m(t + 1) = Vij,m(t) + crout1]ij((m Om), (6) where crout is the learning rate of the percept ron weights, 1]ij is the LISSOM map unit activity, (m is the target activation for unit m. ((m = 1 if the correct digit = m, 0 otherwise). Laterally Interconnected Self-organizing Maps in Handwritten Digit Recognition 739 I Representation I Training Test LISSOM 93.0/ 0.76 88.1/ 3.10 SOM 84.5/ 0.68 84.1/ 1.71 Raw Input 99.2/ 0.06 72.3/ 5.06 Table 1: Final Recognition Results. The average recognition percentage and its variance over the 10 different splits are shown for the training and test sets. The differences in each set are statistically significant with p > .9999. 3 Experiments A subset of 2992 patterns from the NIST Database 3 was used as training and testing data. 1 The patterns were normalized to make sure taht each example had an equal effect on the LISSOM map (Sirosh and Miikkulainen 1994). LISSOM was trained with 2000 patterns. Of these, 1700 were used to train the perceptron layer, and the remaining 300 were used as the validation set to determine when to stop training the perceptrons. The final recognition performance of the whole system was measured on the remaining 992 patterns, which neither LISSOM nor the perceptrons had seen during training. The experiment was repeated 10 times with different random splits of the 2992 input patterns into training, validation, and testing sets. The LISSOM map can be organized starting from initially random weights. However, if the input dimensionality is large, as it is in case of the 32 X 32 bitmaps, each unit on the map is activated roughly to the same degree, and it is difficult to bootstrap the self-organizing process (Sirosh and Miikkulainen 1994, 1996). The standard Self-Organizing Map algorithm can be used to preorganize the map in this case. The SOM performs preliminary feature analysis of the input, and forms a coarse topological map of the input space. This map can then be used as the starting point for the LISSOM algorithm, which modifies the topological organization and learns lateral connections that decorrelate and represent a more clear categorization of the input patterns. The initial self-organizing map was formed in 8 epochs over the training set, gradually reducing the neighborhood radius from 20 to 8. The lateral connections were then added to the system, and over another 30 epochs, the afferent and lateral weights of the map were adapted according to equations 3 and 4. In the beginning, the excitation radius was set to 8 and the inhibition radius to 20. The excitation radius was gradually decreased to 1 making the activity patterns more concentrated and causing the units to become more selective to particular types of input patterns. For comparison, the initial self-organized map was also trained for another 30 epochs, gradually decreasing the neighborhood size to 1 as well. The final afferent weights for the SOM and LISSOM maps are shown in figures 2 and 3. After the SOM and LISSOM maps were organized, a complete set of activation patterns on the two maps were collected. These patterns then formed the training input for the perceptron layer. Two separate versions were each trained for 500 epochs, one with SOM and the other with LISSOM patterns. A third perceptron layer was trained directly with the input bitmaps as well. Recognition performance was measured by counting how often the most highly active perceptron unit was the correct one. The results were averaged over the 10 different splits. On average, the final LISSOM+perceptron system correctly recognized 88.1% of the 992 pattern test sets. This is significantly better than the 84.1% 1 Downloadable at ftp:j jsequoyah.ncsl.nist.gov jpubjdatabasesj. 740 Y. CHOE, J. SIROSH, R. MIIKKULAINEN Iliiji '·~1,;i;:!il , '8 ....... ···· Slll .... ". "1111 "Q" .. '11 .111/1 ""' <·1,1111 Figure 2: Final Afferent Weights of the SOM map. The digit-like patterns represent the afferent weights of each map unit projected on the input layer. For example, the lower left corner represents the afferent weights of unit (0,0). High weight values are shown in black and low in white. The pattern of weights shows the input pattern to which this unit is most sensitive (6 in this case). There are local clusters sensitive to each digit category. of the SOM+perceptron system, and the 72.3% achieved by the perceptron layer alone (Table 1). These results suggest that the internal representations generated by the LISSOM map are more distinct and easier to recognize than the raw input patterns and the representations generated by the SOM map. 4 Discussion The architecture was motivated by the hypothesis that the lateral inhibitory connections of the LISSOM map would decorrelate and force the map activity patterns to become more distinct. The recognition could then be performed by even the simplest classification architectures, such as the perceptron. Indeed, the LISSOM representations were easier to recognize than the SOM patterns, which lends evidential support to the hypothesis. In additional experiments, the percept ron output layer was replaced by a two-weight-Iayer backpropagation network and a Hebbian associator net, and trained with the same patterns as the perceptrons. The recognition results were practically the same for the perceptron, backpropagation, and Hebbian output networks, indicating that the internal representations formed by the LISSOM map are the crucially important part of the recognition system. A comparison of the learning curves reveals two interesting effects (figure 4). First, even though the perceptron net trained with the raw input patterns initially performs well on the test set, its generalization decreases dramatically during training. This is because the net only learns to memorize the training examples, which does not help much with new noisy patterns. Good internal representations are therefore crucial for generalization. Second, even though initially the settling process of the LISSOM map forms patterns that are significantly easier to recognize than Laterally Interconnected Self-organizing Maps in Handwritten Digit Recognition 741 Figure 3: Final Afferent Weights of the LISSOM map. The squares identify the above-average inhibitory lateral connections to unit (10,4) (indicated by the thick square). Note that inhibition comes mostly from areas of similar functionality (i.e. areas sensitive to similar input), thereby decorrelating the map activity and forming a sparser representation of the input. the initial, unsettled patterns (formed through the afferent connections only), this difference becomes insignificant later during training. The afferent connections are modified according to the final, settled patterns, and gradually learn to anticipate the decorrelated internal representations that the lateral connections form. 5 Conclusion The experiments reported in this paper show that LISSOM forms internal representations of the input patterns that are easier to categorize than the raw inputs and the patterns on the SOM map, and suggest that LISSOM can form a useful front end for character recognition systems, and perhaps for other pattern recognition systems as well (such as speech) . The main direction of future work is to apply the approach to larger data sets, including the full NIST 3 database, to use a more powerful recognition network instead of the perceptron, and to increase the map size to obtain a richer representation of the input space. Acknowledgements This research was supported in part by National Science Foundation under grant #IRI-9309273. Computer time for the simulations was provided by the Pittsburgh Supercomputing Center under grants IRI930005P and IRI940004P, and by a High Performance Computer Time Grant from the University of Texas at Austin. References Allinson, N. M., Johnson, M. J., and Moon, K. J. (1994). Digital realisation of selforganising maps. In Touretzky, D. S., editor, Advances in Neural Information Processing Systems 6. San Mateo, CA: Morgan Kaufmann. 742 "0 ~ 0 () -oe. 100 95 90 85 80 75 Y. CHOE. J. SIROSH. R. MIIKKULAINEN Comparison:Test 'SettIEi<CLlSSOU' 'Unsettled LISSOM' ----. . 'SOM' .... . :.Rawj~~~t' ... . ~---... ------------_.-----~------ -- . ... j . . .... -.---_ ..... --.-......... . .. . . . . ... ,.. . ", .. ~ . '" ..... - ~. ... .. .................... --... .. 7 0 ~ __ ~ ____ L-__ -L ____ L-__ -L ____ L-__ -L ____ L-__ ~ __ ~ o 50 100 150 200 250 300 350 400 450 500 Epochs Figure 4: Comparison of the learning curves, A perceptron network was trained to recognize four different kinds of internal representations: the settled LISSOM patterns, the LISSOM patterns before settling, the patterns on the final SOM network, and raw input bitmaps. The recognition accuracy on the test set was then measured and averaged over 10 simulations. The generalization of the raw input + perceptron system decreases rapidly as the net learns to memorize the training patterns. The difference of using settled and unsettled LISSOM patterns diminishes as the afferent weights of LISSOM learn to take into account the decorrelation performed by the lateral weights. Denker, J. S., Gardner, W. R., Graf, H. P., Henderson, D., Howard, R. E., Hubbard, W., Jackel, L. D., Baird, H. S., and Guyon, I. (1989). Neural network recognizer for hand-written zip code digits. In Touretzky, D. S., editor, Advances in Neural Information Processing Systems 1. San Mateo, CA: Morgan Kaufmann. Fukushima, K., and Wake, N. (1990). Alphanumeric character recognition by neocognitron. In Advanced Neural Computers, 263- 270. Elsevier Science Publishers B.V. (North-Holland). Ie Cun, Y., Boser, B., Denker, J. S., Henderson, D., Howard, R. E., Hubbard, W., and Jackel, 1. D. (1990) . Handwritten digit recognition with a backpropagation network. In Touretzky, D. S., editor, Advances in Neural Information Processing Systems 2. San Mateo, CA: Morgan Kaufmann. Martin, G. L., and Pittman, J. A. (1990). Recognizing hand-printed letters and digits. In Touretzky, D. S., editor, Advances in Neural Information Processing Systems 2. San Mateo, CA: Morgan Kaufmann. Sirosh, J. , and Miikkulainen, R. (1994). Cooperative self-organization of afferent and lateral connections in cortical maps. Biological Cybernetics, 71:66- 78. Sirosh, J., and Miikkulainen, R. (1995). Ocular dominance and patterned lateral connections in a self-organizing model of the primary visual cortex. In Tesauro, G ., Touretzky, D. S., and Leen, T . K., editors, Advances in Neural Information Processing Systems 7. Cambridge, MA: MIT Press. Sirosh, J., and Miikkulainen, R. (1996). Topographic receptive fields and patterned lateral interaction in a self-organizing model of the primary visual cortex. Neural Computation (in press).	1995	6
1,106	Does the Wake-sleep Algorithm Produce Good Density Estimators? Peter Dayan Brendan J. Frey, Geoffrey E. Hinton Department of Computer Science University of Toronto Toronto, ON M5S 1A4, Canada {frey, hinton} @cs.toronto.edu Department of Brain and Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139, USA dayan@ai.mit.edu Abstract The wake-sleep algorithm (Hinton, Dayan, Frey and Neal 1995) is a relatively efficient method of fitting a multilayer stochastic generative model to high-dimensional data. In addition to the top-down connections in the generative model, it makes use of bottom-up connections for approximating the probability distribution over the hidden units given the data, and it trains these bottom-up connections using a simple delta rule. We use a variety of synthetic and real data sets to compare the performance of the wake-sleep algorithm with Monte Carlo and mean field methods for fitting the same generative model and also compare it with other models that are less powerful but easier to fit. 1 INTRODUCTION Neural networks are often used as bottom-up recognition devices that transform input vectors into representations of those vectors in one or more hidden layers. But multilayer networks of stochastic neurons can also be used as top-down generative models that produce patterns with complicated correlational structure in the bottom visible layer. In this paper we consider generative models composed of layers of stochastic binary logistic units. Given a generative model parameterized by top-down weights, there is an obvious way to perform unsupervised learning. The generative weights are adjusted to maximize the probability that the visible vectors generated by the model would match the observed data. Unfortunately, to compute the derivatives of the log probability of a visible vector, d, with respect to the generative weights, e, it is necessary to consider all possible ways in which d could be generated. For each possible binary representation a in the hidden units the derivative needs to be weighted by the posterior probability of a given d and e: P(ald, e) = P(ale)p(dla, e)ILP(~le)p(dl~, e). 13 (1) 662 B. J. FREY. G. E. HINTON, P. DAYAN It is intractable to compute P(ald, 9), so instead of minimizing -logP(dI9), we minimize an easily computed upper bound on this quantity that depends on some additional parameters, <1>: -logP(dI9) ~ F(dI9, <1» = - I, Q(al d, <I»logP(a, d19) + I,Q(ald, <I»logQ(ald, <1». (2) a a F(dI9, <1» is a Helmholtz free energy and is equal to -logP(dI9) when the distribution Q(-Id, <1» is the same as the posterior distribution P(-Id, 9). Otherwise, F(dI9, <1» exceeds -logP(dI9) by the asymmetric divergence: D = I,Q(ald, <I»log (Q(ald, <I»IP(ald, 9» . (3) a We restrict Q( -I d, <1» to be a product distribution within each layer that is conditional on the binary states in the layer below and we can therefore compute it efficiently using a bottom-up recognition network. We call a model that uses bottom-up connections to minimize the bound in equation 2 in this way a Helmholtz machine (Dayan, Hinton. Neal and Zemel 1995). The recognition weights <I> take the binary activities in one layer and stochastically produce binary activities in the layer above using a logistic function. So, for a given visible vector, the recognition weights may produce many different representations in the hidden layers, but we can get an unbiased sample from the distribution Q(-Id, <1» in a single bottom-up pass through the recognition network. The highly restricted form of Q( -I d, <1» means that even if we use the optimal recognition weights, the gap between F(dI9, <1» and -logP(dI9) is large for some generative models. However, when F(dI9, <1» is minimized with respect to the generative weights, these models will generally be avoided. F(dI9, <1» can be viewed as the expected number of bits required to communicate a visible vector to a receiver. First we use the recognition model to get a sample from the distribution Q( -I d, <1». Then, starting at the top layer, we communicate the activities in each layer using the top-down expectations generated from the already communicated activities in the layer above. It can be shown that the number of bits required for communicating the state of each binary unit is sklog(qk1pk) + (l-sk)log[(1-qk)/(1-Pk)], where Pk is the top-down probability that Sk is on and qk is the bottom-up probability that Sk is on. There is a very simple on-line algorithm that minimizes F(dI9, <1» with respect to the generative weights. We simply use the recognition network to generate a sample from the distribution Q(-Id, <1» and then we increment each top-down weight 9kj by ESk(SrPj), where 9kj connects unit k to unit j. It is much more difficult to exactly follow the gradient of F(dI9, <1» with respect to the recognition weights, but there is a simple approximate method (Hinton, Dayan, Frey and Neal 1995). We generate a stochastic sample from the generative model and then we increment each bottom-up weight <l>ij by ESi(Sj- f/j) to increase the log probability that the recognition weights would produce the correct activities in the layer above. This way of fitting a Helmholtz machine is called the "wake-sleep" algorithm and the purpose of this paper is to assess how effective it is at performing highdimensional density estimation on a variety of synthetically constructed data sets and two real-world ones. We compare it with other methods of fitting the same type of generative model and also with simpler models for which there are efficient fitting algorithms. 2 COMPETITORS We compare the wake-sleep algorithm with six other density estimation methods. All data units are binary and can take on values dk = 1 (on) and dk = 0 (off). Gzip. Gzip (Gailly, 1993) is a practical compression method based on Lempel-Ziv coding. This sequential data compression technique encodes future segments of data by transmitDoes the Wake-sleep Algorithm Produce Good Density Estimators? 663 ting codewords that consist of a pointer into a buffer of recent past output together with the length of the segment being coded. Gzip's perfonnance is measured by subtracting the length of the compressed training set from the length of the compressed training set plus a subset of the test set. Taking all disjoint test subsets into account gives an overall test set code cost. Since we are interested in estimating the expected perfonnance on one test case, to get a tight lower bound on gzip's perfonnance, the subset size should be kept as small as possible in order to prevent gzip from using early test data to compress later test data. Base Rate Model. Each visible unit k is assumed to be independent of the others with a probability Pk of being on. The probability of vector d is p(d) = Ilk Pkdk (1 - Pk)l- dk . The arithmetic mean of unit k's activity is used to estimate Pk' except in order to avoid serious overfitting, one extra on and one extra off case are included in the estimate. Binary Mixture Model. This method is a hierarchical extension of the base rate model which uses more than one set of base rates. Each set is called a component. Component j has probability 1tj and awards each visible unit k a probability Pjk of being on. The net probability of dis p(d) = Lj 1tj Ilk Pj/k (1 - Pjk)l-dk . For a given training datum, we consider the component identity to be a missing value which must be filled in before the parameters can be adjusted. To accomplish this, we use the expectation maximization algorithm (Dempster, Laird and Rubin 1977) to maximize the log-likelihood of the training set, using the same method as above to avoid serious overfitting. Gibbs Machine (GM). This machine uses the same generative model as the Helmholtz machine, but employs a Monte Carlo method called Gibbs sampling to find the posterior in equation 1 (Neal, 1992). Unlike the Helmholtz machine it does not require a separate recognition model and with sufficiently prolonged sampling it inverts the generative model perfectly. Each hidden unit is sampled in fixed order from a probability distribution conditional on the states of the other hidden and visible units. To reduce the time required to approach equilibrium, the network is annealed during sampling. Mean Field Method (MF). Instead of using a separate recognition model to approximate the posterior in equation 1, we can assume that the distribution over hidden units is factorial for a given visible vector. Obtaining a good approximation to the posterior is then a matter of minimizing free energy with respect to the mean activities. In our experiments, we use the on-line mean field learning algorithm due to Saul, Jaakkola, and Jordan (1996). Fully Visible Belief Network (FVBN). This method is a special case of the Helmholtz machine where the top-down network is fully connected and there are no hidden units. No recognition model is needed since there is no posterior to be approximated. 3 DATA SETS The perfonnances of these methods were compared on five synthetic data sets and two real ones. The synthetic data sets had matched complexities: the generative models that produced them had 100 visible units and between 1000 and 2500 parameters. A data set with 100,000 examples was generated from each model and then partitioned into 10,000 for training, 10,000 for validation and 80,000 for testing. For tractable cases, each data set entropy was approximated by the negative log-likelihood of the training set under its generative model. These entropies are approximate lower bounds on the perfonnance. The first synthetic data set was generated by a mixture model with 20 components. Each component is a vector of 100 base rates for the 100 visible units. To make the data more realistic, we arranged for there to be many different components whose base rates are all extreme (near 0 or 1) representing well-defined clusters and a few components with most base rates near 0.5 representing much broader clusters. For componentj, we selected base rate Pjk from a beta distribution with mean Ilt and variance 1lt(1-1lt)/40 (we chose this variance to keep the entropy of visible units low for Ilt near 0 or 1, representing well-defined clusters). Then, as often as not we randomly replaced each Pjk with 1-Pjk to 664 B. 1. FREY, G. E. HINTON, P. DAY AN make each component different (without doing this, all components would favor all units off). In order to obtain many well-defined clusters, the component means Il.i were themselves sampled from a beta distribution with mean 0.1 and variance 0.02. The next two synthetic data sets were produced using sigmoidal belief networks (Neal 1992) which are just the generative parts of binary stochastic Helrnhol tz machines. These networks had full connectivity between layers, one with a 20~100 architecture and one with a 5~10~15~2~100 architecture. The biases were set to 0 and the weights were sampled uniformly from [-2,2), a range chosen to keep the networks from being deterministic. The final two synthetic data sets were produced using Markov random fields. These networks had full bidirectional connections between layers. One had a 10<=>20<=>100 architecture, and the other was a concatenation of ten independent 10<=>10 fields. The biases were set to 0 and the weights were sampled from the set {-4, 0, 4} with probabilities {0.4, 0.4, 0.2}. To find data sets with high-order structure, versions of these networks were sampled until data sets were found for which the base rate method performed badly. We also compiled two versions of a data set to which the wake-sleep algorithm has previously been applied (Hinton et al. 1995). These data consist of normalized and quantized 8x8 binary images of handwritten digits made available by the US Postal Service Office of Advanced Technology. The first version consists of a total of 13,000 images partitioned as 6000 for training, 2000 for validation and 5000 for testing. The second version consists of pairs of 8x8 images (ie. 128 visible units) made by concatenating vectors from each of the above data sets with those from a random reordering of the respective data set. 4 TRAINING DETAILS The exact log-likelihoods for the base rate and mixture models can be computed, because these methods have no or few hidden variables. For the other methods, computing the exact log-likelihood is usually intractable. However, these methods provide an approximate upper bound on the negative log-likelihood in the form of a coding cost or Helmholtz free energy, and results are therefore presented as coding costs in bits. Because gzip performed poorly on the synthetic tasks, we did not break up the test and validation sets into subsets. On the digit tasks, we broke the validation and test sets up to make subsets of 100 visible vectors. Since the "-9" gzip option did not improve performance significantly, we used the default configuration. To obtain fair results, we tried to automate the model selection process subject to the constraint of obtaining results in a reasonable amount of time. For the mixture model, the Gibbs machine, the mean field method, and the Helmholtz machine, a single learning run was performed with each of four different architectures using performance on a validation set to avoid wasted effort. Performance on the validation set was computed every five epochs, and if two successive validation performances were not better than the previous one by more than 0.2%, learning was terminated. The network corresponding to the best validation performance was selected for test set analysis. Although it would be desirable to explore a wide range of architectures, it would be computationally ruinous. The architectures used are given in tables 3 and 4 in the appendix. The Gibbs machine was annealed from an initial temperature of 5.0. Between each sweep of the network, during which each hidden unit was sampled once, the temperature was multiplied by 0.9227 so that after 20 sweeps the temperature was 1.0. Then, the generative weights were updated using the delta rule. To bound the datum probability, the network is annealed as above and then 40 sweeps at unity temperature are performed while summing the probability over one-nearest-neighbor configurations, checking for overlap. A learning rate of 0.01 was used for the Gibbs machine, the mean field method, the Helmholtz machine, and the fully visible belief network. For each of these methods, this value was found to be roughly the largest possible learning rate that safely avoided oscillations. Does the Wake-sleep Algorithm Produce Good Density Estimators? 70r-----------------------------------------------------, 60 50 40 20 Gzip Base rate model -Mixture model -e-Gibbs machine Mean field method -4-Fully visible belief network Entropy • l:~m - 10~--------------------------------------------------~ Mixture 2~1()() BN BN 5~IO~ MRF MRF Single 2~I()() 15~20~I()() 1~2~l()() lOx (IO~IO) digits Tasks Digit pairs 665 Figure 1. Compression performance relative to the Helmholtz machine. Lines connecting the data points are for visualization only, since there is no meaningful interpolant. 5 RESULTS The learning times and the validation performances are given in tables 3 and 4 of the appendix. Test set appraisals and total learning times are given in table 1 for the synthetic tasks and in table 2 for the digit tasks. Because there were relatively many training cases in each simulation, the validation procedure serves to provide timing information more than to prevent overfitting. Gzip and the base rate model were very fast, followed by the fully visible belief network, the mixture model, the Helmholtz machine, the mean field method, and finally the Gibbs machine. Test set appraisals are summarized by compression performance relative to the Helmholtz machine in figure 1 above. Greater compression sizes correspond to lower test set likelihoods and imply worse density estimation. When available, the data set entropies indicate how close to optimum each method comes. The Helmholtz machine yields a much lower cost compared to gzip and base rates on all tasks. Compared to the mixture model, it gives a lower cost on both BN tasks and the MRF 10 x (1O~1O) task. The latter case shows that the Helmholtz machine was able to take advantage of the independence of the ten concatenated input segments, whereas the mixture method was not. Simply to represent a problem where there are only two distinct clusters present in each of the ten segments, the mixture model would require 210 components. Results on the two BN tasks indicate the Helmholtz machine is better able to model multiple simultaneous causes than the mixture method, which requires that only one component (cause) is active at a time. On the other hand, compared to the mixture model, the Helmholtz machine performs poorly on the Mixture 20~100 task. It is not able to learn that only one cause should be active at a time. This problem can be avoided by hard-wiring softmax groups into the Helmholtz machine. On the five synthetic tasks, the Helmholtz machine performs about the same as or better than the Gibbs machine, and runs two orders of magnitude faster. (The Gibbs machine was too slow to run on the digit tasks.) While the quality of density estimation produced by the mean field method is indistinguishable from the Helmholtz machine, the latter runs an order of magnitude faster than the mean field algorithm we used. The fully visible belief network performs significantly better than the Helmholtz machine on the two digit tasks and significantly worse on two of the synthetic tasks. It is trained roughly two orders of magnitude faster than the Helmholtz machine. 666 B. J. FREY, G. E. HINTON, P. DAYAN Table 1. Test set cost (bits) and total training time (hrs) for the synthetic tasks. Model used to produce synthetic data Mixture BN BN5~1O~ MRF MRF 20=>100 20=>100 15~20~100 10~20~100 10 x (1O~1O) Entropy 36.5 63.5 unknown 19.2 36.8 gzip 61.4 o 98.0 0 92.1 o 35.6 0 59.9 0 Base rates 96.6 o 80.7 0 69.2 o 42.2 0 68.1 0 Mixture 36.7 o 74.0 0 62.6 1 19.3 1 49.6 1 GM 44.1 131 63.9 240 58.1 251 26.1 195 40.3 145 MF 42.2 68 64.7 80 58.4 68 19.3 75 38.7 89 HM 42.7 8 65.2 3 58.5 4 19.4 2 38.6 4 FVBN 50.9 o 67.8 0 60.6 0 19.8 0 38.2 0 Table 2. Test set cost (bits) and training time (hrs) for the digit tasks. Method Single digits Method Digit pairs gzip 44.3 0 gZlp 89.2 0 Base rates 59.2 o Base rates 118.4 0 Mixture 37.5 o Mixture 92.7 1 MF 39.5 38 MF 80.7 104 HM 39.1 2 HM 80.4 7 FVBN 35.9 o FVBN 72.9 0 6 CONCLUSIONS If we were given a new data set and asked to leave our research biases aside and do efficient density estimation, how would we proceed? Evidently it would not be worth trying gzip and the base rate model. We'd first try the fully visible belief network and the mixture model, since these are fast and sometimes give good estimates. Hoping to extract extra higher-order structure, we would then proceed to use the Helmholtz machine or the mean field method (keeping in mind that our implementation of the Helmholtz machine is considerably faster than Saul et al. 's implementation of the mean field method). Because it is so slow, we would avoid using the Gibbs machine unless the data set was very small. Acknowledgments We greatly appreciate the mean field software provided by Tommi Jaakkola and Lawrence Saul. We thank members of the Neural Network Research Group at the University of Toronto for helpful advice. The financial support from ITRC, IRIS, and NSERC is appreciated. References Dayan, P., Hinton, G. E., Neal, R. M., and Zemel, R. S. 1995. The Helmholtz machine. Neural Computation 7, 889-904. Dempster, A. P., Laird, N. M. and Rubin, D. B. 1977. Maximum likelihood from incomplete data via the EM algorithm. J. Royal Statistical Society, Series B 34, 1-38. Gailly, J. 1993. gzip program for unix. Hinton, G. E., Dayan, P., Frey, B. J., Neal, R. M. 1995. The wake-sleep algorithm for unsupervised neural networks. Science 268, 1158-1161. Neal, R. M. 1992. Connectionist learning of belief networks. Artificial Intelligence 56,71-113. Saul, L. K., Jaakkola, T., and Jordan, M.I. 1996. Mean field theory for sigmoid belief networks. Submitted to Journal of Artificial Intelligence. Does the Wake-sleep Algorithm Produce Good Density Estimators? 667 Appendix The average validation set cost per example and the associated learning time for each simulation are listed in tables 3 and 4. Architectures judged to be optimal according to validation performance are indicated by ",, and were used to produce the test results given in the body of this paper. Table 3. Validation set cost (bits) and learning time (min) for the synthetic tasks. Model used to produce synthetic data Mixture BN BN 5~lO~ MRF MRF 20~100 20~100 15~20~100 10<=>20<=> 1 00 10 x (1O<=> 10) gzip 61.6 o 98.1 0 92.3 o 35.6 o 60.0 0 Base rates 96.7 o 80.7 0 69.4 o 42.1 o 68.1 0 Mixture 20~100 44.6 3 75.6 3 63.9 4 19.2 3 54.8 5 Mixture 40~100 36.8* 5 74.8 5 63.2 7 19.2 7 52.4 15 Mixture 60~100 36.8 7 74.4 7 62.9 8 19.2 8 51.0 17 Mixture lOO~lOO 37.0 14 74.0* 12 62.7* 13 19.3 12 49.6* 22 OM 20~lOO 50.6 1187 63.9* 1639 58.1* 2084 26.1* 934 40.3* 1425 OM 50~lOO 68.8 2328 80.4 3481 76.4 5234 49.2 6472 56.5 3472 OM 1O~20~100 44.1* 872 66.4 1771 59.8 3084 28.0 767 42.3 1033 OM 20~50~100 52.7 3476 91.3 7504 88.0 4647 55.3 3529 63.5 2781 MF 20~100 49.5 518 64.6 427 58.4* 497 19.4 862 39.2 471 MF 50~100 49.9 1644 64.8 1945 58.6 1465 20.4 1264 38.7* 2427 MF 1O~20~100 46.0 306 64.6* 658 58.5 543 19.3* 569 38.9 882 MF 20~50~100 42.1* 1623 65.0 1798 58.6 1553 19.3 1778 38.8 1575 HM 20~lOO 50.0 41 65.2 28 58.8 41 19.7 15 38.6* 30 HM 50~lOO 50.7 81 65.5 66 59.4 78 20.2 27 38.9 46 HM lO~20~100 43.4 32 65.1* 38 58.5* 45 19.4* 21 38.9 46 HM 20~50~lOO 42.6* 308 67.2 69 59.2 93 19.5 64 39.4 102 FVBN 51.0 7 67.8 7 60.7 6 19.8 8 38.3 6 Table 4. Validation set cost (bits) and learning time (min) for the digit tasks. Method Single digits Method Digit pairs gzip 44.2 0 gzip 88.8 1 Base rates 59.0 0 Base rates 117.9 0 Mixture 16~64 43.2 1 Mixture 32~128 96.9 6 Mixture 32~64 40.0 4 Mixture 64~128 93.8 8 Mixture 64~64 38.0 5 Mixture 128~128 92.4* 14 Mixture 128~64 37.1* 6 Mixture 256~128 92.8 27 MF 16~24~64 39.9 341 MF 16~24~32~128 82.7 1335 MF24~32~64 39.1* 845 MF 16~32~64~128 81.2 1441 MF 12~16~24~64 39.8 475 MF 12~16~24~32~128 82.8 896 MF 16~24~32~64 39.1 603 MF 12~16~32~64~128 80.1* 2586 HM 16~24~64 39.7 24 HM 16~24~32~128 83.8 76 HM 24~32~64 39.4 34 HM 16~32~64~128 80.1* 138 HM 12~16~24~64 40.4 16 HM 12~16~24~32~128 84.6 74 HM 16~24~32~64 38.9* 52 HM 12~16~32~64~128 80.1 135 FVBN 35.8 1 FVBN 72.5 7 PART V IMPLEMENTATIONS	1995	60
1,107	EM Optimization of Latent-Variable Density Models Christopher M Bishop, Markus Svensen and Christopher K I Williams Neural Computing Research Group Aston University, Birmingham, B4 7ET, UK c.m.bishop~aston.ac.uk svensjfm~aston.ac.uk c.k.i.williams~aston.ac.uk Abstract There is currently considerable interest in developing general nonlinear density models based on latent, or hidden, variables. Such models have the ability to discover the presence of a relatively small number of underlying 'causes' which, acting in combination, give rise to the apparent complexity of the observed data set. Unfortunately, to train such models generally requires large computational effort. In this paper we introduce a novel latent variable algorithm which retains the general non-linear capabilities of previous models but which uses a training procedure based on the EM algorithm. We demonstrate the performance of the model on a toy problem and on data from flow diagnostics for a multi-phase oil pipeline. 1 INTRODUCTION Many conventional approaches to density estimation, such as mixture models, rely on linear superpositions of basis functions to represent the data density. Such approaches are unable to discover structure within the data whereby a relatively small number of 'causes' act in combination to account for apparent complexity in the data. There is therefore considerable interest in latent variable models in which the density function is expressed in terms of of hidden variables. These include density networks (MacKay, 1995) and Helmholtz machines (Dayan et al., 1995). Much of this work has been concerned with predicting binary variables. In this paper we focus on continuous data. 466 c. M. BISHOP, M. SVENSEN, C. K. I. WILLIAMS y(x;W) Figure 1: The latent variable density model constructs a distribution function in t-space in terms of a non-linear mapping y(x; W) from a latent variable x-space. 2 THE LATENT VARIABLE MODEL Suppose we wish to model the distribution of data which lives in aD-dimensional space t = (tl, ... , tD). We first introduce a transformation from the hidden variable space x = (Xl, ... , xL) to the data space, governed by a non:-linear function y(x; W) which is parametrized by a matrix of weight parameters W. Typically we are interested in the situation in which the dimensionality L of the latent variable space is less than the dimensionality D of the data space, since we wish to capture the fact that the data itself has an intrinsic dimensionality which is less than D. The transformation y(x; W) then maps the hidden variable space into an L-dimensional non-Euclidean subspace embedded within the data space. This is illustrated schematically for the case of L = 2 and D = 3 in Figure 1. If we define a probability distribution p(x) on the latent variable space, this will induce a corresponding distribution p(y) in the data space. We shall refer to p(x) as the prior distribution of x for reasons which will become clear shortly. Since L < D, the distribution in t-space would be confined to a manifold of dimension L and hence would be singular. Since in reality data will only approximately live on a lower-dimensional space, it is appropriate to include a noise model for the t vector. We therefore define the distribution of t, for given x and W, given by a spherical Gaussian centred on y(x; W) having variance {3-1 so that ( 1) The distribution in t-space, for a given value of the weight matrix W, lS then obtained by integration over the x-distribution p(tIW) = J p(tlx, W)p(x) dx. (2) For a given data set V = (t l , ... , t N ) of N data points, we can determine the weight matrix W using maximum likelihood. For convenience we introduce an error function given by the negative log likelihood: N N E(W) = -In 11 p(tn IW) = - ~ In {J p(tn Ixn, W)p(xn) dxn } . (3) EM Optimization of Latent-Variable Density Models 467 In principle we can now seek the maximum likelihood solution for the weight matrix, once we have specified the prior distribution p(x) and the functional form of the mapping y(x; W), by minimizing E(W). However, the integrals over x occuring in (3), and in the corresponding expression for 'iJ E, will, in general, be analytically intractable. MacKay (1995) uses Monte Carlo techniques to evaluate these integrals and conjugate gradients to find the weights. This is computationally very intensive, however, since a Monte Carlo integration must be performed every time the conjugate gradient algorithm requests a value for E(W) or 'iJ E(W). We now show how, by a suitable choice of model, it is possible to find an EM algorithm for determining the weights. 2.1 EM ALGORITHM There are three key steps to finding a tractable EM algorithm for evaluating the weights. The first is to use a generalized linear network model for the mappmg function y(x; W). Thus we write y(x; W) = W ¢(x) (4) where the elements of ¢(x) consist of M fixed basis functions cPj(x), and W is a D x M matrix with elements Wkj' Generalized linear networks possess the same universal approximation capabilities as multi-layer adaptive networks. The price which has to be paid, however, is that the number of basis functions must typically grow exponentially with the dimensionality L of the input space. In the present context this is not a serious problem since the dimensionality is governed by the latent variable space and will typically be small. In fact we are particularly interested in visualization applications, for which L = 2. The second important step is to use a simple Monte Carlo approximation for the integrals over x. In general, for a function Q(x) we can write J 1 K Q(x)p(x) dx ~ f{ ~ Q(xi ) z=l (5) where xi represents a sample drawn from the distribution p(x). If we apply this to (3) we obtain E(W) = - t,ln{ ~ tp(tnlxni,w)} (6) The third key step to choose the sample of points {xni} to be the same for each term in the summation over n. Thus we can drop the index n on x ni to give N {I K } E(W) = - ~ In f{ ~p(tnlxi, W) (7) We now note that (7) represents the negative log likelihood under a distribution consisting of a mixture of f{ kernel functions. This allows us to apply the EM algorithm to find the maximum likelihood solution for the weights. Furthermore, as a consequence of our choice (4) for the non-linear mapping function, it will turn out that the M-step can be performed explicitly, leading to a solution in terms of a set 468 c. M. BISHOP, M. SYENSEN, C. K. I. WILLIAMS of linear equations. We note that this model corresponds to a constrained Gaussian mixture distribution of the kind discussed in Hinton et al. (1992). We can formulate the EM algorithm for this system as follows. Setting the derivatives of (7) with respect to Wkj to zero we obtain t, t, R;n(W) {t, w"f,(x;) - t~ } f;(x;) = 0 (8) where we have used Bayes' theorem to introduce the posterior probabilities, or responsibilities, for the mixture components given by R- (W) = p(tnlxi, W) m L:~=1 p(tnlxil, W) (9) Similarly, maximizing with respect to (3 we obtain K N ~ = N1D I: I: Rni(W) lIy(xn; W) - t n ll 2 . i=l n=l (10) The EM algorithm is obtained by supposing that, at some point in the algorithm, the current weight matrix is given by wold and the current value of (3 is (30ld. Then we can evaluate the responsibilities using these values for Wand (3 (the E-step), and then solve (8) for the weights to give W new and subsequently solve (10) to give (3new (the M-step). The two steps are repeated until a suitable convergence criterion is reached. In practice the algorithm converges after a relatively small number of iterations. A more formal justification for the EM algorithm can be given by introducing auxiliary variables to label which component is responsible for generating each data point, and then computing the expectation with respect to the distribution of these variables. Application of Jensen's inequality then shows that, at each iteration of the algorithm, the error function will decrease unless it is already at a (local) minimum, as discussed for example in Bishop (1995). If desired, a regularization term can be added to the error function to control the complexity of the model y(x; W). From a Bayesian viewpoint, this corresponds to a prior distribution over weights. For a regularizer which is a quadratic function of the weight parameters, this leads to a straightforward modification to the weight update equations. It is convenient to write the condition (8) in matrix notation as (~TGold~ + AI)(Wnew)T = ~TTold (11) where we have included a regularization term with coefficient A, and I denotes the unit matrix. In (11) ~ is a f{ x M matrix with elements <l>ij = (/Jj(xi ), T is a I< x D matrix, and G is a I< x I< diagonal matrix, with elements N N Tik = I: Rin(W)t~ Gjj = I: ~n(W). (12) n=l n=l We can now solve (11) for w new using standard linear matrix inversion techniques, based on singular value decomposition to allow for possible ill-conditioning. Note that the matrix ~ is constant throughout the algorithm, and so need only be evaluated once at the start. EM Optimization of Latent-Variable Density Models 469 4~----------------------~ 4~----------------------~ 3 3 • , : 1' . • 2 2 o 11 o • -_11~--~----~--~----~--~ -1~--~----~------~~--~ o 2 3 4 -1 0 2 3 4 Figure 2: Results from a toy problem involving data (' x') generated from a 1-dimensional curve embedded in 2 dimensions, together with the projected sample points ('+') and their Gaussian noise distributions (filled circles). The initial configuration, determined by principal component analysis, is shown on the left, and an intermediate configuration, obtained after 4 iterations of EM, is shown on the right. 3 RESULTS We now present results from the application of this algorithm first to a toy problem involving data in three dimensions, and then to a more realistic problem involving 12-dimensional data arising from diagnostic measurements of oil flows along multiphase pipelines. For simplicity we choose the distribution p(x) to be uniform over the unit square. The basis functions ¢j (x) are taken to be spherically symmetric Gaussian functions whose centres are distributed on a uniform grid in x-space, with a common width parameter chosen so that the standard deviation is equal to the separation of neighbouring basis functions. For both problems the weights in the network were initialized by performing principal components analysis on the data and then finding the least-squares solution for the weights which best approximates the linear transformation which maps latent space to target space while generating the correct mean and variance in target space. As a simple demonstration of this algorithm, we consider data generated from a one-dimensional distribution embedded in two dimensions, as shown in Figure 2. 3.1 OIL FLOW DATA Our second example arises in the problem of determining the fraction of oil in a multi-phase pipeline carrying a mixture of oil, water and gas (Bishop and James, 1993). Each data point consists of 12 measurements taken from dual-energy gamma densitometers measuring the attenuation of gamma beams passing through the pipe. Synthetically generated data is used which models accurately the attenuation processes in the pipe, as well as the presence of noise (arising from photon statistics). The three phases in the pipe (oil, water and gas) can belong to one of three different geometrical configurations, corresponding to stratified, homogeneous, and annular flows, and the data set consists of 1000 points distributed equally between the 3 470 c. M. BISHOP, M. SVENSEN, C. K. I. W1LUAMS 2~----~------~-------------, 1.5 ~ ..,..~ • • ..... .,,;' .. .." 0.5 00 ...", 0 " 0 ~ ~iI:" • , ~,J1 :"." +~+ .,. • ~ ........ 0 -0.5 .. ,.,..t:'" .. -, ++ , . ... (~ ~+ -1 • ~O ~ #+ .. 0 6 C -1.5 : .. ~ ~_" ".a:. 0 .'IIIt. .. • aa.; ..... -32 -1 0 2 -2 0 2 4 Figure 3: The left plot shows the posterior-mean projection of the oil data in the latent space of the non-linear model. The plot on the right shows the same data set projected onto the first two principal components. In both plots, crosses, circles and plus-signs represent the stratified, annular and homogeneous configurations respectively. classes. We take the latent variable space to be two-dimensional. This is appropriate for this problem as we know that, locally, the data must have an intrinsic dimensionality of two (neglecting noise on the data) since, for any given geometrical configuration of the three phases, there are two degrees of freedom corresponding to the fractions of oil and water in the pipe (the fraction of gas being redundant since the three fractions must sum to one). It also allows us to use the latent variable model to visualize the data by projection onto x-space. For the purposes of visualization, we note that a data point t n induces a posterior distribution p(xltn, W) in x-space, where W denotes the value of the weight matrix for the trained network. This provides considerably more information in the visualization space than many simple techniques (which generally project each data point onto a single point in the visualization space). For example, the posterior distribution may be multi-modal, indicating that there is more than one region of x-space which can claim significant responsibility for generating the data point. However, it is often convenient to project each data point down to a unique point in x-space. This can be done by finding the mean of the posterior distribution, which itself can be evaluated by a simple Monte Carlo integration using quantities already calculated in the evaluation of W* . Figure 3 shows the oil data visualized in the latent-variable space in which, for each data point, we have plotted the posterior mean vector. Again the points have been labelled according to their multi-phase configuration. We have compared these results with those from a number of conventional techniques including factor analysis and principal component analysis. Note that factor analysis is precisely the model which results if a linear mapping is assumed for y(x; W), a Gaussian distribution p(x) is chosen in the latent space, and the noise distribution in data space is taken to be Gaussian with a diagonal covariance matrix. Of these techniques, principal component analysis gave the best class separation (assessed subjectively) and is illustrated in Figure 3. Comparison with the results from the non-linear model clearly shows that the latter gives much better separation of the three classes, as a consequence of the non-linearity permitted by the latent variable mapping. EM Optimization of Latent-Variable Density Models 471 4 DISCUSSION There are interesting relationships between the model discussed here and a number of well-known algorithms for unsupervised learning. We have already commented that factor analysis is a special case of this model, involving a linear mapping from latent space to data space. The Kohonen topographic map algorithm (Kohonen, 1995) can be regarded as an approximation to a latent variable density model of the kind outlined here. Finally, there are interesting similarities to a 'soft' version of the 'principal curves' algorithm (Tibshirani, 1992). The model we have described can readily be extended to deal with the problem of missing data, provided we assume that the missing data is ignorable and missing at random (Little and Rubin, 1987). This involves maximizing the likelihood function in which the missing values have been integrated out. For the model discussed here, the integrations can be performed analytically, leading to a modified form of the EM algorithm. Currently we are extending the model to allow for mixed continuous and categorical variables. We are also exploring Bayesian approaches, based on Markov chain Monte Carlo, to replace the maximum likelihood procedure. Acknowledgements This work was partially supported by EPSRC grant GR/J75425: Novel Developments in Learning Theory. Markus Svensen would like to thank the staff of the SANS group in Stockholm for their hospitality during part of this project. References Bishop, C. M. (1995). Neural Networks for Pattern Recognition. Oxford University Press. Bishop, C. M. and G. D. James (1993). Analysis of multiphase flows using dualenergy gamma densitometry and neural networks. Nuclear Instruments and Methods in Physics Research A327, 580-593. Dayan, P., G. E. Hinton, R. M. Neal, and R. S. Zemel (1995). The HelmQoltz machine. Neural Computation 7 (5), 889- 904. Hinton, G. E., C. K. 1. Williams, and M. D. Revow (1992). Adaptive elastic models for hand-printed character recognition. In J. E. Moody, S. J. Hanson, and R. P. Lippmann (Eds.), Advances in Neural Information Processing Systems 4. Morgan Kauffmann. Kohonen, T. (1995). Self-Organizing Maps. Berlin: Springer-Verlag. Little, R. J. A. and D. B. Rubin (1987). Statistical Analysis with Missing Data. New York: John Wiley. MacKay, D. J. C. (1995). Bayesian neural networks and density networks. Nuclear Instruments and Methods in Physics Research, A 354 (1), 73- 80. Tibshirani, R. (1992). Principal curves revisited. Statistics and Computing 2, 183-190.	1995	61
1,108	Modern Analytic Techniques to Solve the Dynamics of Recurrent Neural Networks A.C.C. Coolen Dept. of Mathematics King's College London Strand, London WC2R 2LS, U.K. S.N. Laughton Dept. of Physics - Theoretical Physics University of Oxford 1 Keble Road, Oxford OX1 3NP, U.K. D. Sherrington .. Center for Non-linear Studies Los Alamos National Laboratory Los Alamos, New Mexico 87545 Abstract We describe the use of modern analytical techniques in solving the dynamics of symmetric and nonsymmetric recurrent neural networks near saturation. These explicitly take into account the correlations between the post-synaptic potentials, and thereby allow for a reliable prediction of transients. 1 INTRODUCTION Recurrent neural networks have been rather popular in the physics community, because they lend themselves so naturally to analysis with tools from equilibrium statistical mechanics. This was the main theme of physicists between, say, 1985 and 1990. Less familiar to the neural network community is a subsequent wave of theoretical physical studies, dealing with the dynamics of symmetric and nonsymmetric recurrent networks. The strategy here is to try to describe the processes at a reduced level of an appropriate small set of dynamic macroscopic observables. At first, progress was made in solving the dynamics of extremely diluted models (Derrida et al, 1987) and of fully connected models away from saturation (for a review see (Coolen and Sherrington, 1993)). This paper is concerned with more recent approaches, which take the form of dynamical replica theories, that allow for a reliable prediction of transients, even near saturation. Transients provide the link between initial states and final states (equilibrium calculations only provide ·On leave from Department of Physics - Theoretical Physics, University of Oxford 254 A. C. C. COOLEN, S. N. LAUGHTON, D. SHERRINGTON information on the possible final states). In view of the technical nature of the subject, we will describe only basic ideas and results for simple models (full details and applications to more complicated models can be found elsewhere). 2 RECURRENT NETWORKS NEAR SATURATION Let us consider networks of N binary neurons ai E {-I, I}, where neuron states are updated sequentially and stochastically, driven by the values of post-synaptic potentials hi. The probability to find the system at time t in state 0' = (a1,' .. , aN) is denoted by Pt(O'). For the rates Wi(O') of the transitions ai -t -(7i and for the potentials hi (0') we make the usual choice 1 Wi (0') = - [1-ai tanh [,Bhi (0')]] 2 hi(O') = L Jijaj j:f:i The parameter ,B controls the degree of stochasticity: the ,B = 0 dynamics is completely random, whereas for ,B = 00 we find the deterministic rule ai -t sgn[hi(O')]. The evolution in time of Pt(O') is given by the master equation d N dtPt (0') = l: [Pt (FkO' )Wk (FkO') - Pt (0' )Wk (0')] k=l (1) with Fk<P(O') = <P(a1 , ... ,-(7k, ... ,aN)' For symmetric models, where Jij = Jji for all (ij), the dynamics (1) leads asymptotically to the Boltzmann equilibrium distribution Peq(O') '" exp [-,BE(O')], with the energy E(O') = - Li<j adijaj. For associative memory models with Hebbian-type synapses, required to store a set of P random binary patterns e/.1 = (€i, ... , €~ ), the relevant macroscopic observable is the overlap m between the current microscopic state 0' and the pattern to be retrieved (say, pattern 1): m = -Iv Li €lai. Each post-synaptic potential can now be written as the sum of a simple signal term and an interference-noise term, e.g. 1 p=o:N Jij = N L €f€j /.1=1 hi(O') = m€l + ~ l: €f l: €jaj /.1>1 j:f:i (2) All complications arise from the noise terms. The 'Local Chaos Hypothesis' (LCH) consists of assuming the noise terms to be independently distributed Gaussian variables. The macroscopic description then consists of the overlap m and the width ~ of the noise distribution (Amari and Maginu, 1987). This, however, works only for states near the nominated pattern, see also (Nishimori and Ozeki, 1993). In reality the noise components in the potentials have far more complicated statisticsl . Due to the build up of correlations between the system state and the non-nominated patterns, the noise components can be highly correlated and described by bi-modal distributions. Another approach involves a description in terms of correlation- and response functions (with two timearguments). Here one builds a generating functional, which is a sum over all possible trajectories in state space, averaged over the distribution of the non-nominated patterns. One finds equations which are exact for N -t 00 , but, unfortunately, also rather complicated. For the typical neural network models solutions are known only in equilibrium (Rieger et aI, 1988); information on transients has so far only been obtained through cumbersome approximation schemes (Horner et aI, 1989). We now turn to a theory that takes into account the non-trivial statistics of the post-synaptic potentials, yet involves observables with one time-argument only. lCorrelations are negligible only in extremely diluted (asymmetric) networks (Derrida et aI, 1987), and in networks with independently drawn (asymmetric) random synapses Modem Analytic Techniques to Solve the Dynamics of Recurrent Neural Networks 255 3 DYNAMICAL REPLICA THEORIES The evolution of macroscopic observables n( 0') = (01 (0'), ... , OK (0')) can be described by the so-called Kramers-Moyal expansion for the corresponding probability distribution pt(n) (derived directly from (1)). Under certain conditions on the sensitivity of n to single-neuron transitions (7i -t -1J'i, one finds on finite time-scales and for N -t 00 the macroscopic state n to evolve deterministically according to: ~n = EO' pt(O')8 [n-n(O')] Ei Wi(O') [n(FiO')-n(O')] (3) dt EO' pt(O')8 [n-n(O')] This equation depends explicitly on time through Pt(O'). However, there are two natural ways for (3) to become autonomous: (i) by the term Ei Wi(O') [n(FiO') -n(O')] depending on u only through n(O') (as for attractor networks away from saturation), or (ii) by (1) allowing for solutions of the form Pt(O') = fdn(O')] (as for extremely diluted networks). In both cases Pt(O') drops out of (3). Simulations further indicate that for N -t 00 the macroscopic evolution usually depends only on the statistical properties of the patterns {ell}, not on their microscopic realisation ('self-averaging'). This leads us to the following closure assumptions: 1. Probability equipartitioning in the n subshells of the ensemble: Pt(O') '" 8 [nt-n(O')]. If n indeed obeys closed equations, this assumption is safe. 2. Self-averaging of the n flow with resfect to the microscopic details of the non-nominated patterns: tt n -t (dt n)patt. Our equations (3) are hereby transformed into the closed set: ~n _ (EO' 8 [n-n(O')] Ei Wi(O') [n(FiO') - n(O')]) dt EO' 8[n-n(O')] patt The final observation is that the tool for averaging fractions is replica theory: dd n = lim lim ~ (~Wi(O'l) [n(FiO'1)-n(O'1)] rrn 8[n-n(O'O )])patt (4) t n--tO N --too ~ ~ O'I ···O' n i 0=1 The choice to be made for the observables n(O'), crucial for the closure assumptions to make sense, is constrained by requiring the theory to be exact in specific limits: exactness for a -t 0 : n = (m, ... ) exactness for t -t 00: n = (E, ... ) (for symmetric models only) 4 SIMPLE VERSION OF THE THEORY For the Hopfield model (2) the simplest two-parameter theory which is exact for a -t o and for t -t 00 is consequently obtained by choosing n = (m,E). Equivalently we can choose n = (m,r), where r(O') measures the 'interference energy': m = ~ L~I(7i i The result of working out (4) for n = (m, r) is: !m = J dz Dm,r[z] tanh,B (m+z) - m 1 d 1 J "2 dt r =; dz Dm,r[z]z tanh,B (m+z) + 1 - r 256 A. C. C. COOLEN, S. N. LAUGHTON, D. SHERRINGTON 15 ~----------------------------~ r / I / / / / o L-____________________________ ~ o m 1 Figure 1: Simulations (N = 32000, dots) versus simple RS theory (solid lines), for a = 0.1 and j3 = 00. Upper dashed line: upper boundary of the physical region. Lower dashed line: upper boundary of the RS region (the AT instability). in which Dm,r[z] is the distribution of 'interference-noise' terms in the PSP's, for which the replica calculation gives the outcome (in so-called RS ansatz): Dm,r[z] = e-~2 {l-jDY tanh [>.y [~] t+(~+Z) -~+{tl} 2 27rar apr apr + e-~)2 {1-jDY tanh [>.y [~] t +(~-Z)~-{tl} 2 27rar apr apr with Dy = [27rj-te- h2dy, ~ = apr->.2jp and>' = pyaq[l-p(l-q)]-l, and with the remaining parameters {q, {t, p} to be solved from the coupled equations: j j 1-p(1-q)2 m = Dy tanh[>'y+{tj q = Dy tanh2 [>.y+{t] r = [1-p(1-q)]2 Here we only give (partly new) results of the calculation; details can be found in (Coolen and Sherrington, 1994). The noise distribution is not Gaussian (in agreement with simulations, in contrast to LCH). Our simple two-parameter theory is found to be exact for t '" 0, t -7 00 and for a -7 O. Solving numerically the dynamic equations leads to the results shown in figures 1 and 2. We find a nice agreement with numerical simulations in terms of the flow in the (m, r) plane. However, for trajectories leading away from the recall state m '" 1, the theory fails to reproduce an overall slowing down. These deviations can be quantified by comparing cumulants of the noise distributions (Ozeki and Nishimori, 1994), or by applying the theory to exactly solvable models (Coolen and Franz, 1994). Other recent applications include spin-glass models (Coolen and Sherrington, 1994) and more general classes of attractor neural network models (Laughton and Coolen, 1995). The simple two-parameter theory always predicts adequately the location of the transients in the order parameter plane, but overestimates the relaxation speed. In fact, figure 2 shows a remarkable resemblance to the results obtained for this model in (Horner et al, 1989) with the functional integral formalism; the graphs of m(t) are almost identical, but here they are derived in a much simpler way. Modem Analytic Techniques to Solve the Dynamics of Recurrent Neural Networks 257 1 .8 10 2 ·6 --..., ..., ~ '-' ..... !-..... ..... ..... .4 ..... .... .... .... 5 -.... .... ........ .2 -----0 0 0 2 4 6 B 10 0 2 4 6 B 10 t t Figure 2: Simulations (N = 32000, dots) versus simple RS theory (RS stable: solid lines, RS unstable: dashed lines), now as functions of time, for Q; = 0.1 and f3 = 00. 5 ADVANCED VERSION OF THE THEORY Improving upon the simple theory means expanding the set n beyond n = (m,E). Adding a finite number of observables will only have a minor impact; a qualitative step forward, on the other hand, results from introducing a dynamic order parameter function. Since the microscopic dynamics (1) is formulated entirely in terms of neuron states and post-synaptic potentials we choose for n (u) the joint distribution: 1 D[(, h](u) = N L <5 [( -O"i] <5 [h-hi(U)] i This choice has the advantages that (a) both m and (for symmetric systems) E are integrals over D[(, h], so the advanced theory automatically inherits the exactness at t = 0 and t = 00 of the simple one, (b) it applies equally well to symmetric and nonsymmetric models and (c) as with the simple version, generalisation to models with continuous neural variables is straightforward. Here we show the result of applying the theory to a model of the type (1) with synaptic interactions: Jij = ~ ~i~j + .iN [cos(~ )Xij +sin(~ )Yij ] Xij = Xji, Yij = -Yji (independent random Gaussian variables) (describing a nominated pattern being stored on a 'messy' synaptic background). The parameter w controls the degree of synaptic symmetry (e.g. w = 0: symmetric, w = 7r: anti-symmetric). Equation (4) applied to the observable D[(, h](u) gives: 8 ~ 8 mDt[C h] = J2[1-(O"tanh(f3H))Dt] 8h2Dt[(,h] + 8h A [(,h;Dt] + :h {DdCh] [h-Jo(tanh(f3H ))Dt]} 1 1 +2 [l+(tanh(f3h)] Dd--(, h] - 2 [l-(tanh(f3h)] DdC h] 258 E A. C. C. COOLEN, S. N. LAUGHfON, D. SHERRINGTON o .------,------.------.------.------.------~ - .2 -.4 -.6 "- "- .8 '~ ~---------- -_ 1 L-____ -L ______ L-____ ~ ______ ~ ____ ~ ______ ~ o 2 4 6 t Figure 3: Comparison of simulations (N = 8000, solid line), simple two-parameter theory (RS stable: dotted line, RS unstable: dashed line) and advanced theory (solid line), for the w = a (symmetric background) model, with Jo = 0, f3 = 00. Note that the two solid lines are almost on top of each other at the scale shown. ". 0.5 0.0 E -0.5 -0.5 o 2 4 6 o 2 4 6 t t Figure 4: Advanced theory versus N = 5600 simulations in the w = ~7r (asymmetric background) model, with f3 = 00 and J = 1. Solid: simulations; dotted: solving the RS diffusion equation. Modem Analytic Techniques to Solve the Dynamics of Recurrent Neural Networks 259 with (f(a,H))D = L:". JdH D[a,H]J(a, H). All complications are concentrated in the kernel A[C h; DJ, which is to be solved from a nontrivial set of equations emerging from the replica formalism. Some results of solving these equations numerically are shown in figures 3 and 4 (for details of the calculations and more elaborate comparisons with simulations we refer to (Laughton, Coolen and Sherrington, 1995; Coolen, Laughton and Sherrington, 1995)). It is clear that the advanced theory quite convincingly describes the transients of the simulation experiments, including the hitherto unexplained slowing down, for symmetric and nonsymmetric models. 6 DISCUSSION In this paper we have described novel techniques for studying the dynamics of recurrent neural networks near saturation. The simplest two-parameter theory (exact for t = 0, for t --+ 00 and for 0: --+ 0), which employs as dynamic order parameters the overlap with a pattern to be recalled and the total 'energy' per neuron, already describes quite accurately the location of the transients in the order parameter plane. The price paid for simplicity is that it overestimates the relaxation speed. A more advanced version of the theory, which describes the evolution of the joint distribution for neuron states and post-synaptic potentials, is mathematically more involved, but predicts the dynamical data essentially perfectly, as far as present applications allow us conclude. Whether this latter version is either exact, or just a very good approximation, still remains to be seen. In this paper we have restricted ourselves to models with binary neural variables, for reasons of simplicity. The theories generalise in a natural way to models with analogue neurons (here, however, already the simple version will generally involve order parameter functions as opposed to a finite number of order parameters). Ongoing work along these lines includes, for instance, the analysis of analogue and spherical attractor networks and networks of coupled oscillators near saturation. References B. Derrida, E. Gardner and A. Zippelius (1987), Europhys. Lett. 4: 167-173 A.C.C. Coolen and D. Sherrington (1993), in J.G. Taylor (ed.), Mathematical Approaches to Neural Networks, 293-305. Amsterdam: Elsevier. S. Amari and K. Maginu (1988), Neural Networks 1: 63-73 H. Nishimori and T. Ozeki (1993), J. Phys. A 26: 859-871 H. Rieger, M. Schreckenberg and J. Zittartz (1988), Z. Phys. B 72: 523-533 H. Horner, D. Bormann, M. Frick, H. Kinzelbach and A. Schmidt (1989), Z. Phys. B 76: 381-398 A.C.C. Coolen and D. Sherrington (1994), Phys. Rev. E 49(3): 1921-1934 H. Nishimori and T. Ozeki (1994), J. Phys. A 27: 7061-7068 A.C.C. Coolen and S. Franz (1994), J. Phys. A 27: 6947-9954 A.C.C. Coolen and D. Sherrington (1994), J. Phys. A 27: 7687-7707 S.N. Laughton and A.C.C. Coolen (1995), Phys. Rev. E 51: 2581-2599 S.N. Laughton, A.C.C. Coolen and D. Sherrington (1995), J. Phys. A (in press) A.C.C. Coolen, S.N. Laughton and D. Sherrington (1995), Phys. Rev. B (in press)	1995	62
1,109	A Realizable Learning Task which Exhibits Overfitting Siegfried Bos Laboratory for Information Representation, RIKEN, Hirosawa 2-1, Wako-shi, Saitama, 351-01, Japan email: boes@zoo.riken.go.jp Abstract In this paper we examine a perceptron learning task. The task is realizable since it is provided by another perceptron with identical architecture. Both perceptrons have nonlinear sigmoid output functions. The gain of the output function determines the level of nonlinearity of the learning task. It is observed that a high level of nonlinearity leads to overfitting. We give an explanation for this rather surprising observation and develop a method to avoid the overfitting. This method has two possible interpretations, one is learning with noise, the other cross-validated early stopping. 1 Learning Rules from Examples The property which makes feedforward neural nets interesting for many practical applications is their ability to approximate functions, which are given only by examples. Feed-forward networks with at least one hidden layer of nonlinear units are able to approximate each continuous function on a N-dimensional hypercube arbitrarily well. While the existence of neural function approximators is already established, there is still a lack of knowledge about their practical realizations. Also major problems, which complicate a good realization, like overfitting, need a better understanding. In this work we study overfitting in a one-layer percept ron model. The model allows a good theoretical description while it exhibits already a qualitatively similar behavior as the multilayer perceptron. A one-layer perceptron has N input units and one output unit. Between input and output it has one layer of adjustable weights Wi, (i = 1, ... ,N). The output z is a possibly nonlinear function of the weighted sum of inputs Xi, i.e. z = g(h) , with 1 N h = I1tT L Wi Xi . vN i=l (1) A Realizable Learning Task Which Exhibits Overfitting 219 The quality of the function approximation is measured by the difference between the correct output z* and the net's output z averaged over all possible inputs. In the supervised learning scheme one trains the network using a set of examples ;fll (JL = 1, ... , P), for which the correct output is known. It is the learning task to minimize a certain cost function, which measures the difference between the correct output z~ and the net's output Zll averaged over all examples. Using the mean squared error as a suitable measure for the difference between the outputs, we can define the training error ET and the generalization error Ea as (2) The development of both errors as a function of the number P of trained examples is given by the learning curves. Training is conventionally done by gradient descend. For theoretical purposes it is very useful to study learning tasks, which are provided by a second network, the so-called teacher network. This concept allows a more transparent definition of the difficulty of the learning task. Also the monitoring of the training process becomes clearer, since it is always possible to compare the student network and the teacher network directly. Suitable quantities for such a comparison are, in the perceptron case, the following order parameters, N q:= IIWII = 2:(Wi )2. (3) i=l Both have a very transparent interpretation, r is the normalized overlap between the weight vectors of teacher and student, and q is the norm of the student's weight vector. These order parameters can also be used in multilayer learning, but their number increases with the number of all possible permutations between the hidden units of teacher and student. 2 The Learning Task Here we concentrate on the case in which a student perceptron has to learn a mapping provided by another perceptron. We choose identical networks for teacher and student. Both have the same sigmoid output function, i.e. g(h) = g(h) = tanh( "Ih). Identical network architectures of teacher and student are realizable tasks. In principle the student is able to learn the task provided by the teacher exactly. Unrealizable tasks can not be learnt exactly, there remains always a finite error. If we use uniformally distributed random inputs ;f and weights W, the weighted sum h in (1) can be assumed as Gaussian distributed. Then we can express the generalization error (2) by the order parameters (3), Ea= JDZ1 JDz2~{tanh["IZll-tanh[q(rzl+~Z2)]r, (4) with the Gaussian measure J 1 +00 dz (Z2) Dz:= -- exp -- 00 ../2i 2 (5) From equation (4) we can see how the student learns the gain "I of the teachers output function. It adjusts the norm q of its weights. The gain "I plays an important role since it allows to tune the function tanhbh) between a linear function b « 1) and a highly nonlinear function b » 1). Now we want to determine the learning curves of this task. 220 s.B6s 3 Emergence of Overfitting 3.1 Explicit Expression for the Weights Below the storage capacity of the perceptron, i.e. a = 1, the minimum of the training error ET is zero. A zero training error implies that every example has been learnt exactly, thus (6) The weights with minimal norm that fulfill this condition are given by the Pseudoinverse (see Hertz et al. 1991), P Wi = 2: h~ (C-l)~v xf, (7) ~,v=l Note, that the weights are completely independent of the output function g(h) = g(h). They are the same as in the simplest realizable case, linear perceptron learns linear perceptron. 3.2 Statistical Mechanics The calculation of the order parameters can be done by a method from statistical mechanics which applies the commonly used replica method. For details about the replica approach see Hertz et al. (1991). The solution of the continuous perceptron problem can be found in Bas et al. (1993). Since the results of the statistical mechanics calculations are exact only in the thermodynamic limit, i.e. N ~ 00, the variable a is the more natural measure. It is defined as the fraction of the number of patterns P over the system size N, i.e. a := PIN. In the thermodynamic limit N and P are infinite, but a is still finite. Normally, reasonable system sizes, such as N ~ 100, are already well described by this theory. Usually one concentrates on the zero temperature limit, because this implies that the training error ET accepts its absolute minimum for every number of presented examples P. The corresponding order parameters for the case, linear perceptron learns linear student, are q='Yva, r =va. (8) The zero temperature limit can also be called exhaustive training, since the student net is trained until the absolute minimum of ET is reached. For small a and high gains 'Y, i.e levels of nonlinearity, exhaustive training leads to overfitting. That means the generalization error Ea(a) is not, as it should, monotonously decreasing with a. It is one reason for overfitting, that the training follows too strongly the examples. The critical gain 'Yc, which determines whether the generalization error Ea ( a) is increasing or decreasing function for small values of a, can be determined by a linear approximation. For small a, both order parameters (3) are small, and the student's tanh-function in (4) can be approximated by a linear function. This simplifies the equation (4) to the following expression, Ea(f) = Ea(O) - i [2H(r) - 'Y 1, with H( 'Y):= J Dz tanh(rz) z. (9) Since the function H(r) has an upper bound, i.e. J2/7r, the critical gain is reached if 'Yc = 2H{rc). The numerical solution gives 'Yc = 1.3371. If 'Y is higher, the slope of Ea(a) is positive for small a. In the following considerations we will use always the gain 'Y = 5 as an example, since this is an intermediate level of nonlinearity. A Realizable Learning Task Which Exhibits Overfitting 1.0 0.8 0.6 0.4 0.2 ...... -- '- .- . -'-.- --.---.- .-. -'-. - '- ' - . -.-. -'-. w _ _ • _ .. __ -- -- -.. -- -.. -- -- ----- .. _- --0.0 0.0 0.2 0.4 0.6 PIN -'-.100.0 10.0 5.0 2.0 1.0 0.5 0.8 221 1.0 Figure 1: Learning curves E ( 0:) for the problem, tanh- perceptron learns tanhperceptron, for different values of the gain,. Even in this realizable case, exhaustive training can lead to overfitting, if the gain , is high enough. 3.3 How to Understand the Emergence of Overfitting Here the evaluation of the generalization error in dependence of the order parameters rand q is helpful. Fig. 2 shows the function EG(r, q) for r between 0 and 1 and q between 0 and 1.2,. The exhaustive training in realizable cases follows always the line q( r) = ,r independent of the actual output function. That means, training is guided only by the training error and not by the generalization error. If the gain , is higher than ,e, the line EG = EG(O, 0) starts with a lower slope than q(r) = ,r, which results in overfitting. 4 How to Avoid Overfitting From Fig. 2 we can guess already that q increases too fast compared to r. Maybe the ratio between q and r is better during the training process. So we have to develop a description for the training process first. 4.1 Training Process We found already that the order parameters for finite temperatures (T > 0) of the statistical mechanics approach are a good description of the training process in an unrealizable learning task (Bos 1995). So we use the finite temperature order parameters also in this task. These are, again taken from the task 'linear perceptron learns linear percept ron' , ( ) = J(~) (1 + 0:) a - 20: q 0:, a, 2 ' a a - 0: r(o:, a) = (0:) a2 0: a (1+0:)a-20:' (10) with the temperature dependent variable a:= 1 + [,8(Q - q)]-l . (11) 222 6.0 5.0 4.0 q 3.0 2.0 ".1.0 . . . . /local minZ i abs. mici. ./ local m~. ........ ... ... ........ ........ S.BOS --- ...... ~: ...... -. 0.0 ~~==~-~-:=::: .... !:. ==±~===--L..::===~·= ··· ···~ ·· ·· ·3· ·· ·· · 0.0 0.2 0.4 0.6 0.8 1.0 r Figure 2: Contour plot of EG(r,q) defined by (4), the generalization error as a function of the two order parameters. Starting from the minimum EG = 0 at (r, q) = (1,5) the contour lines for EG = 0.1,0.2, ... , 0.8 are given (dotted lines). The dashed line corresponds to EG(O,O) = 0.42. The solid lines are parametric curves of the order parameters (r, q) for certain training strategies. The straight line illustrates exhaustive training, the lower ones the optimal training, which will be explained in Fig. 3. Here the gain I = 5. The zero temperature limit corresponds to a = 1. We will show now that the decrease of the temperature dependent parameter a from 00 to 1, describes the evolution of the order parameters during the training process. In the training process the natural parameter is the number of parallel training steps t. In each parallel training step all patterns are presented once and all weights are updated. Fig. 3 shows the evolution of the order parameters (10) as parametric curves (r,q). The exhaustive learning curve is defined by a = 1 with the parameter 0: (solid line). For each 0: the training ends on this curve. The dotted lines illustrate the training process, a runs from infinity to 1. Simulations of the training process have shown that this theoretical curve is a good description, at least after some training steps. We will now use this description of the training process for the definition of an optimized training strategy. 4.2 Optimal temperature The optimized training strategy chooses not a = 1 or the corresponding temperature T = 0, but the value of a (Le. temperature), which minimizes the generalization error EG. In the lower solid curve indicating the parametric curve (r, q) the value of a is chosen for every 0:, which minimizes EG. The function EG(a) has two minima between 0: = 0.5 and 0.7. The solid line indicates always the absolute minimum. The parametric curves corresponding to the local minima are given by the double dashed and dash-dotted lines. Note, that the optimized value a is always related to an optimized temperature through equation (11). But the parameter a is also related to the number of training steps t. A Realizable Learning Task Which Exhibits Overfilling 6.0 5.0 4.0 q 3.0 2.0 1.0 0.0 0.0 local min. abs. min. local min. simulation I--t--l 0.2 0.4 r 223 0.6 0.8 1.0 Figure 3: Training process. The order parameters (10) as parametric curves (r,q) with the parameters a and a. The straight solid line corresponds to exhaustive learning, i.e. a = 1 (marks at a = 0.1,0.2, ... 1.0). The dotted lines describe the training process for fixed a. Iterative training reduces the parameter a from 00 to 1. Examples for a = 0.1,0.2,0.3,0.4,0.9,0.99 are given. The lower solid line is an optimized learning curve. To achieve this curve the value of a is chosen, which minimizes EG absolutely. Between a ~ 0.5 and 0.7 the error EG has two minima; the double- dashed and dash-dotted lines indicate the second, local minimum of EG. Compare with Fig. 2, to see which is the absolute and which the local minimum of EG. A naive early stopping procedure ends always in the minimum with the smaller q, since it is the first minimum during the training process (see simulation indicated with errorbars). 4.3 Early Stopping Fig. 3 and Fig. 2 together indicate that an earlier stopping of the training process can avoid the overfitting. But in order to determine the stopping point one has to know the actual generalization error during the training. Cross-validation tries to provide an approximation for the real generalization error. The cross-validation error Ecv is defined like ET , see (2), on a set of examples, which are not used during the training. Here we calculate the optimum using the real generalization error, given by rand q, to determine the optimal point for early stopping. It is a lower bound for training with finite cross-validation sets. Some preliminary tests have shown that already small cross- validation sets approximate the real EG quite well. Training is stopped, when EG increases. The resulting curve is given by the error bars in Fig. 3. The errorbars indicate the standard deviation of a simulation with N = 100 averaged over 50 trials. In Fig. 4 the same results are shown as learning curves EG(a). There one can see clearly that the early stopping strategy avoids the overfitting. 5 Summary and outlook In this paper we have shown that overfitting can also emerge in realizable learning tasks. The calculation of a critical gain and the contour lines in Fig. 2 imply, that 224 0.5 0.4 0.3 EO 0.2 0.1 0.0 0.0 exh. local min. abs. min. local min. simulation ~ 0.2 S.BOS 0.4 0.6 0.8 1.0 PIN Figure 4: Learning curves corresponding to the parametric curves in Fig. 3. The upper solid line shows again exhaustive training. The optimized finite temperature curve is the lower solid line. From 0: = 0.6 exhaustive and optimal training lead to identical results (see marks). The simulation for early stopping (errorbars) finds the first minimum of EG. the reason for the overfitting is the nonlinearity of the problem. The network adjusts slowly to the nonlinearity of the task. We have developed a method to avoid the overfitting, it can be interpreted in two ways. Training at a finite temperature reduces overfitting. It can be realized, if one trains with noisy examples. In the other interpretation one learns without noise, but stops the training earlier. The early stopping is guided by cross-validation. It was observed that early stopping is not completely simple, since it can lead to a local minimum of the generalization error. One should be aware of this possibility, before one applies early stopping. Since multilayer perceptrons are built of nonlinear perceptrons, the same effects are important for multilayer learning. A study with large scale simulations (Miiller et al. 1995) has shown that overfitting occurs also in realizable multilayer learning tasks. Acknowledgments I would like to thank S. Amari and M. Opper for stimulating discussions, and M. Herrmann for hints concerning the presentation. References S. Bos. (1995) Avoiding overfitting by finite temperature learning and crossvalidation. International Conference on Artificial Neural Networks '95 Vo1.2, p.111. S. Bos, W. Kinzel & M. Opper. (1993) Generalization ability of perceptrons with continuous outputs. Phys. Rev. E 47:1384-1391. J. Hertz, A. Krogh & R. G. Palmer. (1991) Introduction to the Theory of Neural Computation. Reading: Addison-Wesley. K. R. Miiller, M. Finke, N. Murata, K. Schulten & S. Amari. (1995) On large scale simulations for learning curves, Neural Computation in press.	1995	63
1,110	Simulation of a Thalamocortical Circuit for Computing Directional Heading in the Rat Hugh T. Blair* Department of Psychology Yale University New Haven, CT 06520-8205 tadb@minerva.cis.yale.edu Abstract Several regions of the rat brain contain neurons known as head-direction celis, which encode the animal's directional heading during spatial navigation. This paper presents a biophysical model of head-direction cell acti vity, which suggests that a thalamocortical circuit might compute the rat's head direction by integrating the angular velocity of the head over time. The model was implemented using the neural simulator NEURON, and makes testable predictions about the structure and function of the rat head-direction circuit. 1 HEAD-DIRECTION CELLS As a rat navigates through space, neurons called head-direction celis encode the animal's directional heading in the horizontal plane (Ranck, 1984; Taube, Muller, & Ranck, 1990). Head-direction cells have been recorded in several brain areas, including the postsubiculum (Ranck, 1984) and anterior thalamus (Taube, 1995). A variety of theories have proposed that head-direction cells might play an important role in spatial learning and navigation (Brown & Sharp, 1995; Burgess, Recce, & O'Keefe, 1994; McNaughton, Knierim, & Wilson, 1995; Wan, Touretzky, & Redish, 1994; Zhang, 1995). 1.1 BASIC FIRING PROPERTIES A head-direction cell fires action potentials only when the rat's head is facing in a particular direction with respect to the static surrounding environment, regardless of the animal's location within that environment. Head-direction cells are not influenced by the position of the rat's head with respect to its body, they are only influenced by the direction of the Also at the Yale Neuroengineering and Neuroscience Center (NNC), 5 Science Park North, New Haven, CT 06511 Simulation of Thalamocortical Circuit for Computing Directional Heading in Rats 153 G> ~ g> , 0 ~ 05 )( '" E ;! o 0 0~~90~--:-::180:-"--::27:70---~360 Head Direction 360,0 270 90 180 Figure I: Directional Tuning Curve of a Head-Direction Cell head with respect to the stationary reference frame of the spatial environment. Each headdirection cell has its own directional preference, so that together, the entire population of cells can encode any direction that the animal is facing. Figure 1 shows an example of a head-direction cell's directional tuning curve, which plots the firing rate of the celI as a function of the rat's momentary head direction. The tuning curve shows that this cell fires maximalIy when the rat's head is facing in a preferred direction of about 160 degrees. The cell fires less rapidly for directions close to 160 degrees, and stops firing altogether for directions that are far from 160 degrees. 1.2 THE VELOCITY INTEGRATION HYPOTHESIS McNaughton, Chen, & Markus (1991) have proposed that head-direction cells might rely on a process of dead-reckoning to calculate the rat's current head direction, based on the previous head direction and the angular velocity at which the head is turning. That is, head-direction cells might compute the directional position of the head by integrating the angular velocity of the head over time. This velocity integration hypothesis is supported by three experimental findings. First, several brain regions that are associated with headdirection cells contain angular velocity cells, neurons that fire in proportion to the angular head velocity (McNaughton et al., 1994; Sharp, in press). Second, some head-direction cells in postsubiculum are modulated by angular head velocity, such that their peak firing rate is higher if the head is turning in one direction than in the other (Taube et al., 1990). Third, it has recently been found that head-direction cells in the anterior thalamus, but not the postsubiculum, anticipate the future direction of the rat's head (Blair & Sharp, 1995). 1.3 ANTICIPATORY HEAD-DIRECTION CELLS Blair and Sharp (1995) discovered that head-direction cells in the anterior thalamus shift their directional preference to the left during clockwise turns, and to the right during counterclockwise turns. They showed that this shift occurs systematically as a function of head velocity, in a way that alIows these cells anticipate the future direction of the rat's head. To illustrate this, consider a cell that fires whenever the head will be facing a specific direction, 9, in the near future. How would such a cell behave? There are three cases to consider. First, imagine that the rat's head is turning clockwise, approaching the direction 9 from the left side. In this case, the anticipatory cell must fire when the head is facing to the left of 9, because being to the left of 9 and turning clockwise predicts arrival at 9 in the near future. Second, when the head is turning counterclockwise and approaching 9 from the right side, the anticipatory cell must fire when the head is to the right of 9. Third, if the head is still, then the cell should only fire if the head is presently facing 9. In summary, an anticipatory head direction cell should shift its directional preference to the left during clockwise turns, to the right during counterclockwise turns, and not at all when the head is still. This behavior can be formalized by the equation !leV) = 9 - V't, [1] 154 H. T. BLAIR where ~ denotes the cell's preferred present head direction. v denotes the angular velocity of the head. 8 denotes the future head direction that the cell anticipates, and 't is a constant time delay by which the cell's activity anticipates arrival at 8. Equation 1 assumes that ~ is measured in degrees. which increase in the clockwise direction. and that v is positive for clockwise head turns. and negative for counterclockwise head turns. Blair & Sharp (1995) have demonstrated that Equation 1 provides a good approximation of head-direction cell behavior in the anterior thalamus. 1.3 ANTICIPATORY TIME DELAY (r) Initial reports suggested that head-direction cells in the anterior thalamus anticipate the future head direction by an average time delay of't = 40 msec, whereas postsubicular cells encode the present head direction, and therefore "anticipate" by 't = 0 msec (Blair & Sharp, 1995; Taube & Muller, 1995). However, recent evidence suggests that individual neurons in the anterior thalamus may be temporally tuned to anticipate the rat's future head-direction by different time delays between 0-100 msec, and that postsubicular cells may "lag behind" the present head-direction by about to msec (Blair & Sharp, 1996). 2 A BIOPHYSICAL MODEL This section describes a biophysical model that accounts for the properties of head-direction cells in postsubiculum and anterior thalamus. by proposing that they might be connected to form a thalamocortical circuit. The next section presents simulation results from an implementation of the model, using the neural simulator NEURON (Hines, 1993). 2.1 NEURAL ELEMENTS Figure 2 illustrates a basic circuit for computing the rat's head-direction. The circuit consists of five types of cells: 1) Present Head-Direction (PHD) Cells encode the present direction of the rat's head, 2) Anticipatory Head-Direction (AHD) Cells encode the future direction of the rat's head, 3) Angular-Velocity (AV) Cells encode the angular velocity of the rat's head (the CLK AV Cell is active during clockwise turns, and the CNT AV Cell is active during counterclockwise turns), 4) the Angular Speed (AS) Cell fires in inverse proportion to the angular speed of the head, regardless of the turning direction (that is, the AS Cell fires at a lower rate during fast turns, and at a higher rate during slow turns), 5) Angular-Velocity Modulated Head-Direction (AVHD) Cells are head-direction cells that fire AHDCells RTN A~~~IS Excitatory ~ Inhibitory --~,;;",I".· AS Cell MB I ABBREVIADONS AT = Anterior Thalamus MB. Mammillary Bodi .. PS = P08tsubiculum RS • Rempl"'ill Cortex R1N = Reticul.11III. Nu. Figure 2: A Model of the Rat Head-Direction System Simulation of Thalamocortical Circuit for Computing Directional Heading in Rats 155 only when the head is turning in one direction and not the other (the CLK AVHD Cell fires in its preferred direction only when the head is turning clockwise, and the CNT AVHD Cell fires in its preferred direction only when the head turns counterclockwise). 2.2 FUNCTIONAL CHARACTERISTICS In the model, AHD Cells directly excite their neighbors on either side, but indirectly inhibit these same neighbors via the AVHD Cells, which act as inhibitory interneurons. AHD Cells also send excitatory feedback connections to themselves (omitted from Figure 2 for clarity), so that once they become active. they remain active until they are turned off by inhibitory input (the rate of firing can also be modulated by inhibitory input). When the rat is not turning its head. the cell representing the current head direction fires constantly, both exciting and inhibiting its neighbors. In the steady-state condition (Le., when the rat is not turning its head), lateral inhibition exceeds lateral excitation, and therefore activity does not spread in either direction through the layer of AHD Cells. However. when the rat begins turning its head, some of the AVHD Cells are turned off, allowing activity to spread in one direction. For example. during a clockwise head tum. the CLK AV Cell becomes active, and inhibits the layer of CNT AVHD Cells. As a result, AHD Cells stop inhibiting their right neighbors, so activity spreads to the right through the layer of AHD Cells. Because AHD Cells continue to inhibit their neighbors to the left, activity is shut down in the leftward direction, in the wake of the activity spreading to the right. The speed of propagation through the AHD layer is governed by the AS Cell. During slow head turns, the AS Cell fires at a high rate, strongly inhibiting the AHD Cells, and thereby slowing the speed of propagation. During fast head turns, the AS Cell fires at a low rate, weakly inhibiting the AHD Cells, allowing activity to propagate more quickly. Because of inhibition from AS cells, AHD cells fire faster when the head is turning than when it is still (see Figure 4), in agreement with experimental data (Blair & Sharp, 1995). AHD Cells send a topographic projection to PHD Cells, such that each PHD Cell receives excitatory input from an AHD Cell that anticipates when the head will soon be facing in the PHD Cell's preferred direction. AHD Cell activity anticipates PHD Cell activity because there is a transmission delay between the AHD and PHD Cells (assumed to be 5 msec in the simulations presented below). Also, the weights of the connections from AHD Cells to PHD Cells are small, so each AHD Cell must fire several action potentials before its targeted PHD Cell can begin to fire. The time delay between AHD and PHD Cells accounts for anticipatory firing, and corresponds to the 1: parameter in Equation I. 2.3 ANATOMICAL CHARACTERISTICS Each component of the model is assumed to reside in a specific brain region. AHD and PHD Cells are assumed to reside in anterior thalamus (AT) and postsubiculum (PS), respectively. AS Cells have been observed in PS (Sharp, in press) and retrosplenial cortex (RS) (McNaughton, Green, & Mizumori, 1986), but the model predicts that they may also be found in the mammillary bodies (MB), since MB receives input from PS and RS (Shibata, 1989), and MB projects to ATN. AVHD Cells have been observed in PS (Taube et ai., 1990), but the model predicts that they may aiso be found in the reticular thalamic nucleus (RTN), because RTN receives input from PSIRS (Lozsadi, 1994), and RTN inhibits AT. It should be noted that lateral excitation between ATN cells has not been shown, so this feature of the model may be incorrect. Table 1 summarizes anatomical evidence. 3 SIMULATION RESULTS The model illustrated in Figure 2 has been implemented using the neural simulator NEURON (Hines. 1993). Each neural element was represented as a single spherical compart156 H. T. BLAIR Table 1: Anatomical Features of the Model FEATURE OF MODEL PHD Cells in PSIRS AHD Cells in AT AV Cells in PSIRS AT projects to PS AT projects to RTN PSIRS projects to RTN AVHD Cells in RTN AS Cells in MB REFERENCE Chen et aI., 1990; Ranck, 1984 Blair & Sharp, 1995 McNaughton et aI., 1994; Sharp, in press van Groen & Wyss, 1990 Shibata, 1992 Lozsadi, 1994 PREDICTION OF MODEL PREDICTION OF MODEL ment, 30 Jlm in diameter, with RC time constants ranging between 15 and 30 msec. Synaptic connections were simulated using triggered alpha-function conductances. The results presented here demonstrate the behavior of the model, and compare the properties of the model with experimental data. To begin each simulation, a small current was injected in to one of the AHD Cells, causing it to initiate sustained firing. This cell represented the simulated rat's initial head direction. Head-turning behavior was simulated by injecting current into the AV and AS Cells, with an amplitude that yielded firing proportional to the desired angular head velocity. 3.1 ACTIVITY OF HEAD-DIRECTION CELLS Figure 3 presents a simple simulation, which illustrates the behavior of head-direction cells in the model. The simulated rat begins by facing in the direction of 0 degrees. Over the course of 250 msec, the rat quickly turns its head 60 degrees to the right, and then returns to the initial starting position of 0 degrees. The average velocity of the head in this simulation was 480 degrees/sec, which is similar to the speed at which an actual rat performs a fast head tum (Blair & Sharp, 1995). Over the course of the simulation, neural activation propagates from the O-degree cell to the 60-degree cell, and then back to the 0degree cell. 3.2 COMPARISON WITH EXPERIMENTAL DATA To examine how well the model reproduces firing properties of PS and AT cells, another simple simulation was performed. The firing rate the model's PHD and AHD Cells was examined while the simulated rat performed several 360-degree revolutions in both the clockwise and counterclockwise directions. Results are summarized in Figure 4, which ACTIVITY OF PHD CELLS ANIMAL 15c.lll_WlM"-__ -----"'~~ 30CelII----MMIM'-_--M\.W~ oW CelII----~~ ___ -M~~ 60c.III=:::;~~~~~:::;::==::, 50 100 1 Turning Right A'--;Tu....,mIng~Le~1t ---:, Time (msec) , BEHAVIOR WSOIIIt ••.......... ". AVlrlgl Angular Velocity = 480" 'sec Figure 3: Simulation Example Simulation of Thalamocortical Circuit for Computing Directional Heading in Rats 157 Cil12.0 ----------., ~ 10.0 . Z' ~ 8.0 ' CD 6.0 I 15» .:i 40 : g 2.0 i ,; 0.0 i O<>"T (exper.,..ntaI Getal 0-0 Ps (.)(~'""". ,*a) .n _AT (modol dolo' .••••• .. Ps (model ~. ) ••••• [} ...•..........•.•....• -o • • io .2.0 ,'--_ ___ _ _ -----' N 0 100 200 300 400 500 Angular Head Velocity (deglsec) N'25.0 r-, --~----e. ~ ~ 20.0 ~ o.-/' ..... ·····~ 0)15.0 , .= Li: 10.0 , • g, I Cii 5.0 , ~ 0.0 i~ _______ -' o 100 200 300 400 500 [}----------------.----.-o Angular Head Velocity (degJsec) Figure 4: Compared Properties of Real and Simulated Head-Direction Cells compares simulation data with experimental data. The experimental data in Figure 4 shows averaged results for 21 cells recorded in AT, and 19 cells recorded in PS. Because AT cells anticipate the future head direction, they exhibit an angular separation between their clockwise and counterclockwise directiQnal preference, whereas as no such separation occurs for PS cells (see section 2.4). For AT cells, the magnitude of the angular separation is proportional to angular head velocity, with greater separation occurring for fast turns, and less separation for slow turns (see Eq. 1). The left panel of Figure 4 shows that the model's PHD and AHD Cells exhibit a similar pattern of angular separation. Blair & Sharp (1995) reported that the firing rates of AT and PS cells differ in two ways: 1) AT cells fire at a higher rate than PS cells, and 2) AT cells have a higher rate during fast turns than during slow turns, whereas PS cells fire at the same rate, regardless of turning speed. In Figure 4 (right panel), it can be seen that the model reproduces these findings. 4 DISCUSSION AND CONCLUSIONS In this paper, I have presented a neural model of the rat head-direction system. The model includes neural elements whose firing properties are similar to those of actual neurons in the rat brain. The model suggests that a thalamocortical circuit might compute the directional position of the rat's head, by integrating angular head velocity over time. 4.1 COMPARISON WITH OTHER MODELS McNaughton et al. (1991) proposed that neurons encoding head-direction and angular velocity might be connected to form a linear associative mapping network. Skaggs et al. (1995) have refined this idea into a theoretical circuit, which incorporates head-direction and angular velocity cells. However, the Skaggs et al. (1995) circuit does not incorporate anticipatory head-direction cells, like those found in AT. A model that does incorporate anticipatory cells has been developed by Elga, Redish, & Touretzky (unpublished manuscript). Zhang (1995) has recently presented a theoretical analysis of the head-direction circuit, which suggests that anticipatory head-direction cells might be influenced by both the angular velocity and angular acceleration of the head, whereas non-anticipatory cells may be influenced by the angular velocity only, and not the angular acceleration. 4.2 LIMITATIONS OF THE MODEL In its current form, the model suffers some significant limitations. For example, the directional tuning curves of the model's head-direction cells are much narrower than those of actual head-direction cells. Also, in its present form, the model can accurately track the rat's head-direction over a rather limited range of angular head velocities. These limitations are presently being addressed in a more advanced version of the model. 158 H. T. BLAIR Acknowledgments This work was supported by NRSA fellowship number 1 F31 MH11102-01Al from NIMH. a Yale Fellowship. and the Yale Neuroengineering and Neuroscience Center (NNC). I thank Michael Hines. Patricia Sharp. and Steve Fisher for their assistance. References Blair. H.T .. & Sharp. P.E. (1995). Anticipatory head-direction cells in anterior thalamus: Evidence for a thalamocortical circuit that mtegrates angular head velocity to compute head direction. Journal of Neuroscience, IS, 6260-6270. Blair, H.T .• & Sharp (1996). Temporal Tuning of Anticipatory Head-Direction Cells in the Anterior Thalamus of the Rat. Submitted. Brown. M. & Sharp. P.E. (1995). Simulation of spatial learning in the morris water maze by a. neural network model of the hippocampal formation and nucleus accumbens. Hippocampus, 5. 171-188. Burgess, N .• Recce. M .• & O'Keefe. J. (1994). A model of hippocampal function. Neural Networks, 7. 1065-1081. Elga. AN .• Redish, AD .• & Touretzky. D.S. (1995). A model of the rodent head-direction system. Unyublished Manuscript. Hines. M. (1993). NEURON: A program for simulation of nerve equations. In F. Eckman (Ed.). Neural Systems: Analysis and Modeling, Norwell. MA : Kluwer Academic Publishers. pp. 127-136. Lozsadi. D.A. (1994). Organization of cortical afferents to the rostral, limbic sector of the rat thalamic reticular nucleus. The Journal of Comparative Neurology, 341, 520-533. McNaughton. B.L.. Chen. L.L.. & Markus. E.1. (1991). Dead reckoning, landmark learning. and the sense of direction: a neurophysiological and computational hypothesis. Journal of Cognitive Neuroscience, 3, 190-202. McNaughton, B.L.. Green. E.1 .• & Mizumori, S.1.y. (1986). Representation of body motion trajectory by rat sensory motor cortex neurons. Society for Neuroscience Abstracts. 12,260. McNaughton, B.L.. Knierim. J.J .• & Wilson. M.A (1995). Vector encoding and the vestibular foundations of spatial cognition: neurophysiological and computational mechanisms. In M. Gazzaniga (Ed.). The Cognitive Neurosciences. Cambndge: MIT Press. McNaughton. B.L.. Mizumori, S.Y.1 .• Barnes. C.A .• Leonard. B.J .• MarqUiS. M .• & Green. B.J. (1994). Coritcal representation of motion during unrestrained spatial navigaton in the rat. Cerebral Cortex, 4, 27-39. Ranck, J.B. (1984). Head-direction cells in the deep ceUlayers of dorsal presubiculum in freely moving rats. Society for Neuroscience Abstracts, 12, 1524. Shibata. H. (1989). Descending projections to the mammillary nuclei in the rat. as studied by retrograde and anterograde transport of wheat germ agglutinin-horseradish peroxidase. The Journal of Comparative Neurology, 285. 436-452. Shibata. H. (1992). Topographic organization of subcortical projections to the anterior thalamic nuclei in the rat. The Journal of Comparative Neurology, 323, 117-127. Sharp, P.E. (in press). Multiple spatiallbehavioral corrrelates for cells in the rat postsubiculum: multiple regression analysis and comparison to other hippocampal areas. Cerebral Cortex. Skaggs, W.E .• Knierim. J.1 .• Kudrimoti. H.S., & McNaughton, B.L. (1995). A model of the neural basis of the rat's sense of direction. In G. Tesauro. D.S. Touretzky, & T.K. Leen (Eds.), Advances in Neural Information Processing Systems 7. MIT Press. Taube. 1.S. (1995). Head-direction cells recorded in the anterior thalamic nuclei of freelymoving rats. Journal of Neuroscience, 15, 70-86. Taube. J.S .• & Muller. R.O. (1995). Head-direction cell activity in the anterior thalamus. but not the postsubiculum, predicts the animal's future directional heading. Society for Neuroscience Abstracts. 21. 946. Taube. J.S., Muller. R.U .• & Ranck, J.B. (1990). Head-direction cells recorded from the postsubiculum in freely moving rats, I. Description and quantitative analysis. Jounral of Neuroscience, 10, 420-435. van Groen. T., & Wyss, J.M. (1990). The postsubicular cortex in the rat: characterization of the fourth region of subicular cortex and its connections. Journal of Comparative Neurology, 216. 192-210. Wan, H.S .• Touretzky. D.S .• & Redish. D.S. (1994). A rodent navigation model that combines place code. head-direction, and path integration information. Society for Neuroscience Abstracts, 20, 1205. Zhang, K. (1995). Representation of spatial orientation by the intrinsic dynamics of the head-direction cell ensemble: A theory. Submitted.	1995	64
1,111	Clustering data through an analogy to the Potts model Marcelo Blatt, Shai Wiseman and Eytan Domany Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot 76100, Israel Abstract A new approach for clustering is proposed. This method is based on an analogy to a physical model; the ferromagnetic Potts model at thermal equilibrium is used as an analog computer for this hard optimization problem. We do not assume any structure of the underlying distribution of the data. Phase space of the Potts model is divided into three regions; ferromagnetic, super-paramagnetic and paramagnetic phases. The region of interest is that corresponding to the super-paramagnetic one, where domains of aligned spins appear. The range of temperatures where these structures are stable is indicated by a non-vanishing magnetic susceptibility. We use a very efficient Monte Carlo algorithm to measure the susceptibility and the spin spin correlation function. The values of the spin spin correlation function, at the super-paramagnetic phase, serve to identify the partition of the data points into clusters. Many natural phenomena can be viewed as optimization processes, and the drive to understand and analyze them yielded powerful mathematical methods. Thus when wishing to solve a hard optimization problem, it may be advantageous to apply these methods through a physical analogy. Indeed, recently techniques from statistical physics have been adapted for solving hard optimization problems (see e.g. Yuille and Kosowsky, 1994). In this work we formulate the problem of clustering in terms of a ferromagnetic Potts spin model. Using the Monte Carlo method we estimate physical quantities such as the spin spin correlation function and the susceptibility, and deduce from them the number of clusters and cluster sizes. Cluster analysis is an important technique in exploratory data analysis and is applied in a variety of engineering and scientific disciplines. The problem of partitionaZ clustering can be formally stated as follows. With everyone of i = 1,2, ... N patterns represented as a point Xi in a d-dimensional metric space, determine the partition of these N points into M groups, called clusters, such that points in a cluster are more similar to each other than to points in different clusters. The value of M also has to be determined. Clustering Data through an Analogy to the Potts Model 417 The two main approaches to partitional clustering are called parametric and nonparametric. In parametric approaches some knowledge of the clusters' structure is assumed (e.g. each cluster can be represented by a center and a spread around it). This assumption is incorporated in a global criterion. The goal is to assign the data points so that the criterion is minimized. A typical example is variance minimization (Rose, Gurewitz, and Fox, 1993). On the other hand, in non-parametric approaches a local criterion is used to build clusters by utilizing local structure of the data. For example, clusters can be formed by identifying high-density regions in the data space or by assigning a point and its K -nearest neighbors to the same cluster. In recent years many parametric partitional clustering algorithms rooted in statistical physics were presented (see e.g. Buhmann and Kiihnel , 1993). In the present work we use methods of statistical physics in non-parametric clustering. Our aim is to use a physical problem as an analog to the clustering problem. The notion of clusters comes very naturally in Potts spin models (Wang and Swendsen, 1990) where clusters are closely related to ordered regions of spins. We place a Potts spin variable Si at each point Xi (that represents one of the patterns), and introduce a short range ferromagnetic interaction Jij between pairs of spins, whose strength decreases as the inter-spin distance Ilxi - Xj" increases. The system is governed by the Hamiltonian (energy function) 1i = - L hj D8,,8j <i,j> Si = 1 . .. q , (1) where the notation < i, j > stands for neighboring points i and j in a sense that is defined later. Then we study the ordering properties of this inhomogeneous Potts model. As a concrete example, place a Potts spin at each of the data points of fig. 1. ~~--~------~--------~--------~------~--------~------~--~ ·30 ·20 -10 10 20 30 Figure 1: This data set is made of three rectangles, each consisting of 800 points uniformly distributed, and a uniform rectangular background of lower density, also consisting of 800 points. Points classified (with Tclus = 0.08 and () = 0.5) as belonging to the three largest clusters are marked by crosses, squares and x's. The fourth cluster is of size 2 and all others are single point clusters marked by triangles. At high temperatures the system is in a disordered (paramagnetic) phase. As the temperature is lowered, larger and larger regions of high density of points (or spins) exhibit local ordering, until a phase transition occurs and spins in the three rectangular high density regions become completely aligned (i. e. within each region all Si take the same value - super-paramagnetic phase). The aligned regions define the clusters which we wish to identify. As the temperature 418 M. BLATT, S. WISEMAN, E. DOMANY is further lowered, a pseudo-transition occurs and the system becomes completely ordered (ferromagnetic). 1 A mean field model To support our main idea, we analyze an idealized set of points where the division into natural classes is distinct. The points are divided into M groups. The distance between any two points within the same group is d1 while the distance between any two points belonging to different groups is d2 > d1 (d can be regarded as a similarity index). Following our main idea, we associate a Potts spin with each point and an interaction J1 between points separated by distance d1 and an h between points separated by d2 , where a ~ J2 < J1• Hence the Hamiltonian (1) becomes; 1{ = - ~ L L 6~; ,~j - ~ L L 6s; ,sj si = 1, ... , q , (2) /10 i<j /1o<V i ,j where si denotes the ith spin (i = 1, ... , ~) of the lJth group (lJ = 1, ... , M). From standard mean field theory for the Potts model (Wu, 1982) it is possible to show that the transition from the ferromagnetic phase to the paramagnetic phase is at Tc = 2M (qJ.)~Og(q-l) [J1 + (M - 1)h] . The average spin spin correlation function, 6~,,~ j at the paramagnetic phase is t for all points Xi and Xj; i. e. the spin value at each point is independent of the others. The ferromagnetic phase is further divided into two regions. At low temperatures, with high probability, all spins are aligned; that is 6~.,sJ ~ 1 for all i and j. At intermediate temperatures, between T* and Tc, only spins of the same group lJ are aligned with high probability; 6~" ~'-: ~ 1, .' J while spins belonging to different groups, Jl and lJ, are independent; 6~1" s~ ~ 1 . • ' 1 q The spin spin correlation function at the super-paramagnetic phase can be used to decide whether or not two spins belong to the same cluster. In contrast with the mere inter-point distance, the spin spin correlation function is sensitive to the collective behavior of the system and is therefore a suitable quantity for defining collective structures (clusters). The transition temperature T* may be calculated and shown to be proportional to J2 ; T* = a(N, M, q) h. In figure 2 we present the phase diagram, in the (~, ~) plane, for the case M = 4, N = 1000 and q = 6. paramagnetic /' 1e-01 ~ _____ --,-____ ~~ "1 / super-paramagnet~s-,'-' .., 1e"()2 ferromagnetic / "",,; f:: ./,' , ...... ', .. ' 1e-03 ;" .. , .-' 1e..()4~--~~~~--~~----~~~ 1e..()S 1e"()4 1e-03 le-02 le..()1 1e+OO J2JJl Figure 2: Phase diagram of the mean field Potts model (2) for the case M = 4, N = 1000 and q = 6. The critical temperature Tc is indicated by the solid line, and the transition temperature T, by the dashed line. The phase diagram fig. 2 shows that the existence of natural classes can manifest itself in the thermodynamic properties of the proposed Potts model. Thus our approach is supported, provided that a correct choice of the interaction strengths is made. Clustering Data through an Analogy to the Potts Model 419 2 Definition of local interaction In order to minimize the intra-cluster interaction it is convenient to allow an interaction only bet.ween "neighbors". In common \ .... ith other "local met.hods" , we assume that there is a 'local length scale' '" a, which is defined by the high density regions and is smaller than the typical distance between points in the low density regions. This property can be expressed in the ordering properties of the Potts system by choosing a short range interaction. Therefore we consider that each point interacts only with its neighbors with interaction strength __ 1 (!lXi-Xj!l2) Jij J ji - R exp 2a 2 . (3) Two points, Xi and Xj, are defined as neighbors if they have a mutual neighborhood value J{; that is, if Xi is one of the J{ nearest neighbors of Xj and vice-versa. This definition ensures that hj is symmetric; the number of bonds of any site is less than J{. We chose the "local length scale", a, to be the average of all distances Ilxi - Xj II between pairs i and j with a mutual neighborhood value J{. R is the average number of neighbors per site; i. e it is twice the number of non vanishing interactions, Jij divided by the number of points N (This careful normalization of the interaction strength enables us to estimate the critical temperature Tc for any data sample). 3 Calculation of thermodynanlic quantities The ordering properties of the system are reflected by the susceptibility and the Spill spin correlation functioll D'<"'<J' where -.. -. stands for a thermal average. These quantities can be estimated by averaging over the configurations genel'ated by a Monte Carlo procedure. We use the Swendsen-Wang (Wang and Swendsen, 1990) Monte Carlo algorithm for the Potts model (1) not only because of its high efficiency, but also because it utilizes the SW clusters. As will be explained the SW clusters are strongly connected to the clusters we wish to identify. A layman's explanation of the method is as follows. The SW procedure stochastically identifies clusters of aligned spins, and then flips whole clusters simultaneously. Starting from a given spin configuration, SW go over all the bonds between neighboring points, and either "freeze" or delete them. A bond connecting two neighboring sites i and j, is deleted with probability p~,j = exp( - 63 .. 3 J and frozen with probability p? = 1 p~,j. Having gone over all the bonds, all spins which have a path of frozen bonds connecting them are identified as being in the same SW cluster. Note t.hat, according to the definition of p~,j, only spins of the same value can be frozen in the same SW cluster. Now a new spin configuration is generated by drawing, for each cluster, randomly a value s = 1, ... q, which is assigned to all its spins. This procedure defines one Monte Carlo step and needs to be iterated in order to obtain thermodynamic averages. At temperatures where large regions of correlated spins occur, local methods (e. g. Metropolis), which flip one spin at a time, become very slow. The SVl method overcomes this difficulty by flipping large clusters of aligned spins simult.aneously. Hence the SW method exhibits much smaller autocorrelation times than local methods. The strong connection between the SW clusters and the ordering properties of the Pot.ts spins is manifested in the relation -6-- (<1- 1)710+ 1 _".,8) q (4) 420 M. BLATI, S. WISEMAN, E. DOMANY where nij = 1 whenever Si and Sj belong to the same SW-cluster and nij = 0 otherwise. Thus, nij is the probability that Si and Sj belong to the same SW-cluster. The r.h.s. of (4) has a smaller variance than its l.h.s., so that the probabilities nij provide an improved estimator of the spin spin correlation function. 4 Locating the super-paramagnetic phase In order to locate the temperature range in which the system IS III the superparamagnetic phase we measure the susceptibility of the system which is proportional to the variance of the magnetization N2 X = T (m2 - m ) . (5) The magnetization, m, is defined as qNmax/N -1 m=-----q-1 (6) where NJ.' is the number of spins with the value J.l. In the ferromagnetic phase the fluctuations of the magnetization are negligible, so the susceptibility, X, is small. As the temperature is raised, a sudden increase of the susceptibility occurs at the transition from the ferromagnetic to the super-paramagnetic phase. The susceptibility is non-vanishing only in the superparamagnetic phase, which is the only phase where large fluctuations in the magnetization can occur. The point where the susceptibility vanishes again is an upper bound for the transition temperature from the super-paramagnetic to the paramagnetic phase. 5 The clustering procedure Our method consists of two main steps. First we identify the range of temperatures where the clusters may be observed (that corresponding to the super-paramagnetic phase) and choose a temperature within this range. Secondly, the clusters are identified using the information contained in the spin spin correlation function at this temperature. The procedure is summarized here, leaving discussion concerning the choice of the parameters to a later stage. (a) Assign to each point Xi a q-state Potts spin variable Si. q was chosen equal to 20 in the example that we present in this work. (b) Find all the pairs of points having mutual neighborhood value K. We set K = 10. (c) Calculate the strength of the interactions using equation (3). (d) Use the SW procedure with the Hamiltonian (1) to calculate the susceptibility X for various temperatures. The transition temperature from the paramagnetic phase _ 1 can be roughly estimated by Tc ~ 410;(1~A)' (e) Identify the range of temperatures of non-vanishing X (the super-paramagnetic phase). Identify the temperature Tmax where the susceptibility X is maximal, and the temperature Tvanish, where X vanishes at the high temperature side. The optimal temperature to identify the clusters lies between these two temperatures. As a rule of thumb we chose the "clustering temperature" Tcltl~ = Tvan .. ~+Tma.r but the results depend only weakly on Tclu~, as long as T cltls is in the super-paramagnetic range, Tmax < Tcltl~ < Tvani~h. Clustering Data through an Analogy to the Potts Model 421 (f) At the clustering temperature Tclu s , estimate the spin spin correlation, o s "s J ' for all neigh boring pairs of points Xi and Xj, using (4) . (g) Clusters are identified according to a thresholding procedure. The spin spin correlation function 03. ,3J of points Xi and Xj is compared with a threshold, (); if OS,,3J > () they are defined as "friends". Then all mutual friends (including fl'iends of friends, etc) are assigned to the same cluster. We chose () = 0.5. In order to show how this algorithm works, let us consider the distribution of points presented in figure 1. Because of the overlap of the larger sparse rectangle with the smaller rectangles, and due to statistical fluctuations, the three dense rectangles actually contain 883, 874 and 863 points. Going through steps (a) to (d) we obtained the susceptibility as a function of the temperature as presented in figure 3. The susceptibility X is maximal at T max = 0.03 and vanishes at Tvanish = 0.13. In figure 1 we present the clusters obtained according to steps (f) and (g) at Tclus = 0.08. The size of the largest clusters in descending order is 900, 894, 877, 2 and all the rest are composed of only one point. The three biggest clusters correspond to the clusters we are looking for, while the background is decomposed into clusters of size one. 0.035 0030 0025 0020 0015 0 010 0 005 0000 0.00 0.02 0.04 0.06 0 ,08 0.10 012 o.te 016 T Figure 3: The susceptibility density x;;. as a function of t.he t.emperature. Let us discuss the effect of the parameters on the procedure. The number of Potts states, q, determines mainly the sharpness of the transition and the critical temperature. The higher q, the sharper the transition. On the other hand, it is necessary to perform more statistics (more SW sweeps) as the value of q increases. From our simulations, we conclude that the influence of q is very weak. The maximal number of neighbors, f{, also affects the results very little; we obtained quite similar results for a wide range of f{ (5 ~ f{ ~ 20). No dramatic changes were observed in the classification, when choosing clustering temperatures Tc1u3 other than that suggested in (e). However this choice is clearly ad-hoc and a better choice should be found. Our method does not provide a natural way to choose a threshold () for the spin spin correlation function. In practice though, the classification is not very sensitive to the value of (), and values in the range 0.2 < () < 0.8 yield similar results. The reason is that the frequency distribution of the values of the spin spin correlation function exhibit.s t.wo peaks, one close to 1 and the other close to 1, while for intermediate values it is verv close q v t.o zero. In figure (4) we present the average size of the largest S\V cluster as a function of the temperature, along with the size of the largest cluster obtained by the thresholding procedUl'e (described in (7)) using three different threshold values () = 0.1, 0 . .5, o .~). Not.e the agreement. between the largest clust.er size defined by t.he threshold e = 0.5 and the average size of the largest SW cluster for all t.emperatures (This agreement holds for the smaller clusters as well) . It support.s our thresholding procedure as a sensible one at all temperatUl'es. 422 500 o~~~~~~~~~~~~~ 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 T 6 Discussion M. BLATT, S. WISEMAN, E. DOMANY Figure 4: Average size of the largest SW cluster as a function of the temperature, is denoted by the solid line. The triangles, x's and squares denote the size of the largest cluster obtained with thresholds () = 0.2, 0.5 and 0.9 respectively. Other methods that were proposed previously, such as Fukunaga's (1990) , can be formulated as a Metropolis relaxation of a ferromagnetic Potts model at T = O. The clusters are then determined by the points having the same spin value at the local minima of the energy at which the relaxation process terminates. Clearly this procedure depends strongly on the initial conditions. There is a high probability of getting stuck in a metastable state that does not correspond to the desired answer. Such a T = 0 method does not provide any way to distinguish between "good" and "bad" metastable states. We applied Fukunaga's method on the data set of figure (1) using many different initial conditions. The right answer was never obtained. In all runs, domain walls that broke a cluster into two or more parts appeared. Our method generalizes Fukunaga's method by introducing a finite temperature at which the division into clusters is stable. In addition, the SW dynamics are completely insensitive to the initial conditions and extremely efficient. Work in progress shows that our method is especially suitable for hierarchical clustering. This is done by identifying clusters at several temperatures which are chosen according to features of the susceptibility curve. In particular our method is successful in dealing with "real life" problems such as the Iris data and Landsat data. Acknowledgments We thank 1. Kanter for many useful discussions. This research has been supported by the US-Israel Bi-national Science Foundation (BSF) , and the Germany-Israel Science Foundation (GIF). References J .M. Buhmann and H. Kuhnel (1993); Vector quantization with complexity costs, IEEE Trans. Inf. Theory 39, 1133. K. Fukunaga (1990); Introd. to statistical Pattern Recognition, Academic Press. K. Rose, E. Gurewitz, and G.C. Fox (1993); Constrained clustering as an optimization method, IEEE Trans on Patt. Anal. and Mach. Intel. PAMI 15, 785. S. Wang and R.H. Swendsen (1990); Cluster Monte Carlo alg., Physica A 167,565. F.Y. Wu (1982) , The Potts model, Rev Mod Phys, 54, 235. A.L. Yuille and J.J. Kosowsky (1994); Statistical algorithms that converge, Neural Computation 6, 341 (1994).	1995	65
1,112	Learning Fine Motion by Markov Mixtures of Experts Marina Meilii Dept. of Elec. Eng. and Computer Sci. Massachussetts Inst. of Technology Cambridge, MA 02139 mmp@ai.mit.edu Michael I. J Ol'dan Dept.of Brain and Cognitive Sciences Massachussetts Inst. of Technology Cambridge, MA 02139 jordan@psyche.mit.edu Abstract Compliant control is a standard method for performing fine manipulation tasks, like grasping and assembly, but it requires estimation of the state of contact (s.o.c.) between the robot arm and the objects involved. Here we present a method to learn a model of the movement from measured data. The method requires little or no prior knowledge and the resulting model explicitly estimates the s.o.c. The current s.o.c. is viewed as the hidden state variable of a discrete HMM. The control dependent transition probabilities between states are modeled as parametrized functions of the measurement. We show that their parameters can be estimated from measurements at the same time as the parameters of the movement in each s.o.c. The learning algorithm is a variant of the EM procedure. The E step is computed exactly; solving the M step exactly is not possible in general. Here, gradient ascent is used to produce an increase in likelihood. 1 INTRODUCTION For a large class of robotics tasks, such as assembly tasks or manipulation of relatively light-weight objects, under appropriate damping of the manipulator the dynamics of the objects can be neglected. For these tasks the main difficulty is in having the robot achieve its goal despite uncertainty in its position relative to the surrounding objects. Uncertainty is due to inaccurate knowledge of the geometric shapes and positions of the objects, of their physical properties (surface friction coefficients), or to positioning errors in the manipulator. The standard solution to this problem is controlled compliance first introduced in (Mason, 1981). Under compliant motion, the task is performed in stages; in each stage the robot arm 1004 M. MElLA, M. I. JORDAN maintains contact with a selected surface or feature of the environment; the stage ends when contact with the feature corresponding to the next stage is made. Decomposing the given task into subtasks and specifying each goal or subgoal in terms of contact constraints has proven to be a particularly fertile idea, from which a fair number of approaches have evolved. But each of them have to face and solve the problem of estimating the state of contact (i.e. checking if the contact with the correct surface is achieved), a direct consequence of dealing with noisy measurements. Additionally, most approaches assume prior geometrical and physical knowledge of the environment. In this paper we present a method to learn a model of the environment which will serve to estimate the s.o.c. and to predict future positions from noisy measurements. It associates to each state of contact the coresponding movement model (m.m.); that is: a relationship between positions, nominal and actual velocities that holds over a domain of the position-nominal velocity space. The current m.m. is viewed as the hidden state variable of a discrete Hidden Markov Model (HMM) with transition probabilities that are parametrized functions of the measurement. We call this model Markov Mixture of Experts (MME) and show how its parameters can be estimated. In section 2 the problem is defined, section 3 introduces the learning algorithm, section 4 presents a simulated example and 5 discusses other aspects relevant to the implementation. 2 REACHABILITY GRAPHS AND MARKOV MIXTURES OF EXPERTS For any ensemble of objects, the space of all the relative degrees of freedom of the objects in the ensemble is called the configuration space (C-space). Every possible configuration of the ensemble is represented by a unique point in the C-space and movement in the real space maps into continuous trajectories in the C-space (Lozano-Perez, 1983). The sets of points corresponding to each state of contact create a partition over the C-space. Because trajectories are continuous, a point can move from a s.o.c. only to a neighboring s.o.c. This can be depicted by a directed graph with vertices representing states of contact and arcs for the possible transitions between them, called the reach ability graph. If no constraints on the velocities are imposed, then in the reachability graph each s.o.c. is connected to all its neighbours. But if the range of velocities is restricted, the connectivity of the graph decreases and the connections are generally non-symmetric. Figure 1 shows an example of a C-space and its reachability graph for velocities with only positive components. Ideally, in the absence of noise, the states of contact can be perfectly observed and every transition through the graph is thus deterministic. To deal with the uncertainty in the measurements, we will attach probabilities to the arcs of the graph in the following way: Let us denote by Qi the set of configurations corresponding to s.o.c. i and let the movement of a point x with uniform nominal velocity v for a time aT be given by x( t + aT) = r (x, v, aT); both x and v are vectors of same dimension as the C-space. Now, let x', v' be the noisy measurements of the true values x, v, x E Qj and P[x, vlx', v',j] the posterior distribution of (x , v) given the measurements and the s.o.c. Then, the probability of transition to a state i from a given state j in time T3 can be expressed as: P[ilx',v',j] = r P[x,vlx',v',j]dxdv = aij(x',V') (1) J{x ,vIXEQj ,rex ,v ,T.)EQ.} Defining the transition probability matrix A = [aji]rj=l and assuming measurement Learning Fine Motion by Markov Mixtures of Experts 1005 y x (a) (b) Figure 1: A configuration space (a) and its reachability graph (b). The nodes represent movement models: C is the free space, A and B are surfaces with static and dynamic friction, G represents jamming in the corner. The velocity V has positive components. noise P[x'lq = i, x E Qd leads to an HMM with output x having a continuous emission probability distribution and where the s.o.c. plays the role of a hidden state variable. Our main goal is to estimate this model from observed data. To give a general statement of the problem we will assume that all the position, velocity and force measurements are represented by the input vector u; the output vector y of dimensionality ny contains the future position (which our model will learn to predict). Observations are made at moments which are integer multiples of T$' indexed by t = 0,1, .. , T. If T$ is a constant sampling time the dependency of the transition probability on Ts can be ignored. For the purpose of the parameter estimation, the possible dependence between u(t) and yet + 1) will also be ignored, but it should be considered when the trained model is used for prediction. Throughout the following section we will also assume that the input-output dependence is described by a Gaussian conditional density p(y(t)lu(t), q(t) = k) with mean f(u(t),(h:) and variance E = (1'21. This is equivalent to assuming that given the S.O.c. all noise is additive Gaussian output noise, which is obviously an approximation. But this approximation will allow us to derive certain quantities in closed form in an effective way. The function feu, (he) is the m.m. associated with state of contact k (with Ok its parameter vector) and q is the selector variable representing it. Sometimes we will find it useful to partition the domain of a m.m. into subdomains and to represent it by a different function (i .e. a different set of parameters Ok) on each of the subdomains; then, the name movement model will be extended to them. The evolution of q is controlled by a Markov chain which depends on u and of a set of parameters W: aij(u(t), W) = Pr[q(t + 1) = ilq(t) = j, u(t)] t = 0, 1, ... with L aij(u, W) = 1 \:Iu, W, j = 1, . .. , m. (2) 1006 M. MElLA, M. I. JORDAN y u ,,------.1& q ! 'd 1-------' t .......................... .................. ............. _ ••••••••••••• ~ .. .............................. _ ...................... . Figure 2: The Markov Mixture of Experts architecture Fig. 2 depicts this architecture. It can be easily seen that this model generalizes the mixture of experts (ME) architecture (Jacobs, et al., 1991), to which it reduces in the case where aij are independent of j (the columns of A are all equal). It becomes the model of (Bengio and Frasconi, 1995) when A and f are neural networks. 3 AN EM ALGORITHM FOR MME To estimate the values of the unknown parameters (J"2, Wk, Ok, k = 1, ... ,m given the sequence of observations {(u(t), y(t))};=o, T> 0 the Expectation Maximization (EM) algorithm will be used. The states {q(t)};=o play the role of the unobserved variables. More about EM can be found in (Dempster et al., 1977) while aspects specific to this algorithm are in (Meila and Jordan, 1994). The E step computes the probability of each state and of every transition to occur at t E {O, ... , T} given the observations and an initial parameter set. This can be done efficiently by the forward-backward algorithm (Rabiner and Juang, 1986). Pr[q(t) = k I {(u(t), y(t))};=o, W, 0, (J"2] (3) Pr[q(t) = j, q(t + 1) = i I {(u(t), y(t))};=o , W, 0, (J"2] In the M step the new estimates of the parameters are found by maximizing the average complete log-likelihood J, which in our case has the form T-l m J(O, (J"2, W) = L L eij(t) lnaij(u(t), W)t=o i,j=l Since each parameter appears in only one term of J the maximization is equivalent to: T 0l:ew = argmin L 'n(t) lIy(t) - f( u(t), Ok)11 2 Ih t=o (5) Learning Fine Motion by Markov Mixtures of Experts 1007 T-l wnew = argmax L L~ij(t) In (aij(u(t), w)) W t=o ij (6) 1 T m ny(T + 1) ~ ~ ''}'k(t) Ily(t) - I(u(t), Ok )11 2 (7) There is no general closed form solution to (5) and (6). Their difficulty depends on the form of I and aij. The complexity of the m.m. is determined by the geometrical shape of the objects' surfaces. For planar surfaces and no rotational degrees of freedom I is linear in Ok. Then, (5) becomes a weighted least squares problem which can be solved in closed form. The functions in A depend both on the movement and of the noise models. Because the noise is propagated through non-linearities to the output, an exact form as in (1) may be hard to compute analytically. Moreover, a correct noise model for each of the possible uncertainties is rarely available (Eberman, 1995). A common practical approach is to trade accuracy for computability and to parametrize A in a form which is easy to update but deprived of physical meaning. In all the cases where maximization cannot be performed exactly, one can resort to Generalized EM by merely increasing J. In particular, gradient ascent in parameter space is a technique which can replace maximization. This modification will not affect the overall convergence of the EM iteration but can significantly reduce its speed. Because EM only finds local maxima of the likelihood, the initialization is important. If I( u, Ok) correspond to physical movement models, good initial estimates for their parameters can be available. The same applies to those components of W which bear physical significance. A complementary approach is to reduce the number of parameters by explicitly setting the probabilities of impossible transitions to O. 4 SIMULATION RESULTS Simulations have been run on the C-space shown in fig. 1. The inputs were the 4-dimensional vectors of position (x, y) and nominal velocity (Vx , Vy); the output was the predicted position. The coordinate range was [0, 10] and the admissible velocities were confined to the upper right quadrant (Vmax 2: Vx, Vy 2: Vmin > 0). The restriction in direction implied that the trajectories remain in the coordinate domain; it also appeared in the topology of the reachability graph, which has no transition to the free space from another state. This model was implemented by a MME. The m.m. are linear in the parameters, corresponding to the piecewise linearity of the true model. To implement the transition matrix A we used a bank of gating net-works, one for each s.o.c., consisting of 2 layer perceptrons with softmax1 output. There are 230 free parameters in the gating networks and 64 in the m.m. The training set included N = 5000 data points, in sequences of length T ~ 6, all starting in free space. The starting position of the sequence and the nominal velocities at each step were picked randomly. We found that a more uniform distribution of the data points over the states of contact is necessary for successful learning. Since this is not expected to happen in applications (where, e.g., sticking occurs less often than sliding) , the obtained models were tested also on a distribution that 1 () exp(WTx) The softmax function is given by: softmax. x = Z ! T ,i = 1, .. m with Wj , x jexp(Wj x) vectors of the same dimension. 1008 M. MElLA, M. I. JORDAN Table 1: Performance of MME versus ME (a) Model Prediction Standard Error (MSE) 1/2 Test set Trammg distributIon Umform V distribution noise level 0 .1 .2 .3 .4 0 .1 .2 .3 .4 MME,(1' =.2 .024 .113 .222 .332 .443 .023 .11 .219 .327 .437 MME,(1' =0 .003 .114 .228 .343 .456 .010 .109 .218 .327 .435 ME, (1' = .2 .052 .133 .25 .37 .493 .044 .129 .247 .367 .488 ME, (1' =0 .047 .131 .25 .37 .49 .034 .126 .245 .366 .488 (b) State Misclassification Error [%] Test set Trammg distribution Umform V distribution noise level 0 .1 . ~ .3 .4 U .1 .~ .3 .4 MME, (1' =.2 5.15 5.2 5.5 5.9 6.4 3.45 3.5 3.8 4.2 4.6 MME, (1' =0 .78 1.40 2.35 3.25 4.13 .89 1.19 1.70 2.30 2.88 ME, (1' =.2 6.46 6.60 7.18 7.73 8.13 3.85 3.90 4.38 4.99 5.65 ME, (1' =0 6.25 6.45 6.98 7.61 8.15 3.84 3.98 4.53 5.05 5.70 was uniform over velocities (and consequently, highly non-uniform over states of contact). Gaussian noise with (1'=0.2 or 0 was added to the (x, y) training data. In the M step, the parameters of the gating networks were updated by gradient ascent. For the m.m.least squares estimation was used. To ensure that models and gates are correctly coupled, initial values for () are chosen around the true values. As discussed in the previous section, this is not an unrealistic assumption. W was initialized with small random values. Each simulation was run until convergence. We used two criteria to measure the performance of the learning algorithm: square root of prediction MSE and hidden state misdassificaton. The results are summarized in table 1. The test set size is 50,000 in all cases. Input noise is Gaussian with levels between 0 and 0.4. Comparisons were made with a ME model with the same number of states. The simulations show that the MME architecture is tolerant to input noise, although it is not taking it into account explicitly. The MME consistently outperforms the ME model in both prediction and state estimation accuracy. 5 DISCUSSION An algorithm to estimate the parameters of composite movement models in the presence of noisy measurements has been presented. The algorithm exploits the physical decomposability of the problem and the temporal relationship between the data points to produce estimates of both the model's parameters and the s.o.c. It requires only imprecise initial knowledge about the geometry and physical properties of the system. Prediction via MME The trained model can be used either as an estimator for the state of contact or as a forward model in predicting the next position. For the former goal the forward part of the forward-backward algorithm can be used to implement a recursive estimator or the methods in (Eberman, 1995) can be used. The obtained 'Yk(t) , combined with the outputs of the movement models, will produce a predicted output y. An improved posterior estimate of y can be obtained Learning Fine Motion by Markov Mixtures of Experts 1009 by combining f) with the current measurement. Scaling issues. Simulations have shown that relatively large datasets are required for training even for a small number of states. But, since the states represent physical entities, the model will inherit the geometrical locality properties thereof. Thus, the number of possible transitions from a state will be bounded by a small constant when the number of states grows, keeping the data complexity linear in m. As a version of EM, our algorithm is batch. It follows that parameters are not adapted on line. In particular, the discretization time T& must be fixed prior to training. But small changes in Ts can be accounted for by rescaling the velocities V. For the other changes, inasmuch as they are local, relearning can be confined to those components of the architecture which are affected. References Bengio, Y. and Frasconi, P. (1995). An input output HMM architecture. In G. Tesauro, D. Touretzky, & T. Leen (Eds.), Neural Information Processing Sys. tems 7, Cambridge, MA: MIT Press, pp. 427-435. Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, B, 39:1- 38. Eberman, B. S. (1995). A sequential decision approach to sensing manipulation contact features. PhD thesis, M.I.T., Dept. of Electrical Engineering. Jacobs, R. A., Jordan, M. 1., Nowlan, S., & Hinton, G. E. (1991). Adaptive mixtures of local experts. Neural Computation, 3, 1-12. Lozano-Perez, T. (1983). Spatial planning: a configuration space approach. IEEE Transactions on Computers. Mason, M. T. (1981). Compliance and force control for computer controlled manipulation. IEEE Trans. on Systems, Man and Cybernetics. Meila, M. and Jordan, M. 1. (1994). Learning the parameters of HMMs with auxilliary input. Technical Report 9401, MIT Computational Cognitive Science, Cambridge, MA. Rabiner, R. L. and Juang, B. H. (1986). An introduction to hidden Markov models. ASSP Magazine, 3(1):4-16.	1995	66
1,113	Stable Linear Approximations to Dynamic Programming for Stochastic Control Problems with Local Transitions Benjamin Van Roy and John N. Tsitsiklis Laboratory for Information and Decision Systems Massachusetts Institute of Technology Cambridge, MA 02139 e-mail: bvr@mit.edu, jnt@mit.edu Abstract We consider the solution to large stochastic control problems by means of methods that rely on compact representations and a variant of the value iteration algorithm to compute approximate costto-go functions. While such methods are known to be unstable in general, we identify a new class of problems for which convergence, as well as graceful error bounds, are guaranteed. This class involves linear parameterizations of the cost-to- go function together with an assumption that the dynamic programming operator is a contraction with respect to the Euclidean norm when applied to functions in the parameterized class. We provide a special case where this assumption is satisfied, which relies on the locality of transitions in a state space. Other cases will be discussed in a full length version of this paper. 1 INTRODUCTION Neural networks are well established in the domains of pattern recognition and function approximation, where their properties and training algorithms have been well studied. Recently, however, there have been some successful applications of neural networks in a totally different context - that of sequential decision making under uncertainty (stochastic control). Stochastic control problems have been studied extensively in the operations research and control theory literature for a long time, using the methodology of dynamic programming [Bertsekas, 1995]. In dynamic programming, the most important object is the cost-to-go (or value) junction, which evaluates the expected future 1046 B. V. ROY, 1. N. TSITSIKLIS cost to be incurred, as a function of the current state of a system. Such functions can be used to guide control decisions. Dynamic programming provides a variety of methods for computing cost-to- go functions. Unfortunately, dynamic programming is computationally intractable in the context of many stochastic control problems that arise in practice. This is because a cost-to-go value is computed and stored for each state, and due to the curse of dimensionality, the number of states grows exponentially with the number of variables involved. Due to the limited applicability of dynamic programming, practitioners often rely on ad hoc heuristic strategies when dealing with stochastic control problems. Several recent success stories - most notably, the celebrated Backgammon player of Tesauro (1992) - suggest that neural networks can help in overcoming this limitation. In these applications, neural networks are used as compact representations that approximate cost- to-go functions using far fewer parameters than states. This approach offers the possibility of a systematic and practical methodology for addressing complex stochastic control problems. Despite the success of neural networks in dynamic programming, the algorithms used to tune parameters are poorly understood. Even when used to tune the parameters of linear approximators, algorithms employed in practice can be unstable [Boyan and Moore, 1995; Gordon, 1995; Tsitsiklis and Van Roy, 1994]. Some recent research has focused on establishing classes of algorithms and compact representation that guarantee stability and graceful error bounds. Tsitsiklis and Van Roy (1994) prove results involving algorithms that employ feature extraction and interpolative architectures. Gordon (1995) proves similar results concerning a closely related class of compact representations called averagers. However, there remains a huge gap between these simple approximation schemes that guarantee reasonable behavior and the complex neural network architectures employed in practice. In this paper, we motivate an algorithm for tuning the parameters of linear compact representations, prove its convergence when used in conjunction with a class of approximation architectures, and establish error bounds. Such architectures are not captured by previous results. However, the results in this paper rely on additional assumptions. In particular, we restrict attention to Markov decision problems for which the dynamic programming operator is a contraction with respect to the Euclidean norm when applied to functions in the parameterized class. Though this assumption on the combination of compact representation and Markov decision problem appears restrictive, it is actually satisfied by several cases of practical interest. In this paper, we discuss one special case which employs affine approximations over a state space, and relies on the locality of transitions. Other cases will be discussed in a full length version of this paper. 2 MARKOV DECISION PROBLEMS We consider infinite horizon, discounted Markov decision problems defined on a finite state space S = {I, .. . , n} [Bertsekas, 1995]. For every state i E S, there is a finite set U(i) of possible control actions, and for each pair i,j E S of states and control action u E U (i) there is a probability Pij (u) of a transition from state i to state j given that action u is applied. Furthermore, for every state i and control action u E U (i), there is a random variable Ciu which represents the one-stage cost if action u is applied at state i. Let f3 E [0,1) be a discount factor. Since the state spaces we consider in this paper Stable Linear Approximations Programming for Stochastic Control Problems 1047 are finite, we choose to think of cost-to-go functions mapping states to cost- to-go values in terms of cost-to-go vectors whose components are the cost-to-go values of various states. The optimal cost-to-go vector V* E !Rn is the unique solution to Bellman's equation: Vi= min. (E[CiU]+.BLPij(U)Vj), ViES. (1) uEU(t) jES If the optimal cost-to-go vector is known, optimal decisions can be made at any state i as follows: u=arg min. (E[CiU]+.BLPij(U)l--j), ViES. uEU(t) jES There are several algorithms for computing V* but we only discuss the value iteration algorithm which forms the basis of the approximation algorithm to be considered later on. We start with some notation. We define the dynamic programming operator as the mapping T : !Rn r-t !Rn with components Ti : !Rn r-t !R defined by Ti(V) = min. (E[CiU]+.BLPij(U)Vj), ViES. (2) uEU(t) jES It is well known and easy to prove that T is a maximum norm contraction. In particular , IIT(V) - T(V')lloo :s; .BIIV - V'lIoo, The value iteration algorithm is described by V(t + 1) = T(V(t)), where V (0) is an arbitrary vector in !Rn used to initialize the algorithm. It is easy to see that the sequence {V(t)} converges to V, since T is a contraction. 3 APPROXIMATIONS TO DYNAMIC PROGRAMMING Classical dynamic programming algorithms such as value iteration require that we maintain and update a vector V of dimension n. This is essentially impossible when n is extremely large, as is the norm in practical applications. We set out to overcome this limitation by using compact representations to approximate cost-to-go vectors. In this section, we develop a formal framework for compact representations, describe an algorithm for tuning the parameters of linear compact representations, and prove a theorem concerning the convergence properties of this algorithm. 3.1 COMPACT REPRESENTATIONS A compact representation (or approximation architecture) can be thought of as a scheme for recording a high-dimensional cost-to-go vector V E !Rn using a lowerdimensional parameter vector wE !Rm (m «n). Such a scheme can be described by a mapping V : !Rm r-t !Rn which to any given parameter vector w E !Rm associates a cost-to-go vector V (w). In particular, each component Vi (w) of the mapping is the ith component of a cost-to-go vector represented by the parameter vector w. Note that, although we may wish to represent an arbitrary vector V E !Rn, such a scheme allows for exact representation only of those vectors V which happen to lie in the range of V. In this paper, we are concerned exclusively with linear compact representations of the form V(w) = Mw, where M E !Rnxm is a fixed matrix representing our choice of approximation architecture. In particular, we have Vi(w) = Miw, where Mi (a row vector) is the ith row of the matrix M. 1048 B. V. ROY, J. N. TSITSIKLIS 3.2 A STOCHASTIC APPROXIMATION SCHEME Once an appropriate compact representation is chosen, the next step is to generate a parameter vector w such that V{w) approximates V. One possible objective is to minimize squared error of the form IIMw VII~. If we were given a fixed set of N samples {( iI, ~:), (i2' Vi;), ... , (i N, ~:)} of an optimal cost-to-go vector V, it seems natural to choose a parameter vector w that minimizE's E7=1 (Mij w ~;)2. On the other hand, if we can actively sample as many data pairs as we want, one at a time, we might consider an iterative algorithm which generates a sequence of parameter vectors {w(t)} that converges to the desired parameter vector. One such algorithm works as follows: choose an initial guess w(O), then for each t E {O, 1, ... } sample a state i{t) from a uniform distribution over the state space and apply the iteration (3) where {a(t)} is a sequence of diminishing step sizes and the superscript T denotes a transpose. Such an approximation scheme conforms to the spirit of traditional function approximation - the algorithm is the common stochastic gradient descent method. However, as discussed in the introduction, we do not have access to such samples of the optimal cost-to-go vector. We therefore need more sophisticated methods for tuning parameters. One possibility involves the use of an algorithm similar to that of Equation 3, replacing samples of ~(t) with TiCt) (V(t)). This might be justified by the fact that T(V) can be viewed as an improved approximation to V, relative to V. The modified algorithm takes on the form (4) Intuitively, at each time t this algorithm treats T(Mw(t)) as a "target" and takes a steepest descent step as if the goal were to find a w that would minimize IIMwT(Mw(t))II~. Such an algorithm is closely related to the TD(O) algorithm of Sutton (1988). Unfortunately, as pointed out in Tsitsiklis and Van Roy (1994), such a scheme can produce a diverging sequence {w(t)} of weight vectors even when there exists a parameter vector w that makes the approximation error V* - Mw* zero at every state. However, as we will show in the remainder of this paper, under certain assumptions, such an algorithm converges. 3.3 MAIN CONVERGENCE RESULT Our first assumption concerning the step size sequence {a(t)} is standard to stochastic approximation and is required for the upcoming theorem. Assumption 1 Each step size a(t) is chosen prior to the generation of i(t), and the sequence satisfies E~o a(t) = 00 and E~o a 2 (t) < 00. Our second assumption requires that T : lRn t-+ lRn be a contraction with respect to the Euclidean norm, at least when it operates on value functions that can be represented in the form Mw, for some w. This assumption is not always satisfied, but it appears to hold in some situations of interest, one of which is to be discussed in Section 4. Assumption 2 There exists some {3' E [0, 1) such that IIT(Mw) - T(Mw')112 ::; {3'IIMw - Mw'112, Vw,w' E lRm. Stable Linear Approximations to Programming for Stochastic Control Problems 1049 The following theorem characterizes the stability and error bounds associated with the algorithm when the Markov decision problem satisfies the necessary criteria. Theorem 1 Let Assumptions 1 and 2 hold, and assume that M has full column rank. Let I1 = M(MT M)-l MT denote the projection matrix onto the subspace X = {Mwlw E ~m}. Then, (a) With probability 1, the sequence w(t) converges to w, the unique vector that solves: Mw = I1T(Mw). (b) Let V be the optimal cost-to-go vector. The following error bound holds: IIMw* - V1I2 ~ (1 ;!~ynllI1V - Vlloo. 3.4 OVERVIEW OF PROOF Due to space limitations, we only provide an overview of the proof of Theorem 1. Let s : ~m f-7 ~m be defined by s(w) = E [( Miw - Ti(Mw(t)))MT] , where the expectation is taken over i uniformly distributed among {I, .. . , n}. Hence, E[w(t + l)lw(t), a(t)] = w(t) - a(t)s(w(t)), where the expectation is taken over i(t). We can rewrite s as s(w) = ~(MTMW - MTT(MW)) , and it can be thought of as a vector field over ~m. If the sequence {w(t)} converges to some w, then s ( w) must be zero, and we have MTMw MTT(Mw) Mw = I1T(Mw). Note that III1T(Mw) - I1T(Mw')lb ~ {j'IIMw - Mw'112, Vw,w' E ~m, due to Assumption 2 and the fact that projection is a nonexpansion of the Euclidean norm. It follows that I1Te) has a unique fixed point w E ~m, and this point uniquely satisfies Mw* = I1T(Mw). We can further establish the desired error bound: IIMw - V112 < IIMw - I1T(I1V) 112 + III1T(I1V) - I1V112 + III1V - V112 < {j'IIMw - V112 + IIT(I1V) - V112 + III1V - V1I2 < t3'IIMw - V112 + (1 + mv'nIII1V - Vlloo, and it follows that Consider the potential function U(w) = ~llw wII~. We will establish that (\1U(w))T s(w) 2 ,U(w), for some, > 0, and we are therefore dealing with a 1050 B. V. ROY, J. N. TSITSIKLIS "pseudogradient algorithm" whose convergence follows from standard results on stochastic approximation [Polyak and Tsypkin, 1972J. This is done as follows: (\7U(w)f s(w) ~ (w - w) T MT (Mw - T(Mw)) ~ (w - w) T MT(Mw - IIT(Mw) - (J - II)T(MW)) = ~(MW-Mw)T(MW-IIT(MW)), where the last equality follows because MTrr = MT. Using the contraction assumption on T and the nonexpansion property of projection mappings, we have IlIIT(Mw) - Mw112 IIIIT(Mw) - rrT(Mw)112 ::; ,6'IIMw - Mw1I2' and applying the Cauchy-Schwartz inequality, we obtain (\7U(W))T s(w) > 1 -(IIMw - Mwll~ -IIMw - Mw1121IMw* - IIT(Mw)112) n !:.(l - ,6')IIMw - Mw*II~· n > Since M has full column rank, it follows that (\7U(W))T s(w) ~ 1'U(w), for some fixed l' > 0, and the proof is complete. 4 EXAMPLE: LOCAL TRANSITIONS ON GRIDS Theorem 1 leads us to the next question: are there some interesting cases for which Assumption 2 is satisfied? We describe a particular example here that relies on properties of Markov decision problems that naturally arise in some practical situations. When we encounter real Markov decision problems we often interpret the states in some meaningful way, associating more information with a state than an index value. For example, in the context of a queuing network, where each state is one possible queue configuration, we might think of the state as a vector in which each component records the current length of a particular queue in the network. Hence, if there are d queues and each queue can hold up to k customers, our state space is a finite grid zt (Le., the set of vectors with integer components each in the range {O, ... ,k-l}). Consider a state space where each state i E {I, ... , n} is associated to a point xi E zt (n = k d ), as in the queuing example. We might expect that individual transitions between states in such a state space are local. That is, if we are at a state xi the next visited state x j is probably close to xi in terms of Euclidean distance. For instance, we would not expect the configuration of a queuing network to change drastically in a second. This is because one customer is served at a time so a queue that is full can not suddenly become empty. Note that the number of states in a state space of the form zt grows exponentially with d. Consequently, classical dynamic programming algorithms such as value iteration quickly become impractical. To efficiently generate an approximation to the cost-to-go vector, we might consider tuning the parameters w E Rd and a E R of an affine approximation ~(w, a) = wT xi + a using the algorithm presented in the previous section. It is possible to show that, under the following assumption Stable Linear Approximations to Programming for Stochastic Control Problems 1051 concerning the state space topology and locality of transitions, Assumption 2 holds with f3' = .; f32 + k~3' and thus Theorem 1 characterizes convergence properties of the algorithm. Assumption 3 The Markov decision problem has state space S = {1, ... , k d }, and each state i is uniquely associated with a vector xi E zt with k ~ 6(1 - (32)-1 + 3. A ny pair xi, x j E zt of consecutively visited states either are identical or have exactly one unequal component, which differs by one. While this assumption may seem restrictive, it is only one example. There are many more candidate examples, involving other approximation architectures and particular classes of Markov decision problems, which are currently under investigation. 5 CONCLUSIONS We have proven a new theorem that establishes convergence properties of an algorithm for generating linear approximations to cost-to-go functions for dynamic programming. This theorem applies whenever the dynamic programming operator for a Markov decision problem is a contraction with respect to the Euclidean norm when applied to vectors in the parameterized class. In this paper, we have described one example in which such a condition holds. More examples of practical interest will be discussed in a forthcoming full length version of this paper. Acknowledgments This research was supported by the NSF under grant ECS 9216531, by EPRI under contract 8030-10, and by the ARO. References Bertsekas, D. P. (1995) Dynamic Programming and Optimal Control. Athena Scientific, Belmont, MA. Boyan, J. A. & Moore, A. W. (1995) Generalization in Reinforcement Learning: Safely Approximating the Value Function. In J. D. Cowan, G. Tesauro, and D. Touretzky, editors, Advances in Neural Information Processing Systems 7. Morgan Kaufmann. Gordon, G. J. (1995) Stable Function Approximation in Dynamic Programming. Technical Report: CMU-CS-95-103, Carnegie Mellon University. Polyak, B. T. & Tsypkin, Y. Z., (1972) Pseudogradient Adaptation and Training Algorithms. A vtomatika i Telemekhanika, 3:45-68. Sutton, R. S. (1988) Learning to Predict by the Method of Temporal Differences. Machine Learning, 3:9-44. Tesauro, G. (1992) Practical Issues in Temporal Difference Learning. Machine Learning, 8:257-277. Tsitsiklis, J. & Van Roy, B. (1994) Feature-Based Methods for Large Scale Dynamic Programming. Technical Report: LIDS-P-2277, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology. Also to appear in Machine Learning.	1995	67
1,114	Generalisation of A Class of Continuous Neural Networks John Shawe-Taylor Dept of Computer Science, Royal Holloway, University of London, Egham, Surrey TW20 OEX, UK Email: johnCdcs.rhbnc.ac . uk Jieyu Zhao* IDSIA, Corso Elvezia 36, 6900-Lugano, Switzerland Email: jieyuCcarota.idsia.ch Abstract We propose a way of using boolean circuits to perform real valued computation in a way that naturally extends their boolean functionality. The functionality of multiple fan in threshold gates in this model is shown to mimic that of a hardware implementation of continuous Neural Networks. A Vapnik-Chervonenkis dimension and sample size analysis for the systems is performed giving best known sample sizes for a real valued Neural Network. Experimental results confirm the conclusion that the sample sizes required for the networks are significantly smaller than for sigmoidal networks. 1 Introduction Recent developments in complexity theory have addressed the question of complexity of computation over the real numbers. More recently attempts have been made to introduce some computational cost related to the accuracy of the computations [5]. The model proposed in this paper weakens the computational power still further by relying on classical boolean circuits to perform the computation using a simple encoding of the real values. Using this encoding we also show that Teo circuits interpreted in the model correspond to a Neural Network design referred to as Bit Stream Neural Networks, which have been developed for hardware implementation [8]. With the perspective afforded by the general approach considered here, we are also able to analyse the Bit Stream Neural Networks (or indeed any other adaptive system based on the technique), giving VC dimension and sample size bounds for PAC learning. The sample sizes obtained are very similar to those for threshold networks, Work performed while at Royal Holloway, University of London 268 1. SHAWE-TAYLOR, J. ZHAO despite their being derived by very different techniques. They give the best bounds for neural networks involving smooth activation functions, being significantly lower than the bounds obtained recently for sigmoidal networks [4, 7]. We subsequently present simulation results showing that Bit Stream Neural Networks based on the technique can be used to solve a standard benchmark problem. The results of the simulations support the theoretical finding that for the same sample size generalisation will be better for the Bit Stream Neural Networks than for classical sigmoidal networks. It should also be stressed that the approach is very general - being applicable to any boolean circuit - and by its definition employs compact digital hardware. This fact motivates the introduction of the model, though it will not play an important part in this paper. 2 Definitions and Basic Results A boolean circuit is a directed acyclic graph whose nodes are referred to as gates, with a single output node of out-degree zero. The nodes with in-degree zero are termed input nodes. The nodes that are not input nodes are computational nodes. There is a boolean function associated with each computational node of arity equal to its in-degree. The function computed by a boolean network is determined by assigning (input) values to its input nodes and performing the function at each computational node once its input values are determined. The result is the value at the output node. The class TCo is defined to be those functions that can be computed by a family of polynomially sized Boolean circuits with unrestricted fanin and constant depth, where the gates are either NOT or THRESHOLD. In order to use the boolean circuits to compute with real numbers we use the method of stochastic computing to encode real numbers as bit streams. The encoding we will use is to consider the stream of binary bits, for which the l's are generated independently at random with probability p, as representing the number p. This is referred to as a Bernoulli sequence of probability p. In this representation, the multiplication of two independently generated streams can be achieved by a simple AND gate, since the probability of a Ion the output stream is equal to P1P2, where Pl is the probability of a 1 on the first input stream and P2 is the probability of a 1 on the second input stream. Hence, in this representation the boolean circuit consisting of a single AND gate can compute the product of its two inputs. More background information about stochastic computing can be found in the work of Gaines [1]. The analysis we provide is made by treating the calculations as exact real valued computations. In a practical (hardware) implementation real bit streams would have to be generated [3] and the question of the accuracy of a delivered result arlses. In the applications considered here the output values are used to determine a binary value by comparing with a threshold of 0.5. Unless the actual output is exactly 1 or ° (which can happen), then however many bits are collected at the output there is a slight probability that an incorrect classification will be made. Hence, the number of bits required is a function of the difference between the actual output and 0.5 and the level of confidence required in the correctness of the classification. Definition 1 The real function computed by a boolean circuit C, which computes the boolean function fe : {O, l}n - {O, I}, is the function ge : [0, lr -* [0,1], Generalisation of a Class of Continuous Neural Networks 269 obtained by coding each input independently as a Bernoulli sequence and interpreting the output as a similar sequence. Hence, by the discussion above we have for the circuit C consisting of a single AND gate, the function ge is given by ge(:l:1, :1:2) = :1:1:1:2. We now give a proposition showing that the definition of real computation given above is well-defined and generalises the Boolean computation performed by the circuit. Proposition 2 The bit stream on the output of a boolean circuit computing a real function is a Bernoulli sequence. The real function ge computed by an n input boolean circuit C can be expressed in terms of the corresponding boolean function fe as follows: n o:E{O,1}" i=1 In particular, gel{o,)}" = fe · Proof: The output bit stream is a Bernoulli sequence, since the behaviour at each time step is independent of the behaviour at previous time sequences, assuming the input sequences are independent. Let the probability of a 1 in the output sequence be p. Hence, ge (:I:) = p. At any given time the input to the circuit must be one of the 2n possible binary vectors a. P:l(a) gives the probability of the vector a occurring. Hence, the expected value of the output of the circuit is given in the proposition statement, but by the properties of a Bernoulli sequence this value is also p. The final claim holds since Po: (a) = 1, w hile Po: (a') = 0 for a # a' .• Hence, the function computed by a circuit can be denoted by a polynomial of degree n, though the representation given above may involve exponentially many terms. This representation will therefore only be used for theoretical analysis. 3 Bit Stream Neural Networks In this section we describe a neural network model based on stochastic computing and show that it corresponds to taking TCo circuits in the framework considered in Section 2. A Stochastic Bit Stream Neuron is a processing unit which carries out very simple operations on its input bit streams. All input bit streams are combined with their corresponding weight bit streams and then the weighted bits are summed up. The final total is compared to a threshold value. If the sum is larger than the threshold the neuron gives an output 1, otherwise O. There are two different versions of the Stochastic Bit Stream Neuron corresponding to the different data representations. The definitions are given as follows. Definition 3 (AND-SBSN): A n-input AND version Stochastic Bit Stream Neuron has n weights in the range [-1,1 j and n inputs in the range [0,1 j, which are all unipolar representations of Bernoulli sequences. An extra sign bit is attached to each weight Bernoulli sequence. The threshold 9 is an integer lying between -n to n which is randomly generated according to the threshold probability density function ¢( 9). The computations performed during each operational cycle are 270 J. SHAWE-TA YLOR, J. ZHAO (1) combining respectively the n bits from n input Bernoulli sequences with the corresponding n bits from n weight Bernoulli sequences using the AND operation. (2) assigning n weight sign bits to the corresponding output bits of the AND gate, summing up all the n signed output bits and then comparing the total with the randomly generated threshold value. If the total is not less than the threshold value, the AND-SBSN outputs 1, otherwise it outputs O. We can now present the main result characterising the functionality of a Stochastic Bit Stream Neural Network as the real function of an Teo circuit. Theorem 4 The functionality of a family of feedforward networks of Bit Stream Neurons with constant depth organised into layers with interconnections only between adjacent layers corresponds to the function gc for an TCo circuit C of depth twice that of the network. The number of input streams is equal to the number of network inputs while the number of parameters is at most twice the number of weights. Proof: Consider first an individual neuron. We construct a circuit whose real functionality matches that of the neuron. The circuit has two layers. The first consists of a series of AND gates. Each gate links one input line of the neuron with its corresponding weight input. The outputs of these gates are linked into a threshold gate with fixed threshold 2d for the AND-SBSN, where d is the number of input lines to the neuron. The threshold distribution of the AND SBSN is now simulated by having a series of 2d additional inputs to the threshold gate. The number of additional input streams required to simulate the threshold depends on how general a distribution is allowed for the threshold. We consider three cases: 1. If the threshold is fixed (i.e. not programmable), then no additional inputs are required, since the actual threshold can be suitably adapted. 2. If the threshold distribution is always focussed on one value (which can be varied), then an additional flog2(2d)1 (rlog2(d)l) inputs are required to specify the binary value of this number. A circuit feeding the corresponding number of 1 's to the threshold gate is not hard to construct. 3. In the fully general case any series of 2d + 1 (d + 1) numbers summing to one can be assigned as the probabilities of the possible values 4>(0),4>(1), ... , 4>(t), where t 2d for the AND SBSN. We now construct a circuit which takes t input streams and passes the I-bits to the threshold gate of all the inputs up to the first input stream carrying a O. No fUrther input is passed to the threshold gate. In other words Threshold gate receives s q. Input streams 1, ... , s have bit 1 and bits of input either s = t or input stream s + 1 has input o. We now set the probability p, of stream s as follows; PI p, 1 - 4>(0) 1 2:;~~ 4>( i) 1 2:;~g 4>( i) for s = 2, ... , t With these values the probability of the threshold gate receiving s bits is 4>( s) as required. Generalisation of a Class of Continuous Neural Networks 271 This completes the replacement of a single neuron. Clearly, we can replace all neurons in a network in the same manner and construct a network with the required properties provided connections do not 'shortcut' layers, since this would create interactions between bits in different time slots. _ 4 VC Dimension and Sample Sizes In order to perform a VC Dimension and sample size analysis of the Bit Stream Neural Networks described in the previous section we introduce the following general framework. Definition 5 For a set Q of smooth functions f : R n x Rl -+ R, the class F is defined as F = Fg = {fw Ifw{x) = f{x, w), f E Q}. The corresponding classification class obtained by taking a fixed set of s of the functions from Q, thresholding the corresponding functions from F at 0 and combining them (with the same parameter vector) in some logical formula will be denoted H,{F). We will denote H1{F) by H{F). In our case we will consider a set of circuits C each with n + l input connections, n labelled as the input vector and l identified as parameter input connections. Note that if circuits have too few input connections, we can pad them with dummy ones. The set g will then be the set Q=Qe={gc!CEC}, while Fgc will be denoted by Fe. We now quote some of the results of [7] which uses the techniques of Karpinski and MacIntyre [4] to derive sample sizes for classes of smoothly parametrised functions. Proposition 6 [7} Let Q be the set of polynomials P of degree at most d with P : R n x Rl -+ Rand F = Fg = {PwIPw{x) = p{x, w),p E g}. Hence, there are l adjustable parameters and the input dimension is n . Then the VC-dimension of the class H,{Fe) is bounded above by log2{2{2d)l) + 1711og2{s). Corollary 7 For a set of circuits C, with n input connections and l parameter connections, the VC-dimension of the class H,{Fe) is bounded above by Proof: By Proposition 2 the function gc computed by a circuit C with t input connections has the form t gc{x) = L P;e(a)fc{a), where P;e{a) = II xfi{l- xd1- cxi ). i=l Hence, gc( x) is a polynomial of degree t. In the case considered the number t of input connections is n + l. The result follows from the proposition. _ 272 J. SHAWE-TAYLOR. 1. ZHAO Proposition 8 [7] Let 9 be the set of polynomials P of degree at most d with p: 'Rn X 'Rl -+ 'R and F = Fg = {PwIPw(x) = p(x, w),p E g}. Hence, there are l adjustable parameters and the input dimension is n. If a function h E H.(F) correctly computes a function on a sample of m inputs drawn independently according to a fixed probability distribution, where m ~ "",(e, 0) = e(1 ~ y'€) [Uln ( 4e~) + In (2l/(~ - 1)) 1 then with probability at least 1 - 0 the error rate of h will be less than E on inputs drawn according to the same distribution. Corollary 9 For a set of circuits C, with n input connections and l parameter connections, If a function h E H.(Fc) correctly computes a function on a sample of m inputs drawn independently according to a fixed probability distribution, where m ~ "",(e, 0) = e(1 ~ y€) [Uln ( 4eJs~n +l)) + In Cl/(~ - 1)) 1 then with probability at least 1 - 0 the error rate of h will be less than E on inputs drawn according to the same distribution. Proof: As in the proof of the previous corollary, we need only observe that the functions gC for C E C are polynomials of degree at most n + l. • Note that the best known sample sizes for threshold networks are given in [6]: m ~ "",(e, 0) = e(1 ~ y'€) [2Wln (6~) + In (l/(lo- 1)) 1 ' where W is the number of adaptable weights (parameters) and N is the number of computational nodes in the network. Hence, the bounds given above are almost identical to those for threshold networks, despite the underlying techniques used to derive them being entirely different. One surprising fact about the above results is that the VC dimension and sample sizes are independent of the complexity of the circuit (except in as much as it must have the required number of inputs). Hence, additional layers of fixed computation cannot increase the sample complexity above the bound given). 5 Simulation Results The Monk's problems which were the basis of a first international comparison of learning algorithms, are derived from a domain in which each training example is represented by six discrete-valued attributes. Each problem involves learning a binary function defined over this domain, from a sample of training examples of this function. The 'true' concepts underlying each Monk's problem are given by: MONK-I: (attributet = attribute2) or (attribute5 = 1) MONK-2: (attributei = 1) for EXACTLY TWO i E {I, 2, ... , 6} MONK-3: (attribute5 = 3 and attribute4 = 1) or (attribute5 =1= 4 and attribute2 =1= 3) Generalisation of a Class of Continuous Neural Networks 273 There are 124, 169 and 122 samples in the training sets of MONK-I, MONK-2 and MONK-3 respectively. The testing set has 432 patterns. The network had 17 input units, 10 hidden units, 1 output unit, and was fully connected. Two networks were used for each problem. The first was a standard multi-layer perceptron with sigmoid activation function trained using the backpropagation algorithm (BP Network). The second network had the same architecture, but used bit stream neurons in place of sigmoid ones (BSN Network). The functionality of the neurons was simulated using probability generating functions to compute the probability values of the bit streams output at each neuron. The backpropagation algorithm was adapted to train these networks by computing the derivative of the output probability value with respect to the individual inputs to that neuron [8]. Experiments were performed with and without noise in the training examples. There is 5% additional noise (misclassifications) in the training set of MONK-3. The results for the Monk's problems using the moment generating function simulation are shown as follows: BP Network BSN Network training testing training testing MONK-l 100% 86.6% 100% 97.7% MONK-2 100% 84.2% 100% 100% MONK-3 97.1% 83.3% 98.4% 98.6% It can be seen that the generalisation of the BSN network is much better than that of a general multilayer backpropagation network. The results on MONK-3 problem is extremely good. The results reported by Hassibi and Stork [2] using a sophisticated weight pruning technique are only 93.4% correct for the training set and 97.2% correct for the testing set. References [1] B. R. Gaines, Stochastic Computing Systems, Advances in Information Systems Science 2 (1969) pp37-172. [2] B. Hassibi and D.G. Stork, Second order derivatives for network pruning: Optimal brain surgeon, Advances in Neural Information Processing System, Vol 5 (1993) 164-171. [3] P. Jeavons, D.A. Cohen and J. Shawe-Taylor, Generating Binary Sequences for Stochastic Computing, IEEE Trans on Information Theory, 40 (3) (1994) 716-720. [4] M. Karpinski and A. MacIntyre, Bounding VC-Dimension for Neural Networks: Progress and Prospects, Proceedings of EuroCOLT'95, 1995, pp. 337-341, Springer Lecture Notes in Artificial Intelligence, 904. [5] P. Koiran, A Weak Version of the Blum, Shub and Smale Model, ESPRIT Working Group NeuroCOLT Technical Report Series, NC-TR-94-5, 1994. [6] J. Shawe-Taylor, Threshold Network Learning in the Presence of Equivalences, Proceedings of NIPS 4, 1991, pp. 879-886. [7] J. Shawe-Taylor, Sample Sizes for Sigmoidal Networks, to appear in the Proceedings of Eighth Conference on Computational Learning Theory, COLT'95, 1995. [8] John Shawe-Taylor, Peter Jeavons and Max van Daalen, "Probabilistic Bit Stream Neural Chip: Theory", Connection Science, Vol 3, No 3, 1991.	1995	68
1,115	Visual gesture-based robot guidance with a modular neural system E. Littmann, Abt. Neuroinformatik, Fak. f. Informatik Universitat Ulm, D-89069 Ulm, FRG enno@neuro.informatik.uni-ulm.de A. Drees, and H. Ritter AG Neuroinformatik, Techn. Fakultat Univ. Bielefeld, D-33615 Bielefeld, FRG andrea,helge@techfak.uni-bielefeld.de Abstract We report on the development of the modular neural system "SEEEAGLE" for the visual guidance of robot pick-and-place actions. Several neural networks are integrated to a single system that visually recognizes human hand pointing gestures from stereo pairs of color video images. The output of the hand recognition stage is processed by a set of color-sensitive neural networks to determine the cartesian location of the target object that is referenced by the pointing gesture. Finally, this information is used to guide a robot to grab the target object and put it at another location that can be specified by a second pointing gesture. The accuracy of the current system allows to identify the location of the referenced target object to an accuracy of 1 cm in a workspace area of 50x50 cm. In our current environment, this is sufficient to pick and place arbitrarily positioned target objects within the workspace. The system consists of neural networks that perform the tasks of image segmentation, estimation of hand location, estimation of 3D-pointing direction, object recognition, and necessary coordinate transforms. Drawing heavily on the use of learning algorithms, the functions of all network modules were created from data examples only. 1 Introduction The rapidly developing technology in the fields of robotics and virtual reality requires the development of new and more powerful interfaces for configuration and control of such devices. These interfaces should be intuitive for the human advisor and comfortable to use. Practical solutions so far require the human to wear a device that can transfer the necessary information. One typical example is the data glove [14, 12]. Clearly, in the long run solutions that are contactless will be much more desirable, and vision is one of the major modalities that appears especially suited for the realization of such solutions. In the present paper, we focus on a still restricted but very important task in robot control, the guidance of robot pick-and-place actions by unconstrained human pointing gestures in a realistic laboratory environment. The input of target locations by 904 E. LITTMANN, A. DREES, H. RITTER pointing gestures provides a powerful, very intuitive and comfortable functionality for a vision-based man-machine interface for guiding robots and extends previous work that focused on the detection of hand location or the discrimination of a small, discrete number of hand gestures only [10, 1, 2, 8]. Besides two color cameras, no special device is necessary to evaluate the gesture of the human operator. A second goal of our approach is to investigate how to build a neural system for such a complex task from several neural modules. The development of advanced artificial neural systems challenges us with the task of finding architect.ures for the cooperat.ion of multiple functional modules such that. part of the structure of the overall system can be designed at a useful level of abstraction, but at the same t.ime learning can be used to create or fine-tune the functionality of parts of t.he system on the basis of suit.able training examples. To approach this goal requires to shift the focus from exploring t.he properties of single networks to exploring the propert.ies of entire systems of neural networks. The work on "mixtures of experts" [3, 4] is one important contribution along these lines. While this is a widely applicable and powerful approach, there clearly is a need to go beyond the exploration of strictly hierarchical systems and to gain experience with architectures t.hat admit more complex types of information flow as required e.g. by the inclusion of feat.ures such as control of focal attention or reent.rant processing branches. The need for such features arose very naturally in the context of the task described above, and in the following sect.ion we will report our results wit.h a system architecture that is crucially based on the exploitation of such elements. 2 System architecture Our system, described in fig. 1, is situated in a complex laboratory environment. A robot arm with manipulator is mounted at one side of a table with several objects of different color placed on it. A human operator is positioned at the next side to the right of the robot. This scenery is watched by two cameras from the other two sides from high above. The cameras yield a stereo color image of t.he scene (images 10). The operator points with one hand at one of the objects on the table. On the basis of the image information, the object is located and the robot grabs it. Then, the operator points at another location, where the robot releases the object. 1 The syst.em consists of several hardware components: a PUMA 560 robot arm with six axes and a three-fingered manipulator 2; two single-chip PULNIX color cameras; two ANDRox vision boards with software for data acquisition and processing; a work space consisting of a table with a black grid on a yellow surface. Robot and person refer to the same work space. Bot.h cameras must show both the human hand and the table with the objects. Within this constraint, the position of the cameras can be chosen freely as long as they yield significantly different views. An important prerequisite for the recognition of the pointing direction is the segmentation of the human hand from the background scenery. This task is solved by a LLM network (Sl) trained to yield a probability value for each image pixel to belong to the hand region. The training is based on t.he local color information. This procedure has been investigated in [7]. An important feature of the chosen method is the great reliability and robustness of both the classification performance and the localization accuracy of the searched object. Furthermore, the performance is quite constant over a wide range of image resolutions. This allows a fast two-step procedure: First, the images are segmented in low resolution (Sl: 11 -+ A1) and the hand position is extracted. Then, a small 1 In analogy to the sea eagle who watches its prey from high above, shoots down to grab the prey, and then flies to a safe place to feed, we nicknamed our system "SEE-EAGLE". 2Development by Prof. Pfeiffer, TV Munich Visual Gesture-based Robot Guidance with a Modular Neural System 905 Fig. 1: System architecture. From two color camera images 10 we extract the hand position (11 I> Sl I> A1 (pixel coord.) I> P1 I> cartesian hand coord.). In a subframe centered on the hand location (12) we determine the pointing direction (12 I> S2 I> A2 (pixel coord.) I> G I> D I> pointing angles). Pointing direction and hand location define a cartesian target location that is mapped to image coord. that define the centers of object subframes (10 I> P2 I> 13). There we determine the target object (13 I> S3 I> A3) and map the pixel coord. of its centers to world coord. (A3 I> P3 I> world target loc.). These coordinates are used to guide the robot R to the target object. 906 E. LITTMANN. A. DREES. H. RlTIER subframe (12) around the estimated hand position is processed in high resolution by another dedicated LLM network (S2: 12 -t A2). For details of the segmentation process, refer to [6]. The extraction of hand information by LLMs on the basis of Gabor masks has already been studied for hand posture [9] and orientation [5]. The method is based on a segmented image containing the hand only (A2). This image is filtered by 36 Gabor masks that are arranged on a 3x3 grid with 4 directions per grid position and centered on the hand. The filter kernels have a radius of 10 pixels, the distance between the grid points is 20 pixels. The 36 filter responses (G) form the input vector for a LLM network (D). Further details of the processing are reported in [6]. The network yields the pointing direction of the hand (D: 12 -t G -t pointing direction). Together with the hand position which is computed by a parametrized self-organizing map ("PSOM", see below and [11, 13]) (P1: Al -t cartesian hand position), a (cartesian) target location in the workspace can be calculated. This location can be retransformed by the PSOM into pixel coordinates (P2: cartesian target location -t target pixel coordinates). These coordinates define the center of an "attention region" (13) that is searched for a set of predefined target objects. This object recognition is performed by a set of LLM color segmentation networks (S3: 13 -t A3), each previously trained for one of the defined targets. A ranking procedure is used to determine the target object. The pixel coordinates ofthe target in the segmented image are mapped by the PSOM to world coordinates (P3: A3 -t cartesian target position). The robot R now moves to above these world coordinates, moves vertically down, grabs whatever is there, and moves upward again. Now, the system evaluates a second pointing gesture that specifies the place where to place the object. This time, the world coordinates calculated on the basis of the pointing direction from network D and the cartesian hand location from PSOM PI serve directly as target location for the robot. For our processing we must map corresponding pixels in the stereo images to cartesian world coordinates. For these transformations, training data was generated with aid of the robot on a precise sampling grid. We automatically extract the pixel coordinates of a LED at the tip of the robot manipulator from both images. The seven-dimensional feature vector serves as training input for an PSOM network [11]. By virtue of its capability to represent a transformation in a symmetric, "multiway" -fashion, this offers the additional benefit that both the camera-to-world mapping and its inverse can be obtained with a single network trained only once on a data set of 27 calibration positions of the robot. A detailed description for such a procedure can be found in [13]. 3 Results 3.1 System performance The accuracy of the current system allows to estimate the pointing target to an accuracy of 1 ± 0.4 cm (average over N = 7 objects at randomly chosen locations in the workspace) in a workspace area of 50x50 cm. In our current environment, this is sufficient to pick and place any of the seven defined target objects at any location in the workspace. This accuracy can only be achieved if we use the object recognition module described in sec. 2. The output of the pointing direction module approximates the target location with an considerably lower accuracy of 3.6± 1.6 cm. 3.2 Image segmentation The problem to evaluate these preprocessing steps has been discussed previously [7], especially the relation of specifity and sensitivity of the network for the given task. As the pointing recognition is based on a subframe centered on the hand center, it is very sensitive to deviations from this center so that a good localization accuracy Visual Gesture-based Robot Guidance with a Modular Neural System 907 is even more important than the classification rate. The localization accuracy is calculated by measuring the pixel distance between the centers determined manually on the original image and as the center of mass in the image obtained after application of the neural network. Table 1 provides quantitative results. On the whole) the two-step cascade of LLM networks yields for 399 out of 4 00 images an activity image precisely centered on the human hand. Only in one image) the first LLM net missed the hand completely) due to a second hand in the image that could be clearly seen in this view. This image was excluded from further processing and from the evaluation of the localization accuracy. Camera A Camera B Pixel deviatIOn NRMSE Pixel deViatIOn NRMSE Person A 0.8 ± 1.2 0.03 ± 0.06 0.8 ± 2.2 0.03 ± 0.09 Person H 1.3 ± 1.4 0.06 ± 0.11 2.2 ± 2.8 0.11 ± 0.21 Table 1: Estimation error of the hand localization on the test set. Absolute error in pixels and normalized error for both persons and both camera images. 3.3 Recognition performance One major problem in recognizing human pointing gestures is the variability of these gestures and their measurement for the acquisition of reliable training information. Different persons follow different strategies where and how to point (fig. 2 (center) and (right». Therefore) we calculate this information indirectly. The person is told to point at a certain grid position with known world coordinates. From the camera images we extract the pixel positions of the hand center and map them to world coordinates using the PSOM net (PI in fig. 1). Given these coordinates the angles of the intended pointing vector with the basis vectors of the world coordinate system can be calculated trigonometrically. These angles form the target vector for the supervised training of a LLM network (D in fig. 1). After training) the output of the net is used to calculate the point where the pointing vector intersects the table surface. For evaluation of the network performance we measure the Euclidian distance between this point and the actual grid point where the person intended to point at. Fig. 3 (left) shows the mean euclidean error MEE of the estimated target position as a function of the number of learning steps. The error on the training set can be considerably reduced) whereas on the test set the improvement stagnates after some 500 training steps. If we perform even more training steps the performance might actually suffer from overfitting. The graph compares training and test results achieved on images obtained by two different ways of determining the hand center. The "manual" curves show the performance that can be achieved if the Gabor masks are manually centered on the hand. For the "neuronal)) curves) the center of mass calculated in the fine-segmented and postprocessed subframe was used. This allows us to study the influence of the error of the segmentation and localization steps on the pointing recognition. This influence is rather small. The MEE increases from 17 mm for the optimal method to 19 mm for the neural method) which is hardly visible in practice. The curves in fig. 3 (center) are obtained if we apply the networks to images of another person. The MEE is considerably larger but a detailed analysis' shows that part of this deviation is due to systematic differences in the pointing strategy as shown in fig. 2 (right). Over a wide range, the number of nodes used for the LLM network has only minor influence on the performance. While obviously the performance on the training set can be arbitrarily improved by spending more nodes, the differences in the MEE on the test set are negligible in a range of 5 to 15 nodes. Using more nodes is problematic as the training data consists of 50 examples only. If not indicated otherwise) we use LLM networks with 10 nodes. Further results) 908 E. LIITMANN. A. DREES. H. RIITER Fig. 2: The table grid points can be reconstructed according to the network output. The target grid is dotted. Reconstruction of training grid (left) and test grid (center) for one person, and of the test grid for another person (right). MER MEB on test oet of unknown perron 30 m ..... aI,trainn :l~ neuronal, train 70 4 Fig. 3: The euclidean error of manual, test 20 ~ 68 ----~--.--66 estimated target point calcue I~ e £ ~ £ 64 lated using the network out10 62 -~. 60 put depends on the prepro~58 cessing (left), and the person 0 56 100 250 sao 1000 2SOO SOOO 100 :l~ sao 1000 2SOO SOOO (center). train.., itHabonr trairq IteratioN comparing the pointing recognition based on only one of the camera images, indicate that the method works better if the camera takes a lateral view rather than a frontal view. All evaluations were done for both persons. The performance was always very similar. 4 Discussion While we begin to understand many properties of neural networks at the single network level, our insight into principled ways of how to build neural systems is still rather limited. Due to the complexity of this task, theoretical progress is (and probably will continue to be) very slow. What we can do in the mean time, however, is to experiment with different design strategies for neural systems and try to "evolve" useful approaches by carefully chosen case studies. The current work is an effort along these lines. It is focused on a challenging, practically important vision task with a number of generic features that are shared with vision tasks for which biological vision systems were evolved. One important issue is how to achieve robustness at the different processing levels of the system. There are only very limited possibilities to study this issue in simulations, since practically nothing is known about the statistical properties of the various sources of error that occur when dealing with real world data. Thus, a real implementation that works with actual data is practically the only way to study the robustness issue in a realistic fashion. Therefore, the demonstrated integration of several functional modules that we had developed previously in more restricted settings [7, 6] was a non-trivial test of the feasability of having these functions cooperate in a larger, modular system. It also gives confidence that the scaling problem can be dealt with successfully if we apply modular neural nets. A related and equally important issue was the use of a processing strategy in which earlier processing stages incrementally restrict the search space for the subsequent stages. Thus, the responsibility for achieving the goal is not centralized in any single module and subsequent modules have always the chance to compensate for limited errors of earlier stages. This appears to be a generally useful strategy for achieving Visual Gesture-based Robot Guidance with a Modular Neural System 909 robustness and for cutting computational costs that is related to the use of "focal attention" , which is clearly an important element of many biological vision systems. A third important point is the extensive use of learning to build the essential constituent functions of the system from data examples. We are not yet able to train the assembled system as a whole. Instead, different modules are trained separately and are integrated only later. Still, the experience gained with assembling a complex system via this "engineering-type" of approach will be extremely valuable for gradually developing the capability of crafting larger functional building blocks by learning methods. We conclude that carefully designed experiments with modular neural systems that are based on the use of real world data and that focus on similar tasks for which also biological neural systems were evolved can make a significant contribution in tackling the challenge that lies ahead of us: to develop a reliable technology for the construction of large-scale artificial neural systems that can solve complex tasks in real world environments. Acknowledgements We want to thank Th. Wengerek (robot control), J. Walter (PSOM implementation), and P. Ziemeck (image acquisition software). This work was supported by BMFT Grant No. ITN9104AO. References [1] T. J. Darell and A. P. Pentland. Classifying hand gestures with a view-based distributed representation. In J. D. Cowan, G. Tesauro, and J. Alspector, editors, Neural Information Processing Systems 6, pages 945-952. Morgan Kaufman, 1994. [2] J. Davis and M. Shah. Recognizing hand gestures. In J.-O. Eklundh, editor, Computer Vision ECCV '94, volume 800 of Lecture Notes in Computer Science, pages 331340. Springer-Verlag, Berlin Heidelberg New York, 1994. [3] R.A. Jacobs, M.1. Jordan, S.J. Nowlan, and G.E. Hinton. Adaptive mixtures of local experts. Neural Computation, 3:79- 87, 1991. [4] M.1. Jordan and R.A. Jacobs. Hierarchical mixtures of experts and the EM algorithm. Neural Computation, 6(2):181-214, 1994. [5] F. Kummert, E. Littmann, A. Meyering, S. Posch, H. Ritter, and G. Sagerer. A hybrid approach to signal interpretation using neural and semantic networks. In Mustererkennung 1993, pages 245-252. Springer, 1993. [6] E. Littmann, A. Drees, and H. Ritter. Neural recognition of human pointing gestures in real images. Submitted to Neural Processing Letters, 1996. [7] E. Littmann and H. Ritter. Neural and statistical methods for adaptive color segmentation a comparison. In G. Sagerer, S. Posch, and F. Kummert, editors, Mustererkennung 1995, pages 84-93. Springer-Verlag, Heidelberg, 1995. [8] C. Maggioni. A novel device for using the hand as a human-computer interface. In Proceedings HC1'93 Human Control Interface, Loughborough, Great Britain, 1993. [9] A. Meyering and H. Ritter. Learning 3D shape perception with local linear maps. In Proc. of the lJCNN, volume IV, pages 432-436, Baltimore, MD, 1992. [10] Steven J. Nowlan and John C. Platt. A convolutional neural network hand tracker. In Neural Information Processing Systems 7. Morgan Kaufman Publishers, 1995. [11] H. Ritter. Parametrized self-organizing maps for vision learning tasks. In P. Morasso, editor, ICANN '94. Springer-Verlag, Berlin Heidelberg New York, 1994. [12] K. Viiiina.nen and K. Bohm. Gesture driven interaction as a human factor in virtual environments - an approach with neural networks. In R. Earnshaw, M. Gigante, and H. Jones, editors, Virtual reality systems, pages 93-106. Academic Press, 1993. [13] J. Walter and H. Ritter. Rapid learning with parametrized self-organizing maps. Neural Computing, 1995. Submitted. [14] T. G. Zimmermann, J. Lanier, C. Blanchard, S. Bryson, and Y. Harvill. A hand gesture interface device. In Proc. CHI+GI, pages 189-192, 1987.	1995	69
1,116	hnproved Silicon Cochlea • uSIng Compatible Lateral Bipolar Transistors Andre van Schalk, Eric Fragniere, Eric Vittoz MANTRA Center for Neuromimetic Systems Swiss Federal Institute of Technology CH-IOI5 Lausanne email: vschaik@di.epfl.ch Abstract Analog electronic cochlear models need exponentially scaled filters. CMOS Compatible Lateral Bipolar Transistors (CLBTs) can create exponentially scaled currents when biased using a resistive line with a voltage difference between both ends of the line. Since these CLBTs are independent of the CMOS threshold voltage, current sources implemented with CLBTs are much better matched than current sources created with MOS transistors operated in weak inversion. Measurements from integrated test chips are shown to verify the improved matching. 1. INTRODUCTION Since the original publication of the "analog electronic cochlea" by Lyon and Mead in 1988 [I], several other analog VLSI models have been proposed which try to capture more of the details of the biological cochlear function [2],[3],[4]. In spite of the differences in their design, all these models use filters with exponentially decreasing cutoff frequencies. This exponential dependency is generally obtained using a linear decreasing voltage on the gates of MOS transistors operating in weak-inversion. In weak-inversion, the drain current of a saturated MOS transistor depends exponentially on its gate voltage. The linear decreasing voltage is easily created using a resistive poly silicon line; if there is a voltage difference between the two ends of the line, the voltage on the line will decrease linearly all along its length. 672 A. V AN SCHAlK. E. FRAGNIl1RE. E. VlrrOZ The problem of using MOS transistors in weak-inversion as current sources is that their drain currents are badly matched. An RMS mismatch of 12% in the drain current of two identical transistors with equal gate and source voltages is not exceptional [5], even when sufficient precautions, such as a good layout, are taken. The main cause of this mismatch is a variation of the threshold voltage between the two transistors. Since the threshold voltage and its variance are technology parameters, there is no good way to reduce the mismatch once the chip has been fabricated. One can avoid this problem using Compatible Lateral Bipolar Transistors (CLBTs) [6] for the current sources. They can be readily made in a CMOS substrate, and their collector current also depends exponentially on their base voltage, while this current is completely independent of the CMOS technology's threshold Voltage. The remaining mismatch is due to geometry mismatch of the devices, a parameter which is much better controlled than the variance of the threshold voltage. Therefore, the use of CLBTs can yield a large improvement in the regularity of the spacing of the cochlear filters. This regularity is especially important in a cascade of filters like the cochlea, since one filter can distort the input signal of all the following filters. We have integrated an analog electronic cochlea as a cascade of second-order lOW-pass filters, using CLBTs as exponentially scaled current sources. The design of this cochlea is based on the silicon cochlea described in [7], since a number of important design issues, such as stability, dynamic range, device mismatch and compactness, have already been addressed in this design. In this paper, the design of [7] is briefly presented and some remaining possible improvements are identified. These improvements, notably the use of Compatible Lateral Bipolar Transistors as current sources, a differentiation that does not need gain correction and temperature independent biasing of the cut-off frequency, are then discussed in more detail. Finally, measurement results of a test chip will be presented and compared to the design without CLBTs. 2. THE ANALOG ELECTRONIC COCHLEA The basic building block for the filters in all analog electronic cochlear models is the transconductance amplifier, operated in weak inversion. For input voltages smaller than about 60 mV pp, the amplifier can be approximated as a linear transconductance: with transconductance gm given by: 10 gm = 2nUT (1) (2) where Io is the bias current, n is the slope factor, and the thermal voltage UT = kT/q = 25.6 mV at room temperature. This linear range is usually the input range used in the cochlear filters, yielding linear filters. In [7], a transconductance amplifier having a wider linear input range is proposed. This allows larger input signals to be used, up to about 140 m Vpp. Furthermore, the wide range transconductance amplifier can be used to eliminate the large-signal instability shown to be present in the original second-order section [7]. This second-order section will be discussed in more detail in section 3.2. Improved Silicon Cochlea Using Compatible Lateral Bipolar Transistors 673 The traditional techniques to improve matching [5], as for instance larger device sizes for critical devices and placing identical devices close together with identical orientation, are also discussed in [7] with respect to the implementation of the cochlear filter cascade. The transistors generating the bias current 10 of the transconductance amplifiers in the second-order sections were identified as the most critical devices, since they have the largest effect on the cut-off frequency and the quality factor of each section. Therefore, extra area had to be devoted to these bias transistors. A further improvement is obtained in [7] by using a single resistive line to bias both the transconductance amplifiers controlling the cut-off frequency and the transconductance amplifier controlling the quality factor. The quality factor Q is then changed by varying the source of the transistor which biases the Q control amplifier. Instead of using two tilted resistive lines, this scheme uses only one tilted resistive line and a non-tilted Q control line, and therefore doesn't need to rely on an identical tilt on both resistive lines. 3. IMPROVED ANALOG ELECTRONIC COCHLEA The design discussed in the previous section already showed a substantial improvement over the first analog electronic cochlea by Lyon and Mead. However, several improvements remain possible. 3.1 VT VARIATION The bias transistors have been identified as the major source of mismatch of the cochlea's parameters. This mismatch is mainly due to variation of the threshold voltage VT of the MOS transistors. Since the drain current of a saturated MOS transistor in weak-inversion depends exponentially on the difference between its gate-source voltage and its threshold voltage, small variations in VT introduce large variations in the drain current of these transistors, and since both the cut-off frequency and the quality factor of the filters are proportional to these drain currents, large parameter variations are generated by small V T variations. This problem can be circumvented by the use of CMOS Compatible Lateral Bipolar transistors as bias transistors. A CMOS Compatible Lateral Bipolar Transistor is obtained if the drain or source junction of a MOS transistor is forward-biased in order to inject minority carriers into the local substrate. If the gate voltage is negative enough (for an n-channel device), then no current can flow at the surface and the operation is purely bipolar [6]. Fig. 1 shows the major flows of current carriers in this mode of operation, with the source, drain and well terminals renamed emitter E, collector C and base B. VBC<O :fG C ISub ~ holes -... p ........ electrons n Fig. 1. : Bipolar operation of the MOS transistor: carrier flows and symbol. 674 A. V AN SCHAlK. E. FRAGNIERE. E. VITIOZ Since there is no p+ buried layer to prevent injection to the substrate, this lateral npn bipolar transistor is combined with a vertical npn. The emitter current IE is thus split into a base current IB, a lateral collector current Ic and a substrate collector current Isub• Therefore, the common-base current gain ex. = -IdlE cannot be close to 1. However, due to the very small rate of recombination inside the well and to the high emitter efficiency, the common-emitter current gain ~ = IeIlB can be large. Maximum values of ex. and ~ are obtained in concentric structures using a minimum size emitter surrounded by the collector and a minimum lateral base width. For VCE = VBE-VBC larger than a few hundred millivolts, this transistor is in active mode and the collector current is given, as for a normal bipolar transistor, by fu k=~e~ W where ISb is the specific current in bipolar mode, proportional to the cross-section of the emitter to collector flow of carriers. Since k is independent of the MOS transistor threshold voltage V T, the main source of mismatch of distributed MOS current sources is suppressed, when o....BTs are used to create the current sources. VC.c D __ ...... --'=-_B c::::J lEI _ 0+ poIy-Si p+ (b) Fig. 2. o....BT cascode circuit (a) and its layout (b). A disadvantage of the CLBT is its low Early voltage, i.e., the device has a low output resistance. Therefore, it is preferable to use a cascode circuit as shown in fig. 2. This yields an output resistance several hundred times larger than that of the single o....BT, whereas the area penalty, in a layout as shown in fig 2b, is acceptable. Another disadvantage of the CLBTs, when biased using a resistive line, is their base current, which introduces an additional voltage drop on the resistive line. However, since the cut-off frequencies in the cochlea are controlled by the output current of the CLBTs and since these cut-off frequencies are relatively small, typically 20 kHz, the output current of the CLBTs will be small. If the common-emitter current gain ~ is much larger than 1, the base current of these o....BTs will be very small, and the voltage error introduced by the small base currents will be negligible. Furthermore, since the cut-off frequencies of the cochlea will typically span 2 decades with an exponentially decreasing cut-off frequency from the beginning to the end, only the first few filters will have any noticeable influence on the current drawn from the resistive line. 3.2 DIFFERENTIATION The stabilized second-order section of [7] uses two wide range transconductance amplifiers (A 1 and A2 in fig. 3) with equal bias current and equal capacitive load, to control the cut-off frequency. A basic transconductance amplifier (A3) is used in a Improved Silicon Cochlea Using Compatible Lateral Bipolar Transistors 675 feedback path to control the quality factor of the filter. The voltage VOU1 at the output of each second-order stage represents the basilar membrane displacement. Since the output of the biological cochlea is proportional to the velocity of the basilar membrane, the output of each second-order stage has to be differentiated. In [7] this is done by creating a copy of the output current Lru- of amplifier A2 at every stage. Since the voltage on a capacitor is proportional to the integral of the current onto the capacitor, Idit is effectively proportional to the basilar membrane velocity. Yet, with equal displacement amplitudes, velocity will be much larger for high frequencies than for low frequencies, yielding output signals with an amplitude that decreases from the beginning of the cochlea to the end. This can be corrected by normalizing Lru- to give equal amplitude at every output. A second resistive line with identical tilt controlling the gain of the current mirrors that create the copies of Idit at each stage is used for this purpose in [7]. However, if using a single resistive line for the control of the cut-off frequencies and the quality factor improves the performance of the chip, the same is true for the control of the current mirror gain. fromprev. section Fig. 3. One section of the cochlear cascade, with differentiator. An alternative solution, which does not need normalization, is to take the difference between VOuI and VI (see fig. 3). This can be shown to be equivalent to differentiating V Out. with OdB gain at the cut-off frequency for all stages. This can be easily done with a combination of 2 transconductance amplifiers. These amplifiers can have a large bias current, so they can also be used to buffer the cascade voltages before connecting them to the output pins of the chip, to avoid charging the cochlear cascade with the extra capacitance introduced by the output pins. 3.3 TEMPERATURE SENSITIVITY The cut-off frequency of the first and the last low-pass filter in the cascade can be set by applying voltages to both ends of the resistive line, and the intermediate filters will have a cut-off frequency decreasing exponentially from the beginning to the end. Yet, if we apply directly a voltage to the ends of the resistive line, the actual cut-off frequency obtained will depend on the temperature, since the current depends exponentially on the applied voltage normalized to the thermal voltage Ur (see(3). It is therefore better to create the voltages at both ends of the resistive line on-chip using a current biasing a CLBT with its base connected to its collector (or its drain connected to its gate if aMOS transistor is used). If this gate voltage is buffered, so that the current through the resistive line is not drawn from the input current, the bias currents of the first and last filter, and thus the cut-off frequency of all filters can be set, independent of temperature. 676 A. V AN SCHAlK, E. FRAGNIERE, E. VITTOZ 3.4 THE IMPROVED SILICON COCHLEA The improved silicon cochlea is shown in figure 4. It uses the cochlear sections shown in figure 3, CLBTs as the bias transistors of each filter, and one resistive line to bias all CLBTs. The resistive line is biased using two bipolar current mirror structures and two voltage buffers, which allow temperature independent biasing of the cut-off frequencies of the cochlea. A similar structure is used to create the voltage source V q to control, independent of temperature, the actual quality factor of each section. The actual bipolar current mirror implemented uses the cascode structure shown in figure 2a, however this is not shown in figure 4 for clarity. Vdiffl Fig 4. The improved silicon cochlea 4. TEST RESULTS The proposed silicon cochlea has been integrated using the ECPD15 technology at ES2 (Grenoble, France), containing 104 second-order stages, on a 4.77mm X 3.21mm die. Every second stage is connected to a pin, so its output voltage can be measured. In fig. 5, the frequency response curves after on-chip derivation are shown for the output taps of both the cochlea described in [7] (left), and our version (right). This clearly shows the improved regularity of the cut-off frequencies and the gain obtained using CLBTs. The drop-off in gain for the higher frequency stages (right) is a border effect, since at the beginning of the cochlea no accumulation of gain has yet taken place. In the figure on the left this is not visible, since the first nine outputs are not presented. -20 ·30 F~(Hz) 10000 10 ~ ~ 0 ·10 ·20 ·30 F~(Hz) Fig.5. Measured frequency responses at the different taps. 20000 In fig. 6 we show the cut-off frequency versus tap number of both chips. Ideally, this should be a straight line on a log-linear scale, since the cut-off frequency decreases Improved Silicon Cochlea Using Compatible Lateral Bipolar Transistors 677 exponentially with tap number. This also clearly shows the improved regularity using CLBTs as current sources. lOOOO·r-------------------, 200·~----------------~~ 10 15 20 25 30 o 10 20 30 40 50 Fig.6. Cut-off frequency (Hz) versus tap number for both silicon cochleae. 5. CONCLUSIONS Since the biological cochlea functions as a distributed filter, where the natural frequency decreases exponentially with the position along the basilar membrane, analog electronic cochlear models need exponentially scaled filters. The output current of a Compatible Lateral Bipolar Transistor depends exponentially on the base-emitter voltage. It is therefore easy to create exponentially scaled current sources using CLBTs biased with a resistive polysilicon line. Because the CLBTs are insensitive to variations of the CMOS threshold voltage VT, current sources implemented with CLBTs are much better matched than current sources using MaS transistors in weak inversion. Regularity is further improved using an on-chip differentiation that does not need a second resistive line to correct its gain, and therefore doesn't depend on identical tilt on both resistive lines. Better independence of temperature can be obtained by fixing the frequency domain of the cochlea using bias currents instead of voltages. Acknowledgments The authors would like to thank Felix Lustenberger for simulation and layout of the chip. We are also indebted to Lloyd Watts for allowing us to use his measurement data. References [1] R.F. Lyon and C.A. Mead, "An analog electronic cochlea," IEEE Trans. Acoust .• Speech. Signal Processing, vol. 36, pp. 1119-1134, July 1988. [2] R.F. Lyon, "Analog implementations of auditory models," Proc. DARPA Workshop Speech and Natural Language. San Mateo, CA:Morgan Kaufmann, 1991. [3] W. Liu, et. al., "Analog VLSI implementation of an auditory periphery model," Advances Res. VLSI, Proc. 1991 Santa Cruz Con/., MIT Press, 1991, pp. 153-163. [4] L. Watts, "Cochlear Mechanics: Analysis and Analog VLSI," Ph.D. thesis, California Institute of Technology, Pasadena, 1992. [5] E. Vittoz, "The design of high-performance analog circuits on digital CMOS chips," IEEE 1. Solid-State Circuits, vol. SC-20, pp. 657-665, June 1985. [6] E. Vittoz, "MaS transistors operated in the lateral bipolar mode and their application in CMOS technology," IEEE 1. Solid-State Circuits, vol. SC-24, pp. 273-279, June 1983. [7] L. Watts, et. al., "Improved implementation of the silicon cochlea," IEEE 1. SolidState Circuits, vol. SC-27, pp. 692-700, May 1992.	1995	7
1,117	Symplectic Nonlinear Component Analysis Lucas C. Parra Siemens Corporate Research 755 College Road East, Princeton, NJ 08540 lucas@scr.siemens.com Abstract Statistically independent features can be extracted by finding a factorial representation of a signal distribution. Principal Component Analysis (PCA) accomplishes this for linear correlated and Gaussian distributed signals. Independent Component Analysis (ICA), formalized by Comon (1994), extracts features in the case of linear statistical dependent but not necessarily Gaussian distributed signals. Nonlinear Component Analysis finally should find a factorial representation for nonlinear statistical dependent distributed signals. This paper proposes for this task a novel feed-forward, information conserving, nonlinear map - the explicit symplectic transformations. It also solves the problem of non-Gaussian output distributions by considering single coordinate higher order statistics. 1 Introduction In previous papers Deco and Brauer (1994) and Parra, Deco, and Miesbach (1995) suggest volume conserving transformations and factorization as the key elements for a nonlinear version of Independent Component Analysis. As a general class of volume conserving transformations Parra et al. (1995) propose the symplectic transformation. It was defined by an implicit nonlinear equation, which leads to a complex relaxation procedure for the function recall. In this paper an explicit form of the symplectic map is proposed, overcoming thus the computational problems. 438 L. C.PARRA In order to correctly measure the factorization criterion for non-Gaussian output distributions, higher order statistics has to be considered. Comon (1994) includes in the linear case higher order cumulants of the output distribution. Deco and Brauer (1994) consider multi-variate, higher order moments and use them in the case of nonlinear volume conserving transformations. But the calculation of multicoordinate higher moments is computational expensive. The factorization criterion for statistical independence can be expressed in terms of minimal mutual information. Considering only volume conserving transformations allows to concentrate on single coordinate statistics, which leads to an important reduction of computational complexity. So far, this approach (Deco & Schurman, 1994; Parra et aI., 1995) has been restricted to second order statistic. The present paper discusses the use of higher order cumulants for the estimation of the single coordinate output distributions. The single coordinate entropies measured by the proposed technique match the entropies of the sampled data more accurately. This leads in turns to better factorization results. 2 Statistical Independence More general than decorrelation used in PCA the goal is to extract statistical independent features from a signal distribution p(x). We look for a deterministic transformation on ~n: y = f(x) which generates a factorial representation p(y) = It p(Yd, or at least a representation where the individual coordinates P(Yi) of the output variable yare "as factorial as possible". This can be accomplished by minimizing the mutual information M I[P(y)]. n o ::; M I[P(y)] = L H[P(Yi)] - H[P(y)], (1) i=l since M I[P(y)] = 0 holds if p(y) is factorial. The mutual information can be used as a measure of "independence". The entropies H in the definition (1) are defined as usual by H[P(y)] = - J~oop(y)lnp(y)dy. As in linear PCA we select volume conserving transformations, but now without restricting ourselves to linearity. In the noise-free case of reversible transformations volume conservation implies conservation of entropy from the input x to the output y, i.e. H[P(y)] = H[P(x)] = canst (see Papoulis, 1991). The minimization of mutual information (1) reduces then to the minimization of the single coordinate output entropies H[P(Yi)]. This substantially simplifies the complexity of the problem, since no multi-coordinate statistics is required. 2.1 Measuring the Entropy with Cumulants With an upper bound minimization criterion the task of measuring entropies can be avoided (Parra et aI., 1995): (2) Symplectic Nonlinear Component Analysis Edgeworth appIOlClmatlOr'l to second and fanh order O.B,----~--~-~--~-___, 0.7 0.6 ~0 .5 ~ 04 >~ 03 ~ Q.. 0.2 0., o ~ . : .O.~~--=---,!------=---~----: dQ(y1)/dY1 -----~> )i 439 1 Figure 1: LEFT: Doted line: exponential distribution with additive Gaussian noise sampled with 1000 data points. (noise-variance/decay-constant = 0.2). Dashed line: Gaussian approximation equivalent to the Edgeworth approximation to second order. Solid line: Edgeworth approximation including terms up to fourth order. RIGHT: Structure of the volume conserving explicit symplectic map. The minimization of the individual output coordinate entropies H(P(Yi)] simplifies to the minimization of output variances (Ti. For the validity of that approach it is crucial that the map y = f(x) transforms the arbitrary input distribution p(x) into a Gaussian output distribution. But volume conserving and continuous maps can not transform arbitrary distributions into Gaussians. To overcome this problem one includes statistics - higher than second order - to the optimization criterion. Comon (1994) suggests to use the Edgeworth expansion of a probability distribution. This leads to an analytic expression of the entropy in terms of measurable higher order cumulants. Edgeworth expands the multiplicative correction to the best Gaussian approximation of the distribution in the orthonormal basis of Hermite polynomials hcr(y). The expansion coefficients are basically given by the cumulants Ccr of distribution p~y). The Edgeworth expansions reads for a zero-mean distribution with variance (T , (see Kendall & Stuart, 1969) p(y) 2 -l-e-~ f(y) -j2;(J (3) Note, that by truncating this expansion at a certain order, we obtain an approximation Papp(Y), which is not strictly positive. Figure 1, left shows a sampled exponential distribution with additive Gaussian noise. By cutting expansion (3) at fourth order, and further expanding the logarithm in definition of entropy up to sixth order, Comon (1994) approximates the entropy by, 440 L.C.PARRA 1 1 c§ 1 c~ 7 c~ 1 c~ C4 H(P(Y)app] ~ 2"ln(271'e) + In 0' - 120'6 - 480'8 - 480'12 + 8" 0'60'4 (4) We suggest to use this expression to minimize the single coordinate entropies in the definition of the mutual information (1). 2.2 Measuring the Entropy by Estimating an Approximation Note that (4) could only be obtained by truncating the expansion (3). It is therefore limited to fourth order statistic, which might be not enough for a satisfactory approximation. Besides, the additional approximation of the logarithm is accurate only for small corrections to the best Gaussian approximation, i.e. for fey) ~ 1. For distributions with non-Gaussian tails the correction terms might be rather large and even negative as noted above. We therefore suggest alternatively, to measure the entropy by estimating the logarithm of the approximated distribution In Papp (y) with the given data points Yv and using Edgeworth approximation (3) for Papp (y), 1 N 1 N H(P(y)] ~ - N L lnpapp (Yv) = canst + In 0' - N LIn f(yv) (5) v=1 v=1 Furthermore, we suggest to correct the truncated expansion Papp by setting fapp (y) -+ 0 for all fapp (y) < O. For the entropy measurement (5) there is in principle no limitation to any specific order. In table 1 the different measures of entropy are compared. The values in the row labeled 'partition' are measured by counting the numbers n(i) of data points falling in equidistant intervals i of width D.y and summing -pC i)D.y lnp(i) over all intervals, with p(i)D.y = n(i)IN. This gives good results compared to the theoretical values only because of the relatively large sampling size. These values are presented here in order to have an reliable estimate for the case of the exponential distribution, where cumulant methods tend to fail. The results for the exponential distribution show the difficulty of the measurement proposed by Comon, whereas the estimation measurement given by equation (5) is stable even when considering (for this case) unreliable 5th and 6th order cumulants. The results for the symmetric-triangular and uniform distribution demonstrate the insensibility of the Gaussian upper bound for the example of figure 2. A uniform squared distribution is rotated by an angle a. On the abscissa and ordinate a triangular or uniform distribution are observed for the different angles a = II/4 or a = 0 respectively. The approximation of the single coordinate entropies with a Gaussian measure is in both cases the same. Whereas measurements including higher order statistics correctly detect minimal entropy (by fixed total information) for the uniform distribution at a = O. 3 Explicit Symplectic Transformation Different ways of realizing a volume conserving transformation that guarantees H(P(x)] = H(P(x)] have been proposed (Deco & Schurman, 1994; Parra et aI., Symplectic Nonlinear Component Analysis 441 11easured entropy of Gauss uniform triangular exponential sampled distributions symmetric + Gauss noise partition 1.35 ± .02 .024 ± .006 .14 ± .02 1.31 ± .03 Gaussian upper bound (2) 1.415 ± .02 .18 ± .016 .18 ± .02 1.53 ± .04 Coman, eq. (4) 1.414 ± .02 .14 ± .015 .17 ± .02 3.0 ± 2.5 Estimate (5) - 4th order 1.414 ± .02 .13 ± .015 .17±.02 1.39 ± .05 Estimate (5) - 6th order 1.414 ± .02 .092 ± .001 .16 ± .02 1.3 ± .5 theoretical value 1.419 .0 .153 Table 1: Entropy values for different distributions sampled with N = 1000 data points and the different estimation methods explained in the text. The standard deviations are obtained by multiple repetition of the experiment. 1995). A general class of volume conserving transformations are the symplectic maps (Abraham & Marsden, 1978). An interesting and for our purpose important fact is that any symplectic transformation can be expressed in terms of a scalar function. And in turn any scalar function defines a symplectic map. In (Parra et al., 1995) a non-reflecting symplectic transformation has been presented. But its implicit definition results in the need of solving a nonlinear equation for each data point. This leads to time consuming computations which limit in practice the applications to low dimensional problems (n~ 10). In this work reflecting symplectic transformations with an explicit definition are used to define a "feed-forward" volume conserving maps. The input and output space is divided in two partitions x = (Xl, X2) and Y = (Yl, Y2), with Xl, X2, Yl , Y2 E ?Rn / 2 . (6) The structure of this symplectic map is represented in figure 1, right. Two scalar functions P : ?Rn / 2 1-+ ?R and Q : ?Rn / 2 1-+ ?R can be chosen arbitrarily. Note that for quadratic functions equation (6) represents a linear transformation. In order to have a general transformation we introduce for each of these scalar functions a 3-layer perceptron with nonlinear hidden units and a single linear output unit: (7) The scalar functions P and Q are parameterized by the network parameters Wl, W2 E Rm and Wl, W 2 E Rm x Rn/2. The hidden-unit, nonlinear activation function 9 applies to each component of the vectors WlYl and W2X2 respectively. Because of the structure of equation (6) the output coordinates Yl depend only additively on the input coordinates Xl. To obtain a more general nonlinear dependence a second symplectic layer has to be added. To obtain factorial distributions the parameters of the map have to be trained. The approximations of the single coordinate entropies (4) or (5) are inserted in the mutual information optimization criterion (1). These approximations are expressed through moments in terms of the measured output data points. Therefore, the 442 O,B,.---~-~-~-~-~-~-~---, 0,6 0,4 0,2 -0.2 -0.4 -0.6 , , . :.': :' ... " , ..... :.,' , . , -~~,B---0~,6-~-0~.4---0~,2-~--0~.2--0~.4--0~.6-~0,B L.C.PARRA Figure 2: Sampled 2-dimensional squared uniform distribution rotated by 7l" /4. Solid lines represent the directions found by any of the higher order techniques explained in the text. Dashed lines represent directions calculated by linear PCA. (This result is arbitrary and varies with noise) . gradient of these expressions with respect to parameters ofthe map can be computed in principle. For that matter different kinds of averages need to be computed. Even though, the computational complexity is not substantially increased compared with the efficient minimum variances criterion (2), the complexity of the algorithm increases considerably. Therefore, we applied an optimization algorithm that does not require any gradient information. The simple stochastic and parallel update algorithm ALOPEX (Unnikrishnan & Venugopal, 1994) was used. 4 Experiments As explained above, finding the correct statistical independent directions of a rotated two dimensional uniform distribution causes problems for techniques which include only second order statistic. The statistical independent coordinates are simply the axes parallel to the edges of the distribution (see figure 2). A rotation i. e. a linear transformation suffices for this task. The covariance matrix of the data is diagonal for any rotation of the squared distribution and, hence, does not provide any information about the correct orientation of the square. It is well known, that PCA fails to find in the case of non-Gaussian distributions the statistical independent coordinates. Similarly the Gaussian upper bound technique (2)is not capable to minimize the mutual information in this case. Instead, with anyone of the higher order criteria explained in the previous section one finds the appropriate coordinates for any linearly transformed multi-dimensional uniform distribution. This has been observed empirically for a series of setups. The symplectic map was restricted in this experiments to linea1;ity by using square scalar functions. The second example shows that the proposed technique in fact finds nonlinear relations between the input coordinates. An one-dimensional signal distributed according to the distribution of figure 1 was nonlinearly transformed into a twoSymplectic Nonlinear Component Analysis . '.: <~., .' . 443 . : ' .. ; Figure 3: Symplectic map trained with 4th and 2nd order statistics corresponding to the equations (5) and (2) respectively. Left: input distribution. The line at the center of the distribution gives the nonlinear transformed noiseless signal distributed according to the distribution shown in figure 1. Center and Right: Output distribution of the symplectic map corresponding to the 4th order (right) and 2nd order (center) criterion. dimensional signal and corrupted with additive noise, leading to the distribution shown in figure 3, left. The task of finding statistical independent coordinates has been tackled by an explicit symplectic transformation with. n = 2 and m = 6. On figure 3 the different results for the optimization according to the Gaussian upper bound criterion (2) and the approximated entropy criterion (5) are shown. Obviously considering higher order statistics in fact improves the result by finding the better representation of the nonlinear dependency. Reference Abraham, R., & Marsden, J . (1978). Foundations of Mechanics The BenjaminCummings Publishing Company, Inc., London. Comon, P. (1994). Independent component analysis, A new concept Signal Processing, 36, 287- 314. Deco, G., & Brauer, W. (1994). Higher Order Statistical Decorrelation by Volume Concerving Nonlinear Maps. Neural Networks, ? submitted. Deco, G., & Schurman, B. (1994). Learning Time Series Evolution by Unsupervised Extraction of Correlations. Physical Review E, ? submitted. Kendall, M. G., & Stuart, A. (1969). The Advanced Theory of Statistics (3 edition)., Vol. 1. Charles Griffin and Company Limited, London. Papoulis, A. (1991). Probability, Random Variables, and Stochastic Processes. Third Edition, McGraw-Hill, New York. Parra, L., Deco, G., & Miesbach, S. (1995). Redundancy reduction with information-preserving nonlinear maps. Network, 6(1), 61-72. Unnikrishnan, K., P., & Venugopal, K., P. (1994). Alopex: A Correlation-Based Learning Algorithm for Feedforward and Recurrent Neural Networks. Neural Computation, 6(3), 469- 490.	1995	70
1,118	Independent Component Analysis of Electroencephalographic Data Scott Makeig Naval Health Research Center P.O. Box 85122 San Diego CA 92186-5122 scott~cplJmmag.nhrc.navy.mil Tzyy-Ping Jung Naval Health Research Center and Computational Neurobiology Lab The Salk Institute, P.O. Box 85800 San Diego, CA 92186-5800 jung~salk.edu Anthony J. Bell Computational Neurobiology Lab The Salk Institute, P.O. Box 85800 San Diego, CA 92186-5800 tony~salk.edu Terrence J. Sejnowski Howard Hughes Medical Institute and Computational Neurobiology Lab The Salk Institute, P.O. Box 85800 San Diego, CA 92186-5800 terry~salk.edu Abstract Because of the distance between the skull and brain and their different resistivities, electroencephalographic (EEG) data collected from any point on the human scalp includes activity generated within a large brain area. This spatial smearing of EEG data by volume conduction does not involve significant time delays, however, suggesting that the Independent Component Analysis (ICA) algorithm of Bell and Sejnowski [1] is suitable for performing blind source separation on EEG data. The ICA algorithm separates the problem of source identification from that of source localization. First results of applying the ICA algorithm to EEG and event-related potential (ERP) data collected during a sustained auditory detection task show: (1) ICA training is insensitive to different random seeds. (2) ICA may be used to segregate obvious artifactual EEG components (line and muscle noise, eye movements) from other sources. (3) ICA is capable of isolating overlapping EEG phenomena, including alpha and theta bursts and spatially-separable ERP components, to separate ICA channels. (4) N onstationarities in EEG and behavioral state can be tracked using ICA via changes in the amount of residual correlation between ICA-filtered output channels. 146 S. MAKEIG, A. l . BELL, T.-P. lUNG, T. l. SEJNOWSKI 1 Introduction 1.1 Separating What from Where in EEG Source Analysis The joint problems of EEG source segregation, identification, and localization are very difficult, since the problem of determining brain electrical sources from potential patterns recorded on the scalp surface is mathematically underdetermined. Recent efforts to identify EEG sources have focused mostly on verforming spatial segregation and localization of source activity [4]. By applying the leA algorithm of Bell and Sejnowski [1], we attempt to completely separate the twin problems of source identification (What) and source localization (Where). The leA algorithm derives independent sources from highly correlated EEG signals statistically and without regard to the physical location or configuration of the source generators. Rather than modeling the EEG as a unitary output of a multidimensional dynamical system, or as "the roar of the crowd" of independent microscopic generators, we suppose that the EEG is the output of a number of statistically independent but spatially fixed potential-generating systems which may either be spatially restricted or widely distributed. 1.2 Independent Component Analysis Independent Component Analysis (leA) [1, 3] is the name given to techniques for finding a matrix, Wand a vector, w, so that the elements, u = (Ul .. . uNF, of the linear transform u = Wx + W of the random vector, x = [Xl ... xNF, are statistically independent. In contrast with decorrelation techniques such as Principal Components Analysis (peA) which ensure that {UiUj} = 0, Vij, ICA imposes the much stronger criterion that the multivariate probability density function (p .d.f.) of u factorizes: fu(u) = n::l fu.(ud . Finding such a factorization involves making the mutual information between the Ui go to zero: I(ui,uj) = O,Vij. Mutual information is a measure which depends on all higher-order statistics of the Ui while decorrelation only takes account of 2nd-order statistics. In (1], a new algorithm was proposed for carrying out leA. The only prior assumption is that the unknown independent components, Ui, each have the same form of cumulative density function (c.d.f.) after scaling and shifting, and that we know this form, call it Fu(u). ICA can then be performed by maximizing the entropy, H(y), of a non-linearly transformed vector: y = Fu(u) . This yields stochastic gradient ascent rules for adjusting Wand w: where y = (:ih ... YN F, the elements of which are: , a 0Yi (h ( )] Yi = -whic if y = Fu U 0Yi OUi _ Ofu(Ui) OFu(Ui) (1) (2) It can be shown that an leA solution is a stable point of the relaxation of eqs.(1-2). In practical tests on separating mixed speech signals, good results were found when using the logistic function, Yi = (1 + e-u• )-1, instead of the known c.d.f., Fu, of the speech signals. In this case Yi = 1 - 2Yi, and the algorithm has a simple form. These results were obtained despite the fact that the p.d.f. of the speech signals was not exactly matched by the gradient of the logistic function. In the experiments in this paper, we also used the speedup technique of prewhitening described in [2]. Independent Component Analysis of Electroencephalographic Data 147 1.3 Applying leA to EEG Data The leA technique appears ideally suited for performing source separation in domains where, (1) the sources are independent, (2) the propagation delays of the 'mixing medium' are negligible, (3) the sources are analog and have p.d.f.'s not too unlike the gradient of a logistic sigmoid, and (4) the number of independent signal sources is the same as the number of sensors, meaning if we employ N sensors, using the ICA algorithm we can separate N sources. In the case of EEG signals, N scalp electrodes pick up correlated signals and we would like to know what effectively 'independent brain sources' generated these mixtures. If we assume that the complexity of EEG dynamics can be modeled, at least in part, as a collection of a modest number of statistically independent brain processes, the EEG source analysis problem satisfies leA assumption (1). Since volume conduction in brain tissue is effectively instantaneous, leA assumption (2) is also satisfied. Assumption (3) is plausible, but assumption (4), that the EEG is a linear mixtures of exactly N sources, is questionable, since we do not know the effective number of statistically independent brain signals contributing to the EEG recorded from the scalp. The foremost problem in interpreting the output of leA is, therefore, determining the proper dimension of input channels, and the physiological and/or psychophysiological significance of the derived leA source channels. Although the leA model of the EEG ignores the known variable synchronization of separate EEG generators by common subcortical or corticocortical influences [5], it appears promising for identifying concurrent signal sources that are either situated too close together, or are too widely distributed to be separated by current localization techniques. Here, we report a first application of the ICA algorithm to analysis of 14-channel EEG and ERP recordings during sustained eyes-closed performance of an auditory detection task, and give evidence suggesting that the leA algorithm may be useful for identifying psychophysiological state transitions. 2 Methods EEG and behavioral data were collected to develop a method of objectively monitoring the alertness of operators of complex systems [8] . 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. Auditory 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, and superimposed on a continuous 39-Hz click train evoking a 39-Hz steady-state response (SSR). Short, and task-irrelevant probe tones of two 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 International 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. Target Hits were defined as targets responded to within a 100-3000 ms window, while Lapses were targets not responded to. Two sessions each from three of the subjects were selected for analysis based on their containing at least 50 response Lapses. A continuous performance measure, local error rate, was computed by convolving the irregularly-sampled performance index time series (Hit=O/Lapse=l) with a 95 s smoothing window advanced for 1.64 s steps. 148 S. MAKEIG, A. l. BELL, T.-P. lUNG, T. 1. SEJNOWSKI The leA algorithm in eqs.(1-2) was applied to the 14 EEG recordings. The time index was permuted to ensure signal stationarity, and the 14-dimensional time point vectors were presented to a 14 ---. 14 leA network one at a time. To speed convergence, we first pre-whitened the data to remove first- and second-order statistics. The learning rate was annealed from 0.03 to 0.0001 during convergence. After each pass through the whole training set, we checked the amount of correlation between the leA output channels and the amount of change in weight matrix, and stopped the training procedure when, (1) the mean correlation among all channel pairs was below 0.05, and (2) the leA weights had stopped changing appreciably. 3 Results A small (4.5 s) portion of the resulting leA-transformed EEG time series is shown in Figure 1. As expected, correlations between the leA traces are close to zero. The dominant theta wave (near 7 Hz) spread across many EEG channels (left paneQ is more or less isolated to leA trace 1 (upper right), both in the epoch shown and throughout the session. Alpha activity (near 10 Hz) not obvious in the EEG data is uncovered in leA trace 2, which here and throughout the session contains alpha bursts interspersed with quiescent periods. Other leA traces (3-8) contain brief oscillatory bursts which are not easy to characterize, but clearly display different dynamics from the activity in leA trace 1 which dominates the raw EEG record. ICA trace 10 contains near-De changes associated with eye slow movements in the EOG and most frontal (Fpz) EEG channels. leA trace 13 contains mostly line noise (60 Hz), while ICA traces 9 and 14 have a broader high frequency (50-100 Hz) spectrum, suggesting that their source is likely to be high-frequency activity generated by scalp muscles. Apparently, the ICA source solution for this data does not depend strongly on learning rate or initial conditions. When the same portion of one session was used to train two leA networks with different random starting weights, data presentation orders, and learning rates, the two final ICA weight matrices were very close to one another. Filtering another segment of EEG data from the same session using each ICA matrix produced two ICA source transforms in which 11 of the 14 bestcorrelated output channel pairs correlated above 0.95 and none correlated less than 0.894. While ICA training minimized mutual information, and therefore also correlations between output channels during the initial (alert) leA training period, output data channels filtered by the same leA weight matrix became more correlated during the drowsy portion of the session, and then reverted to their initial levels of (de)correlation when the subject again became alert. Conversely, filtering the same session's data with an leA weight matrix trained on the drowsy portion of the session produced output channels that were more correlated during the alert portions of the session than during the drowsy training period. Presumably, these changes in residual correlation among ICA outputs reflect changes in the dynamics and topographic structure of the EEG signals in alert and drowsy brain states. An important problem in human electrophysiology is to determine a means of objectively identifying overlapping ERP subcomponents. Figure 3 (right paneQ shows an leA decomposition of (left paneQ ERPs to detected (Hit) and undetected (Lapse) targets by the same subject. leA spatial filtering produces two channels (S[I-2]) separating out the 39-Hz steady-state response (SSR) produced by the continuous 39-Hz click stimulation during the session. Note the stimulus-induced perturbation in SSR amplitude previously identified in [6]. Three channels (H[I-3]) pass time-limited components of the detected target response, while four others (1[1-4]) Independent Component Analysis of Electroencephalographic Data J 49 EEG leA Fz ~V~~~hI'o/A. Cz ~~MvN'N{\~v<wv'yJ\J\r"~ pz V&V\fM~IIjJ-r~ F3 .;vwvwvvvrv~~WV'JI~ F4 ~\0.,fvo/lf'1Vlf\,~~~ C3 VIV"'vVWv!l!vWN/W\'~~ C4 \MIV{lAtv!ifVV{\AfJV0~~ T4 ~~~~ P3 v"V'v""Nv\~"'-'V P4 M'-VVI/<{WY'V0,Ww~ Fpz ~\~I"IVlV';.~~ EOG~~~ 3 VfJV\.'\I\~~~'~ 4 'rIvV\.JJvvV'-r~·~, 5 i~'MI'\'V1fV{\tNN~10~~ 6 {.I'VVVVVvw....;;~~rwvr'ri(.,'r·Nvf 7 /<¥Yl1~'riiwNV~~~~ 8 ~v.AJvJw-~\Jn~ 9 >I~vw~Y!"'~.fW'Iwi'fr."'I 10 ~~~~~ 11 1~IVVV~\fvv{iJYlw~~ 12/¥v\~~ 13~~~~~~~ 14 ~~\"W!~~~~~ -- 1 sec. Figure 1: Left: 4.5 seconds of 14-channel EEG data. Right: an le A transform of the same data, using weights trained on 6.5 minutes of similar data from the same seSSlOn. 150 Scalp ERPs Fz ~~~~...., + Cz ;., ~~P •. ..-.woiQfiS$i + pz -... + "-,;r::> Gtr~ --,.~ + Oz ~ .,. , Q ... ¢I,;;" '''iIi 04"'. + F3 ~ ""~~n;; + F4 :1 ~;f'O"IV'~~ 'tV; + C3 -., ' 1;so~ ; _ .... '''''''S + C4 ;;., ,~, coA'C>"",~ .... ,W Ii ::v; + T3 ~ 1"·" ,,~"" ~;t;!¥ ;Nft~h'" + T4 ;'+~~t ~~~."w + - .J P3 1 c ......... r::c;.,.~4&f' ! c 44c;c;e + P4 ~." 0« '4 ~~;. '" • j" ""+ + FPZ~n~~~ •. + EOG;""1 :: -............ ~ ... '1* it~ + S. MAKEIG, A. J. BELL, T.-P. JUNG, T. J. SEJNOWSKI ICAERPs - I H2;c _'_~4W. ;11 + H3 ~ • ...,p.~~ Rep ~ L1 -..,~. v.. ..At '~"lf"'o ~ + L2 - +~~"4 = e ..... ~ + L3 ::"I-·~~ ~ + L4 -1$ f- ...... · A.E:::::;? <:::!> <P ~ + 81 ~'f'r4'i"'Qf4;.~TIN'I"m"'''''~''i + 82 ;,"1Y¥".-U'lt ~'tt~'.'~'~'~"'fH','t1 + U1 ;'~t.,,=,,,~~FW"" ~ .. _~ + U2 ~ I-- iiIIe<>"" ,;I6'O'~' t'4ac*'" , + U3 -.... ro1~C--_ 'k. • ... + U4 ~~ 111(ok i. tdt .. 'IWoio'" "M I~ C + US :+~[P'~Q""'~ "' ..... ~ .. _ o 0.2 0.4 0.6 0.8 1 sec Detected targets Undetected targets Figure 2: Left panel: Event-related potentials (ERPs) in response to undetected (bold traces) and detected (faint traces) noise targets during two half-hour sessions. Right panel: Same ERP signals filtered using an leA weight matrix trained on the ERP data. Independent Component Analysis of Electroencephalographic Data 151 components of the (larger) undetected target response. We suggest these represent the time course of the locus (either focal or distributed) of brain response activity, and may represent a solution to the longstanding problem of objectively dividing evoked responses into neurobiologically meaningful, temporally overlapping subcomponents. 4 Conclusions ICA appears to be a promising new analysis tool for human EEG and ERP research. It can isolate a wide range of artifacts to a few output channels while removing them from remaining channels. These may in turn represent the time course of activity in longlasting or transient independent 'brain sources' on which the algorithm converges reliably. By incorporating higher-order statistical information, ICA avoids the non-uniqueness associated with decorrelating decompositions. The algorithm also appears to be useful for decomposing evoked response data into spatially distinct subcomponents, while measures of nonstationarity in the ICA source solution may be useful for observing brain state changes. Acknowledgments This report was supported in part 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. Dr. Bell is supported by grants from the Office of Naval Research and the Howard Hughes Medical Institute. References [1] A.J. Bell & T.J. Sejnowski (1995). An information-maximization approach to blind separation and blind deconvolution, Neural Computation 7:1129-1159. [2] A.J. Bell & T.J. Sejnowski (1995). Fast blind separation based on information theory, in Proc. Intern. Symp. on Nonlinear Theory and Applications (NOLTA), Las Vegas, Dec. 1995. [3] P. Comon (1994) Independent component analysis, a new concept? Signal processing 36:287-314. [4] A.M. Dale & M.1. Sereno (1993) EEG and MEG source localization: a linear approach. J. Cogn. Neurosci. 5:162. [5] R. Galambos & S. Makeig. (1989) Dynamic changes in steady-state potentials. In Erol Basar (ed.), Dynamics of Sensory and Cognitive Processing of the Brain, 102-122. Berlin:Springer-Verlag. [6] S. Makeig & R. Galambos. (1989) The CERP: Event-related perturbations in steady-state responses. In E. Basar & T.H. Bullock (ed.), Brain Dynamics: Progress and Perspectives, 375-400. Berlin:Springer-Verlag. [7] T-P. Jung, S. Makeig, M. Stensmo, & T. Sejnowski. Estimating alertness from the EEG power spectrum. Submitted for publication. [8] S. Makeig & M. Inlow (1993) Lapses in alertness: Coherence of fluctuations in performance and EEG spectrum. Electroencephalog. din. N europhysiolog. 86:23-35.	1995	71
1,119	Competence Acquisition in an Autonomous Mobile Robot using Hardware Neural Techniques. Geoff Jackson and Alan F. Murray Department of Electrical Engineering Edinburgh University Edinburgh, ER9 3JL Scotland, UK gbj@ee.ed.ac.uk,afm@ee.ed.ac.uk Abstract In this paper we examine the practical use of hardware neural networks in an autonomous mobile robot. We have developed a hardware neural system based around a custom VLSI chip, EPSILON III, designed specifically for embedded hardware neural applications. We present here a demonstration application of an autonomous mobile robot that highlights the flexibility of this system. This robot gains basic mobility competence in very few training epochs using an "instinct-rule" training methodology. 1 INTRODUCTION Though neural networks have been shown as an effective solution for a diverse range of real-world problems, applications and especially hardware implementations have been few and slow to emerge. For example in the DARPA neural networks study of 1988; of the 77 neural network applications investigated only 4 had resulted in field tested systems [Widrow, 1988]. Furthermore, none of these used dedicated neural network hardware. It is our view that this lack of tangible successes can be summarised by the following points: • Most neural applications will be served optimally by fast, generic digital computers . • Dedicated digital neural accelerators have a limited lifetime as "the fastest" , as standard computers develop so rapidly. lEdinburgh Pulse Stream Implemenation of a Learning Oriented Network. 1032 G. JACKSON, A. F. MURRAY • Analog neural VLSI is a niche technology, optimally applied at the interface between the real world and higher-level digital processing. This attitude has some profound implications with respect to the size, nature and constraints we place on new hardware neural designs. After several years of research into hardware neural network implementation, we have now concentrated on the areas in which analog neural network technology has an "edge" over well established digital technology. Within the pulse stream neural network research at the University of Edinburgh, the EPSILON chip's areas of strength can be summarised as: • Analog or digital inputs, digital outputs. • Modest size. • Scaleable and cascadeable design. • Compact, low power. This list points naturally and strongly to problems on the boundary of the real, analog world and digital processing, such as pre-processing/interpretation of analog sensor data. Here a modest neural network can act as an intelligent analog-to-digital converter presenting preprocessed information to its host. We are now engaged in a two pronged approach, whereby development of technology to improve the performance of pulse stream neural network chips is occurring concurrently with a search and development of applications to which this technology can be applied. The key requirements of this technological development are that devices must: • Work directly with analog signals. • Provide a moderate size network. • Have the potential for a fully integrated solution. In working with the above constraints and goals we have developed a new chip, EPSILON II, and a bus based processor card incorporating it. It is our aim to use this system to develop applications. As our first demonstration the EPSILON processor card has been mounted on an autonomous mobile robot. In this case the network utilises a mixture of analog and digital sensor information and performs a mapping between input/sensor space, a mixture of analog and digital signals, and output motor control. 2 THE EPSILON II CHIP The EPSILON II chip has been designed around the requirements of an application based system. It follows on from an earlier generation of pulse stream neural network chip, the EPSILON chip [Murray, 1992]. The EPSILON II chip represents neural states as a pulse encoded signal. These pulse encoded signals have digital signal levels which make them highly immune to noise and ideal for inter and intra-chip communication, facilitating efficient cascading of chips to form larger systems. The EPSILON II chip can take as inputs either pulse encoded signals or analog voltage levels, thus facilitating the fusing of analog and digital data in one system. Internally the chip is analog in nature allowing the synaptic multiplication function to be carried out in compact and efficient analog cells [J ackson, 1994]. Table 1 shows the principal specifications of the EPSILON II chip. The EPSILON II chip is based around a 32x32 synaptic matrix allowing efficient interfacing to digital systems. Several features of the device have been developed specifically for applications based usage. The first of these is a programmable input mode. This Competence Acquisition in an Autonomous Mobile Robot 1033 Table 1: EPSILON II Specifications EPSILON II Chip Specifications No. of state input pins 32 Input modes Analogt PW or PF Input mode programmability Bit programmable No. of state outputs 32 pinned out Output modes PW or PF Digital recovery of analog liP Yes - PW encoded No. of Synapses 1024 Additional autobias synapses 4 per output neuron Weight storage Dynamic Programmable activity voltage Yes Die size 6.9mm x 7mm allows each of the network inputs to be programmed as either a direct analog input or a digital pulse encoded input. We believe that this is vital for application based usage where it is often necessary to fuse real-world analog data with historical or control data generated digitally. The second major feature is a pulse recovery mode. This allows conversion of any analog input into a digital value for direct use by the host system. Both these features are utilised in the robotics application described in section 4 of this paper. 3 EPSILON PROCESSOR CARD The need to embed the EPSILON chip in a processor card is driven by several considerations. FirstlYt working with pulse encoded signals requires substantial processing to interface directly to digital systems. If the neural processor is to be transparent to the host system and is not to become a substantial processing overheadt then all pulse support operations must be carried out independently of the host system. SecondlYt to respond to further chip level advances and allow rapid prototyping of new applications as they emerget a certain amount of flexibility is needed in the system. It is with these points in mind that the design of the flexible EPSILON Processor Card (EPC) was undertaken. 3.1 DESIGN SPECIFICATION The EPC has been designed to meet the following specifications. The card must: • Operate on a conventional digital bus system. • Be transparent to the host processor t that is carry out all the necessary pulse encoding and decoding. • Carry out the refresh operations of the dynamic weights stored on the EPSILON chip. • Generate the ramp waveforms necessary for pulse width coding. • Support the operation of multiple EPCts. • Allow direct input of analog signals. As all data used and generated by the chip is effectively of 8-bit resolutiont the STE bUSt an industry standard 8-bit bUSt was chosen for the bus system. This is also cost 1034 G. JACKSON, A. F. MURRAY effective and allows the use of readily available support cards such as processors, DSP cards and analog and digital signal conditioning cards. To allow the transparency of operation the card must perform a variety of functions. A block diagram indicating these functions is shown in figure 1. FPGA ..................... ........................... __ ........ . · . · . · . · . · . · . 1--""':---1 Pulse to Dig. Conv. Dig. to Pulse Cony. Weight refresh Ctrl. Weight RAM Figure 1: EPSILON Processor Card A substantial amount of digital processing is required by the card, especially in the pulse conversion circuitry. To conform to the Eurocard standard size of the STE specification an FPGA device is used to "absorb" most of the digital logic. A twin mother/daughter board design is also used to isolate sensitive analog circuitry from the digital logic. The use of the FPGA makes the card extremely versatile as it is now easily reconfigurable to adapt to specialist application. The dotted box of figure 1 shows functions implemented by the FPGA device. An on board EPROM can hold multiple FPGA configurations such that the board can be reconfigured "on the fly" . All EPSILON support functions, such as ramp generation, weight refresh, pulse conversion and interface control are carried out on the card. Also the use of the FPGA means that new ideas are easily tested as all digital signal paths go via this device. Thus a card of new functionality can be designed without the need to design a new PCB. 3.2 SPECIALIST BUSES The digital pulse bus is buffered out under control of the FPGA to the neural bus along with two control signals. Handshaking between EPC's is done over these lines to allow the transfer of pulse stream data between processors. This implies that larger networks can be implemented with little or no increase in computation time or overhead. A separate analog bus is included to bring analog inputs directly onto the chip. 4 APPLICATIONS DEVELOPMENT The over-riding reason for the development of the EPC is to allow the easy development of hardware neural network applications. We have already indicated that we believe that this form of neural technology will find its niche where its advantages of direct sensor interface, compactness and cost-effectiveness are of prime importance. As a good and intrinsically interesting example of this genre of applications, we have chosen autonomous mobile robotic control as a first test for EPSILON II. The object of this demonstrator is not to advance the state-of-the-art in robotics. Competence Acquisition in an Autonomous Mobile Robot 1035 Rather it is to demonstrate analog neural VLSI in an appropriate and stimulating context. 4.1 "INSTINCT-RULE" ROBOT The "instinct-rule" robotic control philosophy is based on a software-controlled exemplarfrom the University's Department of Artificial Intelligence [Nehmzow, 1992]. The robot incorporates an EPC which interfaces all the analog sensor signals and provides the programmable neural link between sensor/input space and the motor drive actuators. ~ -"*,::::--,,, '" o en ffi -..,...,\--,,L en a) Controller Architecture. b) Instinct rule robot. Figure 2: "Instinct Rule" Robot The controller architecture is shown in figure 2. The neural network implemented on the EPC is the plastic element that determines the mapping between sensory data and motor actions. The majority of the monitor section is currently implemented on a host processor and monitors the performance of the neural network. It does this by regularly evaluating a set of instinct rules. These rules are simple behaviour based axioms. For example, we use two rules to promote simple obstacle avoidance competence in the robot, as listed in column one of table 2 Table 2: Instinct Rules Simple obstacle avoidance. Wall following l. Keep crash sensors inactive. l. Keep crash sensors inactive. 2. Move forward. 2. Keep side sensors active. 3. Move forward. If an instinct rule is violated the drive selector then chooses the next strongest output (motor action) from the neural network. This action is then performed to see if it relieves the violation. If it does, it is used as targets to train the neural network. If it does not, the next strongest action is tried. The mechanism to accomplish this will be described in more detail in section 4.2. Using this scheme the robot can be initialised with random weights (i.e. no mapping between sensors and motor control) and within a few epochs obtains basic obstacle avoidance competence. It is a relatively easy matter to promote more complex behaviour with the addition of other rules. For example to achieve a wall following behaviour a third 1036 G. JACKSON, A. F. MURRAY rule is introduced as shown in column two of table 2. Navigational tasks can be accomplished with the addition of a "maximise navigational signal" rule. An example of this is a light sensor mounted on the robot producing a behaviour to move towards a light source. Equally, a signal from a more complex, higher level, navigational system could be used. Thus the instinct rule controller handles basic obstacle avoidance competence and motor/sensory interface tasks leaving other resources free for intensive navigational tasks. 4.2 INSTINCT RULE EVALUATION USING SOMATIC TENSION The original instinct rule robot used binary sensor signals and evaluated performance of alternative actions for fixed, and progressively longer, periods of time [Nehmzow, 1992]. With the EPC interfacing directly to analog sensors an improved scheme has been developed. If we sum all sensors onto a neuron with fixed and equal weights we gain a measure of total sensory activity. Let us call this somatic tension as an analogy to biological signal aggregation on the soma. If we have an instinct violation and an alternative action is performed we can monitor this somatic tension to gauge the performance of this action. If tension decreases significantly we continue the action. If it increases significantly we choose an alternative action. If tension remains high and roughly the same, we are in a tight situation, for example say a corner. In this case we perform actions for progressively longer periods continuing to monitor somatic tension for a drop. 4.3 RESULTS AND DISCUSSION The instinct rule robot has been constructed and its performance is comparable with software-controlled predecessors. Unfortunately direct comparisons are not possible due to unavailability of the original exemplars and differing physical characteristics of the robots themselves. In developing the application several observations were made concerning the behaviour of the system that would not have come to light in a simulated environment. In any system including real mechanics and real analog signals, imperfections and noise are present. For example, in a real robot we cannot guarantee that a forward motion directive will result in perfect forward motion due to inherent asymmetries in the system. The instinct rule architecture does not assume a-priori knowledge such as this so behaviour is not affected adversely. This was tested by retarding one drive motor of the robot to give it a bias to one side. In early development, as the monitor was being tuned, the robot showed a tendency to oscillatory motion, thus exhibiting undesirable behaviour that satisfies its instincts. It could, for example, oscillate back and forth at a corner. In a simulated environment this continues indefinitely. However, with real mechanics and noisy analog sensors the robot breaks out of this undesirable behaviour. These observations strengthen the arguments for hardware development aimed at embedded systems. The robot application is but an example of the different, and often surprising conditions that pertain in a "real" system. If neural networks are to find applications in real-world, low-cost and analog-interface applications, these are the conditions we must deal with, and appropriate, analog hardware is the optimal medium for a solution. Competence Acquisition in an Autonomous Mobile Robot 1037 5 CONCLUSIONS This paper has described pulse stream neural networks that have been developed to a system level to aid development of applications. We have therefore defined areas of strengths of this technology along with suggestions of where this is best applied. The strengths of this system include: 1. Direct interfacing to analog signals. 2. The ability to fuse direct analog sensor data with digital sensor data processed elsewhere in the system. 3. Distributed processing. Several EPC's may be embedded in a system to allow multiple networks and/or multi layer networks. 4. The EPC represents a flexible system level development environment. It is easily reconfigured for new applications or improved chip technology. 5. The EPC requires very little computational overhead from the host system and can operate independently if needed. A demonstration application of an instinct rule robot has been presented highlighting the use of neural networks as an interface between real-world analog signals and digital control. In conclusion we believe that the immediate future of neural analog VLSI is in small applications based systems that interface directly to the real-world. We see this as the primary niche area where analog VLSI neural networks will replace conventional digital systems. Acknow ledgements Thanks are due to Ulrich Nehmzow, University of Manchester, for discussions and information on the instinct-rule controller and the loan of his original robot - Alder. References [Caudell, 1990] Caudell, M. and Butler, C. (1990). Naturally Intelligent Systems. MIT Press, Cambridge, Ma. [Jackson, 1994] Jackson, G., Hamilton, A., and Murray, A. F. (1994). Pulse stream VLSI neural systems: into robotics. In Proceedings ISCAS'94, volume 6, pages 375-378. IEEE Press. [Maren, 1990] Maren, A., Harston, C., and Pap, R. (1990). Handbook of Neural Computing Applications. Academic Press, San Diego, Ca. [Murray,1992] Murray, A. F., Baxter, D. J., Churcher, S., Hamilton, A., Reekie, H. M., and Tarassenko, L. (1992). The Edinburgh pulse stream implementation of a learning-oriented network (EPSILON) chip. In Neural Information Processing Systems (NIPS) Conference. [Nehmzow, 1992] Nehmzow, U. (1992). Experiments in Competence Acquisition for Autonomous Mobile Robots. PhD thesis, University of Edinburgh. [Widrow, 1988] Widrow, B. (1988). DARPA Neural Network Study. AFCEA International Press.	1995	72
1,120	Gaussian Processes for Regression Christopher K. I. Williams Neural Computing Research Group Aston University Birmingham B4 7ET, UK c.k.i.williams~aston.ac.uk Carl Edward Rasmussen Department of Computer ,Science University of Toronto Toronto, ONT, M5S lA4, Canada carl~cs.toronto.edu Abstract The Bayesian analysis of neural networks is difficult because a simple prior over weights implies a complex prior distribution over functions. In this paper we investigate the use of Gaussian process priors over functions, which permit the predictive Bayesian analysis for fixed values of hyperparameters to be carried out exactly using matrix operations. Two methods, using optimization and averaging (via Hybrid Monte Carlo) over hyperparameters have been tested on a number of challenging problems and have produced excellent results. 1 INTRODUCTION In the Bayesian approach to neural networks a prior distribution over the weights induces a prior distribution over functions. This prior is combined with a noise model, which specifies the probability of observing the targets t given function values y, to yield a posterior over functions which can then be used for predictions. For neural networks the prior over functions has a complex form which means that implementations must either make approximations (e.g. MacKay, 1992) or use Monte Carlo approaches to evaluating integrals (Neal, 1993). As Neal (1995) has argued, there is no reason to believe that, for real-world problems, neural network models should be limited to nets containing only a "small" number of hidden units. He has shown that it is sensible to consider a limit where the number of hidden units in a net tends to infinity, and that good predictions can be obtained from such models using the Bayesian machinery. He has also shown that a large class of neural network models will converge to a Gaussian process prior over functions in the limit of an infinite number of hidden units. In this paper we use Gaussian processes specified parametrically for regression problems. The advantage of the Gaussian process formulation is that the combination of Gaussian Processes for Regression 515 the prior and noise models can be carried out exactly using matrix operations. We also show how the hyperparameters which control the form of the Gaussian process can be estimated from the data, using either a maximum likelihood or Bayesian approach, and that this leads to a form of "Automatic Relevance Determination" (Mackay 1993j Neal 1995). 2 PREDICTION WITH GAUSSIAN PROCESSES A stochastic process is a collection of random variables {Y (x) Ix EX} indexed by a set X. In our case X will be the input space with dimension d, the number of irlputs. The stochastic process is specified by giving the probability distribution for every finite subset of variables Y(x(1)), ... , Y(x(k)) in a consistent manner. A Gaussian process is a stochastic process which can be fully specified by its mean function J.1.(:x:) = E[Y(x)] and its covariance function C(X ,X/) = E[(Y(x) - J.1.(x))(Y(x/)J.1.( Xl))]; any finite set of points will have a joint multivariate Gaussian distribution. Below we consider Gaussian processes which have J.1.( x) == O. In section 2.1 we will show how to parameterise covariances using hyperparametersj for now we consider the form of the covariance C as given. The training data consists of n pairs of inputs and targets {( xCi) , t(i)) , i = 1 .. . n} . The input vector for a test case is denoted x (with no superscript). The inputs are d-dimensional Xl, . .. , Xd and the targets are scalar. The predictive distribution for a test case x is obtained from the n + 1 dimensional joint Gaussian distribution for the outputs of the n training cases and the test case, by conditioning on the observed targets in the training set. This procedure is illustrated in Figure 1, for the case where there is one training point and one test point. In general, the predictive distribution is Gaussian with mean and variance kT (x)K- 1t C(x,x) - kT(x)K- 1k(x), (1) (2) where k(x) = (C(x, x(1)), ... , C(x, x(n))f , K is the covariance matrix for the training cases Kij = C(x(i), x(j)), and t = (t(l), ... , t(n))T . The matrix inversion step in equations (1) and (2) implies that the algorithm has O( n3 ) time complexity (if standard methods of matrix inversion are employed); for a few hundred data points this is certainly feasible on workstation computers, although for larger problems some iterative methods or approximations may be needed. 2.1 PARAMETERIZING THE COVARIANCE FUNCTION There are many choices of covariance functions which may be reasonable. Formally, we are required to specify functions which will generate a non-negative definite covariance matrix for any set of points (x(1 ), ... , x(k )). From a modelling point of view we wish to specify covariances so that points with nearby inputs will give rise to similar predictions. We find that the following covariance function works well: d Vo exp{ -t L WI(x~i) x~j))2} 1=1 d +ao + a1 Lx~i)x~j) + V18(i ,j), 1=1 (3) 516 / / y / / y1 c. K. I. WILLIAMS, C. E. RASMUSSEN y p(y) Figure 1: An illustration of prediction using a Gaussian process. There is one training case (x(1), t(1)) and one test case for which we wish to predict y. The ellipse in the lefthand plot is the one standard deviation contour plot of the joint distribution of Yl and y. The dotted line represents an observation Yl = t(1). In the right-hand plot we see the distribution of the output for the test case, obtained by conditioning on the observed target. The y axes have the same scale in both plots. where (} = log(vo, V1, W1, . . . , Wd, ao, ad plays the role of hyperparameters1. We define the hyperparameters to be the log of the variables in equation (4) since these are positive scale-parameters. The covariance function is made up of three parts; the first term, a linear regression term (involving ao and aI) and a noise term V1b(i, j). The first term expresses the idea that cases with nearby inputs will have highly correlated outputs; the WI parameters allow a different distance measure for each input dimension. For irrelevant inputs, the corresponding WI will become small, and the model will ignore that input. This is closely related to the Automatic Relevance Determination (ARD) idea of MacKay and Neal (MacKay, 1993; Neal 1995). The Vo variable gives the overall scale of the local correlations. This covariance function is valid for all input dimensionalities as compared to splines, where the integrated squared mth derivative is only a valid regularizer for 2m > d (see Wahba, 1990). ao and a1 are variables controlling the scale the of bias and linear contributions to the covariance. The last term accounts for the noise on the data; VI is the variance of the noise. Given a covariance function, the log likelihood of the training data is given by 1= - ~ logdet I< ~tT I<-lt - !!.log27r. (4) 222 In section 3 we will discuss how the hyperparameters III C can be adapted, in response to the training data. 2.2 RELATIONSHIP TO PREVIOUS WORK The Gaussian process view provides a unifying framework for many regression methods. ARMA models used in time series analysis and spline smoothing (e.g. Wahba, 1990 and earlier references therein) correspond to Gaussian process prediction with 1 We call () the hyperparameters as they correspond closely to hyperparameters in neural networks; in effect the weights have been integrated out exactly. Gaussian Processes for Regression 517 a particular choice of covariance function2 . Gaussian processes have also been used in the geostatistics field (e.g. Cressie, 1993), and are known there as "kriging", but this literature has concentrated on the case where the input space is two or three dimensional, rather than considering more general input spaces. This work is similar to Regularization Networks (Poggio and Girosi, 1990; Girosi, Jones and Poggio, 1995), except that their derivation uses a smoothness functional rather than the equivalent covariance function. Poggio et al suggested that the hyperparameters be set by cross-validation. The main contributions of this paper are to emphasize that a maximum likelihood solution for 8 is possible, to recognize the connections to ARD and to use the Hybrid Monte Carlo method in the Bayesian treatment (see section 3). 3 TRAINING A GAUSSIAN PROCESS The partial derivative of the log likelihood of the training data I with respect to all the hyperparameters can be computed using matrix operations, and takes time O( n 3 ) . In this section we present two methods which can be used to adapt the hyperparameters using these derivatives. 3.1 MAXIMUM LIKELIHOOD In a maximum likelihood framework, we adjust the hyperparameters so as to maximize that likelihood of the training data. We initialize the hyperparameters to random values (in a reasonable range) and then use an iterative method, for example conjugate gradient, to search for optimal values of the hyperparameters. Since there are only a small number of hyperparameters (d + 4) a relatively small number of iterations are usually sufficient for convergence. However, we have found that this approach is sometimes susceptible to local minima, so it is advisable to try a number of random starting positions in hyperparameter space. 3.2 INTEGRATION VIA HYBRID MONTE CARLO According to the Bayesian formalism, we should start with a prior distribution P( 8) over the hyperparameters which is modified using the training data D to produce a posterior distribution P(8ID). To make predictions we then integrate over the posterior; for example, the predicted mean y( x) for test input x is given by y(x) = J Y8(x)P(8I D)d8 (5) where Y8( x) is the predicted mean (as given by equation 1) for a particular value of 8. It is not feasible to do this integration analytically, but the Markov chain Monte Carlo method of Hybrid Monte Carlo (HMC) (Duane et ai, 1987) seems promising for this application. We assign broad Gaussians priors to the hyperparameters, and use Hybrid Monte Carlo to give us samples from the posterior. HMC works by creating a fictitious dynamical system in which the hyperparameters are regarded as position variables, and augmenting these with momentum variables p. The purpose of the dynamical system is to give the hyperparameters "inertia" so that random-walk behaviour in 8-space can be avoided. The total energy, H, of the system is the sum of the kinetic energy, J{, (a function of the momenta) and the potential energy, E. The potential energy is defined such that p(8ID) ex: exp(-E). We sample from the joint distribution for 8 and p given by p(8,p) ex: exp(-E2Technically splines require generalized covariance functions. 518 C. K. I. WILUAMS, C. E. RASMUSSEN I<); the marginal of this distribution for 8 is the required posterior. A sample of hyperparameters from the posterior can therefore be obtained by simply ignoring the momenta. Sampling from the joint distribution is achieved by two steps: (i) finding new points in phase space with near-identical energies H by simulating the dynamical system using a discretised approximation to Hamiltonian dynamics, and (ii) changing the energy H by doing Gibbs sampling for the momentum variables. Hamiltonian Dynamics Hamilton's first order differential equations for H are approximated by a discrete step (specifically using the leapfrog method). The derivatives of the likelihood (equation 4) enter through the derivative of the potential energy. This proposed state is then accepted or rejected using the Metropolis rule depending on the final energy H* (which is not necessarily equal to the initial energy H because of the discretization). The same step size c is used for all hyperparameters, and should be as large as possible while keeping the rejection rate low. Gibbs Sampling for Momentum Variables The momentum variables are updated using a modified version of Gibbs sampling, thereby allowing the energy H to change. A "persistence" of 0.95 is used; the new value of the momentum is a weighted sum of the previous value (with weight 0.95) and the value obtained by Gibbs sampling (weight (1 - 0.952)1/2). With this form of persistence, the momenta change approximately twenty times more slowly, thus increasing the "inertia" of the hyperparameters, so as to further help in avoiding random walks. Larger values of the persistence will further increase the inertia, but reduce the rate of exploration of H . Practical Details The priors over hyperparameters are set to be Gaussian with a mean of -3 and a standard deviation of 3. In all our simulations a step size c = 0.05 produced a very low rejection rate « 1 %). The hyperparameters corresponding to V1 and to the WI ' S were initialised to -2 and the rest to O. To apply the method we first rescale the inputs and outputs so that they have mean of zero and a variance of one on the training set. The sampling procedure is run for the desired amount of time, saving the values of the hyperparameters 200 times during the last two-thirds of the run. The first third of the run is discarded; this "burn-in" is intended to give the hyperparameters time to come close to their equilibrium distribution. The predictive distribution is then a mixture of 200 Gaussians. For a squared error loss, we use the mean of this distribution as a point estimate. The width of the predictive distribution tells us the uncertainty of the prediction. 4 EXPERIMENTAL RESULTS We report the results of prediction with Gaussian process on (i) a modified version of MacKay's robot arm problem and (ii) five real-world data sets. 4.1 THE ROBOT ARM PROBLEM We consider a version of MacKay's robot arm problem introduced by Neal (1995). The standard robot arm problem is concerned with the mappings Y1 = r1 cos Xl + r2 COS(X1 + X2) Y2 = r1 sin Xl + r2 sin(x1 + X2) (6) Gaussian Processes for Regression 519 Method No. of inputs sum squared test error Gaussian process 2 1.126 Gaussian process 6 1.138 MacKay 2 1.146 Neal 2 1.094 Neal 6 1.098 Table 1: Results on the robot arm task. The bottom three lines of data were obtained from Neal (1995) . The MacKay result is the test error for the net with highest "evidence". The data was generated by picking Xl uniformly from [-1.932, -0.453] and [0.453, 1.932] and picking X2 uniformly from [0.534, 3.142]. Neal added four further inputs, two of which were copies of Xl and X2 corrupted by additive Gaussian noise of standard deviation 0.02, and two further irrelevant Gaussian-noise inputs with zero mean and unit variance. Independent zero-mean Gaussian noise of variance 0.0025 was then added to the outputs YI and Y2 . We used the same datasets as Neal and MacKay, with 200 examples in the training set and 200 in the test set. The theory described in section 2 deals only with the prediction of a scalar quantity Y , so predictors were constructed for the two outputs separately, although a joint prediction is possible within the Gaussian process framework (see co-kriging, §3.2.3 in Cressie, 1993). Two experiments were conducted, the first using only the two "true" inputs, and the second one using all six inputs. In this section we report results using maximum likelihood training; similar results were obtained with HMC. The log( v),s and loge w )'s were all initialized to values chosen uniformly from [-3.0, 0.0], and were adapted separately for the prediction of YI and Y2 (in these early experiments the linear regression terms in the covariance function involving aa and al were not present) . The conjugate gradient search algorithm was allowed to run for 100 iterations, by which time the likelihood was changing very slowly. Results are reported for the run which gave the highest likelihood of the training data, although in fact all runs performed very similarly. The results are shown in Table 1 and are encouraging, as they indicate that the Gaussian process approach is giving very similar performance to two well-respected techniques. All of the methods obtain a level of performance which is quite close to the theoretical minimum error level of 1.0 . ...Jt is interesting to look at the values of the w's obtained after the optimization; for the Y2 task the values were 0.243,0.237,0.0639,7.0 x 10-4 , 2.32 x 10- 6 ,1.70 x 10- 6 , and Va and VI were 7.5278 and 0.0022 respectively. The w values show nicely that the first two inputs are the most important, followed by the corrupted inputs and then the irrelevant inputs. During training the irrelevant inputs are detected quite quickly, but the w's for the corrupted inputs shrink more slowly, implying that the input noise has relatively little effect on the likelihood. 4.2 FIVE REAL-WORLD PROBLEMS Gaussian Processes as described above were compared to several other regression algorithms on five real-world data sets in (Rasmussen, 1996; in this volume). The data sets had between 80 and 256 training examples, and the input dimension ranged from 6 to 16. The length of the HMC sampling for the Gaussian processes was from 7.5 minutes for the smallest training set size up to 1 hour for the largest ones on a R4400 machine. The results rank the methods in the order (lowest error first) a full-blown Bayesian treatment of neural networks using HMC, Gaussian 520 C. K. I. WILLIAMS, C. E. RASMUSSEN processes, ensembles of neural networks trained using cross validation and weight decay, the Evidence framework for neural networks (MacKay, 1992), and MARS. We are currently working on assessing the statistical significance of this ordering. 5 DISCUSSION We have presented the method of regression with Gaussian processes, and shown that it performs well on a suite of real-world problems. We have also conducted some experiments on the approximation of neural nets (with a finite number of hidden units) by Gaussian processes, although space limitations do not allow these to be described here. Some other directions currently under investigation include (i) the use of Gaussian processes for classification problems by softmaxing the outputs of k regression surfaces (for a k-class classification problem), (ii) using non-stationary covariance functions, so that C(x, Xl) f:- C(lx - XII) and (iii) using a covariance function containing a sum of two or more terms of the form given in line 1 of equation 3. We hope to make our code for Gaussian process prediction publically available in the near future. Check http://www.cs.utoronto.ca/neuron/delve/delve.html for details. Acknowledgements We thank Radford Neal for many useful discussions, David MacKay for generously providing the robot arm data used in this paper, and Chris Bishop, Peter Dayan, Radford Neal and Huaiyu Zhu for comments on earlier drafts. CW was partially supported by EPSRC grant GRjJ75425. References Cressie, N. A. C. (1993). Statistics for Spatial Data. Wiley. Duane, S., Kennedy, A. D., Pendleton, B. J., and Roweth, D. (1987). Hybrid Monte Carlo. Physics Letters B, 195:216-222. Girosi, F., Jones, M., and Poggio, T. (1995). Regularization Theory and Neural Networks Architectures. Neural Computation, 7(2):219-269. MacKay, D. J. C. (1992). A Practical Bayesian Framework for Backpropagation Networks. Neural Computation, 4(3):448-472. MacKay, D. J. C. (1993). Bayesian Methods for Backpropagation Networks. In van Hemmen, J. L., Domany, E., and Schulten, K., editors, Models of Neural Networks II. Springer. Neal, R. M. (1993). Bayesian Learning via Stochastic Dynamics. In Hanson, S. J., Cowan, J. D., and Giles, C. L., editors, Neural Information Processing Systems, Vol. 5, pages 475-482. Morgan Kaufmann, San Mateo, CA. Neal, R. M. (1995). Bayesian Learning for Neural Networks. PhD thesis, Dept. of Computer Science, University of Toronto. Poggio, T. and Girosi, F. (1990). Networks for approximation and learning. Proceedings of IEEE, 78:1481-1497. Rasmussen, C. E. (1996). A Practical Monte Carlo Implementation of Bayesian Learning. In Touretzky, D. S., Mozer, M. C., and Hasselmo, M. E., editors, Advances in Neural Information Processing Systems 8. MIT Press. Wahba, G. (1990). Spline Models for Observational Data. Society for Industrial and Applied Mathematics. CBMS-NSF Regional Conference series in applied mathematics.	1995	73
1,121	Reorganisation of Somatosensory Cortex after Tactile Training Rasmus S. Petersen John G. Taylor Centre for Neural Networks, King's College London Strand, London WC2R 2LS, UK Abstract Topographic maps in primary areas of mammalian cerebral cortex reorganise as a result of behavioural training. The nature of this reorganisation seems consistent with the behaviour of competitive neural networks, as has been demonstrated in the past by computer simulation. We model tactile training on the hand representation in primate somatosensory cortex, using the Neural Field Theory of Amari and his colleagues. Expressions for changes in both receptive field size and magnification factor are derived, which are consistent with owl monkey experiments and make a prediction which goes beyond them. 1. INTRODUCTION The primary cortical areas of mammals are now known to be plastic throughout life; reviewed recently by Kaas(1995). The problem of how and why the underlying learning processes work is an exciting one, for which neural network modelling appears well suited. In this contribution, we model the long-term effects of tactile training (Jenkins et ai, 1990) on the functional organisation of monkey primary somatosensory cortex, by perturbing a topographic net (Takeuchi and Amari, 1979). 1.1 ADAPTATION IN ADULT SOMATOSENSORY CORTEX Light touch activates skin receptors which in primates are mapped, largely topographically, in area 3b. In a series of papers, Merzenich and colleagues describe how area 3b becomes reorganised following peripheral nerve damage (Merzenich et ai, 1983a; 1983b) or digit amputation (Merzenich et ai, 1984). The underlying learning processes may also explain the phenomenon of phantom limb "telescoping" (Haber, 1955). Recent advances in brain scanning are beginning to make them observable even in the human brain (Mogilner et ai, 1993). 1.2 ADAPTATION ASSOCIATED WITH TACTILE TRAINING Jenkins et al trained owl monkeys to maintain contact with a rotating disk. The apparatus was arranged so that success eventually involved touching the disk with only the digit tips. Hence these regions received selective stimulation. Some time after training had been completed electro-physiological recordings were made from area 3b. These revealed an increase in Magnification Factor (MF) for the stimulated skin and a decrease in Reorganization of Somatosensory Cortex after Tactile Training 83 the size of Receptive Fields (RFs) for that region. The net territory gained for light touch of the digit tips came from area 3a and/or the face region of area 3b, but details of any changes in these representations were not reported. 2. THEORETICAL FRAMEWORK 2.1 PREVIOUS WORK Takeuchi and Amari(1979), Ritter and Schulten(1986), Pearson et al(1987) and Grajski and Merzenich( 1990) have all modelled amputationldenervation by computer simulation of competitive neural networks with various Hebbian weight dynamics. Grajski and Merzenich(1990) also modelled the data of Jenkins et al. We build on this research within the Neural Field Theory framework (Amari, 1977; Takeuchi and Amari, 1979; Amari, 1980) of the Neural Activity Model of Willshaw and von der Malsburg(1976). 2.2 NEURAL ACTIVITY MODEL Consider a "cortical" network of simple, laterally connected neurons. Neurons sum inputs linearly and output a sigmoidal function of this sum. The lateral connections are excitatory at short distances and inhibitory at longer ones. Such a network is competitive: the steady state consists of blobs of activity centred around those neurons locally receiving the greatest afferent input (Amari, 1977). The range of the competition is limited by the range of the lateral inhibition. Suppose now that the afferent synapses adapt in a Hebbian manner to stimuli that are localised in the sensory array; the lateral ones are fixed. Willshaw and von der Malsburg(1976) showed by computer simulation that this network is able to form a topographic map of the sensory array. Takeuchi and Amari( 1979) amended the WillshawMalsburg model slightly: neurons possess an adaptive firing threshold in order to prevent synaptic weight explosion, rather than the more usual mechanism of weight normalisation. They proved that a topographic mapping is stable under certain conditions. 2.3 TAKEUCHI-AMARI THEORY Consider a one-dimensional model. The membrane dynamics are: au(~y,t) = -u(x,y,t)+ f s(x,y' ,t)a(y- y')dy'so(x,t)ao + f w(x-x')f[u(x' ,y,t)]dx'-h (1) Here u(x,y,t) is the membrane potential at time I for point x when a stimulus centred at y is being presented; h is a positive resting potential; w(z) is the lateral inhibitory weight between two points in the neural field separated by a distance z - positive for small Izl and negative for larger Izl; s(x,y,t) is the excitatory synaptic weight from y to x at time I and sr/X,I) is an inhibitory weight from a tonically active inhibitory input aD to x at time t - it is the adaptive firing threshold. f[u] is a binary threshold function that maps positive membrane potentials to 1 and non-positive ones to O. Idealised, point-like stimuli are assumed, which "spread out" somewhat on the sensory surface or subcortically. The spreading process is assumed to be independent of y and is described in the same coordinates. It is represented by the function a(y-y'), which describes the effect of a point input at y spreading to the point y'. This is a decreasing, positive, symmetric function of Iy-y'l. With this type of input, the steady-state activity of the network is a single blob, localised around the neuron with maximum afferent input. 84 R. S. PETERSEN, J. O. TAYLOR The afferent synaptic weights adapt in a leaky Hebbian manner but with a time constant much larger than that of the membrane dynamics (1). Effectively this means that learning occurs on the steady state of the membrane dynamics. The following averaged weight dynamics can be justified (Takeuchi and Amari, 1979; Geman 1979): as( x, y, t) ( J) ( [A )] at =-s x,y,t)+b p(y' a Y-Y')f u(x,y' dy' aso(~y,t) =-so(x,y,t)+b' aoJ p(y')f[u(x,y')]dy' (2) where r1(x,y') is the steady-state of the membrane dynamics at x given a stimulus at y' and p(y') is the probability of a stimulus at y '; b, b' are constants. Empirically, the "classical" Receptive Field (RF) of a neuron is defined as the region of the input field within which localised stimulation causes change in its activity. This concept can be modelled in neural field theory as: the RF of a neuron at x is the portion of the input field within which a stimulus evokes a positive membrane potential (inhibitory RFs are not considered). If the neural field is a continuous map of the sensory surface then the RF of a neuron is fully described by its two borders rdx), rix), defined formally: i = 1,2 (3) which are illustrated in figure 1. Let RF size and RF position be denoted respectively by the functions rex) and m(x), which represent experimentally measurable quantities. In terms of the border functions they can be expressed: r(x) = r2 (x) - r1 (x) m(x) = -} (rl {x} + r2 (x)) y ~--------------------------- x (4) Figure 1: RF boundaries as a function of position in the neural field, for a topographically ordered network. Only the region in-between rdx) and rix) has positive steadystate membrane potential r1(x,y). rdx) and rix) are defined by the condition r1(x,r;(x))=O i=J,2. for Using (1), (2) and the definition (3), Takeuchi and Amari(1979) derived dynamical equations for the change in RF borders due to learning. In the case of uniform stimulus probability, they found solutions for the steady-state RF border functions. With periodic boundary conditions, the basic solution is a linear map with constant RF size: Reorganization of Somatosensory Cortex after Tactile Training r(x) = ro = const m(x) = px ++ro uni ( ) rl x = px r~tni (x) = px+ ro 85 (5) This means that both RF size and activity blob size are uniform across the network and that RF position m(x) is a linear function of network location. (The value of p is determined by boundary conditions; ro is then determined from the joint equilibrium of (I), (2». The inverse of the RF position function, denoted by m-l(y), is the centre of the cortical active region caused by a stimulus centred at y. The change in m-l(y) over a unit interval in the input field is, by empirical definition, the cortical magnification factor (MF). Here we model MF as the rate of change of m-l(y). The MF for the system described by (5) is: d _I () -I -m y =p dy (6) 3. ANALYSIS OF TACTILE TRAINING 3.1 TRAINING MODEL AND ASSUMPTIONS Jenkins et aI's training sessions caused an increase in the relative frequency of stimulation to the finger tips, and hence a decrease in relative frequency of stimulation elsewhere. Over a long time, we can express this fact as a localised change in stimulus probability (figure 2). (This is not sufficient to cause cortical reorganisation - Recanzone et al( 1992) showed that attention to the stimulation is vital. We consider only attended stimulation in this model). To account for such data it is clearly necessary to analyse non-uniform stimulus probabilities, which demands extending the results of Takeuchi and Amari. Unfortunately, it seems to be hard to obtain general results. However, a perturbation analysis around the uniform probability solution (5) is possible. To proceed in this way, we must be able to assume that the change in the stimulus probability density function away from uniformity is small. This reasoning is expressed by the following equation: p(y) = Po + E p(y) (7) where pry) is the new stimulus probability in terms of the uniform one and a perturbation due to training: E is a small constant. The effect of the perturbation is to ease the weight dynamics (2) away from the solution (5) to a new steady-state. Our goal is to discover the effect of this on the RF border functions, and hence for RF size and MF. p(y) o y Figure 2: The type of change in stimulus probability density that we assume to model the effects of behavioural training. 86 R. S. PETERSEN, J. G. TAYLOR 3.2 PERTURBATION ANALYSIS 3.2.1 General Case For a small enough perturbation, the effect on the RF borders and on the activity blob size ought also to be small. We consider effects to first order in E, seeking new solutions of the form: i = 1,2 ,{x} = r; {x} - ~ {x} m{x} = +(~ (X}+'2 (x}) (8) where the superscript peT denotes the new, perturbed equilibrium and uni denotes the unperturbed, uniform probability equilibrium. Using (1) and (2) in (3) for the post-training RF borders, expanding to first order in E, a pair of difference equations may be obtained for the changes in RF borders. It is convenient to define the following terms: ro rt '(x) At (x) = J p(y+ px)k(y)dy-b' a~ J p(y)dy o r,"no (x) o r;-n' (x) A2 {x} = J p(y + px + TO )k(y)dy - b' a~ J p(y)dy k(y) = b J a(y - y' )a(y' )dy' (9) B = b' a~p() -k(ro)po > 0 C= w(p-tTo)p-t <0 where the signs of Band C arise due to stability conditions (Amari, 1977; Takeuchi and Amari, 1979). In terms of RF size and RF position (4), the general result is: B~2 ,(X} = ~(~ + I)At (x) - M2 (x) BC~2m{X) = (B- C -+ C~)(~+ I}At (x) + (C- B++(C -2B)~)A2 (x) (10) where ~ is the difference operator: ~ f{ x) = f( x + p - t To) - f( x) (11 ) 3.2.2 Particular Case The second order difference equations (l0) are rather opaque. This is partly due to coupling in y caused by the auto-correlation function key): (10) simplifies considerably if very narrow stimuli are assumed - a(y)=O(y) (see also Amari, 1980). For periodic boundary conditions: (12) where: Reorganization of Somatosensory Cortex after Tactile Training 87 m -I P(W (y) = m -I pre (y) + Em -I (y) = p-l(y_+ro)+Em-l(y) (13) and we have used the crude approximation: d _() 1 ( 1 _I ) dx m x "" t;: ~m x - 2" P ro (14) which demands smoothness on the scale of 10. However, for perturbations like that sketched in figure 2, this is sufficient to tell us about the constant regions of MF. (We would not expect to be able to model the data in the transition region in any case, as its form is too dependent upon fine detail of the model). Our results (12) show that the change in RF size of a neuron is simply minus the total change in stimulus probability over its RF. Hence RF size decreases where p(y) increases and vice versa. Conversely, the change in MF at a given stimulus location is roughly the local average change in stimulus probability there. Note that changes in RF size correlate inversely with changes in MF. Figure 3 is a sketch of these results for the perturbation of figure 2. MF RF o \ I I o L.J y Figure 3: Results of perturbation analysis for how behavioural training (figure 2) changes RF size and MF respectively, in the case where stimulus width can be neglected. For MF - due to the approximation (14) - predictions do not apply near the transitions. 4. DISCUSSION Equations (12) are the results of our model for RF size and MF after area 3b has fully adapted to the behavioural task, in the case where stimulus width can be neglected. They appear to be fully consistent with the data of Jenkins et al described above: RF size decreases in the region of cortex selective for the stimulated body part and the MF for this body part increases. Our analysis also makes a specific prediction that goes beyond Jenkins et aI's data, directly due to the inverse relationship between changes in RF size and those in MF. Within the regions that surrender territory to the entrained finger tips (sometimes the face region), for which MF decreases, RF sizes should increase. Surprisingly perhaps, these changes in RF size are not due to adaptation of the afferent weights s(x,y). The changes are rather due to the adaptive threshold term six). This point will be discussed more fully elsewhere. A limitation of our analysis is the assumption that the change in stimulus probability is in some sense small. Such an approximation may be reasonable for behavioural training but seems less so as regards important experimental protocols like amputation or denervation. Evidently a more general analysis would be highly desirable. 88 R. S. PETERSEN,J. O. TAYLOR 5. CONCLUSION We have analysed a system with three interacting features: lateral inhibitory interactions; Hebbian adaptivity of afferent synapses and an adaptive firing threshold. Our results indicate that such a system can account for the data of Jenkins et aI, concerning the response of adult somatosensory cortex to the changing environmental demands imposed by tactile training. The analysis also brings out a prediction of the model, that may be testable. Acknowledgements RSP is very grateful for a travel stipend from the NIPS Foundation and for a Nick Hughes bursary from the School of Physical Sciences and Engineering, King's College London, that enabled him to participate in the conference. References Amari S. (1977) BioI. Cybern. 2777-87 Amari S. (1980) Bull. Math. Biology 42339-364 Geman S. (1979) SIAM 1. App. Math. 36 86-105 Grajski K.A., Merzenich M.M. (1990) in Neural Information Processing Systems 2 Touretzky D.S. (Ed) 52-59 HaberW.B. (1955)1. Psychol. 40115-123 Jenkins W.M., Merzenich M.M., Ochs M.T., Allard T., Gufc-Robles E. (1990) 1. Neurophysiol. 63 82-104 Kaas J.H. (1995) in The Cognitive Neurosciences Gazzaniga M.S. (Ed ic) 51-71 Merzenich M.M., Kaas J.H., Wall J.T., Nelson R.J., Sur M., Felleman DJ. (1983a) Neuroscience 8 35-55 Merzenich M.M., Kaas J.H., Wall J.T., Sur M., Nelson R.I., Felleman DJ. (1983b) Neuroscience 10639-665 Merzenich M.M., Nelson R.I., Stryker M.P., Cynader M.S., Schoppmann A., Zook J.M. (1984) 1. Compo Neural. 224591-605 Mogilner A., Grossman A.T., Ribrary V., Joliot M., Vol mann J., Rapaport D., Beasley R., L1inas R. (1993) Proc. Natl. Acad. Sci. USA 90 3593-3597 Pearson J.e., Finkel L.H., Edelman G.M. (1987) 1. Neurosci. 124209-4223 Recanzone G.H., Merzenich M.M., Jenkins W.M., Grajski K.A., Dinse H.R. (1992) 1. Neurophysiol. 67 1031-1056 Ritter H., Schulten K. (1986) BioI. Cybern. 5499-106 Takeuchi A., Amari S. (1979) BioI. Cybern. 35 63-72 Willshaw DJ., von der Malsburg e. (1976) Proc. R. Soc. Lond. B194 203-243	1995	74
1,122	Prediction of Beta Sheets in Proteins Anders Krogh The Sanger Centre Hinxton, Carobs CBIO IRQ, UK. Email: krogh@sanger.ac. uk S~ren Kamaric Riis Electronics Institute, Building 349 Technical University of Denmark 2800 Lyngby, Denmark Email: riis@ei.dtu.dk Abstract Most current methods for prediction of protein secondary structure use a small window of the protein sequence to predict the structure of the central amino acid. We describe a new method for prediction of the non-local structure called ,8-sheet, which consists of two or more ,8-strands that are connected by hydrogen bonds. Since,8strands are often widely separated in the protein chain, a network with two windows is introduced. After training on a set of proteins the network predicts the sheets well, but there are many false positives. By using a global energy function the ,8-sheet prediction is combined with a local prediction of the three secondary structures a-helix, ,8-strand and coil. The energy function is minimized using simulated annealing to give a final prediction. 1 INTRODUCTION Proteins are long sequences of amino acids. There are 20 different amino acids with varying chemical properties, e. g. , some are hydrophobic (dislikes water) and some are hydrophilic [1]. It is convenient to represent each amino acid by a letter and the sequence of amino acids in a protein (the primary structure) can be written as a string with a typical length of 100 to 500 letters. A protein chain folds back on itself, and the resulting 3D structure (the tertiary structure) is highly correlated to the function of the protein. The prediction of the 3D structure from the primary structure is one of the long-standing unsolved problems in molecular biology. As an important step on the way a lot of work has been devoted to predicting the local conformation of the protein chain, which is called the secondary structure. Neural network methods are currently the most successful for predicting secondary structure. The approach was pioneered by Qian and Sejnowski [2] and Bohr et al. [3], but later extended in various ways, see e.g. [4] for an overview. In most of this work, only the two regular secondary structure elements a-helix and ,8-strand are being distinguished, and everything else is labeled coil. Thus, the methods based 918 A. KROGH, S. K. RIIS H-\ H-\ t !" o=c ,,=c {-~ {-H \ ' /=0 HH-\ f fa o=c /o=c ~-o ~-: Figure 1: Left: Anti-parallel,B-sheet. The vertical lines correspond to the backbone of the protein. An amino acid consists of N-Ca-C and a side chain on the Ca that is not shown (the 20 amino acids are distinguished by different side chains). In the anti-parallel sheet the directions of the strands alternate, which is here indicated quite explicitly by showing the middle strand up-side down. The H-bonds between the strands are shown by 11111111. A sheet has two or more strands, here the antiparallel sheet is shown with three strands. Right: Parallel ,B-sheet consisting of two strands. on a local window of amino acids give a three-state prediction of the secondary structure of the central amino acid in the window. Current predictions of secondary structure based on single sequences as input have accuracies of about 65-66%. It is widely believed that this accuracy is close to the limit of what can be done from a local window (using only single sequences as input) [5], because interactions between amino acids far apart in the protein chain are important to the structure. A good example of such non-local interactions are the ,B-sheets consisting of two or more ,B-strands interconnected by H-bonds, see fig. 1. Often the ,B-strands in a sheet are widely separated in the sequence, implying that only part of the available sequence information about a ,B-sheet can be contained in a window of, say, 13 amino acids. This is one of the reasons why the accuracy of ,B-strand predictions are generally lower than the accuracy of a-helix predictions. The aim of this work is to improve prediction of secondary structures by combining local predictions of a-helix, ,B-strand and coil with a non-local method predicting ,B-sheets. Other work along the same directions include [6] in which ,B-sheet predictions are done by linear methods and [7] where a so-called density network is applied to the problem. 2 A NEURAL NETWORK WITH TWO WINDOWS We aim at capturing correlations in the ,B-sheets by using a neural network with two windows, see fig. 2. While window 1 is centered around amino acid number i (ai), window 2 slides along the rest of the chain. When the amino acids centered in each of the two windows sit opposite each other in a ,B-sheet the target output is 1, and otherwise O. After the whole protein has been traversed by window 2, window 1 is moved to the next position (i + 1) and the procedure is repeated. If the protein is L amino acids long this procedure yields an output value for each of the L(L -1)/2 Prediction of Beta Sheets in Proteins Figure 2: Neural network for predicting ,B-sheets. The network employs weight sharing to improve the encoding of the amino acids and to reduce the number of adjustable parameters. 919 pairs of amino acids. We display the output in a L x L gray-scale image as shown in fig. 3. We assume symmetry of sheets, i.e., if the two windows are interchanged, the output does not change. This symmetry is ensured (approximately) during training by presenting all inputs in both directions. Each window of the network sees K amino acids. An amino acid is represented by a vector of20 binary numbers all being zero, except one, which is 1. That is, the amino acid A is represented by the vector 1,0,0, ... ,0 and so on. This coding ensures that the input representations are un correlated , but it is a very inefficient coding, since 20 amino acids could in principle be represented by only 5 bit. Therefore, we use weight sharing [8] to learn a better encoding [4]. The 20 input units corresponding to one window position are fully connected to three hidden units. The 3 x (20 + 1) weights to these units are shared by all window positions, i.e., the activation of the 3 hidden units is a new learned encoding of the amino acids, so instead of being represented by 20 binary values they are represented by 3 real values. Of course the number of units for this encoding can be varied, but initial experiments showed that 3 was optimal [4]. The two windows of the network are made the same way with the same number of inputs etc .. The first layer of hidden units in the two windows are fully connected to a hidden layer which is fully connected to the output unit, see fig. 2. Furthermore, two structurally identical networks are used: one for parallel and one for anti-parallel ,B-sheets. The basis for the training set in this study is the set of 126 non-homologous protein chains used in [9], but chains forming ,B-sheets with other chains are excluded. This leaves us with 85 proteins in our data set. For a protein of length L only a very small fraction of the L(L - 1)/2 pairs are positive examples of ,B-sheet pairs. Therefore it is very important to balance the positive and negative examples to avoid the situation where the network always predicts no ,B-sheet. Furthermore, there are several types of negative examples with quite different occurrences: 1) two amino acids of which none belong to a ,B-sheet; 2) one in a ,B-sheet and one which is not in a ,B-sheet; 3) two sitting in ,B-sheets, but not opposite to each other. The balancing was done in the following way. For each positive example selected at random a negative example from each of the three categories were selected at random. If the network does not have a second layer of hidden units, it turns out that the result is no better than a network with only one input window, i.e., the network cannot capture correlations between the two windows. Initial experiments indicated that about 10 units in the second hidden layer and two identical input windows of size K = 9 gave the best results. In fig. 3(left) the prediction of anti-parallel sheets is shown for the protein identified as 1acx in the Brookhaven Protein Data Bank 920 120 100 .. 80 :g '" o .!: ~ 60 / 40 ". 20 A. KROGH, S. K. RIIS Figure 3: Left: The prediction of anti-parallel ,8-sheets in the protein laex. In the upper triangle the correct structure is shown by a black square for each ,8-sheet pair. The lower triangle shows the prediction by the two-window network. For any pair of amino acids the network output is a number between zero (white) and one (black), and it is displayed by a linear gray-scale. The diagonal shows the prediction of a-helices. Right: The same display for parallel ,8-sheets in the protein 4fxn. Notice that the correct structure are lines parallel to the diagonal, whereas they are perpendicular for anti-parallel sheets. For both cases the network was trained on a training set that did not contain the protein for which the result is shown. [10]. First of all, one notices the checker board structure of the prediction of ,8sheets. This is related to the structure of ,8-sheets. Many sheets are hydrophobic on one side and hydrophilic on the other. The side chains of the amino acids in a strand alternates between the two sides of the sheet, and this gives rise to the periodicity responsible for the pattern. Another network was trained on parallel ,8-sheets. These are rare compared to the anti-parallel ones, so the amount of training data is limited. In fig. 3(right) the result is shown for protein 4fxn. This prediction seems better than the one obtained for anti-parallel sheets, although false positive predictions still occurs at some positions with strands that do not pair. Strands that bind in parallel ,8-sheets are generally more widely separated in the sequence than strands in anti-parallel sheets. Therefore, one can imagine that the strands in parallel sheets have to be more correlated to find each other in the folding process, which would explain the better prediction accuracy. The results shown in fig. 3 are fairly representative. The network misses some of the sheets, but false positives present a more severe problem. By calculating correlation coefficients we can show that the network doe!> capture some correlations, but they seem to be weak. Based on these results, we hypothesize that the formation of ,8sheets is only weakly dependent on correlations between corresponding ,8-strands. This is quite surprising. However weak these correlations are, we believe they can still improve the accuracy of the three state secondary structure prediction. In order to combine local methods with the non-local ,8-sheet prediction, we introduce a global energy function as described below. Prediction of Beta Sheets in Proteins 921 3 A GLOBAL ENERGY FUNCTION We use a newly developed local neural network method based on one input window [4] to give an initial prediction of the three possible structures. The output from this network is constrained by soft max [11], and can thus be interpreted as the probabilities for each of the three structures. That is, for amino acid ai, it yields three numbers Pi,n, n = 1,2 or 3 indicating the probability of a-helix (Pi,l) , (3sheet (pi,2), or coil (pi,3). Define Si,n = 1 if amino acid i is assigned structure n and Si,n = 0 otherwise. Also define hi,n = 10gPi,n. We now construct the 'energy function' (1) i n where weights Un are introduced for later usage. Assuming the probabilities Pi,n are independent for any two amino acids in a sequence, this is the negative log likelihood of the assigned secondary structure represented by s, provided that Un = 1. As it stands, alone, it is a fairly trivial energy function, because the minimum is the assignment which corresponds to the prediction with the maximum Pi,n at each position i-the assignment of secondary structure that one would probably use anyway. For amino acids ai and aj the logarithm of the output of the (3-sheet network described previously is called qfj for parallel (3-sheets and qfj for anti-parallel sheets. We interpret these numbers as the gain in energy if a (3-sheet pair is formed. (As more terms are added to the energy, the interpretation as a log-likelihood function is gradually fading.) If the two amino acids form a pair in a parallel (3-sheet, we set the variable T~ equal to 1, and otherwise to 0, and similarly with Tii for antiparallel sheets. Thus the Tii and T~ are sparse binary matrices. Now the total energy of the (3-sheets can be expressed as Hf3(s, Ta, TP) = ~[CaqfjTij + CpqfjT~], (2) 'J where Ca and Cp determine the weights of the two terms in the function. Since an amino acid can only be in one structure, the dynamic T and S variables are constrained: Only Tii or T~ can be 1 for the same (i, j), and if any of them is 1 the amino acids involved must be in a (3-sheet, so Si,2 = Sj,2 = 1. Also, Si ,2 can only be 1 if there exists a j with either Iii or T~ equal to 1. Because of these constraints we have indicated an S dependence of H f3. The last term in our energy function introduces correlations between neighboring amino acids. The above assumption that the secondary structure of the amino acids are independent is of course a bad assumption, and we try to repair it with a term Hn(s) = L: L: Jnm Si,n Si+l,m, i nm (3) that introduces nearest neighbor interactions in the chain. A negative J11, for instance, means that a following a is favored, and e.g., a positive h2 discourages a (3 following an a. Now the total energy is (4) Since (3-sheets are introduced in two ways, through hi ,2 and qij, we need the weights Un in (1) to be different from 1. The total energy function (4) has some resemblance with a so-called Potts glass in an external field [12]. The crucial difference is that the couplings between the 922 A. KROGH, S. K. RIIS 'spins' Si are dependent on the dynamic variables T. Another analogy of the energy function is to image analysis, where couplings like the T's are sometimes used as edge elements. 3.1 PARAMETER ESTIMATION The energy function contains a number of parameters, Un, Ca , Cp and Jnm . These parameters were estimated by a method inspired by Boltzmann learning [13]. In the Boltzmann machine the estimation of the weights can be formulated as a minimization of the difference between the free energy of the 'clamped' system and that of the 'free-running' system [14]. If we think of our energy function as a free energy (at zero temperature), it corresponds to minimizing the difference between the energy of the correct protein structure and the minimum energy, where p is the total number of proteins in the training set. Here the correct structure of protein J-l is called S(J-l) , Ta(J-l), TP(p), whereas s(J-l), Ta(J-l) , TP(J-l) represents the structure that minimizes the energy Htotal. By definition the second term of C is less than the first, so C is bounded from below by zero. The cost function C is minimized by gradient descent in the parameters. This is in principle straightforward, because all the parameters appear linearly in Htotal. However, a problem with this approach is that C is minimal when all the parameters are set to zero, because then the energy is zero. It is cured by constraining some of the parameters in Htotal. We chose the constraint l:n Un = 1. This may not be the perfect solution from a theoretical point of view, but it works well. Another problem with this approach is that one has to find the minimum of the energy Htotal in the dynamic variables in each iteration of the gradient descent procedure. To globally minimize the function by simulated annealing each time would be very costly in terms of computer time. Instead of using the (global) minimum of the energy for each protein, we use the energy obtained by minimizing the energy from the correct structure. This minimization is done by a greedy algorithm in the following way. In each iteration the change in s, Ta, TP which results in the largest decrease in Htotal is carried out. This is repeated until any change will increase Htotal. This algorithm works towards a local stability of the protein structures in the training set. We believe it is not only an efficient way of doing it, but also a very sensible way. In fact, the method may well be applicable in other models, such as Boltzmann machines. 3.2 STRUCTURE PREDICTION BY SIMULATED ANNEALING After estimation of the parameters on which the energy function Htotal depends, we can proceed to predict the structure of new proteins. This was done using simulated annealing and the EBSA package [15]. The total procedure for prediction is, 1. A neural net predicts a-helix, ,8-strand or coil. The logarithm of these predictions give all the hi,n for that protein. 2. The two-window neural networks predict the ,8-sheets. The result is the qfj from one network and the qfj from the other. 3. A random configuration of S, Ta, TP variables is generated from which the simulated annealing minimization of Htotal was started. During annealing, all constraints on s, Ta, TP variables are strictly enforced. Prediction of Beta Sheets in Proteins 923 4. The final minimum configuration s is the prediction of the secondary structure. The ,B-sheets are predicted by t a and tv. Using the above scheme, an average secondary structure accuracy of 66.5% is obtained by seven-fold cross validation. This should be compared to 66.3% obtained by the local neural network based method [4] on the same data set. Although these preliminary results do not represent a significant improvement, we consider them very encouraging for future work. Because the method not only predicts the secondary structure, but also which strands actually binds to form ,B-sheets, even a modest result may be an important step on the way to full 3D predictions. 4 CONCLUSION In this paper we introduced several novel ideas which may be applicable in other contexts than prediction of protein structure. Firstly, we described a neural network with two input windows that was used for predicting the non-local structure called ,B-sheets. Secondly, we combined local predictions of a-helix, ,B-strand and coil with the ,B-sheet prediction by minimization of a global energy function. Thirdly, we showed how the adjustable parameters in the energy function could be estimated by a method similar to Boltzmann learning. We found that correlations between ,B-strands in ,B-sheets are surprisingly weak. Using the energy function to combine predictions improves performance a little. Although we have not solved the protein folding problem, we consider the results very encouraging for future work. This will include attempts to improve the performance of the two-window network as well as experimenting with the energy function, and maybe add more terms to incorporate new constraints. Acknowledgments: We would like to thank Tim Hubbard, Richard Durbin and Benny Lautrup for interesting comments on this work and Peter Salamon and Richard Frost for assisting with simulated annealing. This work was supported by a grant from the Novo Nordisk Foundation. References [1] C. Branden and J. Tooze, Introduction to Protein Structure (Garland Publishing, Inc., New York, 1991). [2] N. Qian and T. Sejnowski, Journal of Molecular Biology 202, 865 (1988). [3] H. Bohr et al., FEBS Letters 241, 223 (1988). [4] S. Riis and A. Krogh, Nordita Preprint 95/34 S, submitted to J. Compo BioI. [5] B. Rost, C. Sander, and R. Schneider, J Mol. BioI. 235, 13 (1994). [6] T. Hubbard, in Proc. of the 27th HICSS, edited by R. Lathrop (IEEE Computer Soc. Press, 1994), pp. 336-354. [7] D. J. C. MacKay, in Maximum Entropy and Bayesian Methods, Cambridge 1994, edited by J. Skilling and S. Sibisi (Kluwer, Dordrecht, 1995). [8] Y. Le Cun et al., Neural Computation 1, 541 (1989). [9] B. Rost and C. Sander, Proteins 19, 55 (1994). [10] F. Bernstein et al., J Mol. BioI. 112,535 (1977). [11] J. Bridle, in Neural Information Processing Systems 2, edited by D. Touretzky (Morgan Kaufmann, San Mateo, CA, 1990), pp. 211-217. [12] K. Fisher and J. Hertz, Spin glasses (Cambridge University Press, 1991). [13] D. Ackley, G. Hinton, and T. Sejnowski, Cognitive Science 9, 147 (1985). [14] J. Hertz, A. Krogh, and R. Palmer, Introduction to the Theory of Neural Computation (Addison-Wesley, Redwood City, 1991). [15] R. Frost, SDSC EBSA, C Library Documentation, version 2.1. SDSC Techreport.	1995	75
1,123	Active Learning in Multilayer Perceptrons Kenji Fukumizu Information and Communication R&D Center, Ricoh Co., Ltd. 3-2-3, Shin-yokohama, Yokohama, 222 Japan E-mail: fuku@ic.rdc.ricoh.co.jp Abstract We propose an active learning method with hidden-unit reduction. which is devised specially for multilayer perceptrons (MLP). First, we review our active learning method, and point out that many Fisher-information-based methods applied to MLP have a critical problem: the information matrix may be singular. To solve this problem, we derive the singularity condition of an information matrix, and propose an active learning technique that is applicable to MLP. Its effectiveness is verified through experiments. 1 INTRODUCTION When one trains a learning machine using a set of data given by the true system, its ability can be improved if one selects the training data actively. In this paper, we consider the problem of active learning in multilayer perceptrons (MLP). First, we review our method of active learning (Fukumizu el al., 1994), in which we prepare a probability distribution and obtain training data as samples from the distribution. This methodology leads us to an information-matrix-based criterion similar to other existing ones (Fedorov, 1972; Pukelsheim, 1993). Active learning techniques have been recently used with neural networks (MacKay, 1992; Cohn, 1994). Our method, however, as well as many other ones has a crucial problem: the required inverse of an information matrix may not exist (White, 1989). We propose an active learning technique which is applicable to three-layer perceptrons. Developing a theory on the singularity of a Fisher information matrix, we present an active learning algorithm which keeps the information matrix nonsingular. We demonstrate the effectiveness of the algorithm through experiments. 296 K. FUKUMIZU 2 STATISTICALLY OPTIMAL TRAINING DATA 2.1 A CRITERION OF OPTIMALITY We review the criterion of statistically optimal training data (Fukumizu et al., 1994). We consider the regression problem in which the target system maps a given input z to y according to y = I(z) + Z, where I( z) is a deterministic function from R L to R M , and Z is a random variable whose law is a normal distribution N(O,(12IM ), (IM is the unit M x M matrix). Our objective is to estimate the true function 1 as accurately as possible. Let {/( z; O)} be a parametric model for estimation. We use the maximum likelihood estimator (MLE) 0 for training data ((z(v), y(v»)}~=l' which minimizes the sum of squared errors in this case. In theoretical derivations, we assume that the target function 1 is included in the model and equal to 1(,; (0 ). We make a training example by choosing z(v) to try, observing the resulting output y(v), and pairing them. The problem of active learning is how to determine input data {z(v)} ~=l to minimize the estimation error after training. Our approach is a statistical one using a probability for training, r( z), and choosing {z(v) }:Y"=l as independent samples from r(z) to minimize the expectation of the MSE in the actual environment: In the above equation, Q is the environmental probability which gives input vectors to the true system in the actual environment, and E{(zlv"yIV')} means the expectation on training data. Eq.(I), therefore, shows the average error of the trained machine that is used as a substitute of the true function in the actual environment. 2.2 REVIEW OF AN ACTIVE LEARNING METHOD Using statistical a.~ymptotic theory, Eq. (1) is approximated a.~ follows: 2 EMSE = (12 + ~ Tr [I(Oo)J-1(Oo)] + O(N-3j2), (2) where the matrixes I and J are (Fisher) illformation matrixes defined by 1(0) = J I(z;O)dQ(z). J(O) = J I(z;O)r(z)dz. The essential part of Eq.(2) is Tr[I(Oo)J-1(Oo»), computed by the unavailable parameter 00 • We have proposed a practical algorithm in which we replace 00 with O. prepare a family of probability {r( z; 'lI) I 'U : paramater} to choose training samples, and optimize 'U and {) iteratively (Fllkumizll et al., 1994). Active Learning Algorithm 1. Select an initial training data set D[o] from r( z; 'lI[O])' and compute 0[0]' 2. k:= 1. 3. Compute the optimal v = V[k] to minimize Tr[I(O[k_l])J-1(O[k_l]»)' Active Learning in Multilayer Perceptrons 297 4. Choose ~ new training data from r(z;V[k]) and let D[k] be a union of D[k-l] and the new data. 5. Compute the MLE 9[k] based on the training data set D[k]. 6. k := k + 1 and go to 3. The above method utilizes a probability to generate training data. It has the advantage of making many data in one step compared to existing ones in which only one data is chosen in each step, though their criterions are similar to each other. 3 SINGULARITY OF AN INFORMATION MATRIX 3.1 A PROBLEM ON ACTIVE LEARNING IN MLP Hereafter, we focus on active learning in three-layer perceptrons with H hidden units, NH = {!(z, O)}. The map !(z; 0) is defined by H L h(z; 0) = L Wij s(L UjkXk + (j) + 7]i, (1~i~M), (3) j=1 k=1 where s(t) is the sigmoidal function: s(t) = 1/(1 + e-t ). Our active learning method as well as many other ones requires the inverse of an information matrix J. The information matrix of MLP, however, is not always invertible (White, 1989). Any statistical algorithms utilizing the inverse, then, cannot be applied directly to MLP (Hagiwara et al., 1993). Such problems do not arise in linear models, which almost always have a nonsingular information matrix. 3.2 SINGULARITY OF AN INFORMATION MATRIX OF MLP The following theorem shows that the information matrix of a three-layer perceptron is singular if and only if the network has redundant hidden units. We can deduce tha.t if the information matrix is singular, we can make it nonsingular by eliminating redundant hidden units without changing the input-output map. Theorem 1 Assume r(z) is continuous and positive at any z. Then. the Fisher information matrix J is singular if and only if at least one of the follo'wing three con(litions is satisfied: (1) u,j := (Ujl, ... , UjL)T = 0, for some j. (2) Wj:= (Wlj, ... ,WMj) = OT , for some j. (3) For difJerenth andh, (U,h,(jt) = (U,1,(h) or (U,h,(it) = -(U,h,(h)· The rough sketch of the proof is shown below. The complete proof will appear in a forthcoming pa.per ,(Fukumizu, 1996). Rough sketch of the proof. We know easily that an information matrix is singular if and ouly if {()fJ:~(J)}a are linearly dependent. The sufficiency can be proved easily. To show the necessity, we show that the derivatives are linearly independent if none of the three conditions is satisfied. Assume a linear relation: 298 K. FUKUMIZU We can show there exists a basis of R L , (Z(l), ... , Z(L», such that Uj . z(l) i- 0 for 'Vj, 'VI, and Uj! . z(l) + (h i- ±(u12 . z(l) + (h) for jl i- h,'VI. We replace z in eq.(4) by z(l)t (t E R). Let my) := Uj· z(l), Sjl) := {z E C I z = ((2n+ 1)1T/=1(j)/m~l), n E Z}, and D(l) := C - UjSY). The points in S~l) are the singularities of s(m~l) z + (j). We define holomorphic functions on D(l) as q,~l)(z) ._ 'Ef=l aijs(my> z + (j) + aiO + 'E~l 'E~=l,BjkWijS'(my) z + (j)x~l> z +'E~l,BjOWijS'(my)z+(j), (1 ~ i ~ M). From eq.( 4), we have q,~l) (t) = 0 for all t E R. Using standard arguments on isolated singularities of holomorphic functions, we know SY) are removable singularities of q,~l)(z), and finally obtain Wij 'E~=l,BjkX~I) = 0, Wij,BjO = 0, aij = 0, aiO = o. It is easy to see ,Bjk = O. This completes the proof. 3.3 REDUCTION PROCEDURE We introduce the following reduction procedure based on Theorem 1. Used during BP training, it eliminates redundant hidden units and keeps the information matrix nonsingular. The criterion of elimination is very important, because excessive elimination of hidden units degrades the approximation capacity. We propose an algorithm which does not increase the mean squared error on average. In the following, let Sj := s( itj . z + llj) and £( N) == A/ N for a positive number A. Reduction Procedure 1. If IIWjll2 J(Sj - s((j))2dQ < £(N), then eliminate the jth hidden unit, and lli -. lli + WijS((j) for all i. 2. If IIwjll2 J(sj)2dQ < €(N), then eliminate the jth hidden unit. 3. If IIwhll2 J(sh - sjJ2dQ < €(N) for different it and h, then eliminate the hth hidden unit and Wij! -. wih + Wijz for all i. 4. If IIwhll2 J(1 - sh - sjJ 2dQ < €(N) for different jl and h, ~hen eliminate the j2th hidden unit and wih -. Wij! - wih, ili -. ili + wih for all 'i. From Theorem 1, we know that Wj, itj, (ith' (h) - (it'};, (j!), or (ith, (h )+( it]:, (h) can be reduced to 0 if the information matrix is singular. Let 0 E NK denote the reduced parameter from iJ according to the above procedure. The above four conditions are, then, given by calculating J II/(x; 0) -/(x; iJ)WdQ. We briefly explain how the procedure keeps the information matrix nonsingular and does not increase EMSE in high probability. First, suppose detJ(Oo) = 0, then there exists Off E NK (K < H) such that f(x;Oo) = f(x;Off) and detJ(Of) i- 0 in N K. The elimination of hidden units up to K, of course, does not increase the EMSE. Therefore, we have only to consider the case in which detJ(Oo) i- 0 and hidden units are eliminated. Suppose J II/(z; Off) -/(z; Oo)1I2dQ > O(N- 1 ) for any reduced parameter Off from 00 • The probability of satisfying J II/(z;iJ) -/(z;O)WdQ < A/N is very small for Active Learning in Multilayer Perceptrons 299 a sufficiently small A. Thus, the elimination of hidden units occurs in very tiny probability. Next, suppose J 1I!(x; (Jff) - !(x; (Jo)1I 2dQ = O(N-l). Let 0 E NK be a reduced parameter made from 9 with the same procedure as we obtain (Jff from (Jo. We will show for a sufficiently small A, where OK is MLE computed in NK. We write (J = ((J(l),(J(2») in which (J(2) is changed to 0 in reduction, changing the coordinate system if necessary. The Taylor expansion and asymptotic theory give E [JII!(x; OK) - !(x; (Jo)1I2dQ] ~ JII!(x; (Jf)-!(x; (Jo)11 2dQ+ ~ Tr[In((Jf)Jil1((Jf)), 2 E [JII!(x; 9) - !(x; O)WdQ] ~ JII!(x; (Jf)-!(x; (Jo)1I 2dQ+ ;, Tr[h2 ((Jf)J2;l ((Jo)], where Iii and Jii denote the local information matrixes w.r.t. (J(i) ('i = 1,2). Thus, E [JII!(x; 0) - !(x; (Jo)1I 2dQ] - E [JII!(x; OK) - !(x; (Jo)1I 2dQ] 2 ~ -E [JII!(x;o) - !(X;O)1I 2dQ] +;' Tr[h2((Jf)J;1((Jo)) 2 - ;, Tr[Ill((Jf)Jil1((Jf)] + E [JII!(x; 0) - !(x; (Jo)1I 2dQ] . Since the sum of the last two terms is positive, the 1.h.s is positive if E[f II!( x; OK)_ !(x; 0)1I 2dQ) < BIN for a sufficiently small B. Although we cannot know the value of this expectation, we can make the probability of holding this enequality very high by taking a small A. 4 ACTIVE LEARNING WITH REDUCTION PROCEDURE The reduction procedure keeps the information matrix nonsingular and makes the active learning algorithm applicable to MLP even with surplus hidden units. Active Learning with Hidden Unit Reduction 1. Select initial training data set Do from r( x; V[O]). and compute 0[0]' 2. k:= 1, and do REDUCTION PROCEDURE. 3. Compute the optimal v = 1'[k] to minimize Tr[I(9[k_l])J-l (9[k-l] )). using the steepest descent method. 4. Choose Nk new training data from r( x; V[k]) and let D[k] be a union of D[k-l] and the new data. 5. Compute the MLE 9[kbbased on the training data D[k] using BP with REDUCTION PROCE URE. 6. k:= k + 1 and go to 3. The BP with reduction procedure is applicable not only to active learning, but to a variety of statistical techniques that require the inverse of an information matrix. We do not discuss it in this paper. however. 300 • • -- Active Learning • Active Learning [Av·Sd,Av+Sd] - .. - . Passive Learning + Passive Learning [Av·Sd,Av+Sd] ~ + + + + • • + • • • 200 400 600 800 100> The Number of Training nata K. FUKUMIZU 0.00001 4 -- Learning Curve ..•.. It of hidden units O.IXXXlOOI 0 100 200 300 400 soo 600 700 800 900 100> The Number of Training nata Figure 1: Active/Passive Learning: f(x) = s(x) 5 EXPERIMENTS We demonstrate the effect of the proposed active learning algorithm through experiments. First we use a three-layer model with 1 input unit, 3 hidden units, and 1 output unit. The true function f is a MLP network with 1 hidden unit. The information matrix is singular at 0o, then. The environmental probability, Q, is a normal distribution N(O,4). We evaluate the generalization error in the actual environment using the following mean squared error of the function values: ! 1If(:l!; 0) - f(:l!)11 2dQ. We set the deviation in the true system II = 0.01. As a family of distributions for training {r(:l!;v)}, a mixture model of 4 normal distributions is used. In each step of active learning, 100 new samples are added. A network is trained using online BP, presented with all training data 10000 times in each step, and operated the reduction procedure once a 100 cycles between 5000th and 10000th cycle. We try 30 trainings changing the seed of random numbers. In comparison, we train a network passively based on training samples given by the probability Q. Fig.1 shows the averaged learning curves of active/passive learning and the number of hidden units in a typical learning curve. The advantage of the proposed active learning algorithm is clear. We can find that the algorithm has expected effects on a simple, ideal approximation problem. Second, we apply the algorithm to a problem in which the true function is not included in the MLP model. We use MLP with 4 input units, 7 hidden units, and 1 output unit. The true function is given by f(:l!) = erf(xt), where erf(t) is the error function. The graph of the error function resembles that of the sigmoidal function, while they never coincide by any affine transforms. We set Q = N(0,25 X 14). We train a network actively/passively based on 10 data sets, and evaluate MSE's of function values. Other conditions are the same as those of the first experiment. Fig.2 shows the averaged learning curves and the number of hidden units in a typical learning curve. We find tha.t the active learning algorithm reduces the errors though the theoretical condition is not perfectly satisfied in this case. It suggests the robustness of our active learning algorithm. Active Learning in Multilayer Perceptrons -- Active Learning O.IXXlI - .. - . Passive Learning 200 400 600 800 IIXXl The Number ofTraining nata -- Learning Curve ..•.. # of hidden units L .... .... . ... ......... : 301 8 ;; 7 It :s c :; ~ 6 e. :r is: ~ :s .. 5; r-~~~r-~-r~--r-~-+4 100 200 300 400 500 600 700 800 900 IIXXl The Number of Training nata ~ Figure 2: Active/Passive Learning: f(z) = erf(xI) 6 CONCLUSION We review statistical active learning methods and point out a problem in their application to MLP: the required inverse of an information matrix does not exist if the network has redundant hidden units. We characterize the singularity condition of an information matrix and propose an active learning algorithm which is applicable to MLP with any number of hidden units. The effectiveness of the algorithm is verified through computer simulations, even when the theoretical assumptions are not perfectly satisfied. References D. A. Cohn. (1994) Neural network exploration using optimal experiment design. In J. Cowan et al. (ed.), A d'vances in Neural Information Processing SYHtems 6, 679-686. San Mateo, CA: Morgan Kaufmann. V. V. Fedorov. (1972) Theory of Optimal Experiments. NY: Academic Press. K. Fukumizu. (1996) A Regularity Condition of the Information Matrix of a Multilayer Percept ron Network. Neural Networks, to appear. K. Fukumizu, & S. Watanabe. (1994) Error Estimation and Learning Data Arrangement for Neural Networks. Proc. IEEE Int. Conf. Neural Networks :777-780. K. Hagiwara, N. Toda, & S. Usui. (1993) On the problem of applying AIC to determine the structure of a layered feed-forward neural network. Proc. 1993 Int. Joint ConI. Neural Networks :2263-2266. D. MacKay. (1992) Information-based objective functions for active data selection, Ne'ural Computation 4(4):305-318. F. Pukelsheim. (1993) Optimal Design of Experiments. NY: John Wiley & Sons. H. White. (1989) Learning in artificial neural networks: A statistical perspective Neural Computation 1 ( 4 ):425-464.	1995	76
1,124	A model of transparent motion and non-transparent motion aftereffects Alexander Grunewald* Max-Planck Institut fur biologische Kybernetik Spemannstrafie 38 D-72076 Tubingen, Germany Abstract A model of human motion perception is presented. The model contains two stages of direction selective units. The first stage contains broadly tuned units, while the second stage contains units that are narrowly tuned. The model accounts for the motion aftereffect through adapting units at the first stage and inhibitory interactions at the second stage. The model explains how two populations of dots moving in slightly different directions are perceived as a single population moving in the direction of the vector sum, and how two populations moving in strongly different directions are perceived as transparent motion. The model also explains why the motion aftereffect in both cases appears as non-transparent motion. 1 INTRODUCTION Transparent motion can be studied using displays which contain two populations of moving dots. The dots within each population have the same direction of motion, but directions can differ between the two populations. When the two directions are very similar, subjects report seeing dots moving in the average direction (Williams & Sekuler, 1984). However, when the difference between the two directions gets large, subjects perceive two overlapping sheets of moving dots. This percept is called transparent motion. The occurrence of transparent motion cannot be explained by direction averaging, since that would result in a single direction of perceived motion. Rather than just being a quirk of the human visual system, transparent motion is an important issue in motion processing. For example, when a robot is moving its • Present address: Caltech, Mail Code 216-76, Pasadena, CA 91125. 838 A. GRUNEWALD motion leads to a velocity field. The ability to detect transparent motion within that velocity field enables the robot to detect other moving objects at the same time that the velocity field can be used to estimate the heading direction of the robot. Without the ability to code mUltiple directions of motion at the same location, i.e. without the provision for transparent motion, this capacity is not available. Traditional algorithms have failed to properly process transparent motion, mainly because they assigned a unique velocity signal to each location, instead of allowing the possibility for multiple motion signals at a single location. Consequently, the study of transparent motion has recently enjoyed widespread interest. STIMULUS PERCEPT Test Figure 1: Two populations of dots moving in different directions during an adaptation phase are perceived as transparent motion. Subsequent viewing of randomly moving dots during a test phase leads to an illusory percept of unidirectional motion, the motion aftereffect (MAE). Stimulus and percept in both phases are shown. After prolonged exposure to an adaptation display containing dots moving in one direction, randomly moving dots in a test display appear to be moving in the opposite direction (Hiris & Blake, 1992; Wohlgemuth, 1911). This illusory percept of motion is called the motion aftereffect (MAE). Traditionally this is explained by assuming that pairs of oppositely tuned direction selective units together code the presence of motion. When both are equally active, no motion is seen. Visual motion leads to stronger activation of one unit, and thus an imbalance in the activity of the two units. Consequently, motion is perceived. Activation of that unit causes it to fatigue, which means its response weakens. After motion offset, the previously active unit sends out a reduced signal compared to its partner due to adaptation. Thus adaptation generates an imbalance between the two units, and therefore illusory motion, the MAE, is perceived. This is the ratio model (Sutherland, 1961). Recent psychophysical results show that after prolonged exposure to transparent motion, observers perceive a MAE of a single direction of motion, pointing in the vector average of the adaptation directions (Mather, 1980; Verstraten, Fredericksen, & van de Grind, 1994). Thus adaptation to transparent motion leads to a non-transparent MAE. This is illustrated in Figure 1. This result cannot be accounted for by the ratio model, since the non-transparent MAE does not point in the direction opposite to either of the adaptation directions. Instead, this result suggests that direction selective units of all directions interact and thus contribute to the MAE. This explanation is called the distribution-shift model (Mather, 1980). However, thus far it has only been vaguely defined, and no demonstration has been given that shows how this mechanism might work. A Model of Transparent Motion and Non-transparent Motion Aftereffects 839 This study develops a model of human motion perception based on elements from both the ratio and the distribution-shift models for the MAE. The model is also applicable to the situation where two directions of motion are present. When the directions differ slightly, only a single direction is perceived. When the directions differ a lot, transparent motion is perceived. Both cases lead to a unitary MAE. 2 OUTLINE OF THE MODEL The model consists of two stages. Both stages contain units that are direction selective. The architecture of the model is shown in Figure 2. ~----~--~,~r---~---'---\ -, Stage 2 CD080CD 86) +----+~--+~--~----~--~--~--~~ Figure 2: The model contains two stages of direction selective units. Units at stage 1 excite units of like direction selectivity at stage 2, and inhibit units of opposite directions. At stage 2 recurrent inhibition sharpens directional motion responses. The grey level indicates the strength of interaction between units. Strong influence is indicated by black arrows, weak influence is indicated by light grey arrows. Units in stage 1 are broadly tuned motion detectors. In the present study the precise mechanism of motion detection is not central, and hence it has not been modeled. It is assumed that the bandwidth of motion detectors at this stage is about 30 degrees (Raymond, 1993; Williams, Tweten, & Sekuler, 1991). In the absence of any visual motion, all units are active at a baseline level; this is equivalent to neuronal noise. Whenever motion of a particular direction is present in the input, the activity of the corresponding unit (Vi) is activated maximally (Vi = 9), and units of similar direction selectivity are weakly activated (Vi = 3). The activities of all other units decrease to zero. Associated with each unit i at stage 1 is a weight Wi that denotes the adaptational state of unit i to fire a unit at stage 2. During prolonged exposure to motion these weights adapt, and their strength decreases. The equation governing the strength of the weights is given below: dWi = R(1- w·) - V·W · dt ~ ~~, where R = 0.5 denotes the rate of recovery to the baseline weight. When Wi = 1 the corresponding unit is not adapted. The further Wi is reduced from 1, the more 840 A. GRUNEWALD the corresponding unit is adapted. The products ViWi are transmitted to stage 2. Each unit of stage 1 excites units coding similar directions at stage 2, and inhibits units coding opposite directions of motion. The excitatory and inhibitory effects between units at stages 1 and 2 are caused by kernels, shown in Figure 3. Feedforward kernels Feedback kernels 1 1 I 1excitatory excitatory 0.8 ------0.8 ---~--- inhibitory inhibitory 0.6 0 . 6 0.4 0.4 r0.2 0.2 f---------0 -0 ------+ --------180 0 180 -180 o 180 Figure 3: Kernels used in the model. Left: excitatory and inhibitory kernels between stages 1 and 2; right: excitatory and inhibitory feedback kernels within stage 2. Activities at stage 2 are highly tuned for the direction of motion. The broad activation of motion signals at stage 1 is directionally sharpened at stage 2 through the interactions between recurrent excitation and inhibition. Each unit in stage 2 excites itself, and interacts with other units at stage 2 through recurrent inhibition. This inhibition is maximal for close directions, and falls off as the directions become more dissimilar. The kernels mediating excitatory and inhibitory interactions within stage 2 are shown in Figure 3. Through these inhibitory interactions the directional tuning of units at stage 2 is sharpened; through the excitatory feedback it is ensured that one unit will be maximally active. Activities of units at stage 2 are given by Mi = max4(mi' 0), where the behavior of mi is governed by: F/ and Fi- denote the result of convolving the products of the activities at stage 1 and the corresponding adaptation level, VjWj , with excitatory and inhibitory feedforward kernels respectively. Similarly, Bt and Bj denote the convolution of the activities M j at stage 2 with the feedback kernels. 3 SIMULATIONS OF PSYCHOPHYSICAL RESULTS In the simulations there were 24 units at each stage. The model was simulated dynamically by integrating the differential equations using a fourth order RungeKutta method with stepsize H = 0.01 time units. The spacing of units in direction space was 15 degrees at both stages. Spatial interactions were not modeled. In the simulations shown, a motion stimulus is present until t = 3. Then the motion stimulus ceases. Activity at stage 2 after t = 3 corresponds to a MAE. A Model of Transparent Motion and Non-transparent Motion Aftereffects 841 3.1 UNIDIRECTIONAL MOTION When adapting to a single direction of motion, the model correctly generates a motion signal for that particular direction of motion. After offset of the motion input, the unit coding the opposite direction of motion is activated, as in the MAE. A simulation of this is shown in Figure 4. Stage 1 Stage 2 act act 360 360 Figure 4: Simulation of single motion input and resulting MAE. Motion input is presented until t = 3. During adaptation the motion stimulus excites the corresponding units at stage 1, which in turn activate units at stage 2. Due to recurrent inhibition only one unit at stage 2 remains active (Grossberg, 1973), and thus a very sharp motion signal is registered at stage 2. During adaptation the weights associated with the units that receive a motion input decrease. After motion offset, all units receive the same baseline input. Since the weights of the previously active units are decreased, the corresponding cells at stage 2 receive less feedforward excitation. At the same time, the previously active units receive strong feedforward inhibition, since they receive inhibition from units tuned to very different directions of motion and whose weights did not decay during adaptation. Similarly, the units coding the opposite direction of motion as those previously active receive more excitation and less inhibition. Through recurrent inhibition the unit at stage 2 coding the opposite direction to that which was active during adaptation is activated after motion offset: this activity corresponds to the MAE. Thus the MAE is primarily an effect of disinhibition. 3.2 TRANSPARENT MOTION: SIMILAR DIRECTIONS Two populations of dots moving in different, but very similar, directions lead to bimodal activation at stage 1. Since the feedforward excitatory kernel is broadly tuned, and since the directions of motion are similar, the ensuing distribution of activities at stage 2 is unimodal, peaking halfway between the two directions of motion. This corresponds to the vector average of the directions of motion of the two populations of dots. A simulation of this is shown in Figure 5. During adaptation the units at stage 1 corresponding to the input adapt. As before this means that after motion offset the previously active units receive less excitatory input and more inhibitory input. As during adaptation this signal is unimodal. Also, the unit at stage 2 coding the opposite direction to that of the stimulus receives 842 act Stage 1 60 120 180 direction 240 Stage 2 60 120 180 direction 240 A. GRUNEWALD Figure 5: Simulation of two close directions of motion. Stage 2 of the network model registers unitary motion and a unitary MAE. less inhibition and more excitation. Through the recurrent activities within stage 2, that unit gets maximally activated. A unimodal MAE results. 3.3 TRANSPARENT MOTION: DIFFERENT DIRECTIONS When the directions of the two populations of dots in a transparent motion display are sufficiently distinct, the distribution of activities at stage 2 is no longer unimodal, but bimodal. Thus, recurrent inhibition leads to activation of two units at stage 2. They correspond to the two stimulus directions. A simulation is shown in Figure 6. Stage 1 act Stage 2 60 120 180 direction 240 Figure 6: Simulation of two distinct directions of motion. Stage 2 of the model registers transparent motion during adaptation, but the MAE is unidirectional. Feedforward inhibition is tuned much broader than feedforward excitation, and as a consequence the inhibitory signal during adaptation is unimodal, peaking at the unit of stage 2 coding the opposite direction of the average of the two previously active directions. Therefore that unit receives the least amount of inhibition after motion offset. It receives the same activity from stage 1 as units coding nearby directions, since the corresponding weights at stage 1 did not adapt. Due to recurrent activities at stage 2 that unit becomes active: non-transparent motion is registered. A Model of Transparent Motion and Non-transparent Motion Aftereffects 843 4 DISCUSSION Recently Snowden, Treue, Erickson, and Andersen (1991) have studied the effect of transparent motion stimuli on neurons in areas VI and MT of macaque monkey. They simultaneously presented two populations of dots, one of which was moving in the preferred direction of the neuron under study, and the other population was moving in a different direction. They found that neurons in VI were barely affected by the second population of dots. Neurons in MT, on the other hand, were inhibited when the direction of the second population differed from the preferred direction, and inhibition was maximal when the second population was moving opposite to the preferred direction. These results support key mechanisms of the model. At stage 1 there is no interaction between opposing directions of motion. The feedforward inhibition between stages 1 and 2 is maximal between opposite directions. Thus activities of units at stage 1 parallel neural activities recorded at VI, and activities of units at stage 2 parallels those neural activities recorded in area MT. Acknowledgments This research was carried out under HFSP grant SF-354/94. Reference Grossberg, S. (1973). Contour enhancement, short term memory, and constancies in reverberating neural networks. Studies in Applied Mathematics, LII, 213-257. Hiris, E., & Blake, R. (1992). Another perspective in the visual motion aftereffect. Proceedings of the National Academy of Sciences USA, 89, 9025-9028. Mather, G. (1980). The movement aftereffect and a distribution-shift model for coding the direction of visual movement. Perception, 9, 379-392. Raymond, J. E. (1993). Movement direction analysers: independence and bandwidth. Vision Research, 33(5/6), 767-775. Snowden, R. J., Treue, S., Erickson, R. G., & Andersen, R. A. (1991). The response of area MT and VI neurons to transparent motion. Journal of Neuroscience, 11 (9), 2768-2785. Sutherland, N. S. (1961). Figural after-effects and apparent size. Quarterly Journal of Experimental Psychology, 13, 222-228. Verstraten, F. A. J., Fredericksen, R. E., & van de Grind, W. A. (1994). Movement aftereffect of bi-vectorial transparent motion. Vision Research, 34, 349-358. Williams, D., Tweten, S., & Sekuler, R. (1991). Using metamers to explore motion perception. Vision Research, 31 (2), 275-286. Williams, D. W., & Sekuler, R. (1984). Coherent global motion percept from stochastic local motions. Vision Research, 24 (1), 55-62. Wohlgemuth, A. (1911). On the aftereffect of seen movement. British Journal of Psychology (Monograph Supplement), 1, 1-117.	1995	77
1,125	Improved Gaussian Mixture Density Estimates Using Bayesian Penalty Terms and Network Averaging Dirk Ormoneit Institut fur Informatik (H2) Technische Universitat Munchen 80290 Munchen, Germany ormoneit@inJormatik.tu-muenchen.de Abstract Volker Tresp Siemens AG Central Research 81730 Munchen, Germany Volker. Tresp@zJe.siemens.de We compare two regularization methods which can be used to improve the generalization capabilities of Gaussian mixture density estimates. The first method uses a Bayesian prior on the parameter space. We derive EM (Expectation Maximization) update rules which maximize the a posterior parameter probability. In the second approach we apply ensemble averaging to density estimation. This includes Breiman's "bagging" , which recently has been found to produce impressive results for classification networks. 1 Introduction Gaussian mixture models have recently attracted wide attention in the neural network community. Important examples of their application include the training of radial basis function classifiers, learning from patterns with missing features, and active learning. The appeal of Gaussian mixtures is based to a high degree on the applicability of the EM (Expectation Maximization) learning algorithm, which may be implemented as a fast neural network learning rule ([Now91], [Orm93]). Severe problems arise, however, due to singularities and local maxima in the log-likelihood function. Particularly in high-dimensional spaces these problems frequently cause the computed density estimates to possess only relatively limited generalization capabilities in terms of predicting the densities of new data points. As shown in this paper, considerably better generalization can be achieved using regularization. Improved Gaussian Mixture Density Estimates Using Bayesian Penalty Terms 543 We will compare two regularization methods. The first one uses a Bayesian prior on the parameters. By using conjugate priors we can derive EM learning rules for finding the MAP (maximum a posteriori probability) parameter estimate. The second approach consists of averaging the outputs of ensembles of Gaussian mixture density estimators trained on identical or resampled data sets. The latter is a form of "bagging" which was introduced by Breiman ([Bre94]) and which has recently been found to produce impressive results for classification networks. By using the regularized density estimators in a Bayes classifier ([THA93], [HT94], [KL95]), we demonstrate that both methods lead to density estimates which are superior to the unregularized Gaussian mixture estimate. 2 Gaussian Mixtures and the EM Algorithm Consider the lroblem of estimating the probability density of a continuous random vector x E 'R based on a set x* = {xk 11 S k S m} of iid. realizations of x. As a density model we choose the class of Gaussian mixtures p(xle) = L:7=1 Kip(xli, pi, Ei ), where the restrictions Ki ~ 0 and L:7=1 Kj = 1 apply. e denotes the parameter vector (Ki' Iti, Ei)i=1. The p(xli, Pi, Ei ) are multivariate normal densities: p( xli, Pi , Ei) = (271")- 41Ei 1- 1/ 2 exp [-1/2(x - Pi)tEi 1 (x - Iti)] . The Gaussian mixture model is well suited to approximate a wide class of continuous probability densities. Based on the model and given the data x, we may formulate the log-likelihood as lee) = log [rrm p(xkle)] = ",m log "'~ Kip(xkli, Pi, Ei). k=l .L...".k=1 .L...".J=l Maximum likelihood parameter estimates e may efficiently be computed with the EM (Expectation Maximization) algorithm ([DLR77]). It consists of the iterative application of the following two steps: 1. In the E-step, based on the current parameter estimates, the posterior probability that unit i is responsible for the generation of pattern xk is estimated as (1) 2. In the M-step, we obtain new parameter estimates (denoted by the prime): ~m hk k , wk-l i X Pi = ~m hi wl=l i , 1 L m k K · = h· J m k=1 J (2) (3) ~.' _ L:~1 hf(xk - pD(xk - pDt L.JJ m I L:l=l hi (4) Note that K~ is a scalar, whereas p~ denotes a d-dimensional vector and E/ is a d x d matrix. It is well known that training neural networks as predictors using the maximum likelihood parameter estimate leads to overfitting. The problem of overfitting is even more severe in density estimation due to singularities in the log-likelihood function. Obviously, the model likelihood becomes infinite in a trivial way if we concentrate all the probability mass on one or several samples of the training set. 544 D. ORMONEIT, V. TRESP In a Gaussian mixture this is just the case if the center of a unit coincides with one of the data points and E approaches the zero matrix. Figure 1 compares the true and the estimated probability density in a toy problem. As may be seen, the contraction of the Gaussians results in (possibly infinitely) high peaks in the Gaussian mixture density estimate. A simple way to achieve numerical stability is to artificially enforce a lower bound on the diagonal elements of E. This is a very rude way of regularization, however, and usually results in low generalization capabilities. The problem becomes even more severe in high-dimensional spaces. To yield reasonable approximations, we will apply two methods of regularization, which will be discussed in the following two sections. Figure 1: True density (left) and unregularized density estimation (right). 3 Bayesian Regularization In this section we propose a Bayesian prior distribution on the Gaussian mixture parameters, which leads to a numerically stable version of the EM algorithm. We first select a family of prior distributions on the parameters which is conjugate. Selecting a conjugate prior has a number of advantages. In particular, we obtain analytic solutions for the posterior density and the predictive density. In our case, the posterior density is a complex mixture of densitiest . It is possible, however, to derive EM-update rules to obtain the MAP parameter estimates. A conjugate prior of a single multivariate normal density is a product of a normal density N(JLilft,1]-lEi) and a Wishart density Wi(E;lla,,8) ([Bun94]). A proper conjugate prior for the the mixture weightings '" = ("'1, ... , "'n) is a Dirichlet density D("'hV. Consequently, the prior of the overall Gaussian mixture is the product D(",lr) il7=1 N(JLilil, 71-1Ei)Wi(E;1Ia, ,8). Our goal is to find the MAP parameter estimate, that is parameters which assume the maximum of the log-posterior Ip(S) 2:=~=1 log 2:=;=1 "'iP(X k Ii, JLi, Ei ) + log D("'lr) + 2:=;=1 [logN(JLilft, 71-1Ei) + log Wi(E;lla, ,8)]. As in the unregularized case, we may use the EM-algorithm to find a local maximum • A family F of probability distributions on 0 is said to be conjugate if, for every 1r E F, the posterior 1r(0Ix) also belongs to F ([Rob94]). tThe posterior distribution can be written as a sum of nm simple terms. tThose densities are defined as follows (b and c are normalizing constants): D(1I:17) N(Il.lp,1,-IE.) W i(Ei l la,,8) bIIn 11:7,-1, with 11:, ~ 0 and ",n 11:. = 1 .=1 ~.=l (21r)-i 11,-IE;I-l/2 exp [-~(Il' - Mt Ei1(1l' - M] = cIEillo-Cd+l)/2 exp [-tr(,8Ei 1)] • Improved Gaussian Mixture Density Estimates Using Bayesian Penalty Terms 545 of Ip(8). The E-step is identical to (1). The M-step becomes "m hk + 1 "m hk k + A , L.."k-l i ri (5) ,L.."k=l i x '1J1. "'i = "n J1.i = "m hi m + L.."i=l ri - n L..,,1=1 i + 11 (6) E~ = 2:;-1 hf(xk - J1.D(xk - J1.Dt + 11(J1.i - jJ.)(J1.i - jJ.)t + 2f3 I 2:~1 h~ + 20: - d (7) As typical for conjugate priors, prior knowledge corresponds to a set of artificial training data which is also reflected in the EM-update equations. In our experiments, we focus on a prior on the variances which is implemented by f3 =F 0, where o denotes the d x d zero matrix. All other parameters we set to "neutral" values: ri=l'v'i : l::;i::;n, 0:= (d+I)/2, 11=0, f3=iJld ld is the d x d unity matrix. The choice of 0: introdu~es a bias which favors large variances§. The effect of various values of the scalar f3 on the density estimate is illustrated in figure 2. Note that if iJ is chosen too small, overfitting still occurs. If it is chosen to large, on the other hand, the model is too constraint to recognize the underlying structure. Figure 2: Regularized density estimates (left: iJ = 0.05, right: 'iJ = 0.1). Typically, the optimal value for iJ is not known a priori. The simplest procedure consists of using that iJ which leads to the best performance on a validation set, analogous to the determination of the optimal weight decay parameter in neural network training. Alternatively, iJ might be determined according to appropriate Bayesian methods ([Mac9I]). Either way, only few additional computations are required for this method if compared with standard EM. 4 Averaging Gaussian Mixtures In this section we discuss the averaging of several Gaussian mixtures to yield improved probability density estimation. The averaging over neural network ensembles has been applied previously to regression and classification tasks ([PC93]). There are several different variants on the simple averaging idea. First, one may train all networks on the complete set of training data. The only source of disagreement between the individual predictions consists in different local solutions found by the likelihood maximization procedure due to different starting points. Disagreement is essential to yield an improvement by averaging, however, so that this proceeding only seems advantageous in cases where the relation between training data and weights is extremely non-deterministic in the sense that in training, §If A is distributed according to Wi(AIO', (3), then E[A- 1 ] = (0' - (d + 1)/2)-1 {3. In our case A is B;-I, so that E[Bi] -+ 00 • {3 for 0' -+ (d + 1)/2. 546 D. ORMONEIT, V. TRESP different solutions are found from different random starting points. A straightforward way to increase the disagreement is to train each network on a resampled version of the original data set. If we resample the data without replacement, the size of each training set is reduced, in our experiments to 70% of the original. The averaging of neural network predictions based on resampling with replacement has recently been proposed under the notation "bagging" by Breiman ([Bre94]), who has achieved dramatic.ally improved results in several classification tasks. He also notes, however, that an actual improvement of the prediction can only result if the estimation procedure is relatively unstable. As discussed, this is particularly the case for Gaussian mixture training. We therefore expect bagging to be well suited for our task. 5 Experiments and Results To assess the practical advantage resulting from regularization, we used the density estimates to construct classifiers and compared the resulting prediction accuracies using a toy problem and a real-world problem. The reason is that the generalization error of density estimates in terms of the likelihood based on the test data is rather unintuitive whereas performance on a classification problem provides a good impression of the degree of improvement. Assume we have a set of N labeled data z* = {(xk, lk)lk = 1, ... , N}, where lk E Y = {I, ... , C} denotes the class label of each input xk . A classifier of new inputs x is yielded by choosing the class I with the maximum posterior class-probability p(llx). The posterior probabilities may be derived from the class-conditional data likelihood p(xll) via Bayes theorem: p(llx) = p(xll)p(l)/p(x) ex p(xll)p(l). The resulting partitions ofthe input space are optimal for the true p(llx). A viable way to approximate the posterior p(llx) is to estimate p(xll) and p(l) from the sample data. 5.1 Toy Problem In the toy classification problem the task is to discriminate the two classes of circulatory arranged data shown in figure 3. We generated 200 data points for each class and subdivided them into two sets of 100 data points. The first was used for training, the second to test the generalization performance. As a network architecture we chose a Gaussian mixture with 20 units. Table 1 summarizes the results, beginning with the unregularized Gaussian mixture which is followed by the averaging and the Bayesian penalty approaches. The three rows for averaging correspond to the results yielded without applying resampling (local max.), with resampling withFigure 3: Toy Classification Task. Improved Gaussian Mixture Density Estimates Using Bayesian Penalty Terms 547 out replacement (70% subsets), and with resampling with replacement (bagging). The performances on training and test set are measured in terms of the model loglikelihood. Larger values indicate a better performance. We report separate results for dass A and B, since the densities of both were estimated separately. The final column shows the prediction accuracy in terms of the percentage of correctly classified data in the test set. We report the average results from 20 experiments. The numbers in brackets denote the standard deviations u of the results. Multiplying u with T19;95%/v'20 = 0.4680 yields 95% confidence intervals. The best result in each category is underlined. Algorithm Log-Likelihood Training Test Accuracy A B A B I unreg. -120.8 (13.3) -120.4 (10.8) -224.9 (32.6) -241.9 (34.1) 80.6'70 (2.8) Averaging: local max. -115.6 (6.0) -112.6 (6.6) -200.9 (13.9) -209.1 (16.3) 81.8% (3.1) 70% subset -106.8 (5.8) -105.1 (6.7) -188.8 (9.5) -196.4 (11.3) 83.2% (2.9) bagging -83.8 (4.9) -83.1 (7.1) -194.2 (7.3) -200.1 (11.3) 82.6% (3.4) Penalty: [3 = 0.01 -149.3 (18.5) -146.5 (5.9) -186.2 (13.9) -182.9 (11.6) 83.1% (2.9) [3 = 0.02 -156.0 (16.5) -153.0 (4.8) -177.1 (11.8) -174.9 (7.0) 84.4% (6.3) [3 = 0.05 -173.9 (24.3) -167.0 (15.8) -182.0 (20.1) -173.9 (14.3) 81.5% (5.9) [3 = 0.1 -183.0 (21.9) -181.9 (21.1) -184.6 (21.0) -182.5 (21.1) 78.5% (5.1) Table 1: Performances in the toy classification problem . As expected, all regularization methods outperform the maximum likelihood approach in terms of correct classification. The performance of the Bayesian regularization is hereby very sensitive to the appropriate choice of the regularization parameter (3. Optimality of (3 with respect to the density prediction and oytimality with respect to prediction accuracy on the test set roughly coincide (for (3 = 0.02). A veraging is inferior to the Bayesian approach if an optimal {3 is chosen. 5.2 BUPA Liver Disorder Classification As a second task we applied our methods to a real-world decision problem from the medical environment. The problem is to detect liver disorders which might arise from excessive alcohol consumption. Available information consists of five blood tests as well as a measure of the patients' daily alcohol consumption. We subdivided the 345 available samples into a training set of 200 and a test set of 145 samples. Due to the relatively few data we did not try to determine the optimal regularization parameter using a validation process and will report results on the test set for different parameter values. Algorithm unregularized Bayesian penalty ({3 = 0.05) Bayesian penalty «(3 = 0.10) Bayesian penal ty (3 = 0.20 averaging local maxima averaging (70 % subset) averaging (bagging) Accuracy 64.8 % 65.5 % 66.9 % 61.4 % 65.5 0 72.4 % 71.0 % Table 2: Performances in the liver disorder classification problem. 548 D. ORMONEIT. V. TRESP The results of our experiments are shown in table 2. Again, both regularization methods led to an improvement in prediction accuracy. In contrast to the toy problem, the averaged predictor was superior to the Bayesian approach here. Note that the resampling led to an improvement of more than five percent points compared to unresampled averaging. 6 Conclusion We proposed a Bayesian and an averaging approach to regularize Gaussian mixture density estimates. In comparison with the maximum likelihood solution both approaches led to considerably improved results as demonstrated using a toy problem and a real-world classification task. Interestingly, none of the methods outperformed the other in both tasks. This might be explained with the fact that Gaussian mixture density estimates are particularly unstable in high-dimensional spaces with relatively few data. The benefit of averaging might thus be greater in this case. A veraging proved to be particularly effective if applied in connection with resampIing of the training data, which agrees with results in regression and classification tasks. If compared to Bayesian regularization, averaging is computationally expensive. On the other hand, Baysian approaches typically require the determination of hyper parameters (in our case 13), which is not the case for averaging approaches. References [Bre94] L. Breiman. Bagging predictors. Technical report, UC Berkeley, 1994. [Bun94] W. Buntine. Operations for learning with graphical models. Journal of Artificial Intelligence Research, 2:159-225, 1994. [DLR77] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. J. Royal Statistical Society B, 1977. [HT94] T. Hastie and R. Tibshirani. Discriminant analysis by gaussian mixtures. Technical report, AT&T Bell Labs and University of Toronto, 1994. [KL95] N. Kambhatla and T. K. Leen. Classifying with gaussian mixtures and clusters. In Advances in Neural Information Processing Systems 7. Morgan Kaufman, 1995. [Mac91] D. MacKay. Bayesian Modelling and Neural Networks. PhD thesis, California Institute of Technology, Pasadena, 1991. [Now91] S. J. Nowlan. Soft Competitive Adaption: Neural Network Learning Algorithms based on Fitting Statistical Mixtures. PhD thesis, School of Computer Science, Carnegie Mellon University, Pittsburgh, 1991. [Orm93] D. Ormoneit. Estimation of probability densities using neural networks. Master's thesis, Technische Universitiit Munchen, 1993. [PC93] M. P. Perrone and L. N. Cooper. When networks disagree: Ensemble methods for hybrid Neural networks. In Neural Networks for Speech and Image Processing. Chapman Hall, 1993. [Rob94] C. P. Robert. The Bayesian Choice. Springer-Verlag, 1994. [THA93] V. Tresp, J. Hollatz, and S. Ahmad. Network structuring and training using rule-based knowledge. In Advances in Neural Information Processing Systems 5. Morgan Kaufman, 1993.	1995	78
1,126	A Practical Monte Carlo Implementation of Bayesian Learning Carl Edward Rasmussen Department of Computer Science University of Toronto Toronto, Ontario, M5S 1A4, Canada carl@cs.toronto.edu Abstract A practical method for Bayesian training of feed-forward neural networks using sophisticated Monte Carlo methods is presented and evaluated. In reasonably small amounts of computer time this approach outperforms other state-of-the-art methods on 5 datalimited tasks from real world domains. 1 INTRODUCTION Bayesian learning uses a prior on model parameters, combines this with information from a training set, and then integrates over the resulting posterior to make predictions. With this approach, we can use large networks without fear of overfitting, allowing us to capture more structure in the data, thus improving prediction accuracy and eliminating the tedious search (often performed using cross validation) for the model complexity that optimises the bias/variance tradeoff. In this approach the size of the model is limited only by computational considerations. The application of Bayesian learning to neural networks has been pioneered by MacKay (1992), who uses a Gaussian approximation to the posterior weight distribution. However, the Gaussian approximation is poor because of multiple modes in the posterior. Even locally around a mode the accuracy of the Gaussian approximation is questionable, especially when the model is large compared to the amount of training data. Here I present and test a Monte Carlo method (Neal, 1995) which avoids the Gaussian approximation. The implementation is complicated, but the user is not required to have extensive knowledge about the algorithm. Thus, the implementation represents a practical tool for learning in neural nets. A Practical Monte Carlo Implementation of Bayesian Learning 599 1.1 THE PREDICTION TASK The training data consists of n examples in the form of inputs x = {x(i)} and corresponding outputs y = {y(i)} where i = 1 ... n. For simplicity we consider only real-valued scalar outputs. The network is parametrised by weights w, and hyperparameters h that control the distributions for weights, playing a role similar to that of conventional weight decay. Weights and hyperparameters are collectively termed 0, and the network function is written as F/I (x), although the function value is only indirectly dependent on the hyperparameters (through the weights). Bayes' rule gives the posterior distribution for the parameters in terms of the likelihood, p(ylx, 0), and prior, p(O): (Olx ) = p(O)p(ylx, O) p ,y p(ylx) To minimize the expected squared error on an unseen test case with input x(n+l), we use the mean prediction (1) 2 MONTE CARLO SAMPLING The following implementation is due to Neal (1995). The network weights are updated using the hybrid Monte Carlo method (Duane et al. 1987). This method combines the Metropolis algorithm with dynamical simulation. This helps to avoid the random walk behavior of simple forms of Metropolis, which is essential if we wish to explore weight space efficiently. The hyperparameters are updated using Gibbs sampling. 2.1 NETWORK SPECIFICATION The networks used here are always of the same form: a single linear output unit, a single hidden layer of tanh units and a task dependent number of input units. All layers are fully connected in a feed forward manner (including direct connections from input to output). The output and hidden units have biases. The network priors are specified in a hierarchical manner in terms of hyperparameters; weights of different kinds are divided into groups, each group having it's own prior. The output-bias is given a zero-mean Gaussian prior with a std. dev. of u = 1000, so it is effectively unconstrained. The hidden-biases are given a two layer prior: the bias b is given a zero-mean Gaussian prior b '" N(O, ( 2 ); the value of u is specified in terms of precision r = u- 2 , which is given a Gamma prior with mean p = 400 (corresponding to u = 0.05) and shape parameter a = 0.5; the Gamma density is given by p(r) '" Gamma(p, a) ex: r Ol / 2- 1 exp( -ra/2p). Note that this type of prior introduces a dependency between the biases for different hidden units through the common r. The prior for the hidden-to-output weights is identical to the prior for the hidden-biases, except that the variance of these weights under the prior is scaled down by the square root of the number of hidden units, such that the network output magnitude becomes independent of the number of hidden units. The noise variance is also given a Gamma prior with these parameters. 600 C. E. RASMUSSEN The input-to-hidden weights are given a three layer prior: again each weight is given a zero-mean Gaussian prior w rv N(O, (12); the corresponding precision for the weights out of input unit i is given a Gamma prior with a mean J.l and a shape parameter a1 = 0.5: Ti rv Gamma(J.l, a1). The mean J.l is determined on the top level by a Gamma distribution with mean and shape parameter ao = 1: J.li rv Gamma(400,ao). The direct input-to-output connections are also given this prior. The above-mentioned 3 layer prior incorporates the idea of Automatic Relevance Determination (ARD), due to MacKay and Neal, and discussed in Neal (1995) . The hyperparameters, Ti, associated with individual inputs can adapt according to the relevance of the input; for an unimportant input, Ti can grow very large (governed by the top level prior), thus forcing (1i and the associated weights to vanish. 2.2 MONTE CARLO SPECIFICATION Sampling from the posterior weight distribution is performed by iteratively updating the values of the network weights and hyperparameters. Each iteration involves two components: weight updates and hyperparameter updates. A cursory description of these steps follows. 2.2.1 Weight Updates Weight updates are done using the hybrid Monte Carlo method. A fictitious dynamical system is generated by interpreting weights as positions, and augmenting the weights w with momentum variables p. The purpose of the dynamical system is to give the weights "inertia" so that slow random walk behaviour can be avoided during exploration of weight space. The total energy, H, of the system is the sum of the kinetic energy, I<, (a function of the momenta) and the potential energy, E. The potential energy is defined such that p(w) ex exp( -E). We sample from the joint distribution for wand p given by p(w,p) ex exp(-E - I<), under which the marginal distribution for w is given by the posterior. A sample of weights from the posterior can therefore be obtained by simply ignoring the momenta. Sampling from the joint distribution is achieved by two steps: 1) finding new points in phase space with near-identical energies H by simulating the dynamical system using a discretised approximation to Hamiltonian dynamics, and 2) changing the energy H by doing Gibbs sampling for the momentum variables. Hamiltonian Dynamics. Hamilton's first order differential equations for Hare approximated by a series of discrete first order steps (specifically by the leapfrog method). The first derivatives of the network error function enter through the derivative of the potential energy, and are computed using backpropagation. In the original version of the hybrid Monte Carlo method the final position is then accepted or rejected depending on the final energy H'" (which is not necessarily equal to the initial energy H because of the discretisation). Here we use a modified version that uses an average over a window of states instead. The step size of the discrete dynamics should be as large as possible while keeping the rejection rate low. The step sizes are set individually using several heuristic approximations, and scaled by an overall parameter c. We use L = 200 iterations, a window size of 20 and a step size of c = 0.2 for all simulations. Gibbs Sampling for Momentum Variables. The momentum variables are updated using a modified version of Gibbs sampling, allowing the energy H to change. A "persistence" of 0.95 is used; the new value of the momentum is a weighted sum of the previous value (weight 0.95) and the value obtained by Gibbs sampling (weight (1 - 0.952)1/2). With this form of persistence, the momenta A Practical Monte Carlo Implementation of Bayesian Learning 601 changes approx. 20 times more slowly, thus increasing the "inertia" of the weights, so as to further help in avoiding random walks. Larger values of the persistence will further increase the weight inertia, but reduce the rate of exploration of H. The advantage of increasing the weight inertia in this way rather than by increasing L is that the hyperparameters are updated at shorter intervals, allowing them to adapt to the rapidly changing weights. 2.2.2 Hyperparameter Updates The hyperparameters are updated using Gibbs sampling. The conditional distributions for the hyperparameters given the weights are of the Gamma form, for which efficient generators exist, except for the top-level hyperparameter in the case of the 3 layer priors used for the weights from the inputs; in this case the conditional distribution is more complicated and a form of rejection sampling is employed. 2.3 NETWORK TRAINING AND PREDICTION The network training consists of two levels of initialisation before sampling for networks used for prediction. At the first level of initialisation the hyperparameters (variance of the Gaussians) are kept constant at 1, allowing the weights to grow during 1000 leapfrog iterations. Neglecting this phase can cause the network to get caught for a long time in a state where weights and hyperparameters are both very small. The scheme described above is then invoked and run for as long as desired, eventually producing networks from the posterior distribution. The initial 1/3 of these nets are discarded, since the algorithm may need time to reach regions of high posterior probability. Networks sampled during the remainder of the run are saved for making predictions. The predictions are made using an average of the networks sampled from the posterior as an approximation to the integral in eq. (1). Since the output unit is linear the final prediction can be seen as coming from a huge (fully connected) ensemble net with appropriately scaled output weights. All the results reported here were for ensemble nets with 4000 hidden units. The size of the individual nets is given by the rule that we want at least as many network parameters as we have training examples (with a lower limit of 4 hidden units). We hope thereby to be well out of the underfitting region. Using even larger nets would probably not gain us much (in the face of the limited training data) and is avoided for computational reasons. All runs used the parameter values given above. The only check that is necessary is that the rejection rate stays low, say below 5%; if not, the step size should be lowered. In all runs reported here, c = 0.2 was adequate. The parameters concerning the Monte Carlo method and the network priors were all selected based on intuition and on experience with toy problems. Thus no parameters need to be set by the user. 3 TESTS The performance of the algorithm was evaluated by comparing it to other state-ofthe-art methods on 5 real-world regression tasks. All 5 data sets have previously been studied using a 10-way cross-validation scheme (Quinlan 1993). The tasks in these domains is to predict price or performance of an object from various discrete and real-valued attributes. For each domain the data is split into two sets of roughly equal size, one for training and one for testing. The training data is 602 C. E. RASMUSSEN further subdivided into full-, half-, quarter- and eighth-sized subsets, 15 subsets in total. Networks are trained on each of these partitions, and evaluated on the large common test set. On the small training sets, the average performance and one std. dev. error bars on this estimate are computed. 3.1 ALGORITHMS The Monte Carlo method was compared to four other algorithms. For the three neural network methods nets with a single hidden layer and direct input-output connections were used. The Monte Carlo method was run for 1 hour on each of the small training sets, and 2,4 and 8 hours respectively on the larger training sets. All simulations were done on a 200 MHz MIPS R4400 processor. The Gaussian Process method is described in a companion paper (Williams & Rasmussen 1996). The Evidence method (MacKay 1992) was used for a network with separate hyperparameters for the direct connections, the weights from individual inputs (ARD), hidden biases, and output biases. Nets were trained using a conjugate gradient method, allowing 10000 gradient evaluations (batch) before each of 6 updates of the hyperparameters. The network Hessian was computed analytically. The value of the evidence was computed without compensating for network symmetries, since this can lead to a vastly over-estimated evidence for big networks where the posterior Gaussians from different modes overlap. A large number of nets were trained for each task, with the number of hidden units computed from the results of previous nets by the following heuristics: The min and max number of hidden units in the 20% nets with the highest evidences were found. The new architecture is picked from a Gaussian (truncated at 0) with mean (max - min)/2 and std. dev. 2 + max - min, which is thought to give a reasonable trade-off between exploration and exploitation. This procedure is run for 1 hour of cpu time or until more than 1000 nets have been trained. The final predictions are made from an ensemble of the 20% (but a maximum of 100) nets with the highest evidence. An ensemble method using cross-validation to search over a 2-dimensional grid for the number of hidden units and the value of a single weight decay parameter has been included, as an attempt to have a thorough version of "common practise". The weight decay parameter takes on the values 0, 0.01, 0.04, 0.16, 0.64 and 2.56. Up to 6 sizes of nets are used, from 0 hidden units (a linear model) up to a number that gives as many weights as training examples. Networks are trained with a conjugent gradient method for 10000 epochs on each of these up to 36 networks, and performance was monitored on a validation set containing 1/3 of the examples, selected at random. This was repeated 5 times with different random validation sets, and the architecture and weight decay that did best on average was selected. The predictions are made from an ensemble of 10 nets with this architecture, trained on the full training set. This algorithm took several hours of cpu time for the largest training sets. The Multivariate Adaptive Regression Splines (MARS) method (Friedman 1991) was included as a non-neural network approach. It is possible to vary the maximum number of variables allowed to interact in the additive components of the model. It is common to allow either pairwise or full interactions. I do not have sufficient experience with MARS to make this choice. Therefore, I tried both options and reported for each partition on each domain the best performance based on the test error, so results as good as the ones reported here might not be obtainable in practise. All other parameters of MARS were left at their default values. MARS always required less than 1 minute of cpu time. A Practical Monte Carlo Implementation of Bayesian Learning 603 2 1.5 1 0.5 Auto price 0* + x o~------~----~----~--0.6 0.5 0.4 0.3 0.2 0.1 10 20 40 House t >«1>* + IS! 80 o~~----~------~----~-32 64 128 256 Servo 1 0.8 0.6 0.4 0.2 OtIS! X * o~~------~----~----~--11 22 44 88 Cpu 0.6 0.5 0.4 + 0.3 o IS! 0.2 X * 0.1 OL-~----~------~----~-13 0.25 0.2 0.15 0.1 0.05 26 52 Mpg 104 * Xo+ IS! OL-~----~----~----~-24 48 96 192 Geometric mean x Monte Carlo o Gaussian Evidence + Backprop * MARS IS! Gaussian Process 0.283 0.364 0.339 0.371 0.304 Figure 1: Squared error on test cases for the five algorithms applied to the five problems. Errors are normalized with respect to the variance on the test cases. The x-axis gives the number of training examples; four different set sizes were used on each domain. The error bars give one std. dev. for the distribution of the mean over training sets. No error bar is given for the largest size, for which only a single training set was available. Some of the large error bars are cut of at the top. MARS was unable to run on the smallest partitions from the Auto price and the servo domains; in these cases the means of the four other methods were used in the reported geometric mean for MARS. 604 C. E. RASMUSSEN Table 1: Data Sets domain # training cases # test cases # binary inputs # real inputs Auto Price 80 79 0 16 Cpu 104 105 0 6 House 256 250 1 12 Mpg 192 200 6 3 Servo 88 79 10 2 3.2 PERFORMANCE The test results are presented in fig. 1. On the servo domain the Monte Carlo method is uniformly better than all other methods, although the difference should probably not always be considered statistically significant. The Monte Carlo method generally does well for the smallest training sets. Note that no single method does well on all these tasks. The Monte Carlo method is never vastly out-performed by the other methods. The geometric mean of the performances over all 5 domains for the the 4 different training set sizes is computed. Assuming a Gaussian distribution of prediction errors, the log of the error variance can (apart from normalising constants) be interpreted as the amount of information unexplained by the models. Thus, the log of the geometric means in fig. 1 give the average information unexplained by the models. According to this measure the Monte Carlo method does best, closely followed by the Gaussian Process method. Note that MARS is the worst, even though the decision between pairwise and full interactions were made on the basis of the test errors. 4 CONCLUSIONS I have outlined a black-box Monte Carlo implementation of Bayesian learning in neural networks, and shown that it has an excellent performance. These results suggest that Monte Carlo based Bayesian methods are serious competitors for practical prediction tasks on data limited domains. Acknowledgements I am grateful to Radford Neal for his generosity with insight and software. This research was funded by a grant to G. Hinton from the Institute for Robotics and Intelligent Systems. References S. Duane, A. D. Kennedy, B. J. Pendleton & D. Roweth (1987) "Hybrid Monte Carlo", Physics Letters B, vol. 195, pp. 216-222. J. H. Friedman (1991) "Multivariate adaptive regression splines" (with discussion), Annals of Statistics, 19,1-141 (March). Source: http://lib.stat.cmu.edu/general/mars3.5. D. J. C. MacKay (1992) "A practical Bayesian framework for backpropagation networks", Neural Computation, vol. 4, pp. 448- 472. R. M. Neal (1995) Bayesian Learning for Neural Networks, PhD thesis, Dept. of Computer Science, University of Toronto, ftp: pub/radford/thesis. ps. Z from ftp. cs . toronto. edu. J. R. Quinlan (1993) "Combining instance-based and model-based learning", Proc. ML '93 (ed P.E. Utgoff), San Mateo: Morgan Kaufmann. C. K. I. Williams & C. E. Rasmussen (1996). "Regression with Gaussian processes", NIPS 8, editors D. Touretzky, M. Mozer and M. Hesselmo. (this volume).	1995	79
1,127	On the Computational Power of Noisy Spiking Neurons Wolfgang Maass Institute for Theoretical Computer Science, Technische Universitaet Graz Klosterwiesgasse 32/2, A-8010 Graz, Austria, e-mail: maass@igi.tu-graz.ac.at Abstract It has remained unknown whether one can in principle carry out reliable digital computations with networks of biologically realistic models for neurons. This article presents rigorous constructions for simulating in real-time arbitrary given boolean circuits and finite automata with arbitrarily high reliability by networks of noisy spiking neurons. In addition we show that with the help of "shunting inhibition" even networks of very unreliable spiking neurons can simulate in real-time any McCulloch-Pitts neuron (or "threshold gate"), and therefore any multilayer perceptron (or "threshold circuit") in a reliable manner. These constructions provide a possible explanation for the fact that biological neural systems can carry out quite complex computations within 100 msec. It turns out that the assumption that these constructions require about the shape of the EPSP's and the behaviour of the noise are surprisingly weak. 1 Introduction We consider networks that consist of a finite set V of neurons, a set E ~ V x V of synapses, a weightwu,v ~ 0 and a response junctioncu,v : R+ -+ R for each synapse 212 W.MAASS (u,v) E E (where R+ := {x E R: x ~ O}), and a threshold/unction Sv : R+ --t R+ for each neuron v E V. If F u ~ R + is the set of firing times of a neuron u, then the potential at the trigger zone of neuron v at time t is given by Pv(t) := L L wu,v' u : (u, v) E EsE Fu : s < t eu,v(t - s). The threshold function Sv(t - t') quantifies the "reluctance" of v to fire again at time t, if its last previous firing was at time t'. We assume that Sv(O) E (0,00), Sv(x) = 00 for x E (0, 'TreJ] (for some constant 'TreJ > 0, the "absolute refractory period"), and sup{Sv(x) : X ~ 'T} < 00 for any'T > 'TreJ. In a deterministic model for a spiking neuron (Maass, 1995a, 1996) one can assume that a neuron v fires exactly at those time points t when Pv(t) reaches (from below) the value Sv(t - t'). We consider in this article a biologically more realistic model, where as in (Gerstner, van Hemmen, 1994) the size of the difference Pv(t)-Sv(t-t') just governs the probability that neuron v fires. The choice of the exact firing times is left up to some unknown stochastic processes, and it may for example occur that v does not fire in a time intervall during which Pv (t) - Sv(t - t') > 0, or that v fires "spontaneously" at a time t when Pv(t) -Sv(t-t') < O. We assume that (apart from their communication via potential changes) the stochastic processes for different neurons v are independent. It turns out that the assumptions that one has to make about this stochastic firing mechanism in order to prove our results are surprisingly weak. We assume that there exist two arbitrary functions L, U : R X R+ ----1 [0,1] so that L(~, i) provides a lower bound (and U(~, i) provides an upper bound) for the probability that neuron v fires during a time intervall of length e with the property that Pv(t)-Sv(t-t') ~ ~ (respectively Pv(t)-Sv(t-t') ~ ~) for all tEl up to the next firing of v (t' denotes the last firing time of v be/ore I). We just assume about these functions Land U that they are non-decreasing in each of their two arguments (for any fixed value of the other argument), that lim U(~, i) = ° for any fixed ~~-oo i > 0, and that lim L(~, e) > 0 for allY fixed e ~ R/6 (where R is the assumed ~~OO length of the rising segment of an EPSP, see below). The neurons are allowed to be "arbitrarily noisy" in the sense that the difference lim L(~, i) lim U(~, i) ~~OO ~~-oo can be arbitrarily small. Hence our constructions also apply to neurons that exhibit persistent firing failures, and they also allow for synapses that fail with a rather high probability. Furthermore a detailed analysis of our constructions shows that we can relax the somewhat dubious assumption that the noise-distributions for different neurons are independent. Thus we are also able to deal with "systematic noise" in the distribution of firing times of neurons in a pool (e.g. caused by changes in the biochemical environment that simultaneously affect many neurons in a pool). It turns out that it suffices to assume only the following rather weak properties of the other functions involved in our model: 1) Each response function CU , I ) : R+ ----1 R is either excitatory or inhibitory (and for the sake of biological realism one may assume that each neuron u induces only one type of response). All excitatory response functions eu,v(x) have the value On the Computational Power of Noisy Spiking Neurons 213 o for x E [O,~u,v), and the value eE(X ~u ,v) for x ~ ~u,v, where ~u,v ~ 0 is the delay for this synapse between neurons u and v, and e E is the common shape of all excitatory response functions ("EPSP's))). Corresponding assumptions are made about the inhibitory response functions ("IPSP's))), whose common shape is described by some function eI : R+ -+ {x E R : x ~ O}. 2) eE is continuous, eE(O) = 0, eE(X) = 0 for all sufficiently large x, and there exists some parameter R > 0 such that eE is non-decreasing in [0, R], and some parameter p > 0 such that eE(X + R/6) ~ p + eE (x) for all x E [O,2R/3]. 3) _eI satisfies the same conditions as eE . 4) There exists a source BN- of negative "background noise", that contributes to the potential Pv(t) of each neuron v an additive term that deviates for an arbitrarily long time interval by an arbitrarily small percentage from its average value w; ~ 0 (which we can choose). One can delete this assumption if one assumes that the firing threshold of neurons can be shifted by some other mechanism. In section 3 we will assume in addition the availability of a corresponding positive background noise BN+ with average value wt ~ O. In a biological neuron tI one can interpret BN- and BN+ as the combined effect of a continuous bombardment with a very large number of IPSP's (EPSP's) from randomly firing neurons that arrive at remote synapses on the dendritic tree of v. We assume that we can choose the values of delays ~u , v and weights Wu,v, wt ,w; . We refer to all assumptions specified in this section as our "weak assumptions" about noisy spiking neurons. It is easy to see that the most frequently studied concrete model for noisy spiking neurons, the spike response model (Gerstner and van Hemmen, 1994) satisfies these weak assumptions, and is hence a special case. However not even for the more concrete spike response model (or any other model for noisy spiking neurons) there exist any rigorous results about computations in these models. In fact, one may view this article as being the first that provides results about the computational complexity of neural networks for a neuron model that is acceptable to many neurobiologistis as being reasonably realistic. In this article we only address the problem of reliable digital computing with noisy spiking neurons. For details of the proofs we refer to the forthcoming journal-version of this extended abstract. For results about analog computations with noisy spiking neurons we refer to Maass, 1995b. 2 Simulation of Boolean Circuits and Finite Automata with Noisy Spiking Neurons Theorem 1: For any deterministic finite automaton D one can construct a network N(D) consisting of any type of noisy spiking neurons that satisfy our weak assumptions, so that N(D) can simulate computations of D of any given length with arbitrarily high probability of correctness. 214 W.MAASS Idea of the proof: Since the behaviour of a single noisy spiking neuron is completely unreliable, we use instead pools A, B, ... of neurons as the basic building blocks in our construction, where all neurons v in the same pool receive approximately the same "input potential" Pv(t). The intricacies of our stochastic neuron model allow us only to employ a "weak coding" of bits, where a "1" is represented by a pool A during a time interval I, if at least PI ·IAI neurons in A fire (at least once) during I (where PI > 0 is a suitable constant), and "0" is represented if at most Po ·IAI firings of neurons occur in A during I, where Po with 0 < Po < PI is another constant (that can be chosen arbitrarily small in our construction). The described coding scheme is weak since it provides no useful upper bound (e.g. 1.5·Pl ·IAI) on the number of neurons that fire during I if A represents a "1" (nor on the number of firings of a single neuron in A). It also does not impose constraints on the exact timing of firings in A within I. However a "0" can be represented more precisely in our model, by choosing po sufficiently small. The proof of Theorem 1 shows that this weak coding of bits suffices for reliable digital computations. The idea of these simulations is to introduce artificial negations into the computation, which allow us to exploit that "0" has a more precise representation than "1". It is apparently impossible to simulate an AND-gate in a straightforward fashion for a weak coding of bits, but one can simulate a NOR-gate in a reliable manner. • Corollary 2: Any boolean function can be computed by a sufficiently large network of noisy spiking neurons (that satisfy our weak assumptions) with arbitrarily high probability of correctness. 3 Fast Simulation of Threshold Circuits via Shunting Inhibition For biologically realistic parameters, each computation step in the previously constructed network takes around 25 msec (see point b) in section 4}. However it is well-known that biological neural systems can carry out complex computations within just 100 msec (Churchland, Sejnowski, 1992). A closer inspection of the preceding construction shows, that one can simulate with the same speed also OR- and NOR-gates with a much larger fan-in than just 2. However wellknown results from theoretical computer science (see the results about the complexity class ACo in the survey article by Johnson in (van Leeuwen, 1990)) imply that for any fixed number of layers the computational power of circuits with gates for OR, NOR, AND, NOT remains very weak, even if one allows any polynomial size fan-in for such gates. In contrast to that, the construction in this section will show that by using a biologically more realistic model for a noisy spiking neuron, one can in principle simulate within 100 msec 3 or more layers of a boolean circuit that employs substantially more powerful boolean gates: threshold gates (Le. "Mc Culloch-Pitts neurons", also called "perceptrons"). The use of these gates provides a giant leap in computational On the Computational Power of Noisy Spiking Neurons 215 power for boolean circuits with a small number of layers: In spite of many years of intensive research, one has not been able to exhibit a single concrete computational problem in the complexity classes P or NP that can be shown to be not computable by a polynomial size threshold circuit with 3 layers (for threshold circuits with integer weights of unbounded size the same holds already for just 2 layers). In the neuron model that we have employed so far in this article, we have assumed (as it is common in the spike response model) that the potential Pv(t) at the trigger zone of neuron v depends linearly on all the terms Wu,v . cu,v(t - s). There exists however ample biological evidence that this assumption is not appropriate for certain types of synapses. An example are synapses that carry out shunting inhibition (see. e.g. (Abeles, 1991) and (Shepherd, 1990)). When a synapse of this type (located on the dendritic tree of a neuron v) is activated, it basically erases (through a short circuit mechanism) for a short time all EPSP's that pass the location of this synapse on their way to the trigger zone of v. However in contrast to those IPSP's that occur linearly in the formula for Pv(t) , the activation of such synapse for shunting inhibition has no impact on those EPSP's that travel to the trigger ZOne of v through another part of its dendritic tree. We model shunting inhibition in our framework as follows. We write r for the subset of all neurons 'Y in V that can "veto" other synapses (u, v) via shunting inhibition (we assume that the neurons in r have no other role apart from that). We allow in our formal model that certain 'Y in r are assigned as label to certain synapses (u, v) that have an excitatory response function cu,v. If'Y is a label of (u, v), then this models the situation that 'Y can intercept EPSP's from u on their way to the trigger zone of v via shunting inhibition. We then define Pv(t) = L (L wtt,tJ . Ett,v(t - s) . II s...,(t)) , u E V : (u, v) E EsE Ftt : s < t 'Y is label of (u, v) where we assume that S...,(t) E [0,1] is arbitrarily close to 0 for a short time interval after neuron 'Y has fired, and else equal to 1. The firing mechanism for neurons 'Y E r is defined like for all other neurons. Theorem 3: One can simulate any threshold circuit T by a sufficiently large network N(T) of noisy spiking neurons with shunting inhibition (with arbitrarily high probability of correctness). The computation time of N(T) does not depend on the number of gates in each layer, and is proportional to the number of layers in the threshold circuit T. Idea of the proof of Theorem 3: It is already impossible to simulate in a straightforward manner an AND-gate with weak coding of bits. The same difficulties arise in an even more drastic way if one wants to simulate a threshold gate with large fan-in. The left part of Figure 1 indicates that with the help of shunting inhibition one can transform via an intermediate pool of neurons Bl the bit that is weakly encoded by 216 W.MAASS Al into a contribution to Pv(t) for neurons v E C that is throughout a time interval J arbitrarily close to 0 if Al encodes a "0", and arbitrarily close to some constant P* > 0 if Al encodes a "I" (we will call this a "strong coding" of a bit). Obviously it is rather easy to realize a threshold gate if one can make use of such strong coding of bits. 8 r--------------------I I , , ® ~ 11 : E E , , , , : , E ) IAII I )IB11 SI ,- , lE :) ) '------+ C I )~-4[!] I E ) : -----+ , : H' ~ : : I ----------------------~ Figure 1: Realization of a threshold gate G via shunting inhibition (SI). The task of the module in Figure 1 is to simulate with noisy spiking neurons a n given boolean threshold gate G that outputs 1 if L: Q:iXi ~ e, and 0 else. For i=I simplicity Figure 1 shows only the pool Al whose firing activity encodes (in weak coding) the first input bit Xl. The other input bits are represented (in weak coding) simultaneously in pools A:l> ... , An parallel to AI. If Xl = 0, then the firing activity in pool Al is low, hence the shunting inhibition from pool Bl intercepts those EPSP's that are sent from BN+ to each neuron v in pool C. More precisely, we assume that each pool Bi associated with a different input bit Xi carries out shunting inhibition on a different subtree of the dendritic tree of such neurOn v (where each such subtree receives EPSP's from BN+). If Xl = 1, the higher firing activity in pool Al inhibits the neurons in BI for some time period. Hence during the relevant time interval BN+ contributes an almost constant positive summand to the potential Pv(t) of neurons v in C. By choosing wt and w; appropriately, one can achieve that during this time interval the potential Pv(t) of neurons v in 11 C is arbitrarily much positive if L: Q:iXi ~ e, and arbitrarily much negative if i=1 n L: Q:iXi < e. Hence the activity level of C encodes the output bit of the threshold i=l gate G (in weak coding). The purpose of the subsequent pools D and F is to synchronize (with the help of "double-negation") the output of this module via a pacemaker or synfire chain PM. In this way one can achieve that all input "bits" to another module that simulates a threshold gate On the next layer of circuit T arrive simultaneously. • On the Computational Power of Noisy Spiking Neurons 217 4 ConcI usion Our constructions throw new light on various experimental data, and on our attempts to understand neural computation and coding: a) If One would record all firing times of a few arbitrarily chosen neurons in our networks during many repetitions of the same computation, one is likely to see that each run yields quite different seemingly random firing sequences, where however a few firing patterns will occur more frequently than could be explained by mere chance. This is consistent with the experimental results reported in (Abeles, 1991), and one should also note that the synfire chains of (Abeles, 1991) have many features in common with the here constructed networks. b) If one plugs in biologically realistic values (see (Shepherd, 1990), (Churchland, Sejnowski, 1992)) for the length of transmission delays (around 5 msec) and the duration of EPSP's and IPSP's (around 15 msec for fast PSP's), then the computation time of our modules for NOR- and threshold gates comes out to be not more than 25 msec. Hence in principle a multi-layer perceptron with up to 4 layers can be simulated within 100 msec. c) Our constructions provide new hypotheses about the computational roles of regular and shunting inh'ibition, that go far beyond their usually assumed roles. d) We provide new hypotheses regarding the computational role of randomly firing neurons, and of EPSP's and IPSP's that arrive through synapses at distal parts of biological neurons (see the use of BN+ and BN- in our constructions). References: M. Abeles. (1991) Corticonics: Neural Circuits of the Cerebral Cortex. Cambridge University Press. P. S. Churchland, T. J. Sejnowski. (1992) The Computational Brain. MIT-Press. W. Gerstner, J. L. van Hemmen. (1994) How to describe neuronal activity: spikes, rates, or assemblies? Advances in Neural Information Processing Systems, vol. 6, Morgan Kaufmann: 463-470. W. Maass. (1995a) On the computational complexity of networks of spiking neuronS (extended abstract). Advances in Neural Information Processing Systems, vol. 7 (Proceedings of NIPS '94), MIT-Press, 183-190. W. Maass. (1995b) An efficient implementation of sigmoidal neural nets in temporal coding with noisy spiking neurons. IGI-Report 422 der Technischen Universitiit Graz, submitted for publication. W. Maass. (1996) Lower bounds for the computational power of networks of spiking neurons. N eu.ral Computation 8: 1, to appear. G. M. Shepherd. (1990) The Synaptic Organization of the Brain. Oxford University Press. J. van Leeuwen, ed. (1990) Handbook of Theoretical Computer Science, vol. A: Algorithms and Complexity. MIT-Press.	1995	8
1,128	A Smoothing Regularizer for Recurrent Neural Networks Lizhong Wu and John Moody Oregon Graduate Institute, Computer Science Dept., Portland, OR 97291-1000 Abstract We derive a smoothing regularizer for recurrent network models by requiring robustness in prediction performance to perturbations of the training data. The regularizer can be viewed as a generalization of the first order Tikhonov stabilizer to dynamic models. The closed-form expression of the regularizer covers both time-lagged and simultaneous recurrent nets, with feedforward nets and onelayer linear nets as special cases. We have successfully tested this regularizer in a number of case studies and found that it performs better than standard quadratic weight decay. 1 Introd uction One technique for preventing a neural network from overfitting noisy data is to add a regularizer to the error function being minimized. Regularizers typically smooth the fit to noisy data. Well-established techniques include ridge regression, see (Hoerl & Kennard 1970), and more generally spline smoothing functions or Tikhonov stabilizers that penalize the mth-order squared derivatives of the function being fit, as in (Tikhonov & Arsenin 1977), (Eubank 1988), (Hastie & Tibshirani 1990) and (Wahba 1990). Thes( -ilethods have recently been extended to networks of radial basis functions (Girosi, Jones & Poggio 1995), and several heuristic approaches have been developed for sigmoidal neural networks, for example, quadratic weight decay (Plaut, Nowlan & Hinton 1986), weight elimination (Scalettar & Zee 1988),(Chauvin 1990),(Weigend, Rumelhart & Huberman 1990) and soft weight sharing (Nowlan & Hinton 1992).1 All previous studies on regularization have concentrated on feedforward neural networks. To our knowledge, recurrent learning with regularization has not been reported before. ITwo additional papers related to ours, but dealing only with feed forward networks, came to our attention or were written after our work was completed. These are (Bishop 1995) and (Leen 1995). Also, Moody & Rognvaldsson (1995) have recently proposed several new classes of smoothing regularizers for feed forward nets. A Smoothing Regularizer for Recurrent Neural Networks 459 In Section 2 of this paper, we develop a smoothing regularizer for general dynamic models which is derived by considering perturbations of the training data. We present a closed-form expression for our regularizer for two layer feedforward and recurrent neural networks, with standard weight decay being a special case. In Section 3, we evaluate our regularizer's performance on predicting the U.S. Index of Industrial Production. The advantage of our regularizer is demonstrated by comparing to standard weight decay in both feedforward and recurrent modeling. Finally, we conclude our paper in Section 4. 2 Smoothing Regularization 2.1 Prediction Error for Perturbed Data Sets Consider a training data set {P: Z(t),X(t)}, where the targets Z(t) are assumed to be generated by an unknown dynamical system F(I(t)) and an unobserved noise process: Z(t) = F(I(t» + E(t) with I(t) = {X(s), s = 1,2,···, t} . (1) Here, I(t) is, the information set containing both current and past inputs X(s), and the E(t) are independent random noise variables with zero mean and variance (F2. Consider next a dynamic network model Z(t) = F(~, I(t)) to be trained on data set P, where ~ represents a set of network parameters, and F( ) is a network transfer function which is assumed to be nonlinear and dynamic. We assume that F( ) has good approximation capabilities, such that F(~p,I(t)) ~ F(I(t)) for learnable parameters ~ p. Our goal is to derive a smoothing regularizer for a network trained on the actual data set P that in effect optimizes the expected network performance (prediction risk) on perturbed test data sets of form {Q : Z(t),X(t)}. The elements of Q are related to the elements of P via small random perturbations Ez(t) and Ez(t), so that Z(t) = Z(t) + Ez(t) , (2) X(t) = X(t) + Ez(t) . (3) The Ez(t) and Ez(t) have zero mean and variances (Fz2 and (Fz2 respectively. The training and test errors for the data sets P and Q are N Dp = ~ L [Z(t) - F(~p,I(t))]2 (4) t=l N DQ = ~ L[Z(t) - F(~p,i(t)W , t=l (5) where ~ p denotes the network parameters obtained by training on data set P, and l(t) = {X(s),s = 1,2,··· ,t} is the perturbed information set of Q. With this notation, our goal is to minimize the expected value of DQ, while training on D p. Consider the prediction error for the perturbed data point at time t: d(t) = [Z(t) F(~p,i(t)W . (6) With Eqn (2), we obtain d(t) = [Z(t) + Ez(t) F(~p,I(t)) + F(~p,I(t)) F(~p,i(t)W, [Z(t) F(~p,I(t)W + [F(~p,I(t)) F(~p,l(t)W + [Ez(t)]2 +2[Z(t) F(~p,I(t))JIF(~p,I(t)) F(~p,i(t))] +2Ez(t)lZ(t) F(~p,l(t))]. (7) 460 L. WU. 1. MOODY Assuming that C:z(t) is uncorrelated with [Z(t) F(~p,i(t»] and averaging over the exemplars of data sets P and Q, Eqn(7) becomes 1 N 1 N Dp+ NL[F(~p,I(t»-F(~p,i(t)W+ NL[c:z(t)]2 t=1 t=1 DQ = 2 N + N L[Z(t) F(~p,I(t»)][F(~p,I(t» F(~p,i(t»]. t=l (8) The third term, 2:::'1 [C:z (t)]2, in Eqn(8) is independent of the weights, so it can be neglected during the learning process. The fourth term in Eqn(8) is the crosscovariance between [Z~t) F(~p,I(t»] and [F(~p,I(t» F(~p,i(t»]. Using the inequality 2ab ~ a + b2 , we can see that minimizing the first term D p and the second term ~ 2:~I[F(~p,I(t» F(~p,i(t»]2 in Eqn (8) during training will automatically decrease the effect of the cross-covariance term. Therefore, we exclude the cross-covariance term from the training criterion. The above analysis shows that the expected test error DQ can be minimized by minimizing the objective function D: 1 N 1 N D = N L[Z(t) F(~, I(t»]2 + N L[F(~p, I(t» - F(~ p,i(t»]2. (9) t=l t=l In Eqn (9), the second term is the time average of the squared disturbance IIZ(t) - Z(t)1I2 of the trained network output due to the input perturbation lIi(t) - I(t)W. Minimizing this term demands that small changes in the input variables yield correspondingly small changes in the output. This is the standard smoothness prior, nanlely that if nothing else is known about the function to be approximated, a good option is to assume a high degree of smoothness. Without knowing the correct functional form of the dynamical system F- or using such prior assumptions, the data fitting problem is ill-posed. In (Wu & Moody 1996), we have shown that the second term in Eqn (9) is a dynamic generalization of the first order Tikhonov stabilizer. 2.2 Form of the Proposed Smoothing Regularizer Consider a general, two layer, nonlinear, dynamic network with recurrent connections on the internal layer 2 as described by Yet) = f (WY(t - T) + V X(t» ,Z(t) = UY(t) (10) where X(t), Yet) and Z(t) are respectively the network input vector, the hidden output vector and the network output; ~ = {U, V, W} is the output, input and recurrent connections of the network; f( ) is the vector-valued nonlinear transfer function of the hidden units; and T is a time delay in the feedback connections of hidden layer which is pre-defined by a user and will not be changed during learning. T can be zero, a fraction, or an integer, but we are interested in the cases with a small T.3 20ur derivation can easily be extended to other network structures. 3When the time delay T exceeds some critical value, a recurrent network becomes unstable and lies in oscillatory modes. See, for example, (Marcus & Westervelt 1989). A Smoothing Regularizer for Recurrent Neural Networks 461 When T = 1, our model is a recurrent network as described by (Elman 1990) and (Rumelhart, Hinton & Williams 1986) (see Figure 17 on page 355). When T is equal to some fraction smaller than one, the network evolves ~ times within each input time interval. When T decreases and approaches zero, our model is the same as the network studied by (Pineda 1989), and earlier, widely-studied additive networks. In (Pineda 1989), T was referred to as the network relaxation time scale. (Werbos 1992) distinguished the recurrent networks with zero T and non-zero T by calling them simultaneous recurrent networks and time-lagged recurrent networks respectively. We have found that minimizing the second term of Eqn(9) can be obtained by smoothing the output response to an input perturbation at every time step. This yields, see (Wu & Moody 1996): IIZ(t)-Z(t)W~p/(~p)IIX(t)-X(t)W for t=1,2, ... ,N. (11) We call PT 2 (~ p) the output sensitivity of the trained network ~ p to an input perturbation. PT 2 ( ~ p) is determined by the network parameters only and is independent of the time variable t. We obtain our new regularizer by training directly on the expected prediction error for perturbed data sets Q. Based on the analysis leading to Eqns (9) and (11), the training criterion thus becomes 1 N D = N 2:[Z(t) F(~,I(t)W + .\p/(~) . t=l (12) The coefficient .\ in Eqn(12) is a regularization parameter that measures the degree of input perturbation lIi(t) - I(t)W. The algebraic form for PT(~) as derived in (Wu & Moody 1996) is: P (~)- ,IIUIIIIVII {1(,IIWIl-l)} T 1 _ ,IIWII exp T ' (13) for time-lagged recurrent networks (T > 0). Here, 1111 denotes the Euclidean matrix norm. The factor, depends upon the maximal value of the first derivatives of the activation functions of the hidden units and is given by: , = m~ II/(oj(t)) I , (14) t ,] where j is the index of hidden units and OJ(t) is the input to the ph unit. In general, , ~ 1. 4 To insure stability and that the effects of small input perturbations are damped out, it is required, see (Wu & Moody 1996), that ,IIWII < 1 . (15) The regularizer Eqn(13) can be deduced for the simultaneous recurrent networks in the limit THO by: p(~) = P (~) = ,IIUIIIIVII (16) 0 1 - ,IIWII . If the network is feedforward, W = 0 and T = 0, Eqns (13) and (16) become p(~) = ,11U1I11V1l . (17) Moreover, if there is no hidden layer and the inputs are directly connected to the outputs via U, the network is an ordinary linear model, and we obtain p(~) = IIUII , (18) 4For instance, f'(x} = [1- f(x})f(x} if f(x) = l+!-z. Then, "'{ = max 1 f'(x}} 1= t. 462 L. WU, J. MOODY which is standard quadratic weight decay (Plaut et al. 1986) as is used in ridge regression (Hoerl & Kennard 1970). The regularizer (Eqn(17) for feedforward networks and Eqn (13) for recurrent networks) was obtained by requiring smoothness of the network output to perturbations of data. We therefore refer to it as a smoothing regularizer. Several approaches can be applied to estimate the regularization parameter..x, as in (Eubank 1988), (Hastie & Tibshirani 1990) and (Wahba 1990). We will not discuss this subject in this paper. In the next section, we evaluate the new regularizer for the task of predicting the U.S. Index of Industrial Production. Additional empirical tests can be found in (Wu & Moody 1996). 3 Predicting the U.S. Index of Industrial Production The Index of Industrial Production (IP) is one of the key measures of economic activity. It is computed and published monthly. Our task is to predict the onemonth rate of change of the index from January 1980 to December 1989 for models trained from January 1950 to December 1979. The exogenous inputs we have used include 8 time series such as the index of leading indicators, housing starts, the money supply M2, the S&P 500 Index. These 8 series are also recorded monthly. In previous studies by (Moody, Levin & Rehfuss 1993), with the same defined training and test data sets, the normalized prediction errors of the one month rate of change were 0.81 with the neuz neural network simulator, and 0.75 with the proj neural network simulator. We have simulated feedforward and recurrent neural network models. Both models consist of two layers. There are 9 input units in the recurrent model, which receive the 8 exogenous series and the previous month IP index change. We set the time-delayed length in the recurrent connections T = 1. The feedforward model is constructed with 36 input units, which receive 4 time-delayed versions of each input series. The time-delay lengths a,re 1, 3, 6 and 12, respectively. The activation functions of hidden units in both feedforward and recurrent models are tanh functions. The number of hidden units varies from 2 to 6. Each model has one linear output unit. We have divided the data from January 1950 to December 1979 into four nonoverlapping sub-sets. One sub-set consists of 70% of the original data and each of the other three subsets consists of 10% of the original data. The larger sub-set is used as training data and the three smaller sub-sets are used as validation data. These three validation data sets are respectively used for determination of early stopped training, selecting the regularization parameter and selecting the number of hidden units. We have formed 10 random training-validation partitions. For each trainingvalidation partition, three networks with different initial weight parameters are trained. Therefore, our prediction committee is formed by 30 networks. The committee error is the average of the errors of all committee members. All networks in the committee are trained simultaneously and stopped at the same time based on the committee error of a validation set. The value of the regularization parameter and the number of hidden units are determined by minimizing the committee error on separate validation sets. Table 1 compares the out-of-sample performance of recurrent networks and feedforA Smoothing Regularizer for Recurrent Neural Networks 463 Table 1: Nonnalized prediction errors for the one-month rate of return on the U.S. Index of Industrial Production (Jan. 1980 - Dec. 1989). Each result is based on 30 networks. Model Regularizer Mean ± Std Median Max Min Committee Recurrent Smoothing 0.646±0.008 0.647 0.657 0.632 0.639 Networks Weight Decay 0.734±0.018 0.737 0.767 0.704 0.734 Feedforward Smoothing 0.700±0.023 0.707 0.729 0.654 0.693 Networks Weight Decay 0.745±0.043 0.748 0.805 0.676 0.731 ward networks trained with our smoothing regularizer to that of networks trained with standard weight decay. The results are based on 30 networks. As shown, the smoothing regularizer again outperfonns standard weight decay with 95% confidence (in t-distribution hypothesis) in both cases of recurrent networks and feedforward networks. We also list the median, maximal and minimal prediction errors over 30 predictors. The last column gives the committee results, which are based on the simple average of 30 network predictions. We see that the median, maximal and minimal values and the committee results obtained with the smoothing regularizer are all smaller than those obtained with standard weight decay, in both recurrent and feedforward network models. 4 Concluding Remarks Regularization in learning can prevent a network from overtraining. Several techniques have been developed in recent years, but all these are specialized for feedforward networks. To our best knowledge, a regularizer for a recurrent network has not been reported previously. We have developed a smoothing regularizer for recurrent neural networks that captures the dependencies of input, output, and feedback weight values on each other. The regularizer covers both simultaneous and time-lagged recurrent networks, with feedforward networks and single layer, linear networks as special cases. Our smoothing regularizer for linear networks has the same fonn as standard weight decay. The regularizer developed depends on only the network parameters, and can easily be used. A more detailed description of this work appears in (Wu & Moody 1996). References Bishop, C. (1995), 'Training with noise is equivalent to Tikhonov regularization', Neural Computation 7(1), 108-116. Chauvin, Y. (1990), Dynamic behavior of constrained back-propagation networks, in D. Touretzky, ed., 'Advances in Neural Infonnation Processing Systems 2', Morgan Kaufmann Publishers, San Francisco, CA, pp. 642-649. Elman, J. (1990), 'Finding structure in time', Cognition Science 14, 179-211. Eubank, R. L. (1988), Spline Smoothing and Nonparametric Regression, Marcel Dekker, Inc. Girosi, F., Jones, M. & Poggio, T. (1995), 'Regularization theory and neural networks architectures', Neural Computation 7, 219-269. 464 L. WU, J. MOODY Hastie, T. J. & Tibshirani, R. J. (1990), Generalized Additive Models, Vol. 43 of Monographs on Statistics and Applied Probability, Chapman and Hall. Hoerl, A. & Kennard, R. (1970), 'Ridge regression: biased estimation for nonorthogonal problems', Technometrics 12, 55-67. Leen, T. (1995), 'From data distributions to regularization in invariant learning', Neural Computation 7(5), 974-98l. Marcus, C. & Westervelt, R. (1989), Dynamics of analog neural networks with time delay, in D. Touretzky, ed., 'Advances in Neural Information Processing Systems 1', Morgan Kaufmann Publishers, San Francisco, CA. Moody, J. & Rognvaldsson, T. (1995), Smoothing regularizers for feed-forward neural networks, Oregon Graduate Institute Computer Science Dept. Technical Report, submitted for publication, 1995. Moody, J., Levin, U. & Rehfuss, S. (1993), 'Predicting the U.S. index of industrial production', In proceedings of the 1993 Parallel Applications in Statistics and Economics Conference, Zeist, The Netherlands. Special issue of Neural Network World 3(6), 791-794. Nowlan, S. & Hinton, G. (1992), 'Simplifying neural networks by soft weightsharing', Neural Computation 4(4), 473-493. Pineda, F. (1989), 'Recurrent backpropagation and the dynamical approach to adaptive neural computation', Neural Computation 1(2), 161-172. Plaut, D., Nowlan, S. & Hinton, G. (1986), Experiments on learning by back propagation, Technical Report CMU-CS-86-126, Carnegie-Mellon University. Rumelhart, D., Hinton, G. & Williams, R. (1986), Learning internal representations by error propagation, in D. Rumelhart & J. McClelland, eds, 'Parallel Distributed Processing: Exploration in the microstructure of cognition', MIT Press, Cambridge, MA, chapter 8, pp. 319-362. Scalettar, R. & Zee, A. (1988), Emergence of grandmother memory in feed forward networks: learning with noise and forgetfulness, in D. Waltz & J. Feldman, eds, 'Connectionist Models and Their Implications: Readings from Cognitive Science', Ablex Pub. Corp. Tikhonov, A. N. & Arsenin, V. 1. (1977), Solutions of Ill-posed Problems, Winston; New York: distributed solely by Halsted Press. Scripta series in mathematics. Translation editor, Fritz John. Wahba, G. (1990), Spline models for observational data, CBMS-NSF Regional Conference Series in Applied Mathematics. Weigend, A., Rumelhart, D. & Huberman, B. (1990), Back-propagation, weightelimination and time series prediction, in T. Sejnowski, G. Hinton & D. Touretzky, eds, 'Proceedings of the connectionist models summer school', Morgan Kaufmann Publishers, San Mateo, CA, pp. 105-116. Werbos, P. (1992), Neurocontrol and supervised learning: An overview and evaluation, in D. White & D. Sofge, eds, 'Handbook of Intelligent Control', Van Nostrand Reinhold, New York. Wu, L. & Moody, J. (1996), 'A smoothing regularizer for feedforward and recurrent neural networks', Neural Computation 8(3), 463-491.	1995	80
1,129	REMAP: Recursive Estimation and Maximization of A Posteriori Probabilities - Application to Transition-Based Connectionist Speech Recognition Yochai Konig, Herve Bourlard~ and Nelson Morgan {konig, bourlard,morgan }@icsi.berkeley.edu International Computer Science Institute 1947 Center Street Berkeley, CA 94704, USA. Abstract In this paper, we introduce REMAP, an approach for the training and estimation of posterior probabilities using a recursive algorithm that is reminiscent of the EM-based Forward-Backward (Liporace 1982) algorithm for the estimation of sequence likelihoods. Although very general, the method is developed in the context of a statistical model for transition-based speech recognition using Artificial Neural Networks (ANN) to generate probabilities for Hidden Markov Models (HMMs). In the new approach, we use local conditional posterior probabilities of transitions to estimate global posterior probabilities of word sequences. Although we still use ANNs to estimate posterior probabilities, the network is trained with targets that are themselves estimates of local posterior probabilities. An initial experimental result shows a significant decrease in error-rate in comparison to a baseline system. 1 INTRODUCTION The ultimate goal in speech recognition is to determine the sequence of words that has been uttered. Classical pattern recognition theory shows that the best possible system (in the sense of minimum probability of error) is the one that chooses the word sequence with the maximum a posteriori probability (conditioned on the * Also affiliated with with Faculte Poly technique de Mons, Mons, Belgium REMAP: Recursive Estimation and Maximization of A Posteriori Probabilities 389 evidence). If word sequence i is represented by the statistical model M i , and the evidence (which, for the application reported here, is acoustical) is represented by a sequence X = {Xl, ... , X n , ... , X N }, then we wish to choose the sequence that corresponds to the largest P(MiIX). In (Bourlard & Morgan 1994), summarizing earlier work (such as (Bourlard & Wellekens 1989)), we showed that it was possible to compute the global a posteriori probability P(MIX) of a discriminant form of Hidden Markov Model (Discriminant HMM), M, given a sequence of acoustic vectors X. In Discriminant HMMs, the global a posteriori probability P(MIX) is computed as follows: if r represents all legal paths (state sequences ql, q2, ... , qN) in Mi, N being the length of the sequence, then P(Mi IX) = L P(Mi, ql, q2, ... , qNIX) r in which ~n represents the specific state hypothesized at time n, from the set Q = {ql, ... , q , qk, ... , qK} of all possible HMM states making up all possible models Mi. We can further decompose this into: P(Mi, ql, q2,···, qNIX) = P(ql, q2,···, qNIX)P(Milql, q2,···, qN, X) Under the assumptions stated in (Bourlard & Morgan 1994) we can compute N P(ql, q2,···, qNIX) = II p(qnlqn-l, xn) n=l The Discriminant HMM is thus described in terms of conditional transition probabilities p(q~lq~-l' xn), in which q~ stands for the specific state ql of Q hypothesized at time n and can be schematically represented as in Figure 1. P(IkIIIkI, x) p(/aell/ael, x) P(ltIlltI, x) P(/aelllkl, x) P(ltll/ael, x) Figure 1: An example Discriminant HMM for the word "cat". The variable X refers to a specific acoustic observation Xn at time n. Finally, given a state sequence we assume the following approximation: P(Milql, q2,···, qN, X) :::::::: P(Milql, q2,···, qN) We can estimate the right side of this last equation from a phonological model (in the case that a given state sequence can belong to two different models). All the required (local) conditional transition probabilities p(q~lq~-l> xn) can be estimated by the Multi-Layer Perceptron (MLP) shown in Figure 2. Recent work at lesl has provided us with further insight into the discriminant HMM, particularly in light of recent work on transition-based models (Konig & Morgan 1994j Morgan et al. 1994). This new perspective has motivated us to further develop the original Discriminant HMM theory. The new approach uses posterior probabilities at both local and global levels and is more discriminant in nature. In this paper, we introduce the Recursive Estimation-Maximization of A posteriori 390 Y. KONIG, H. BOURLARD, N. MORGAN P(CurrenCstlte I Acoustics, Prevlous_stlte) t ···· .. t 0.1 •• 0 Previous Stlte t t t t Acoustics Figure 2: An MLP that estimates local conditional transition probabilities. Probabilities (REMAP) training algorithm for hybrid HMM/MLP systems. The proposed algorithm models a probability distribution over all possible transitions (from all possible states and for all possible time frames n) rather than picking a single time point as a transition target. Furthermore, the algorithm incrementally increases the posterior probability of the correct model, while reducing the posterior probabilities of all other models. Thus, it brings the overall system closer to the optimal Bayes classifier. A wide range of discriminant approaches to speech recognition have been studied by researchers (Katagiri et al. 1991; Bengio et al. 1992; Bourlard et al. 1994). A significant difficulty that has remained in applying these approaches to continuous speech recognition has been the requirement to run computationally intensive algorithms on all of the rival sentences. Since this is not generally feasible, compromises must always be made in practice. For instance, estimates for all rival sentences can be derived from a list of the "N-best" utterance hypotheses, or by using a fully connected word model composed of all phonemes. 2 REMAP TRAINING OF THE DISCRIMINANT HMM 2.1 MOTIVATIONS The discriminant HMM/MLP theory as described above uses transition-based probabilities as the key building block for acoustic recognition. However, it is well known that estimating transitions accurately is a difficult problem (Glass 1988). Due to the inertia of the articulators, the boundaries between phones are blurred and overlapped in continuous speech. In our previous hybrid HMM/MLP system, targets were typically obtained by using a standard forced Viterbi alignment (segmentation). For a transition-based system as defined above, this procedure would thus yield rigid transition targets, which is not realistic. Another problem related to the Viterbi-based training of the MLP presented in Figure 2 and used in Discriminant HMMs, is the lack of coverage of the input space during training. Indeed, during training (based on hard transitions), the MLP only processes inputs consisting of "correct" pairs of acoustic vectors and correct previous state, while in recognition the net should generalize to all possible combinations of REMAP: Recursive Estimation and Maximization of A Posteriori Probabilities 391 acoustic vectors and previous states, since all possible models and transitions will be hypothesized for each acoustic input. For example, some hypothesized inputs may correspond to an impossible condition that has thus never been observed, such as the acoustics of the temporal center of a vowel in combination with a previous state that corresponds to a plosive. It is unfortunately possible that the interpolative capabilities of the network may not be sufficient to give these "impossible" pairs a sufficiently low probability during recognition. One possible solution to these problems is to use a full MAP algorithm to find transition probabilities at each frame for all possible transitions by a forward-backwardlike algorithm (Liporace 1982), taking all possible paths into account. 2.2 PROBLEM FORMULATION As described above, global maximum a posteriori training of HMMs should find the optimal parameter set e maximizing J II P(Mj IXj, e) (1) j=1 in which Mj represents the Markov model associated with each training utterance Xj, with j = 1, ... , J. Although in principle we could use a generalized back-propagation-like gradient procedure in e to maximize (1) (Bengio et al. 1992), an EM-like algorithm should have better convergence properties, and could preserve the statistical interpretation of the ANN outputs. In this case, training of the discriminant HMM by a global MAP criterion requires a solution to the following problem: given a trained MLP at iteration t providing a parameter set et and, consequently, estimates of P(q~lxn' q~-I' et ), how can we determine new MLP targets that: 1. will be smooth estimates of conditional transition probabilities q~-1 -+ q~, Vk,f E [1, K] and "In E [1, N], 2. when training the MLP for iteration t+ 1, will lead to new estimates of et+l and P(q~lxn' q~-I' et+1) that are guaranteed to incrementally increase the global posterior probability P(MiIX, e)? In (Bourlard et al. 1994), we prove that a re-estimate of MLP targets that guarantee convergence to a local maximum of (1) is given by1: (2) where we have estimated the left-hand side using a mapping from the previous state and the local acoustic data to the current state, thus making the estimator realizable by an MLP with a local acoustic window. 2 Thus, we will want to estimate 1 In most of the following, we consider only one particular training sequence X associated with one particular model M. It is, however, easy to see that all of our conclusions remain valid for the case of several training sequences Xj, j = 1, ... , J. A simple way to look at the problem is to consider all training sequences as a single training sequence obtained by concatenating all the X,'s with boundary conditions at every possible beginning and ending point. 2Note that, as done in our previous hybrid HMM/MLP systems, all conditional on Xn can be replaced by X;::!: = {x n - c , ., •. , X n , .•• , Xn+d} to take some acoustic context into account. 392 Y. KONIG, H. BOURLARD, N. MORGAN the transition probability conditioned on the local data (as MLP targets) by using the transition probability conditioned on all of the data. In (Bourlard et al. 1994), we further prove that alternating MLP target estimation (the "estimation" step) and MLP training (the" maximization" step) is guaranteed to incrementally increase (1) over t.3 The remaining problem is to find an efficient algorithm to express P(q~IX, q~-l' M) in terms of P(q~lxn, q~-l) so that the next iteration targets can be found. We have developed several approaches to this estimation, some of which are described in (Bourlard et al. 1994). Currently, we are implementing this with an efficient recursion that estimates the sum of all possible paths in a model, for every possible transition at each possible time. From these values we can compute the desired targets (2) for network training by P( t IX M k ) = P(M, q~, ~~_lIX) qn , , qn-l ~ . P(M J k IX) DJ ,qn, qn-l (3) 2.3 REMAP TRAINING ALGORITHM The general scheme of the REMAP training of hybrid HMM/MLP systems can be summarized as follow: 1. Start from some initial net providing P(q~lxn' q~-l' et ), t = 0, V possible (k,£)-pairs4. 2. Compute MLP targets P(q~IXj,q~_l,et,Mj) according to (3), V training sentences Xj associated with HMM Mj, V possible (k, £) state transition pairs in Mj and V X n, n = 1, ... , N in Xj (see next point). 3. For every Xn in the training database, train the MLP to minimize the relative entropy between the outputs and targets. See (Bourlard et ai, 1994) for more details. This provides us with a new set of parameters et , for t = t + 1. 4. Iterate from 2 until convergence. This procedure is thus composed of two steps: an Estimation (E) step, corresponding to step 2 above, and a Maximization (M) step, corresponding to step 3 above. In this regards, it is reminiscent of the Estimation-Maximization (EM) algorithm as discussed in (Dempster et al. 1977). However, in the standard EM algorithm, the M step involves the actual maximization of the likelihood function. In a related approach, usually referred to as Generalized EM (GEM) algorithm, the M step does not actually maximize the likelihood but simply increases it (by using, e.g., a gradient procedure). Similarly, REMAP increases the global posterior function during the M step (in the direction of targets that actually maximize that global function), rather than actually maximizing it. Recently, a similar approach was suggested for mapping input sequences to output sequences (Bengio & Frasconi 1995). 3Note here that one "iteration" does not stand for one iteration of the MLP training but for one estimation-maximization iteration for which a complete MLP training will be required. 4This can be done, for instance, by training up such a net from a hand-labeled database like TIMIT or from some initial forward-backward estimator of equivalent local probabilities (usually referred to as "gamma" probabilities in the Baum-Welch procedure). REMAP: Recursive Estimation and Maximization of A Posteriori Probabilities 393 System Error Rate DHMM, pre-REMAP 14.9% 1 REMAP iteration 13.6% 2 REMAP iterations 13.2% Table 1: Training and testing on continuous numbers, no syntax, no durational models. 3 EXPERIMENTS AND RESULTS For testing our theory we chose the Numbers'93 corpus. It is a continuous speech database collected by CSLU at the Oregon Graduate Institute. It consists of numbers spoken naturally over telephone lines on the public-switched network (Cole et al. 1994). The Numbers'93 database consists of 2167 speech files of spoken numbers produced by 1132 callers. We used 877 of these utterances for training and 657 for cross-validation and testing (200 for cross-validation) saving the remaining utterances for final testing purposes. There are 36 words in the vocabulary, namely zero, oh, 1, 2, 3, ... ,20, 30, 40, 50, ... ,100, 1000, a, and, dash, hyphen, and double. All our nets have 214 inputs: 153 inputs for the acoustic features, and 61 to represent the previous state (one unit for every possible previous state, one state per phoneme in our case). The acoustic features are combined from 9 frames with 17 features each (RASTA-PLP8 + delta features + delta log gain) computed with an analysis window of 25 ms computed every 12.5 ms (overlapping windows) and with a sampling rate of 8 Khz. The nets have 200 hidden units and 61 outputs. Our results are summarized in Table 1. The row entitled "DHMM, pre-REMAP" corresponds to a Discriminant HMM using the same training approach, with hard targets determined by the first system, and additional inputs to represent the previous state The improvement in the recognition rate as a result of REMAP iterations is significant at p < 0.05. However all the experiments were done using acoustic information alone. Using our (baseline) hybrid system under equal conditions, i.e., no duration information and no language information, we get 31.6% word error; adding the duration information back we get 12.4% word error. We are currently experimenting with enforcing minimum duration constraints in our framework. 4 CONCLUSIONS In summary: • We have a method for MAP training and estimation of sequences. • This can be used in a new form of hybrid HMM/MLP. Note that recurrent nets or TDNNs could also be used. As with standard HMM/MLP hybrids, the network is used to estimate local posterior probabilities (though in this case they are conditional transition probabilities, that is, state probabilities conditioned on the acoustic data and the previous state). However, in the case of REMAP these nets are trained with probabilistic targets that are themselves estimates of local posterior probabilities. • Initial experiments demonstrate a significant reduction in error rate for this process. 394 Y. KONIG, H. BOURLARD, N. MORGAN Acknowledgments We would like to thank Kristine Ma and Su-Lin Wu for their help with the Numbers'93 database. We also thank OGI, in particular to Ron Cole, for providing the database. We gratefully acknowledge the support of the Office of Naval Research, URI No. N00014-92-J-1617 (via UCB), the European Commission via ESPRIT project 20077 (SPRACH), and ICSI and FPMs in general for supporting this work. References BENGIO, Y., & P. FRASCONI. 1995. An input output HMM architecture. In Advances in Neural Information Processing Systems, ed. by G. Tesauro, D. Touretzky, & T. Leen, volume 7. Cambridge: MIT press. --, R. DE MORI, G. FLAMMIA, & R. KOMPE. 1992. Global optimization of a neural network-hidden Markov model hybrid. IEEE trans. on Neural Networks 3.252-258. BOURLARD, H., Y. KONIG, & N. MORGAN. 1994. REMAP: Recursive estimation and maximization of a posteriori probabilities, application to transition-based connectionist speech recognition. Technical Report TR-94-064, International Computer Science Institute, Berkeley, CA. --, & N. MORGAN. 1994. Connectionist Speech Recognition - A Hybrid Approach. Kluwer Academic Publishers. --, & C. J. WELLEKENS. 1989. Links between Markov models and multilayer perceptrons. In Advances in Neural Information Processing Systems 1, ed. by D.J. Touretzky, 502-510, San Mateo. Morgan Kaufmann. COLE, R.A., M. FANTY, & T. LANDER. 1994. Telephone speech corpus development at CSL U. In Proceedings Int 'I Conference on Spoken Language Processing, Yokohama, Japan. DEMPSTER, A. P., N. M. LAIRD, & D. B. RUBIN. 1977. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B 34.1-38. GLASS, J. R., 1988. Finding Acoustic Regularities in Speech Applications to Phonetic Recognition. M.LT dissertation. KATAGIRI, S., C.H. LEE, & JUANG B.H. 1991. New discriminative training algorithms based on the generalized probabilistic decent method. In Proc. of the IEEE Workshop on Neural Netwroks for Signal Processing, ed. by RH. Juang, S.Y. Kung, & C.A. Kamm, 299-308. KONIG, Y., & N. MORGAN. 1994. Modeling dynamics in connectionist speech recognition - the time index model. In Proceedings Int'l Conference on Spoken Language Processing, 1523-1526, Yokohama, Japan. LIPORACE, L. A. 1982. Maximum likelihood estimation for multivariate observations of markov sources. IEEE Trans. on Information Theory IT-28.729-734. MORGAN, N., H. BOURLARD, S. GREENBERG, & H. HERMANSKY. 1994. Stochastic perceptual auditory-event-based models for speech recognition. In Proceedings Int'l Conference on Spoken Language Processing, 1943-1946, Yokohama, Japan.	1995	81
1,130	Harmony Networks Do Not Work Rene Gourley School of Computing Science Simon Fraser University Burnaby, B.C., V5A 1S6, Canada gourley@mprgate.mpr.ca Abstract Harmony networks have been proposed as a means by which connectionist models can perform symbolic computation. Indeed, proponents claim that a harmony network can be built that constructs parse trees for strings in a context free language. This paper shows that harmony networks do not work in the following sense: they construct many outputs that are not valid parse trees. In order to show that the notion of systematicity is compatible with connectionism, Paul Smolensky, Geraldine Legendre and Yoshiro Miyata (Smolensky, Legendre, and Miyata 1992; Smolen sky 1993; Smolen sky, Legendre, and Miyata 1994) proposed a mechanism, "Harmony Theory," by which connectionist models purportedly perform structure sensitive operations without implementing classical algorithms. Harmony theory describes a "harmony network" which, in the course of reaching a stable equilibrium, apparently computes parse trees that are valid according to the rules of a particular context-free grammar. Harmony networks consist of four major components which will be explained in detail in Section 1. The four components are, Tensor Representation: A means to interpret the activation vector of a connectionist system as a parse tree for a string in a context-free language. Harmony: A function that maps all possible parse trees to the non-positive integers so that a parse tree is valid if and only if its harmony is zero. Energy: A function that maps the set of activation vectors to the real numbers and which is minimized by certain connectionist networks!. Recursive Construction: A system for determining the weight matrix of a connectionist network so that if its activation vector is interpreted as a parse 1 Smolensky, Legendre and Miyata use the term "harmony" to refer to both energy and harmony. To distinguish between them, we will use the term that is often used to describe the Lyapunov function of dynamic systems, "energy" (see for example Golden 1986). 32 R. GOURLEY tree, then the network's energy is the negation of the harmony of that parse tree. Smolen sky et al. contend that, in the process of minimizing their energy values, harmony networks implicitly maximize the harmony of the parse tree represented by their activation vector. Thus, if the harmony network reaches a stable equilibrium where the energy is equal to zero, the parse tree that is represented by the activation vector must be a valid parse tree: When the lower-level description of the activation-spreading process satisfies certain mathematical properties, this process can be analyzed on a higher level as the construction of that structure including the given input structure which maximizes Harmony. (Smolensky 1993, p848, emphasis is original) Unfortunately, harmony networks do not work they do not always construct maximum-harmony parse trees. The problem is that the energy function is defined on the values of the activation vector. By contrast, the harmony function is defined on possible parse trees. Section 2 of this paper shows that these two domains are not equal, that is, there are some activation vectors that do not represent any parse tree. The recursive construction merely guarantees that the energy function passes through zero at the appropriate points; its minima are unrestricted. So, while it may be the case that the energy and harmony functions are negations of one another, it is not always the case that a local minimum of one is a local maximum of the other. More succinctly, the harmony network will find minima that are not even trees, let alone valid parse trees. The reason why harmony networks do not work is straightforward. Section 3 shows that the weight matrix must have only negative eigenvalues, for otherwise the network constructs structures which are not valid trees. Section 4 shows that if the weight matrix has only negative eigenvalues, then the energy function admits only a single zero the origin. Furthermore, we show that the origin cannot be interpreted as a valid parse tree. Thus, the stable points of a harmony network are not valid parse trees. 1 HARMONY NETWORKS 1.1 TENSOR REPRESENTATION Harmony theory makes use of tensor products (Smolensky 1990; Smolensky, Legendre, and Miyata 1992; Legendre, Miyata, and Smolensky 1991) to convolve symbols with their roles. The resulting products are then added to represent a labelled tree using the harmony network's activation vector. The particular tensor product used is very simple: (aI, a2,· · ·, an) <8> (bl , b2,.·., bm ) = (albl , alb2, ... , a}bm , a2bl, a2b2, ... , a2bm, .. . , anbm ) If two tensors of differing dimensions are to be added, then they are essentially concatenated. Binary trees are represented with this tensor product using the following recursive rules: 1. The tensor representation of a tree containing no vertices is O. Harmony Networks Do Not Work 33 Table 1: Rules for determining harmony and the weight matrix. Let G = (V, E, P, S) be a context-free grammar of the type suggested in section 1.2. The rules for determining the harmony of a tree labelled with V and E are shown in the second column. The rules for determining the system of equations for recursive construction are shown in the third column. (Smolensky, Legendre, and Miyata 1992; Smolensky 1993) Grammar Harmony Rule Energy Equation Element S For every node labelled Include (S+00r,)Wroot(S+00rr) = 2 S add -1 to H(T). in the system of equations xEE For every node labelled Include (x +60r,)Wroot (x +60r,) = 2 x add -1 to H(T). in the system of equations For every node labelled x add -2 or -3 to H(T) Include (x+60r,)Wroot(x+00r,) = 4 x E V\ depending on whether or 6 in the system of equations, depend{S} or not x appears on ing on whether or not x appears on the the left of a producleft of a production with two symbols tion with two symbols on the right. on the right. For every edge where Include in the system of equations, x yz x is the parent and y (x + 60 r,)Wroot (6 + y 0 r,) = -2 or x is the left child add 2. (0 + y 0 r,)Wroot(x + 60 r,) = -2 yE P Similarly, add 2 every (x + 60 r,)Wroot(O + z 0 r,) = -2 time z is the right child of x. (6 + z 0 r,)Wroot(x + 6® r,) = -2 2. If A is the root of a tree, and TL, TR are the tensor product representations of its left subtree and right subtree respectively, then A + TL 0 r, + TR 0 rr is the tensor representation of the whole tree. The vectors, r" and rr are called "role vectors" and indicate the roles of left child and right child. 1.2 HARMONY Harmony (Legendre, Miyata, and Smolensky 1990; Smolensky, Legendre, and Miyata 1992) describes a way to determine the well-formedness of a potential parse tree with respect to a particular context free grammar. Without loss of generality, we can assume that the right-hand side of each production has at most two symbols, and if a production has two symbols on the right, then it is the only production for the variable on its left side. For a given binary tree, T, we compute the harmony of T, H(T) by first adding the negative contributions of all the nodes according to their labels, and then adding the contributions of the edges (see first two columns of table 1). 34 R.GOURLEY 1.3 ENERGY Under certain conditions, some connectionist models are known to admit the following energy or Lyapunov function (see Legendre, Miyata, and Smolensky 1991): 1 E(a) = --atWa 2 Here, W is the weight matrix of the connectionist network, and a is its activation vector. Every non-equilibrium change in the activation vector results in a strict decrease in the network's energy. In effect, the connectionist network serves to minimize its energy as it moves towards equilibrium. 1.4 RECURSIVE CONSTRUCTION Smolensky, Legendre, and Miyata (1992) proposed that the recursive structure of their tensor representations together with the local nature of the harmony calculation could be used to construct the weight matrix for a network whose energy function is the negation of the harmony of the tree represented by the activation vector. First construct a matrix W root which satisfies a system of equations. The system of equations is found by including equations for every symbol and production in the grammar, as shown in column three of table 1. Gourley (1995) shows that if W is constructed from copies of W root according to a particular formula, and if aT is a tensor representation for a tree, T, then E(aT) = -H(T). 2 SOME ACTIVATIONS ARE NOT TREES As noted above, the reason why harmony networks do not work is that they seek minima in their state space which may not coincide with parse tree representations. One way to amelioarate this would be to make every possible activation vector represent some parse tree. If every activation vector represents some parse tree, then the rules that determine the weight matrix will ensure that the energy minima agree with the valid parse trees. Unfortunately, in that case, the system of equations used to determine W root has no solution. If every activation vector is to represent some parse tree, and the symbols of the grammar are two dimensional, then there are symbols represented by each vector, (Xl, xt), (Xl, X2), (X2' xt), and (X2' X2), where Xl 1= X2 . These symbols must satisfy the equations given in table 1 , and so, Xi{Wrootll + Wroot12 + Wroot~l + Wroot~~) XiWrootll + XIX2 W root12 + XIX2 W root:n + x~Wroot:n X~Wrootll + XIX2Wrootl~ + XIX2 W root:n + xiWroot~2 x~(Wrootll + Wroot12 + Wroot~l + Wrootn) Because hi E {2, 4, 6}, there must be a pair hi, hj which are equal. In that case, it can be shown using Gaussian elimination that there is no solution for Wrootll , Wrootl~' Wroot~l , Wroot~~. Similarly, if the symbols are represented by vectors of dimension three or greater, the same contradiction occurs. Thus there are some activation vectors that do not represent any tree valid or invalid. The question now becomes one of determining whether all of the harmony network's stable equilibria are valid parse trees. Harmony Networks Do Not Work 35 a b Figure 1: Energy functions of two-dimensional harmony networks. In each case, the points i and f respectively represent an initial and a final state of the network. In a, one eigenvector is positive and the other is negative; the hashed plane represents the plane E = 0 which intersects the energy function and the vertical axis at the origin. In b, one eigenvalue is negative while the other is zero; The heavy line represents the intersection of the surface with the plane E = 0 and it intersects the vertical axis at the origin. 3 NON-NEGATIVE EIGENVECTORS YIELD NON-TREES If any of the eigenvalues of the weight matrix, W, is positive, then it is easy to show that the harmony network will seek a stable equilibrium that does not represent a parse tree at all. Let A > 0 be a positive eigenvalue of W, and let e be an eigenvector, corresponding to A, that falls within the state space. Then, 1 1 E(e) = --etWe = --Aete < O. 2 2 Because the energy drops below zero, the harmony network would have to undergo an energy increase in order to find a zero-energy stable equilibrium. This cannot happen, and so, the network reaches an equilibrium with energy strictly less than zero. Figure la illustrates the energy function of a harmony network where one eigenvalue is positive. Because harmony is the negation of energy, in this figure all the valid parse trees rest on the hashed plane, and all the invalid parse trees are above it. As we can see, the harmony network with positive eigenvalues will certainly find stable equilibria which are not valid parse tree representations. Now, suppose W, the weight matrix, has a zero eigenvalue. If e is an eigenvector corresponding to that eigenvalue, then for every real a, aWe = O. Consequently, one of the following must be true: 1. ae is not a stable equilibrium. In that case, the energy function must drop below zero, yielding a sub-zero stable equilibrium a stable equilibrium that does not represent any tree. 2. ae is a stable equilibrium. Then for every a, ae must be a valid tree representation. Such a situation is represented in fig36 R. GOURLEY Figure 2: The energy function of a two-dimensional harmony network where both eigenvalues are negative. The vertical axis pierces the surface at the origin, and the points i and f respectively represent an initial and a final state of the network. ure Ib where the set of all points ae is represented by the heavy line. This implies that there is a symbol, (al, a2, . . . , an), such that Ckl(al , a2, .. . ,an),Ck2(al,a2, . . . ,an), .. . ,an2+l(al,a2, ... , an) are also all symbols. As before, this implies that Wroot must satisfy the equation, t hi hi E «al, ... , an) + 0 ® r,) Wroot«al, ... , an) + 0 0 r,) 2" ' {2 4 6} a · " , for i = 1 ... n2 + 1. Again using Gaussian elimination, it can be shown that there is no solution to this system of equations. In either case, the harmony network admits stable equilibria that do not represent any tree. Thus, the eigenvalues must all be negative. 4 NEGATIVE EIGENVECTORS YIELD NON-TREES If all the eigenvalues of the weight matrix are negative, then the energy function has a very special shape: it is a paraboloid centered on the origin and concave in the direction of positive energy. This is easily seen by considering the first and second derivatives of E: 8E(x) __ ~ W, .. x . 8 2 E(x) - -W, . . 8x; L..j '.1' 8x;8x; '.1 Clearly, all the first derivatives are zero at the origin, and so, it is a critical point. Now the origin is a strict minimum if all the roots of the following well-known equation are positive: 0= det = det I-W - All det 1- W - All is the characteristic polynomial of -W. If A is a root then it is an eigenvalue of - W, or equivalently, it is the negative of an eigenvalue of W . Because all of W's eigenvalues are negative, the origin is a strict minimum, and indeed it is the only minimum. Such a harmony network is illustrated in Figure 2. Hannony Networks Do Not Work 37 Thus the origin is the only stable point where the energy is zero, but it cannot represent a parse tree which is valid for the grammar. If it does, then S + TL 0 r, + TR (9 rr = (0, . . . ,0) where TL, TR are appropriate left and right subtree representations, and S is the start symbol of the grammar. Because each of the subtrees is multiplied by either r, or rr, they are not the same dimension as S, and are consequently concatenated instead of added. Therefore S = O. But then, Wroot must satisfy the equation (0 + 0 (9 r,)Wroot(O + 0 (9 r,) =-2 This is impossible, and so, the origin is not a valid tree representation. 5 CONCLUSION This paper has shown that in every case, a harmony network will reach stable equilibria that are not valid parse trees. This is not unexpected. Because the energy function is a very simple function, it would be more surprising if such a connectionist system could construct complicated structures such as parse trees for a context free grammar. Acknowledgements The author thanks Dr. Robert Hadley and Dr. Arvind Gupta, both of Simon Fraser University, for their invaluable comments on a draft of this paper. References Golden, R. (1986). The 'brain-state-in-a-box' neural model is a gradient descent algorithm. Journal of Mathematical Psychology 30, 73-80. Gourley, R. (1995). Tensor represenations and harmony theory: A critical analysis. Master's thesis, Simon Fraser University, Burnaby, Canada. In preparation. Legendre, G., Y. Miyata, and P. Smolensky (1990). Harmonic grammar - a formal multi-level connectionist theory of linguistic well-formedness: Theoretical foundations. In Proceedings of the Twelfth National Conference on Cognitive Science, Cambridge, MA, pp. 385- 395. Lawrence Erlbaum. Legendre, G., Y. Miyata, and P. Smolensky (1991). Distributedrecursive structure processing. In B. Mayoh (Ed.), Proceedings of the 1991 Scandinavian Conference on Artificial Intelligence, Amsterdam, pp. 47-53. lOS Press. Smolensky, P. (1990). Tensor product variable binding and the representation of symbolic structures in connectionist systems. Artificial Intelligence 46, 159-216. Smolensky, P. (1993). Harmonic grammars for formal languages. In S. Hanson, J. Cowan, and C. Giles (Eds.), Advances in Neural Information Processing Systems 5, pp. 847-854. San Mateo: Morgan Kauffman. Smolensky, P., G. Legendre, and Y. Miyata (1992). Principles for an integrated connectionist/symbolic theory of higher cognition. Technical Report CU-CS-60092, University of Colorado Computer Science Department. Smolensky, P., G. Legendre, and Y. Miyata (1994) . Integrating connectionist and symbolic computation for the theory of language. In V. Honavar and L. Uhr (Eds.), Artificial Intelligence and Neural Networks: Steps Toward Principled Integration, pp. 509-530. Boston: Academic Press.	1995	82
1,131	Learning Sparse Perceptrons Jeffrey C. Jackson Mathematics & Computer Science Dept. Duquesne University 600 Forbes Ave Pittsburgh, PA 15282 jackson@mathcs.duq.edu Abstract Mark W. Craven Computer Sciences Dept. University of Wisconsin-Madison 1210 West Dayton St. Madison, WI 53706 craven@cs.wisc.edu We introduce a new algorithm designed to learn sparse perceptrons over input representations which include high-order features. Our algorithm, which is based on a hypothesis-boosting method, is able to PAC-learn a relatively natural class of target concepts. Moreover, the algorithm appears to work well in practice: on a set of three problem domains, the algorithm produces classifiers that utilize small numbers of features yet exhibit good generalization performance. Perhaps most importantly, our algorithm generates concept descriptions that are easy for humans to understand. 1 Introd uction Multi-layer perceptron (MLP) learning is a powerful method for tasks such as concept classification. However, in many applications, such as those that may involve scientific discovery, it is crucial to be able to explain predictions. Multi-layer perceptrons are limited in this regard, since their representations are notoriously difficult for humans to understand. We present an approach to learning understandable, yet accurate, classifiers. Specifically, our algorithm constructs sparse perceptrons, i.e., single-layer perceptrons that have relatively few non-zero weights. Our algorithm for learning sparse perceptrons is based on a new hypothesis boosting algorithm (Freund & Schapire, 1995). Although our algorithm was initially developed from a learning-theoretic point of view and retains certain theoretical guarantees (it PAC-learns the class of sparse perceptrons), it also works well in practice. Our experiments in a number of real-world domains indicate that our algorithm produces perceptrons that are relatively comprehensible, and that exhibit generalization performance comparable to that of backprop-trained MLP's (Rumelhart et al., 1986) and better than decision trees learned using C4.5 (Quinlan, 1993). Learning Sparse Perceptrons 655 We contend that sparse perceptrons, unlike MLP's, are comprehensible because they have relatively few parameters, and each parameter describes a simple (Le. linear) relationship. As evidence that sparse perceptrons are comprehensible, consider that such linear functions are commonly used to express domain knowledge in fields such as medicine (Spackman, 1988) and molecular biology (Stormo, 1987). 2 Sparse Perceptrons A perceptron is a weighted threshold over the set of input features and over higherorder features consisting of functions operating on only a limited number of the input features. Informally, a sparse perceptron is any perceptron that has relatively few non-zero weights. For our later theoretical results we will need a more precise definition of sparseness which we develop now. Consider a Boolean function I : {O, 1 } n -t { -1, + 1 }. Let Ck be the set of all conjunctions of at most k of the inputs to I. Ck includes the "conjunction" of 0 inputs, which we take as the identically 1 function. All of the functions in Ck map to {-1,+1}, and every conjunction in Ck occurs in both a positive sense (+1 represents true) and a negated sense (-1 represents true). Then the function I is a k-perceptron if there is some integer s such that I(x) = sign(L::=1 hi(x)), where for all i, hi E Ck, and sign(y) is undefined if y = 0 and is y/lyl otherwise. Note that while we have not explicitly shown any weights in our definition of a k-perceptron I, integer weights are implicitly present in that we allow a particular hi E Ck to appear more than once in the sum defining I. In fact, it is often convenient to think of a k-perceptron as a simple linear discriminant function with integer weights defined over a feature space with O(nk) features, one feature for each element of Ck • We call a given collection of s conjunctions hi E Ck a k-perceptron representation of the corresponding function I, and we call s the size of the representation. We define the size of a given k-perceptron function I as the minimal size of any k-perceptron representation of I. An s-sparse k-perceptron is a k-perceptron I such that the size of I is at most s. We denote by PI: the set of Boolean functions over {O, 1}n which can be represented as k-perceptrons, and we define Pk = Un Pi:. The subclass of s-sparse k-perceptrons is denoted by Pk,/l" We are also interested in the class P~ of k-perceptrons with real-valued weights, at most r of which are non-zero. 3 The Learning Algorithm In this section we develop our learning algorithm and prove certain performance guarantees. Our algorithm is based on a recent "hypothesis boosting" algorithm that we describe after reviewing some basic learning-theory terminology. 3.1 PAC Learning and Hypothesis Boosting Following Valiant (1984), we say that a function class :F (such as Pk for fixed k) is (strongly) PAC-learnable if there is an algorithm A and a polynomial function PI such that for any positive f and 8, any I E :F (the target junction), and any probability distribution D over the domain of I, with probability at least 1 8, algorithm A(EX(f, D), f, 8) produces a function h (the hypothesis) such that Pr[PrD[/(x) I- hex)] > f] < 8. The outermost probability is over the random choices made by the EX oracle and any random choices made by A. Here EX(f, D) denotes an oracle that, when queried, chooses a vector of input values x with probability D and returns the pair (x,/(x)) to A. The learning algorithm A must run in time PI (n, s, c 1 , 8-1 ), where n is the length of the input vector to I and s is the size of 656 J. C. JACKSON, M. W. CRAVEN AdaBoost Input: training set S of m examples of function f, weak learning algorithm WL that is (~ - 'Y)-approximate, l' Algorithm: 1. T +-- ~ In(m) 2. for all xES, w(x) +-- l/m 3. for i = 1 to T do 4. for all XES, Di(X) +-- w(x)/ L:l=l w(x). 5. invoke WL on S and distribution Di, producing weak hypothesis hi 6. €i +-- L:z.h;(z);oI:/(z) Di(X) 7. (3i +-- €i/ (1 - €i) 8. for all XES, if h(x) = f(x) then w(x) +-- w(x) . (3i 9. enddo Output: h(x) == sign (L::=l -In((3i) . hi{x)) Figure 1: The AdaBoost algorithm. f; the algorithm is charged one unit of time for each call to EX. We sometimes call the function h output by A an €-approximator (or strong approximator) to f with respect to D. If F is PAC-learnable by an algorithm A that outputs only hypotheses in class 1£ then we say that F is PAC-learnable by 1£. If F is PAClearnable for € = 1/2 - 1/'P2(n, s), where'P2 is a polynomial function, then :F is weakly PA C-learnable, and the output hypothesis h in this case is called a weak approximator. Our algorithm for finding sparse perceptrons is, as indicated earlier, based on the notion of hypothesis boosting. The specific boosting algorithm we use (Figure 1) is a version of the recent AdaBoost algorithm (Freund & Schapire, 1995). In the next section we apply AdaBoost to "boost" a weak learning algorithm for Pk,8 into a strong learner for Pk,8' AdaBoost is given a set S of m examples of a function f : {O,1}n ---+ {-1, +1} and a weak learning algorithm WL which takes € = ! - l' for a given l' b must be bounded by an inverse polynomial in nand s). Adaf300st runs for T = In(m)/(2'Y2) stages. At each stage it creates a probability distribution Di over the training set and invokes WL to find a weak hypothesis hi with respect to Di (note that an example oracle EX(j, Di) can be simulated given Di and S). At the end of the T stages a final hypothesis h is output; this is just a weighted threshold over the weak hypotheses {hi I 1 ~ i ~ T}. If the weak learner succeeds in producing a (~-'Y)-approximator at each stage then AdaBoost's final hypothesis is guaranteed to be consistent with the training set (Freund & Schapire, 1995). 3.2 PAC-Learning Sparse k-Perceptrons We now show that sparse k-perceptrons are PAC learnable by real-weighted kperceptrons having relatively few nonzero weights. Specifically, ignoring log factors, Pk,8 is learnable by P~O(82) for any constant k. We first show that, given a training set for any f E Pk,8' we can efficiently find a consistent h E p~( 8 2)' This consistency algorithm is the basis of the algorithm we later apply to empirical learning problems. We then show how to turn the consistency algorithm into a PAC learning algorithm. Our proof is implicit in somewhat more general work by Freund (1993), although he did not actually present a learning algorithm for this class or analyze Learning Sparse Perceptrons 657 the sample size needed to ensure f-approximation, as we do. Following Freund, we begin our development with the following lemma (Goldmann et al., 1992): Lemma 1 (Goldmann Hastad Razhorov) For I: {0,1}n -+ {-1,+1} and H, any set 01 functions with the same domain and range, il I can be represented as I(x) = sign(L::=l hi(X», where hi E H, then lor any probability distribution D over {O, 1}n there is some hi such that PrD[f(x) ¥- hi(x)] ~ ~ 218 ' If we specialize this lemma by taking H = Ck (recall that Ck is the set of conjunctions of at most k input features of f) then this implies that for any I E Pk,8 and any probability distribution D over the input features of I there is some hi E Ck that weakly approximates I with respect to D. Therefore, given a training set S and distribution D that has nonzero weight only on instances in S, the following simple algorithm is a weak learning algorithm for Pk: exhaustively test each of the O(nk) possible conjunctions of at most k features until we find a conjunction that a - 218 )-approximates I with respect to D (we can efficiently compute the approximation of a conjunction hi by summing the values of D over those inputs where hi and I agree). Any such conjunction can be returned as the weak hypothesis. The above lemma proves that if I is a k-perceptron then this exhaustive search must succeed at finding such a hypothesis. Therefore, given a training set of m examples of any s-sparse k-perceptron I, AdaBoost run with the above weak learner will, after 2s2In(m) stages, produce a hypothesis consistent with the training set. Because each stage adds one weak hypothesis to the output hypothesis, the final hypothesis will be a real-weighted k-perceptron with at most 2s2In(m) nonzero weights. We can convert this consistency algorithm to a PAC learning algorithm as follows. First, given a finite set of functions F, it is straightforward to show the following (see, e.g., Haussler, 1988): Lemma 2 Let F be a finite set ollunctions over a domain X. For any function lover X, any probability distribution D over X, and any positive f and ~, given a set S ofm examples drawn consecutively from EX(f, D), where m ~ f-1(ln~-1 + In IFI), then Pr[3h E F I "Ix E S f(x) = h(x) & Prv[/(x) ¥- h(x)] > f] < ~, where the outer probability is over the random choices made by EX(f,D). The consistency algorithm above finds a consistent hypothesis in P~, where r = 2s2 In(m). Also, based on a result of Bruck (1990), it can be shown that In IP~I = o (r2 + kr log n). Therefore, ignoring log factors, a randomly-generated training set of size O(kS4 If) is sufficient to guarantee that, with high probability, our algorithm will produce an f-approximator for any s-sparse k-perceptron target. In other words, the following is a PAC algorithm for Pk,8: compute sufficiently large (but polynomial in the PAC parameters) m, draw m examples from EX(f, D) to create a training set, and run the consistency algorithm on this training set. So far we have shown that sparse k-perceptrons are learnable by sparse perceptron hypotheses (with potentially polynomially-many more weights). In practice, of course, we expect that many real-world classification tasks cannot be performed exactly by sparse perceptrons. In fact, it can be shown that for certain (reasonable) definitions of "noisy" sparse perceptrons (loosely, functions that are approximated reasonably well by sparse perceptrons), the class of noisy sparse k-perceptrons is still PAC-learnable. This claim is based on results of Aslam and Decatur (1993), who present a noise-tolerant boosting algorithm. In fact, several different boosting algorithms could be used to learn Pk,s (e.g., Freund, 1993). We have chosen to use AdaBoost because it seems to offer significant practical advantages, particularly in terms of efficiency. Also, our empirical results to date indicate that our algorithm 658 J. C. JACKSON, M. W. CRAVEN works very well on difficult (presumably "noisy") real-world problems. However, one potential advantage of basing the algorithm on one of these earlier boosters instead of AdaBoost is that the algorithm would then produce a perceptron with integer weights while still maintaining the sparseness guarantee of the AdaBoostbased algorithm. 3.3 Practical Considerations We turn now to the practical details of our algorithm, which is based on the consistency algorithm above. First, it should be noted that the theory developed above works over discrete input domains (Boolean or nominal-valued features). Thus, in this paper, we consider only tasks with discrete input features. Also, because the algorithm uses exhaustive search over all conjunctions of size k, learning time depends exponentially on the choice of k. In this study we to use k = 2 throughout, since this choice results in reasonable learning times. Another implementation concern involves deciding when the learning algorithm should terminate. The consistency algorithm uses the size of the target function in calculating the number of boosting stages. Of course, such size information is not available in real-world applications, and in fact, the target function may not be exactly representable as a sparse perceptron. In practice, we use cross validation to determine an appropriate termination point. To facilitate comprehensibility, we also limit the number of boosting stages to at most the number of weights that would occur in an ordinary perceptron for the task. For similar reasons, we also modify the criteria used to select the weak hypothesis at each stage so that simple features are preferred over conjunctive features. In particular, given distribution D at some stage j, for each hi E Ck we compute a correlation Ev[/ . hi]. We then mUltiply each high-order feature's correlation by i. The hi with the largest resulting correlation serves as the weak hypothesis for stage j. 4 Empirical Evaluation In our experiments, we are interested in assessing both the generalization ability and the complexity of the hypotheses produced by our algorithm. We compare our algorithm to ordinary perceptrons trained using backpropagation (Rumelhart et al., 1986), multi-layer perceptrons trained using backpropagation, and decision trees induced using the C4.5 system (Quinlan, 1993). We use C4.5 in our experiments as a representative of "symbolic" learning algorithms. Symbolic algorithms are widely believed to learn hypotheses that are more comprehensible than neural networks. Additionally, to test the hypothesis that the performance of our algorithm can be explained solely by its use of second-order features, we train ordinary perceptrons using feature sets that include all pairwise conjunctions, as well as the ordinary features. To test the hypothesis that the performance of our algorithm can be explained by its use of relatively few weights, we consider ordinary perceptrons which have been pruned using a variant of the Optimal Brain Damage (OBD) algorithm (Le Cun et al., 1989). In our version of OBD, we train a perceptron until the stopping criteria are met, prune the weight with the smallest salience, and then iterate the process. We use a validation set to decide when to stop pruning weights. For each training set, we use cross-validation to select the number of hidden units (5, 10, 20, 40 or 80) for the MLP's, and the pruning confidence level for the C4.5 trees. We use a validation set to decide when to stop training for the MLP's. We evaluate our algorithm using three real-world domains: the voting data set from the UC-Irvine database; a promoter data set which is a more complex superset of Learning Sparse Perceptrons 659 a e : es -se accuracy. T bl 1 11 t t perceptrons domain boosting C4.5 multi-layer ordinary 2nd-order pruned voting 91.5% 89.2% * 92.2% 90.8% 89.2% * 87.6% * promoter 92.7 84.4 * 90.6 90.0 * 88.7 * 88.2 * coding 72.9 62.6 * 71.6 * 70.7 * 69.8 * 70.3 * Table 2: Hypothesis complexity (# weights). perceptrons domain boosting multi-layer ordinary 2nd-order pruned voting 12 651 30 450 12 promoters 41 2267 228 25764 59 protein coding 52 4270 60 1740 37 U C-Irvine one; and a data set in which the task is to recognize protein-coding regions in DNA (Craven & Shavlik, 1993). We remove the physician-fee-freeze feature from the voting data set to make the problem more difficult. We conduct our experiments using a lO-fold cross validation methodology, except for in the protein-coding domain. Because of certain domain-specific characteristics of this data set, we use 4-fold cross-validation for our experiments with it. Table 1 reports test-set accuracy for each method on all three domains. We measure the statistical significance of accuracy differences using a paired, two-tailed t-test. The symbol '*' marks results in cases where another algorithm is less accurate than our boosting algorithm at the p ::; 0.05 level of significance. No other algorithm is significantly better than our boosting method in any of the domains. From these results we conclude that (1) our algorithm exhibits good generalization performance on number of interesting real-world problems, and (2) the generalization performance of our algorithm is not explained solely by its use of second-order features, nor is it solely explained by the sparseness of the perceptrons it produces. An interesting open question is whether perceptrons trained with both pruning and second-order features are able to match the accuracy of our algorithm; we plan to investigate this question in future work. Table 2 reports the average number of weights for all of the perceptrons. For all three problems, our algorithm produces perceptrons with fewer weights than the MLP's, the ordinary perceptrons, and the perceptrons with second-order features. The sizes of the OBD-pruned perceptrons and those produced by our algorithm are comparable for all three domains. Recall, however, that for all three tasks, the perceptrons learned by our algorithm had significantly better generalization performance than their similar-sized OBD-pruned counterparts. We contend that the sizes of the perceptrons produced by our algorithm are within the bounds of what humans can readily understand. In the biological literature, for example, linear discriminant functions are frequently used to communicate domain knowledge about sequences of interest. These functions frequently involve more weights than the perceptrons produced by our algorithm. We conclude, therefore, that our algorithm produces hypotheses that are not only accurate, but also comprehensible. We believe that the results on the protein-coding domain are especially interesting. The input representation for this problem consists of 15 nominal features representing 15 consecutive bases in a DNA sequence. In the regions of DNA that encode proteins (the positive examples in our task), non-overlapping triplets of consecu660 J. C. JACKSON, M. W. eRA VEN tive bases represent meaningful "words" called codons. In previous work (Craven & Shavlik, 1993), it has been found that a feature set that explicitly represents codons results in better generalization than a representation of just bases. However, we used the bases representation in our experiments in order to investigate the ability of our algorithm to select the "right" second-order features. Interestingly, nearly all of the second-order features included in our sparse perceptrons represent conjunctions of bases that are in the same codon. This result suggests that our algorithm is especially good at selecting relevant features from large feature sets. 5 Future Work Our present algorithm has a number of limitations which we plan to address. Two areas of current research are generalizing the algorithm for application to problems with real-valued features and developing methods for automatically suggesting highorder features to be included in our algorithm's feature set. Acknowledgements Mark Craven was partially supported by ONR grant N00014-93-1-0998. Jeff Jackson was partially supported by NSF grant CCR-9119319. References Aslam, J. A. & Decatur, S. E. (1993). General bounds on statistical query learning and PAC learning with noise via hypothesis boosting. In Proc. of the 34th Annual Annual Symposium on Foundations of Computer Science, (pp. 282-291). Bruck, J . (1990). Harmonic analysis of polynomial threshold functions. SIAM Journal of Discrete Mathematics, 3(2):168-177. Craven, M. W. & Shavlik, J. W. (1993). Learning to represent codons: A challenge problem for constructive induction. In Proc. of the 13th International Joint Conf. on Artificial Intelligence, (pp. 1319-1324), Chambery, France. Freund, Y. (1993). Data Filtering and Distribution Modeling Algorithms for Machine Learning. PhD thesis, University of California at Santa Cruz. Freund, Y. & Schapire, R. E. (1995). A decision-theoretic generalization of on-line learning and an application to boosting. In Proc. of the ~nd Annual European Conf. on Computational Learning Theory. Goldmann, M., Hastad, J., & Razborov, A. (1992). Majority gates vs. general weighted threshold gates. In Proc. of the 7th IEEE Conf. on Structure in Complexity Theory. Haussler, D. (1988). Quantifying inductive bias: AI learning algorithms and Valiant's learning framework. Artificial Intelligence, (pp. 177-221). Le Cun, Y., Denker, J. S., & Solla, S. A. (1989). Optimal brain damage. In Touretzky, D., editor, Advances in Neural Information Processing Systems (volume ~) . Quinlan, J. R. (1993). C4.5: Programs for Machine Learning. Morgan Kaufmann. Rumelhart, D., Hinton, G., & Williams, R. (1986). Learning internal representations by error propagation. In Rumelhart, D. & McClelland, J., editors, Parallel Distributed Processing: Explorations in the microstructure of cognition. Volume 1. MIT Press. Spackman, K. A. (1988). Learning categorical decision criteria. In Proc. of the 5th International Conf. on Machine Learning, (pp. 36-46), Ann Arbor, MI. Stormo, G. (1987). Identifying coding sequences. In Bishop, M. J. & Rawlings, C. J., editors, Nucleic Acid and Protein Sequence Analysis: A Practical Approach. IRL Press. Valiant,1. G. (1984). A theory of the learnable. Comm. of the ACM, 27(11):1134-1142.	1995	83
1,132	The Role of Activity in Synaptic Competition at the Neuromuscular Junction Samuel R. H. Joseph Centre for Cognitive Science Edinburgh University Edinburgh, U.K. email: sam@cns.ed.ac.uk David J. Willshaw Centre for Cognitive Science Edinburgh University Edinburgh, U.K. email: david@cns.ed.ac.uk Abstract An extended version of the dual constraint model of motor endplate morphogenesis is presented that includes activity dependent and independent competition. It is supported by a wide range of recent neurophysiological evidence that indicates a strong relationship between synaptic efficacy and survival. The computational model is justified at the molecular level and its predictions match the developmental and regenerative behaviour of real synapses. 1 INTRODUCTION The neuromuscular junction (NMJ) of mammalian skeletal muscle is one of the most extensively studied areas of the nervous system. One aspect of its development that it shares with many other parts of the nervous system is its achievement of single innervation, one axon terminal connecting to one muscle fibre, after an initial state of polyinnervation. The presence of electrical activity is associated with this transition, but the exact relationship is far from clear. Understanding how activity interacts with the morphogenesis of neural systems could provide us with insights into methods for constructing artificial neural networks. With that in mind, this paper examines how some of the conflicting ideas about the development of neuromuscular connections can be resolved. The Role of Activity in Synaptic Competition at the Neuromuscular Junction 97 2 EXPERIMENTAL FINDINGS The extent to which a muscle is innervated can be expressed in terms of the motor unit size - the number of fibres contacted by a given motor axon. Following removal of some motor axons at birth, the average size of the remaining motor units after withdrawal of poly innervation is larger than normal (Fladby & Jansen, 1987). This strongly suggests that individual motor axons successfully innervate more fibres as a result of the absence of their neighbours. It is appealing to interpret this as a competitive process where terminals from different axons compete for the same muscle endplate. Since each terminal is made up of a number of synapses the process can be viewed as the co-existence of synapses from the same terminal and the elimination of synapses from different terminals on the same end plate. 2.1 THE EFFECTS OF ELECTRICAL ACTIVITY There is a strong activity dependent component to synapse elimination. Paralysis or stimulation of selected motor units appears to favour the more active motor terminals (Colman & Lichtman, 1992), while inactive axon terminals tend to coexist. Recent work also shows that active synaptic sites can destabilise inactive synapses in their vicinity (Balice-Gordon & Lichtman, 1994). These findings support the idea that more active terminals have a competitive advantage over their inactive fellows, and that this competition takes place at a synaptic level. Activity independent competition has been demonstrated in the rat lumbrical muscle (Ribchester, 1993). This muscle is innervated by the sural and the lateral plantar nerves. If the sural nerve is damaged the lateral plantar nerve will expand its territory to the extent that it innervates the entire muscle. On subsequent reinnervation the regenerating sural nerve may displace some of the lateral plantar nerve terminals. If the muscle is paralysed during reinnervation more lateral plantar nerve terminals are displaced than in the normal case, indicating that competition between inactive terminals does take place, and that paralysis can give an advantage to some terminals. 3 MODELS AND MECHANISMS If the nerve terminals are competing with each other for dominance of motor endplates, what is the mechanism behind it? As mentioned above, activity is thought to play an important role in affecting the competitive chances of a terminal, but in most models the terminals compete for some kind of trophic resource (Gouze et aI., 1983; Willshaw, 1981). It is possible to create models that use competition for either a postsynaptic (endplate) resource or a presynaptic (motor axon) resource. Both types of model have advantages and disadvantages, which leads naturally to the possibility of combining the two into a single model. 3.1 BENNET AND ROBINSON'S DUAL CONSTRAINT MODEL The dual constraint model (DCM) (Bennet & Robinson, 1989), as extended by Rasmussen & Willshaw (1993), is based on a reversible reaction between molecules from a presynaptic resource A and a postsynaptic resource B. This reaction takes place in the synaptic cleft and produces a binding complex C which is essential for 98 S. R. H. JOSEPH, D. J. Wll...LSHAW the terminal's survival. Each motor axon and muscle fibre has a limited amount of their particular resource and the size of each terminal is proportional to the amount of the binding complex at that terminaL The model achieves single innervation and a perturbation analysis performed by Rasmussen & Willshaw (1993) showed that this single innervation state is stable. However, for the DCM to function the forward rate of the reaction had to be made proportional to the size of the terminal, which was difficult to justify other than suggesting it was related to electrical activity. 3.2 SELECTIVE MECHANISMS While the synapses in the surviving presynaptic terminal are allowed to coexist, synapses from other axons are eliminated. How do synapses make a distinction between synapses in their own terminal and those in others? There are two possibilities: (i) Synchronous transmitter release in the synaptic boutons of a motor neuron could distinguish synapses, allowing them to compete as cartels rather than individuals (Colman & Lichtman, 1992). (ii) The synapses could be employing selective recognition mechanisms, e.g the 'induced-fit' model (Rib chester & Barry, 1994). A selective mechanism implies that all the synapses of a given motor neuron can be identified by a molecular substrate. In the induced-fit model each motor neuron is associated with a specific isoform of a cellular adhesion molecule (CAM); the synapses compete by attempting to induce all the CAMs on the end plate into the conformation associated with their neuron. This kind of model can be used to account for much of the developmental and regenerative processes of the NMJ. However, it has difficulty explaining Balice-Gordon & Lichtman's (1994) focal blockade experiments which show competition between synapses distinguished only by the presence of activity. If, instead, activity is responsible for the distinction of friend from foe, how can competition take place at the terminal level when activity is not present? Could we resolve this dilemma by extending the dual constraint model? 4 EXTENDING THE DUAL CONSTRAINT MODEL Tentative suggestions can be made for the identity of the 'mystery molecules' in the DCM. According to McMahan (1990) a protein called agrin is synthesised in the cell bodies of motor neurons and transported down their axons to the muscle. When this protein binds to the surface of the developing muscle, it causes acetylcholine receptors (AChRs), and other components of the postsynaptic apparatus, to aggregate on the myotube surface in the vicinity of the activated agrin. Other work (Wallace, 1988) has provided insights into the mechanism used by agrin to cause the aggregation of the postsynaptic apparatus. Initially, AChR aggregates, or 'speckles', are free to diffuse laterally in the myotube plasma membrane (Axelrod et aI., 1976). When agrin binds to an agrin-specific receptor, AChR speckles in the immediate vicinity of the agrin-receptor complex are immobilised. As more speckles are trapped larger patches are formed, until a steady state is reached. Such a patch will remain so long as agrin is bound to its receptor and Ca++ and energy supplies are available. Following AChR activation by acetylcholine, Ca++ enters the postsynaptic cell. Since Ca++ is required for both the formation and maintenance of AChR aggregates, The Role of Activity in Synaptic Competition at the Neuromuscular Junction 99 a feedback loop is possible whereby the bigger a patch is the more Ca++ it will have available when the receptors are activated. Crucially, depolarisation of nonjunctional regions blocks AChR expression (Andreose et al., 1995) and it is AChR activation at the NMJ that causes depolarisation of the postsynaptic cell. So it seems that agrin is a candidate for molecule A, but what about B or C? It is tempting to posit AChR as molecule B since it is the critical postsynaptic resource. However, since agrin does not bind directly to the acetylcholine receptor, a different sort of reaction is required. 4.1 A DIFFERENT SORT OF REACTION If AChR is molecule B, and one agrin molecule can attract at least 160 AChRs (Nitkin et al., 1987) the simple reversible reaction of the DCM is ruled out. Alternatively, AChR could exist in either free, B f' or bound, Bb states, being converted through the mediation of A. Bb would now play the role of C in the DCM. It is possible to devise a rate equation for the change in the number of receptors at a nerve terminal over time: dBb = nABf - (3Bb dt (1) where n and (3 are rate constants. The increase in bound AChR over time is proportional to the amount of agrin at a junction and the number of free receptors in the endplate area, while the decrease is proportional to the amount of bound AChRs. The rate equation (1) can be used as the basis of an extended DCM if four other factors are considered: (i) Agrin stays active as receptors accumulate, so the conservation equations for A and Bare: M Ao = An + LAnj j=1 N Bo = Bmf + LBimb i=1 (2) where the subscript 0 indicates the fixed resource available to each muscle or neuron, the lettered subscripts indicate the amount of that substance that is present in the neuron n, muscle fibre m and terminal nm, and there are N motor neurons and M muscle fibres. (ii) The size of a terminal is proportional to the number of bound AChRs, so if we assume the anterograde flow is evenly divided between the lin terminals of neuron n, the transport equation for agrin is: (3) where>. and IS are transport rate constants and the retrograde flow is assumed proportional to the amount of agrin at the terminal and inversely proportional to the size of the terminal. (iii) AChRs are free to diffuse laterally across the surface of the muscle, so the forward reaction rate will be related to the probability of an AChR speckle intersecting a terminal, which is itself proportional to the terminal diameter. (iv) The influx of Ca++ through AChRs on the surface of the endplate will also affect the forward reaction rate in proportion to the area of the terminal. Taking Bb to be proportional to the volume of the postsynaptic apparatus, these last two terms are proportional to B~/3 and B;/3 respectively. This gives the final rate equation: (4) 100 S. R. H. JOSEPH, D. J. WILLSHA W Equations (3) and (4) are similar to those in the original DCM, only now we have been able to justify the dependence of the forward reaction rate on the size of the terminal, Bnmb . We can also resolve the distinction paradox, as follows. 4.2 RESOLVING THE DISTINCTION PARADOX In terms of distinguishing between synapses it seems plausible that concurrently active synapses (Le. those belonging to the same neuron) will protect themselves from the negative effects of depolarisation. In paralysed systems, synapses will benefit from the AChR accumulating affects of the agrin molecules in those synapses nearby (i.e. those in the same terminal). It was suggested (Jennings, 1994) that competition between synapses of the same terminal was seen after focal blockade because active AChRs help stabilise the receptors around them and suppress those further away. This fits in with the stabilisation role of Ca++ in this model and the suppressive effects of depolarisation, as well as the physical range of these effects during 'heterosynaptic suppression' (Lo & Poo, 1991). It seems that Jenning's mechanism, although originally speculative, is actually quite a plausible explanation and one that fits in well with the extended DCM. The critical effect in the XDCM is that if the system is paralysed during development there is a change in the dependency of the forward reaction rate on the size of an individual terminal. This gives the reinnervating terminals a small initial advantage due to their more competitive diameter/volume ratios. As we shall see in the next section, this allows us to demonstrate activity independent competition. 5 SIMULATING THE EXTENDED DCM In terms of achieving single innervation the extended DCM performs just as well as the original, and when subjected to the same perturbation analysis it has been demonstrated to be stable. Simulating a number of systems with as many muscle fibres and motor neurons as found in real muscles allowed a direct comparison of model findings with experimental data (figure 1) . ..... -~ ... '; ,-\-. . t\_ '. .~\\ " " .... • E""pcrimental + Simulation ~,~ •• 1 ............. __ ..... . ~.----~----+. .. ----~--~~ Days after birth Figure 1: Elimination of Polyinnervation in Rat soleus muscle and Simulation Figure 2 shows nerve dominance histograms of reinnervation in both the rat lumbrical muscle and its extended DCM simulation. Both compare the results produced when the system is paralysed from the outset of reinnervation (removal of B~~b The Role of Activity in Synaptic Competition at the Neuromuscular Junction 101 term from equation (4)) with the normal situation. Note that in both the simulation and the experiment the percentage of fibres singly innervated by the reinnervating sural nerve is increased in the paralysis case. Inactive sural nerve terminals are displacing more inactive lateral plantar nerve terminals (activity independent competition). They can achieve this because during paralysis the terminals with the largest diameters capture more receptors, while the terminals with the largest volumes lose more agrin; so small reinnervating terminals do a little better. However, if activity is present the receptors are captured in proportion to a terminal's volume, so there's no advantage to a small terminal's larger diameter/volume ratio. I Nerve Dominance Histogram (Experimental) I I I, , I 1:::11 I I I Nerve Dominance Histogram (Simulation) I SingleLPN Multi SingleSN I SingieLPN Mull! SmglcSN ________________________ ~ ~ I ____________ __ Figure 2: Types of Innervation by Lateral Plantar and Sural Nerves 6 DISCUSSION The extensions to the DCM outlined here demonstrate both activity dependent and independent competition and provide greater biochemical plausibility. However this is still only a phenomenological demonstration and further experimental work is required to ascertain its validity. There is a need for illumination concerning the specific chemical mechanisms that underlie agrin's aggregational effects and the roles that both Ca++ and depolarisation play in junctional dynamics. An important connection made here is one between synaptic efficiency and junctional survival. Ca++ and NO have both been implicated in Hebbian mechanisms (Bliss & Collingridge, 1993) and perhaps some of the principles uncovered here may be applicable to neuroneuronic synapses. This work should be followed up with a direct model of synaptic interaction at the NMJ that includes the presynaptic effects of depolarisation, allowing the efficacy of the synapse to be related to its biochemistry; an important step forward in our understanding of nervous system plasticity. Relating changes in synaptic efficiency to neural morphogenesis may also give insights into the construction of artificial neural networks. Acknowledgements We are grateful to Michael Joseph and Bruce Graham for critical reading of the manuscript and to the M.R.C. for funding this work. 102 S. R. H. JOSEPH, D. J. WILLS HAW References Andreose J. S., Fumagalli G. & L0mo T. (1995) Number of junctional acetylcholine receptors: control by neural and muscular influences in the rat. Journal of Physiology 483.2:397-406. Axelrod D., Ravdin P., Koppel D. E., Schlessinger J., Webb W. W., Elson E. L. & Podleski T. R. (1976) Lateral motion offluorescently labelled acetylcholine receptors in membranes of developing muscle fibers. Proc. Natl. Acad. Sci. USA 73:45944598. Balice-Gordon R. J. & Lichtman J. W. (1994) Long-term synapse loss induced by focal blockade of postsynaptic receptors. Nature 372:519-524. Bennett M. R. & Robinson J. (1989) Growth and elimination of nerve terminals during polyneuronal innervation of muscle cells: a trophic hypothesis. Proc. Royal Soc. Lond. [Biol] 235:299-320. Bliss T. V. P. & Collingridge G. L. (1993) A synaptic model of memory: long-term potentiation in the hippocampus. Nature 361:31-39. Colman H. & Lichtman J. W. (1992) 'Cartellian' competition at the neuromuscular junction. Trends in Neuroscience 15, 6:197-199. Fladby T. & Jansen J. K. S. (1987) Postnatal loss of synaptic terminals in the partially denervated mouse soleus muscle. Acta. Physiol. Scand 129:239-246. Gouze J. L., Lasry J. M. & Changeux J. -Po (1983) Selective stabilization of muscle innervation during development: A mathematical model. Biol Cybern. 46:207-215. Jennings C. (1994) Death of a synapse. Nature 372:498-499. Lo Y. J. & Poo M. M. (1991) Activity-dependent synapse competition in vitro: heterosynaptic suppression of developing synapses. Science 254:1019-1022. McMahan U. J. (1990) The Agrin Hypothesis. Cold Spring Harbour Symp. Quant. Biol. 55:407-419. Nitkin R. M., Smith M. A., Magill C., Fallon J. R., Yao Y. -M. M., Wallace B. G. & McMahan U. J. (1987) Identification of agrin, a synaptic organising protein from Torpedo electric organ. Journal Cell Biology 105:2471-2478. Rasmussen C. E. & Willshaw D. J. (1993) Presynaptic and postsynatic competition in models for the development of neuromuscular connections. B. Cyb. 68:409-419. Ribchester R. R. (1993) Co-existence and elimination of convergent motor nerve terminals in reinnervated and paralysed adult rat skeletal muscle. J. Phys. 466: 421-441. Ribchester R. R. & Barry J. A. (1994) Spatial Versus Consumptive Competition at Polyneuronally Innervated Neuromuscular Junctions. Exp. Physiology 79:465-494. Wallace B. G. (1988) Regulation of agrin-induced acetylcholine receptor aggregation by Ca++ and phorbol ester. Journal of Cell Biol. 107:267-278. Willshaw D. J. (1981) The establishment and the subsequent elimination of polyneuronal innervation of developing muscle: theoretical considerations. Proc. Royal Soc. Lond. B212: 233-252.	1995	84
1,133	! #" $ %'& !( ) * +-,/.103254 687:95,/;<0=43>3?@0BADC3E 7 ,503,GF50HJI<? 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Ä u a U Õ ! / Ya 565 U 7 U:. q aD .RÜ41/. U7¨ a Ø !5 V © x ¼ ¾hî Áª 2 x!ò ½ òôó 4 Á å ø = Î ¥¬« Å ±% ü ^ Ê j Ô c Ô 7 s u?C 5 « u Ô¯® ¯ . 5 U:. Ø b . U i @° Ó Ê ±!I Í > W 79 u 5 a =)° U . s 7 U . 4± A3 } . s 3 . U a n !5 Î`² 6 èRÃ ó ³ ö ½ Rµ´´º ¶· ¸ º À{fÄ ü ¹ ú Åo °?º ü Æ jV» Ð . U . s 5 a /O« ý <Iu i _ / 4 .n 5:a /#"m@ Ò ±!Ê!Ê I)¼ . s 7 /.U a s c5-7 4 u a 8:.´a q U 9 ¥ 9 r7 U 9 Î a / q n a _ o . 5 Ì ]7 q .n = a s 7!/ v .ì5 U C 4 > a / U yY X Ø 7 y _ 9 " 5 .½ U ) a /Nq s aY_ . ú î)Y é 2¾ º?¾ ¿ èRÃò:ºé¶Ã] B < ÅÀÁ Ê Ó i Í u ~ .Ü . i MÒ ±!Ê!±ÃÂ à ¾ º é º w /?ÃìÁON ó Ä º ¼ º å ¿b3 Å ½!ð? º¯5èå ½é 3 Æ ö Ã ½ å;ZÈÇ)É . x¼ö ½ ¿Có . YÊË6ºÁ¼ è i _u 74 . ! uQ s 5 5 ø;Ì / u Î ¹ ¯ 7 / XO« V ¢ Æ ª Ü i M ÒÓ Ó B F Í 9 i 5 u s . U . CU 9 V§Íg. b. s 5 5 u a /U J / a 5bÎ U 9 .Ï/ . s 7 o / RU ¨ a s c 5 i L Á N[î Áé 2 x!ò:º ¼ ã å;Z ç èRÃòCºé ç ö g Å º å öºÃ ÄDÏ ± m Ò+Ð °Ç ú `Ñ u i ¯ 7 X y «Òu Ô Î ¯ .5 U . s .ÜU @ ÒÓ ±] F1Ó /7 a X /. ~ s 7!Ü+/ . U a s c5m9 VU ^O au 7½ aYÔ . U 9 · U:9 a / ÖÕ Í > /7 9 ;u 5 7 / 5 U 7 _ 9 ° ×U > " 6 èÃó ö ½¾ ´´º ¶Ró . ºRÀÈØ Ä üYË üÅ B ü û ü Å] = < ! " #	1995	85
1,134	Explorations with the Dynamic Wave Model Thomas P. Rebotier Department of Cognitive Science UCSD, 9500 Gilman Dr LA JOLLA CA 92093-0515 rebotier@cogsci.ucsd.edu Jeffrey L. Elman Department of Cognitive Science UCSD, 9500 Gilman Dr LA JOLLA CA 92093-0515 elman@cogsci.ucsd.edu Abstract Following Shrager and Johnson (1995) we study growth of logical function complexity in a network swept by two overlapping waves: one of pruning, and the other of Hebbian reinforcement of connections. Results indicate a significant spatial gradient in the appearance of both linearly separable and non linearly separable functions of the two inputs of the network; the n.l.s. cells are much sparser and their slope of appearance is sensitive to parameters in a highly non-linear way. 1 INTRODUCTION Both the complexity of the brain (and concomittant difficulty encoding that C0111plexity through any direct genetic mapping). as well as the apparently high degree of cortical plasticity suggest that a great deal of cortical structure is emergent rather than pre-specified. Several neural models have explored the emergence of complexity. Von der Marlsburg (1973) studied the grouping of orientation selectivity by competitive Hebbian synaptic modification. Linsker (1986.a, 1986.b and 1986.c) showed how spatial selection cells (off-center on-surround), orientation selective cells, and finally orientation columns, emerge in successive layers from random input by simple, Hebbian-like learning rules. ~[iller (1992, 1994) studied the emergence of orientation selective columns from activity dependant competition between on-center and off-center inputs. Kerzsberg, Changeux and Dehaene (1992) studied a model with a dual-aspect learning mechanism: Hebbian reinforcement of the connection strengths in case of correlated activity, and gradual pruning of immature connections. Cells in this model were organized on a 2D grid , connected to each other according to a probability exponentially decreasing with distance, and received inputs from two different sources, 550 T. P. REBOTIER, J. L. ELMAN A and B, which might or might not be correlated. The analysis of the network revealed 17 different kinds of cells: those whose ou tpu t after several cycles depended on the network's initial state, and the 16 possible logical functions of two inputs. Kerzsberg et al. found that learning and pruning created different patches of cells implementing common logical functions, with strong excitation within the patches and inhibition between patches. Shrager and Johnson (1995) extended that work by giving the network structure in space (structuring the inputs in intricated stripes) or in time, by having a Hebbian learning occur in a spatiotemporal wave that passed through the network rather than occurring everywhere simultaneously. Their motivation was to see if these learning conditions might create a cascade of increasingly complex functions. The approach was also motivated by developmental findings in humans and monkeys suggesting a move of the peak of maximal plasticity from the primary sensory and motor areas to\vards parietal and then frontal regions. Shrager and Johnson classified the logical functions into three groups: the constants (order 0), those that depend on one input only (order 1), those that depend on both inputs (order 2). They found that a slow wave favored the growth of order 2 cells, whereas a fast wave favored order 1 cells. However, they only varied the connection reinforcement (the growth Trophic Factor), so that the still diffuse pruning affected the rightmost connections before they could stabilize, resulting in an overall decrease which had to be compensated for in the analysis. In this work, v,,'e followed Shrager and Johnson in their study of the effect of a dynamic wave of learning. We present three novel features. Firstly, both the growth trophic factor (hereafter, TF) and the probability of pruning (by analogy, "death factor", DF) travel in gaussian-shaped waves. Second, we classify the cells in 4, not 3, orders: order 3 is made of the non-linearly separable logical functions, whereas the order 2 is now restricted to linearly separable logical functions of both inputs. Third. we use an overall measure of network performance: the slope of appearance of units of a given order. The density is neglected as a measure not related to the specific effects we are looking for, namely, spatial changes in complexity. Thus, each run of our network can be analyzed using 4 values: the slopes for units of order 0, 1, 2 and 3 (See Table 1.). This extreme summarization of functional information allows us to explore systematically many parameters and to study their influence over how complexity grows in space. Table 1: Orders of logical complexity ORDER FUNCTIONS o True False 1 A !A B !B 2 A.B !A.B A.!B !A.!B AvB !AvB Av!B !Av!B 3 A xor B, A==B 2 METHODS Our basic network consisted of 4 columns of 50 units (one simulation verified the scaling up of results, see section 3.2). Internal connections had a gaussian bandwidth and did not wrap around. All initial connections were of weight 1, so that the connectivity weights given as parameters specified a number of labile connections. Early investigations were made with a set of manually chosen parameters (" MANExplorations with the Dynamic Wave Model 551 UAL"). Afterwards, two sets of parameters were determined by a Genetic Algorithm (see Goldberg 1989): the first, "SYM", by maximizing the slope of appearance of order 3 units only, the second, " ASY" , byoptimizing jointly the appearance of order 2 and order 3 units (" ASY"). The "SYM" network keeps a symmetrical rate of presentation between inputs A and B. In contrast, the" ASY" net presents input B much more often than input A. Parameters are specified in Table 1 and, are in "natural" units: bandwidths and distances are in "cells apart", trophic factor is homogenous to a weight, pruning is a total probability. Initial values and pruning necessited random number generation. \Ve used a linear congruence generator (see p284 in Press 1988), so that given the same seed, two different machines could produce exactly the same run. All the points of each Figure are means of several (usually 40) runs with different random seeds and share the same series of random seeds. Table 2: Default parameters MAN. SYM. ASY. name description 8.5 6.20 12 Wae mean ini. weight of A excitatory connections 6.5 5.2 9.7 Wai mean ini. weight of A inhibitory connections 8.5 8.5 13.4 Wbe mean ini. weight of B excitatory connections 6.5 6.5 14.1 \Vbi mean ini. weight of B inhibitory connections 5.0 6.5 9.9 Wne m.ini. density of internal excitatory connections 3.5 1.24 12.4 Wni m.ini. density of internal inhibitory connections 0.2 0.20 0.28 DW relative variation in initial weights 7.0 1.26 0.65 Bne bandwidth of internal excitatory connections 7.0 2.86 0.03 Bni bandwidth of internal inhibitory connections 0.7 0.68 0.98 Cdw celerity of dynamic wave 1.5 3.0 -3.2 Ddw distance between the peaks of both waves 9.87 17.6 16.4 Wtf base level of TF (=highest available weight) 0.6 0.6 0.6 Btf bandwidth of TF dynamic wave 3.5 1.87 3.3 Tst Threshold of stabilisation (pruning stop) 0.6 0.64 0.5 Bdf band .. vidth of DF dynamic wave 0.65 0.62 0.12 Pdf base level of DF (total proba. of degeneration) 0.5 0.5 0.06 Pa probability of A alone in the stimulus set 0.5 0.5 0.81 Pb probability of B alone in the stimulus set 0.00 0.00 0.00 Pab probability of simultaneous s A and B 3 RESULTS 3.1 RESULTS FORMAT All Figures have the same format and summarize 40 runs per point unless otherwise specified. The top graph presents the mean slope of appearance of all 4 orders of complexity (see Table 1) on the y axis, as a function of different values of the experimentally manipulated parameter, on the x axis. The bottom left graph shows the mean slope for order 2, surrounded by a gray area one standard deviation below and above. The bottom right graph shows the mean slope for order 3, also with a I-s.d. surrounding area. The slopes have not been normalized, and come from networks whose columns are 50 units high, so that a slope of 1.0 indicates that the number of such units increase in average by one unit per columns, ie, by 3 units	1995	86
1,135	Generalization in Reinforcement Learning: Successful Examples Using Sparse Coarse Coding Richard S. Sutton University of Massachusetts Amherst, MA 01003 USA richOcs.umass.edu Abstract On large problems, reinforcement learning systems must use parameterized function approximators such as neural networks in order to generalize between similar situations and actions. In these cases there are no strong theoretical results on the accuracy of convergence, and computational results have been mixed. In particular, Boyan and Moore reported at last year's meeting a series of negative results in attempting to apply dynamic programming together with function approximation to simple control problems with continuous state spaces. In this paper, we present positive results for all the control tasks they attempted, and for one that is significantly larger. The most important differences are that we used sparse-coarse-coded function approximators (CMACs) whereas they used mostly global function approximators, and that we learned online whereas they learned offline. Boyan and Moore and others have suggested that the problems they encountered could be solved by using actual outcomes ("rollouts"), as in classical Monte Carlo methods, and as in the TD().) algorithm when). = 1. However, in our experiments this always resulted in substantially poorer performance. We conclude that reinforcement learning can work robustly in conjunction with function approximators, and that there is little justification at present for avoiding the case of general ).. 1 Reinforcement Learning and Function Approximation Reinforcement learning is a broad class of optimal control methods based on estimating value functions from experience, simulation, or search (Barto, Bradtke &; Singh, 1995; Sutton, 1988; Watkins, 1989). Many of these methods, e.g., dynamic programming and temporal-difference learning, build their estimates in part on the basis of other Generalization in Reinforcement Learning 1039 estimates. This may be worrisome because, in practice, the estimates never become exact; on large problems, parameterized function approximators such as neural networks must be used. Because the estimates are imperfect, and because they in turn are used as the targets for other estimates, it seems possible that the ultimate result might be very poor estimates, or even divergence. Indeed some such methods have been shown to be unstable in theory (Baird, 1995; Gordon, 1995; Tsitsiklis & Van Roy, 1994) and in practice (Boyan & Moore, 1995). On the other hand, other methods have been proven stable in theory (Sutton, 1988; Dayan, 1992) and very effective in practice (Lin, 1991; Tesauro, 1992; Zhang & Diett erich , 1995; Crites & Barto, 1996). What are the key requirements of a method or task in order to obtain good performance? The experiments in this paper are part of narrowing the answer to this question. The reinforcement learning methods we use are variations of the sarsa algorithm (Rummery & Niranjan, 1994; Singh & Sutton, 1996). This method is the same as the TD(>.) algorithm (Sutton, 1988), except applied to state-action pairs instead of states, and where the predictions are used as the basis for selecting actions. The learning agent estimates action-values, Q""(s, a), defined as the expected future reward starting in state s, taking action a, and thereafter following policy 71'. These are estimated for all states and actions, and for the policy currently being followed by the agent. The policy is chosen dependent on the current estimates in such a way that they jointly improve, ideally approaching an optimal policy and the optimal action-values. In our experiments, actions were selected according to what we call the £-greedy policy. Most of the time, the action selected when in state s was the action for which the estimate Q(s,a) was the largest (with ties broken randomly). However, a small fraction, £, ofthe time, the action was instead selected randomly uniformly from the action set (which was always discrete and finite). There are two variations of the sarsa algorithm, one using conventional accumulate traces and one using replace traces (Singh & Sutton, 1996). This and other details of the algorithm we used are given in Figure 1. To apply the sarsa algorithm to tasks with a continuous state space, we combined it with a sparse, coarse-coded function approximator known as the CMAC (Albus, 1980; Miller, Gordon & Kraft, 1990; Watkins, 1989; Lin & Kim, 1991; Dean et al., 1992; Tham, 1994). A CMAC uses multiple overlapping tilings of the state space to produce a feature ~epresentation for a final linear mapping where all the learning takes place. See Figure 2. The overall effect is much like a network with fixed radial basis functions, except that it is particularly efficient computationally (in other respects one would expect RBF networks and similar methods (see Sutton & Whitehead, 1993) to work just as well). It is important to note that the tilings need not be simple grids. For example, to avoid the "curse of dimensionality," a common trick is to ignore some dimensions in some tilings, i.e., to use hyperplanar slices instead of boxes. A second major trick is "hashing"-a consistent random collapsing of a large set of tiles into a much smaller set. Through hashing, memory requirements are often reduced by large factors with little loss of performance. This is possible because high resolution is needed in only a small fraction of the state space. Hashing frees us from the curse of dimensionality in the sense that memory requirements need not be exponential in the number of dimensions, but need merely match the real demands of the task. 2 Good Convergence on Control Problems We applied the sarsa and CMAC combination to the three continuous-state control problems studied by Boyan and Moore (1995): 2D gridworld, puddle world, and mountain car. Whereas they used a model of the task dynamics and applied dynamic programming backups offline to a fixed set of states, we learned online, without a model, and backed up whatever states were encountered during complete trials. Unlike Boyan 1040 1. Initially: wa(f) := ~, ea(f) := 0, 'ria E Actions, 'rifE CMAC-tiles. 2. Start of Trial: s:= random-stateO; F := features(s); a := E-greedy-policy(F). 3. Eligibility Traces: e,,(f) := )..e,,(f), 'rib, 'rIf; 3a. Accumulate algorithm: ea(f) := ea(f) + 1, 'rIf E F. R. S. SUTION 3b. Replace algorithm: ea(f) := 1, e,,(f) := 0, 'rIf E F, 'rib t a. 4. Environment Step: Take action a; observe resultant reward, r, and next state, s'. 5. Choose Next Action: F' := features(s'), unless s' is the terminal state, then F' := 0; a' := £-greedy-policy(F'). 7. Loop: a := a'; s := s'; F := F'; if s' is the terminal state, go to 2; else go to 3. Figure 1: The sarsa algorithm for finite-horizon (trial based) tasks. The function £greedy-policy( F) returns, with probability E, a random action or, with probability 1- £, computes L:JEF Wa for each action a and returns the action for which the sum is largest, resolving any ties randomly. The function features( s) returns the set of CMAC tiles corresponding to the state s. The number of tiles returned is the constant c. Qo, a, and)" are scalar parameters. C\I =It: C o <Ii C Q) E o . ................... ; .......... .,... ... _-- Tiling #1 ~ Tiling #2 ----f---- ·····i-··· ... -.~ ... - _· _-t -·_· ... .. + ........ ! ......... < .. .. ..... ~ .... •... : .... ; ........ -f.." .. ---L. ·····1···· .... : Dimension #1 Figure 2: CMACs involve multiple overlapping tilings of the state space. Here we show two 5 x 5 regular tilings offset and overlaid over a continuous, two-dimensional state space. Any state, such as that shown by the dot, is in exactly one tile of each tiling. A state's tiles are used to represent it in the sarsa algorithm described above. The tilings need not be regular grids such as shown here. In particular, they are often hyperplanar slices, the number of which grows sub-exponentially with dimensionality of the space. CMACs have been widely used in conjunction with reinforcement learning systems (e.g., Watkins, 1989; Lin &. Kim, 1991; Dean, Basye &. Shewchuk, 1992; Tham, 1994). Generalization in Reinforcement Learning 1041 and Moore, we found ro bust good performance on all tasks. We report here results for the puddle world and the mountain car, the more difficult of the tasks they considered. Training consisted of a series of trials, each starting from a randomly selected nongoal state and continuing until the goal region was reached. On each step a penalty (negative reward) of -1 was incurred. In the puddle-world task, an additional penalty was incurred when the state was within the "puddle" regions. The details are given in the appendix. The 3D plots below show the estimated cost-to-goal of each state, i.e., maXa Q(8, a). In the puddle-world task, the CMACs consisted of 5 tilings, each 5 x 5, as in Figure 2. In the mountain-car task we used 10 tilings, each 9 x 9. Puddle World Learned State Values 60 68 Trial 12 Trial 100 Figure 3: The puddle task and the cost-to-goal function learned during one run. Mountain Car Goal Step 428 Figure 4: The mountain-car task and the cost-to-goal function learned during one run. The engine is too weak to accelerate directly up the slope; to reach the goal, the car must first move away from it. The first plot shows the value function learned before the goal was reached even once. We also experimented with a larger and more difficult task not attempted by Boyan and Moore. The acrobot is a two-link under-actuated robot (Figure 5) roughly analogous to a gymnast swinging on a highbar (Dejong & Spong, 1994; Spong & Vidyasagar, 1989). The first joint (corresponding to the gymnast's hands on the bar) cannot exert 1042 R.S. SUTIQN The object is to swing the endpoint (the feet) above the bar by an amount equal to one of the links. As in the mountain-car task, there are three actions, positive torque, negative torque, and no torque, and reward is -Ion all steps. (See the appendix.) The Acrobat 100~~-----------------------------, Acrobot Learning Curves Goal: Raise tiP above line StepsfTrial (log scale) 10 100 Typical :/Sln910 Run 200 300 Trials Figure 5: The Acrobot and its learning curves. 3 The Effect of A Smoothod Average of 10 Runs 400 500 A key question in reinforcement learning is whether it is better to learn on the basis of actual outcomes, as in Monte Carlo methods and as in TD(A) with A = 1, or to learn on the basis of interim estimates, as in TD(A) with A < 1. Theoretically, the former has asymptotic advantages when function approximators are used (Dayan, 1992; Bertsekas, 1995), but empirically the latter is thought to achieve better learning rates (Sutton, 1988). However, hitherto this question has not been put to an empirical test using function approximators. Figures 6 shows the results of such a test. StepslTrial A veragtd over fLlSI 20 tnals and 30 runs Mountain Car rr .... --.-------------. "",---,r---r---...,....,-----, 700 600 Accunrulate I--~--__ ~--__ ~---} .,; " -f--~---.--~~-__ -l o 2 0 4 0 6 II 8 1 J U 0 2 0 4 0 6 0. 11 1 2. IX Puddle World Replare CostfTrial Avengod over tu'Sl 40 lna15 2')1) and 30 ruDS 1--__ --__ ~ ____ ~--+ l S I I u 2 0 4 06 08 12 IX Figure 6: The effects of A and Ct in the Mountain-Car and Puddle-World tasks. Figure 7 summarizes this data, and that from two other systematic studies with different tasks, to present an overall picture of the effect of A. In all cases performance is an inverted-U shaped function of A, and performance degrades rapidly as A approaches I, where the worst performance is obtained. The fact that performance improves as A is increased from 0 argues for the use of eligibility traces and against I-step methods such as TD(O) and 1-step Q-Iearning. The fact that performance improves rapidly as A is reduced below 1 argues against the use of Monte Carlo or "rollout" methods. Despite the theoretical asymptotic advantages of these methods, they are appear to be inferior in practice. Acknowledgments The author gratefully acknowledges the assistance of Justin Boyan, Andrew Moore, Satinder Singh, and Peter Dayan in evaluating these results. Generalization in Reinforcement Learning Mountain Car 700,-----...... --~ 650 600 Stepsffrial 550 500 .50 240 230 220 210 Costffrial 200 190 180 T 170 .L 160 150 0 0"1 0.4 0.6 0. 8 Puddle World Replace ~_;:i ~ 0.2 0.4 0.6 0.8 A. 1043 Random Walk I> 0.5 · · · 6 Accumulate: 0 .• Root Mean 03 Squared Error Replace 0.2 0.2 0.4 0.6 0.8 A. Cart and Pole 300 ~ 250 200 Failures per 100,000 steps i--_~ Accumulate · 150 ---~· · ........ ~ .... ~ ... ~ 100 0" ~ 50 0.2 0 .• 0.6 0.' A. Figure 7: Performance versus A, at best Q, for four different tasks. The left panels summarize data from Figure 6. The upper right panel concerns a 21-state Markov chain, the objective being to predict, for each state, the probability of terminating in one terminal state as opposed to the other (Singh & Sutton, 1996). The lower left panel concerns the pole balancing task studied by Barto, Sutton and Anderson (1983). This is previously unpublished data from an earlier study (Sutton, 1984). References Albus, J. S. (1981) Brain, Behavior, and RoboticI, chapter 6, pages 139-179. Byte Books. Baird, L. C. (1995) Residual Algorithms: Reinforcement Learning with Function Approximation. Proc. ML95. Morgan Kaufman, San Francisco, CA. Barto, A. G., Bradtke, S. J., & Singh, S. P. (1995) Real-time learning and control using asynchronous dynamic programming. Artificial Intelligence. Barto, A. G., Sutton, R. S., & Anderson, C. W. (1983) Neuronlike elements that can solve difficult learning control problems. TranI. IEEE SMC, 13, 835-846. Bertsekas, D. P. (1995) A counterexample to temporal differences learning. Neural Computation, 7, 270-279. Boyan, J. A. & Moore, A. W. (1995) Generalization in reinforcement learning: Safelyapproximating the value function. NIPS-7. San Mateo, CA: Morgan Kaufmann. Crites, R. H. & Barto, A. G. (1996) Improving elevator performance using reinforcement learning. NIPS-8. Cambridge, MA: MIT Press. Dayan, P. (1992) The convergence of TD(~) for general ~. Machine Learning, 8,341-362. Dean, T., Basye, K. & Shewchuk, J. (1992) Reinforcement learning for planning and control. In S. Minton, Machine Learning Methodl for Planning and Scheduling. Morgan Kaufmann. Dejong, G. & Spong, M. W. (1994) Swinging up the acrobot: An example of intelligent control. In Proceedingl of the American Control Conference, pagel 1.158-1.161.. Gordon, G. (1995) Stable function approximation in dynamic programming. Proc. ML95. Lin, L. J. (1992) Self-improving reactive agents based on reinforcement learning, planning and teaching. Machine Learning, 8(3/4), 293-321. Lin, CoS. & Kim, H. (1991) CMAC-based adaptive critic self-learning control. IEEE TranI. Neural Networkl, I., 530-533. Miller, W. T., Glanz, F. H., & Kraft, L. G. (1990) CMAC: An associative neural network alternative to backpropagation. Proc. of the IEEE, 78, 1561-1567. 1044 R.S. SUTION Rummery, G. A. & Niranjan, M. (1994) On-line Q-Iearning using connectionist systems. Technical Report CUED /F-INFENG /TR 166, Cambridge University Engineering Dept. Singh, S. P. & Sutton, R. S. (1996) Reinforcement learning with replacing eligibility traces. Machine Learning. Spong, M. W. & Vidyasagar, M. (1989) Robot Dynamic, and Control. New York: Wiley. Sutton, R. S. (1984) Temporal Credit A"ignment in Reinforcement Learning. PhD thesis, University of Massachusetts, Amherst, MA. Sutton, R. S. (1988) Learning to predict by the methods of temporal differences. Machine Learning, 3, 9-44. Sutton, R. S. & Whitehead, S. D. (1993) Online learning with random representations. Proc. ML93, pages 314-321. Morgan Kaufmann. Tham, C. K. (1994) Modular On-Line Function Approximation for Scaling up Reinforcement Learning. PhD thesis, Cambridge Univ., Cambridge, England. Tesauro, G. J. (1992) Practical issues in temporal difference learning. Machine Learning, 8{3/4),257-277. Tsitsiklis, J. N. & Van Roy, B. (1994) Feature-based methods for large-scale dynamic programming. Techical Report LIDS-P2277, MIT, Cambridge, MA 02139. Watkins, C. J. C. H. (1989) Learning from Delayed Reward,. PhD thesis, Cambridge Univ. Zhang, W. & Dietterich, T. G., (1995) A reinforcement learning approach to job-shop scheduling. Proc. IJCAI95. Appendix: Details of the Experiments In the puddle world, there were four actions, up, down, right, and left, which moved approximately 0.05 in these directions unless the movement would cause the agent to leave the limits of the space. A random gaussian noise with standard deviation 0.01 was also added to the motion along both dimensions. The costs (negative rewards) on this task were -1 for each time step plus additional penalties if either or both of the two oval "puddles" were entered. These penalties were -400 times the distance into the puddle (distance to the nearest edge). The puddles were 0.1 in radius and were located at center points (.1, .75) to (A5, .75) and (A5, A) to (045, .8). The initial state of each trial was selected randomly uniformly from the non-goal states. For the run in Figure 3, a == 0.5, >. == 0.9, c == 5, f == 0.1, and Qo == O. For Figure 6, Qo == -20. Details of the mountain-car task are given in Singh & Sutton (1996). For the run in Figure 4, a == 0.5, >. == 0.9, c == 10, f == 0, and Qo == O. For Figure 6, c == 5 and Qo == -100. In the acrobot task, the CMACs used 48 tilings. Each of the four dimensions were divided into 6 intervals. 12 tilings depended in the usual way on all 4 dimensions. 12 other tilings depended only on 3 dimensions (3 tilings for each of the four sets of 3 dimensions). 12 others depended only on two dimensions (2 tilings for each of the 6 sets of two dimensions. And finally 12 tilings depended each on only one dimension (3 tilings for each dimension). This resulted in a total of 12 .64 + 12 . 63 + 12 .62 + 12 ·6 == 18,648 tiles. The equations of motion were: 91 = _d~l (d29. + rPl) 9. == ( m.I~2 + I. - ~:) -1 (T + ~: rPl rP2 ) d1 == mll~l + m.(l~ + 1~2 + 2hlc. cosO.) + II + I.) d. == m.(l~. + hie. cosO.) + I. .. . . rPl = -m.lde.O.,inO. - 2m.ldc.0.Ol,inO. + (ml lel + m.h)gcos(Ol - 7r/2) + rP. rP. == m 2 Ie.gcos(01 +0. - 7r/2) where T E {+1, -1,0} was the torque applied at the second joint, and .6. == 0.05 was the time increment. Actions were chosen after every four of the state updates given by the above equations, corresponding to 5 Hz. The angular velocities were bounded by 91 E [-47r, 47r] and 9. E [-97r,97r]. Finally, the remaining constants were m1 == m2 == 1 (masses of the links), 11 == h == 1 (lengths of links), lei == 'e2 == 0.5 (lengths to center of mass of links), II == 12 == 1 (moments of inertia of links), and g == 9.8 (gravity). The parameters were a == 0.2, >. == 0.9, c == 48, f == 0, Qo == O. The starting state on each trial was 01 == O. == O.	1995	87
1,136	A Multiscale Attentional Framework for Relaxation Neural Networks Dimitris I. Tsioutsias Dept. of Electrical Engineering Yale University New Haven, CT 06520-8285 tsioutsias~cs.yale.edu Eric Mjolsness Dept. of Computer Science & Engineering University of California, San Diego La Jolla, CA 92093-0114 emj~cs.ucsd.edu Abstract We investigate the optimization of neural networks governed by general objective functions. Practical formulations of such objectives are notoriously difficult to solve; a common problem is the poor local extrema that result by any of the applied methods. In this paper, a novel framework is introduced for the solution oflargescale optimization problems. It assumes little about the objective function and can be applied to general nonlinear, non-convex functions; objectives in thousand of variables are thus efficiently minimized by a combination of techniques - deterministic annealing, multiscale optimization, attention mechanisms and trust region optimization methods. 1 INTRODUCTION Many practical problems in computer vision, pattern recognition, robotics and other areas can be described in terms of constrained optimization. In the past decade, researchers have proposed means of solving such problems with the use of neural networks [Hopfield & Tank, 1985; Koch et ai., 1986], which are thus derived as relaxation dynamics for the objective functions codifying the optimization task. One disturbing aspect of the approach soon became obvious, namely the apparent inability of the methods to scale up to practical problems, the principal reason being the rapid increase in the number of local minima present in the objectives as the dimension of the problem increases. Moreover most objectives, E( v), are highly nonlinear, non-convex functions of v , and simple techniques (e.g. steepest descent) 634 D. I. TSIOUTSIAS, E. MJOLSNESS will, in general, locate the first minimum from the starting point. In this work, we propose a framework for solving large-scale instances of such optimization problems. We discuss several techniques which assist in avoiding spurious minima and whose combined result is an objective function solution that is computationallyefficient, while at the same time being globally convergent. In section 2.1 we discuss the use of deterministic annealing as a means of avoiding getting trapped into local minima. Section 2.2 describes multiscale representations of the original objective in reduced spatial domains. In section 2.3 we present a scheme for reducing the computational requirements of the optimization method used, by means of a focus of attention mechanism. Then, in section 2.4 we introduce a trust region method for the relaxation phase of the framework, which uses second order information (i.e. curvature) of the objective function. In section 3 we present experimental results on the application of our framework to a 2-D region segmentation objective with discontinuities. Finally, section 4 summarizes our presentation. 2 THEORETWALFRAMEWORK Our optimization framework takes the form of a list of nested loops indicating the order of conceptual (and computational) phases that occur: from the outer to the inner loop we make use of deterministic annealing, a multiscale representation, an attentional mechanism and a trust region optimization method. 2.1 ANNEALING NETS The usefulness of statistical mechanics for designing optimization procedures has recently been established; prime examples are simulated annealing and its various mean field theory approximations [Hopfield & Tank, 1985; Durbin & Willshaw, 1987]. The success of such methods is primarily due to entropic terms included in the objective (i.e. syntactic terms), but the price to pay is their highly nonlinear form. Interestingly, those terms can effectively be convexified by the use of a "temperature" parameter, T , allowing for a reduction in the number of minima and the ability to track the solution through "temperature". 2.2 MULTISCALE REPRESENTATION To solve large-scale problems in thousands of variables, we need to speed up the convergence of the method while still retaining valid state-space trajectories. To accomplish this we introduce smaller, approximate versions of the problem at coarser spatial scales [Mjolsness et al. , 1991]; the nonlinearity of the original objective is maintained at all scales, as opposed to other approaches where the objectives and their derivatives are either approximated by the use of finite difference methods, or solved for by multigrid techniques where a quadratic objective is still assumed. Consequently, the multiscale representation exploits the effective smoothness in the objectives: by alternating relaxation phases between coarser and finer scales, we use the former to identify extrema and the latter to localise them. 2.3 FOCUS OF ATTENTION To further reduce the computational requirements of larg~scale optimization (and indirectly control its temporal behavior), we use a focus of attention (FoA) mechanism [Mjolsness & Miranker, 1993], reminiscent of the spotlight hypothesis argued A Multiscale Attentional Framework for Relaxation Neural Networks 635 to exist in early vision systems [Koch & Ullman, 1985; Olshausen et al., 1993]. The effect of a FoA is to support efficient, responsive analysis: it allows resources to be focused on selected areas of a computation and can rapidly redirect them as the task requirements evolve. Specifically, the FoA becomes a characteristic function, 7l'(X) , determining which of the N neurons are active and which are clamped during relaxation, by use of a discrete-valued vector, X, and by the rule: 7l'i(X) = 1 if neuron Vi is in the FoA, and zero otherwise. Moreover, a limited number, n, of neurons Vi are active at any given instant: I:i 7l'i(X) = n, with n« Nand n chosen as an optimal FoA size. To tie the attentional mechanism to the multiscale representation, we introduce a partition of the neurons Vi into blocks indexed by a (corresponding to coarse-scale blockneurons), via a sparse rectangular matrix Bia E {O, I} such that I:a Bia = 1, Vi, with i = 1, ... ,N, a = 1,oo.,K and K«N. Then 7l'i(X) = I:aBiaXa, and we use each component of X for switching a different block of the partition; thus, a neuron Vi is in the FoA iff its coarse scale block a is in the FoA, as indicated by Xa. As a result, our FoA need not necessarily have a single region of activity: it may well have a distributed activity pattern as determined by the partitions Bia. 1 Clocked objective function notation [Mjolsness & Miranker, 1993] makes the task more apparent: during the active-x phase the FoA is computed for the next activev phase, determining the subset of neurons Vi on which optimization is to be carried out. We introduce the quantity E ;dv] == g~ ~ (Ti is a time axis for Vi) [Mjolsness & Miranker, 1993] as an estimate of the predicted dE arising from each Vi if it joins the FoA. For HopfieldjGrossberg dynamics this measure becomes: E ;d v ] = _g~(gi1(Vi)) (~~) 2 == -gHUi)(E,i)2 (1) wi th E,i ~f 'V'i E, and gi the transfer function for neuron Vi (e.g. a sigmoid function). Eq. (1) is used here analogously to saliency measures introduced into neurophysiological work [Koch & Ullman, 1985]; we propose it as a global measure of conspicuousness. As a result, attention becomes a k-winner-take-all (kWTA) network: a a where I refers to the scale for which the FoA is being determined (I = 1, ... , L), EEl conforms with the clocked objective notation, and the last summand corresponds to the subspace on which optimization is to be performed, as determined by the current FoA.2 Periodically, an analogous FoA through spatial scales is run, allowing re-direction of system resources to the scale which seems to be having the largest combined benefit and cost effect on the optimization [Tsioutsias & Mjolsness, 1995]. The combined effect of multiscale optimization and FoA is depicted schematically in Fig. 1: reduced-dimension functionals are created and a FoA beam "shines" through scales picking the neurons to work on. 1 Preferably, Bia will be chosen to minimize the number of inter-block connections. 2 Before computing a new FoA we update the neighbors of all neurons that were included in the last focus; this has a similar effect to an implicit spreading of activation. 636 D. I. TSIOUTSIAS, E. MJOLSNESS Layer 3 Layer 1 Figure 1: Multiscale Attentional Neural Nets: FoA on a layer (e.g. L=l) competes with another FoA (e.g. L=2) to determine both preferable scale and subspace. 2.4 OPTIMIZATION PHASE To overcome the problems generally associated with the steepest descent method, other techniques have been devised. Newton's method, although successful in small to medium-sized problems, does not scale well in large non-convex instances and is computationally intensive. Quasi-Newton methods are efficient to compute, have quadratic termination but are not globally convergent for general nonlinear, nonconvex functions. A method that guarantees global convergence is the trust region method [Conn et al., 1993]. The idea is summarized as follows : Newton's method suffers from non-positive definite Hessians; in such a case, the underlying function m(k)(6) obtained from the 2nd order Taylor expansion of E(Vk + 6) does not have a minimum and the method is not defined, or equivalently, the region around the current point Vk in which the Taylor series is adequate does not include a minimizing point of m(k)(6). To resolve this, we can define a neighborhood Ok of Vk such that m(k)(6) agrees with E(Vk + 6) in some sense; then, we pick Vk+l = Vk + 6k , where 6 k minimizes m(k)(6) , V(Vk + 6) E Ok . Thus, we seek a solution to the resulting subproblem: (3) where 1I ·lIp is any kind of norm (for instance, the L2 norm leads to the LevenbergMarquardt methods) , and ~k is the radius of Ok, adaptively modified based on an accuracy ratio Tk = (~E(k)/~m(k) = (E(k ) E(Vk + 6k»/(m(k)(O) m(k)(6k»; ~E(k) is the "actual reduction" in E(k) when step 6 k is taken, and ~m(k) the "predicted reduction" . The closer Tk is to unity, the better the agreement between the local quadratic model of E (k) and the objective itself is, and ~k is modified adaptively to reflect this [Conn et al., 1993]. We need to make some brief points here (a complete discussion will be given elsewhere [Tsioutsias & Mjolsness, 1995]): A Multiscale Attentional Framework for Relaxation Neural Networks 637 • At each spatial scale of our multiscale representation, we optimize the corresponding objective by applying a trust region method. To obtain sufficient relaxation progress as we move through scales we have to maintain meaningful region sizes, Llk; to that end we use a criterion based on the curvature of the functionals along a searching direction. • The dominant relaxation computation within the algorithm is the solution of eq. (3). We have chosen to solve this subproblem with a preconditioned conjugate gradient method (PCG) that uses a truncated Newton step to speed up the computation; steps are accepted when a sufficiently good approximation to the quasi-Newton step is found. 3 In our case, the norm in eq. (3) becomes the elliptical norm 1I~llc = ~tc~, where a diagonal preconditioner to the Hessian is used as the scaling matrix C. • If the neuronal connectivity pattern of the original objective is sparse (as happens for most practical combinatorial optimization problems), the pattern of the resulting Hessian can readily be represented by sparse static data structures,4 as we have done within our framework. Moreover, the partition matrices, Bia, introduce a moderate fill-in in the coarser objectives and the sparsity of the corresponding Hessians is again taken into account. 3 EXPERIMENTS We have applied our proposed optimization framework to a spatially structured objective from low-level vision, namely smooth 2-D region segmentation with the inclusion of discontinuity detection processes: ij ij ij ij ij where d is the set of image intensities, j is the real-valued smooth surface to be fit to the data, lV and lh are the discrete-valued line processes indicating a non-zero value in the intensity gradient, and ¢(x) = -(2go)-1[lnx+ln(1-x)] is a barrier function restricting each variable into (0,1) by infinite barriers at the borders. Eq. (4) is a mixed-nonlinear objective involving both continuous and binary variables; our framework optimizes vectors j, lh and lV simultaneously at any given scale as continuous variables, instead of earlier two-step, alternate continuous/discrete-phase approaches [Terzopoulos, 1986]. We have tested our method on gradually increasing objectives, from a "small" size of N=12,288 variables for a 64x64 image, up to a large size of N=786 ,432 variables for a 512x512 image; the results seem to coincide with our theoretical expectations: a significant reduction in computational cost was observed and consistent convergence towards the optimum of the objective was found for various numbers of coarse scales and FoA sizes. The dimension of the objective at any scale I was chosen via a power law: N(L-l+1)! L, where L is the total number of scales and N the size of 3 The algorithm can also handle directions of negative curvature. 4 This property becomes important in a neural net implementation. 638 D. I. TSIOUTSIAS, E. MJOLSNESS the original objective. The effect of our multiscale optimization with and without a FoA is shown in Fig. 2 for the 128x128 and the 512x512 nets, where E( v) is the best final configuration with a one-level no-FoA net, and cumulative cost is an accumulated measure in the number of connection updates at each scale; a consistent scale-up in computational efficiency can be noted when L > 1, while the cost measure also reflects the relative total wall-clock times needed for convergence. Fig. 3 shows part of a comparative study we made for saliency measures alternative to eq. (1) (e.g. g~IE,il), in order to investigate the validity of eq. (1) as a predictor of l:!..E: the more prominent "linearity" in the left scatterplot seems to justify our choice of saliency. 104 .--___ M-'S-'-/_A_T_N_e_t_s ,..,,(_12_8_t2-,)_: _L_=--,1 ,_2'-,3 ___ ---, 10' MS/ AT Nets (512t2) : L=1,2,3,4 10' 10' 10' 10-' 10" 10-110 10' 10' 10' 10' ~ 10 l ~ '" 2 Nl I 10' #1 >' g10- 1 10-' 10" 10-4 10-' 2000 10-' 0 Figure 2: Multiscale Optimization (curves labeled by number of scales used): #numbered curves correspond to nets without a FoA , simply-numbered ones to nets with a FoA used at all scales. The lowest costs result from the combined use of multiscale optimization and FoA. 4 CONCLUSION We have presented a framework for the optimization of large-scale objective functions using neural networks that incorporate a multiscale attentional mechanism. Our method allows for a continuous adaptation of the system resources to the computational requirements of the relaxation problem through the combined use of several techniques. The framework was applied to a 2-D image segmentation objective with discontinuities; formulations of this problem with tens to hundreds of thousands of variables were then successfully solved. Acknow ledgements This work was supported partly by AFOSR-F49620-92-J-0465 and the Yale Center of Theoretical and Applied Neuroscience. 60000 A Multiscale Attentional Framework for Relaxation Neural Networks 639 10' (128t2) : Focus on 1st level - proposed saliency 10' (128t2) : Focus on 1st level - absolute gradient 10' ~ 10° o o :0 ~ 10-' .!! .. 8-,.,10-' o c . !! OJ 11l 10-3 " .. ~ !10- 4 o o o o .. .. 8 8 00 o 0 o o 00 ",00 0 10' ,. o o :0 .. o : 10' " 0. ~ c .!! .. ~10-1 ~ ~ 10-' 0 0 0 0 0 0 0 3 8 o 8 .. ~0o; /.0 0 "1:,00 0 0" .. 0 " I 10 -~0~-.-'-'-'u.tl~Oo.,J....Ll.J"!'1=0-.-'-'-~1 O:=.-.l.....L..Ll.';t'!loO r-'-~.tO!:.-r-u~1~0-:r'-'~ 100 10-~0b.--'"-U.~I~"ol_:r'-'.w.m ~I"O~ I _r-u.li;~ lo.L_:r'-'-,-"~uI"O~I_ • .-'-'-l.l..lLU~l ul o,,," _ • .l.....L..Lu.;I"~ol-cr'-'-~ 1 00 (Average Della-E per block) (Average Della-E per block) Figure 3: Saliency Comparison: (left), saliency as in eq. (1); (right), the absolute gradient was used instead. References A. Conn, N. Gould, A. Sartanaer, & Ph. Toint. (1993) Global Convergence of a Class of Trust Region Algorithms for Optimization Using Inexact Projections on Convex Constraints. SIAM J. of Optimization, 3(1):164-221. R. Durbin & D. Willshaw. (1987) An Analogue Approach to the TSP Problem Using an Elastic Net Method. Nature, 326:689-691. J. Hopfield & D. W. Tank. (1985) Neural Computation of Decisions in Optimization Problems. Bioi. Cybernei., 52:141-152. C. Koch, J. Marroquin & A. Yuille. (1986) Analog 'Neuronal' Networks in Early Vision. Proc. of the National Academy of Sciences USA, 83:4263-4267. C. Koch, & S. Ullman. (1985) Shifts in Selective Visual Attention: Towards the Underlying Neural Circuitry. Human Neurobiology, 4 :219-227. E. Mjolsness, C. Garrett, & W. Miranker. (1991) Multiscale Optimization in Neural Nets. IEEE Trans. on Neural Networks, 2(2):263-274. E. Mjolsness & W. Miranker. (1993) Greedy Lagrangians for Neural Networks: Three Levels of Optimization in Relaxation Dynamics. YALEU/DCS/TR-945. (URL file:!!cs.ucsd.edu!pub!emj!papers!yale-TR-945.ps.Z) B. Olshausen, C. Anderson, & D. Van Essen. (1993) A Neurobiological Model of Visual Attention and Invariant Pattern Recognition Based on Dynamic Routing of Information. The Journal of Neuroscience , 13(11):4700-4719. D. Terzopoulos. (1986) Regularization of Inverse Visual Problems Involving Discontinuities. IEEE Trans. PAMI, 8:419-429. D. I. Tsioutsias & E. Mjolsness. (1995) Global Optimization in Neural Nets: A Novel Relaxation Framework. To appear as a UCSD-CSE-TR, Dec. 1995.	1995	88
1,137	VLSI Model of Primate Visual Smooth Pursuit Ralph Etienne-Cummings Department of Electrical Engineering, Southern Illinois University, Carbondale, IL 62901 Jan Van der Spiegel Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, PA 19104 Paul Mueller Corticon, Incorporated, 3624 Market Str, Philadelphia, PA 19104 Abstract A one dimensional model of primate smooth pursuit mechanism has been implemented in 2 11m CMOS VLSI. The model consolidates Robinson's negative feedback model with Wyatt and Pola's positive feedback scheme, to produce a smooth pursuit system which zero's the velocity of a target on the retina. Furthermore, the system uses the current eye motion as a predictor for future target motion. Analysis, stability and biological correspondence of the system are discussed. For implementation at the focal plane, a local correlation based visual motion detection technique is used. Velocity measurements, ranging over 4 orders of magnitude with < 15% variation, provides the input to the smooth pursuit system. The system performed successful velocity tracking for high contrast scenes. Circuit design and performance of the complete smooth pursuit system is presented. 1 INTRODUCTION The smooth pursuit mechanism of primate visual systems is vital for stabilizing a region of the visual field on the retina. The ability to stabilize the image of the world on the retina has profound architectural and computational consequences on the retina and visual cortex, such as reducing the required size, computational speed and communication hardware and bandwidth of the visual system (Bandera, 1990; Eckert and Buchsbaum, 1993). To obtain similar benefits in active machine vision, primate smooth pursuit can be a powerful model for gaze control. The mechanism for smooth pursuit in primates was initially believed to be composed of a simple negative feedback system which attempts to zero the motion of targets on the fovea, figure I (a) (Robinson, 1965). However, this scheme does not account for many psychophysical properties of smooth VLSI Model of Primate Visual Smooth Pursuit 707 pursuit, which led Wyatt and Pola (1979) to proposed figure l(b), where the eye movement signal is added to the target motion in a positive feed back loop. This mechanism results from their observation that eye motion or apparent target motion increases the magnitude of pursuit motion even when retinal motion is zero or constant. Their scheme also exhibited predictive qualities, as reported by Steinbach (1976). The smooth pursuit model presented in this paper attempts the consolidate the two models into a single system which explains the findings of both approaches. Target Retinal Moticn Motion Eye Motion Target Motion Eye Motion e~ G lee I > ee = e t ~; G ~ co G r = 0 G+l (a) e~~ (b) Figure I: System Diagrams of Primate Smooth Pursuit Mechanism. (a) Negative feedback model by Robinson (1965). (b) Positive feedback model by Wyatt and Pola (1979). > The velocity based smooth pursuit implemented here attempts to zero the relative velocity of the retina and target. The measured retinal velocity, is zeroed by using positive feedback to accumulate relative velocity error between the target and the retina, where the accumulated value is the current eye velocity. Hence, this model uses the Robinson approach to match target motion, and the Wyatt and Pola positive feed back loop to achieve matching and to predict the future velocity of the target. Figure 2 shows the system diagram of the velocity based smooth pursuit system. This system is analyzed and the stability criterion is derived. Possible computational blocks for the elements in figure I (b) are also discussed. Furthermore, since this entire scheme is implemented on a single 2 /lm CMOS chip, the method for motion detection, the complete tracking circuits and the measured results are presented. Retinal Motion Eye Motion Figure 2: System Diagram of VLSI Smooth Pursuit Mechanism. er is target velocity in space, Bt is projected target velocity, Be is the eye velocity and Br is the measured retinal velocity. 2 VELOCITY BASED SMOOTH PURSUIT Although figure I (b) does not indicate how retinal motion is used in smooth pursuit, it provides the only measurement of the projected target motion. The very process of calculating retinal motion realizes negative feed back between the eye movement and the target motion, since retinal motion is the difference between project target and eye motion. If Robinson's model is followed, then the eye movement is simply the amplified version of the retinal motion. If the target disappears from the retina, the eye motion would be zero. However, Steinbach showed that eye movement does not cea~ when the target fades off and on, indicating that memory is used to predict target motion. Wyatt and Palo showed a direct additive influence of eye movement on pursuit. However, the computational blocks G' and a of their model are left unfilled. 708 R. ETIENNE-CUMMINGS, J. V AN DER SPIEGEL, P. MUELLER In figure 2, the gain G models the internal gain of the motion detection system, and the internal representation of retinal velocity is then V r. Under zero-slip tracking, the retinal velocity is zero. This is obtained by using positive feed back to correct the velocity error between target, er, and eye, ee. The delay element represents a memory of the last eye velocity while the current retinal motion is measured. If the target disappears, the eye motion continues with the last value, as recorded by Steinbach, thus anticipating the position of the target in space. The memory also stores the current eye velocity during perfect pursuit. The internal representation of eye velocity, Ve, is subsequently amplified by H and used to drive the eye muscles. The impulse response of the system is given in equations (I). Hence, the relationship between eye velocity and target velocity is recursive and given by equations (2). To prove the stability of this system, the retinal velocity can be expressed in terms of the target motion as given in equations (3a). The ideal condition for accurate performance is for GH = 1. However, in practice, gains of different amplifiers () z-) () -.f..(z) = GH--_-) (a); ~(I1) = GH[-8(11) + u(n)] (b) (}r 1 - Z (}r n-) (}e(n) = (},(n) - (}r(n) = GH[-8(n) + u(n)] * (}r(n) = GHL(},.(k) k=O 11 () r ( 11) = (),( n ) (1 - GH) => () r( 1l ) = 0 if GH = 1 => () in) = (),( 11 ) ( a) () (n) 11 ~ 00 ) 0 if 11 - GH I < 1 => 0 < GH < 2 for stability ( b) r (I) (2) (3) are rarely perfectly matched. Equations (3b) shows that stability is assured for O<GH< 2. Figure 3 shows a plot of eye motion versus updates for various choices of GH. At each update, the retinal motion is computed. Figure 3(a) shows the eye's motion at the on-set of smooth pursuit. For GH = 1, the eye movement tracks the target's motion exactly, and lags slightly only when the target accelerates. On the other hand, if GH« I, the eye's motion always lags the target's. If GH -> 2, the system becomes increasing unstable, but converges for GH < 2. The three cases presented correspond to the smooth pursuit system being critically, over and under damped, respectively. 3 HARDWARE IMPLEMENTATION Using the smooth pursuit mechanism described, a single chip one dimensional tracking system has been implemented. The chip has a multi-layered computational architecture, similar to the primate's visual system. Phototransduction, logarithmic compression, edge detection, motion detection and smooth pursuit control has been integrated at the focal-plane. The computational layers can be partitioned into three blocks, where each block is based on a segment of biological oculomotor systems. 3.1 IMAGING AND PREPROCESSING The first three layers of the system mimics the photoreceptors, horizontal cells arx:l bipolar cells of biological retinas. Similar to previous implementations of silicon retinas, the chip uses parasitic bipolar transistors as the photoreceptors. The dynamic range of photoreceptor current is compressed with a logarithmic response in low light arx:l square root response in bright light. The range compress circuit represents 5-6 orders of magnitude of light intensity with 3 orders of magnitude of output current dynamic range. Subsequently, a passive resistive network is used to realize a discrete implementation of a Laplacian edge detector. Similar to the rods and cones system in primate retinas, the response time, hence the maximum detectable target speed, is ambient intensity dependent (160 (12.5) Ils in 2.5 (250) IlW/cm2). However, this does prevent the system from handling fast targets even in dim ambient lighting. VLSI Model of Primate Visual Smooth Pursuit 20 15 10 ~ 5 g 0 u > -5 -10 -15 -20 0 20 15 10 5 ~ ] 0 " > -5 • Target · 10 - -Eye: GH=I 99 -Eye GH=IOO __ .Eye: GH=O_IO · 15 -20 50 100 150 500 600 700 800 900 Updates Updates (a) (b) Figure 3: (a) The On-Set of Smooth Pursuit for Various GH Values. (b) Steady-State Smooth Pursuit. 3.2 MOTION MEASUREMENT 709 1000 This computational layer measures retinal motion. The motion detection technique implemented here differs from those believed to exist in areas V 1 and MT of the primate visual cortex. Alternatively, it resembles the fly's and rabbit's retinal motion detection system (Reichardt, 1961; Barlow and Levick, 1965; Delbruck, 1993). This is not coincidental, since efficient motion detection at the focal plane must be performed in a small areas and using simple computational elements in both systems. The motion detection scheme is a combination of local correlation for direction determination, and pixel transfer time measurement for speed. In this framework, motion is defined as the disappearance of an object, represented as the zero-crossings of its edges, at a pixel, followed by its re-appearance at a neighboring pixel. The (dis)appearance of the zero-crossing is determined using the (negative) positive temporal derivative at the pixel. Hence, motion is detected by AND gating the positive derivative of the zerocrossing of the edge at one pixel with the negative derivative at a neighboring pixel. The direction of motion is given by the neighboring pixel from which the edge disappeared. Provided that motion has been detected at a pixel, the transfer time of the edge over the pixel's finite geometry is inversely proportional to its speed. Equation (4) gives the mathematical representation of the motion detection process for an object moving in +x direction. In the equation. f,(l.'k,y.t) is the temporal response of pixel k as the zero crossing of an edge of an object passes over its 2a aperture. Equation (4) gives the direction of motion, while equation (5) gives the speed. The schematic of motion _ x = [ f f,( l: k, y, t) > 0] [ f f t(l.' k + J, y, t) < 0] = 0 ( a) motion+x=[~f,(l.'k-J,y,t)<O][~f/l.'k , y,t»O] (b) (4) 2a(k-n)-a = 8[t v ]8[x - 2ak] x Motion.' t = m 2a(k -n) -a v x J v x Speed + x = t - t 2 a d m 2a(k -n) +a Disappear .' t d = --~-- vx (5) the VLSI circuit of the motion detection model is shown in figure 4(a). Figure 4(b) shows reciprocal of the measured motion pulse-width for 1 D motion. The on-chip speed, et, is the projected target speed. The measured pulse-widths span 3-4 orders magnitude, 710 R. ETIENNE-CUMMINGS, J. VAN DER SPIEGEL, P. MUELLER One-Over Pulse-Width vs On-Chip Speed O.R • ~ 0.4 " ~ -0.0 +--------::II~-----__+ M ~ -0.4 -0.8 ---e-- \IPW_Lefi - -. - - IIPW_ Rlght -1.2 +-'----'--''-+--'--'--'--t---'--''--'-+-'--'--'-t-'--'--'-t---'--'--'-+ -12.0 -R.O -40 00 4.0 8.0 12.0 Left Right On-Chip Speed rcml~J (a) (b) Figure 4: (a) Schematic of the Motion Detection Circuit. Measured Output of the Motion Detection Circuit. (b) depending on the ambient lighting, and show less than 15% variation between chips, pixels, and directions (Etienne-Cummings, 1993). 3.3 THE SMOOTH PURSUIT CONTROL SYSTEM The one dimensional smooth pursuit system is implemented using a 9 x I array of motion detectors. Figure 5 shows the organization of the smooth pursuit chip. In this system, only diverging motion is computed to reduce the size of each pixel. The outputs of the motion detectors are grouped into one global motion signal per direction. This grouping is performed with a simple, but delayed, OR, which prevents pulses from neighboring motion cells from overlapping. The motion pulse trains for each direction are XOR gated, which allows a single integrator to be used for both directions, thus limiting mis-match_ The final value of the integrator is inversely proportional to the target's speed. The OR gates conserve the direction of motion. The reciprocal of the integrator voltage is next computed using the linear mode operation of a MOS transistor (Etienne-Cummings, 1993). The unipolar integrated pulse allows a single inversion circuit to be used for both directions of motion, again limiting mis-match. The output of the "one-over" circuit is amplified, and the polarity of the measured speed is restored. This analog voltage is proportional to retinal speed. The measured retinal speed is subsequently ailed to the stored velocity. Figure 6 shows the schematic for the retinal velocity accumulation (positive feedback) and storage (analog Wave Forms Motion Pulse Integration and "One-Over" V = GIRetinal Velocityl Polarity Restoration Retinal Velocity Accumulation and Sample/Hold Figure 5: Architecture of the VLSI Smooth Pursuit System. Sketches of the wave forms for a fast leftward followed by a slow rightward retinal motion are shown. VLSI Model of Primate Visual Smooth Pursuit 711 memory). The output of the XOR gate in figure 5 is used by the sample-and-hold circuit to control sampling switches S I and S2. During accumulation, the old stored velocity value, which is the current eye velocity, is isolated from the summed value. At the falling edge of the XOR output, the stored value on C2 is replaced by the new value on Cl. This stored value is amplified using an off chip motor driver circuit, and used to move the chip. The gain of the motor driver can be finely controlled for optimal operation. Motor System Retinal Velocity Accumulatiun Two Phase Sample/Hold Target Velocity Figure 6: Schematic Retinal Velocity Error Accumulation, Storage and Motor Driver Systems. Figure 7(a) shows a plot of one-over the measured integrated voltage as a function of on chip target speed. Due to noise in the integrator circuit, the dynamic range of the motion detection system is reduced to 2 orders of magnitude. However, the matching between left and right motion is unaffected by the integrator. The MaS "one-over" circuit, used to compute the analog reciprocal of the integrated voltage, exhibits only 0.06% deviation from a fitted line (Etienne-Cummings, 1993b). Figure 7(b) shows the measured increments in stored target velocity as a function of retinal (on-chip) speed. This is a test of all the circuit components of the tracking system. Linearity between retinal velocity increments and target velocity is observed, however matching between opposite motion has degraded. This is caused by the polarity restoration circuit since it is the only location where different circuits are used for opposite motion. On average, positive increments are a factor of 1.2 times larger than negative increments. The error bars shows the variation in velocity increments for different motion cells and different Chips. The deviation is less than 15 %. The analog memory has a leakage of 10 mV/min and an asymmetric swing of 2 to -1 V, caused by the buffers. The dynamic range of the complete smooth pursuit system is measured to be 1.5 orders magnitude. The maximum speed of the system is adjustable by varying the integrator charging time. The maximum speed is ambient intensity dependent and ranges from 93 cmls to 7 cm/s on-chip speed in Integrated Pulse vs On-Chip Speed 24 ~ 16 .' ~ 8 ._ ~ 0 -t--------",/II!...------+ il £ -8 oS :: -16 -24 -e--lnlPuI~_l..xft _ _ • _ JntPlllo;e_Rl~hl -32 -t-'---'---'-'--+-'--~~-t--'"-'-~_t_--"--''---'---"-t -100 -5.0 0.0 5.0 10.0 On-Chip Speed lemlsl 1.4 1.2 ~ l'! 1.0 " e ~ O.R u .s g 0.6 LLl .::;. g 04 OJ > 02 0.0 0 Velocity Error Increment vs On-Chip Speed ----. Nc~_ Jn crt~nl __ • _ _ Po,,_Incremclll 4 6 On-Chip Speed lem/s) (a) (b) Figure 7. (a) Measured integrated motion pulse voltage. (b) Measured output for the complete smooth pursuit system. 10 712 R. ETIENNE-CUMMINGS, J. VAN DER SPIEGEL, P. MUELLER bright (250 JlW/cm2) and dim (2.5 JlW/cm2) lighting, respectively. However, for any maximum speed chosen, the minimum speed is a factor of 0.03 slower. The minimum speed is limited by the discharge time of the temporal differentiators in the motion detection circuit to 0.004 cmls on chip. The contrast sensitivity of this system proved to be the stumbling block, and it can not track objects in normal indoor lighting. However, all circuits components tested successfully when a light source is used as the target. Additional measured data can be found in (Etienne-Cummings, 1995). Further work will improve the contrast sensitivity, combat noise and also consider two dimensional implementations with target acquisition (saccades) capabilities. 4 CONCLUSION A model for biological and silicon smooth pursuit has been presented. It combines the negative feed back and positive feedback models of Robinson and Wyatt and Pola. The smooth pursuit system is stable if the gain product of the retinal velocity detection system and the eye movement system is less than 2. VLSI implementation of this system has been performed and tested. The performance of the system suggests that wide range (92.9 - 0.004 cmls retinal speed) target tracking is possible with a single chip focal plane system. To improve this chip's performance, care must be taken to limit noise, improve matching and increase contrast sensitivity. Future design should also include a saccadic component to re-capture escaped targets, similar to biological systems. References C. Bandera, "Foveal Machine Vision Systems", Ph.D. Thesis, SUNY Buffalo, New York, ]990 H. Barlow and W. Levick, 'The Mechanism for Directional Selective Units in Rabbit's Retina", Journal of Physiology, Vol. 178, pp. 477-504, ]965 T. Delbruck, "Silicon Retina with Correlation-Based, Velocity-Tuned Pixels ", IEEE Transactions on Neural Networks, Vol. 4:3, pp. 529-41, 1993 M. Eckert and G. Buchsbaum, "Effect of Tracking Strategies on the Velocity Structure of Two-Dimensional Image Sequences", J. Opt. Soc. Am., Vol. AIO:7, pp. 1582-85, 1993 R. Etienne-Cummings et at., "A New Temporal Domain Optical Flow Measurement Technique for Focal Plane VLSI Implementation", Proceedings of CAMP 93, M. Bayoumi, L. Davis and K. Valavanis (Eds.), pp. 24]-25] , 1993 R. Etienne-Cummings, R. Hathaway and J. Van der Spiegel, "An Accurate and Simple CMOS 'One-Over' Circuit", Electronic Letters, Vol. 29-18, pp. ]618-]620, 1993b R. Etienne-Cummings et aI., "Real-Time Visual Target Tracking: Two Implementations of Velocity Based Smooth Pursuit", Visual Information Processing IV, SPIE Vol. 2488, Orlando, 17-18 April 1995 W. Reichardt, "Autocorrelation, A Principle for the Evaluation of Sensory Information by the Central Nervous System", Sensory Communication, Wiley, New York, 1961 D. Robinson, "The Mechanism of Human Smooth Pursuit Eye Movement", Journal of Physiology ( London) Vol. 180, pp. 569-591 , 1965 M. Steinbach, "Pursuing the Perceptual Rather than the Retinal Stimuli", Vision Research, Vol. 16, pp. 1371-1376,1976 H. Wyatt and J. Pola, "The Role of Perceived Motion in Smooth Pursuit Eye Movements", Vision Research, Vol. 19, pp. 613-618, 1979	1995	89
1,138	Parallel Optimization of Motion Controllers via Policy Iteration J. A. Coelho Jr., R. Sitaraman, and R. A. Grupen Department of Computer Science University of Massachusetts, Amherst, 01003 Abstract This paper describes a policy iteration algorithm for optimizing the performance of a harmonic function-based controller with respect to a user-defined index. Value functions are represented as potential distributions over the problem domain, being control policies represented as gradient fields over the same domain. All intermediate policies are intrinsically safe, i.e. collisions are not promoted during the adaptation process. The algorithm has efficient implementation in parallel SIMD architectures. One potential application - travel distance minimization - illustrates its usefulness. 1 INTRODUCTION Harmonic functions have been proposed as a uniform framework for the solution of several versions of the motion planning problem. Connolly and Grupen [Connolly and Grupen, 1993] have demonstrated how harmonic functions can be used to construct smooth, complete artificial potentials with no local minima. In addition, these potentials meet the criteria established in [Rimon and Koditschek, 1990] for navigation functions. This implies that the gradient of harmonic functions yields smooth ("realizable") motion controllers. By construction, harmonic function-based motion controllers will always command the robot from any initial configuration to a goal configuration. The intermediate configurations adopted by the robot are determined by the boundary constraints and conductance properties set for the domain. Therefore, it is possible to tune both factors so as to extremize user-specified performance indices (e.g. travel time or energy) without affecting controller completeness. Based on this idea, Singh et al. [Singh et al., 1994] devised a policy iteration method for combining two harmonic function-based control policies into a controller that minimized travel time on a given environment. The two initial control policies were Parallel Optimization of Motion Controllers via Policy Iteration 997 derived from solutions to two distinct boundary constraints (Neumann and Dirichlet constraints). The policy space spawned by the two control policies was parameterized by a mixing coefficient, that ultimately determined the obstacle avoidance behavior adopted by the robot. The resulting controller preserved obstacle avoidance, ensuring safety at every iteration of the learning procedure. This paper addresses the question of how to adjust the conductance properties associated with the problem domain 0, such as to extremize an user-specified performance index. Initially, conductance properties are homogeneous across 0, and the resulting controller is optimal in the sense that it minimizes collision probabilities at every step [Connolly, 1994]1. The method proposed is a policy iteration algorithm, in which the policy space is parameterized by the set of node conductances. 2 PROBLEM CHARACTERIZATION The problem consists in constructing a path controller ifo that maximizes an integral performance index 'P defined over the set of all possible paths on a lattice for a closed domain 0 C Rn, subjected to boundary constraints. The controller ifo is responsible for generating the sequence of configurations from an initial configuration qo on the lattice to the goal configuration qG, therefore determining the performance index 'P. In formal terms, the performance index 'P can be defined as follows: Def. 1 Performance indez 'P : qa 'P for all q E L(O), where 'Pqo.* = L f(q)· q=qo L(O) is a lattice over the domain 0, qo denotes an arbitrary configuration on L(O), qG is the goal configuration, and f(q) is a function of the configuration q. For example, one can define f(q) to be the available joint range associated with the configuration q of a manipulator; in this case, 'P would be measuring the available joint range associated with all paths generated within a given domain. 2.1 DERIVATION OF REFERENCE CONTROLLER The derivation of ifo is very laborious, requiring the exploration of the set of all possible paths. Out of this set, one is primarily interested in the subset of smooth paths. We propose to solve a simpler problem, in which the derived controller if is a numerical approximation to the optimal controller ifo, and (1) generates smooth paths, (2) is admissible, and (3) locally maximizes P. To guarantee (1) and (2), it is assumed that the control actions of if are proportional to the gradient of a harmonic function f/J, represented as the voltage distribution across a resistive lattice that tessellates the domain O. The condition (3) is achieved through incremental changes in the set G of internodal conductancesj such changes maximize P locally. Necessary condition for optimality: Note that 'Pqo.* defines a scalar field over L(O). It is assumed that there exists a well-defined neighborhood .N(q) for node qj in fact, it is assumed that every node q has two neighbors across each dimension. Therefore, it is possible to compute the gradient over the scalar field Pqo.ff by locally approximating its rate of change across all dimensions. The gradient VP qo defines lThis is exactly the control policy derived by the TD(O) reinforcement learning method, for the particular case of an agent travelling in a grid world with absorbing obstacle and goal states, and being rewarded only for getting to the goal states (see [Connolly, 1994]). 998 J. A. COELHO Jr., R. SITARAMAN, R. A. GRUPEN a reference controller; in the optimal situation, the actions of the controller if will parallel the actions of the reference controller. One can now formulate a policy iteration algorithm for the synthesis of the reference controller: 1. Compute if = -V~, given conductances G; 2. Evaluate VP q: - for each cell, compute 'P ;;.. 1'''' - for each cell, compute V'P q. 3. Change G incrementally, minimizing the approz. error € = f(if, VPq); 4. If € is below a threshold €o, stop. Otherwise, return to (1). On convergence, the policy iteration algorithm will have derived a control policy that maximizes l' globally, and is capable of generating smooth paths to the goal configuration. The key step on the algorithm is step (3), or how to reduce the current approximation error by changing the conductances G. 3 APPROXIMATION ALGORITHM Given a set of internodal conductances, the approximation error € is defined as € - L cos (if, VP) (1) qEL(n) or the sum over L(n) of the cosine of the angle between vectors if and VP. The approximation error € is therefore a function ofthe set G of internodal conductances. There exist 0(nd") conductances in a n-dimensional grid, where d is the discretization adopted for each dimension. Discrete search methods for the set of conductance values that minimizes € are ruled out by the cardinality of the search space: 0(knd"), if k is the number of distinct values each conductance can assume. We will represent conductances as real values and use gradient descent to minimize €, according to the approximation algorithm below: 1. Evaluate the apprommation error €j 2. Compute the gradient V€ = g;; j 3. Update conductances, making G = G - aVE; 4. Normalize conductances, such that minimum conductance gmin = 1; Step (4) guarantees that every conductance g E G will be strictly positive. The conductances in a resistive grid can be normalized without constraining the voltage distribution across it, due to the linear nature of the underlying circuit. The complexity of the approximation algorithm is dominated by the computation of the gradient V€(G). Each component of the vector V€(G) can be expressed as 8€ = _ '" 8cos(ifq, VPq). (2) 8g· L...J 8g· 'qEL(n) , By assumption, if is itself the gradient of a harmonic function ¢> that describes the voltage distribution across a resistive lattice. Therefore, the calculation of :;. involves the evaluation of ~ over all domain L(n), or how the voltage ¢>q is affected by changes in a certain conductance gi. For n-dimensional grids, *£ is a matrix with d" rows and 0( nd") columns. We posit that the computation of every element of it is unnecessary: the effects of changing Parallel Optimization of Motion Controllers via Policy Iteration 999 g, will be more pronounced in a certain grid neighborhood of it, and essentially negligible for nodes beyond that neighborhood. Furthermore, this simplification allows for breaking up the original problem into smaller, independent sub-problems suitable to simultaneous solution in parallel architectures. 3.1 THE LOCALITY ASSUMPTION The first simplifying assumption considered in this work establishes bounds on the neighborhood affected by changes on conductances at node ij specifically, we will assume that changes in elements of g, affect only the voltage at nodes in J/(i) , being J/(i) the set composed of node i and its direct neighbors. See [Coelho Jr. et al., 1995] for a discussion on the validity of this assumption. In particular, it is demonstrated that the effects of changing one conductance decay exponentially with grid distance, for infinite 2D grids. Local changes in resistive grids with higher dimensionality will be confined to even smaller neighborhoods. The locality assumption simplifies the calculation of :;. to But ~ [ if . V!, 1 = 1.... [8if. VP _ if· VP 8if. if l. 8g, lifllV'P1 lifllV'P1 8g, lifl2 (8g, ) Note that in the derivation above it is assumed that changes in G affects primarily the control policy if, leaving VP relatively unaffected, at least in a first order approximation. Given that if = - V~, it follows that the component 7r; at node q can be approximated by the change of potential across the dimension j, as measured by the potential on the corresponding neighboring nodes: 7r"1 = ¢q- - ¢q+, and 87r; = _1_ [8¢q _ _ 8¢q+] , q 2b. 2 8g, 2b. 2 8gi 8gi' where b. is the internodal distance on the lattice L(n). 3.2 DERIVATION OF G; The derivation of ~ involves computing the Thevenin equiValent circuit for the resistive lattice, when every conductance 9 connected to node i is removed. For clarity, a 2D resistive grid was chosen to illustrate the procedure. Figure 1 depicts the equivalence warranted by Thevenin's theorem [Chua et al., 1987] and the relevant variables for the derivation of ~. As shown, the equivalent circuit for the resistive grid consists of a four-port resistor, driven by four independent voltage sources. The relation between the voltage vector i = [¢t ¢4Y and the current vector r = [it ... i 4]T is expressed as Rf+w, (3) where R is the impedance matrix for the grid equivalent circuit and w is the vector of open-circuit voltage sources. The grid equivalent circuit behaves exactly like the whole resistive gridj there is no approximation error. 1000 J. A. COELHO Jr., R. SITARAMAN, R. A. GRUPEN ... + .............. . ! cJl2 i cJl 3 1 i 3 i cJl4 ... + .............. . Grid Equivalent Circuit cJl o Figure 1: Equivalence established by Thevenin's theorem. The derivation of the 20 parameters (the elements of Rand w) of the equivalent circuit is detailed in [Coelho Jr. et al., 1995]j it involves a series ofrelaxation operations that can be efficiently implemented in SIMD architectures. The total number of relaxations for a grid with n l nodes is exactly 6n - 12, or an average of 1/2n relaxations per link. In the context of this paper, it is assumed that Rand w are known. Our primary interest is to compute how changes in conductances g1c affect the voltage vector i, or the matrix 84> = I 84>j I, for {Jk· 8g 8g1c 1, . .. ,4 1, ... ,4. The elements of ~ can be computed by derivating each of the four equality relations in Equation 3 with respect to g1c, resulting in a system of 16 linear equations, and 16 variables - the elements of ~. Notice that each element of i can be expressed as a linear function of the potentials i, by applying Kirchhoff's laws [Chua et al., 1987]: 4 APPLICATION EXAMPLE A robot moves repeatedly toward a goal configuration. Its initial configuration is not known in advance, and every configuration is equally likely of being the initial configuration. The problem is to construct a motion controller that minimizes the overall travel distance for the whole configuration space. If the configuration space o is discretized into a number of cells, define the combined travel distance D(?T) as D(?T) L dq,if, (4) qEL(O) where dq,if is the travel distance from cell q to the goal configuration qG, and robot displacements are determined by the controller?T. Figure 2 depicts an instance of the travel distance minimization problem, and the paths corresponding to its optimal solution, given the obstacle distribution and the goal configuration shown. A resistive grid with 17 x 17 nodes was chosen to represent the control policies generated by our algorithm. Initially, the resistive grid is homogeneous, with all internodal resistances set to 10. Figure 3 indicates the paths the robot takes when commanded by ifO, the initial control policy derived from an homogeneous resistive grid. Parallel Optimization of Motion Controllers via Policy Iteration 1001 16r----,....---,....----,r-----, 12 12 16 Figure 2: Paths for optimal solution of the travel distance minimization problem. 16,....----,r-----,-----,----, 12 Figure 3: Paths for the initial solution of the same problem. The conductances in the resistive grid were then adjusted over 400 steps of the policy iteration algorithm, and Figure 4 is a plot of the overall travel distance as a function of the number of steps. It also shows the optimal travel distance (horizontal line), corresponding to the optimal solution depicted in Figure 2. The plot shows that convergence is initially fast; in fact, the first 140 iterations are responsible for 90% of the overall improvement. After 400 iterations, the travel distance is within 2.8% of its optimal value. This residual error may be explained by the approximation incurred in using a discrete resistive grid to represent the potential distribution. Figure 5 shows the paths taken by the robot after convergence. The final paths are straightened versions of the paths in Figure 3. Notice also that some of the final paths originating on the left of the I-shaped obstacle take the robot south of the obstacle, resembling the optimal paths depicted in Figure 2. 5 CONCLUSION This paper presented a policy iteration algorithm for the synthesis of provably correct navigation functions that also extremize user-specified performance indices. The algorithm proposed solves the optimal feedback control problem, in which the final control policy optimizes the performance index over the whole domain, assuming that every state in the domain is as likely of being the initial state as any other state. The algorithm modifies an existing harmonic function-based path controller by incrementally changing the conductances in a resistive grid. Departing from an homogeneous grid, the algorithm transforms an optimal controller (i.e. a controller that minimizes collision probabilities) into another optimal controller, that extremizes locally the performance index of interest. The tradeoff may require reducing the safety margin between the robot and obstacles, but collision avoidance is preserved at each step of the algorithm. Other Applications: The algorithm presented can be used (1) in the synthesis of time-optimal velocity controllers, and (2) in the optimization of non-holonomic path controllers. The algorithm can also be a component technology for Intelligent Vehicle Highway Systems (IVHS), by combining (1) and (2). 1002 1170....----,----,----r----, 17.~ --------------~ 1710 1680 16500L---IOO~-~200'":--~300~-~400 Figure 4: Overall travel distance, as a function of iteration steps. J. A. COELHO Jr .• R. SITARAMAN. R. A. GRUPEN 16r-----.r-----.....---.,---., 12 12 16 Figure 5: Final paths, after 800 policy iteration steps. Performance on Parallel Architectures: The proposed algorithm is computationally demandingj however, it is suitable for implementation on parallel architectures. Its sequential implementation on a SPARC 10 workstation requires ~ 30 sec. per iteration, for the example presented. We estimate that a parallel implementation of the proposed example would require ~ 4.3 ms per iteration, or 1. 7 seconds for 400 iterations, given conservative speedups available on parallel architectures [Coelho Jr. et al., 1995]. Acknowledgements This work was supported in part by grants NSF CCR-9410077, IRI-9116297, IRI9208920, and CNPq 202107/90.6. References [Chua et aI., 1987] Chua, L., Desoer, C., and Kuh, E. (1987). Linear and Nonlinear Circuits. McGraw-Hill, Inc., New York, NY. [Coelho Jr. et al., 1995] Coelho Jr., J., Sitaraman, R., and Grupen, R. (1995). Control-oriented tuning of harmonic functions. Technical Report CMPSCI Technical Report 95-112, Dept. Computer Science, University of Massachusetts. [Connolly, 1994] Connolly, C. I. (1994). Harmonic functions and collision probabilities. In Proc. 1994 IEEE Int. Conf. Robotics Automat., pages 3015-3019. IEEE. [Connolly and Grupen, 1993] Connolly, C. I. and Grupen, R. (1993). The applications of harmonic functions to robotics. Journal of Robotic Systems, 10(7):931946. [Rimon and Koditschek, 1990] Rimon, E. and Koditschek, D. (1990). Exact robot navigation in geometrically complicated but topologically simple spaces. In Proc. 1990 IEEE Int. Conf. Robotics Automat., volume 3, pages 1937-1942, Cincinnati, OH. [Singh et aI., 1994] Singh, S., Barto, A., Grupen, R., and Connolly, C. (1994). Robust reinforcement learning in motion planning. In Advances in Neural Information Processing Systems 6, pages 655-662, San Francisco, CA. Morgan Kaufmann Publishers.	1995	9
1,139	Reinforcement Learning by Probability Matching Philip N. Sabes sabes~psyche.mit.edu Michael I. Jordan jordan~psyche.mit.edu Department of Brain and Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 Abstract We present a new algorithm for associative reinforcement learning. The algorithm is based upon the idea of matching a network's output probability with a probability distribution derived from the environment's reward signal. This Probability Matching algorithm is shown to perform faster and be less susceptible to local minima than previously existing algorithms. We use Probability Matching to train mixture of experts networks, an architecture for which other reinforcement learning rules fail to converge reliably on even simple problems. This architecture is particularly well suited for our algorithm as it can compute arbitrarily complex functions yet calculation of the output probability is simple. 1 INTRODUCTION The problem of learning associative networks from scalar reinforcement signals is notoriously difficult. Although general purpose algorithms such as REINFORCE (Williams, 1992) and Generalized Learning Automata (Phansalkar, 1991) exist, they are generally slow and have trouble with local minima. As an example, when we attempted to apply these algorithms to mixture of experts networks (Jacobs et al. , 1991), the algorithms typically converged to the local minimum which places the entire burden of the task on one expert. Here we present a new reinforcement learning algorithm which has faster and more reliable convergence properties than previous algorithms. The next section describes the algorithm and draws comparisons between it and existing algorithms. The following section details its application to Gaussian units and mixtures of Gaussian experts. Finally, we present empirical results. Reinforcement Learning by Probability Matching 1081 2 REINFORCEMENT PROBABILITY MATCHING We begin by formalizing the learning problem. Given an input x E X from the environment, the network must select an output y E y. The network then receives a scalar reward signal r, with a mean r and distribution that depend on x and y. The goal of the learner is to choose an output which maximizes the expected reward. Due to the lack of an explicit error signal, the learner must choose its output stochastically, exploring for better rewards. Typically the learner starts with a parameterized form for the conditional output density P8(ylx), and the learning problem becomes one of finding the parameters 0 which maximize the expected reward: Jr(O) = 1 p(x)p8(ylx)r(x, y)dydx. X,Y We present an alternative route to the maximum expected reward cost function, and in doing so derive a novel learning rule for updating the network's parameters. The learner's task is to choose from a set of conditional output distributions based on the reward it receives from the environment. These rewards can be thought of as inverse energies; input/output pairs that receive high rewards are low energy and are preferred by the environment. Energies can always be converted into probabilities through the Boltzmann distribution, and so we can define the environment's conditional distribution on Y given x, ( I) exp( _T- 1 E(x, y» exp(T-1r(x, y» pyx = ZT(X) = ZT(X) , where T is a temperature parameter and ZT(X) is a normalization constant which depends on T . This distribution can be thought of as representing the environment's ideal input-output mapping, high reward input-output pairs being more typical or likely than low reward pairs. The temperature parameter determines the strength of this preference: when T is infinity all outputs are equally likely; when T is zero only the highest reward output is chosen. This new distribution is a purely theoretical construct, but it can be used as a target distribution for the learner. If the 0 are adjusted so that P8(ylx) is nearly equal to p(ylx), then the network's output will typically result in high rewards. The agreement between the network and environment conditional output densities can be measured with the Kullback-Liebler (KL) divergence: K L(p II P) = -1 p(x)p8(ylx) [logp(Ylx) -logp8(ylx)] dydx (1) X,Y = -~ 1 p(x)p8(ylx)[r(x,y) - Tr8(X,y)]dydx+ f p(x)logZT(x)dx, X,Y Jx where r8(x, y) is defined as the logarithm of the conditional output probability and can be thought of as the network's estimate of the mean reward. This cost function is always greater than or equal to zero, with equality only when the two probability distributions are identical. Keeping only the part of Equation 1 which depends on 0, we define the Probability Matching (PM) cost function: JpM(O) = - f p(x)p8(ylx)[r(x, y) - Tr8(X, y)] dydx = -Jr(O) - TS(P8) Jx,y The PM cost function is analogous to a free energy, balancing the energy, in the form of the negative of the average reward, and the entropy S(P8) of the output 1082 P. N. SABES, M. I. JORDAN -1 -1 -0.5 0 0.5 T=l T=.5 T= .2 T= .05 Figure 1: p's (dashed) and PM optimal Gaussians (solid) for the same bimodal reward function and various temperatures. Note the differences in scale. distribution. A higher T corresponds to a smoother target distribution and tilts the balance of the cost function in favor of the entropy term, making diffuse output distributions more favorable. Likewise, a small T results in a sharp target distribution placing most of the weight on the reward dependent term of cost function, which is always optimized by the singular solution of a spike at the highest reward output. Although minimizing the PM cost function will result in sampling most often at high reward outputs, it will not optimize the overall expected reward if T > O. There are two reasons for this. First, the output y which maximizes ro(x, y) may not maximize rex, y). Such an example is seen in the first panel of Figure 1: the network's conditional output density is a Gaussian with adjustable mean and variance, and the environment has a bimodal reward function and T = 1. Even in the realizable case, however, the network will choose outputs which are suboptimal with respect to its own predicted reward, with the probability of choosing output y falling off exponentially with ro(x, y). The key point here is that early in learning this non-optimality is exactly what is desired. The PM cost function forces the learner to maintain output density everywhere the reward, as measure by pl/T, is not much smaller than its maximum. When T is high, the rewards are effectively flattened and even fairly small rewards look big. This means that a high temperature ensures that the learner will explore the output space. Once the network is nearly PM optimal, it would be advantageous to "sharpen up" the conditional output distribution, sampling more often at outputs with higher predicted rewards. This translates to decreasing the entropy of the output distribution or lowering T. Figure 1 shows how the PM optimal Gaussian changes as the temperature is lowered in the example discussed above; at very low temperatures the output is almost always near the mode of the target distribution. In the limit of T = 0, J PM becomes original reward maximization criterion Jr. The idea of the Probability Matching algorithm is to begin training with a large T, say unity, and gradually decrease it as the performance improves, effectively shifting the bias of the learner from exploration to exploitation. We now must find an update rule for 0 which minimizes JpM(O). We proceed by looking for a stochastic gradient descent step. Differentiating the cost function gives \T OJpM(O) = -1 p(x)po(Ylx) [rex, y) - Tro(x, y)] \T oro(x, y)dydx. X,Y Thus, if after every action the parameters are updated by the step t:.o = a [r - Tro(x, y)] \T oro (x, y), (2) where alpha is a constant which can vary over time, then the parameters will on average move down the gradient of the PM cost function. Note that any quantity Reinforcement Learning by Probability Matching 1083 which does not depend on Y or r can be added to the difference in the update rule, and the expected step will still point along the direction of the gradient. The form of Equation 2 is similar to the REINFORCE algorithm (Williams, 1992), whose update rule is t:.() = a(r - b)V' elogpe(Ylx), where b, the reinforcement baseline, is a quantity which does not depend on Y or r. Note that these two update rules are identical when T is zero.! The advantage of the PM rule is that it allows for an early training phase which encourages exploration without forcing the output distribution to converge on suboptimal outputs. This will lead to striking qualitative differences in the performance of the algorithm for training mixtures of Gaussian experts. 3 UPDATE RULES FOR GAUSSIAN UNITS AND MIXTURES OF GAUSSIAN EXPERTS We employ Gaussian units with mean I' = w T x and covariance 0"21. The learner must select the matrix wand scalar 0" which minimize JpM(W, 0"). Applying the update rule in Equation 2, we get t:.w " 1 a[r - Tr(x,y)] 2"(Y -I'?x 0" " 1 ("Y -I'W ) a [r - Tr(x, y)] 0"2 0"2 - 1 . In practice, for both single Gaussian units and the mixtures presented below we avoid the issue of constraining 0" > 0 by updating log 0" directly. We can generalize the linear model by considering a conditional output distribution in the form of a mixture of Gaussian experts (Jacobs et al., 1991), N 1 1 p(Ylx) = Lgi(x)(27r0"1)-~ exp(--2I1y -l'iW)· i=! 20"i Expert i has mean I'i = w r x and covariance 0"[1. The prior probability given x of choosing expert i, gi(X), is determined by a single layer gating network with weight matrix v and softmax output units. The gating network learns a soft partitioning of the input space into regions for which each expert is responsible. Again, we can apply Equation 2 to get the PM update rules: t:.Vi a [r - Tf(x,y)] (hi - gi)X t:.Wi a [r - Tf(x, y)] hi~(Y - l'i?X O"i 6.O"i a[r-Tf(x,Y)]hi : 1 ("Y~riW -1), where hi = giPi(ylx)jp(ylx) is the posterior probability of choosing expert i given y. We note that the PM update rules are equivalent to the supervised learning gradient descent update rules in (Jacobs et al., 1991) modulated by the difference between the actual and expected rewards. lThis fact implies that the REINFORCE step is in the direction of the gradient of JR(B), as shown by (Williams, 1992). See Williams and Peng, 1991, for a similar REINFORCE plus entropy update rule. 1084 P. N. SABES, M. I. JORDAN Table 1: Convergence times and gate entropies for the linear example (standard errors in parentheses). Convergence times: An experiment consisting of 50 runs was conducted for each algorithm, with a wide range of learning rates and both reward functions. Best results for each algorithm are reported. Entropy: Values are averages over the last 5,000 time steps of each run. 20 runs of 50,000 time steps were conducted. Algorithm I Convergence Time I Entropy PM, T= 1 1088 (43) .993 .0011 PM, T=.5 .97 .02 PM, T =.1 .48 .04 REINFORCE 2998 (183) .21 .03 REINF-COMP 1622 (46) .21 .03 Both the hi and r depend on the overall conditional probability p(ylx), which in turn depends on each Pi(ylx). This adds an extra step to the training procedure. After receiving the input x, the network chooses an expert based on the priors gi(X) and an output y from the selected expert's output distribution. The output is then . sent back to each of the experts in order to compute the likelihood of their having generated it. Given the set of Pi'S, the network can update its parameters as above. 4 SIMULATIONS We present three examples designed to explore the behavior of the Probability Matching algorithm. In each case, networks were trained using Probability Matching, REINFORCE, and REINFORCE with reinforcement comparison (REINFCOMP), where a running average of the reward is used as a reinforcement baseline (Sutton, 1984). In the first two examples an optimal output function y(x) was chosen and used to calculate a noisy error, c = Ily - y(x) - zll, where z was i.i.d. zero-mean Gaussian with u = .1. The error signal determined the reward by one of two functions, r = -c2/2 or exp( _c2 /2). When the RMSE between the network mean and the optimal output was less that .05 the network was said to have converged. 4.1 A Linear Example In this example x was chosen uniformly from [-1,1]2 x {I}, and the optimal output was y* = Ax, for a 2 x 3 matrix A. A mixture of three Gaussian experts was trained. The details of the simulation and results for each algorithm are shown in Table 1. Probability Matching with constant T = 1 shows almost a threefold reduction in training time compared to REINFORCE and about a 50% improvement over REINF-COMP. The important point of this example is the manner in which the extra Gaussian units were employed. We calculated the entropy of the gating network, normalized so that a value of one means that each expert has equal probability of being chosen and a value of zero means that only one expert is ever chosen. The values after 50,000 time steps are shown in the second column of Table 1. When T ~ 1, the Probability Matching algorithm gives the three experts roughly equal priors. This is due to the fact that small differences in the experts' parameters lead to increased output entropy if all experts are used. REINFORCE on the other hand always converges to a solution which employs only one expert. This difference in the behavior of the algorithms will have a large qualitative effect in the next example. Reinforcement Learning by Probability Matching J085 Table 2: Results for absolute value. The percentage of trials that converged and the average time to convergence for those trials. Standard errors are in parentheses. 50 trials were conducted for a range of learning rates and with both reward functions; the best results for each algorithm are shown. Algorithm I Successful Trials I Convergence Time I PM 100% 6,052 313) REINFORCE 48% 76,775 3,329) REINF-COMP 38% 42,105 3,869) 8.0 110 100 10 60 2.0 .0 0 .0 10 -2.0 0.0 1.0 1.0 '.0 ' .0 ' .0 0.0 1.0 2.0 '.0 '.0 (a) (b) (c) (d) Figure 2: Example 4.3. The environment's probability distribution for T = 1: (a) density plot of p. vs. y / x, (b) cross-sectional view with Y2 = o. Locally weighted mean and variance of Y2/X over representative runs: (c) T = 1, (d) T = 0 (i.e. REINFORCE). 4.2 Absolute Value We used a mixture of two Gaussian units to learn the absolute value function. The details of the simulation and the best results for each algorithm are shown in Table 2. Probability Matching with constant T = 1 converged to criterion on every trial, in marked contrast to the REINFORCE algorithm. With no reinforcement baseline, REINFORCE converged to criterion in only about half of the cases, less with reinforcement comparison. In almost all of the trials that didn't converge, only one expert was active on the domain of the input. Neither version of REINFORCE ever converged to criterion in less than 14,000 time steps. This example highlights the advantage of the Probability Matching algorithm. During training, all three algorithms initially use both experts to capture the overall mean of the data. REINFORCE converges on this local minimum, cutting one expert off before it has a chance to explore the rest of the parameter space. The Probability Matching algorithm keeps both experts in use. Here, the more conservative approach leads to a stark improvement in performance. 4.3 An Example with Many Local Maxima In this example, the learner's conditional output distribution was a bivariate Gaussian with It = [Wl, W2]T x, and the environment's rewards were a function of y/x. The optimal output distribution p(y/x) is shown in Figures 2(a,b). These figures can also be interpreted as the expected value of p for a given w. The weight vector is initially chosen from a uniform distribution over [-.2, .2]2, depicted as the very small while dot in Figure 2(a). There are a series of larger and larger local maxima off to the right, with a peak of height 2n at Wl = 2n. The results are shown in Table 3. REINFORCE, both with and without reinforcement comparison, never got past third peak; the variance of the Gaussian unit would '.0 1086 P. N. SABES, M. I. JORDAN Table 3: Results for Example 4.3. These values represent 20 runs for 50,000 time steps each. The first and second columns correspond to number of the peak the learner reached. Mean Final Range of Final Mean Final Algorithm log2 Wl log2 Wl'S (T PM, T= 2 28,8 [19.1,51.0] > 101> PM, T = 1 6.34 5.09,8.08 13.1 PM, T=.5 3.06 3.04,3.07 .40 REINFORCE 2.17 2.00,2.90 .019 REINF-COMP 2.05 2.05,2.06 .18 very quickly close down to a small value making further exploration of the output space impossible. Probability Matching, on the other hand, was able to find greater and greater maxima, with the variance growing adaptively to match the local scale of the reward function. These differences can be clearly seen in Figures 2( c,d), which show typical behavior for the Probability Matching algorithm with T = 1 and T = O. 5 CONCLUSION We have presented a new reinforcement learning algorithm for associative networks which converges faster and more reliably than existing algorithms. The strength of the Probability Matching algorithm is that it allows for a better balance between exploration of the output space and and exploitation of good outputs. The parameter T can be adjusted during learning to allow broader output distributions early in training and then to force the network to sharpen up its distribution once nearly optimal parameters have been found. Although the applications in this paper were restricted to networks with Gaussian units, the Probability Matching algorithm can be applied to any reinforcement learning task and any conditional output distribution. It could easily be employed, for example, on classification problems using logistic or multinomial (softmax) output units or mixtures of such units. Finally, the simulations presented in this paper are of simple examples. Preliminary results indicate that the advantages of the Probability Matching algorithm scale up to larger, more interesting problems. References Jacobs, R. A., Jordan, M. I., Nowlan, S. J., and Hinton, G. E. (1991). Adaptive mixtures of local experts. Neural Computation, 3:79-87. Phansalkar, V. V. (1991). Learning automata algorithms for connectionist systems - local and global convergence. PhD Thesis, Dept. of Electrical Engineering, India Institute of Science, Bangalore. Sutton, R. S. (1984). Temporal credit assignment in reinforcement learning. PhD Thesis, Dept. of Computer and Information Science, University of Massachusetts, Amherst, MA. Williams, R. J. (1992). Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8:229-256. Williams, R. J. and Peng, J. (1991). Function optimization using connectionist reinforcement learning algorithms. Connection Science, 3:241-268.	1995	90
1,140	Generalized Learning Vector Quantization Atsushi Sato & Keiji Yamada Information Technology Research Laboratories, NEC Corporation 1-1, Miyazaki 4-chome, Miyamae-ku, Kawasaki, Kanagawa 216, Japan E-mail: {asato.yamada}@pat.cl.nec.co.jp Abstract We propose a new learning method, "Generalized Learning Vector Quantization (GLVQ)," in which reference vectors are updated based on the steepest descent method in order to minimize the cost function. The cost function is determined so that the obtained learning rule satisfies the convergence condition. We prove that Kohonen's rule as used in LVQ does not satisfy the convergence condition and thus degrades recognition ability. Experimental results for printed Chinese character recognition reveal that GLVQ is superior to LVQ in recognition ability. 1 INTRODUCTION Artificial neural network models have been applied to character recognition with good results for small-set characters such as alphanumerics (Le Cun et aI., 1989) (Yamada et al., 1989). However, applying the models to large-set characters such as Japanese or Chinese characters is difficult because most of the models are based on Multi-Layer Perceptron (MLP) with the back propagation algorithm, which has a problem in regard to local minima as well as requiring a lot of calculation. Classification methods based on pattern matching have commonly been used for large-set character recognition. Learning Vector Quantization (LVQ) has been studied to generate optimal reference vectors because of its simple and fast learning algorithm (Kohonen, 1989; 1995). However, one problem with LVQ is that reference vectors diverge and thus degrade recognition ability. Much work has been done on improving LVQ (Lee & Song, 1993) (Miyahara & Yoda, 1993) (Sato & Tsukumo, 1994), but the problem remains unsolved. Recently, a generalization of the Simple Competitive Learning (SCL) has been under 424 A. SATO, K. YAMADA study (Pal et al., 1993) (Gonzalez et al., 1995), and one unsupervised learning rule has been derived based on the steepest descent method to minimize the cost function. Pal et al. call their model "Generalized Learning Vector Quantization," but it is not a generalization of Kohonen's LVQ. In this paper, we propose a new learning method for supervised learning, in which reference vectors are updated based on the steepest descent method, to minimize the cost function. This is a generalization of Kohonen's LVQ, so we call it "Generalized Learning Vector Quantization (GLVQ)." The cost function is determined so that the obtained learning rule satisfies the convergence condition. We prove that Kohonen's rule as used in LVQ does not satisfy the convergence condition and thus degrades recognition ability. Preliminary experiments revealed that non-linearity in the cost function is very effective for improving recognition ability. Printed Chinese character recognition experiments were carried out, and we can show that the recognition ability of GLVQ is very high compared with LVQ. 2 REVIEW OF LVQ Assume that a number of reference vectors Wk are placed in the input space. Usually, several reference vectors are assigned to each class. An input vector x is decided to belong to the same class to which the nearest reference vector belongs. Let Wk(t) represent sequences of the Wk in the discrete-time domain. Heretofore, several LVQ algorithms have been proposed (Kohonen, 1995), but in this section, we will focus on LVQ2.1. Starting with properly defined initial values, the reference vectors are updated as follows by the LVQ2.1 algorithm: Wi(t + 1) = Wi(t) - a(t)(x - Wi(t)), (1) Wj(t + 1) = Wj(t) + a(t)(x - Wj(t)), (2) where 0 < aCt) < 1, and aCt) may decrease monotonically with time. The two reference vectors Wi and Wj are the nearest to x; x and Wj belong to the same class, while x and Wi belong to different classes. Furthermore, x must fall into the "window," which is defined around the midplane of Wi and Wj. That is, if the following condition is satisfied, Wi and Wj are updated: min (~> ~~) > s, (3) where di = Ix - wd, dj = Ix - wjl. The LVQ2.1 algorithm is based on the idea of shifting the decision boundaries toward the Bayes limits with attractive and repulsive forces from x. However, no attention is given to what might happen to the location of the Wk, so the reference vectors diverge in the long run. LVQ3 has been proposed to ensure that the reference vectors continue approximating the class distributions, but it must be noted that if only one reference vector is assigned to each class, LVQ3 is the same as LVQ2.1, and the problem of reference vector divergence remains unsolved. 3 GENERALIZED LVQ To ensure that the reference vectors continue approximating the class distributions, we propose a new learning method based on minimizing the cost function. Let Wl be the nearest reference vector that belongs to the same class of x, and likewise let W2 be the nearest reference vector that belongs to a different class from x. Let us consider the relative distance difference p,( x) defined as follows: dl - d2 P,(x)=d1 +d2 ' (4) Generalized Learning Vector Quantization 425 where dl and d2 are the distances of:B from WI and W2, respectively. ft(x) ranges between -1 and + 1, and if ft( x) is negative, x is classified correctly; otherwise, x is classified incorrectly. In order to improve error rates, 1£( x) should decrease for all input vectors. Thus, a criterion for learning is formulated as the minimizing of a cost function S defined by (5) i=l where N is the number of input vectors for training, and f(ft) is a monotonically increasing function. To minimize S, WI and W2 are updated based on the steepest descent method with a small positive constant a as follows: as Wi Wj - a--, i = 1,2 (6) aWj If squared Euclid distance, dj = Ix - wd 2 , is used, we can obtain the following. as = as aft adl = _ of 4d2 (x _ WI) (7) aWl aft adl aWl aft (dl + d2)2 as = as aft ad2 = + of 4dl (x _ W2) (8) aW2 aft ad2 aW2 01£ (dl + d2)2 Therefore, the GLVQ's learning rule can be described as follows: of d2 WI WI + a aft (dl + d2)2 (x - wt) (9) of dl W2 W2 - a aft (dl + d2)2 (x - W2) (10) Let us discuss the meaning of f(ft). of/aft is a kind of gain factor for updating, and its value depends on x. In other words, of/aft is a weight for each x. To decrease the error rate, it is effective to update reference vectors mainly by input vectors around class boundaries, so that the decision boundaries are shifted toward the Bayes limits. Accordingly, f(ft) should be a non-linear monotonically increasing function, and it is considered that classification ability depends on the definition of f(ft). In this paper, of/aft = f(ft,t){l- f(ft,t)} was used in the experiments, where t is learning time and f(ft, t) is a sigmoid function of 1/(1 + e-lJt). In this case, of / aft has a single peak at ft = 0, and the peak width becomes narrower as t increases, so the input vectors that affect learning are gradually restricted to those around the decision boundaries. Let us discuss the meaning of ft. WI and W2 are updated by attractive and repulsive forces from x, respectively, as shown in Eqs. (9) and (10), and the quantities of updating, ILlwd and ILlw21, depend on derivatives of ft. Reference vectors will converge to the equilibrium states defined by attractive and repulsive forces, so it is considered that convergence property depends on the definition of ft. 4 DISCUSSION First, we show that the conventional LVQ algorithms can be derived based on the framework of GLVQ. If ft = dl for dl < d2, ft = -d2 for dl > d2, and f(ft) = ft, the cost f~nction is written as S = ~dl <d2 dl ~dl >d2 d2 . Then, we can obtain the followmg: WI WI + a(x - WI), W2 W2 W2 W2 - a(x - W2), WI WI for dl < d2 for dl > d2 (11) (12) 426 A. SATO, K. YAMADA This learning algorithm is the same as LVQ1. If It = dI -d2 and f(lt) = It for Iltl < s, f(lt) = const for Iltl > s, the cost function is written as S = 2: IJJ1<s(di - d2 ) + C. Then, we can obtain the following: if Iltl < s (x falls into the window) WI WI + a(x - W2) (13) W2 W2 - a(x - W2) (14) In this case, WI and W2 are updated simultaneously, and this learning algorithm is the same as LVQ2.1. SO it can be said that GLVQ is a generalized model that includes the conventional LVQs. Next, we discuss the convergence condition. We can obtain other learning algorithms by defining a different cost function, but it must be noted that the convergence property depends on the definition of the cost function. The main difference between GLVQ and LVQ2.1 is the definition of It; It = (dI -d2)/(di +d2) in GLVQ, It = dl - d2 in LVQ2.1. Why do the reference vectors diverge in LVQ2.1, while they converge in GLVQ, as shown later? In order to clarify the convergence condition, let us consider the following learning rule: WI WI + alx - w2lk(x - wt} (15) W2 W2 - alx - wIlk(x - W2) (16) Here, I~Wll and I~W21 are the quantities of updating by the attractive and the repulsive forces, respectively. The ratio of these two is calculated as follows: I~WII alx - w21klx - wII Ix - w2lk- I I~W21 = alx - wIlklx - w21 = Ix - wll k - I (17) If the initial values of reference vectors are properly defined, most x's will satisfy Ix - wd < Ix - w21. Therefore, if k > 1, the attractive force is greater than the repulsive force, and the reference vectors will converge, because the attractive forces come from x's that belong to the same class of WI. In GLVQ, k = 2 as shown in Eqs. (9) and (10), and the vectors will converge, while they will diverge in LVQ2.1 because k = 0. According to the above discussion, we can use di/(d1 + d2) or just dj, instead of di/(d1 + d2)2 in Eqs. (9) and (10). This correction does not affect the convergence condition. The essential problem in LVQ2.1 results from the drawback in Kohonen's rule with k = 0. In other words, the cost function used in LVQ is not determined so that the obtained learning rule satisfies the convergence condition. 5 EXPERIMENTS 5.1 PRELIMINARY EXPERIMENTS The experimental results using Eqs. (15) and (16) with a = 0.001, shown in Fig. 1, support the above discussion on the convergence condition. Two-dimensional input vectors with two classes shown in Fig. 1( a) were used in the experiments. The ideal decision boundary that minimizes the error rate is shown by the broken line. One reference vector was assigned to each class with initial values (x, y) = (0.3,0.5) for Class A and (x,y) = (0.7,0.5) for Class B. Figure l(b) shows the distance between the two reference vectors during learning. The distance remains the same value for k > 1, while it increases with time for k ~ 1; that is, the reference vectors diverge. Figure 2 shows the experimental results from GLVQ for linearly non-separable patterns compared with LVQ2.1. The input vectors shown in Fig. 2(a) were obtained by shifting all input vectors shown in Fig. l(a) to the right by Iy - 0.51. The ideal Generalized Learning Vector Quantization 427 1.0 6.0 Class A 0 ! k = 0.0 -+-0 Class B x 5.0 t k=0.5 -f--0.8 k = 1.0 · 13· · · f k= 1.5 .. )( _ .. 4.0 i k = 2.0 -6-.c: g 0.6 ~ ! 'iii c: 3.0 0 .l!! a. .!!! >0.4 0 t 2.0 ! 0.2 1.0 ,I.I 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 50 X Position Iteration (a) (b) Figure 1: Experimental results that support the discussion on the convergence condition with one reference vector for each class. (a) Input vectors used in the experiments. The broken line shows the ideal decision boundary. (b) Distance between two reference vectors for each k value during learning. The distance remains the same value for k > 1, while it diverges for k $ 1. decision boundary that minimizes the error rate is shown by the broken line. Two reference vectors were assigned to each class with initial values (x, y) = (0.3,0.4) and (0.3, 0.6) for Class A, and (x,y) = (0.7,0.4) and (0.7,0.6) for Class B. The gain factor 0: was 0.004 in GLVQ and LVQ2.1, and the window parameter sin LVQ2.1 was 0.8 in the experiments. Figure 2(b) shows the number of error counts for all the input vectors during learning. GLVQ(NL) shows results by GLVQ with a non-linear function; that is, af lap = f(p, t){1 - f(p, t)}. The number of error counts decreased with time to the minimum determined by the Bayes limit. GLVQ(L) shows results by GLVQ with a linear function; that is, a flap = 1. The number of error counts did not decrease to the minimum. This indicates that non-linearity of the cost function is very effective for improving recognition ability. Results using LVQ2.1 show that the number of error counts decreased in the beginning, but overall increased gradually with time. The degradation in the recognition ability results from the divergence of the reference vectors, as we have mentioned earlier. 5.2 CHARACTER RECOGNITION EXPERIMENTS Printed Chinese character recognition experiments were carried out to examine the performance of GLVQ. Thirteen kinds of printed fonts with 500 classes were used in the experiments. The total number of characters was 13,000; half of which were used as training data, and the other half were used as test data. As input vectors, 256-dimensional orientation features were used (Hamanaka et al., 1993). Only one reference vector was assigned to each class, and their initial values were defined by averaging training data for each class. Recognition results for test data are tabulated in Table 1 compared with other methods. TM is the template matching method using mean vectors. LVQ2 is the earlier version of LVQ2.1. The learning algorithm is the same as LVQ2.1 described in Section 2, but di must be less than dj. The gain factor 0: was 0.05, and the window parameter s was 0.65 in the experiments. The experimental result by LVQ3 was 428 1,0 0,8 c: 0,6 ,Q .;i 0 a. 0,4 ~ 0,2 0,0 '---__ -'--__ ..1..-__ ....1...-__ --'--__ -"-__ -' 0,0 0,2 0,4 0,6 0,8 1.0 1,2 X Position (a) A,SATO.K.Y~DA 1OO~---r----r----r----r---, 140 GLVQ(NL) --GL VQ(L) -+--LVQ2,1 ·13 .. • 4OL-~~~~~~~ __ --~ o 20 40 00 80 100 Iteration (b) Figure 2: Experimental results for linearly non-separable patterns with two reference vectors for each class. (a) Input vectors used in the experiments. The broken line shows the ideal decision boundary. (b) The number of error counts during learning. GLVQ (NL) and GLVQ (L) denote the proposed method using a non-linear and linear function in the cost function, respectively. This shows that non-linearity of the cost function is very effective for improving classification ability. Table 1: Experimental results for printed Chinese character recognition compared with other methods. Methods TMI LVQ22 LVQ2.1 IVQ3 GLVQ Error rates(%) 0.23 0.18 0.11 0.08 0.05 1 Template matching using mean vectors, 2The earlier version of LVQ2,l. 30ur previous model (Improved Vector Quantization), the same as that by LVQ2.1, because only one reference vector was assigned to each class. IVQ (Improved Vector Quantization) is our previous model based on Kohonen's rule (Sato & Tsukumo, 1994). The error rate was extremely low for GLVQ, and a recognition rate of 99.95% was obtained. Ambiguous results can be rejected by thresholding the value of J,t(x). If input vectors with J,t(x) ~ -0.02 were rejected, a recognition rate of 100% would be obtained, with a rejection rate of 0.08% for this experiment. 6 CONCLUSION We proposed the Generalized Learning Vector Quantization as a new learning method. We formulated the criterion for learning as the minimizing of the cost function, and obtained the learning rule based on the steepest descent method. GLVQ is a generalized method that includes LVQ. We discussed the convergence condition and showed that the convergence property depends on the definition of Generalized Learning Vector Quantization 429 the cost function. We proved that the essential problem of the divergence of the reference vectors in LVQ2.1 results from a drawback of Kohonen's rule that does not satisfy the convergence condition. Preliminary experiments revealed that nonlinearity in the cost function is very effective for improving recognition ability. We carried out printed Chinese character recognition experiments and obtained a recognition rate of 99.95%. The experimental results revealed that GLVQ is superior to the conventional LVQ algorithms. Acknowledgements We are indebted to Mr. Jun Tsukumo and our colleagues in the Pattern Recognition Research Laboratory for their helpful cooperation. References Y. Le Cun, B. Bose, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard, and L. D. Jackel, "Handwritten Digit Recognition with a Back-Propagation Network," Neural Information Processing Systems 2, pp. 396-404 (1989). K. Yamada, H. Kami, J. Tsukumo, and T. Temma, "Handwritten Numeral Recognition by Multi-Layered Neural Network with Improved Learning Algorithm," Proc. of the International Joint Conference on Neural Networks 89, Vol. 2, pp. 259-266 (1989). T. Kohonen, S elf-Organization and Associative Memory, 3rd ed. , Springer-Verlag (1989). T. Kohonen, "LVQ-.PAK Version 3.1 The Learning Vector Quantization Program Package," LVQ Programming Team of the Helsinki University of Technology, (1995). S. W. Lee and H. H. Song, "Optimal Design of Reference Models Using Simulated Annealing Combined with an Improved LVQ3," Proc. of the International Conference on Document Analysis and Recognition, pp. 244-249 (1993). K. Miyahara and F. Yoda, "Printed Japanese Character Recognition Based on Multiple Modified LVQ Neural Network," Proc. of the International Conference on Document Analysis and Recognition, pp. 250- 253 (1993). A. Sato and J. Tsukumo, "A Criterion for Training Reference Vectors and Improved Vector Quantization," Proc. of the International Conference on Neural Networks, Vol. 1, pp.161-166 (1994). N. R. Pal, J. C. Bezdek, and E. C.-IC Tsao, "Generalized Clustering Networks and Kohonen's Self-organizing Scheme," IEEE Trans. of Neural Networks, Vol. 4, No.4, pp. 549-557 (1993). A. I. Gonzalez, M. Grana, and A. D'Anjou, "An Analysis ofthe GLVQ Algorithm," IEEE Trans. of Neural Networks, Vol. 6, No.4, pp. 1012-1016 (1995). M. Hamanaka, K. Yamada, and J. Tsukumo, "On-Line Japanese Character Recognition Experiments by an Off-Line Method Based on Normalization-Cooperated Feature Extraction," Proc. of the International Conference on Document Analysis and Recognition, pp. 204-207 (1993).	1995	91
1,141	Bayesian Methods for Mixtures of Experts Steve Waterhouse Cambridge University Engineering Department Cambridge CB2 1PZ England Tel: [+44] 1223 332754 srw1001@eng.cam.ac.uk David MacKay Cavendish Laboratory Madingley Rd. Cambridge CB3 OHE England Tel: [+44] 1223 337238 mackay@mrao.cam.ac.uk ABSTRACT Tony Robinson Cambridge University Engineering Department Cambridge CB2 1PZ England. Tel: [+44] 1223 332815 ajr@eng.cam.ac.uk We present a Bayesian framework for inferring the parameters of a mixture of experts model based on ensemble learning by variational free energy minimisation. The Bayesian approach avoids the over-fitting and noise level under-estimation problems of traditional maximum likelihood inference. We demonstrate these methods on artificial problems and sunspot time series prediction. INTRODUCTION The task of estimating the parameters of adaptive models such as artificial neural networks using Maximum Likelihood (ML) is well documented ego Geman, Bienenstock & Doursat (1992). ML estimates typically lead to models with high variance, a process known as "over-fitting". ML also yields over-confident predictions; in regression problems for example, ML underestimates the noise level. This problem is particularly dominant in models where the ratio of the number of data points in the training set to the number of parameters in the model is low. In this paper we consider inference of the parameters of the hierarchical mixture of experts (HME) architecture (Jordan & Jacobs 1994). This model consists of a series of "experts," each modelling different processes assumed to be underlying causes of the data. Since each expert may focus on a different subset of the data which may be arbitrarily small, the possibility of over-fitting of each process is increased. We use Bayesian methods (MacKay 1992a) to avoid over-fitting by specifying prior belief in various aspects of the model and marginalising over parameter uncertainty. The use of regularisation or "weight decay" corresponds to the prior assumption that the model should have smooth outputs. This is equivalent to a prior p(ela) on the parameters e of the model, where a are the hyperparameters of the prior. Given a set of priors we may specify a posterior distribution of the parameters given data D, p(eID, a, 11) ex: p(Dle, 1l)p(e\a, 11), (1) where the variable 11 encompasses the assumptions of model architecture, type of regularisation used and assumed noise model. Maximising the posterior gives us the most probable parameters eMP. We may then set the hyperparameters either by cross-validation, or by finding the maximum of the posterior distribution of the 352 S. WATERHOUSE, D. MACKAY, T. ROBINSON hyperparameters P(aID), also known as the "evidence" (Gull 1989). In this paper we describe a method, motivated by the Expectation Maximisation (EM) algorithm of Dempster, Laird & Rubin (1977) and the principle of ensemble learning by variational free energy minimisation (Hinton & van Camp 1993, Neal & Hinton 1993) which achieves simultaneous optimisation of the parameters and hyper parameters of the HME. We then demonstrate this algorithm on two simulated examples and a time series prediction task. In each task the use of the Bayesian methods prevents over-fitting of the data and gives better prediction performance. Before we describe this algorithm, we will specify the model and its associated priors. MIXTURES OF EXPERTS The mixture of experts architecture (Jordan & Jacobs 1994) consists of a set of "experts" which perform local function approximation. The expert outputs are combined by a "gate" to form the overall output. In the hierarchical case, the experts are themselves mixtures of further experts, thus extending the network in a tree structured fashion. The model is a generative one in which we assume that data are generated in the domain by a series of J independent processes which are selected in a stochastic manner. We specify a set of indicator variables Z = {zt) : j = 1 ... J, n = 1 ... N}, where zt) is 1 if the output y(n) was generated by expert j and zero otherwise. Consider the case of regression over a data set D = {x(n) E 9tk, y(n) E 9tP, n = 1 ... N} with p = 1. We specify that the conditional probability of the scalar output y(n) given the input vector x(n) at exemplar (n) is J p(y(n)lx(n), 8) = L P(zt)lx(n), ~j)p(y(n)lx(n), Wj, {3j), (2) j=1 where {~j E 9tk} is the set of gate parameters, and {(Wj E 9tk), {3J the set of expert parameters. In this case, p(y(n)lx(n), Wj,{3j) is a Gaussian: (3) where 1/ {3j is the variance of expert j, I and Jt) = !}(x(n), Wj) is the output of expert j, giving a probabilistic mixture model. In this paper we restrict the expert output to be a linear function of the input, !}(x(n>, Wj) = w"f x(n). We model the action of selecting process j with the gate, the outputs of which are given by the softmax function of the inner products of the input vector2 and the gate parameter vectors. The conditional probability of selecting expert j given input x(n) is thus: (4) A straightforward extension of this model also gives us the conditional probability h;n) of expert j having been selected given input x(n) and output y(n) , hj") = P( zj') = 11/'), z ('), 9) = gj") ~j") / t. g~') ~j(') . (5) 1 Although {Jj is a parameter of expert j, in common with MacKay (1992a) we consider it as a hyperparameter on the Gaussian noise prior. 2In all notation, we assume that the input vector is augmented by a constant term, which avoids the need to specify a "bias" term in the parameter vectors. Bayesian Methods for Mixtures of Experts PRIORS We assume a separable prior on the parameters e of the model: J p(ela) = IT P(~jltl)P(wjlaj) i=l 353 (6) where {aj} and {tl} are the hyperparameters for the parameter vectors of the experts and the gate respectively. We assume Gaussian priors on the parameters of the experts {Wj} and the gate {~j}' for example: (7) For simplicity of notation, we shall refer to the set of all smoothness hyperparameters as a = {tl, aj} and the set of all noise level hyperparameters as /3 = {/3j}. Finally, we assume Gamma priors on the hyperparameters {tl, aj, /3j} of the priors, for example: 1 (/3,) p~ P(log /3j IPlh up) = r(pp) u~ exp( - /3j / up), (8) where up, Pp are the hyper-hyperparameters which specify the range in which we expect the noise levels /3j to lie. INFERRING PARAMETERS USING ENSEMBLE LEARNING The EM algorithm was used by Jordan & Jacobs (1994) to train the HME in a maximum likelihood framework. In the EM algorithm we specify a complete data set {D,Zl which includes the observed ~ataD and the set ~fi~dic~tor variables Z. Given e(m- 15, the E step of the EM algOrIthm computes a dIstrIbutIOn P(ZID, e(m-l» over Z. The M step then maximises the expected value of the complete data likelihood P(D, ZI e) over this distribution. In the case of the HME, the indicator variables Z = {{zt)}f=tl~=l specify which expert was responsible for generating the data at each time. We now outline an algorithm for the simultaneous optimisation of the parameters e and hyperparameters a and /3, using the framework of ensemble learning by variational free energy minimisation (Hinton & van Camp 1993). Rather than optimising a point estimate of e, a and /3, we optimise a distribution over these parameters. This builds on Neal & Hinton's (1993) description ofthe EM algorithm in terms of variational free energy minimisation. We first specify an approximating ensemble Q(w,~, a, /3, Z) which we optimise so that it approximates the posterior distribution P(w,~, a, /3, ZID, J{) well. The objective function chosen to measure the quality of the approximation is the variational free energy, F(Q) = J dw d~ da d/3 dZ Q(w,~, a,/3,Z) log P( Q(;, ~,;,/3'iJ{)' (9) w, ,a, ,Z,D where the joint probability of parameters {w,~}, hyperparameters, {a,/3}, missing data Z and observed data D is given by, 354 S. WATERHOUSE, D. MACKAY, T. ROBINSON P(w,~,a,{3,Z,DIJI) = J N ~ P(,u) n P(~jl,u)P(aj)P(wjlaj)P({3jlpj, Vj) n (P(Z}") = ll:r:(n), ~j)p(y(n)I:r:(n), Wj, (3j») J (10) FI _I The free energy can be viewed as the sum of the negative log evidence -log P(DIJI) and the Kullback-Leibler divergence between Q and P(w,~, a,{3,ZID,JI). F is bounded below by -log P(DIJ{), with equality when Q = P(w,~, a, (3, ZID, JI). We constrain the approximating ensemble Q to be separable in the form Q(w,~,a,{3,Z) = Q(w)Q(~)Q(a)Q({3)Q(Z). We find the optimal separable distribution Q by considering separately the optimisation of F over each separate ensemble component QO with all other components fixed. Optimising Qw(w) and Q~(~). As a functional of Qw(w), F is F = J dwQw(w) [L iwJWj+ ti/n)~(y(n)_yj"»2+10gQw(W)l +const (ll) ] n=1 where for any variable a, a denotes I da Q(a) a . Noting that the w dependent terms are the log of a posterior distribution and that a divergence I Q log(QIP) is minimised by setting Q = P, we can write down the distribution Qw(w) that minimises this expression. For given data and Qa, Q/J' Qz, Q~, the optimising distribution Q~Pt(w) is This is a set of J Gaussian distributions with means {Wj}, which can be found exactly by quadratic optimisation. We denote the variance covariance matrices of Q~ft(Wj) by {Lwj } . The analogous expression for the gates Q?\~) is obtained in a similar fashion and is given by We approximate each Q~P\~j) by a Gaussian distribution fitted at its maximum ~j = ~j with variance covariance matrix l:~j. Optimising Qz(Z) By a similar procedure, the optimal distribution Q~P\z) is given by (14) where (15) Bayesian Methods for Mixtures of Experts 355 and ~j is the value of l;j computed above. The standard E-step gives us a distribution of Z given a fixed value of parameters and the data, as shown in equation (5). In this case, by finding the optimal Qz(Z) we obtain the alternative expression of (15), with dependencies on the uncertainty of the experts' predictions. Ideally (if we did not made the assumption of a separable distribution Q) Qz might be expected to contain an additional effect of the uncertainty in the gate parameters. We can introduce this by the method of MacKay (1992b) for marginalising classifiers, in the case of binary gates. Optimising Qa(a) and QfJ(f3) Finally, for the hyperparameter distributions, the optimal values of ensemble functions give values for aj and {3j as 1 _ wJ Wj + 21 utlj + Trace1:Wj lij k+ 2paj An analogous procedure is used to set the hyperparameters {Jl} of the gate. MAKING PREDICTIONS In order to make predictions using the model, we must maryinaiise over the parameters and hyper parameters to get the predictive distribution. We use the optimal distributions QoptO to approximate the posterior distribution. For the experts, the marginalised outputs are given by ~(N+I) = h(x(N+l), w7P), with variance a 2 = x(N+I)T l:wx(N+I) + a 2 where a 2 = 1 / it. We may also marginalise yl aj • .Bj J J ' J PJ over the gate parameters (MacKay 1992b) to give marginalised outputs for the gates. The predictive distribution is then a mixture of Gaussians, with mean and variance given by its first and second moments, J y(N+1) = Lg/N+l)y/N+I); i=1 SIMULATIONS Artificial Data J a2 = ""' g .(N+I)(a2 + (",(N+I»2) _ (,,(N+I»2 yla . .B L..J I ylai . .B, VI V · (17) i=1 In order to test the performance of the Bayesian method, we constructed two artificial data sets. Both data sets consist of a known function corrupted by additive zero mean Gaussian noise. The first data set, shown in Figure (1a) consists of 100 points from a piecewise linear function in which the leftmost portion is corrupted with noise of variance 3 times greater than the rightmost portion. The second data set, shown in Figure (1b) consists of 100 points from the function get) = 4. 26(e- t - 4e- 2t + 3e- 3t), corrupted by Gaussian noise of constant variance 0.44. We trained a number of models on these data sets, and they provide a typical set of results for the maximum likelihood and Bayesian methods, together with the error bars on the Bayesian solutions. The model architecture used was a 6 deep binary hierarchy of linear experts. In both cases, the ML solutions tend to overfit the noise in the data set. The Bayesian solutions, on the other hand, are both smooth functions which are better approximations to the underlying functions. Time Series Prediction The Bayesian method was also evaluated on a time series prediction problem. This consists of yearly readings of sunspot activity from 1700 to 1979, and was first 356 -- "' ' . --I -~. ~ ~ ~i -\~~ ~ ' " . " I . "1, . ,1 ~ j~ .. . • i "~ ii " ~i. 'I j i .,' '. ; .. ..... , , "~.. ! ~ ,l . ••• ~,i / •• \, ' .. / . ...• (a) S. WATERHOUSE. D. MACKAY. T. ROBINSON (b) ..... ... " : ... .. ... . ..... . ... -Original function • Original + Noise . -ML solution -- Bayesian solution ... Error bars Figure 1: The effect of regularisation on fitting known functions corrupted with noise. considered in the connectionist community by Weigend, Huberman & Rumelhart (1990), who used an MLP with 8 hidden tanh units, to predict the coming year's activity based on the activities of the previous 12 years. This data set was chosen since it consists of a relatively small number of examples and thus the probability of over-fitting sizeable models is large. In previous work, we considered the use of a mixture of 7 experts on this problem. Due to the problems of over-fitting inherent in ML however, we were constrained to using cross validation to stop the training early. This also constrained the selection of the model order, since the branches of deep networks tend to become "pinched off" during ML training, resulting in local minima during training. The Bayesian method avoids this over-fitting of the gates and allows us to use very large models. Table 1: Single step prediction on the Sunspots data set using a lag vector of 12 years. NMSE is the mean squared prediction error normalised by the variance of the entire record from 1700 to 1979. The models used were; WHR: Weigend et al's MLP result; 1HME_7_CV: mixture of 7 experts trained via maximum likelihood and using a 10 % cross validation scheme; 8HME2-ML & 8HME2J3ayes: 8 deep binary HME,trained via maximum likelihood (ML) and Bayesian method (Bayes). MODEL Train NMSE Test NMSE 1700-1920 1921-1955 1956-1979 WHR 0.082 0.086 0.35 1HME7_CV 0.061 0.089 0.27 8HME2_ML 0.052 0.162 0.41 8HME2..Bayes 0.079 0.089 0.26 Table 1 shows the results obtained using a variety of methods on the sunspots task. The Bayesian method performs significantly better on the test sets than the maximum likelihood method (8HME2_ML), and is competitive with the MLP of Weigend et al (WHR). It should be noted that even though the number of parameters in the 8 deep binary HME (4992) used is much larger than the number of training examples (209), the Bayesian method still avoids over-fitting of the data. This allows us to specify large models and avoids the need for prior architecture selection, although in some cases such selection may be advantageous, for example if the number of processes inherent in the data is known a-priori. Bayesian Methods for Mixtures of Experts 357 In our experience with linear experts, the smoothness prior on the output function of the expert does not have an important effect; the prior on the gates and the Bayesian inference of the noise level are the important factors. We expect that the smoothness prior would become more important if the experts used more complex basis functions. DISCUSSION The EM algorithm is a special case of the ensemble learning algorithm presented here: the EM algorithm is obtained if we constrain Qe(8) and Qf3(f3) to be delta functions and fix a = O. The Bayesian ensemble works better because it includes regularization and because the uncertainty of the parameters is taken into account when predictions are made. It could be of interest in future work to investigate how other models trained by EM could benefit from the ensemble learning approach such as hidden Markov models. The Bayesian method of avoiding over-fitting has been shown to lend itself naturally to the mixture of experts architecture. The Bayesian approach can be implemented practically with only a small computational overhead and gives significantly better performance than the ML model. References Dempster, A. P., Laird, N. M. & Rubin, D. B. (1977), 'Maximum likelihood from incomplete data via the EM algorithm', Journal of the Royal Statistical Society, Series B 39, 1- 38. Geman, S., Bienenstock, E. & Doursat, R. (1992), 'Neural networks and the bias / variance dilemma', Neural Computation 5, 1-58. Gull, S. F. (1989), Developments in maximum entropy data analysis, in J . Skilling, ed., 'Maximum Entropy and Bayesian Methods, Cambridge 1988', Kluwer, Dordrecht, pp. 53-71. Hinton, G. E. & van Camp, D. (1993), Keeping neural networks simple by minimizing the description length of the weights, To appear in: Proceedings of COLT-93. Jordan, M. I. & Jacobs, R. A. (1994), 'Hierarchical Mixtures of Experts and the EM algorithm', Neural Computation 6, 181- 214. MacKay, D. J . C. (1992a), 'Bayesian interpolation', Neural Computation 4(3),415447. MacKay, D. J . C. (1992b), 'The evidence framework applied to classification networks', Neural Computation 4(5), 698- 714. Neal, R. M. & Hinton, G. E. (1993), 'A new view of the EM algorithm that justifies incremental and other variants'. Submitted to Biometrika. Available at URL:ftp:/ /ftp.cs.toronto.edu/pub/radford/www. Weigend, A. S., Huberman, B. A. & Rumelhart, D. E. (1990), 'Predicting the future: a connectionist approach', International Journal of Neural Systems 1, 193- 209.	1995	92
1,142	High-Speed Airborne Particle Monitoring Using Artificial Neural Networks Alistair Ferguson ERDC, Univ. of Hertfordshire A.Ferguson@herts.ac.uk Paul Kaye ERDC, Univ. of Hertfordshire Theo Sabisch Dept. Electrical and Electronic Eng. U niv. of Hertfordshire Laurence C. Dixon NOC, Univ. of Hertfordshire Hamid Bolouri ERDC, Univ. of Hertfordshire, Herts, ALtO 9AB, UK Abstract Current environmental monitoring systems assume particles to be spherical, and do not attempt to classify them. A laser-based system developed at the University of Hertfordshire aims at classifying airborne particles through the generation of two-dimensional scattering profiles. The pedormances of template matching, and two types of neural network (HyperNet and semi-linear units) are compared for image classification. The neural network approach is shown to be capable of comparable recognition pedormance, while offering a number of advantages over template matching. 1 Introduction Reliable identification of low concentrations of airborne particles requires high speed monitoring of large volumes of air, and incurs heavy computational overheads. An instrument to detect particle shape and size from spatial light scattering profiles has High-speed Airborne Particle Monitoring Using Artificial Neural Networks 981 previously been described [6]. The system constrains individual particles to traverse a laser beam. Thus, spatial distributions of the light scattered by individual particles may be recorded as two dimensional grey-scale images. Due to their highly distributed nature, Artificial Neural Networks (ANNs) offer the possibility of high-speed non-linear pattern classification. Their use in particulate classification has already been investigated. The work by Kohlus [7] used contour data extracted from microscopic images of particles, and so was not real-time. While using laser scattering data to allow real-time analysis, Bevan [2] used only three photomultipliers, from which very little shape information can be collected. This paper demonstrates the plausibility of particle classification based on shape recognition using an ANN. While capable of similar recognition rates, the neural networks are shown to offer a number of advantages over template matching. 2 The Hyper N et Architecture HyperNet is the term used to denote the hardware model of a RAM-based sigma-pi neural architecture developed by Gurney [5]. The architecture is similar in nature to the pRAM of Gorse and Taylor (references in [4]). The amenability of these nodes to hardware realisation has been extensively investigated, leading to custom VLSI implementations of both nodes [3, 4]. Each HyperNet node is termed a multicube unit (MeU), and consists of a number of subunits, each with an arbitrary number of inputs. j references the nodes, with i = 1, ... ,Ii indexing the subunits. IJ denotes the site addresses, and is the set of bit strings 1J1, ... ,lJn wl1ere n denotes the number of inputs to the subunit. Zc refers to the cth real-valued input, with Zc E [0,1] and Zc == (1 - zc). For each of the 2n site store locations, two sets are defined: c E M:!o if IJc = 0; c E M:!t if IJc = 1. The access probability p(lJii) for location IJ in subunit i of hidden layer node j is therefore (1) The activation (ai ) is formed by accumulating the proportional site values (SIS';) from every subunit. The activation is then passed through a sigmoidal transfer function to yield the node output (yi). (2) . . 1 y' = u(a1 ) = . 1 + ea1 / p (3) where p is a positive parameter determining the steepness of the sigmoidal curve. By combining equations (1) and (2), it becomes apparent that the node is a higherorder or sigma-pi node [9]. A wide variety of learning algorithms have been tailored for these nodes, notably reward-penalty and back-propagation [5]. 982 A. FERGUSON, T. SABISCH, P. KAYE. L. C. DIXON. H. BOWURJ 3 Description of the Particle Monitoring System The instrument draws air through the laser scattering chamber at approximately 1.5 min-1 , and is constrained to a column of approximately 0.8mm diameter at the intersection with the laser beam. Light scattered into angles between 300 and 1410 to the beam direction is reflected through the optics and onto the photocathode of an intensified CCD (charge-coupled device), thus giving rise to the scattering profile. The imaging device used has a pixel resolution of 385 x 288, which is quantised into 2562 8-bit pixels by the frame grabbing processor card of the host computer. Data was collected on eight particle types, namely: long and short caffeine fibres; 31lm and 121lm micro-machined silicon dioxide fibres; copper flakes (2- 5Ilm in length and O.lllm thick); 31lm and 4.31lm polystyrene spheres; and salt crystals. An exemplar profile for each class is given in figure 1. Almost all the image types are highly variable. In particular, the scattering profile obtained for a fibrous particle is affected by its orientation as it passes through the laser beam. The scattering profiles are intrinsically centred, with the scaling giving important information regarding the size of the particle. The experiments reported here use 100 example scattering profiles for each of the eight particle classes. For each class, 50 randomly selected images were used to construct the templates or train the neural network (training set), and the remainder used to test the performance of the pattern classifiers. 4 Experimental Results The performance of template matching is compared to both HyperNet and networks of semi-linear units. In all experiments, high-speed classification is emphasised by • ~. ?'., ~ . .. ~.., l t.. .', " , t : ~ , . . , '. , 'l.J; Figure 1: Exemplar Image Profile For Each Of The Eight Benchmark Classes High-speed Airborne Particle Monitoring Using Artificial Neural Networks 983 avoiding image preprocessing operations such as transformation to the frequency domain, histogram equalisation, and other filtering operations. Furthermore, all experiments use the scatter profile image as input, and include no other information. The current monitoring system produces a 2562 8-bit pixel image. The sensitivity of the camera is such that a single pixel can represent the registration of a single photon of light. Two possible methods of reducing computation, implementable through the use of a cheaper, less sensitive camera were investigated. The first grouped neighbouring pixels to form a single average intensity value. The neighbourhood size was restricted to powers of two, producing images ranging in size from 2562 to 42 pixels. The second banded grey levels into groups, again in powers of two. Each pixel could therefore range from eight bits down to one. 4.1 Template Matching Results The construction of reference templates is crucial to successful classification. Two approaches to template construction were investigated <D Single reference image for each class. Various techniques were applied ranging from individual images, to mode, median, and mean averaged templates. Mean averaged templates were found to lead to the highest classification rates. In this approach, each pixel location in the template takes on the averaged value of that location across the 50 training images. ® Multiple templates per class. A K-means clustering algorithm [1] was used to identify clusters of highly correlated images within each class. The initial cluster centres were hand selected. The maximum number of clusters within each class was limited to six. For each cluster, the reference template was constructed using the mean averaging approach above. Tables 1 and 2 summarise the recognition rates achieved using single, and multiple mean averaged templates for each particle class. In both cases, the best average recognition rate using this approach was gained with 1282 3-bit pixel images. With a single template this lead to a recognition rate of 78.2%, increasing to 85.2% for multiple templates. However, the results for both 162 and 82 pixel images are reasonable approximations of the best performance, and represent an acceptable trade-off between computational cost and performance. With few exceptions, multiple templates per class led to higher recognition rates than for the corresponding single template results. This is attributable to the variability of the particles within a class. As expected, the effect of grey level quantisation is inversely proportional to that of local averaging. In order to evaluate the efficiency of the template construction methods, every image in the training set was used as a reference template. 2562 8-bit, 1282 3-bit, and 642 2-bit pixel images were used for these experiments. However, the recognition rate did not exceed 85%, demonstrating the success of the template generation schemes previously employed. 984 A. FERGUSON, T. SABISCH, P. KAYE, L. C. DIXON, H. BOLOURI Table 1: Single Template Per Class % Recognition Rates grey levels image size 2562 1282 642 322 162 82 42 256 73.5 75.0 74.7 74.7 74.7 75.0 67.2 128 73.5 75.0 74.7 74.7 74.5 75.0 68.5 64 73.0 75.0 74.5 74.5 74.2 74.7 66.2 32 73.0 74.7 75.2 75.5 74.7 74.2 66.5 16 74.0 76.0 76.7 76.0 75.0 15.5 56.0 8 15.5 18.2 11.5 11.5 16.0 73.7 38.7 4 68.4 69.7 71.0 70.7 69.7 58.5 18.7 2 69.7 68.7 65.5 66.2 46.2 23.0 16.6 Table 2: Multiple Templates Per Class % Recognition Rates grey levels image size 2562 1282 642 322 162 82 42 256 78.0 80.0 80.2 80.5 79.0 76.7 10.2 128 78.5 80.2 80.5 80.5 79.0 77.0 69.7 64 78.7 80.2 80.2 80.5 79.2 76.0 69.2 32 78.2 81.2 81.7 80.0 78.7 76.7 67.7 16 80.2 83.5 83.0 81.2 79.5 78.5 56.0 8 82.2 85.2 84.5 84.1 81.0 80.0 43.5 4 72.7 74.5 72.2 72.2 69.5 61.2 39.2 2 69.7 70.2 70.7 62.7 51.7 51.7 0.03 4.2 Neural Network Results A fully connected three layer feed-forward network was used in all experiments. The number of hidden layer neurons was equal to the square root of the number of pixels. The target patterns were chosen to minimise the number of output layer nodes, while ensuring an equitable distribution of zeros and ones. Six output layer neurons were used to give a minimum Hamming distance of two between target patterns. The classification of a pattern was judged to be the particle class whose target pattern was closest (lowest difference error). The HyperNet architecture was trained using steepest descent, though the line search was hardware based and inexact. The semilinear network was trained using a variety of back-propagation type algorithms, with the best results obtained reported. Both networks were randomly initialised. Due to the enormous training overhead, only 162 and 82 pixel images were tried. The recognition rates achieved are given in table 3. Both neural networks are significantly better than the single, and some of the multiple template matching results. With optimisation of the network structures, it is likely that the ANNs could exceed the performance of multiple templates. High-speed Airborne Particle Monitoring Using Artificial Neural Networks 985 Table 3: Neural Network % Recognition Rates Classifier Quantisation Levels 162 4 bit 1162 3-bit 82 4-bit 82 3-bit HyperNet 83.8 I 82.3 83.0 76.8 Semi-linear 84.5 86.3 77.8 76.0 le+09 , , , , , ,-.. le+08 , '" c:: '-" ~ Ie+{)? o~ ll! 0;; le+06 '" 8 ~ le+05 le+04 82 162 322 642 1282 2562 Number of pixels Figure 2: Hardware classification speeds for a single pattern against image size 5 Speed Considerations Single processor, pipelined hardware implementations of the three classification techniques have been considered. A fast (45ns) multiply-accumulate chip (Logic Devices Ltd, LMA201O) was utilised for semi-linear units. Both template matching and HyperNet were implemented using the Logic Devices LGC381 ALU (26ns per accumulate). The cost of these devices is approximately the same (£10-20). The HyperNet implementation uses a bit-stream approach to eliminate the probability multiplications [8], with a stream length of 256 bits. Figure 2 plots single pattern processing time for each classifier against image size. For small image resolutions, the semi-linear network offers the best performance, being almost three times faster than template matching. However, template matching and HyperNet yield faster performance at higher image resolutions. At the optimum (indicated by template matching results (§4.1); 1282 pixels), HyperNet is almost seven times faster than the comparable implementation of semi-linear units. While the hardware performance of template matching is similar to HyperNet, it suffers from a number of disadvantages to which the neural approaches are immune CD Recognition rate is dependent on the choice of reference images. ® Multiple reference images must be used to achieve good recognition rates 986 A. FERGUSON, T. SABISCH, P. KAYE, L. C. DIXON, H. BOLOURI which drastically increases the amount of computation required. @ New reference images must be found whenever a new class is introduced. @ Difficult to make behaviour adaptive, ie. respond to changing conditions. 6 Conclusions The feasibility of constructing an airborne particle monitoring system capable of reliable particle identification at high speeds has been demonstrated. Template matching requires multiple reference images and is cumbersome to develop. The neural networks offer easier training procedures and equivalent recognition rates. In addition, HyperNet has the advantage of high speed operation at large image sizes. Acknowledgements The authors would like to thank Dr. Eric Dykes and Dr. Edwin Hirst at the University of Hertfordshire, Dr. Kevin Gurney at BruneI University, and the EPSRC and the Royal Society for financial support. References [1] Stephen Banks. Signal Processing, Image Processing, and Pattern Recognition. Prentice Hall, 1990. [2] A V Bevan et al. The application of neural networks to particle shape classification. Journal of Aerosol Science, 23(Suppl. 1):329-332, 1992. [3] Hamid Bolouri et al. Design, manufacture, and evaluation of a scalable highperformance neural system. Electronics Letters, 30(5):426-427, 3 March 1994. [4] T G Clarkson et al. The pRAM: An adaptive VLSI chip. IEEE 'Ihmsactions on Neural Networks, 4(3):408-412, May 1993. [5] Kevin N Gurney. Learning in networks of structured hypercubes. PhD thesis, Department of Electrical Engineering, UK, 1995. [6] Paul H Kaye et al. Airborne particle shape and size classification from spatial light scattering profiles. Journal of Aerosol Science, 23(6):597--611, 1992. [7] R Kohlus et al. Particle shape analysis as an example of knowledge extraction by neural nets. Part. Part. Syst. Charact., 10:275-278, 1993. [8] Paul Morgan et al. Hardware implementation of a real-valued sigma-pi network. In Artificial Neural Networks 5, volume 2, pages 351-356, North-Holland, 1995. [9] David E Rumelhart et al. Parallel Distributed Processing: Explorations in the Macrostructure of Cognition, volume 1. MIT Press, 1986. PART IX CONTROL	1995	93
1,143	Adaptive Mixture of Probabilistic Transducers Yoram Singer AT&T Bell Laboratories singer@research.att.com Abstract We introduce and analyze a mixture model for supervised learning of probabilistic transducers. We devise an online learning algorithm that efficiently infers the structure and estimates the parameters of each model in the mixture. Theoretical analysis and comparative simulations indicate that the learning algorithm tracks the best model from an arbitrarily large (possibly infinite) pool of models. We also present an application of the model for inducing a noun phrase recognizer. 1 Introduction Supervised learning of a probabilistic mapping between temporal sequences is an important goal of natural sequences analysis and classification with a broad range of applications such as handwriting and speech recognition, natural language processing and DNA analysis. Research efforts in supervised learning of probabilistic mappings have been almost exclusively focused on estimating the parameters of a predefined model. For example, in [5] a second order recurrent neural network was used to induce a finite state automata that classifies input sequences and in [1] an input-output HMM architecture was used for similar tasks. In this paper we introduce and analyze an alternative approach based on a mixture model of a new subclass of probabilistic transducers, which we call suffix tree transducers. The mixture of experts architecture has been proved to be a powerful approach both theoretically and experimentally. See [4,8,6, 10,2, 7] for analyses and applications of mixture models, from different perspectives such as connectionism, Bayesian inference and computational learning theory. By combining techniques used for compression [13] and unsupervised learning [12], we devise an online algorithm that efficiently updates the mixture weights and the parameters of all the possible models from an arbitrarily large (possibly infinite) pool of suffix tree transducers. Moreover, we employ the mixture estimation paradigm to the estimation of the parameters of each model in the pool and achieve an efficient estimate of the free parameters of each model. We present theoretical analysis, simulations and experiments with real data which show that the learning algorithm indeed tracks the best model in a growing pool of models, yielding an accurate approximation of the source. All proofs are omitted due to the lack of space 2 Mixture of Suffix Tree Transducers Let ~in and ~Ot.lt be two finite alphabets. A Suffix Tree Transducer T over (~in, ~Ot.lt) is a rooted,l~jn I-ary tree where every internal node of T has one child for each symbol in ~in. The nodes of the tree are labeled by pairs (s, l' ~), where s is the string associated with the path (sequence of symbols in ~n) that leads from the root to that node, and 1'~ : ~Ot.lt -+ [0,1] is the output probability function. A suffix tree transducer (stochastically) maps arbitrarily long input sequences over ~in to output sequences over ~Ot.lt as follows. The probability 382 Y. SINGER that T will output a string Yl, Y2, ... ,Yn in I:~ut given an input string Xl, X2, ... , Xn in I:in, denoted by PT(Yl, Y2, ... , YnlXl, X2 ,"" xn), is n~=li8. (Yk), where sl = Xl, and for 1 ::; j ::; n - 1, si is the string labeling the deepest node reached by taking the path corresponding to xi, xi -1, Xi -2, ... starting at the root of T. A suffix tree transducer is therefore a probabilistic mapping that induces a measure over the possible output strings given an input string. Examples of suffix tree transducers are given in Fig. 1. Figure 1: A suffix tree transducer (left) over (Lin, LQut) = ({O, 1} , {a, b, c}) and two ofits possible sub-models (subtrees). The strings labeling the nodes are the suffixes of the input string used to predict the output string. At each node there is an output probability function defined for each of the possible output symbols. For instance, using the suffix tree transducer depicted on the left, the probability of observing the symbol b given that the input sequence is ... ,0, 1,0, is 0.1. The probability of the current output, when each transducer is associated with a weight (prior), is the weighted sum of the predictions of each transducer. For example, assume that the weights of the trees are 0.7 (left tree), 0.2 (middle), and 0.1. then the probability thattheoutputYn = a given that (X n -2, Xn-l, Xn) = (0,1,0) is 0.7· P7j (aIOl0) + 0.2· P7i(aIIO) + 0.1 . P7)(aIO) = 0.7 . 0.8 + 0.2 . 0.7 + 0.1 . 0.5 = 0.75. Given a suffix tree transducer T we are interested in the prediction of the mixture of all possible subtrees of T. We associate with each subtree (including T) a weight which can be interpreted as its prior probability. We later show how the learning algorithm of a mixture of suffix tree transducers adapts these weights with accordance to the performance (the evidence in Bayesian terms) of each subtree on past observations. Direct calculation of the mixture probability is infeasible since there might be exponentially many such subtrees. However, the technique introduced in [13] can be generalized and applied to our setting. Let T' be a subtree of T. Denote by nl the number of the internal nodes of T' and by n2 the number of leaves of T' which are not leaves of T. For example, nl = 2 and n2 = I, for the tree depicted on the right part of Fig. 1, assuming that T is the tree depicted on the left part of the figure. The prior weight of a tree T'. denoted by Po(T') is defined to be (1 Q-)n\ an2 , where a E (0, 1). Denote by Sub(T) the set of all possible subtrees of T including T itself. It can be easily verified that this definition of the weights is a proper measure, i.e., LT/ESUb(T) Po(T') = 1. This distribution over trees can be extended to unbounded trees assuming that the largest tree is an infinite lI:in I-ary suffix tree transducer and using the following randomized recursive process. We start with a suffix tree that includes only the root node. With probability a we stop the process and with probability 1 - a we add all the possible lI:in I sons of the node and continue the process recursively for each of the sons. Using this recursive prior the suffix tree transducers, we can calculate the prediction of the mixture at step n in time that is linear in n, as follows, aie(Yn) + (1 - a) (aixn(Yn) + (1- a) (aixn_\xn(Yn) + (1 - a) ... Therefore, the prediction time of a single symbol is bounded by the maximal depth of T, or the length of the input sequence if T is infinite. Denote by 1'8 (Yn) the prediction of the mixture of subtrees rooted at s, and let Leaves(T) be the set of leaves of T . The above Adaptive Mixture of Probabilistic Transducers 383 sum equals to 'Ye(Yn), and can be evaluated recursively as follows,1 'Y3(Yn) = { 13(Yn) _ S E Le~ves(T) ( ) a I3 (Yn) + (I - a)r(X n_I.I.3)(Yn) otherwise I For example, given that the input sequence is ... ,0, 1, 1,0, then the probabilities of the mixtures of subtrees for the tree depicted on the left part of Fig. 1, for Yn = b and given that a = 1/2, are, 'Yllo(b) = 0.4 , 'YlO(b) = 0.5 . 11O(b) + 0.5 . 0.4 = 0.3 , 'Yo(b) = 0.5 . lo(b) + 0.5 ·0.3 = 0.25, 'Ye(b) = 0.5 . le(b) + 0.5 ·0.25 = 0.25. 3 An Online Learning Algorithm We now describe an efficient learning algorithm for a mixture of suffix tree transducers. The learning algorithm uses the recursive priors and the evidence to efficiently update the posterior weight of each possible subtree. In this section we assume that the output probability functions are known. Hence, we need to evaluate the following, ~ P(Yn IT')P(T'I(XI, YI), ... ,(Xn_l, Yn-t) T'ESub(T) def ~ P(Yn IT')Pn (T') (2) T'ESub(T) where Pn(T') is the posterior weight of T'. Direct calculation of the above sum requires exponential time.. However, using the idea of recursive calculation as in Equ. (1) we can efficiently calculate the prediction of the mixture. Similar to the definition of the recursive prior a, we define qn (s) to be the posterior weight of a node S compared to the mixture of all nodes below s. We can compute the prediction of the mixture of suffix tree transducers rooted at s by simply replacing the prior weight a with the posterior weight, qn-l (s), as follows, _ ( ) _ { 13(Yn) S E Leaves(T) 13 Yn qn-I(S)r3(Yn) + (1 - qn-l(S»'Y(X n _I.I.3)(Yn) otherwise In order to update qn(s) we introduce one more variable, denoted by rn(s). ro(s) = 10g(a/(1 - a» for all s, rn(s) is updated as follows, rn(s) = rn-l(s) + log(/3(Yn» -log('YXn_I.13(Yn» . , (3) Setting (4) Therefore, rn( s) is the log-likelihood ratio between the prediction of s and the prediction of the mixture of all nodes below s in T. The new posterior weights qn (s) are calculated from rn(s), (5) In summary, for each new observation pair, we traverse the tree by following the path that corresponds to the input sequence x n X n -I X n _ 2 .. . The predictions of each sub-mixture are calculated using Equ. (3). Given these predictions the posterior weights of each sub-mixture are updated using Equ. (4) and Equ. (5). Finally, the probability of Yn induced by the whole mixture is the prediction propagated out of the root node, as stated by Lemma 3.1. Lemma3.1 LT'ESub(T) P(YnlT')Pn(T') = 'Ye(Yn). Let Lossn (T) be the logarithmic loss (negative log-likelihood) of a suffix tree transducer T after n input-output pairs. That is, Lossn(T) = L7=1 -log(P(YiIT». Similarly, the loss 1 A similar derivation still holds even if there is a different prior 0'. at each node s of T. For the sake of simplicity we assume that 0' is constant. 384 Y. SINGER of the mixture is defined to be, Lossr;:ix = 2:~=1 -log(.ye(yd). The advantage of using a mixture of suffix tree transducers over a single suffix tree is due to the robustness of the solution, in the sense that the prediction of the mixture is almost as good as the prediction of the best suffix tree in the mixture. Theorem 1 Let T be a (possibly infinite) suffix tree transducer, and let (Xl, yd, .. . , (xn, Yn) be any possible sequence of input-output pairs. The loss of the mixture is at most, Lossn(T') -log(Po(T'»,Jor each possible subtree T'. The running time of the algorithm is D n where D is the maximal depth ofT or n2 when T is infinite. The proof is based on a technique introduced in [4]. Note that the additional loss is constant, hence the normalized loss per observation pair is, Po(T')/n, which decreases like O(~). Given a long sequence of input-output pairs or many short sequences, the structure of the suffix tree transducer is inferred as well. This is done by updating the output functions, as described in the next section, while adding new branches to the tree whenever the suffix of the input sequence does not appear in the current tree. The update of the weights, the parameters, and the structure ends when the maximal depth is reached, or when the beginning of the input sequence is encountered. 4 Parameter Estimation In this section we describe how the output probability functions are estimated. Again, we devise an online scheme. Denote by C;'(y) the number of times the output symbol y was observed out of the n times the node s was visited. A commonly used estimator smoothes each count by adding a constant ( as follows, (6) The special case of ( = ~ is termed Laplace's modified rule of succession or the add~ estimator. In [9], Krichevsky and Trofimov proved that the loss of the add~ estimator, when applied sequentially, has a bounded logarithmic loss compared to the best (maximumlikelihood) estimator calculated after observing the entire input-output sequence. The additional loss of the estimator after n observations is, 1/2(II:outl - 1) log(n) + lI:outl-l. When the output alphabet I:out is rather small, we approximate "y 8 (y) by 78 (y) using Equ. (6) and increment the count of the corresponding symbol every time the node s is visited. We predict by replacing "y with its estimate 7 in Equ. (3). The loss of the mixture with estimated output probability functions, compared to any subtree T' with known parameters, is now bounded as follows, LOSS,:ix ~ Lossn(T') -log(Po(T')) + 1/2IT'1 (Il:outl-l) log(n/IT'I) + IT'I (Il:outl-l), where IT'I is the number of leaves in T'. This bound is obtained by combining the bound on the prediction of the mixture from Thm. 1 with the loss of the smoothed estimator while applying Jensen's inequality [3]. When lI:out 1 is fairly large or the sample size if fairly small, the smoothing of the output probabilities is too crude. However, in many real problems, only a small subset of the output alphabet is observed in a given context (a node in the tree). For example, when mapping phonemes to phones [II], for a given sequence of input phonemes the phones that can be pronounced is limited to a few possibilities. Therefore, we would like to devise an estimation scheme that statistically depends on the effective local alphabet and not on the whole alphabet. Such an estimation scheme can be devised by employing again a mixture of models, one model for each possible subset I:~ut of I:out. Although there are 211:0 .. ,1 subsets of I:out, we next show that if the estimators depend only on the size of each subset then the whole mixture can be maintained in time linear in lI:out I. Adaptive Mixture of Probabilistic Transducers 385 Denote by .y~ (YIII:~ut 1 = i) the estimate of 'Y~ (y) after n observations given that the alphabet I:~t is of size i. Using the add~ estimator, .y~(YIII:~utl = i) = (C~(y) + 1/2)/(n + i/2). Let I:~ut(s) be the set of different output symbols observed at node s, i.e. I:~ut(s) = {u 1 u = Yi", s = (xi"-I~I+1' .. . ,Xi,,), 1 $ k $ n} , and define I:0 (s) to be the empty set There are (IIo.'I-II~.,(~)I) possible alphabets of out . i-II:.,(~)I size i. Thus, the prediction of the mixture of all possible subsets of I:out is, An( ) = I~I (lI:outl- 1I:~ut(s)l) ':l An( I·) i~ Y L...J . _ lI:n (s)1 w1 'Y~ Y J , j=II:.,(~)1 J out (7) where wi is the posterior probability of an alphabet of size i. Evaluation of this sum requires O(II:01.lt I) operations (and not 0(2IIo.,1 ». We can compute Equ. (7) in an online fashion as follows. Let, ( lI:01.ltl- 1I:~ut(s)l) 0 TIn Ak-l( . 10) 0_ lI:n ()I W, 'Y8 y,,, ~ 2 out s k=l (8) Without loss of generality, let us assume a uniform prior for the possible alphabet sizes. Then, Po(I:~ut) = Po(II:~1.Itl = i) ~ w? = 1/ (I I:01.lt1 (lI::utl)) 0 Thus, for all i ~(i) = 1/II:01.ltl. ~+1 (i) is updated from ~(i) as follows, m+l (0) _ m ( 0) {~:(';'j;' )+'/2 1 ~ 2 1 ~ 2 X n+I/2 i-II~y,(~)1 ~ IIo.,I-II:.,(8)1 n+i/2 if 1I:~ti/(s)1 > i if 1I:~titl(s)1 $ i and Yi,,+1 E I:~1.It(s) if 1I:~titl(s)1 $ i and Yi,,+1 ¢ I:~ut(s) Informally: If the number of different symbols observed so far exceeds a given size then all alphabets of this size are eliminated from the mixture by slashing their posterior probability to zero. Otherwise, if the next symbol was observed before, the output probability is the prediction of the addi estimator. Lastly, if the next symbol is entirely new, we need to sum the predictions of all the alphabets of size i which agree on the first 1I:~1.It(s)1 and Yi,,+1 is one of their i 1I:~1.It(s)1 (yet) unobserved symbols. Funhermore, we need to multiply by the apriori probability of observing Yi ,,+10 Assuming a uniform prior over the unobserved symbols this probability equals to 1/(II:01.lt 1 1I:~1.It( s)l). Applying Bayes rule again, the prediction of the mixture of all possible subsets of the output alphabet is, IIo." IIo.,1 .y~(Yin+l) = 2: ~+l(i) / 2: ~(i) ° (9) i=l i=l Applying twice the online mixture estimation technique, first for the structure and then for the parameters, yields an efficient and robust online algorithm. For a sample of size n, the time complexity of the algorithm is DII:01.ltln (or lI:01.ltln2 if 7 is infinite). The predictions of the adaptive mixture is almost as good as any suffix tree transducer with any set of parameters. The logarithmic loss of the mixture depends on the number of non-zero parameters as follows, Lossr;:ix $ Lossn (7') -log(Po(7'» + 1/21Nz log(n) + 0(17'III:01.ltl) , where lNz is the number of non-zero parameters of the transducer 7'0 If lNz ~ 17'III:out l then the performance of the above scheme, when employing a mixture model for the parameters as well, is significantly better than using the add~ rule with the full alphabet. 386 Y. SINGER 5 Evaluation and Applications In this section we briefly present evaluation results of the model and its learning algorithm. We also discuss and present results obtained from learning syntactic structure of noun phrases. We start with an evaluation of the estimation scheme for a multinomial source. In order to check the convergence of a mixture model for a multinomial source, we simulated a source whose output symbols belong to an alphabet of size 10 and set the probabilities of observing any of the last five symbols to zero. Therefore, the actual alphabet is of size 5. The posterior probabilities for the sum of all possible subsets of I:out of size i (1 :::; i :::; 10) were calculated after each iteration. The results are plotted on the left part of Fig. 2. The very first observations rule out alphabets of size lower than 5 by slashing their posterior probability to zero. After few observations, the posterior probability is concentrated around the actual size, yielding an accurate online estimate of the multinomial source. The simplicity of the learning algorithm and the online update scheme enable evaluation of the algorithm on millions of input-output pairs in few minutes. For example, the average update time for a suffix tree transducer of a maximal depth 10 when the output alphabet is of size 4 is about 0.2 millisecond on a Silicon Graphics workstation. A typical result is shown in Fig. 2 on the right. In the example, I:out = I:in = {I, 2, 3,4}. The description of the source is as follows. If Xn ~ 3 then Yn is uniformly distributed over I:out. otherwise (xn :::; 2) Yn = Xn-S with probability 0.9 and Yn-S = 4 Xn-S with probability 0.1. The input sequence Xl, X2, .•• was created entirely at random. This source can be implemented by a sparse suffix tree transducer of maximal depth 5. Note that the actual size of the alphabet is only 2 at half of the leaves of the tree. We used a suffix tree transducer of maximal depth 20 to learn the source. The negative of the logarithm of the predictions (normalized per symbol) are shown for (a) the true source, (b) a mixture of suffix tree transducers and their parameters, (c) a mixture of only the possible suffix tree transducers (the parameters are estimated using the addl scheme), and (d) a single (overestimated) model of depth 8. Clearly, the mixture mode? converge to the entropy of the source much faster than the single model. Moreover, employing twice the mixture estimation technique results in an even faster convergence. UbUI "b'~1 - "I - "I .. .. : .:~.: ......... ~.:~.:~ .,. • ,. I ,. • 1. I" I,. 0.1 ..... t ..... t ..... t .... It .... II •••• "~'··b"b"'b"b'·b .. .. .. .. .. -. • • • • • 0 I" I" I" I" I " • " 'b'·L'··LUL"~'·~ . . . .. .. .. ... •• " •• II •• I" • II • " • ... . . . . . • • • I • tI I 1. I" I,. I " I" "I ... "I -. "I -_ ':1 -. "I .. _ "I .. _ ,:~,:~,:~,~,:~,:~ I" I 10 I to .,. I " 5 ,. 1.B .g 1.6 Ii: (d) Single Overestimated Model - (e) Mixture 01 Models .--- (bl MiX1ure 01 Models and Parameters .. ...... (a Source 50 100 150 200 250 300 350 400 450 500 Number 01 Examples Figure 2: Left: Example of the convergence of the posterior probability of a mixture model for a multinomial source with large number of possible outcomes when the actual number of observed symbols is small. Right: performance comparison of the predictions of a single model, two mixture models and the true underlying transducer. We are currently exploring the applicative possibilities of the algorithm. Here we briefly discuss and demonstrate how to induce an English noun phrase recognizer. Recognizing noun phrases is an important task in automatic natural text processing, for applications such as information retrieval, translation tools and data extraction from texts. A common practice is to recognize noun phrases by first analyzing the text with a part-of-speech tagger, which assigns the appropriate part-of-speech (verb, noun, adjective etc.) for each word in Adaptive Mixture of Probabilistic Transducers 387 context. Then, noun phrases are identified by manually defined regular expression patterns that are matched against the part-of-speech sequences. We took an alternative route by building a suffix tree transducer based on a labeled data set from the UPENN tree-bank corpus. We defined I:in to be the set of possible part-of-speech tags and set I:out = {O, I}, where the output symbol given its corresponding input symbol (the part-of-speech tag of the current word) is 1 iff the word is part of a noun phrase. We used over 250, 000 marked tags and tested the performance on more than 37,000 tags. The test phase was performed by freezing the model structure, the mixture weights and the estimated parameters. The suffix tree transducer was of maximal depth 15 hence very long phrases can be statistically identified. By tresholding the output probability we classified the tags in the test data and found that less than 2.4% of the words were misclassified. A typical result is given in Table 1. We are currently investigating methods to incorporate linguistic knowledge into the model and its learning algorithm and compare the performance of the model with traditional techniques. Scmrmc:e Tcm Smith group cru..f cxcc:utiYe of U.K. metal. pos tag PNP PNP NN NN NN IN PNP NNS Class 1 1 0 1 1 1 0 1 1 Predi ction 0.99 0.99 0.01 0.98 0.98 0.98 0.02 0.99 0.99 Sentence ODd industrial material. mabr will bo:cune chainnon P~S tag CC JJ NNS NN MD VB NN Class 1 1 1 1 0 0 0 1 0 Prediction 0.67 0.96 0.99 0.96 0.03 0.03 0.01 0.81 0.01 Table 1: Extraction of noun phrases using a suffix tree transducer. In this typical example, two long noun phrases were identified correctly with high confidence. Acknowledgments Thanks to Y. Bengio, Y. Freund, F. Pereira, D. Ron. R. Schapire. and N. Tishby for helpful discussions. The work on syntactic structure induction is done in collaboration with I. Dagan and S. Engelson. This work was done while the author was at the Hebrew University of Jerusalem. References [1] Y. Bengio and P. Fransconi. An input output HMM architecture. InNIPS-7. 1994. [2] N. Cesa-Bianchi. Y. Freund. D. Haussler. D.P. Helmbold, R.E. Schapire, and M. K. Warmuth. How to use expert advice. In STOC-24, 1993. [3] T.M. Cover and J .A. Thomas. Elements of information theory. Wiley. 1991. [4] A. DeSantis. G. Markowski. and M.N. Wegman. Learning probabilistic prediction functions. In Proc. of the 1st Wksp. on Comp. Learning Theory. pages 312-328,1988. [5] C.L. Giles. C.B. Miller, D. Chen, G.Z. Sun, H.H. Chen. and Y.C. Lee. Learning and extracting finite state automata with second-orderrecurrent neural networks. Neural Computation. 4:393405.1992. [6] D. Haussler and A. Barron. How well do Bayes methods work for on-line prediction of {+ 1, -1 } values? In The3rdNEC Symp. on Comput. andCogn., 1993. [7] D.P. HeImbold and R.E. Schapire. Predicting nearly as well as the best pruning of a decision tree. In COLT-8. 1995. [8] R.A. Jacobs, M.1. Jordan. SJ. NOWlan, and G.E. Hinton. Adaptive mixture of local experts. Neural Computation, 3:79-87. 1991. [9] R.E. Krichevsky and V.K. Trofimov. The performance of universal encoding. IEEE Trans. on Inform. Theory. 1981. [10] Nick Littlestone and Manfred K. Warmuth. The weighted majority algorithm. Information and Computation, 108:212-261,1994. [11] M.D. Riley. A statistical model for generating pronounication networks. In Proc. of IEEE Con/. on Acoustics. Speech and Signal Processing. pages 737-740.1991. [12] D. Ron. Y. Singer, and N. Tishby. The power of amnesia. In NIPS-6. 1993. [13] F.MJ. Willems. Y.M. Shtarkov. and TJ. Tjalkens. The context tree weighting method: Basic properties. IEEE Trans. Inform. Theory. 41(3):653-664.1995.	1995	94
1,144	SEEMORE: A View-Based Approach to 3-D Object Recognition Using Multiple Visual Cues Bartlett W. Mel Department of Biomedical Engineering University of Southern California Los Angeles, CA 90089 mel@quake.usc.edu Abstract A neurally-inspired visual object recognition system is described called SEEMORE, whose goal is to identify common objects from a large known set-independent of 3-D viewiag angle, distance, and non-rigid distortion. SEEMORE's database consists of 100 objects that are rigid (shovel), non-rigid (telephone cord), articulated (book), statistical (shrubbery), and complex (photographs of scenes). Recognition results were obtained using a set of 102 color and shape feature channels within a simple feedforward network architecture. In response to a test set of 600 novel test views (6 of each object) presented individually in color video images, SEEMORE identified the object correctly 97% of the time (chance is 1%) using a nearest neighbor classifier. Similar levels of performance were obtained for the subset of 15 non-rigid objects. Generalization behavior reveals emergence of striking natural category structure not explicit in the input feature dimensions. 1 INTRODUCTION In natural contexts, visual object recognition in humans is remarkably fast, reliable, and viewpoint invariant. The present approach to object recognition is "view-based" (e.g. see [Edelman and Bulthoff, 1992]), and has been guided by three main dogmas. First, the "natural" object recognition problem faced by visual animals involves a large number of objects and scenes, extensive visual experience, and no artificial 866 B. W.MEL distinctions among object classes, such as rigid, non-rigid, articulated, etc. Second, when an object is recognized in the brain, the "heavy lifting" is done by the first wave of action potentials coursing from the retina to the inferotemporal cortex (IT) over a period of 100 ms [Oram and Perrett, 1992]. The computations carried out during this time can be modeled as a shallow but very wide feedforward network of simple image filtering operations. Shallow means few processing levels, wide means a sparse, high-dimensional representation combining cues from multiple visual submodalities, such as color, texture, and contour [Tanaka et al., 1991]. Third, more complicated processing mechanisms, such as those involving focal attention, segmentation, binding, normalization, mental rotation, dynamic links, parts recognition, etc., may exist and may enhance recognition performance but are not necessary to explain rapid, robust recognition with objects in normal visual situations. In this vein, the main goal of this project has been to explore the limits of performance of a shallow-but very wide-feedforward network of simple filtering operations for viewpoint-invariant 3-D object recognition, where the filter "channels" themselves have been loosely modeled after the shape- and color-sensitive visual response properties seen in the higher levels of the primate visual system [Tanaka et al., 1991]. Architecturally similar approaches to vision have been most often applied in the domain of optical character recognition [Fukushima et al., 1983, Le Cun et al., 1990]. SEEMORE'S architecture is also similar in spirit to the color histogramming approach of [Swain and Ballard, 1991], but includes spatially-structured features that provide also for shape-based generalization. Figure 1: The database includes 100 objects of many different types, including rigid (soup can), non-rigid (necktie), statistical (bunch of grapes), and photographs of complex indoor and outdoor scenes. SEEMORE: A View-based Approach to 3-D Object Recognition 867 2 SEEMORE'S VISUAL WORLD SEEMORE's database contains 100 common 3-D objects and photogaphs of scenes, each represented by a set of pre-segmented color video images (fig. 1). The training set consisted of 12-36 views of each object as follows. For rigid objects, 12 training views were chosen at roughly 60° intervals in depth around the viewing sphere, and each view was then scaled to yield a total of three images at 67%, 100%, and 150%. Image plane orientation was allowed to vary arbitrarily. For non-rigid objects, 12 training views were chosen in random poses. During a recognition trial, SEEMORE was required to identify novel test images of the database objects. For rigid objects, test images were drawn from the viewpoint interstices of the training set, excluding highly foreshortened views (e.g. bottom of can). Each test view could therefore be presumed to be correctly recognizable, but never closer than roughly 30-> in orientation in depth or 22% in scale to the nearest training view of the object, while position and orientation in the image plane could vary arbitrarily. For non-rigid objects, test images consisted of novel random poses. Each test view depicted the isolated object on a smooth background. 2.1 FEATURE CHANNELS SEEMORE's internal representation of a view of an object is encoded by a set of feature channels. The ith channel is based on an elemental nonlinear filter fi(z, y, (h, (J2, .• . ), parameterized by position in the visual field and zero or more internal degrees of freedom. Each channel is by design relatively sensitive to changes in the image that are strongly related to object identity, such as the object's shape, color, or texture, while remaining relatively insensitive to changes in the image that are unrelated to object identity, such as are caused by changes in the object's pose. In practice, this invariance is achieved in a straightfOl'ward way for each channel by subsampling and summing the output of the elemental channel filter over the entire visual field and one or more of its internal degrees of freedom, giving a channel output Fi = Lx,y,(h, ... fiO. For example, a particular shape-sensitive channel might "look" for the image-plane projections of right-angle corners, over the entire visual field, 360° of rotation in the image plane, 30° of rotation in depth, one octave in scale, and tolerating partial occlusion and/or slight misorientation of the elemental contours that define the right angle. In general, then, Fi may be viewed as a "cell" with a large receptive field whose output is an estimate of the number of occurences of distal feature i in the workspace over a large range of viewing parameters. SEEMORE'S architecture consists of 102 feature channels, whose outputs form an input vector to a nearest-neighbor classifer. Following the design of the individual channels, the channel vector F = {FI, ... F102} is (1) insensitive to changes in image plane position and orientation of the object, (2) modestly sensitive to changes in object scale, orientation in depth, or non-rigid deformation, but (3) highly sensitive to object "quality" as pertains to object identity. Within this representation, total memory storage for all views of an object ranged from 1,224 to 3,672 integers. As shown in fig. 2, SEEMORE's channels fall into in five groups: (1) 23 color channels, each of which responds to a small blob of color parameterized by "best" hue and saturation, (2) 11 coarse-scale intensity corner channels parameterized by open angle, (3) 12 "blob" features, parameterized by the shape (round and elongated) and 868 B. W.MEL size (small, medium, and large) of bright and dark intensity blobs, (4) 24 contour shape features, including straight angles, curve segments of varying radius, and parallel and oblique line combinations, and (5) 16 shape/texture-related features based on the outputs of Gabor functions at 5 scales and 8 orientations. The implementations of the channel groups were crude, in the interests of achieving a working, multiple-cue system with minimal development time. Images were grabbed using an off-the-shelf Sony S-Video Camcorder and Sun Video digitizing board. Colors o. e. oe .e oe 00 •• •• •• •• oe o Angles Blobs 0.1 o 0 c =:> Gabor-Based Features Contours sin2 +cos2 ./" 1: energy @ scale i .......... 0 2 _ energy variance @scalei 6 0 45 90 <30 >30 Figure 2: SEEMORE's 102 channels fall into 5 groups, sensitive to (1) colors, (2) intensity corners, (3) circular and elongated intensity blobs, (4) contour shape features, and (5) 16 oriented-energy and relative-orientation features based on the outputs of Gabor functions at several scales and orientations. 3 RECOGNITION SEEMORE's recognition performance was assesed quantitatively as follows. A test set consisting of 600 novel views (100 objects x 6 views) was culled from the database, and presented to SEEMORE for identification. It was noted empirically that a compressive transform on the feature dimensions (histogram values) led to improved classification performance; prior to all learning and recognition operations, SEEMORE: A View-based Approach to 3-D Object Recognition 869 Figure 3: Generalization using only shape-related channels. In each row, a novel test view is shown at far left. The sequence of best matching training views (one per object) is shown to right, in order of decreasing similarity. therefore, each feature value was replaced by its natural logarithm (0 values were first replaced with a small positive constant to prevent the logarithm from blowing up). For each test view, the city-block distance was computed to every training view in the database and the nearest neighbor was chosen as the best match. The log transform of the feature dimension:.; thus tied this distance to the ratios of individual feature values in two images rather than their differences. 4 RESULTS Recognition time on a Sparc-20 was 1-2 minutes per view; the bulk of the time was devoted to shape processing, with under 2 seconds required for matching. Recognition results are reported as the proportion of test views that were correctly classified. Performance using all 102 channels for the 600 novel object views in the intact test set was 96.7%; the chance rate of correct classification was 1%. Across recognition conditions, second-best matches usually accounted for approximately half the errors. Results were broken down in terms of the separate contributions to recognition performance of color-related vs. shape-related feature channels. Performance using only the 23 color-related channels was 87.3%, and using only the 79 shape-related channels was 79.7%. Remarkably, very similar performance figures were obtained for the subset of 90 test views of the non-rigid objects, which included several scarves, a bike chain, necklace, belt, sock, necktie, maple-leaf cluster, bunch of grapes, knit bag, and telephone cord. Thus, a novel random configuration of a telephone cord was as easily recognized as a novel view of a shovel. 870 B. W.MEL 5 GENERALIZATION BEHAVIOR Numerical indices of recognition performance are useful, but do not explicitly convey the similarity structure of the underlying feature space. A more qualitative but extremely informative representation of system performance lies in the sequence of images in order of increasing distance from a test view. Records of this kind are shown in fig. 3 for trials in which only shape-related channels were used. In each, a test view is shown at the far left, and the ordered set of nearest neighbors is shown to the right. When a test view's nearest neighbor (second image from left) was not the correct match, the trial was classified as an error. As shown in row (1), a view of a book is judged most similar to a series of other books (or the bottom of a rectangular cardboard box)---each a view of a rectangular object with high-frequency surface markings. A similar sequence can be seen in subsequent rows for (2) a series of cans, each a right cylinder with detailed surface markings, (3) a series of smooth, not-quite-round objects, (4) a series of photographs of complex scenes, and (5) a series of dinosaurs (followed by a teddy bear). In certain cases, SEEMORE'S shape-related similarity metric was more difficult to visually interpret or verbalize (last two rows), or was different from that of a human observer. 6 DISCUSSION The ecology of natural object vision gives rise to an apparent contradiction: (i) generalization in shape-space must in some cases permit an object whose global shape has been grossly perturbed to be matched to itself, such as the various tangled forms of a telephone cord, but (ii) quasi-rigid basic-level shape categories (e.g. chair, shoe, tree) must be preserved as well, and distinguished from each other. A partial It wi uti on to this conundrum lies in the observation tbat locally-cumputed shape statistics are in large part preserved under the global shape deformations that non-rigid common objects (e.g. scarf, bike-chain) typically undergo. A feature-space representation with an emphasis on locally-derived shape channels will therefore exhibit a significant degree of invariance to global nonrigid shape deformations. The definition of shape similarity embodied in the present approach is that two objects are similar if they contain similar profiles (histograms) of their shape measures, which emphasize locality. One way of understanding the emergence of global shape categories, then, such as "book", "can", "dinosaur", etc., is to view each as a set of instances of a single canonical object whose local shape statistics remain quasi-stable as it is warped into various global forms. In many cases, particularly within rigid object categories, exemplars may share longer-range shape statistics as well. It is useful to consider one further aspect of SEEMORE'S shape representation, pertaining to an apparent mismatch between the simplicity of the shape-related feature channels and the complexity of the shape categories that can emerge from them. Specifically, the order of binding of spatial relations within SEEMORE's shape channels is relatively low, i.e. consisting of single simple open or closed curves, or conjunctions of two oriented contours or Gabor patches. The fact that shape categories, such as "photographs of rooms", or "smooth lumpy objects", cluster together in a feature space of such low binding order would therefore at first seem surprising. This phenomenon relates closely to the notion of "wickelfeatures" (see [Rumelhart and McClelland, 1986], ch. 18), in which features (relating to phonemes) SEEMORE: A View-based Approach to 3-D Object Recognition 871 that bind spatial information only locally are nonetheless used to represent global patterns (words) with little or no residual ambiguity. The pre segmentation of objects is a simplifying assumption that is clearly invalid in the real world. The advantage of the assumption from a methodological perspective is that the object similarity structure induced by the feature dimensions can be studied independently from the problem of segmenting or indexing objects imbedded in complex scenes. In continuing work, we are pursuing a leap to sparse very-highdimensional space (e.g. 10,000 dimensions), whose advantages for classification in the presence of noise (or clutter) have been discussed elsewhere [Kanerva, 1988, Califano and Mohan, 1994]. Acknowledgements Thanks to J6zsef Fiser for useful discusf!ions and for development of the Gabor-based channel set, to Dan Lipofsky and Scott Dewinter for helping in the construction of the image database, and to Christof Koch for providing support at Caltech where this work was initiated. This work was funded by the Office of Naval Research, and the McDonnell-Pew Foundation. References [Califano and Mohan, 1994] Califano, A. and Mohan, R. (1994). Multidimensional indexing for recognizing visual shapes. IEEE Trans. on PAMI, 16:373-392. [Edelman and Bulthoff, 1992] Edelman, S. and Bulthoff, H. (1992). Orientation dependence in the recognition of familiar and novel views of three-dimensional objects. Vision Res., 32:2385-2400. [Fukushima et al., 1983] Fukushima, K., Miyake, S., and Ito, T. (1983). Neocognitron: A neural network model for a mechanism of visual pattern recognition. IEEE Trans. Sys. Man & Cybernetics, SMC-13:826-834. [Kanerva, 1988] Kanerva, P. (1988). Sparse distributed memory. MIT Press, Cambridge, MA. [Le Cun et al., 1990] Le Cun, Y., Matan, 0., Boser, B., Denker, J., Henderson, D., Howard, R., Hubbard, W., Jackel, L., and Baird, H. (1990). Handwritten zip code recognition with multilayer networks. In Proc. of the 10th Int. Conf. on Patt. Rec. IEEE Computer Science Press. [Oram and Perrett, 1992] Oram, M. and Perrett, D. (1992). Time course of neural responses discriminating different views of the face and head. J. Neurophysiol., 68(1) :70-84. [Rumelhart and McClelland, 1986] Rumelhart, D. and McClelland, J. (1986). Parallel distributed processing. MIT Press, Cambridge, Massachusetts. [Swain and Ballard, 1991] Swain, M. and Ballard, D. (1991). Color indexing. Int. J. Computer Vision, 7:11-32. [Tanaka et al., 1991] Tanaka, K., Saito, H., Fukada, Y., and Moriya, M. (1991). Coding visual images of objects in the inferotemporal cortex of the macaque monkey. J. Neurophysiol., 66:170-189. PART VIII APPLICATIONS	1995	95
1,145	Using Pairs of Data-Points to Define Splits for Decision Trees Geoffrey E. Hinton Department of Computer Science University of Toronto Toronto, Ontario, M5S lA4, Canada hinton@cs.toronto.edu Michael Revow Department of Computer Science University of Toronto Toronto, Ontario, M5S lA4, Canada revow@cs.toronto.edu Abstract Conventional binary classification trees such as CART either split the data using axis-aligned hyperplanes or they perform a computationally expensive search in the continuous space of hyperplanes with unrestricted orientations. We show that the limitations of the former can be overcome without resorting to the latter. For every pair of training data-points, there is one hyperplane that is orthogonal to the line joining the data-points and bisects this line. Such hyperplanes are plausible candidates for splits. In a comparison on a suite of 12 datasets we found that this method of generating candidate splits outperformed the standard methods, particularly when the training sets were small. 1 Introduction Binary decision trees come in many flavours, but they all rely on splitting the set of k-dimensional data-points at each internal node into two disjoint sets. Each split is usually performed by projecting the data onto some direction in the k-dimensional space and then thresholding the scalar value of the projection. There are two commonly used methods of picking a projection direction. The simplest method is to restrict the allowable directions to the k axes defined by the data. This is the default method used in CART [1]. If this set of directions is too restrictive, the usual alternative is to search general directions in the full k-dimensional space or general directions in a space defined by a subset of the k axes. Projections onto one of the k axes defined by the the data have many advantages 508 G. E. HINTON, M. REVOW over projections onto a more general direction: 1. It is very efficient to perform the projection for each of the data-points. We simply ignore the values of the data-point on the other axes. 2. For N data-points, it is feasible to consider all possible axis-aligned projections and thresholds because there are only k possible projections and for each of these there are at most N - 1 threshold values that yield different splits. Selecting from a fixed set of projections and thresholds is simpler than searching the k-dimensional continuous space of hyperplanes that correspond to unrestricted projections and thresholds. 3. Since a split is selected from only about N k candidates, it takes only about log2 N + log2 k bits to define the split. So it should be possible to use many more of these axis-aligned splits before overfitting occurs than if we use more general hyperplanes. If the data-points are in general position, each subset of size k defines a different hyperplane so there are N!/k!(N - k)! distinctly different hyperplanes and if k < < N it takes approximately k log2 N bits to specify one of them. For some datasets, the restriction to axis-aligned projections is too limiting. This is especially true for high-dimensional data, like images, in which there are strong correlations between the intensities of neighbouring pixels. In such cases, many axis-aligned boundaries may be required to approximate a planar boundary that is not axis-aligned, so it is natural to consider unrestricted projections and some versions of the CART program allow this. Unfortunately this greatly increases the computational burden and the search may get trapped in local minima. Also significant care must be exercised to avoid overfitting. There is, however, an intermediate approach which allows the projections to be non-axis-aligned but preserves all three of the attractive properties of axis-aligned projections: It is trivial to decide which side of the resulting hyperplane a given data-point lies on; the hyperplanes can be selected from a modest-sized set of sensible candidates; and hence many splits can be used before overfitting occurs because only a few bits are required to specify each split. 2 Using two data-points to define a projection Each pair of data-points defines a direction in the data space. This direction is a plausible candidate for a projection to be used in splitting the data, especially if it is a classification task and the two data-points are in different classes. For each such direction, we could consider all of the N - 1 possible thresholds that would give different splits, or, to save time and reduce complexity, we could only consider the threshold value that is halfway between the two data-points that define the projection. If we use this threshold value, each pair of data-points defines exactly one hyperplane and we call the two data-points the "poles" of this hyperplane. For a general k-dimensional hyperplane it requires O( k) operations to decide whether a data-point, C, is on one side or the other. But we can save a factor of k by using hyperplanes defined by pairs of data-points. If we already know the distances of C from each of the two poles, A, B then we only need to compare Using Pairs of Data Points to Define Splits for Decision Trees 509 B A Figure 1: A hyperplane orthogonal to the line joining points A and B. We can quickly determine on which side a test point, G, lies by comparing the distances AG and BG. these two distances (see figure 1).1 So if we are willing to do O(kN2) operations to compute all the pairwise distances between the data-points, we can then decide in constant time which side of the hyperplane a point lies on. As we are building the decision tree, we need to compute the gain in performance from using each possible split at each existing terminal node. Since all the terminal nodes combined contain N data-points and there are N(N - 1)/2 possible splits2 this takes time O(N3) instead of O(kN3). So the work in computing all the pairwise distances is trivial compared with the savings. Using the Minimum Description Length framework, it is clear that pole-pair splits can be described very cheaply, so a lot of them can be used before overfitting occurs. When applying MDL to a supervised learning task we can assume that the receiver gets to see the input vectors for free. It is only the output vectors that need to be communicated. So if splits are selected from a set of N (N -1) /2 possibilities that is determined by the input vectors, it takes only about 210g2 N bits to communicate a split to a receiver. Even if we allow all N - 1 possible threshold values along the projection defined by two data-points, it takes only about 310g2 N bits. So the number of these splits that can be used before overfitting occurs should be greater by a factor of about k/2 or k/3 than for general hyperplanes. Assuming that k « N, the same line of argument suggests that even more axis-aligned planes can be used, but only by a factor of about 2 or 3. To summarize, the hyperplanes planes defined by pairs of data-points are computationally convenient and seem like natural candidates for good splits. They overcome the major weakness of axis-aligned splits and, because they can be specified in a modest number of bits, they may be more effective than fully general hyperplanes when the training set is small. 1 If the threshold value is not midway between the poles, we can still save a factor of k but we need to compute (d~c - d1c )/2dAB instead of just the sign of this expression. 2Since we only consider splits in which the poles are in different classes, this number ignores a factor that is independent of N. 510 G. E. HINTON, M. REVOW 3 Building the decision tree We want to compare the "pole-pair" method of generating candidate hyperplanes with the standard axis-aligned method and the method that uses unrestricted hyperplanes. We can see no reason to expect strong interactions between the method of building the tree and the method of generating the candidate hyperplanes, but to minimize confounding effects we always use exactly the same method of building the decision tree. We faithfully followed the method described in [1], except for a small modification where the code that was kindly supplied by Leo Breiman used a slightly different method for determining the amount of pruning. Training a decision tree involves two distinct stages. In the first stage, nodes are repeatedly split until each terminal node is "pure" which means that all of its datapoints belong to the same class. The pure tree therefore fits the training data perfectly. A node is split by considering all candidate decision planes and choosing the one that maximizes the decrease in impurity. Breiman et. al recommend using the Gini index to measure impurity.3 If pUlt) is the probability of class j at node t, then the Gini index is 1 - 2:j p2(jlt). Clearly the tree obtained at the end of the first stage will overfit the data and so in the second stage the tree is pruned by recombining nodes. For a tree, Ti , with ITil terminal nodes we consider the regularized cost: (1) where E is the classification error and Q is a pruning parameter. In "weakest-link" pruning the terminal nodes are eliminated in the order which keeps (1) minimal as Q increases. This leads to a particular sequence, T = {TI' T2, ... Tk} of subtrees, in which ITII > IT21 ... > ITkl. We call this the "main" sequence of subtrees because they are trained on all of the training data. The last remaining issue to be resolved is which tree in the main sequence to use. The simplest method is to use a separate validation set and choose the tree size that gives best classification on it. Unfortunately, many of the datasets we used were too small to hold back a reserved validation set. So we always used 10-fold cross validation to pick the size of the tree. We first grew 10 different subsidiary trees until their terminal nodes were pure, using 9/10 of the data for training each of them. Then we pruned back each of these pure subsidiary trees, as above, producing 10 sequences of subsidiary subtrees. These subsidiary sequences could then be used for estimating the performance of each subtree in the main sequence. For each of the main subtrees, Ti , we found the largest tree in each subsidiary sequence that was no larger than Ti and estimated the performance of Ti to be the average of the performance achieved by each subsidiary subtree on the 1/10 of the data that was not used for training that subsidiary tree. We then chose the Ti that achieved the best performance estimate and used it on the test set4. Results are expressed as 3Impurity is not an information measure but, like an information measure, it is minimized when all the nodes are pure and maximized when all classes at each node have equal probability. 4This differs from the conventional application of cross validation, where it is used to Using Pairs of Data Points to Define Splits for Decision Trees 511 JR TR LV DB BC GL VW WN VH WV IS SN Size (N) 150 215 345 768 683 163 990 178 846 2100 351 208 Classes (e) 3 3 2 2 2 2 11 3 4 3 2 2 Attributes (k) 4 5 6 8 9 9 10 13 18 21 34 60 Table 1: Summary of the datasets used. the ratio of the test error rate to the baseline rate, which is the error rate of a tree with only a single terminal node. 4 The Datasets Eleven datasets were selected from the database of machine learning tasks maintained by the University of California at Irvine (see the appendix for a list of the datasets used). Except as noted in the appendix, the datasets were used exactly in the form of the distribution as of June 1993. All datasets have only continuous attributes and there are no missing values.5 The synthetic "waves" example [1] was added as a twelfth dataset. Table 1 gives a brief description of the datasets. Datasets are identified by a two letter abbreviation along the top. The rows in the table give the total number of instances, number of classes and number of attributes for each dataset. A few datasets in the original distribution have designated training and testing subsets while others do not. To ensure regularity among datasets, we pooled all usable examples in a given dataset, randomized the order in the pool and then divided the pool into training and testing sets. Two divisions were considered. The large training division had ~ of the pooled examples allocated to the training set and ~ to the test set. The small training division had ~ of the data in the training set and ~ in the test set. 5 Results Table 2 gives the error rates for both the large and small divisions of the data, expressed as a percentage of the error rate obtained by guessing the dominant class. In both the small and large training divisions of the datasets, the pole-pair method had lower error rates than axis-aligned or linear cart in the majority of datasets tested. While these results are interesting, they do not provide any measure of confidence that one method performs better or worse than another. Since all methods were trained and tested on the same data, we can perform a two-tailed McNemar test [2] on the predictions for pairs of methods. The resulting P-values are given in table 3. On most of the tasks, the pole-pair method is significantly better than at least one of the standard methods for at least one of the training set sizes and there are only 2 tasks for which either of the other methods is significantly better on either training set size. determine the best value of ex rather than the tree size 5In the Be dataset we removed the case identification number attribute and had to delete 16 cases with missing values. 512 G. E. HINTON, M. REVOW Database Small Train Large Train cart linear pole cart linear pole IR 14.3 14.3 4.3 5.6 5.6 5.6 TR 36.6 26.8 14.6 33.3 33.3 20.8 LV 88.9 100.0 100.0 108.7 87.0 97.8 DB 85.8 82.2 87.0 69.7 69.7 59.6 BC 12.8 14.1 8.3 15.7 12.0 9.6 GL 62.5 81.3 89.6 46.4 46.4 35.7 VW 31.8 37.7 30.0 21.4 26.2 19.2 WN 17.8 13.7 11.0 14.7 11.8 14.7 VH 42.5 46.5 44.2 36.2 43.9 40.7 WV 28.9 25.8 24.3 30.6 24.8 26.6 IS 44.0 31.0 41.7 21.4 23.8 42.9 SN 65.2 71.2 48.5 48.4 45.2 48.4 Table 2: Relative error rates expressed as a percentage of the baseline rate on the small and large training sets. 6 Discussion We only considered hyperplanes whose poles were in different classes, since these seemed more plausible candidates. An alternative strategy is to disregard class membership, and consider all possible pole-pairs. Another variant of the method arises depending on whether the inputs are scaled. We transformed all inputs so that the training data has zero mean and unit variance. However, using unsealed inputs and/or allowing both poles to have the same class makes little difference to the overall advantage of the pole-pair method. To summarize, we have demonstrated that the pole-pair method is a simple, effective method for generating projection directions at binary tree nodes. The same idea of minimizing complexity by selecting among a sensible fixed set of possibilities rather than searching a continuous space can also be applied to the choice of input-tohidden weights in a neural network. A Databases used in the study IR - Iris plant database. TR - Thyroid gland data. LV - BUPA liver disorders. DB - Pima Indians Diabetes. BC - Breast cancer database from the University of Wisconsin Hospitals. GL - Glass identification database. In these experiments we only considered the classification into float/nonfloat processed glass, ignoring other types of glass. VW - Vowel recognition. WN - Wine recognition. VH - Vehicle silhouettes. WV - Waveform example, the synthetic example from [1]. IS - Johns Hopkins University Ionosphere database. SN - Sonar - mines versus rocks discrimination. We did not control for aspect-angle. Using Pairs of Data Points to Define Splits for Decision Trees 513 Small Training - Large Test IR TR LV DB BC GL VW WN VH WV IS -SN Axis- Pole .02 ~ .18 .46 .06 .02 .24 .15 .33 ,QQ.. .44 .07 Linear- Pole ~ .13 1.0 .26 ~ .30 .00 .41 .27 .17 .09 ~ Axis-Linear 1.0 .06 .18 .30 .40 J>O J>O .31 .08 .03 ~ .32 Large Training - Small Test IR TR LV DB BC GL VW WN VH WV IS SN Axis-Pole .75 .23 .29 :.Q!.. .11 .29 .26 .69 .14 .08 :02 .60 Linear-Pole .75 .23 .26 :.Q!.. .25 .30 J!!... .50 .25 .26 :os .50 Axis-Linear 1.0 1.0 .07 1.0 .29 .69 .06 .50 F3"" :.Q!. .50 .50 Table 3: P-Values using a two-tailed McNemar test on the small (top) and large (bottom) training sets. Each row gives P-values when the methods in the left most column are compared. A significant difference at the P = 0.05 level is indicated with a line above (below) the P-value depending on whether the first (second) mentioned method in the first column had superior performance. For example, in the top most row, the pole-pair method was significantly better than the axis-aligned method on the TR dataset. Acknowledgments We thank Leo Breiman for kindly making his CART code available to us. This research was funded by the Institute for Robotics and Intelligent Systems and by NSERC. Hinton is a fellow of the Canadian Institute for Advanced Research. References [1] L. Breiman, J. H. Freidman, R. A. Olshen, and C. J. Stone. Classification and regression trees. Wadsworth international Group, Belmont, California, 1984. [2] J. L. Fleiss. Statistical methods for rates and proportions. Second edition. Wiley, 1981.	1995	96
1,146	Dynamics of On-Line Gradient Descent Learning for Multilayer Neural Networks David Saad" Dept. of Compo Sci. & App. Math. Sara A. Solla t CONNECT, The Niels Bohr Institute Blegdamsdvej 17 Copenhagen 2100, Denmark Aston University Birmingham B4 7ET, UK Abstract We consider the problem of on-line gradient descent learning for general two-layer neural networks. An analytic solution is presented and used to investigate the role of the learning rate in controlling the evolution and convergence of the learning process. Learning in layered neural networks refers to the modification of internal parameters {J} which specify the strength of the interneuron couplings, so as to bring the map fJ implemented by the network as close as possible to a desired map 1. The degree of success is monitored through the generalization error, a measure of the dissimilarity between fJ and 1. Consider maps from an N-dimensional input space e onto a scalar (, as arise in the formulation of classification and regression tasks. Two-layer networks with an arbitrary number of hidden units have been shown to be universal approximators [1] for such N-to-one dimensional maps. Information about the desired map i is provided through independent examples (e, (1'), with (I' = i(e) for all p . The examples are used to train a student network with N input units, K hidden units, and a single linear output unit; the target map i is defined through a teacher network of similar architecture except for the number M of hidden units. We investigate the emergence of generalization ability in an on-line learning scenario [2], in which the couplings are modified after the presentation of each example so as to minimize the corresponding error. The resulting changes in {J} are described as a dynamical evolution; the number of examples plays the role of time. In this paper we limit our discussion to the case of the soft-committee machine [2], in which all the hidden units are connected to the output unit with positive couplings of unit strength, and only the input-to-hidden couplings are adaptive. D.Saad@aston.ac.uk tOn leave from AT&T Bell Laboratories, Holmdel, NJ 07733, USA Dynamics of On-line Gradient Descent Learning for Multilayer Neural Networks 303 Consider the student network: hidden unit i receives information from input unit r through the weight hr, and its activation under presentation of an input pattern ~ = (6,· .. ,~N) is Xi = J i .~, with J i = (hl, ... ,JiN) defined as the vector of incoming weights onto the i-th hidden unit. The output of the student network is a(J,~) = L:~l 9 (Ji . ~), where 9 is the activation function of the hidden units, taken here to be the error function g(x) == erf(x/V2), and J == {Jdl<i<K is the set of input-to-hidden adaptive weights. - Training examples are of the form (e, (Il) . The components of the independently drawn input vectors ~Il are un correlated random variables with zero mean and unit variance. The corresponding output (Il is given by a deterministic teacher whose internal structure is the same as for the student network but may differ in the number of hidden units. Hidden unit n in the teacher network receives input information through the weight vector Bn = (Bnl , ... , BnN), and its activation under presentation of the input pattern e is Y~ = Bn . e. The corresponding output is (Il = L:~=l 9 (Bn ·e). We will use indices i,j,k,l ... to refer to units in the student network, and n, m, ... for units in the teacher network. The error made by a student with weights J on a given input ~ is given by the quadratic deviation (1) Performance on a typical input defines the generalization error Eg(J) < E(J ,~) >{O through an average over all possible input vectors ~, to be performed implicitly through averages over the activations x = (Xl"'" XK) and Y = (YI I ' •• I YM). Note that both < Xi >=< Yn >= 0; second order correlations are given by the overlaps among the weight vectors associated with the various hidden units: < Xi Xk > = J i . Jk == Qikl < Xi Yn > = J i . Bn == Rin, and < Yn Ym > = Bn . Bm == Tnm. Averages over x and yare performed using the resulting multivariate Gaussian probability distribution, and yield an expression for the generalization error in terms of the parameters Qik l Rin, and Tnm [3]. For g(x) == erf(x/V2) the result is: 1 {,"" Qik ,"" Tnm L...J arCSlll + L...J arCSlll --;:;=:=;:;;;=---;:::=:::::::;:;;;:== 7r'k V1+Qii V1+Qu V1+Tnn V1+Tmm z nm 2 ,"" Rin } L...J arCSlll . . V1 + Qi; V1 + Tnn sn (2) The parameters Tnm are characteristic of the task to be learned and remain fixed. The overlaps Qik and Rin, which characterize the correlations among the various student units and their degree of specialization towards the implementation of the desired task, are determined by the student weights J and evolve during training. A gradient descent rule for the update of the student weights results in Jf+l = Jf + N bf e, where the learning rate TJ has been scaled with the input size N, and or "g'(xf) [~g(y~) - ~g(xj')l (3) is defined in terms of both the activation function 9 and its derivative g'. The time evolution of the overlaps Rin and Qik can be explicitly written in terms of similar 304 D. SAAD. S. A. SOLLA difference equations. In the large N limit the normalized number of examples Q' = piN can be interpreted as a continuous time variable, leading to the equations of motion (4) to be averaged over all possible ways in which an example can be chosen at a given time st.ep. The dependence on the current input e is only through the activations x and y; the corresponding averages can be performed analytically for g(x) = erf( x I v'2), resulting in a set of coupled first-order differential equations [3]. These dynamical equations are exact, and provide a novel tool used here to analyze the learning process for a general soft-committee machine with an arbitrary number ]{ of hidden units, trained to implement a task defined through a teacher of similar architecture except for the number M of hidden units. In what follows we focus on uncorrelated teacher vectors of unit length, Tnm = onm. The time evolution of the overlaps Rin and Qik follows from integrating the equations of motion (4) from initial conditions determined by a random initialization of the student vectors {Jdl<i<K. Random initial norms Qii for the student vectors are taken here from a unIform distribution in the [0,0.5] interval. Overlaps Qik between independently chosen student vectors Ji and Jk, or ~n between J i and an unknown teacher vector Bn are small numbers, of order 1/VN for N ~ ]{, M, and taken here from a uniform distribution in the [0,10- 12] interval. We show in Fig. 1a-c the evolution of the overlaps and generalization error for a realizable case: ]{ = M = 3 and "I = 0.1. This example illustrates the successive regimes of the learning process. The system quickly evolves into a symmetric subspace controlled by an unstable suboptimal solution which exhibits no differentiation among the various student hidden units. Trapping in the symmetric subspace prevents the specialization needed to achieve the optimal solution, and the generalization error remains finite, as shown by the plateau in Fig. 1c. The symmetric solution is unstable, and the perturbation introduced through the random initialization of the overlaps ~n eventually takes over: the student units become specialized and the matrix R of student-teacher overlaps tends towards the matrix T, except for a permutational symmetry associated with the arbitrary labeling of the student hidden units. The generalization error plateau is followed by a monotonic decrease towards zero once the specialization begins and the system evolves towards the optimal solution. The evolution of the overlaps and generalization error for the unrealizable case ]{ < M is characterized by qualitatively similar stages, except that the asymptotic behavior is controlled by a suboptimal solution which reflects the differences between student and teacher architectures. Curves for the time evolution of the generalization error for different values of "I shown in Fig. 1d for ]{ = M = 3 identify trapping in the symmetric subspace as a small "I phenomenon. We therefore consider the equations of motion (4) in the small "I regime. The term proportional to "12 is neglected and the resulting truncated equations of motion are used to investigate a phase characterized by students of similar norms: Qii = Q for all 1 ~ i ~ ]{, similar correlations among themselves: Qik = C for all i 1= k, and similar correlations with the teacher vectors: R in = R for all 1 ~ i ~ ]{, 1 ~ n ~ M. The resulting dynamical equations exhibit a fixed point solution at M M - ]{2 + ..j]{4 ]{2 + M2 rcr Q" = C" = ]{2 2M _ 1 and R" = V M (5) Dynamics of On-line Gradient Descent Learning for Multilayer Neural Networks 305 (a) (b) LOr1.0r-......... R" --R'2 / O.BO.BR, J .. R2, , 1 i 0., --0'2 t ---- R2 2 ........ R2, 5 .!:I:: 0.6t ~ 0.6- - . Rl1 - .-- RJ 2 ..... . .... I ---- 0, J ~~~~- g:: J a ......... O2 J ~ ---- R" R 6 0.40.4? ~. "j' --' ~ 0.2 ~--" 0.20.0 0.0 0 2000 4000 6000 BOOO 0 2000 4000 6000 BOOO ex ex (c) (d) 0.1O.OB11 0.1 O.OB11 0.3 bO 0.06-·- · ·110.5 ~ 0.06bO --110.7 W 0.04-f-----, 0.04 'r--"-' , .. ...... 0.02i ..... \ .. 0.02 \ i I 0.0 0.0 I 0 2000 4000 6000 BOOO 0 2000 4000 6000 ex ex Figure 1: Dependence of the overlaps and the generalization error on the normalized number of examples Q' for a three-node student learning a three-node teacher characterized by Tnm = onm. Results for TJ = 0.1 are shown for (a) student-student overlaps Qik and (b) student-teacher overlaps Rin . The generalization error is shown in (c) , and again in (d) for different values of the learning rate. for the general case, which reduces to Q C* 1 - 2K-1 and R* _ rcr _ 1 - V K - VK(2K -1) (6) in the realizable case (K = M), where the corresponding generalization error is given by E; = ~ {i - K arcsin (2~{ )} . (7) A simple geometrical picture explains the relation Q* = C* = M(R)2 at the symmetric fixed point. The learning process confines the student vectors {Jd to the subspace SB spanned by the set of teacher vectors {Bn} . For Tnm = onm the teacher vectors form an orthonormal set: Bn = en , with en . em = Onm for 1 :::; n , m :::; M , and provide an expansion for the weight vectors of the trained student: Ji = Ln Rinen . The student-teacher overlaps Rin are independent of i in the symmetric phase and independent of n for an isotropic teacher: Rin = R" for all 1 :::; i :::; K and 1 :::; n :::; M. The expansion Ji = R Ln en for all i results in Q* = C* = M(R)2. 306 D. SAAD, S. A. SOLLA The length of the symmetric plateau is controlled by the degree of asymmetry in the initial conditions [2] and by the learning rate "I. The small "I analysis predicts trapping times inversely proportional to "I, in quantitative agreement with the shrinking plateau of Fig. 1d. The increase in the height of the plateau with decreasing "I is a second order effect, as the truncated equations of motion predict a unique value: f; = 0.0203 for K = M = 3. The mechanism for the second order effect is revealed by an examination of Fig. 1a: the student-student overlaps do agree with the prediction C" = 0.2 of the small "I analysis for K = M = 3, but the norms of the student vectors remain larger, at Q = Q" +~ . The gap ~ between diagonal and off-diagonal elements is observed numerically to increase with increasing "I, and is responsible for the excess generalization error. A first order expansion in ~ at R = R", C = C .. , and Q = Q" + ~ yields t = - - l\. arcsm + K{7r , . (1) g 7r 6 2K 2K -1 } 2K + 1 ~ , in agreement with the trend observed in Fig. 1d for the realizable case. (8) The excess norm ~ of the student vectors corresponds to a residual component in J i not confined to the subspace SB. The weight vectors of the trained student can be written as Ji = R" Ln en + Jt, with Jt . en = 0 for all 1 :s n :s M. Student weight vectors are not constrained to be identical; they differ through orthogonal components Jt which are typically uncorrelated: Jt·J t = 0 for i =1= k. Correlations Qik = C do satisfy C = C .. = M(R .. )2, but norms Qii = Q are given by Q = Q"+~, with ~ =11 J.L 112. Learning at very small "I tends to eliminate J.L and confine the student vectors to SB. Escape from the symmetric subspace signals the onset of hidden unit specialization. As shown in Fig. 1b, the process is driven by a breaking of the uniformity of the student-teacher correlations: each student node becomes increasingly specialized to a specific teacher node, while its overlap with the remaining teacher nodes decreases and eventually decays to its asymptotic value. In the realizable case this asymptotic value is zero, while in the unrealizable case two different non-zero asymptotic values distinguish weak overlaps with teacher nodes imitated by other student vectors from more significant overlaps with those teacher nodes not specifically imitated by any of the student vectors. The matrix of student-teacher overlaps can no longer be characterized by a unique parameter, as we need to distinguish between a dominant overlap R between a given student node and the teacher node it begins to imitate, secondary overlaps S between the same student node and the teacher nodes to which other student nodes are being assigned, and residual overlaps U with the remaining teacher nodes. The student hidden nodes can be relabeled so as to bring the matrix of student-teacher overlaps to the form Rin = RDin + S(l - Din)8(K - n) + U(l - 8(K - n)), where the step function 8 is 0 for negative arguments and 1 otherwise. The emerging differentiation among student vectors results in a decrease of the overlaps Qik = C for i =1= k, while their norms Qii = Q increase. The matrix of student-student overlaps takes the form Qik = QDik + C(l - Dik). Here we limit our description of the onset of specialization to the realizable case, for which Rin = RDin +S(l-Din). The small "I analysis is extended to allow for S =1= R in order to describe the escape from the symmetric subspace. The resulting dynamical equations are linearized around the fixed point solution at Q" = C .. = 1/(2K - 1) and R" = S .. = 1/ J K (2K - 1), and the generalization error is expanded around its fixed point value (7) to first order in the corresponding deviations q, c, r, and s. The analysis identifies a relevant perturbation with q = c = 0 and s = -r /(K -1), which Dynamics of On-line Gradient Descent Learning for Multilayer Neural Networks 307 Figure 2: Dependence of the two leading decay eigenvalues on the learning rate 'fJ in the realizable case: A1 (curved line) and A2 (straight line) are shown for M = J{ = 3. 0.3-.--- ------------, 0.2 t<. 0.1 -0.1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 leaves the generalization error unchanged and explains the behavior illustrated in Fig. la-b. It is the differentiation between Rand S which signals the escape from the symmetric subspace; the differentiation between Q and C occurs for larger values of Q'. The relevant perturbation corresponds to an enhancement of the overlap R = R + r between a given student node and the teacher node it is learning to imitate, while the overlap S = S* + 5 between the same student node and the remaining teacher nodes is weakened. The time constant associated with this mode is T = (7r/2J{)(2J{ - 1)1/2(2J{ + 1)3/2, with T""" 27rJ{ in the large J{ limit. It is in the subsequent convergence to an asymptotic solution that the realizable and unrealizable cases exhibit fundamental differences. We examine first the realizable scenario, in which the system converges to an optimal solution with perfect generalization. As the specialization continues, the dominant overlaps R grow, and the secondary overlaps S decay to zero. Further specialization involves the decay to zero of the student-student correlations C and the growth of the norms Q of the student vectors. To investigate the convergence to the optimal solution we linearize the equations of motion around the asymptotic fixed point at S* = C" = 0, R* = Q* = 1, with f; = o. We describe convergence to the optimal solution by applying the full equations of motion (4) to a phase characterized by Rin = Rbin + S(l - bin) and Qik = Qbik + C(l - bin). Linearization of the full equations of motion around the asymptotic fixed point results in four eigenvalues; the dependence of the two largest eigenvalues on 'fJ is shown in Fig. 2 for M = J{ = 3. An initially slow mode corresponds to the eigenvalue A2, which remains negative for all values of 'fJ, while the eigenvalue A1 for the initially fast mode becomes positive as 'fJ exceeds 'fJmax, given by 7r 75 - 42V3 'fJmax = J{ 25V3 - 42 (9) to first order in 1/ J{. The optimal solution with f* = 0 is not accessible for 'fJ > 'fJmax· Exponential convergence of R, S, C, ana Q to their optimal values is guaranteed for all learning rates in the range (0, 'fJmax) ; in this regime the generalization error decays exponentially to f; = 0, with a rate controlled by the slowest decay mode. An expansion of fg in terms of r = 1 - R, 5 , c, and q = 1 - Q reveals that of the leading modes whose eigenvalues are shown in Fig. 2 only the mode associated with A1 contributes to the decay of the linear term, while the decay of the second order term is controlled by the mode associated with A2 and dominates the convergence if 2A2 < A1. The learning rate 'fJopt which guarantees fastest asymptotic decay for the generalization error follows from A1('fJopt) = 2A2('fJopt). The asymptotic convergence of unrealizable learning is an intrinsically more complicated process that cannot be described in closed analytic form. The asymptotic 308 D. SAAD. S. A. SOLLA values of the order parameters and the generalization error depend on the learning rate TJ; convergence to an optimal solution with minimal generalization error requires TJ --+ 0 as a --+ 00. Optimal values for the order parameters follow from a small TJ analysis, equivalent to neglecting J.L and assuming student vectors confined to SB. The resulting expansion J i = 2:~=1 Hinen, with Rii = R, Hin = S for 1 :::; n :::; J{, n 1= i, and Hin = U for J{ + 1 :s n :::; M, leads to Q = R2 + (I< -1)S2 + (M - J{)U 2 , C = 2RS + (I< - 2)S2 = (M - J{)U2 . (10) The equations of motion for the remaining parameters R, S, and U exhibit a fixed point solution which controls the asymptotic behavior. This solution cannot be obtained analytically, but numerical results are well approximated to order (1/ J{3) by R* 6\1'3 - 3 L ( 1 ) 1 8 J{2 1 - J{ , S* (1 --:) :3' w = ~ (1 -2~' ) (11) where L == M - J{. The corresponding fixed point values Q" and C" follow from Eq. (10). Note that R" is lower than for the realizable case, and that correlations U" (significant) and S .. (weaker) between student vectors and the teacher vectors they do not imitate are not eliminated. The asymptotic generalization error is given by (12) to order (1/ J{2). Note its proportionality to the mismatch L between teacher and student architectures. Learning at fixed and sufficiently small TJ results in exponential convergence to an asymptotic solution whose fixed point coordinates are shifted from the values discussed above. The solution is suboptimal; the resulting increase in f; from its optimal value (12) is easily obtained to first order in TJ , and it is also proportional to L. We have investigated convergence to the optimal solution (12) for schedules of the form TJ(a) = TJo/(a-ao)Z for the decay of the learning rate. A constant rate TJo is used for a :::; 0'0; the monotonic decrease of TJ for a > 0'0 is switched on after specialization begins. Asymptotic convergence requires 0 < z :::; 1; fastest decay of the generalization error is achieved for z = 1/2. Specialization as described here and illustrated in Fig.l is a simultaneous process in which each student node acquires a strong correlation with a specific teacher node while correlations to other teacher nodes decrease. Such synchronous escape from the symmetric phase is characteristic of learning scenarios where the target task is defined through an isotropic teacher. In the case of a graded teacher we find that specialization occurs through a sequence of escapes from the symmetric subspace, ordered according to the relevance of the corresponding teacher nodes [3]. Acknowledgement The work was supported by the EU grant CHRX-CT92-0063. References [1] G. Cybenko, Math. Control Signals and Systems 2, 303 (1989). [2] M. Biehl and H. Schwarze, J. Phys. A 28, 643 (1995). [3] D. Saad and S. A. Solla, Phys. Rev. E, 52,4225 (1995).	1995	97
1,147	Geometry of Early Stopping in Linear Networks Robert Dodier * Dept. of Computer Science University of Colorado Boulder, CO 80309 Abstract A theory of early stopping as applied to linear models is presented. The backpropagation learning algorithm is modeled as gradient descent in continuous time. Given a training set and a validation set, all weight vectors found by early stopping must lie on a certain quadric surface, usually an ellipsoid. Given a training set and a candidate early stopping weight vector, all validation sets have least-squares weights lying on a certain plane. This latter fact can be exploited to estimate the probability of stopping at any given point along the trajectory from the initial weight vector to the leastsquares weights derived from the training set, and to estimate the probability that training goes on indefinitely. The prospects for extending this theory to nonlinear models are discussed. 1 INTRODUCTION 'Early stopping' is the following training procedure: Split the available data into a training set and a "validation" set. Start with initial weights close to zero. Apply gradient descent (backpropagation) on the training data. If the error on the validation set increases over time, stop training. This training method, as applied to neural networks, is of relatively recent origin. The earliest references include Morgan and Bourlard [4] and Weigend et al. [7]. * Address correspondence to: dodier~cs. colorado . edu 366 R. DODIER Finnoff et al. [2] studied early stopping empirically. While the goal of a theory of early stopping is to analyze its application to nonlinear approximators such as sigmoidal networks, this paper will deal mainly with linear systems and only marginally with nonlinear systems. Baldi and Chauvin [1] and Wang et al. [6] have also analyzed linear systems. The main result of this paper can be summarized as follows. It can be shown (see Sec. 5) that the most probable stopping point on a given trajectory (fixing the training set and initial weights) is the same no matter what the size of the validation set. That is, the most probable stopping point (considering all possible validation sets) for a finite validation set is the same as for an infinite validation set. (If the validation data is unlimited, then the validation error is the same as the true generalization error.) However, for finite validation sets there is a dispersion of stopping points around the best (most probable and least generalization error) stopping point, and this increases the expected generalization error. See Figure 1 for an illustration of these ideas. 2 MATHEMATICAL PRELIMINARIES In what follows, backpropagation will be modeled as a process in continuous time. This corresponds to letting the learning rate approach zero. This continuum model simplifies the necessary algebra while preserving the important properties of early stopping. Let the inputs be denoted X = (Xij), so that Xij is the j'th component of the i'th observation; there are p components of each of the n observations. Likewise, let y = (Yi) be the (scalar) outputs observed when the inputs are X. Our regression model will be a linear model, Yi = W'Xi + fi, i = 1, ... , n. Here fi represents independent, identically distributed (LLd.) Gaussian noise, fi rv N(O, q2). Let E(w) = !IIXw - Yll2 be one-half the usual sum of squared errors. The error gradient with respect to the weights is \7 E(w) = w'x'x - y'X. The backprop algorithm is modeled as Vi = -\7 E( w). The least-squares solution, at which \7E(w) = 0, is WLS = (X'X)-lX'y. Note the appearence here of the input correlation matrix, X'X = (2:~=1 XkiXkj). The properties of this matrix determine, to a large extent, the properties of the least-squares solutions we find. It turns out that as the number of observations n increases without bound, the matrix q2(X'X)-1 converges with probability one to the population covariance matrix of the weights. We will find that the correlation matrix plays an important role in the analysis of early stopping. We can rewrite the error E using a diagonalization of the correlation matrix X'X = SAS'. Omitting a few steps of algebra, p E(w) = ! L AkV~ + !y'(y - XWLS) (1) k=l where v = S'(W-WLS) and A = diag(Al, .. . , Ap). In this sum we see that the magnitude of the k'th term is proportional to the corresponding characteristic value, so moving w toward w LS in the direction corresponding to the largest characteristic value yields the greatest reduction of error. Likewise, moving in the direction corresponding to the smallest characteristic value gives the least reduction of error. Geometry of Early Stopping in Linear Networks 367 So far, we have implicitly considered only one set of data; we have assumed all data is used for training. Now let us distinguish training data, X t and Yt, from validation data, Xv and Yv ; there are nt training and nv validation data. Now each set of data has its own least-squares weight vector, Wt and Wv , and its own error gradient, \lEt(w) and \lEv(w). Also define M t = X~Xt and Mv = X~Xv for convenience. The early stopping method can be analyzed in terms of the these pairs of matrices, gradients, and least-squares weight vectors. 3 THE MAGIC ELLIPSOID Consider the early stopping criterion, d~v (w) = O. Applying the chain rule, dEv = dEv . dw = \lE . -\lE dt dw dt v t, (2) where the last equality follows from the definition of gradient descent. So the early stopping criterion is the same as saying \lEt' \lEv = 0, (3) that is, at an early stopping point, the training and validation error gradients are perpendicular, if they are not zero. Consider now the set of all points in the weight space such that the training and validation error gradients are perpendicular. These are the points at which early stopping may stop. It turns out that this set of points has an easily described shape. The condition given by Eq. 3 is equivalent to (4) Note that all correlation matrices are symmetric, so MtM~ = MtMv. We see that Eq. 4 gives a quadratic form. Let us put Eq. 4 into a standard form. Toward this end, let us define some useful terms. Let M = MtMv, M = HM + M') = HMtMv + MvMt), Vi HWt + wv ), ~w Wt - Wv , and ~ IM-1(M ')~ w=w-i -M w. Now an important result can be stated. The proof is omitted. Proposition 1. \lEt . \lEv = 0 is equivalent to (5) (6) (7) (8) (9) (W - w)'M(w - w) = t~w[M + t(M' - M)M-1(M - M')l~w. 0 (10) The matrix M of the quadratic form given by Eq. 10 is "usually" positive definite. As the number of observations nt and nv of training and validation data increase without bound, M converges to a positive definite matrix. In what follows it will 368 R. DODIER always be assumed that M is indeed positive definite. Given this, the locus defined by V' Et .1 V' Ev is an ellipsoid. The centroid is W, the orientation is determined by the characteristic vectors of M, and the length of the k'th semiaxis is v' c/ Ak, where c is the constant on the righthand side of Eq. 10 and 'xk is the k'th characteristic value of M. 4 THE MAGIC PLANE Given the least-squares weight vector Wt derived from the training data and a candidate early stopping weight vector Wes, any least-squares weight vector Wv from a validation set must lie on a certain plane, the 'magic plane.' The proof of this statement is omitted. Proposition 2. The condition that Wt, W v, and Wes all lie on the magic ellipsoid, (Wt -w)/M(wt -w) = (wv -w)/M(wv -w) = (wes -wYM(wes -w) = c, (11) implies (Wt - wes)/Mwv = (Wt - wes)/Mwes. 0 (12) This shows that Wv lies on a plane, the magic plane, with normal M/(wt - wes). The reader will note a certain difficulty here, namely that M = MtMv depends on the particular validation set used, as does W v. However, we can make progress by considering only a fixed correlation matrix Mv and letting W v vary. Let us suppose the inputs (Xl, X2, •• . ,Xp) are LLd. Gaussian random variables with mean zero and some covariance E. (Here the inputs are random but they are observed exactly, so the error model y = w/x + € still applies.) Then (Mv) = (X~Xv) = nvE, so in Eq. 12 let us replace Mv with its expected value nv:E. That is, we can approximate Eq. 12 with (13) Now consider the probability that a particular point w(t) on the trajectory from w(O) to Wt is an early stopping point, that is, V' Et(w(t)) . V' Ev(w(t)) = O. This is exactly the probability that Eq. 12 is satisfied, and approximately the probability that Eq. 13 is satisfied. This latter approximation is easy to calculate: it is the mass of an infinitesimally-thin slab cutting through the distribution of least-squares validation weight vectors. Given the usual additive noise model y = w/x + € with € being Li.d. Gaussian distributed noise with mean zero and variance (f2, the leastsquares weights are approximately distributed as (14) when the number of data is large. Consider now the plane n = {w : Wi ft = k}. The probability mass on this plane as it cuts through a Gaussian distribution N(/-t, C) is then pn(k, ft) = (27rft/Cft)-1/2 exp( _~ (k ~~:)2) ds (15) where ds denotes an infinitesimal arc length. (See, for example, Sec. VIII-9.3 of von Mises [3].) Geometry of Early Stopping in Linear Networks 0.2S,------r-~-~-~-~__,_-_r_-_, 0.15 0.' O.Os ~~~~~L-lli3~~.ll-~S~~~~--~ Arc leng1h Along Trajectory 369 Figure 1: Histogram of early stopping points along a trajectory, with bins of equal arc length. An approximation to the probability of stopping (Eq. 16) is superimposed. Altogether 1000 validation sets were generated for a certain training set; of these, 288 gave "don't start" solutions, 701 gave early stopping solutions (which are binned here) somewhere on the trajectory, and 11 gave "don't stop" solutions. 5 PROBABILITY OF STOPPING AT A GIVEN POINT Let us apply Eq. 15 to the problem at hand. Our normal is ft = nv :EMt (w t - Wes ) and the offset is k = ft' W es. A formal statement of the approximation of PO can now be made. Proposition 3. Assuming the validation correlation matrix X~Xv equals the mean correlation matrix nv~, the probability of stopping at a point Wes = w(t) on the trajectory from w(O) to Wt is approximately with (17) How useful is this approximation? Simulations were carried out in which the initial weight vector w(O) and the training data (nt = 20) were fixed, and many validation sets of size nv = 20 were generated (without fixing X~Xv). The trajectory was divided into segments of equal length and histograms of the number of early stopping weights on each segment were constructed. A typical example is shown in Figure 1. It can be seen that the empirical histogram is well-approximated by Eq. 16. If for some w(t) on the trajectory the magic plane cuts through the true weights w·, then Po will have a peak at t. As the number of validation data nv increases, the variance of Wv decreases and the peak narrows, but the position w(t) of the peak does not move. As nv -t 00 the peak becomes a spike at w(t). That is, the peak of Po for a finite validation set is the same as if we had access to the true generalization error. In this sense, early stopping does the right thing. It has been observed that when early stopping is employed, the validation error may decrease forever and never rise - thus the 'early stopping' procedure yields the least-squares weights. How common is this phenomenon? Let us consider a fixed 370 R. DODIER training set and a fixed initial weight vector, so that the trajectory is fixed. Letting the validation set range over all possible realizations, let us denote by Pn(t) = Pn(k(t), n(t)) the probability that training stops at time t or later. 1- Pn(O) is the probability that validation error rises immediately upon beginning training, and let us agree that Pn(oo) denotes the probability that validation error never increases. This Pn(t) is approximately the mass that is "behind" the plane n'wv = n'wes , "behind" meaning the points Wv such that (wv - wes)'ft < O. (The identification of Pn with the mass to one side of the plane is not exact because intersections of magic planes are ignored.) As Eq. 15 has the form of a Gaussian p.dJ., it is easy to show that -nw ( k A' "') Pq(k, ft) = G (n'Cft)1/2 (18) where G denotes the standard Gaussian c.dJ., G(z) = (211')-1/2 J~oo exp( -t2 /2)dt. Recall that we take the normal ft of the magic plane through Wes as ft = EMt(wtwes). For t = 0 there is no problem with Eq. 18 and an approximation for the "never-starting" probability is stated in the next proposition. Proposition 4. The probability that validation error increases immediately upon beginning training ("never starting"), assuming the validation correlation matrix X~Xv equals the mean correlation matrix nv:E, is approximately 1 - Pn(O) = 1 - G (Fv (w'" - w(O))'MtE(wt - w(O)) ). 0 (19) U [(Wt - w(O))'MtEMt(wt - w(0))P/2 With similar arguments we can develop an approximation to the "never-stopping" probability. Proposition 5. The probability that training continues indefinitely ("never stopping"), assuming the validation correlation matrix X~Xv equals the mean correlation matrix nvE, is approximately Pn(oo) = G (Fv (w'" - Wt)'Mt:E(±S"')) . U A"'[(s"')'Es"'j1/2 (20) In Eq. 20 pick +s'" if (Wt - w(O))'s'" > 0, otherwise pick -s"'. 0 Simulations are in good agreement with the estimates given by Propositions 4 and 5. 6 EXTENDING THE THEORY TO NONLINEAR SYSTEMS It may be possible to extend the theory presented in this paper to nonlinear approximators. The elementary concepts carryover unchanged, although it will be more difficult to describe them algebraically. In a nonlinear early stopping problem, there will be a surface corresponding to the magic ellipsoid on which 'VEt ...L 'V Ev , but this surface may be nonconvex or not simply connected. Likewise, corresponding to the magic plane there will be a surface on which least-squares validation weights must fall, but this surface need not be fiat or unbounded. It is customary in the world of statistics to apply results derived for linear systems to nonlinear systems by assuming the number of data is very large and various Geometry of Early Stopping in Linear Networks 371 regularity conditions hold. If the errors £. are additive, the least-squares weights again have a Gaussian distribution. As in the linear case, the Hessian of the total error appears as the inverse of the covariance of the least-squares weights. In this asymptotic (large data) regime, the standard results for linear regression carryover to nonlinear regression mostly unchanged. This suggests that the linear theory of early stopping will also apply to nonlinear regression models, such as sigmoidal networks, when there is much data. However, it should be noted that the asymptotic regression theory is purely local - it describes only what happens in the neighborhood of the least-squares weights. As the outcome of early stopping depends upon the initial weights and the trajectory taken through the weight space, any local theory will not suffice to analyze early stopping. Nonlinear effects such as local minima and non-quadratic basins cannot be accounted for by a linear or asymptotically linear theory, and these may play important roles in nonlinear regression problems. This may invalidate direct extrapolations of linear results to nonlinear networks, such as that given by Wang and Venkatesh [5]. 7 ACKNOWLEDGMENTS This research was supported by NSF Presidential Young Investigator award IRl9058450 and grant 90-21 from the James S. McDonnell Foundation to Michael C. Mozer. References [1] Baldi, P., and Y. Chauvin. "Temporal Evolution of Generalization during Learning in Linear Networks," Neural Computation 3, 589-603 (Winter 1991). [2] Finnoff, W., F. Hergert, and H. G. Zimmermann. "Extended Regularization Methods for Nonconvergent Model Selection," in Advances in NIPS 5, S. Hanson, J. Cowan, and C. L. Giles, eds., pp 228-235. San Mateo, CA: Morgan Kaufmann Publishers. 1993. [3] von Mises, R. Mathematical Theory of Probability and Statistics. New York: Academic Press. 1964. [4] Morgan, N., and H. Bourlard. "Generalization and Parameter Estimation in Feedforward Nets: Some Experiments," in Advances in NIPS 2, D. Touretzky, ed., pp 630-637. San Mateo, CA: Morgan Kaufmann. 1990. [5] Wang, C., and S. Venkatesh. "Temporal Dynamics of Generalization in Neural Networks," in Advances in NIPS 7, G. Tesauro, D. Touretzky, and T. Leen, eds. pp 263-270. Cambridge, MA: MIT Press. 1995. [6] Wang, C., S. Venkatesh, J. S. Judd. "Optimal Stopping and Effective Machine Complexity in Learning," in Advances in NIPS 6, J. Cowan, G. Tesauro, and J. Alspector, eds., pp 303-310. San Francisco: Morgan Kaufmann. 1994. [7] Weigend, A., B. Huberman, and D. Rumelhart. "Predicting the Future: A Connectionist Approach," Int'l J. Neural Systems 1, 193-209 (1990).	1995	98
1,148	A Unified Learning Scheme: Bayesian-Kullback Ying-Yang Machine Lei Xu 1. Computer Science Dept., The Chinese University of HK, Hong Kong 2. National Machine Perception Lab, Peking University, Beijing Abstract A Bayesian-Kullback learning scheme, called Ying-Yang Machine, is proposed based on the two complement but equivalent Bayesian representations for joint density and their Kullback divergence. Not only the scheme unifies existing major supervised and unsupervised learnings, including the classical maximum likelihood or least square learning, the maximum information preservation, the EM & em algorithm and information geometry, the recent popular Helmholtz machine, as well as other learning methods with new variants and new results; but also the scheme provides a number of new learning models. 1 INTRODUCTION Many different learning models have been developed in the literature. We may come to an age of searching a unified scheme for them. With a unified scheme, we may understand deeply the existing models and their relationships, which may cause cross-fertilization on them to obtain new results and variants; We may also be guided to develop new learning models, after we get better understanding on which cases we have already studied or missed, which deserve to be further explored. Recently, a Baysian-Kullback scheme, called the YING-YANG Machine, has been proposed as such an effort(Xu, 1995a). It bases on the Kullback divergence and two complement but equivalent Baysian representations for the joint distribution of the input space and the representation space, instead of merely using Kullback divergence for matching un-structuralized joint densities in information geometry type learnings (Amari, 1995a&b; Byrne, 1992; Csiszar, 1975). The two representations consist of four different components. The different combinations of choices of each component lead the YING-YANG Machine into different learning models. Thus, it acts as a general learning scheme for unifying the existing major unsupervised and supervised learnings. As shown in Xu(1995a), its one special case reduces to the EM algorithm (Dempster et aI, 1977; Hathaway, 1986; Neal & Hinton, 1993) A Unified Learning Scheme: Bayesian-Kullback Ying-Yang Machine 445 and the closely related Information Geometry theory and the em algorithm (Amari, 1995a&b), to MDL autoencoder with a "bits-back" argument by Hinton & Zemel (1994) and its alternative equivalent form that minimizes the bits of uncoded residual errors and the unused bits in the transmission channel's capacity (Xu, 1995d), as well as to Multisets modeling learning (Xu, 1995e)- a unified learning framework for clustering, PCA-type learnings and self-organizing map. It other special case reduces to maximum information preservation (Linsker, 1989; Atick & Redlich, 1990; Bell & Sejnowski, 1995). More interestingly its another special case reduces to Helmholtz machine (Dayan et al,1995; Hinton, 1995) with new understandings. Moreover, the YING-YANG machine includes also maximum likelihood or least square learning. Furthermore, the YING- YANG Machine has also been extended to temporal patterns with a number of new models for signal modeling. Some of them are the extensions of Helmholtz machine or maximum information preservation learning to temporal processing. Some of them include and extend the Hidden Markov Model (HMM), AMAR and AR models (Xu, 1995b). In addition, it has also been shown in Xu(1995a&c, 1996a) that one special case of the YING-YANG machine can provide us three variants for clustering or VQ, particularly with criteria and an automatic procedure developed for solving how to select the number of clusters in clustering analysis or Gaussian mixtures a classical problem that remains open for decades. In this paper, we present a deep and systematical further study. Section 2 redescribes the unified scheme on a more precise and systematical basis via discussing the possible marital status of the two Bayesian representations for joint density. Section 3 summarizes and explains those existing models under the unified scheme, particularly we have clarified some confusion made in the previous papers (Xu, 1995a&b) on maximum information preservation learning. Section 4 proposed and summarizes a number of possible new models suggested by the unified scheme. 2 BAYESIAN-KULLBACK YING-YANG MACHINE As argued in Xu (1995a), unsupervised and supervised learning problems can be summarized into the problem of estimating joint density P(x, y) of patterns in the input space X and the representation space Y, as shown in Fig.I. Under the Bayesian framework, we have two representations for P(x, y). One is PM! (x , y) = PM! (ylx)PM! (x), implemented by a model Ml called YANG/(male) part since it performs the task of transferring a pattern/(a real body) into a code/(a seed). The other is PM2(X, y) = PM2(xly)PM2(Y), implemented by a model M2 called YING part since it performs the task of generating a pattern/(a real body) from a code/(a seed). They are complement to each other and together implement an entire circle x -t y -t x. This compliments to the ancient chinese YING-YANG philosophy. Here we have four components PM! (x), PM! (ylx), PM2 (xly) and PM2(Y). The PM! (x) can be fixed at some density estimate on input data, e.g., we have at least two choices-Parzen window estimate Ph (x) or empirical estimate Po (x) : Ph(X) = N~d I:~l K( X~XI), Po(x) = limh ... O Ph(X) = -b I:~l 8(x - Xi). (1) For PM!(ylx), PM2 (xly), each can have three choices: (1) from a parametric family specified by model Ml or M2 ; (2) free of model with PM!(ylx) = P(ylx) or PM2(xly) = P(xly); (3) broken channel PM! (ylx) = PM!(y) or PM2(xly) = PM2 (X) . Finally, PM2(y) with its y consistent to PM! (ylx) can also being from a parametric family or free of model. Any combinations of the choices of the four components forms a potential YING-YANG pair. We at least have 2 x 3 x 3 x 2 = 36 pairs. A YING-YANG pair has four types of marital status: (a) marry, i.e., YING and 446 ... p.....n .. tIon apace v Symbola. Intea __ • Binary Cod.. ':.2(Y) Encoding ".ooannlon Aepr __ ntatlon Decoding o ........ trng Aeconatruotlon Figure 1 The joint spaces X, Y and the YING-YANG Machine L. XU YANG match each other; (b) divorce, i.e., YING and YANG go away from each other; (c) YING chases YANG, YANG escapes; (d) YANG chases YING, but YING escapes. The four types can be described by a combination of minimization (chasing) and maximization (escaping) on one of the two Kullback divergences below: , f ( \ PM) (ylx) PM) (x) ( R.(MI,M2) = PMlyx)PMl(x)logp (I)P ()dxdy 2a) x,y M2 x Y M2 Y ( ) f ( \) () PM2 (xly) PM2 (y) ( ) K M2,MI = PM2 X Y PM2 Y log P (I)P ()dxdy 2b x,y Ml Y x Ml x We can replace K(MI' M2) by K(M2, MJ) in the table. The 2nd & 3rd columns are for (c) (d) respectively, each has two cases depending on who starts the act and the two are usually not equivalent. Their results are undefined depending on initial condition for MI,M2, except of two special cases: (i) Free PMl(Y\X) and parametric PM2(X\Y), with minM2 maxMl K being the same as (b) with broken PMl (y\x), and with maXM2 minMl K defined but useless. (ii) Free PM2(X\Y) and parametric PMl(y\X), with minMl maXM2 K the same as case (a) with broken PM2 (xly), with minMl maxM2 K defined but useless. Therefore, we will focus on the status marry and divorce. Even so, not all of the above mentioned 2 x 3 x 3 x 2 = 36 YING-YANG pairs provide sensible learning models although minMl ,M2 K and maxMl ,M2 K are always well defined. Fortunately, a quite number of them indeed lead us to useful learning models, as will be shown in the sequent sections. We can implement minM l ,M2 K(Ml, M 2) by the following Alternative Minimization (ALTMIN) procedure: Step 1 Fix M2 = M21d, to get Mrew = arg M inMl K L( MI, M21d) Step 2 Fix MI = Mfld, to get M:;ew = arg MinM2 KL(Mfld, M2) The ALTMIN iteration will finally converge to a local minimum of K(MI , M 2 ). We can have a similar procedure for maXMl ,M2 K(MI, M2) via replacing Min by Max. Since the above scheme bases on the two complement YING and YANG Bayesian representations and their Kullback divergence for their marital status, we call it Bayesian-Kullback YING- YANG learning scheme. Furthermore, under this scheme we call each obtained YING-YANG pair that is sensible for learning purpose as a Bayesian-Kullback YING- YANG Machine or YING- YANG machine shortly. 3 UNIFIED EXISTING LEARNINGS Let PMl(X) = Po(x) by eq.(l) and put it into eq.(2), through certain mathematics we can get K(M1 , M2) = hMl - haMl - QMl,2 + D with D independent of M 1 , M2 and hMll haMl' QMl,2 given by Eqs.(El)(E2)&(E4) in Tab.2 respectively. The larger A Unified Learning Scheme: Bayesian-Kullback Ying-Yang Machine 447 is the hMl , the more discriminative or separable are the representations in Y for the input data set. The larger is the haMl' the more concentrated the representations in Y . The larger is the qMl,2' the better PM2(xIY) fits the input data. Therefore, minMl ,M2 K(M1, M2) consists of (1) best fitting of PM2 (xIY) on input data via maxQMl ,2' which is desirable, (2) producing more concentrated representations in Y to occupy less resource, which is also desirable and is the behind reason for solving the problem of selecting cluster number in clustering analysis Xu(1995a&c, 1996a), (3) but with the cost of less discriminative representations in Y for the input data. Inversely, maxMl ,M2 K(M1 , M2 ) consists of (1) producing best discriminative or separable representation PMl (ylx) in Y for the input data set, which is desirable, in the cost of (2) producing a more uniform representation in Y to fully occupy the resource, and (3) causing PM2(xly) away from fitting input data. Shown in Table 2 are the unified existing unsupervised learnings. For the case H-f- W, we have hMl = h, haMl =ha, QMl,2 =QM2, and minMJ«M1 , M2) results in PM2(y) = PMl (y) =O:y and PM2(xly)PM2(Y) = PM2(X)PMl (ylx) with PM2 (X) =I:~=l PM2 (xly)PM2 (y)· In turn, we get K(M1 , M2) =-LM2 + D with LM2 being the likelihood given by eq.(E5), i.e., we get maximum likelihood estimation on mixture model. In fact, the ALTMIN given in Tab.2 leads us to exactly the EM algorithm by Dempster et al(1977). Also, here PMl(X,y), PM2(X,y) is equivalent to the data submanifold D and model submanifold M in the Information Geometry theory (Amari, 1995a&b), with the ALTMIN being the em algorithm. As shown in Xu(95a), the cases also includes the MDL auto-encoder (Hinton & Zemel, 1994) and Multi-sets modeling (Xu, 1995e). For the case Single-M, the hMl - haMl is actually the information transmitted by the YANG part from x to y. In this case, its minimization produces a non-sensible model for learning. However, its maximization is exactly the Informax learning scheme (Linsker, 1989; Atick & Redlich, 1990; Bell & Sejnowski, 1995). Here, we clear up a confusion made in Xu(95a&b) where the minimization was mistakenly considered. For the case H-m- W, the hMl -haMl -QMl,2 isjust the -F(d; B, Q) used by Dayan et al (1995) and Hinton et al (1995) for Helmholtz machine. We can set up the detailed correspondence that (i) here PMl(ylx;) is their Qa; (ii) logPM2(x,y is their -Ea; and (iii) their Pa is PM2 (ylx) = PM2(xly)PM2(Y)/ I:y PM2(xly)PM2(Y). So, we get a new perspective for Helmholtz machine. Moreover, we know that K(M1, M2) becomes a negative likelihood only when PM2(xly)PM2(Y) = PM2(X)PMl (ylx), which is usually not true when the YANG and YING parts are both parametric. So Helmholtz machine is not equivalent to maximum likelihood learning in general with a gap depending on PM2(xly)PM2 (y) - PM2 (X)PMl (ylx). The equivalence is approximately acceptable only when the family of PM2(xly) or/and PMl (ylx;) is large enough or M2 , Ml are both linear with gaussian density. In Tab.4, the case Single-Munder K(M2, Ml) is the classical maximum likelihood (ML) learning for supervised learning which includes the least square learning by back propagation (BP) for feedfarward net as a special case. Moreover, its counterpart for a backward net as inverse mapping is the case Single-Funder K(Ml, M2). 4 NEW LEARNING MODELS First, a number of variants for the above existing models are given in Table 2. Second, a particular new model can be obtained from the case H-m- Wby changing minMl ,M2 into maxMl ,M2. That is, we have maXMl ,M2 [hMl - haMl - QMl,2]' shortly 448 L.XU Table 2: BKC-YY Machine for Unsupervised Learning ( Part I) : K(MI, M2) Given Data {X;}f:l' Fix PMl (x) = Po(x) by eq.(l), and thus K(MI' M2) = Kb + D, with D irrelevant to M 1 , M2 and K b given by the following formulae and table: h = -N1 ""N,k P(ylx;)logP(ylx;) , hMl = -N 1 "" PMl(ylx;)logPMl(ylx;), (El) ~t"y Ut ,y haMl = 2::yO'~llogO'~l, O'~l = 1:i 2::; PMl(ylx;), ha = 2::yO'ylogO'y, (E2) O'y = 1:i 2::. P(ylx;), P(ylx;) = O'yPM2 (xily)J 2::y O'yPM2 (x;iy), (E3) qM1 ,2 = 1:; 2::;,y PMl (Ylx;) log PM2(x;iy), qM2 = 1:; 2::i,y P(ylx;) log PM2(Xily), (E4) L~2 = 1:i 2::;,y O'y log PM2(x;iy), LM2 = 1:; 2::; log 2::y O'yPM2(X.ly) (E5) Marriage Status H-f-W Single-M Single-F H-m-W W-f-H PM2 (y) Uniform PMl (ylx) = PMl (y) PMl (ylx) PM2(y), Condition free, i.e., PM2(xly) PMl (ylx) and and free PMl:~YJXJ = PM2 (X) = PMl(y) PM2 (xly) PM~~~I~~ = Pyx = Po(xl = P xjy h-ha-qM2 hMl - haMl -LM2 lhMl-haMl haMl Kb = -LM2 (~~l~)] (minl (max) (min) mIn (min) ~1: t'IX ~1: t'IX M2, get Get MI Get M2 M2, get Get P(ylx;) by max by MI by MI by O'y by hMl-haMl max min [hMl mm (E3), O'~l L~2' -haMl - QMl,2] haMl ALTMIN by (E2) 82: get 82: Fix M 1 , M2 by get M2 by max QM2. max QMl 2' Kepeat No No Kepeat No 81,82. Repeat Repeat 81,82. Repeat 1. ML on Mixtures &EM Dupli(Dem77) Informax, cated HelmRelated 2. InformMaximum models holtz to Existing ation mutual by ML machine PCA Equivgeometry Informlearning (Hin95) -lent (Amari95) ation on (Day95) models 3. MDL ~Lin89~ input AutoAti90 data. encoder (BeI95) W in94) 4. ulti-sets modeling (Xu94 ,95) 1. t'or H-f- W type, we have: Three VQ variants when PM2(xly) is Gaussian. Also, criteria for New selecting the correct k for VQ or clustering (Xu95a&c). Results 2. For H-m-W type, we have: Robust PCA + criterion for determining subspace dimension (Xu, 95c). 1. More smooth PMl_(x)given by Parzen window estimate. 2. Factorial coding PM2(y) = ~M2(Y;) with binary y = [YI "', yrn]. Variants 3. Factorial coding PMl (ylx) = . PM2 (Yi Ix) with binary [YI ... , Yrn]. 4. Replace '2::11 .' in all the above items by 'fu ·dy' for real y. Note: H- Husband, W-WIfe, f- follows, M-Male, F-Female, m-matches. X-f-Y stands for X part is free. Single-X stands for the other part broken. H-m-W stands for both parts being parametric. '(min)' stands for min Kb and '(max), stands for max Kb. A Unified Learning Scheme: Bayesian-Kullback Ying-Yang Machine 449 Table 3: BKC-YY Machine for Unsupervised Learning ( Part II) : J(M2, Ml) Given Data {Xi}F:I, Fix PMI (x) = Po(x) by eq.(l), and thus J(M2, Ml) = J(b + D, with D irrelevant to M 1 , M2 and J(b given by the following formulae and table: MarrIage Status H-f-W Single-M Single-F H-m-W C;ondahon The same as those m Table 1. hM2 lha M2 lhaMIhM2 [h~2 + - L MI ,2] LMI ] + haMI J(b (if forcing -qM2,1] PM1(y) = (max) (min) P02 ·(~I) (max) (min) mIn S1: SI: ' SI: Fix MI, Fix M 1 , Fix M 2 , Get M2 get ai:2 get ai:l Get M2 get MI by by (E7). by (E2). by by max in Tab.l mIn hM2 S2: S2: max [haMI update update h~2' - q M2,1] MI by MI by ALTMIN max LMI ,2 max LMI S2: Fix M 1 , get M2 by min h~?-qM~ I 1\l0 ttepeat ttepeat 1\l0 .H.epeat Repeat SI, S2 SI,S2 Repeat SI, S2 t;xlstmg no no no no no models new! new! new! new! new! Vanants Imilar to those m Table 1. Table 4: BKC-YY Machine for Supervised Learning Given Data {Xi,y.}F:I , Fix PMI(X) = Po(x) by eq.(l). (E6) (E7) (ES) (E9) (ElO) W-f-H -LaMI (min) Get MI by max L o MI 1\l0 Repeat no new! h'kl = -kEiPM1(y;JXi)logPMI(Yil x i), h'k2 = -kEiPM2(x;JYi)logPM2(x;Jy.), (Ell) Q'kI ,2 = -k Ei PMI (y;JXi) log PM2 (xdYi), Q'k2 , 1 = -k Ei PM2(X;Jy.) log PMI (Y.lxi) , (El2) L'kl = -k Ei log PMI (y;JXi), L'k2 = -k Ei log PM2(XiIYi) , (El3) K(MI, M2) = Kb + D J((M2, MI) = J(b + D Marnage Status Single-M Single-F H-m-W Single-M Single-F H-m-W J(b hMl -LM2 hMI -QMI2 -LMI hM2 hM2-QM2 1 (max) (min) (min) , (min) (max) (min) , mIrumum ML MIxed ML mlrumum Mixed Feature entropy (ME) F-B entropy B-F F-net B-net net F-net B-net net ~'xastmg no tH' on no tiP on no no models new! B-net new! F-net new! new! 450 L. XU denoted by H-m- W-Max. This model is a dual to the Helmholtz machine in order to focus on getting best discriminative or separable representations PMl (ylx) in Y instead of best fitting of PM2(xly) on input data. Third, by replacing K(M1 , M2) with K(M2, M 1), in Table 3 we can obtain new models that are the counterparts of those given in Table 2. For the case H-J- W, its maxMl,M2 gives minimum entropy estimate on PM2 (X) instead of maximum likelihood estimate on PM2 (X) in Table 2. For the case Single-M, it will function similarly to the case Single-F in Table 2, but with minimum entropy on PMl (ylx) in Table 2 replaced by maximum likelihood on PMl (ylx) here. For the case H-mW, the focus shifts from on getting best fitting of PM2(xly) on input data to on getting best discriminative representations PM 1 (ylx) in Y, which is similar to the just mentioned H-m- W-Max, but with minimum entropy on PMJylx) replaced by maximum likelihood on PM 1 (ylx). The other two cases in Table 3 have been also changed similarly from those in Table 2. Fourth, several new model have also been proposed in Table 4 for supervised learning. Instead of maximum likelihood, the new models suggest learning by minimum entropy or a mix of maximum likelihood and minimum entropy. Finally, further studies on the other status in Table 1 are needed. Heuristically, we can also treat the case H-m- W by two separated steps. We first get Ml by max[hMl - haMl], and then get M2 by maxqMl,2; or we first get M2 by min[h ha qM2] and then get Ml by min[hMl - haMl QMl,2]' The two algorithms attempt to get both a good discriminative representation by PMl (ylx) and a good fitting of PM2 (xly) on input data. However whether they work well needs to be tested experimentally. We are currently conducting experiments on comparison several of the above new models against their existing counterparts. Acknowledgements The work was Supported by the HK RGG Earmarked Grant GUHK250/94E. References Amari, S(1995a) [Amari95] " Information geometry of the EM and em algorithms for neural networks", Neural Networks 8, to appear. Amari, S(1995b), Neural Computation 7 ppI3-18. Atick, J .J. & Redlich, A.N . (1990) [Ati901. Neural Computation Vo1.2, No.3, pp308-320 . Bell A . J . & Sejnowski, T . J.(1995) [Be195], Neural Computation Vo1.7, No.6, 1129-1159. Byrne, W . (1992), IEEE Trans. Neural Networks 3, pp612-620. Csiszar, I. , 11975), Annals of Probabil.ty 3, ppI46-158. Dayan, P. , Hinton , G. E ., & Neal, R. N. (1995) [Day95], Neural Computat.on Vo1.7, No.5 , 889-904. Dempster, A.P., Laird , N .M ., & Rubin, D .B. (1977) [Dem77] , 1. Royal Statist. Soc.ety, 839, 1-38. Hathaway, R.J.(1986), Statistics & Probability Letters 4, pp53-56 . Hinton, G . E ., et ai, (1995) [Hin95], Sc.ence 268, pp1158-1160. Hinton, G. E . & Zemel, R.S. (19M) [Hin94], Advances in NIPS 6, pp3-10. Linsker, R. (1989) [Lin89], Advances in NIPS 1, ppI86-194. Neal , R. N.& Hinton, G. E(1993), A new view of the EM algorithm that Jushfies Incremental and other vanants, pr~rint. Xu, L . (1996 , "How Many Clusters? : A YING-YANG Machine Based Theory For A Classical Open Problem In attern Recognition" , to appear on Proc. IEEE ICNN96. Xu , L. (1995a), "YING-YANG Machine: a Bayesian-Kullback scheme for unified learnings and new results on vector quantization" , Keynote talk, Proc. Inti Conf. on Neural Information Processing (ICONIP95), Oct 30 - Nov . 3, 1995, pp977-988. Xu , L.(1995b), "YING-YANG Machine for Temporal Signals", Keynote talk, Proc IEEE inti Conf. Neural Networks & Signal Processing, Vol.I, pp644-651, Nanjing, 10-13, 1995. Xu , L . (1995c) , "New Advances on The YING-YANG Machine", Invited paper, Proc. of 1995 IntI. Symposium on Artificial Neural Networks, ppIS07-12, Dec. 18-20, Taiwan. Xu , L . (1995d), "Cluster Number Selection, Adaptive EM Algorithms and Competitive Learnings", Invited paper, Proc. Inti Conf. on Neural Information Processing (ICONIP95), Oct 30 - Nov. 3, 1995, Vol. II, ppI499-1502. Xu , L . (1995e), Invited paper, Proc. WCNN95, Vol.I, pp35-42. Also, Invited paper, Proc. IEEE ICNN 1994, ppI315-320 . Xu, L. , & Jordan, M.I . (1993). Proc. of WCNN '93, Portland, OR, Vol. II, 431-434 .	1995	99
1,149	Hebb Learning of Features based on their Information Content Hideki Noda Ferdinand Peper Communications Research Laboratory 588-2, Iwaoka, Iwaoka-cho Nishi-ku, Kobe 651-24 Japan peper@crl.go.jp Kyushu Institute of Technology Dept. Electr., Electro., and Compo Eng. 1-1 Sensui-cho, Tobata-ku Kita-Kyushu 804, Japan noda@kawa.comp.kyutech.ac.jp Abstract This paper investigates the stationary points of a Hebb learning rule with a sigmoid nonlinearity in it. We show mathematically that when the input has a low information content, as measured by the input's variance, this learning rule suppresses learning, that is, forces the weight vector to converge to the zero vector. When the information content exceeds a certain value, the rule will automatically begin to learn a feature in the input. Our analysis suggests that under certain conditions it is the first principal component that is learned. The weight vector length remains bounded, provided the variance of the input is finite. Simulations confirm the theoretical results derived. 1 Introduction Hebb learning, one of the main mechanisms of synaptic strengthening, is induced by cooccurrent activity of pre- and post-synaptic neurons. It is used in artificial neural networks like perceptrons, associative memories, and unsupervised learning neural networks. Unsupervised Hebb learning typically employs rules of the form: J.tw(t) = x(t)y(t) - d(x(t), y(t), w(t)) , (1) where w is the vector of a neuron's synaptic weights, x is a stochastic input vector, y is the output expressed as a function of x T w , and the vector function d is a forgetting term forcing the weights to decay when there is little input. The integration constant J.t determines the learning speed and will be assumed 1 for convenience. The dynamics of rule (1) determines which features are learned, and, with it, the rule's stationary points and the boundedness of the weight vector. In some cases, weight vectors grow to zero or grow unbounded. Either is biologically implausible. Suppression and unbounded growth of weights is related to the characteristics of the input x and to the choice for d. Understanding this relation is important to enable a system, that employs Hebb learning, to learn the right features and avoid implausible weight vectors. Unbounded or zero length of weight vectors is avoided in [5] by keeping the total synaptic strength :Ei Wi constant. Other studies, like [7], conserve the sum-squared Hebb Learning of Features based on their Information Content 247 synaptic strength. Another way to keep the weight vector length bounded is to limit the range of each of the individual weights [4]. The effect of these constraints on the learning dynamics of a linear Hebb rule is studied in [6]. This paper constrains the weight vector length by a nonlinearity in a Hebb rule. It uses a rule of the form (1) with y = S(xT W - h) and d(x, y, w) = c.w, the function S being a smooth sigmoid, h being a constant, and c being a positive constant (see [1] for a similar rule). We prove that the weight vector w assumes a bounded nonzero solution if the largest eigenvalue Al of the input covariance matrix satisfies Ai > cjS'(-h). Furthermore, if Al ~ cjS'(-h) the weight vector converges to the vector O. Since Al equals the variance of the input's first principal component, that is, Ai is a measure for the amount of information in the input, learning is enabled by a high information content and suppressed by a low information content. The next section describes the Hebb neuron and its input in more detail. After characterizing the stationary points of the Hebb learning rule in section 3, we analyze their stability in section 4. Simulations in section 5 confirm that convergence towards a nonzero bounded solution occurs only when the information content of the input is sufficiently high. We finish this paper with a discussion. 2 The Hebb Neuron and its Input Assume that the n-dimensional input vectors x presented to the neuron are generated by a stationary white stochastic process with mean O. The process's covariance matrix :E = E[xxT] has eigenvalues AI, "', An (in order of decreasing size) and corresponding eigenvectors UI, ... , Un. Furthermore, E[llxl12] is finite. This implies that the eigenvalues are finite because E[lIx112] = E[tr[xxT)) = tr[E[xxT)) = L:~=l Ai. It is assumed that the probability density function of x is continuous. Given an input x and a synaptic weight vector w, the neuron produces an output y = S(xTw - h), where S : R -+ R. is a function that satisfies the conditions: Cl. S is smooth, i.e., S is continuous and differentiable and S' is continuous. C2. Sis sublinear, Le., lim S(z)jz = lim S(z)jz = O. z--+oo z--+-oo C3. S is monotonically nondecreasing. C4. S' has one maximum, which is at the point z = -h. Typically, these conditions are satisfied by smooth sigmoidal functions. This includes sigmoids with infinite saturation values, like S(z) = sign(z)lzli/2 (see [9]). The point at which a sigmoid achieves maximal steepness (condition C4) is called its base. Though the step function is discontinuous at its base, thus violating condition Cl, the results in this paper apply to the step function too, because it is the limit of a sequence of continuous sigmoids, and the input density function is continuous and thus Lebesgue-integrable. The learning rule of the neuron is given by VI = xy - cw, (2) c being a positive constant. Use of a linear S(z) = az in this rule gives unstable dynamics: if a > cj Ai, then the length of the weight vector w grows out of bound though ultimately w becomes collinear with Ui' It is proven in the next section that a sublinear S prevents unbounded growth of w. 248 F. Peper and H. Noda 3 Stationary Points of the Learning Rule To get insight into what stationary points the weight vector w ultimately converges to, we average the stochastic equation (2) over the input patterns and obtain (w) = E [xS (XT(W) - h)] - c(w), (3) where (w) is the averaged weight vector and the expectation is taken over x, as with all expectations in this paper. Since the solutions of (2) correspond with the solutions of (3) under conditions described in [2], the averaged (w) will be referred to as w. Learning in accordance to (2) can then be interpreted [lJ as a gradient descent process on an averaged energy function J associated with (3): with T(z) = [~ S(v)dv. To characterize the solutions of (3) we use the following lemma. Lemma 1. Given a unit-length vector u, the function f u : R ~ R is defined by and the constant Au by Au = E[u T xxT u]. The fixed points of fu are as follows. 1. If AuS'( -h) ~ c then fu has one fixed point, i.e., z = o. 2. If AuS'( -h) > c then fu has three fixed points, i.e., z = 0, z = Q:~, and z = Q:~, where Q:~ (Q:~) is a positive (negative) value depending on u. Proof:(Sketch; for a detailed proof see [11]). Function fu is a smooth sigmoid, since conditions C1 to C4 carryover from S to fu. The steepness of fu in its base at z = 0 depends on vector u. If AuS'( -h) ~ c, function fu intersects the line h(z) = z only at the origin, giving z = 0 as the only fixed point. If AuS'( -h) > c, the steepness of f u is so large as to yield two more intersections: z = Q:~ and z = Q:~ • 0 Thus characterizing the fixed points of f u, the lemma allows us to find the fixed points of a vector function g: R n ~ R n that is closely related to (3). Defining 1 g(w) = - E [xS (xTw - h) ] , c we find that a fixed point z = Q:u of f u corresponds to the fixed point w = Q:u u of g. Then, since (3) can be written as w = c.g(w) - c.w, its stationary points are the fixed points of g, that is, w = 0 is a stationary point and for each u for which AuS'( -h) > c there exists one bounded stationary point associated with Q:~ and one associated with Q:~. Consequently, if Al ~ c/ S' (-h) then the only fixed point of g is w = 0, because Al ~ Au for all u. What is the implication of this result? The relation Al ~ c/ S'( -h) indicates a low information content of the input, because AI- equaling the variance of the input's first principal component-is a measure for the input's information content. A low information content thus results in a zero w, suppressing learning. Section 4 shows Hebb Learning of Features based on their Information Content 249 that a high information content results in a nonzero w. The turnover point of what is considered high/low information is adjusted by changing the steepness of the sigmoid in its base or changing constant c in the forgetting term. To show the boundedness of w, we consider an arbitrary point P: w = f3u sufficiently far away from the origin 0 (but at finite distance) and calculate the component of w along the line OP as well as the components orthogonal to OP. Vector u has unit length, and f3 may be assumed positive since its sign can be absorbed by u. Then, the component along 0 P is given by the projection of w on u: This is negative for all f3 exceeding the fixed points of I u because of the sigmoidal shape of lu. So, for any point Pin nn lying far enough from 0 the vector component of win P along the line OP is directed towards 0 and not away from it. This component decreases as we move away from 0, because the value of [f3 - I u (f3)] increases as f3 increases (fu is sublinear). Orthogonal to this is a component given by the projection of w on a unit-length vector v that is orthogonal to u: This component increases as we move away from 0; however, it changes at a slower pace than the component along OP, witness the quotient of both components: lim vTwl = lim cvTg(f3u) = lim vTg(f3u)/f3 = 0 {3-+00 uTw w={3u (3-+00 -c[f3 - lu(f3)] (3-+00 lu(f3)/f3 - 1 . Vector w thus becomes increasingly dominated by the component along 0 P as f3 increases. So, the origin acts as an attractor if we are sufficiently far away from it, implying that w remains bounded during learning. 4 Stability of the Stationary Points To investigate the stability of the stationary points, we use the Hessian of the averaged energy function J described in the last section. The Hessian at point w equals: H(w) = cI - E [xxTS' (xTw - h)]. A stationary point w = w is stable iff H(w) is a positive definite matrix. The latter is satisfied if for every unit-length vector v, (4) that is, if all eigenvalues of the matrix E[xxTS'(XTw - h)] are less than c. First consider the stationary point w = o. The eigenvalues of E[XXT S'( -h)] in decreasing order are AlS'( -h), ... , AnS'( -h). The Hessian H(O) is thus positive definite iff A1S'( -h) < c. In this case w = 0 is stable. It is also stable in the case Al = c/S'( -h), because then (4) holds for all v 1= Ul, preventing growth ofw in directions other than Ul· Moreover, w will not grow in the direction of Ul, because II Ul (f3) I < 1f31 for all f3 1= O. Combined with the results of the last section this implies: Corollary 1. If Al ~ c/S'(-h) then the averaged learning equation (3) will have as its only stationary point w = 0, and this point is stable. If Al > c/ S'( -h) the stationary point w = 0 is not stable, and there will be other stationary points. 250 F. Peper and H. Noda We now investigate the other stationary points. Let w = au u be such a point, u being a unit-length vector and au a nonzero constant. To check whether the Hessian H(auu) is positive definite, we apply the relation E[XYJ = E[XJ E[YJ + Cov[X, Y] to the expression E [ u T xx T uS' (aux T u - h)] and obtain after rewriting: The sigmoidal shape of the function lu implies that lu is less steep than the line h(z) = z at the intersection at z = au, that is, I~(au) < 1. It then follows that E [uTxxTuS'(auxTu - h)] = c/~(au) < c, giving: 1 E [ S' ( aux T u - h)] < Au {c - Cov [ U T xx T U, S' (au x T u - h) ] } . Then, yTE [xxTS'(auxTu - h)] y = AvE [ S' (aux T u - h)] + Cov [ y T xx T y, S' (aux T u - h)] < ~: c ~: Cov [ u T xx T U, S' (aux T u - h)] + Cov [ y T xx T y, S' (aux T u - h) ] . The probability distribution of x unspecified, it is hard to evaluate this upper bound. For certain distributions the upper bound is minimized when Au is maximized, that is, when u = Ul and Au = AI, implying the Hebb neuron to be a nonlinear principal component analyzer. Distributions that are symmetric with respect to the eigenvectors of :E are probably examples of such distributions, as suggested by [11, 12J. For other distributions vector w may assume a solution not collinear with Ul or may periodically traverse (part of) the nonzero fixed-point set of g. 5 Simulations We carry out simulations to test whether learning behaves in accordance with corollary 1. The following difference equation is used as the learning rule: (5) where, is the learning rate and a a constant. The use of a difference .6. in (5) rather than the differential in (2) is computationally easier, and gives identical results if , decreases over training time in accordance with conditions described in [3]. We use ,(t) = 1/(O.Olt + 20). It satisfies these conditions and gives fast convergence without disrupting stability [10J. Its precise choice is not very critical here, though. The neuron is trained on multivariate normally distributed random input samples of dimension 6 with mean 0 and a covariance matrix :E that has the eigenvalues 4.00, 2.25, 1.00, 0.09, 0.04, and 0.01. The degree to which the weight vector and :E's first eigenvector Ul are collinear is measured by the match coefficient [10], defined by: m = cos2 L(Ul' w). In every experiment the neuron is trained for 10000 iterations by (5) with the value of parameter a set to 0.20, 0.25, and 0.30, respectively. This corresponds to the situations in which Al < c/ S'( -h), Al = cj S'( -h), and Al > c/ S'( -h), respectively, since c = 1 and the steepness ofthe sigmoid S(z) = tanh(az) Hebb Learning of Features based on their Information Content 251 in its base z = -h = 0 is 8'(0) = a. We perform each experiment 2000 times, which allows us to obtain the match coefficients beyond iteration 100 within ±0.02 with a confidence coefficient of 95 % (and a smaller confidence coefficient on the first 100 iterations). The random initialization of the weight vector-its initial elements are uniformly distributed in the interval (-1, I)-is different in each experiment. m 1.0.,-------::;;::::;::;;=---, 0.5 a=0.30 ---a=0.25 - - - - - a=0.20 0.0 -'--+----+----1---+-----+--' 1 10 102 103 104 Iterations Figure 1: Match coefficients averaged over 2000 experiments for parameter values a = 0.20, 0.25, and 0.30. Ilwll 1.0 0.0 1 10 102 103 104 Iterations Figure 2: Lengths of the weight vector averaged over 2000 experiments. The curve types are similar to those in Fig. 1. Fig. 1 shows that for all tested values of parameter a the weight vector gradually becomes collinear with Ul over 10000 iterations. The length of the weight vector converges to 0 when a = 0.20 or a = 0.25 (see Fig. 2). In the case a = 0.30, corresponding to Al > cj8'(-h), the length converges to a nonzero bounded value. In conclusion, convergence is as predicted by corollary 1: the weight vector converges to 0 if the information content in the input is too low for climbing the slope of the sigmoid in its base, and otherwise the weight vector becomes nonzero. 6 Discussion Learning by the Hebb rule discussed in this paper is enabled if the input's information content as measured by the variance is sufficiently high, and only then. The results, though valid for a single neuron, have implications for systems consisting of multiple neurons connected by inhibitory connections. A neuron in such a system would have as output y = 8(xT w - h yTy'), where the inhibitory signal yTy' would consist of the vector of output signals y' of the other neurons, weighted by the vector y (see also [1]). Function fu in lemma 1 would, when extended to contain the signal y T y', still pass through the origin because of the zero-meanness of the input, but would have a reduced steepness at the origin caused by the shift in S's argument away from the base. The reduced steepness would make an intersection of f u with the line h(z) = z in a point other than the origin less likely. Consequently, an inhibitory signal would bias the neuron towards suppressing its weights. In a system of neurons this would reduce the emergence of neurons with correlated outputs, because of the mutual presence of their outputs in each other's inhibitory signals. The neurons, then, would extract different features, while suppressing information-poor features. In conclusion, the Hebb learning rule in this paper combines well with inhibitory connections, and can potentially be used to build a system of nonredundant feature extractors, each of which is optimized to extract only information-rich features. 252 F. Peper and H. Noda Moreover, the suppression of weights with a low information content suggests a straightforward way [8J to adaptively control the number of neurons, thus minimizing the necessary neural resources. Acknowledgments We thank Dr. Mahdad N. Shirazi at Communications Research Laboratory (CRL) for the helpful discussions, Prof. Dr. S.-1. Amari for his encouragement, and Dr. Hidefumi Sawai at CRL for providing financial support to present this paper at NIPS'96 from the Council for the Promotion of Advanced Information and Communications Technology. This work was financed by the Japan Ministry of Posts and Telecommunications as part of their Frontier Research Project in Telecommunications. References [1] SA. Amari, "Mathematical Foundations of Neurocomputing," Proceedings of the IEEE, vol. 78, no. 9, pp. 1443-1463, 1990. [2] S. Geman, "Some Averaging and Stability Results for Random Differential Equations," SIAM J. Appl. Math., vol. 36, no. 1, pp. 86-105, 1979. [3] H.J. Kushner and D.S. Clark, "Stochastic Approximation Methods for Constrained and Unconstrained Systems," Applied Mathematical Sciences, vol. 26, New York: Springer-Verlag, 1978. [4] R. Linsker, "Self-Organization in a Perceptual Network," Computer, vol. 21, pp. 105117, 1988. [5] C. von der Malsburg, "Self-Organization of Orientation Sensitive Cells in the Striate Cortex," Kybernetik, vol. 14, pp. 85-100, 1973. [6] K.D. Miller and D.J.C. MacKay, "The Role of Constraints in Hebbian Learning," Neural Computation, vol. 6, pp. 100-126, 1994. [7] E. Oja, "A simplified neuron model as a principal component analyzer," Journal of Mathematics and Biology, vol. 15, pp. 267-273, '1982. [8] F. Peper and H. Noda, "A Mechanism for the Development of Feature Detecting Neurons," Proc. Second New-Zealand Int. Two-Stream Conf. on Artificial Neural Networks and Expert Systems, ANNES'95, Dunedin, New-Zealand, pp. 59-62, 20-23 Nov. 1995. [9] F. Peper and H. Noda, "A Class of Simple Nonlinear I-unit PCA Neural Networks," 1995 IEEE Int. Con/. on Neural Networks, ICNN'95, Perth, Australia, pp. 285-289, 27 Nov.-l Dec. 1995. [10] F. Peper and H. Noda, "A Symmetric Linear Neural Network that Learns Principal Components and their Variances," IEEE Trans. on Neural Networks, vol. 7, pp. 10421047, 1996. [11] F. Peper and H. Noda, "Stationary Points of a Hebb Learning Rule for a Nonlinear Neural Network," Proc. 1996 Int. Symp. Nonlinear Theory and Appl. (NOLTA '96), Kochi, Japan, pp. 241-244, 7-9 Oct 1996. [12] F. Peper and M.N. Shirazi, "On the Eigenstructure of Nonlinearized Covariance Matrices," Proc. 1996 Int. Symp. Nonlinear Theory and Appl. (NOLTA '96), Kochi, Japan, pp. 491-493, 7-9 Oct 1996.	1996	1
1,150	Source Separation and Density Estimation by Faithful Equivariant SOM Juan K. Lin Department of Physics University of Chicago Chicago, IL 60637 jk-lin@uchicago.edu David G. Grier Department of Physics University of Chicago Chicago, IL 60637 d-grier@uchicago.edu Abstract Jack D. Cowan Department of Math University of Chicago Chicago, IL 60637 j-cowan@uchicago.edu We couple the tasks of source separation and density estimation by extracting the local geometrical structure of distributions obtained from mixtures of statistically independent sources. Our modifications of the self-organizing map (SOM) algorithm results in purely digital learning rules which perform non-parametric histogram density estimation. The non-parametric nature of the separation allows for source separation of non-linear mixtures. An anisotropic coupling is introduced into our SOM with the role of aligning the network locally with the independent component contours. This approach provides an exact verification condition for source separation with no prior on the source distributions. 1 INTRODUCTION Much of the current work on visual cortex modeling has focused on the generation of coding which captures statistical independence and sparseness (Bell and Sejnowski 1996, Olshausen and Field 1996). The Bell and Sejnowski model suffers from the parametric and intrinsically non-local nature of their source separation algorithm, while the Olshausen and Field model does not achieve true sparse-distributed coding where each cell has the same response probability (Field 1994). In this paper, we construct an extensively modified SOM with equipartition of activity as a steadystate for the task of local statistical independence processing and sparse-distributed coding. SOFM for Density Approximation and leA 537 Ritter and Schulten (1986) demonstrated that the density of the Kohonen SOM units is not proportional to the input density in the steady-state. In one dimension the Kohonen net under-represents high density and over-represents low density regions. Thus SOM's are generally not used for density estimation. Several modifications for controlling the magnification of the representation have appeared. Recently, Bauer et. al. (1996) used an "adaptive step size" , and Lin and Cowan (1996) used an Lp-norm weighting to control the magnification. Here we concentrate on the later's "faithful representation" algorithms for source separation and density estimation. 2 SHARPLY PEAKED DISTRIBUTIONS Mixtures of sharply peaked source distributions will contain high density contours which correspond to the independent component contours. Blind separation can be performed rapidly for this case in a net with one dimensional branched topology. A digital learning rule where the updates only take on discrete values was used: 1 (1) where K is the learning rate, A(€) the neighborhood function, {w} the SOM unit positions, and ( the input. .", '; ", ', . .. . . . " . " . .. . . . " . ' .. .. \ , . . " " , , " .' ~ ", . . . " _1133 ••.•. Figure 1: Left: linear source separation by branched net. Dashed lines correspond to the independent component axes. Net configuration is shown every 200 points. Dots denote the unit positions after 4000 points. Right: Voronoi partition of the vector space by the SOM units. We performed source separation and coding of two mixed signals in a net with the topology of two cross-linked branches (see Fig. (1)). The neighborhood function IThe sign function sgn(i) takes on a value of 1 for i > 0, 0 for i = 0 and -1 for i < O. Here the sign function acts component-wise on the vector. 538 1. K. Lin, 1. D. Cowan and D. G. Grier A(€) is taken to be Gaussian where € is the distance to the winning unit along the branch structure. Two speech audio files were randomly mixed and pre-whitened first to decorrelate the two mixtures. Since pre-whitening tends to orthogonalize the independent component axes, much of the processing that remains is rotation to find the independent component coordinate system. A typical simulation is shown in Fig. (1). The branches of the net quickly zero in on the high density directions. As seen from the nearest-neighbor Voronoi partition of the distribution (Fig. 1 b), the branched SOM essentially performs a one dimensional equipartition of the mixture. The learning rule Eqn. 1 attempts to place each unit at the componentwise median of the distribution encompassed by its Voronoi partition. For sharply peaked sources, the algorithm will place the units directly on top of the high density ridges. To demonstrate the generality of our non-parametric approach, we perform source separation and density coding of a non-linear mixture. Because our network has local dynamics, with enough units, the network can follow the curved "independent component contours" of the input distribution. The result is shown in Fig. (2). Figure 2: Source separation of non-linear mixture. The mixture is given by ~1 = -2sgn(st} . s~ + 1.1s1 S2, ~2 = -2sgn(s2) . s~ + SI + 1.1s2. Left: the SOM configuration is shown periodically in the figure, with the configuration after 12000 points indicated by the dots. Dashed lines denote two independent component contours. Right: the sources (SI' S2), mixtures (6, 6) and pseudo-histogramequalized representations (01, 02). To unmix the input, a parametric separation approach can be taken where least squares fit to the branch contours is used. For the source separation in Fig. CIa), assuming linear mixing and inserting the branch coordinate system into an unmixing matrix, we find a reduction of the amplitudes of the mixtures to less than one percent of the signal. This is typical of the quality of separation obtained in our simulations. For the non-linear source separation in Fig. (2), parametric unmixing can similarly be accomplished by least squares fit to polynomial contours with SOFM for Density Approximation and leA 539 quadratic terms. Alternatively, taking full advantage of the non-parametric nature of the SOM approach, an approximation of the independent sources can be constructed from the positions Wi. of the winning unit. Or as we show in Fig. (2b), the cell labels i* can be used to give a pseudo-histogram-equalized source representation. This non-parametric approach is thus much more general in the sense that no model is needed of the mixing transformation. Since there is only one winning unit along one branch, only one output channel is active at any given time. For sharply peaked source distributions such as speech, this does not significantly hinder the fidelity of the source representation since the input sources hover around zero most of the time. This property also has the potential for utilization in compression. However, for a full rigorous histogram-equalized source representation, we must turn to a network with a topology that matches the dimensionality of the input. 3 ARBITRARY DISTRIBUTIONS For mixtures of sources with arbitrary distributions, we seek a full N dimensional equipartition. We define an (M, N) partition of !RN to be a partition of !RN into (M + 1)N regions by M parallel cuts normal to each of N distinct directions. The simplest equipartition of a source mixtures is the trivial equipartition along the independent component axes (ICA). Our goal is to achieve this trivial ICA aligned equipartition using a hypercube architecture SOM with M + 1 units per dimension. For an (M, N) equipartition, since the number of degrees of freedom to define the M N hyperplanes grows quadratically in N, while the number of constraints grows exponentially in N, for large enough M the desired trivial equipartition will the unique (M, N) equipartition. We postulate that M = 2 suffices for uniqueness. Complementary to this claim, it is known that a (1, N) equipartition does not exist for arbitrary distributions for N ~ 5 (Ramos 1996). The uniqueness of the (M, N) equipartition of source mixtures thus provides an exact verification condition for noiseless source separation. With? = i-: - i, the digital equipartition learning rule is given by: ~wi ~A(?)· sgn(?) (2) ~Wi· L~Wi' (3) i where A(?) = A( -?). (4) Equipartion of the input distribution can easily be shown to be a steady-state of the dynamics. Let qk be the probability measure of unit k. For the steady-state: < ~wk > = 0 L q; . A({ - k) . sgn(i - k) + qk L A( k - i) . sgn( k - i) L(q; - qk) . A(i - k) . sgn(i - k), i for all units k. By inspection, equipartition, where q; = qk~ for all units i is a solution to the equation above. It has been shown that equipartition is the only 540 J. K. Lin, J. D. Cowan and D. G. Grier steady-state of the learning rule in two dimensional rectangular SOM's (Lin and Cowan 1996), though with the highly overconstrained steady-state equations, the result should be much more general. One further modification of the SOM is required. The desired trivial ICA equipartition is not a proper Voronoi partition except when the independent component axes are orthogonal. To obtain the desired equipartition, it is necessary to change the definition of the winning unit i-. Let (5) be the winning region of the unit at wi' Since a histogram-equalized representation independent of the mixing transformation A is desired, we require that {An(w)} = {n(Aw)} , i.e., n is equivariant under the action of A (see e.g. Golubitsky 1988). ,./ ,./ , , , , , , , , , , .-------------. .------- ------. Voronoi Equ ivariant I I,: I ~i.f ;; ! .' . . :" :,, , ." . (6) .: ; Figure 3: Left: Voronoi and equivariant partitions of the a primitive cell. Right: configuration of the SOM after 4000 points. Initially the units of the SOM were equally spaced and aligned along the two mixture coordinate directions. In two dimensions, we modify the tessellation by dividing up a primitive cell amongst its constituent units along lines joining the midpoints of the sides. For a primitive cell composed of units at ii, b, c and J, the region of the primitive cell represented by ii is the simply connected polygon defined by vertices at ii, (it + b)/2, (it + d)/2 and (it+b+c+d)/4. The two partitions are contrasted in Fig. (3a). Our modified equivariant partition satisfies Eqn. (6) for all non-singular linear transformations. The learning rule given above was shown to have an equipartition steady state. It remains, however, to align the partitions so that it becomes a valid (M, N) partition. The addition of a local anisotropic coupling which physically, in analogy to elastic nets, might correspond to a bending modulus along the network's axes, will tend to align the partitions and enhance convergence to the desired steady state. We SOFMfor Density Approximation and leA 541 supplemented the digital learning rule (Eqs. (2)-(3)) with a movement of the units towards the intersections of least squares line fits to the SOM grid. Numerics are shown in Fig. 3b, where alignment with the independent component coordinate system and density estimation in the form of equipartition can be seen. The aligned equipartition representation formed by the network gives histogramequalized representations of the independent sources, which, because of the equivariant nature of the SOM, will be independent of the mixing matrix. 4 DISCUSSION Most source separation algorithms are parametric density estimation approaches (e.g. Bell and Sejnowski 1995, Pearlmutter and Parra 1996). Alternatively in parallel with this work, the standard SOM was used for the separation of both discrete and uniform sources (Herrmann and Yang 1996, Pajunen et. al. 1996). The source separation approach taken here is very general in the sense that no a priori assumptions about the individual source distributions and mixing transformation are made. Our approach's local non-parametric nature allows for source separation of non-linear mixtures and also possibly the separation of more sharply peaked sources from fewer mixtures. The low to high dimensional map required for the later task will be prohibitively difficult for parametric unmixing approaches. For density estimation in the form of equipartition, we point out the importance of a digital scale-invariant algorithm. Direct dependence on ( and Wi has been extracted out of the learning rule. Because the update depends only upon the partition, the network learns from its own coarse response to stimuli. This along with the equivariant partition modification underscore the dynamic partition nature of the our algorithm. More direct computational geometry partitioning algorithms are currently being pursued. It is also clear that a hybrid local parametric density estimation approach will work for the separation of sharply peaked sources (Bishop et. al. 1996, Utsugi 1996). 5 CONCLUSIONS We have extracted the local geometrical structure of transformations of product distributions. By modifying the SOM algorithm we developed a network with the capability of non-parametrically separating out non-linear source mixtures. Sharply peaked sources allow for quick separation via a branched SOM network. For arbitrary source distributions, we introduce the (M,N) equipartition, the uniqueness of which provides an exact verification condition for source separation. Fundamentally, equipartition of activity is a very sensible resource allocation principle. In this work, the local equipartition coding and source separation processing proceed in tandem, resulting in optimal coding and processing of source mixtures. We believe the digital "counting" aspect of the learning rule, the learning based on the network's own coarse response to stimuli, the local nature of the dynamics, and the coupling of coding and processing make this an attractive approach from both computational and neural modeling perspectives. 542 1. K. Lin, 1. D. Cowan and D. G. Grier References Bauer, H.-U., Der, R., and Herrmann, M. 1996. Controlling the magnification factor of self-organizing feature maps. Neural Compo 8, 757-771. Bell, A. J., and Sejnowski, T. J. 1995. An information-maximization approach to blind separation and blind deconvolution. Neural Compo 7,1129-1159. Bell, A. J., and Sejnowski, T. J. 1996. Edges are the "independent components" of natural scenes. NIPS 9. Bishop, C. M. and Williams, C. 1996. GTM: A principled alternative to the selforganizing map. NIPS 9. Field, D. J. 1994. What is the goal of sensory coding? Neural Compo 6,559-601. Golubitsky, M., Stewart, 1., and Schaeffer, D. G. 1988. Singularities and Groups in Bifurcation Theory. Springer-Verlag, Berlin. Herrmann,M. and Yang, H. H. 1996. Perspectives and limitations of self-organizing maps in blind separation of source signals. Proc. lCONlP'96. Hertz, J., Krogh A., and Palmer, R. G. 1991. Introduction to the Theory of Neural Computation. Addison-Wesley, Redwood City. Kohonen, T. 1995. Self-Organizing Maps. Springer-Verlag, Berlin. Lin, J. K. and Cowan, J. D. 1996. Faithful representation of separable input distributions. To appear in Neural Computation. Olshausen, B. A. and D. J. Field 1996. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature 381, 607-609. Pajunen, P., Hyvarinen, A. and Karhunen, J. 1996. Nonlinear blind source separation by self-organizing maps. Proc. lCONIP'96. Pearlmutter, B. A. and Parra, L. 1996. Maximum likelihood blind source separation: a context-sensitive generalization of lCA. NIPS *9. Ramos, E. A. 1996. Equipartition of mass distributions by hyperplanes. Discrete Comput. Geom. 15, 147-167. Ritter, H., and Schulten, K. 1986. On the stationary state of Kohonen's selforganizing sensory mapping. Bioi. Cybern., 54,99-106. Utsugi, A. 1996. Hyperparameter selection for self-organizing maps. To appear in Neural Computation.	1996	10
1,151	Statistical Mechanics of the Mixture of Experts Kukjin Kang and Jong-Hoon Oh Department of Physics Pohang University of Science and Technology Hyoja San 31, Pohang, Kyongbuk 790-784, Korea E-mail: kkj.jhohOgalaxy.postech.ac.kr Abstract We study generalization capability of the mixture of experts learning from examples generated by another network with the same architecture. When the number of examples is smaller than a critical value, the network shows a symmetric phase where the role of the experts is not specialized. Upon crossing the critical point, the system undergoes a continuous phase transition to a symmetry breaking phase where the gating network partitions the input space effectively and each expert is assigned to an appropriate subspace. We also find that the mixture of experts with multiple level of hierarchy shows multiple phase transitions. 1 Introduction Recently there has been considerable interest among neural network community in techniques that integrate the collective predictions of a set of networks[l, 2, 3, 4]. The mixture of experts [1, 2] is a well known example which implements the philosophy of divide-and-conquer elegantly. Whereas this model are gaining more popularity in various applications, there have been little efforts to evaluate generalization capability of these modular approaches theoretically. Here we present the first analytic study of generalization in the mixture of experts from the statistical 184 K. Kang and 1. Oh physics perspective. Use of statistical mechanics formulation have been focused on the study of feedforward neural network architectures close to the multilayer perceptron[5, 6], together with the VC theory[8]. We expect that the statistical mechanics approach can also be effectively used to evaluate more advanced architectures including mixture models. In this letter we study generalization in the mixture of experts[l] and its variety with two-level hierarchy[2]. The network is trained by examples given by a teacher network with the same architecture. We find an interesting phase transition driven by symmetry breaking among the experts. This phase transition is closely related to the 'division-and-conquer' mechanism which this mixture model was originally designed to accomplish. 2 Statistical Mechanics Formulation for the Mixture of Experts The mixture of experts[2] is a tree consisted of expert networks and gating networks which assign weights to the outputs of the experts. The expert networks sit at the leaves of the tree and the gating networks sit at its branching points of the tree. For the sake of simplicity, we consider a network with one gating network and two experts. Each expert produces its output J,lj as a generalized linear function of the N dimensional input x : J,lj = /(Wj . x), j = 1,2, (1) where Wj is a weight vector of the j th expert with spherical constraint[5]. We consider a transfer function /(x) = sgn(x) which produces binary outputs. The principle of divide-and-conquer is implemented by assigning each expert to a subspace of the input space with different local rules. A gating network makes partitions in the input space and assigns each expert a weighting factor: (2) where the gating function 8(x) is the Heaviside step function. For two experts, this gating function defines a sharp boundary between the two subspace which is perpendicular to the vector V 1 = -V 2 = V, whereas the softmax function used in the original literature [2] yield a soft boundary. Now the weighted output from the mixture of expert is written: 2 J,l(V, W; x) = 2: 9j (x)J,lj (x). (3) j=1 The whole network as well as the individual experts generates binary outputs. Therefore, it can learn only dichotomy rules. The training examples are generated by a teacher with the same architecture as: 2 O'(xlJ) = 2: 8(VJ . x)sgn(WJ . x) , (4) j=1 Statistical Mechanics of the Mixture of Experts 185 where ~o and Wl are the weights of the jth gating network and the expert of the teacher. The learning of the mixture of experts is usually interpreted probabilistically, hence the learning algorithm is considered as a maximum likelihood estimation. Learning algorithms originated from statistical methods such as the EM algorithm are often used. Here we consider Gibbs algorithm with noise level T (= 1/(3) that leads to a Gibbs distribution of the weights after a long time: (5) where Z = J dV dW exp( -(3E(V, Wj)) is the partition function. Training both the experts and the gating network is necessary for a good generalization performance. The energy E of the system is defined as a sum of errors over P examples: p L f(V, W j; xl), (6) 1=1 (7) The performance of the network is measured by the generalization function f(V, W j ) = J dx f(V, Wj; x), where J dx represents an average over the whole input space. The generalization error fg is defined by fg = (((f(W))T)) where ((-.-)) denotes the quenched average over the examples and (- . -)T denotes the thermal average over the probability distribution of Eq. (5). Since the replica calculation turns out to be intractable, we use the annealed approximation: ((log Z)) ~ log((Z)) . (8) The annealed approximation is exact only in the high temperature limit, but it is known that the approximation usually gives qualitatively good results for the case of learning realizable rules[5, 6]. 3 Generalization Curve and the Phase Transition The generalization function f(V, W j) is can be written as a function of overlaps between the weight vectors of the teacher and the student: 2 2 LLPijfij (9) i=l j=l where (10) (11) 186 K. Kang and J. Oh and Rij 1 0 (12) -V··V · N' J' Rij 1 0 N Wi ·Wj . (13) is the overlap order parameters. Here, Pij is a probability that the i th expert of the student learns from examples generated by the j th expert of the teacher. It is a volume fraction in the input space where Vi . x and VJ . x are both positive. For that particular examples, the ith expert of the student gives wrong answer with probability fij with respect to the j th expert of the teacher. We assume that the weight vectors of the teacher, V 0, W~ and W~, are orthogonal to each other, then the overlap order parameters other than the oneS shown above vanish. We use the symmetry properties of the network such as Rv = RYI = R~2 = - RY2, R = Rll = R 22 , and r = R12 = R21 . The free energy also can be written as a function of three order parameters Rv, R, and r . Now we consider a thermodynamic limit where the dimension of the input space N and the number of examples P goes to infinity, keeping the ratio eY = PIN finite. By minimizing the free energy with respect to the order parameters, we find the most probable values ofthe order parameters as well as the generalization error. Fig 1.(a) plots the overlap order parameters Rv, Rand r versus eY at temperature T = 5. Examining the plot, we find an interesting phase transition driven by symmetry breaking among the experts. Below the phase transition point eYe = 51.5, the overlap between the gating networks of the teacher and the student is zero (Rv = 0) and the overlaps between the experts are symmetric (R = r). In the symmetric phase, the gating network does not have enough examples to learn proper partitioning, so its performance is not much better than a random partitioning. Consequently each expert of the student can not specialize for the subspaces with a particular local rule given by an expert of the teacher. Each expert has to learn multiple linear rules with linear structure, which leads to a poor generalization performance. Unless more than a critical amount of examples is provided, the divide-and-conquer strategy does not work. Upon crossing the critical point eYe, the system undergoes a continuous phase transition to the symmetry breaking phase. The order parameter Rv , related to the goodness of partition, begins to increase abruptly and approaches 1 with increasing eY . The gating network now provides a better partition which is close to that of the teacher. The plot of order parameter Rand r, which is overlap between experts of teacher and student, branches at eYe and approaches 1 and 0 respectively. It means that each expert specializes its role by making appropriate pair with a particular expert of the teacher. Fig. l(b) plots the generalization curve (f g versus eY) in the same scale. Though the generalization curve is continuous, the slope of the curve changes discontinuously at the transition point so that the generalization curve has Statistical Mechanics of the Mixture of Experts 187 .,..' O.S 0.6 0.4 /,/-\ -' . , , .. , -' " 0.2 ; , I I ". / " I 0 ; --.. _-- - -----. 0 20 40 60 SO 100 120 140 160 ISO ex (a) 0.5 0.45 0.4 0.35 0.3 ~.25 0.2 0.15 0.1 0.05 0 0 20 40 60 80 100 120 140 160 180 ex (b) Figure 1: (a) The overlap order parameters Rv, R, r versus 0' at T = 5. For 0' < O'c = 51.5, we find Rv = 0 (solid line that follows x axis), and R = r (dashed line). At the transition point, Rv begins to increase abruptly, R (dotted line) and r (dashed line) branches, which approach 1 and 0 respectively. (b) The generalization curve (fg versus 0') for the mixture of experts in the same scale. A cusp at the transition point O'c is shown. 188 0.5 ,...,,------,---,.---,---,--.-------, 0.45 0.4 0.35 0.3 ~.25 0.2 0.15 0.1 0.05 OL-_~ __ ~ __ ~ __ _L __ ~_~ o 50 100 150 a 200 250 300 K. Kang and J. Oh Figure 2: A typical generalization error curve for HME network with continuous weight. T = 5. a cusp. The asymptotic behavior of fg at large 0' is given by: 3 1 f ::::: f3' 1 - e0' (14) where the 1/0' decay is often observed in learning of other feedforward networks. 4 The Mixture of Experts with Two-Level Hierarchy We also study generalization in the hierarchical mixture of experts [2] . Consider a two-level hierarchical mixture of experts consisted of three gating networks and four experts. At the top level the tree is divided into two branch, and they are in turn divided into two branches at the lower level. The experts sit at the four leaves of the tree, and the three gating networks sit at the top and lower-level branching points. The network also learns from the training examples drawn from a teacher network with the same architecture. FIG 2. (b) shows corresponding learning curve which has two cusps related to the phase transitions. For 0' < O'ct, the system is in the fully symmetric phase. The gating networks do not provide correct partition for the experts at both levels of hierarchy and the experts cannot specialize at all. All the overlaps with the weights of the teacher experts have the same value. The first phase transition at the smaller 0'c1 is related to the symmetry breaking by the top-level gating network. For 0'c1 < 0' < O'c2, the top-level gating network partition the input space into two parts, but the lower-level gating network is not functioning properly. The overlap between the gating networks at the lower level of the tree and that of the teacher is still zero. The experts partially specialize into two groups. Specialization among the same group is not accomplished yet. The overlap order parameter Rij can Statistical Mechanics of the Mixture of Experts 189 have two distinct values. The bigger one is the overlap with the two experts of the teacher for which the group is specializing, and the smaller is with the experts of the teacher which belong to the other group. At the second transition point Q'c2, the symmetry related to the lower-level hierarchy breaks. For c¥ > C¥c2, all the gating networks work properly and the input space is divided into four. Each expert makes appropriate pair with an expert of the teacher. Now the overlap order parameters can have three distinct values. The largest is the overlap with matching expert of teacher. The next largest is the overlap with the neighboring teacher expert in the tree hierarchy. The smallest is with the experts of the other group. The two phase transition result in the two cusps of the learning curve. 5 Conclusion Whereas the phase transition of the mixture of experts can be interpreted as a symmetry breaking phenomenon which is similar to the one already observed in the committee machine and the multi-Iayer-perceptron[6, 7], the transition is novel in that it is continuous. This means that symmetry breaking is easier for the mixture of experts than in the multi-layer perceptron. This can be a big advantage in learning of highly nonlinear rules as we do not have to worry about the existence of local minima. We find that the hierarchical mixture of experts can have multiple phase transitions which are related to symmetry breaking at different levels. Note that symmetry breaking comes first from the higher-level branch, which is desirable property of the model. We thank M. I. Jordan, L. K. Saul, H. Sompolinsky, H. S. Seung, H. Yoon and C. K won for useful discussions and comments. This work was partially supported by the Basic Science Special Program of the POSTECH Basic Science Research Institute. References [1] R. A. Jacobs, M. I. Jordan, S. J. Nolwan, and G. E. Hinton, Neural Computation 3, 79 (1991). [2] M. I. Jordan, and R. A. Jacobs, Neural Computation 6, 181 (1994). [3] M.P. Perrone and L. N. Cooper, Neural Networks for Speech and Image Processing, R. J. Mammone. Ed., Chapman-Hill, London, 1993. [4] D. Wolpert, Neural Networks, 5, 241 (1992). [5] H. S. Seung, H. Sompolinsky, and N. Tishby, Phys. Rev. A 45, 6056 (1992) . [6] K. Kang, J.-H. Oh, C. Kwon and Y. Park, Phys. Rev. E 48, 4805 (1993); K. Kang, J .-H. Oh, C. Kwon and Y. Park, Phys. Rev. E 54, 1816 (1996). [7] E. Baum and D. Haussler, Neural Computation 1, 151 (1989).	1996	100
1,152	Predicting Lifetimes in Dynamically Allocated Memory David A. Cohn Adaptive Systems Group Harlequin, Inc. Menlo Park, CA 94025 cohn~harlequin.com Satinder Singh Department of Computer Science University of Colorado Boulder, CO 80309 baveja~cs.colorado.edu Abstract Predictions oflifetimes of dynamically allocated objects can be used to improve time and space efficiency of dynamic memory management in computer programs. Barrett and Zorn [1993] used a simple lifetime predictor and demonstrated this improvement on a variety of computer programs. In this paper, we use decision trees to do lifetime prediction on the same programs and show significantly better prediction. Our method also has the advantage that during training we can use a large number of features and let the decision tree automatically choose the relevant subset. 1 INTELLIGENT MEMORY ALLOCATION Dynamic memory allocation is used in many computer applications. The application requests blocks of memory from the operating system or from a memory manager when needed and explicitly frees them up after use. Typically, all of these requests are handled in the same way, without any regard for how or for how long the requested block will be used. Sometimes programmers use runtime profiles to analyze the typical behavior of their program and write special purpose memory management routines specifically tuned to dominant classes of allocation events. Machine learning methods offer the opportunity to automate the process of tuning memory management systems. In a recent study, Barrett and Zorn [1993] used two allocators: a special allocator for objects that are short-lived, and a default allocator for everything else. They tried a simple prediction method on a number of public-domain , allocation-intensive programs and got mixed results on the lifetime prediction problem. Nevertheless, they showed that for all the cases where they were able to predict well, their strategy of assigning objects predicted to be short-lived to the special allocator led to savings 940 D. A. Cohn and S. Singh in program running times. Their results imply that if we could predict well in all cases we could get similar savings for all programs. We concentrate on the lifetime prediction task in this paper and show that using axis-parallel decision trees does indeed lead to significantly better prediction on all the programs studied by Zorn and Grunwald and some others that we included. Another advantage of our approach is that we can use a large number of features about the allocation requests and let the decision tree decide on their relevance. There are a number of advantages of using lifetime predictions for intelligent memory management. It can improve CPU usage, by using special-purpose allocators, e.g., short-lived objects can be allocated in small spaces by incrementing a pointer and deallocated together when they are all dead. It can decrease memory fragmentation, because the short-lived objects do not pollute the address space of long lived objects. Finally, it can improve program locality, and thus program speed, because the short-lived objects are all allocated in a small part of the heap. The advantages of prediction must be weighed against the time required to examine each request and make that prediction about its intended use. It is frequently argued that, as computers and memory become faster and cheaper, we need to be less concerned about the speed and efficiency of machine learning algorithms. When the purpose of the algorithm is to save space and computation, however, these concerns are paramount. 1.1 RELATED WORK Traditionally, memory management has been relegated to a single, general-purpose allocator. When performance is critical, software developers will frequently build a custom memory manager which they believe is tuned to optimize the performance of the program. Not only is this hand construction inefficient in terms of the programming time required, this "optimization" may seriously degrade the program's performance if it does not accurately reflect the program's use [Wilson et al., 1995]. Customalloc [Grunwald and Zorn, 1992] monitors program runs on benchmark inputs to determine the most commonly requested block sizes. It then produces a set of memory allocation routines which are customized to efficiently allocate those block sizes. Other memory requests are still handled by a general purpose allocator. Barrett and Zorn [1993] studied lifetime prediction based on benchmark inputs. At each allocation request, the call graph (the list of nested procedure/function calls in effect at the time) and the object size was used to identify an allocation site. If all allocations from a particular site were short-lived on the benchmark inputs, their algorithm predicted that future allocations would also be short-lived. Their method produced mixed results at lifetime prediction, but demonstrated the savings that such predictions could bring. In this paper, we discuss an approach to lifetime prediction which uses learned decision trees. In the next section, we first discuss the identification of relevant state features by a decision tree. Section 3 discusses in greater detail the problem of lifetime prediction. Section 4 describes the empirical results of applying this approach to several benchmark programs, and Section 5 discusses the implications of these results and directions for future work. Predicting Lifetimes in Dynamically Allocated Memory 941 2 FEATURE SELECTION WITH A DECISION TREE Barrett and Zorn's approach captures state information in the form of the program's call graph at the time of an allocation request, which is recorded to a fixed predetermined depth. This graph, plus the request size, specifies an allocation "site"; statistics are gathered separately for each site. A drawback of this approach is that it forces a division for each distinct call graph, preventing generalization across irrelevant features. Computationally, it requires maintaining an explicit call graph (information that the program would not normally provide), as well as storing a potentially large table of call sites from which to make predictions. It also ignores other potentially useful information, such as the parameters of the functions on the call stack, and the contents of heap memory and the program registers at the time of the request. Ideally, we would like to examine as much of the program state as possible at the time of each allocation request, and automatically extract those pieces of information that best allow predicting how the requested block will be used. Decision tree algorithms are useful for this sort of task. A decision tree divides inputs on basis of how each input feature improves "purity" of the tree's leaves. Inputs that are statistically irrelevant for prediction are not used in any splits; the tree's final set of decisions examine only input features that improve its predictive performance. Regardless of the parsimony of the final tree however, training a tree with the entire program state as a feature vector is computationally infeasible. In our experiments, detailed below, we arbitrarily used the top 20 words on the stack, along with the request size, as an approximate indicator of program state. On the target machine (a Sparcstation) , we found that including program registers in the feature set made no significant difference, and so dropped them from consideration for efficiency. 3 LIFETIME PREDICTION The characteristic of memory requests that we would like to predict is the lifetime of the block - how long it will be before the requested memory is returned to the central pool. Accurate lifetime prediction lets one segregate memory into shortterm, long-term and permanent storage. To this end, we have used a decision tree learning algorithm to derive rules that distinguish "short-lived" and "permanent" allocations from the general pool of allocation requests. For short-lived blocks, one can create a very simple and efficient allocation scheme [Barrett and Zorn, 1993]. For "permanent" blocks, allocation is also simple and cheap, because the allocator does not need to compute and store any of the information that would normally be required to keep track of the block and return it to the pool when freed. One complication is that of unequal loss for different types of incorrect predictions. An appropriately routed memory request may save dozens of instruction cycles, but an inappropriately routed one may cost hundreds. The cost in terms of memory may also be unequal: a short-lived block that is incorrectly predicted to be "permanent" will permanently tie up the space occupied by the block (if it is allocated via a method that can not be freed). A "permanent" block, however, that is incorrectly predicted to be short-lived may pollute the allocator's short-term space by preventing a large segment of otherwise free memory from being reclaimed (see Barrett and Zorn for examples). These risks translate into a time-space tradeoff that depends on the properties of 942 D. A. Cohn and S. Singh the specific allocators used and the space limitations of the target machine. For our experiments, we arbitrarily defined false positives and false negatives to have equal loss, except where noted otherwise. Other cases may be handled by reweighting the splitting criterion, or by rebalancing the training inputs (as described in the following section). 4 EXPERIMENTS We conducted two types of experiments. The first measured the ability of learned decision trees to predict allocation lifetimes. The second incorporated these learned trees into the target applications and measured the change in runtime performance. 4.1 PREDICTIVE ACCURACY We used the OC1 decision tree software (designed by Murthy et al. [1994]) and considered only axis-parallel splits, in effect, conditioning each decision on a single stack feature. We chose the sum minority criterion for splits, which minimizes the number of training examples misclassified after the split. For tree pruning, we used the cost complexity heuristic, which holds back a fraction (in our case 10%) of the data set for testing, and selects the smallest pruning of the original tree that is within one standard error squared ofthe best tree [Breiman et al. 1984]. The details of these and other criteria may be found in Murthy et al. [1994] and Breiman et al. [1984]. In addition to the automatically-pruned trees, we also examined trees that had been truncated to four leaves, in effect examining no more than two features before making a decision. OC1 includes no provisions for explicitly specifying a loss function for false positive and false negative classifications. It would be straightforward to incorporate this into the sum minority splitting criterion; we chose instead to incorporate the loss function into the training set itself, by duplicating training examples to match the target ratios (in our case, forcing an equal number of positive and negative examples). In our experiments, we used the set of benchmark applications reported on by Barrett and Zorn: Ghostseript, a PostScript interpreter, Espresso, a PLA logic optimizer, and Cfrae, a program for factoring large numbers, Gawk, an AWK programming language interpreter and Perl, a report extraction language. We also examined Gee, a public-domain C compiler, based on our company's specific interest in compiler technology. The experimental procedure was as follows: We linked the application program with a modified mal/oe routine which, in addition to allocating the requested memory, wrote to a trace file the size of the requested block, and the top 20 machine words on the program stack. Calls to free allowed tagging the existing allocations, which, following Barrett and Zorn, were labeled according to how many bytes had been allocated during their lifetime. 1 It is worth noting that these experiments were run on a Sparcstation, which frequently optimizes away the traditional stack frame. While it would have been possible to force the system to maintain a traditional stack, we wished to work from whatever information was available from the program "in the wild" , without overriding system optimizations. 1 We have also examined, with comparable success, predicting lifetimes in terms of the number of intervening calls to malloc; which may be argued as an equally useful measure. We focus on number of bytes for the purposes of comparison with the existing literature. Predicting Lifetimes in Dynamically Allocated Memory 943 Input files were taken from the public ftp archive made available by Zorn and Grunwald [1993]. Our procedure was to take traces of three of the files (typically the largest three for which we could store an entire program trace). Two of the traces were combined to form a training set for the decision tree, and the third was used to test the learned tree. Ghostseript training files: manual.ps and large.ps; test file: ud-doc.ps Espresso training files: cps and mlp4; test file: Z5xp1 Cfrae training inputs: 41757646344123832613190542166099121 and 327905606740421458831903; test input: 417576463441248601459380302877 Gawk training file: adj.awk/words-small.awk; test file: adj.awk/words-Iarge.awk2 Perl training files: endsort.perl (endsort.perl as input), hosts.perl (hosts-data.perl as input); test file: adj.perl(words-small.awk as input) Gee training files: cse.c and combine.c; test file: expr.c 4.1.1 SHORT-LIVED ALLOCATIONS First, we attempted to distinguish short-lived allocations from the general pool. For comparison with Barrett and Zorn [1993], we defined "short-lived" allocations as those that were freed before 32k subsequent bytes had been allocated. The experimental results of this section are summarized in Table 1. Barrett &c Zorn OC1 application false pos % false neg % false pos % false neg % ghostscnpt ° 25.2 0.13 \0.72) 1.7 \13.5) espresso 0.006 72 0.38 (1.39) 6.58 (14.9) cfrac 3.65 52.7 2.5 (0.49) 16.9 (19.4) gawk 0 -3 0.092 (0.092) 0.34 (0.34) perl 1.11 78.6 5.32 (10.8) 33.8 (34.3) gcc 0.85 (2.54) 31.1 (31.0) Table 1: Prediction errors for "short-lived" allocations, in percentages of misallocated bytes. Values in parentheses are for trees that have been truncated to two levels. Barrett and Zorn's results included for comparison where available. 4.1.2 "PERMANENT" ALLOCATIONS We then attempted to distinguish "permanent" allocations from the general pool (Barrett and Zorn only consider the short-lived allocations discussed in the previous section). "Permanent" allocations were those that were not freed until the program terminated. Note that there is some ambiguity in these definitions a "permanent" block that is allocated near the end of the program's lifetime may also be "shortlived". Table 2 summarizes the results of these experiments. We have not had the opportunity to examine the function of each of the "relevant features" in the program stacks; this is a subject for future work. 2For Gawk, we varied the training to match that used by Barrett and Zorn. They used as training input a single gawk program file run with one data set, and tested on the same gawk program run with another. 3We were unable to compute Barrett and Zorn's exact results here, although it appears that their false negative rate was less than 1%. 944 D. A. Cohn and S. Singh application false pos % false neg % ghostscript 0 0.067 espresso 0 1.27 cfrac 0.019 3.3 gcc 0.35 19.5 Table 2: Prediction errors for "permanent" allocations (% misallocated bytes). 4.2 RUNTIME PERFORMANCE The raw error rates we have presented above indicate that it is possible to make accurate predictions about the lifetime of allocation requests, but not whether those predictions are good enough to improve program performance. To address that question, we have incorporated predictive trees into three of the above applications and measured the effect on their runtimes. We used a hybrid implementation, replacing the single monolithic decision tree with a number of simpler, site-specific trees. A "site" in this case was a lexical instance of a call to malloc or its equivalent. When allocations from a site were exclusively short-lived or permanent, we could directly insert a call to one of the specialized allocators (in the manner of Barrett and Zorn). When allocations from a site were mixed, a site-specific tree was put in place to predict the allocation lifetime. Requests predicted to be short-lived were routed to a "quick malloc" routine similar to the one described by Barrett and Zorn; those predicted to be permanent were routed to another routine specialized for the purpose. On tests with random data these specialized routines were approximately four times faster than "malloc". Our experiments targeted three applications with varying degrees of predictive accuracy: ghostscript, gcc, and cfrac. The results are encouraging (see Table 3). For ghostscript and gcc, which have the best predictive accuracies on the benchmark data (from Section 4.1), we had a clear improvement in performance. For cfrac, with much lower accuracy, we had mixed results: for shorter runs, the runtime performance was improved, but on longer runs there were enough missed predictions to pollute the short-lived memory area and degrade performance. 5 DISCUSSION The application of machine learning to computer software and operating systems is a largely untapped field with promises of great benefit. In this paper we have described one such application, producing efficient and accurate predictions of the lifetimes of memory allocations. Our data suggest that, even with a feature set as large as a runtime program stack, it is possible to characterize and predict the memory usage of a program after only a few benchmark runs. The exceptions appear to be programs like Perl and gawk which take both a script and a data file. Their memory usage depends not only upon characterizing typical scripts, but the typical data sets those scripts act upon.4 Our ongoing research in memory management is pursuing a number of other con4Perl's generalization performance is significantly better when tested on the same script with different data. We have reported the results using different scripts for comparison with Barrett and Zorn. Predicting Lifetimes in Dynamically Allocated Memory 945 application benchmark test error run tIme {training set) short I long I permanent normal I predictive ghostscript, trained on ud-doc.ps; 7 sites, 1 tree manual.ps 16/256432 0/3431 0/0 96.29 95.43 large.ps 17.22 16.75 thesis.ps 40.27 37.57 gcc, trained on combine, cse, c-decl; 17 sites, 4 trees expr.c 0/11988 2786/11998 301/536875 12.59 12.40 loop.c 5.16 5.16 reload1.c 7.02 6.81 cfrac, trained on 100 · . ·057; 8 sItes, 4 trees 327 .. ·903 24/7970099 13172/22332 106/271 7.75 7.23 417· · ·771 67.93 74.57 417 .. ·121 225.31 245.64 Table 3: Running times in seconds for applications with site-specific trees. Times shown are averages over 24-40 runs, and with the exception of loop.c, are statistically significant with probability greater than 99%. tinuations of the results described here, including lifetime clustering and intelligent garbage collection. REFERENCES D. Barrett and B. Zorn (1993) Using lifetime predictors to improve memory allocation performance. SIGPLAN'93 - Conference on Programming Language Design and Implementation, June 1993, Albuquerque, New Mexico, pp. 187-196. L. Breiman, J. Friedman, R. Olshen and C. Stone (1984) Classification and Regression Trees, Wadsworth International Group, Belmont, CA. D. Grunwald and B. Zorn (1992) CUSTOMALLOC: Efficient synthesized memory allocators. Technical Report CU-CS-602-92, Dept. of Computer Science, University of Colorado. S. Murthy, S. Kasif and S. Salzberg (1994) A system for induction of oblique decision trees. Journal of Artificial Intelligence Research 2:1-32. P. Wilson, M. Johnstone, M. Neely and D. Boles (1995) Dynamic storage allocation: a survey and critical review. Proc. 1995 Intn'l Workshop on Memory Management, Kinross, Scotland, Sept. 27-29, Springer Verlag. B. Zorn and D. Grunwald (1993) A set of benchmark inputs made publicly available, in ftp archive ftp. cs. colorado . edu: /pub/misc/malloc-benchmarks/.	1996	101
1,153	Reinforcement Learning for Mixed Open-loop and Closed-loop Control Eric A. Hansen, Andrew G. Barto, and Shlorno Zilberstein Department of Computer Science University of Massachusetts Amherst, MA 01003 {hansen.barto.shlomo }<Dcs.umass.edu Abstract Closed-loop control relies on sensory feedback that is usually assumed to be free. But if sensing incurs a cost, it may be costeffective to take sequences of actions in open-loop mode. We describe a reinforcement learning algorithm that learns to combine open-loop and closed-loop control when sensing incurs a cost. Although we assume reliable sensors, use of open-loop control means that actions must sometimes be taken when the current state of the controlled system is uncertain. This is a special case of the hidden-state problem in reinforcement learning, and to cope, our algorithm relies on short-term memory. The main result of the paper is a rule that significantly limits exploration of possible memory states by pruning memory states for which the estimated value of information is greater than its cost. We prove that this rule allows convergence to an optimal policy. 1 Introduction Reinforcement learning (RL) is widely-used for learning closed-loop control policies. Closed-loop control works well if the sensory feedback on which it relies is accurate, fast, and inexpensive. But this is not always the case. In this paper, we address problems in which sensing incurs a cost, either a direct cost for obtaining and processing sensory data or an indirect opportunity cost for dedicating limited sensors to one control task rather than another. If the cost for sensing is significant, exclusive reliance on closed-loop control may make it impossible to optimize a performance measure such as cumulative discounted reward. For such problems, we describe an RL algorithm that learns to combine open-loop and closed-loop control. By learning to take open-loop sequences of actions between sensing, it can optimize a tradeoff'between the cost and value of sensing. Reinforcement Learning for Mixed Open-loop and Closed-loop Control 1027 The problem we address is a special case of the problem of hidden state or partial observability in RL (e.g., Whitehead &. Lin, 1995; McCallum, 1995). Although we assume sensing provides perfect information (a significant limiting assumption), use of open-loop control means that actions must sometimes be taken when the current state of the controlled system is uncertain. Previous work on RL for partially observable environments has focused on coping with sensors that provide imperfect or incomplete information, in contrast to deciding whether or when to sense. Tan (1991) addressed the problem of sensing costs by showing how to use RL to learn a cost-effective sensing procedure for state identification, but his work addressed the question of which sensors to use, not when to sense, and so still assumed closed-loop control. In this paper, we formalize the problem of mixed open-loop and closed-loop control as a Markov decision process and use RL in the form of Q-Iearning to learn an optimal, state-dependent sensing interval. Because there is a combinatorial explosion of open-loop action sequences, we introduce a simple rule for pruning this large search space. Our most significant result is a proof that Q-Iearning converges to an optimal policy even when a fraction of the space of possible open-loop action sequences is explored. 2 Q-learning with sensing costs Q-Iearning (Watkins, 1989) is a well-studied RL algorithm for learning to control a discrete-time, finite state and action Markov decision process (MDP). At each time step, a controller observes the current state x, takes an action a, and receives an immediate reward r with expected value r(x, a). With probability p(x, a, y) the process makes a transition to state y, which becomes the current state on the next time step. A controller using Q-Iearning learns a state-action value function, Q(x, a), that estimates the expected total discounted reward for taking action a in state x and performing optimally thereafter. Each time step, Q is updated for state-action pair (x, a) after receiving reward r and observing resulting state y, as follows: Q(x, a) ~ Q(x, a) + Q: [r + I'V(y) - Q(x, a)] , where Q: E (0,1] is a learning rate parameter, I' E [0,1) is a discount factor, and V(y) = maXb Q(y, b). Watkins and Dayan (1992) prove that Q converges to an optimal state-action value function Q (and V converges to an optimal state value function V) with probability one if all actions continue to be tried from all states, the state-action value function is represented by a lookup-table, and the learning rate is decreased in an appropriate manner. If there is a cost for sensing, acting optimally may require a mixed strategy of openloop and closed-loop control that allows a controller to take open-loop sequences of actions between sensing. This possibility can be modeled by an MDP with two kinds of actions: control actions that have an effect on the current state but do not provide information, and a sensing action that reveals the current state but has no other effect. We let 0 (for observation) denote the sensing action and assume it provides perfect information about the underlying state. Separating control actions and the sensing action gives an agent control over when to receive sensory feedback, and hence, control over sensing costs. When one control action follows another without an intervening sensing action, the second control action is taken without knowing the underlying state. We model this by including "memory states" in the state set of the MDP. Each memory state represents memory of the last observed state and the open-loop sequence of control actions taken since; because we assume sensing provides perfect information, 1028 E. A. Hansen, A. G. Barto and S. Zilberstein Xcl Z xaa xah ~X'~'I xh ~ xhb Figure 1: A tree of memory states rooted at observed state x. The set of control actions is {a, b} and the length bound is 2. remembering this much history provides a sufficient statistic for action selection (Monahan, 1982). Possible memory states can be represented using a tree like the one shown in Figure 1, where the root represents the last observed state and the other nodes represent memory states, one for each possible open-loop action sequence. For example, let xa denote the memory state that results from taking control action a in state x. Similarly, let xab denote the memory state that results from taking control action b in memory state xa. Note that a control action causes a deterministic transition to a subsequent memory state, while a sensing action causes a stochastic transition to an observed state - the root of some tree. There is a tree like the one in figure 1 for each observable state. This problem is a special case of a partially observable MDP and can be formalized in an analogous way (Monahan, 1982). Given a state-transition and reward model for a core MDP with a state set that consists only of the underlying states of a system (which for this problem we also call observable states), we can define a statetransition and reward model for an MDP that includes memory states in its state set. As a convenient notation, let p(x, al .. a", y) denote the probability that taking an open-loop action sequence a} .. a" from state x results in state y, where both x and y are states of the underlying system. These probabilities can be computed recursively from the single-step state-transition probabilities of the core MDP as follows: z State-transition probabilities for the sensing action can then be defined as p(xal .. a" , 0, y) = p(x, al .. a", y), and a reward function for the generalized MDP can be similarly defined as r(xal .. a"_l, a,,) = LP(x, al .. a"_l, y)r(y, a,,). y where the cost of sensing in state x of the core MDP is r(x,o). If we assume a bound on the number of control actions that can be taken between sensing actions (i.e .• a bound on the depth of each tree) and also assume a finite number of underlying states, the number of possible memory states is finite. It follows that the MDP we have constructed is a well-defined finite state and action MDP, and all of the theory developed for Q-Iearning continues to apply, including its convergence proof. (This is not true of partially observable MDPs in general.) Therefore, Q-Iearning can in principle find an optimal policy for interleaving control actions and sensing, assuming sensing provides perfect information. 3 Limiting Exploration A problem with including memory states in the state set of an MDP is that it increases the size of the state set exponentially. The combinatorial explosion of Reinforcement Learningfor Mixed Open-loop and Closed-loop Control 1029 state-action values to be learned raises doubt about the computational feasibility of this generalization of RL. We present a solution in the form of a rule for pruning each tree of memory states, thereby limiting the number of memory states that must be explored. We prove that even if some memory states are never explored, Q-Iearning converges to an optimal state-action value function. Because the state-action value function is left undefined for unexplored memory states, we must carefully define what we mean by an optimal state-action value function. Definition: A state-action value function is optimal if it is sufficient for generating optimal behavior and the values of the state-action pairs visited when behaving optimally are optimal. A state-action value function that is undefined for some states is optimal, by this definition, if a controller that follows it behaves identically to a controller with a complete, optimal state-action value function. This is possible if the states for which the state-action value function is undefined are not encountered when an agent acts optimally. Barto, Bradtke, and Singh (1995) invoke a similar idea for a different class of problems. Let g(xal .. ak) denote the expected reward for taking actions al .. ak in open-loop mode after observing state x: k-l g(xa1 .. ak) = r(x, ad + L -yir(xal .. ai, ai+d· i=l Let h(xa1 .. ak) denote the discounted expected value of perfect information after reaching memory state xa1 .. ak, which is equal to the discounted Q-value for sensing in memory state xal .. ak minus the cost for sensing in this state: h(xa1 .. ak) = -y" LP(xal .. ak,o,y)V(y) = -yk(Q(xa1 .. ak,o) - r(xal .. ak,o)). y Both g and h are easily learned during Q-Iearning, and we refer to the learned estimates as 9 and h. These are used in the pruning rule, as follows: Pruning rule: If g( xal .. ak) + h{ xa1' .ak) ~ V (x), then memory states that descend from xal· .ak do not need to be explored. A controller should immediately execute a sensing action when it reaches one of these memory states. The intuition behind the pruning rule is that a branch of a tree of memory states can be pruned after reaching a memory state for which the value of information is greater than or equal to its cost. Because pruning is based on estimated values, memory states that are pruned at one point during learning may later be explored as learned estimates change. The net effect of pruning, however, is to focus exploration on a subset of memory states, and as Q-Iearning converges, the subset of unpruned memory states becomes stable. The following theorem is proved in an appendix. Theorem: Q-learning converges to an optimal state-action value function with probability one if, in addition to the conditions for convergence given by Watkins and Dayan (1992), exploration is limited by the pruning rule. This result is closely related to a similar result for solving this class of problems using dynamic programming (Hansen, 1997), where it is shown that pruning can assure convergence to an optimal policy even if no bound is placed on the length of open-loop action sequences - under the assumption that it is optimal to sense at finite intervals. This additional result can be extended to Q-Iearning as well, although we do not present the extension in this paper. An artificial length bound can be set as low or high as desired to ensure a finite set of memory states. 1030 E. A. Hansen, A. G. Barto and S. Zilberstein ~ rItli ll I ~ 3 (in.1i Sill" 7 WN:-iNO I~ NWNO I WO X WWNNNO I ~ WWO '4 '5 :'>10 C) NWO I~ wwwo 6 7 X .1 "\10 III WNwn 17 NNNO l) 10 II I~ ~ :>;:>;0 II WW() 11\ WNNNO 13 14 1510 ~ NNWO 12 WWWO 1<) WWNO 17 IX 19 20 6 NNNO L' NNO 20 WWWNO (h) Figure 2: (a) Grid world with numbered states (b) Optimal policy We use the notation 9 and h in our statement of the pruning rule to emphasize its relationship to pruning in heuristic search. If we regard the root of a tree of memory states as the start state and the memory state that corresponds to the best open-loop action sequence as the goal state, then 9 can be regarded as the cost-to-arrive function and the value of perfect information h can be regarded as an upper bound on the cost-to-go function. 4 Example We describe a simple example to illustrate the extent of pruning possible using this rule. Imagine that a "robot" must find its way to a goal location in the upper left-hand corner of the grid shown in Figure 2a. Each cell of the grid corresponds to a state, with the states numbered for convenient reference. The robot has five control actions; it can move north, east, south, or west, one cell at a time, or it can stop. The problem ends when the robot stops. If it stops in the goal state it receives a reward of 100, otherwise it receives no reward. The robot must execute a sequence of actions to reach the goal state, but its move actions are stochastic. If the robot attempts to move in a particular direction, it succeeds with probability o.s. With probability 0.05 it moves in a direction 90 degrees off to one side of its intended direction, with probability 0.05 it moves in a direction 90 degrees off to the other side, and with probability 0.1 it does not move at all. If the robot's movement would take it outside the grid, it remains in the same cell. Because its progress is uncertain, the robot must interleave sensing and control actions to keep track of its location. The reward for sensing is - 1 (i.e., a cost of 1) and for each move action it is -4. To optimize expected total reward, the robot must find its way to the goal while minimizing the combined cost of moving and sensing. Figure 2b shows the optimal open-loop sequence of actions for each observable state. If the bound on the length of an open-loop sequence of control actions is five, the number of possible memory states for this problem is over 64,000, a number that grows explosively as the length bound is increased (to over 16 million when the bound is nine) . Using the pruning rule, Q-Iearning must explore just less than 1000 memory states (and no deeper than nine levels in any tree) to converge to an optimal policy, even when there is no bound on the interval between sensing actions. 5 Conclusion We have described an extension of Q-Iearning for MDPs with sensing costs and a rule for limiting exploration that makes it possible for Q-Iearning to converge to an optimal policy despite exploring a fraction of possible memory states. As already pointed out, the problem we have formalized is a partially observable MDP, Reinforcement Learningfor Mixed Open-loop and Closed-loop Control 1031 although one that is restricted by the assumption that sensing provides perfect information. An interesting direction in which to pursue this work would be to explore its relationship to work on RL for partially observable MDPs, which has so far focused on the problem of sensor uncertainty and hidden state. Because some of this work also makes use of tree representations of the state space and of learned state-action values (e.g., McCallum, 1995), it may be that a similar pruning rule can constrain exploration for such problems. Acknowledgement s Support for this work was provided in part by the National Science Foundation under grants ECS-9214866 and IRI-9409827 and in part by Rome Laboratory, USAF, under grant F30602-95-1-0012. References Barto, A.G.; Bradtke, S.J.; &. Singh, S.P. (1995) Learning to act using real-time dynamic programming. Artificial Intelligence 72(1/2}:81-138. Hansen, E.A. (1997) Markov decision processes with observation costs. University of Massachusetts at Amherst, Computer Science Technical Report 97-01. McCallum, R.A. (1995) Instance-based utile distinctions for reinforcement learning with hidden state. In Proc. 12th Int. Machine Learning Conf. Morgan Kaufmann. Monahan, G.E. (1982) A survey of partially observable Markov decision processes: Theory, models, and algorithms. Management Science 28:1-16. Tan, M. (1991) Cost-sensitive reinforcement learning for adaptive classification and control. In Proc. 9th Nat. Conf. on Artificial Intelligence. AAAI Press/MIT Press. Watkins, C.J.C.H. (1989) Learning from delayed rewards. Ph.D. Thesis, University of Cambridge, England. Watkins, C.J.C.H. &. Dayan, P. (1992) Technical note: Q-Iearning. Machine Learning 8(3/4}:279-292. Whitehad, S.D. &. Lin, L.-J.(1995} Reinforcement learning of non-Markov decision processes. Artificial Intelligence 73:271-306. Appendix Proof of theorem: Consider an MDP with a state set that consists only of the memory states that are not pruned. We call it a "pruned MDP" to distinguish it from the original MDP for which the state set consists of all possible memory states. Because the pruned MDP is a finite state and action MDP, Q-Iearning with pruning converges with probability one. What we must show is that the state-action values to which it converges include every state-action pair visited by an optimal controller for the original MDP, and that for each of these state-action pairs the learned state-action value is equal to the optimal state-action value for the original MDP. Let Q and if denote the values that are learned by Q-Iearning when its exploration is limited by the pruning rule, and let Q and V denote value functions that are optimal when the state set ofthe MDP includes all possible memory states. Because an MDP has an optimal stationary policy and each control action causes a deterministic transition to a subsequent memory state, there is an optimal path through each tree of memory states. The learned value of the root state of each tree is optimal if and only if the learned value of each memory state along this path is also optimal. 1032 E. A. Hansen, A. G. Barto and S. Zilberstein Therefore to show that Q-Iearning with pruning converges to an optimal stateaction value function, it is sufficient to show that V = V for every observable state x. Our proof is by induction on the number of control actions that can be taken between one sensing action and the next. We use the fact that if Q-Iearning has converged, then g(xal .. ai) = g(xal .. ai) and h(xat .. ai) = Eyp(x,al .. ai'Y)V(y) for every memory state xat .. ai . First note that if g(xat) + 1'r(xal' o) + h(xat} > V(x), that is, if V for some observable state x can be improved by exploring a path of a single control action followed by sensing, then it is contradictory to suppose Q-Iearning with pruning has converged because single-depth memory states in a tree are never pruned. Now, make the inductive hypothesis that Q-Iearning with pruning has not converged if V can be improved for some observable state by exploring a path of less than k control actions before sensing. We show that it has not converged if V can be improved for some observable state by exploring a path of k control actions before sensing. Suppose V for some observable state x can be improved by exploring a path that consists of taking the sequence of control actions at .. aA: before sensing, that is, A: A A g(xat .. aA:) + l' r(xat .. aA:, o) + h(xat .. aA:) > V(x), Since only pruning can prevent improvement in this case, let xat .. a, be the memory state at which application of the pruning rule prevents xal .. aA: from being explored. Because the tree has been pruned at this node, V(x) 2:: g(xat .. ai) + h(xat .. ai), and so A: A A g(xat .. aA:) + l' r(xat .. aA:, o) + h(xat .. aA:) > g(xai .. ai) + h(xat .. ai). We can expand this inequality as follows: g(xat .. a,;) + 1" L p(x, al .. ai, y) [g(ya,+1 .. aA:) + 1'A:-ir(yai+t .. aA:, 0) + h(yai+t .. aA:)] y > g(xat .. a,;) + h(:z:at .. ai)' Simplification and expansion of h yields L p(x, at .. ai, y) [g(yai+t .. aA:) + 1'A:-ir(yai+t .. aA:, 0) + 1'A:-i L p(y, ai+1 .. aA:, Z)V(Z)] yES z > L p(x, al .. ai, y)V(y). y Therefore, there is some observable state, y, such that z Because the value of observable state y can be improved by taking less than k control actions before sensing, by the inductive hypothesis Q-Iearning has not yet converged. 0 The proof provides insight into how pruning works. If a state-action pair along some optimal path is temporarily pruned, it must be possible to improve the value of some observable state by exploring a shorter path of memory states that has not been pruned. The resulting improvement of the value function changes the threshold for pruning and the state-action pair that was formerly pruned may no longer be so, making further improvement of the learned value function possible.	1996	102
1,154	Computing with infinite networks Christopher K. I. Williams Neural Computing Research Group Department of Computer Science and Applied Mathematics Aston University, Birmingham B4 7ET, UK c.k.i.williamsGaston.ac.nk Abstract For neural networks with a wide class of weight-priors, it can be shown that in the limit of an infinite number of hidden units the prior over functions tends to a Gaussian process. In this paper analytic forms are derived for the covariance function of the Gaussian processes corresponding to networks with sigmoidal and Gaussian hidden units. This allows predictions to be made efficiently using networks with an infinite number of hidden units, and shows that, somewhat paradoxically, it may be easier to compute with infinite networks than finite ones. 1 Introduction To someone training a neural network by maximizing the likelihood of a finite amount of data it makes no sense to use a network with an infinite number of hidden units; the network will "overfit" the data and so will be expected to generalize poorly. However, the idea of selecting the network size depending on the amount of training data makes little sense to a Bayesian; a model should be chosen that reflects the understanding of the problem, and then application of Bayes' theorem allows inference to be carried out (at least in theory) after the data is observed. In the Bayesian treatment of neural networks, a question immediately arises as to how many hidden units are believed to be appropriate for a task. Neal (1996) has argued compellingly that for real-world problems, there is no reason to believe that neural network models should be limited to nets containing only a "small" number of hidden units. He has shown that it is sensible to consider a limit where the number of hidden units in a net tends to infinity, and that good predictions can be obtained from such models using the Bayesian machinery. He has also shown that for fixed hyperparameters, a large class of neural network models will converge to a Gaussian process prior over functions in the limit of an infinite number of hidden units. 296 C. K. I. Williams Neal's argument is an existence proof-it states that an infinite neural net will converge to a Gaussian process, but does not give the covariance function needed to actually specify the particUlar Gaussian process. In this paper I show that for certain weight priors and transfer functions in the neural network model, the covariance function which describes the behaviour of the corresponding Gaussian process can be calculated analytically. This allows predictions to be made using neural networks with an infinite number of hidden units in time O( n3 ), where n is the number of training examplesl . The only alternative currently available is to use Markov Chain Monte Carlo (MCMC) methods (e.g. Neal, 1996) for networks with a large (but finite) number of hidden units. However, this is likely to be computationally expensive, and we note possible concerns over the time needed for the Markov chain to reach equilibrium. The availability of an analytic form for the covariance function also facilitates the comparison of the properties of neural networks with an infinite number of hidden units as compared to other Gaussian process priors that may be considered. The Gaussian process analysis applies for fixed hyperparameters B. If it were desired to make predictions based on a hyperprior P( B) then the necessary B-space integration could be achieved by MCMC methods. The great advantage of integrating out the weights analytically is that it dramatically reduces the dimensionality of the MCMC integrals, and thus improves their speed of convergence. 1.1 From priors on weights to priors on functions Bayesian neural networks are usually specified in a hierarchical manner, so that the weights ware regarded as being drawn from a distribution P(wIB). For example, the weights might be drawn from a zero-mean Gaussian distribution, where B specifies the variance of groups of weights. A full description of the prior is given by specifying P( B) as well as P( wIB). The hyperprior can be integrated out to give P(w) = J P(wIB)P(B) dB, but in our case it will be advantageous not to do this as it introduces weight correlations which prevent convergence to a Gaussian process. In the Bayesian view of neural networks, predictions for the output value y .. corresponding to a new input value x .. are made by integrating over the posterior in weight space. Let D = ((XI,t1),(xz,tz), ... ,(xn,tn» denote the n training data pairs, t = (tl'" .,tnl and ! .. (w) denote the mapping carried out by the network on input x .. given weights w. P(wlt, B) is the weight posterior given the training dataz. Then the predictive distribution for y .. given the training data and hyperparameters B is (1) We will now show how this can also be viewed as making the prediction using priors over functions rather than weights. Let f(w) denote the vector of outputs corresponding to inputs (Xl, ... , xn) given weights w. Then, using Bayes' theorem we have P(wlt,8) = P(tlw)P(wI8)/ P(tI8), and P(tlw) = J P(tly) o(y - f(w» dy. Hence equation 1 can be rewritten as P(y .. It, 8) = P(~18) J J P(tly) o(Y .. - ! .. (w»o(y - f(w» P(wI8) dw dy (2) However, the prior over (y .. , YI, ... , Yn) is given by P(y .. , y18) = P(y .. Iy, 8)P(yI8) = J o(Y .. - ! .. (w) o(y- f(w»P(wI8) dw and thus the predictive distribution can be 1 For large n, various ap'proximations to the exact solution which avoid the inversion of an n x n matrix are available. 2For notational convenience we suppress the x-dependence of the posterior. Computing with Infinite Networks 297 written as P(y .. lt,8) = P(~18) J P(tly)P(y .. ly, 8)P(yI8) dy = J P(y .. ly, 8)P(ylt, 8) dy (3) Hence in a Bayesian view it is the prior over function values P(y .. , Y18) which is important; specifying this prior by using weight distributions is one valid way to achieve this goal. In general we can use the weight space or function space view, which ever is more convenient, and for infinite neural networks the function space view is more useful. 2 Gaussian processes A stochastic process is a collection of random variables {Y(z)lz E X} indexed by a set X . In our case X will be n d , where d is the number of inputs. The stochastic process is specified by giving the probability distribution for every finite subset of variables Y(zt), ... , Y(Zk) in a consistent manner. A Gaussian process (GP) is a stochastic process which can be fully specified by its mean function jJ( z) = E[Y(z)] and its covariance function C(z, z') = E[(Y(z) - jJ(z»(Y(z') - JJ(z'»]; any finite set ofY-variables will have ajoint multivariate Gaussian distribution. For a multidimensional input space a Gaussian process may also be called a Gaussian random field. Below we consider Gaussian processes which have jJ(z) = 0, as is the case for the neural network priors discussed in section 3. A non-zero JJ(z) can be incorporated into the framework at the expense of a little extra complexity. A widely used class of covariance functions is the stationary covariance functions, whereby C(z, z') = C(z - z'). These are related to the spectral density (or power spectrum) of the process by the Wiener-Khinchine theorem, and are particularly amenable to Fourier analysis as the eigenfunctions of a stationary covariance kernel are exp ik.z. Many commonly used covariance functions are also isotropic, so that C(h) = C(h) where h = z - z' and h = Ihl. For example C(h) = exp(-(h/oy) is a valid covariance function for all d and for 0 < v ~ 2. Note that in this case u sets the correlation length-scale of the random field, although other covariance functions (e.g. those corresponding to power-law spectral densities) may have no preferred length scale. 2.1 Prediction with Gaussian processes The model for the observed data is that it was generated from the prior stochastic process, and that independent Gaussian noise (of variance u~) was then added. Given a prior covariance function CP(Zi,Zj), a noise process CN(Zj,Zj) = U~6ij (i.e. independent noise of variance u~ at each data point) and the training data, the prediction for the distribution of y .. corresponding to a test point z .. is obtained simply by applying equation 3. As the prior and noise model are both Gaussian the integral can be done analytically and P(y .. lt, 8) is Gaussian with mean and variance y(z .. ) = k~(z .. )(Kp + KN)-lt (4) u2(z .. ) = Cp(z .. , z .. ) k~(z .. )(J{p + KN )-lkp(z .. ) (5) where [Ko]ij = Co(Zi, Zj) for a = P, Nand kp(z .. ) = (Cp(z .. , zt), ... , Cp(z .. , zn»T. u~(z .. ) gives the "error bars" of the prediction. Equations 4 and 5 are the analogue for spatial processes of Wiener-Kolmogorov prediction theory. They have appeared in a wide variety of contexts including 298 C. K. I. Williams geostatistics where the method is known as "kriging" (Journel and Huijbregts, 1978; Cressie 1993), multidimensional spline smoothing (Wahba, 1990), in the derivation of radial basis function neural networks (Poggio and Girosi, 1990) and in the work of Whittle (1963). 3 Covariance functions for Neural Networks Consider a network which takes an input z, has one hidden layer with H units and then linearly combines the outputs of the hidden units with a bias to obtain fez). The mapping can be written H fez) = b+ L.:vjh(z;uj) (6) j=l where h(z; u) is the hidden unit transfer function (which we shall assume is bounded) which depends on the input-to-hidden weights u. This architecture is important because it has been shown by Hornik (1993) that networks with one hidden layer are universal approximators as the number of hidden units tends to infinity, for a wide class of transfer functions (but excluding polynomials). Let b and the v's have independent zero-mean distributions of variance O'~ and 0'1) respectively, and let the weights Uj for each hidden unit be independently and identically distributed. Denoting all weights by w, we obtain (following Neal, 1996) Ew[!(z)] 0 Ew[/(z )/(z')] O'~ + L.: O';Eu[hj(z; u)hj(z'; u)] j O'l + HO';Eu[h(z; u)h(z'; u)] (7) (8) (9) where equation 9 follows because all of the hidden units are identically distributed. The final term in equation 9 becomes w 2 Eu[h(z; u)h(z'; u)] by letting 0'; scale as w 2/H. As the transfer function is bounded, all moments of the distribution will be bounded and hence the Central Limit Theorem can be applied, showing that the stochastic process will become a Gaussian process in the limit as H -+ 00. By evaluating Eu[h(z)h(z')] for all z and z' in the training and testing sets we can obtain the covariance function needed to describe the neural network as a Gaussian process. These expectations are, of course, integrals over the relevant probability distributions of the biases and input weights. In the following sections two specific choices for the transfer functions are considered, (1) a sigmoidal function and (2) a Gaussian. Gaussian weight priors are used in both cases. It is interesting to note why this analysis cannot be taken a stage further to integrate out any hyperparameters as well. For example, the variance 0'; of the v weights might be drawn from an inverse Gamma distribution. In this case the distribution P(v) = J P(vIO';)P(O';)dO'; is no longer the product of the marginal distributions for each v weight (in fact it will be a multivariate t-distribution). A similar analysis can be applied to the u weights with a hyperprior. The effect is to make the hidden units non-independent, so that the Central Limit Theorem can no longer be applied. 3.1 Sigmoidal transfer function A sigmoidal transfer function is a very common choice in neural networks research; nets with this architecture are usually called multi-layer perceptrons. Computing with Infinite Networks 299 Below we consider the transfer function h(z; u) = ~(uo+ 'L1=1 UjXi), where ~(z) = 2/ Vii J; e-t2 dt is the error function, closely related to the cumulative distribution function for the Gaussian distribution. Appropriately scaled, the graph of this function is very similar to the tanh function which is more commonly used in the neural networks literature. In calculating V(z, Z/)d;J Eu[h(z; U)h(Z/; u)] we make the usual assumptions (e.g. MacKay, 1992) that u is drawn from a zero-mean Gaussian distribution with covariance matrix E, i.e. u "" N(O, E). Let i = (1, Xl, ... , Xd) be an augmented input vector whose first entry corresponds to the bias. Then Verf(z, Z/) can be written as Verf(z,z/) = ~ J~(uTi)~(uTi/)exp(-!uTE-lu) du (211") 2 IE1 1/ 2 2 (10) This integral can be evaluated analytically3 to give 2 2 -T .... -1 ( 1 • -1 Z .wZ Verf z, z ) = - sm ---;=========== 11" )(1 + 2iTEi)(1 + 2i/TEi/) (11) We observe that this covariance function is not stationary, which makes sense as the distributions for the weights are centered about zero, and hence translational symmetry is not present. Consider a diagonal weight prior so that E = diag(0"5, 0"7, ... ,0"1), so that the inputs i = 1, ... , d have a different weight variance to the bias 0"6. Then for Iz12, Iz/12» (1+20"6)/20"1, we find that Verf(z, Z/) ~ 1-20/11", where 0 is the angle between z and Z/. Again this makes sense intuitively; if the model is made up of a large number of sigmoidal functions in random directions (in z space), then we would expect points that lie diametrically opposite (i.e. at z and -z) to be anti-correlated, because they will lie in the + 1 and -1 regions of the sigmoid function for most directions. 3.2 Gaussian transfer function One other very common transfer function used in neural networks research is the Gaussian, so that h(z; u) = exp[-(z - u)T(z u)/20"~], where 0"; is the width parameter of the Gaussian. Gaussian basis functions are often used in Radial Basis Function (RBF) networks (e.g. Poggio and Girosi, 1990). For a Gaussian prior over the distribution of u so that u "" N(O, O"~I), 1 1 J (z-u)T(z-u) (Z/-u)T(Z/_U) uTu VG(z,z)=( 2)d/2 exp2 exp2 exp---2 G 211"0" u 20" 9 20" 9 20" u (12) By completing the square and integrating out u we obtain ( 0" )d zTz (z - z')T(z - z') zlTz ' VG(Z,Z/) = _e eXP{--2 2 } exp{2 2 }exp{--2 2 } O"U O"m 0"$ O"m (13) where 1/0"2 = 2/0"2 + 1/0"2 0"2 = 20"2 + 0"4/0"2 and 0"2 = 20"2 + 0"2 This formula e 9 u' $ 9 gum u g. can be generalized by allowing covariance matrices Eb and Eu in place of O";! and O"~!; rescaling each input variable Xi independently is a simple example. 3Introduce a dummy parameter A to make the first term in the integrand ~(AUTX). Differentiate the integral with respect to A and then use integration by parts. Finally recognize that dVerfjdA is of the form (1-fP)-1/2d9jdA and hence obtain the sin-1 form of the result, and evaluate it at A = 1. 300 C. K. I. Williams Again this is a non-stationary covariance function, although it is interesting to note that if O"~ 00 (while scaling w 2 appropriately) we find that VG(Z,Z/) ex: exp{-(z - z/)T(z - z/)/40"2} 4. For a finite value of O"~, VG(Z,Z/) is a stationary covariance function "modulated" by the Gaussian decay function exp( _zT z/20"?n) exp( _zIT Zl /20"?n). Clearly if O"?n is much larger than the largest distance in z-space then the predictions made with VG and a Gaussian process with only the stationary part of VG will be very similar. It is also possible to view the infinite network with Gaussian transfer functions as an example of a shot-noise process based on an inhomogeneous Poisson process (see Parzen (1962) §4.5 for details). Points are generated from an inhomogeneous Poisson process with the rate function ex: exp( _zT z/20"~), and Gaussian kernels of height v are centered on each of the points, where v is chosen iid from a distribution with mean zero and variance 0"; . 3.3 Comparing covariance functions The priors over functions specified by sigmoidal and Gaussian neural networks differ from covariance functions that are usually employed in the literature, e.g. splines (Wahba, 1990). How might we characterize the different covariance functions and compare the kinds of priors that they imply? The complex exponential exp ik.z is an eigenfunction of a stationary and isotropic covariance function, and hence the spectral density (or power spectrum) S(k) (k = Ikl) nicely characterizes the corresponding stochastic process. Roughly speaking the spectral density describes the "power" at a given spatial frequency k; for example, splines have S(k) ex: k- f3 . The decay of S(k) as k increases is essential, as it provides a smoothing or damping out of high frequencies. Unfortunately nonstationary processes cannot be analyzed in exactly this fashion because the complex exponentials are not (in general) eigenfunctions of a non-stationary kernel. Instead, we must consider the eigenfunctions defined by J C(z, Z/)¢(Z/)dz l = )..¢(z). However, it may be possible to get some feel for the effect of a non-stationary covariance function by looking at the diagonal elements in its 2d-dimensional Fourier transform, which correspond to the entries in power spectrum for stationary covariance functions. 3.4 Convergence of finite network priors to GPs From general Central Limit Theorem results one would expect a rate of convergence of H-l/2 towards a Gaussian process prior. How many units will be required in practice would seem to depend on the particular values of the weight-variance parameters. For example, for Gaussian transfer functions, O"rn defines the radius over which we expect the process to be significantly different from zero. If this radius is increased (while keeping the variance of the basis functions O"~ fixed) then naturally one would expect to need more hidden units in order to achieve the same level of approximation as before. Similar comments can be made for the sigmoidal case, depending on (1 + 20"6)/20"1I have conducted some experiments for the sigmoidal transfer umction, comparing the predictive performance of a finite neural network with one Input unit to the equivalent Gaussian process on data generated from the GP. The finite network simulations were carried out using a slightly modified version of Neal's MCMC Bayesian neural networks code (Neal, 1996) and the inputs were drawn from a 4Note that this would require w2 00 and hence the Central Limit Theorem would no longer hold, i.e. the process would be non-Gaussian. Computing with Infinite Networks 301 N(O,l) distribution. The hyperparameter settings were UI = 10.0, 0"0 = 2.0, O"v = 1.189 and Ub = 1.0. Roughly speaking the results are that 100's of hidden units are required before similar performance is achieved by the two methods, although there is considerable variability depending on the particular sample drawn from the prior; sometimes 10 hidden units appears sufficient for good agreement. 4 Discussion The work described above shows how to calculate the covariance function for sigmoidal and Gaussian basis functions networks. It is probable similar techniques will allow covariance functions to be derived analytically for networks with other kinds of basis functions as well; these may turn out to be similar in form to covariance functions already used in the Gaussian process literature. In the derivations above the hyperparameters 9 were fixed. However, in a real data analysis problem it would be unlikely that appropriate values of these parameters would be known. Given a prior distribution P(9) predictions should be made by integrating over the posterior distribution P(9It) ()( P(9)P(tI9), where P(tI9) is the likelihood of the training data t under the model; P(tI9) is easily computed for a Gaussian process. The prediction y( z) for test input z is then given by y(z) = J Y9(z)P(9ID)d9 (14) where Y9(z) is the predicted mean (as given by equation 4) for a particular value of 9. This integration is not tractable analytically but Markov Chain Monte Carlo methods such as Hybrid Monte Carlo can be used to approximate it. This strategy was used in Williams and Rasmussen (1996), but for stationary covariance functions, not ones derived from Gaussian processes; it would be interesting to compare results. Acknowledgements I thank David Saad and David Barber for help in obtaining the result in equation 11, and Chris Bishop, Peter Dayan, Ian Nabney, Radford Neal, David Saad and Huaiyu Zhu for comments on an earlier draft of the paper. This work was partially supported by EPSRC grant GR/J75425, "Novel Developments in Learning Theory for Neural Networks". References Cressie, N. A. C. (1993). Statistics for Spatial Data. Wiley. Hornik, K. (1993). Some new results on neural network approximation. Neural Networks 6 (8), 1069-1072. Journel, A. G. and C. J. Huijbregts (1978). Mining Geostatistics. Academic Press. MacKay, D. J. C. (1992). A Practical Bayesian Framework for Backpropagation Networks. Neural Computation 4(3), 448-472. Neal, R. M. (1996). Bayesian Learning for Neural Networks. Springer. Lecture Notes in Statistics 118. Parzen, E. (1962). Stochastic Processes. Holden-Day. Poggio, T. and F. Girosi (1990). Networks for approximation and learning. Proceedings of IEEE 78, 1481-1497. Wahba, G. (1990). Spline Models for Observational Data. Society for Industrial and Applied Mathematics. CBMS-NSF Regional Conference series in applied mathematics. Whittle, P. (1963). Prediction and regulation by linear least-square methods. English Universities Press. Williams, C. K. I. and C. E. Rasmussen (1996). Gaussian processes for regression. In D. S. Touretzky, M. C. Mozer, and M. E. Hasselmo (Eds.), Advances in Neural Information Processing Systems 8, pp. 514-520. MIT Press.	1996	103
1,155	Regression with Input-Dependent Noise: A Bayesian Treatment Christopher M. Bishop C.M.BishopGaston.ac.uk Cazhaow S. Qazaz qazazcsGaston.ac.uk Neural Computing Research Group Aston University, Birmingham, B4 7ET, U.K. http://www.ncrg.aston.ac.uk/ Abstract In most treatments of the regression problem it is assumed that the distribution of target data can be described by a deterministic function of the inputs, together with additive Gaussian noise having constant variance. The use of maximum likelihood to train such models then corresponds to the minimization of a sum-of-squares error function. In many applications a more realistic model would allow the noise variance itself to depend on the input variables. However, the use of maximum likelihood to train such models would give highly biased results. In this paper we show how a Bayesian treatment can allow for an input-dependent variance while overcoming the bias of maximum likelihood. 1 Introduction In regression problems it is important not only to predict the output variables but also to have some estimate of the error bars associated with those predictions. An important contribution to the error bars arises from the intrinsic noise on the data. In most conventional treatments of regression, it is assumed that the noise can be modelled by a Gaussian distribution with a constant variance. However, in many applications it will be more realistic to allow the noise variance itself to depend on the input variables. A general framework for modelling the conditional probability density function of the target data, given the input vector, has been introduced in the form of mixture density networks by Bishop (1994, 1995). This uses a feedforward network to set the parameters of a mixture kernel distribution, following Jacobs et al. (1991). The special case of a single isotropic Gaussian kernel function 348 C. M. Bishop and C. S. Qazaz was discussed by Nix and Weigend (1995), and its generalization to allow for an arbitrary covariance matrix was given by Williams (1996). These approaches, however, are all based on the use of maximum likelihood, which can lead to the noise variance being systematically under-estimated. Here we adopt an approximate hierarchical Bayesian treatment (MacKay, 1991) to find the most probable interpolant and most probable input-dependent noise variance. We compare our results with maximum likelihood and show how this Bayesian approach leads to a significantly reduced bias. In order to gain some insight into the limitations of the maximum likelihood approach, and to see how these limitations can be overcome in a Bayesian treatment, it is useful to consider first a much simpler problem involving a single random variable (Bishop, 1995). Suppose that a variable Z is known to have a Gaussian distribution, but with unknown mean fJ. and unknown variance (J2. Given a sample D == {zn} drawn from that distribution, where n = 1, ... , N, our goal is to infer values for the mean and variance. The likelihood function is given by 2 1 1 2 { N } p(DIfJ., (J ) = (27r(J2)N/2 exp - 2(J2 ?; (Zn - fJ.) . (1) A non-Bayesian approach to finding the mean and variance is to maximize the likelihood jointly over fJ. and (J2, corresponding to the intuitive idea of finding the parameter values which are most likely to have given rise to the observed data set. This yields the standard result N (12 = ~ 2)Zn - Ji)2. n=l (2) It is well known that the estimate (12 for the variance given in (2) is biased since the expectation of this estimate is not equal to the true value C[~2] _ N -1 2 (, (J --(JO N (3) where (J5 is the true variance of the distribution which generated the data, and £[.] denotes an average over data sets of size N. For large N this effect is small. However, in the case of regression problems there are generally much larger number of degrees of freedom in relation to the number of available data points, in which case the effect of this bias can be very substantial. The problem of bias can be regarded as a symptom of the maximum likelihood approach. Because the mean Ji has been estimated from the data, it has fitted some of the noise on the data and this leads to an under-estimate of the variance. If the true mean is used in the expression for (12 in (2) instead of the maximum likelihood expression, then the estimate is unbiased. By adopting a Bayesian viewpoint this bias can be removed. The marginal likelihood of (J2 should be computed by integrating over the mean fJ.. Assuming a 'flat' prior p(fJ.) we obtain (4) Regression with Input-Dependent Noise: A Bayesian Treatment 349 (5) Maximizing (5) with respect to ~2 then gives N -2 1 ~( ~)2 ~ = N _ 1 ~ Z n J.L n=l (6) which is unbiased. This result is illustrated in Figure 1 which shows contours of p(DIJ.L, ~2) together with the marginal likelihood p(DI~2) and the conditional likelihood p(DI;t, ~2) evaluated at J.L = ;t. 2.5 2 0.5 o~~----~----~--~ -2 o mean 2 Q) u c: 2.5 2 .~ 1.5 113 > 1 / 2 4 6 likelihood Figure 1: The left hand plot shows contours of the likelihood function p(DIJ..L, 0-2) given by (1) for 4 data points drawn from a Gaussian distribution having zero mean and unit variance. The right hand plot shows the marginal likelihood function p(DI0-2) (dashed curve) and the conditional likelihood function p(DI{i,0-2) (solid curve). It can be seen that the skewed contours result in a value of 0:2, which maximizes p(DI{i, 0-2), which is smaller than 0:2 which maximizes p(DI0-2). 2 Bayesian Regression Consider a regression problem involving the prediction of a noisy variable t given the value of a vector x of input variablesl . Our goal is to predict both a regression function and an input-dependent noise variance. We shall therefore consider two networks. The first network takes the input vector x and generates an output IFor simplicity we consider a single output variable. The extension of this work to multiple outputs is straightforward. 350 C. M. Bishop and C. S. Qazaz y(x; w) which represents the regression function, and is governed by a vector of weight parameters w. The second network also takes the input vector x, and generates an output function j3(x; u) representing the inverse variance of the noise distribution, and is governed by a vector of weight parameters u. The conditional distribution of target data, given the input vector, is then modelled by a normal distribution p(tlx, w, u) = N(tly, 13-1 ). From this we obtain the likelihood function (7) where j3n = j3(xn; u), N (271') 1/2 ZD = II j3n ' n=l (8) and D == {xn' tn} is the data set. Some simplification of the subsequent analysis is obtained by taking the regression function, and In 13, to be given by linear combinations of fixed basis functions, as in MacKay (1995), so that y(x; w) = wT <j)(x) , j3(x; u) = exp (u T ,p(x)) (9) where choose one basis function in each network to be a constant ¢o = 'l/Jo = 1 so that the corresponding weights Wo and Uo represent bias parameters. The maximum likelihood procedure chooses values wand u by finding a joint maximum over wand u. As we have already indicated, this will give a biased result since the regression function inevitably fits part of the noise on the data, leading to an over-estimate of j3(x). In extreme cases, where the regression curve passes exactly through a data point, the corresponding estimate of 13 can go to infinity, corresponding to an estimated noise variance of zero. The solution to this problem has already been indicated in Section 1 and was first suggested in this context by MacKay (1991, Chapter 6). In order to obtain an unbiased estimate of j3(x) we must find the marginal distribution of 13, or equivalently of u, in which we have integrated out the dependence on w. This leads to a hierarchical Bayesian analysis. We begin by defining priors over the parameters wand u. Here we consider isotropic Gaussian priors of the form (10) p(ulau ) (11) where aw and au are hyper-parameters. At the first stage of the hierarchy, we assume that u is fixed to its most probable value UMP, which will be determined shortly. The most probable value of w, denoted by WMP, is then found by maxiRegression with Input-Dependent Noise: A Bayesian Treatment 351 mizing the posterior distribution2 ( ID ) - p(Dlw, uMP)p(wlow) p w , UMP, Ow (D I ) p UMP, Ow (12) where the denominator in (12) is given by p(DIUMP, ow) = I p(Dlw,uMP)p(wlow)dw. (13) Taking the negative log of (12), and dropping constant terms, we see that WMP is obtained by minimizing N S(w) = L 13nEn + °2w IIwll2 n=l (14) where we have used (7) and (10). For the particular choice of model (9) this minimization represents a linear problem which is easily solved (for a given u) by standard matrix techniques. At the next level of the hierarchy, we find UMP by maximizing the marginal posterior distribution (15) The term p(Dlu, ow) is just the denominator from (12) and is found by integrating over w as in (13). For the model (9) and prior (10) this integral is Gaussian and can be performed analytically without approximation. Again taking logarithms and discarding constants, we have to minimize N 1 N 1 M(u) = L 13nEn + ~u lIuII2 - 2 L In13n + 2ln IAI n=l n=l where IAI denotes the determinant of the Hessian matrix A given by N A = L 13nl/J(Xn)l/J(xn? + Owl n=l (16) (17) and I is the unit matrix. The function M(u) in (16) can be minimized using standard non-linear optimization algorithms. We use scaled conjugate gradients, in which the necessary derivatives of In IAI are easily found in terms of the eigenvalues of A. In summary, the algorithm requires an outer loop in which the most probable value UMP is found by non-linear minimization of (16), using the scaled conjugate gradient algorithm. Each time the optimization code requires a value for M(u) or its gradient, for a new value of u, the optimum value for WMP must be found by minimizing (14). In effect, w is evolving on a fast time-scale, and U on a slow timescale. The corresponding maximum (penalized) likelihood approach consists of a joint non-linear optimization over U and w of the posterior distribution p(w, uID) obtained from (7), (10) and (11). Finally, the hyperparameters are given fixed values Ow = Ou = 0.1 as this allows the maximum likelihood and Bayesian approaches to be treated on an equal footing. 2Note that the result will be dependent on the choice of parametrization since the maximum of a distribution is not invariant under a change of variable. 352 C. M. Bishop and C. S. Qazaz 3 Results and Discussion As an illustration of this algorithm, we consider a toy problem involving one input and one output, with a noise variance which has an x 2 dependence on the input variable. Since the estimated quantities are noisy, due to the finite data set, we consider an averaging procedure as follows. We generate 100 independent data sets each consisting of 10 data points. The model is trained on each of the data sets in turn and then tested on the remaining 99 data sets. Both the Y(Xj w) and (3(Xj u) networks have 4 Gaussian basis functions (plus a bias) with width parameters chosen to equal the spacing of the centres. Results are shown in Figure 2. It is clear that the maximum likelihood results are biased and that the noise variance is systematically underestimated. By contrast, Maximum likelihood Maximum likelihood " 0.8 \ \ 0.5 <1l 0.6 \ C,) t: :5 CI:l c.. a \ ·~0.4 :5 0 <1l en \ / -0.5 ·gO.2 , / " / '/ ~ -1 a -4 -2 a 2 4 -4 -2 a 2 4 x x Bayesian Bayesian 0.8 0.5 <1l 0.6 \ C,) t: :5 CI:l c.. a .~ 0.4 :5 \ 0 <1l en -0.5 ·gO.2 -1 0 -4 -2 a 2 4 -4 -2 a 2 4 x x Figure 2: The left hand plots show the sinusoidal function (dashed curve) from which the data were generated, together with the regression function averaged over 100 training sets. The right hand plots show the true noise variance (dashed curve) together with the estimated. noise variance, again averaged over 100 data sets. the Bayesian results show an improved estimate of the noise variance. This is borne out by evaluating the log likelihood for the test data under the corresponding predictive distributions. The Bayesian approach gives a log likelihood per data point, averaged over the 100 runs, of -1.38. Due to the over-fitting problem, maximum likelihood occasionally gives extremely large negative values for the log likelihood (when (3 has been estimated to be very large, corresponding to a regression curve which passes close to an individual data point). Even omitting these extreme values, the maximum likelihood still gives an average log likelihood per data point of Regression with Input-Dependent Noise: A Bayesian Treatment 353 -17.1 which is substantially smaller than the Bayesian result. We are currently exploring the use of Markov chain Monte Carlo methods (Neal, 1993) to perform the integrations required by the Bayesian analysis numerically, without the need to introduce the Gaussian approximation or the evidence framework. Recently, MacKay (1995) has proposed an alternative technique based on Gibbs sampling. It will be interesting to compare these various approaches. Acknowledgements: This work was supported by EPSRC grant GR/K51792, Validation and Verification of Neural Network Systems. References Bishop, C. M. (1994). Mixture density networks. Technical Report NCRG/94/001, Neural Computing Research Group, Aston University, Birmingham, UK. Bishop, C. M. (1995). Neural Networks for Pattern Recognition. Oxford University Press. Jacobs, R. A., M. I. Jordan, S. J. Nowlan, and G. E. Hinton (1991). Adaptive mixtures of local experts. Neural Computation 3 (1), 79-87. MacKay, D. J. C. (1991). Bayesian Methods for Adaptive Models. Ph.JJ.thesis, California Institute of Technology. MacKay, D. J. C. (1995). Probabilistic networks: new models and new methods. In F. Fogelman-Soulie and P. Gallinari (Eds.), Proceedings ICANN'95 International Conference on Artificial Neural Networks, pp. 331-337. Paris: EC2 & Cie. Neal, R. M. (1993). Probabilistic inference using Markov chain Monte Carlo methods. Technical Report CRG-TR-93-1, Department of Computer Science, University of Toronto, Cananda. Nix, A. D. and A. S. Weigend (1995). Learning local error bars for nonlinear regression. In G. Tesauro, D. S. Touretzky, and T. K. Leen (Eds.), Advances in Neural Information Processing Systems, Volume 7, pp. 489-496. Cambridge, MA: MIT Press. Williams, P. M. (1996). Using neural networks to model conditional multivariate densities. Neural Computation 8 (4), 843-854.	1996	104
1,156	The Generalisation Cost of RAMnets Richard Rohwer and Michal Morciniec rohwerrj~cs.aston.ac.uk morcinim~cs.aston.ac.uk Neural Computing Research Group Aston University Aston Triangle, Birmingham B4 7ET, UK. Abstract Given unlimited computational resources, it is best to use a criterion of minimal expected generalisation error to select a model and determine its parameters. However, it may be worthwhile to sacrifice some generalisation performance for higher learning speed. A method for quantifying sub-optimality is set out here, so that this choice can be made intelligently. Furthermore, the method is applicable to a broad class of models, including the ultra-fast memory-based methods such as RAMnets. This brings the added benefit of providing, for the first time, the means to analyse the generalisation properties of such models in a Bayesian framework . 1 Introduction In order to quantitatively predict the performance of methods such as the ultra-fast RAMnet, which are not trained by minimising a cost function, we develop a Bayesian formalism for estimating the generalisation cost of a wide class of algorithms. We consider the noisy interpolation problem, in which each output data point if results from adding noise to the result y = f(x) of applying unknown function f to input data point x, which is generated from a distribution P (x). We follow a similar approach to (Zhu & Rohwer, to appear 1996) in using a Gaussian process to define a prior over the space of functions, so that the expected generalisation cost under the posterior can be determined. The optimal model is defined in terms of the restriction of this posterior to the subspace defined by the model. The optimum is easily determined for linear models over a set of basis functions. We go on to compute the generalisation cost (with an error bar) for all models of this class, which we demonstrate to include the RAMnets. 254 R. Rohwer and M. Morciniec Section 2 gives a brief overview of RAMnets. Sections 3 and 4 supply the formalism for computing expected generalisation costs under Gaussian process priors. Numerical experiments with this formalism are presented in Section 5. Finally, we discuss the current limitations of this technique and future research directions in Section 6. 2 RAMnets The RAMnet, or n-tuple network is a very fast I-pass learning system that often gives excellent results competitive with slower methods such as Radial Basis Function networks or Multi-layer Perceptrons (Rohwer & Morciniec, 1996). Although a semi-quantitative theory explains how these systems generalise, no formal framework has previously been given to precisely predict the accuracy of n-tuple networks. Essentially, a RAMnet defines a set of "features" which can be regarded as Boolean functions of the input variables. Let the ath feature of x be given by a {O, 1 }-valued function 4>a(x). We will focus on the n-tuple regression network (Allinson & Kolcz, 1995), which outputs (1) in response to input x, if trained on the set of N samples {X(N)Y(N)} = {(xi, !I)}~l' Here U(x , x') = E 4>a(x)4>a(x' ) can be seen to play the role of a smoothing kernel, a provided that it turns out to have a suitable shape. It is well-know that it does, for appropriate choices of feature sets. The strength of this method is that the sums over training data can be done in one pass, producing a table containing two totals for each feature. Only this table is required for recognition. It is interesting to note that there is a familiar way to expand a kernel into the form U(x, x') = E 4>a(x)4>a(x' ), at least when U(x, x') = U(x - x'), if the range of 4> is a not restricted to {O, I}: an eigenfunction expansion l . Indeed, principal component analysis2 applied to a Gaussian with variance V shows that the smallest feature set for a given generalisation cost consists of the (real-valued) projections onto the leading eigenfunctions of V. Be that as it may, the treatment here applies to arbitrary feature sets. 3 Bayesian inference with Gaussian priors Gaussian processes provide a diverse set of priors over function spaces. To avoid mathematical details of peripheral interest, let us approximate the infinitedimensional space of functions by a finite-dimensional space of discretised functions, so that function f is replaced by high-dimensional vector f, and f(x) is replaced by fx, with f(x) ~ fx within a volume Llx around x. We develop the case of scalar functions f, but the generalisation to vector-valued functions is straightforward. 1 In physics, this is essentially the mode function expansion of U-1 , the differential operator with Green's function U. 2V-1 needs to be a compact operator for this to work in the infinite-dimensional limit. The Generalisation Cost of RAMnets 255 We assume a Gaussian prior on f, with zero mean and covariance V /0:: (2) where Za = det(~:V)t. The overall scale of variation of f is controlled by 0:. Illustrative samples of the functions generated from various choices of covariance are given in (Zhu & Rohwer, to appear 1996). With q:c/f3 denoting the (possibly position-dependent) variance of the Gaussian output noise, the likelihood of outputs YeN) given function f and inputs X(N) is p(Y(N)IX(N),f) = (1/Z,8)exp-~L(f:c. _yi)q;.l(f:c' _yi) (3) i where Z~ = n ¥q:c. = det [¥Q] with Qij = q:c.6ij. i Because f and X(N) are independent the joint distribution is ( I) ( I ) ) ( !.bTAb+C)/( ) -!'(f-Ab)TA-1(f-Ab) P Y(N) , f X(N) = P YeN) f , X(N) P (f = e 2 ZaZ,8 e ~ where 6:c:c. is understood to be 1 whenever xi is in the same cell of the discretisation ~ x, and A;;, = o:V;;, + f3'L.q;,I6:c,:c.6x',:c" b:c = f3'L.yiq;.I6:c,:c" and i i C = ~f3 'L. yi q;,Iyi. One can readily verify that i A:c:c' = (ljo:)V:c:c,+ LV:c:ctKtuV:c,,:c, tu where K is the N x N matrix defined by The posterior is readily determined to be * where f = Ab is the posterior mean estimate of the true function f. 4 Calculation of the expected cost and its variance m (5) (6) (4) Let us define the cost of associating an output f:c of the model with an input x that actually produced an output y as m m 2 C(f:c, y) = Hf:c - y) r:c where r:c is a position dependent cost weight. 256 R. Rohwer and M. Morciniec The average of this cost defines a cost functional, given input data X(N): m J m C( f , flx(N» = C(fx, y)P (XIX(N») P (ylx, f) dxdy. (8) This form is obtained by noting that the function f carries no information about the input point x, and the input data X(N) supplies no information about y beyond that supplied by f. The distributions in (8) are unchanged by further conditioning m m on Y(N), so we could write C( f, flx(N» = C( f, flx(N)' YeN»~. This cost functional therefore has the posterior expectation value and variance Plugging in the distributions (2) (applied to a single sample), (3) and (7) leads to: * m T * m tr [QR] (ClX(N),Y(N») = 'ttr[AR] + Hf-f) R(f-f)+ 2/3 (11) where the diagonal matrices Rand Q have the elements Rxx' = P(XIX)TxAxbx,x' and Qxx' = qxbxx'. Similar calculations lead to the expression for the variance (12) m * where the elements of Fare Fxx' = (fx - fx)bx,x'. m . Note that the RAMnet (1) has the form f x = 2: J Xxi yl linear in the output data i YeN), with J xx' = U(x, xi)/ 2:j U(x, Xj). Let us take V to have the form Vex, x') = p(x )G(x - x')p(x'), combining translation-invariant and non-invariant factors in a plausible way. Then with the sums over x replaced by integrals, (11) becomes explicitly 2 (ClX(N), YeN») = ~ J dxP (XIX(N») qxTx + ± J P (XIX(N») Txp;Gxx +.!. LPxtKtuPx" J dxP (XIX(N») Txp;Gx"xGxxt 0" tu +0"2 LyUKutPxt J dxP (XIX(N») Txp;GxtxGxxopx.K$vYv tUV$ +20" LyUKutpxt J dxP (XIX(N») TxPxGxtxJxxvyV tuv + Lyu J dxP (XIX(N») TxJx"xJxxvyv. (13) uv The Generalisation Cost of RAMnets &) True, oplim&1 &nd subOplim&1 functions 1.S,----..----r---,-----,.-----.---r--,------,----r---. • r.: 1 ,,; '. C 0.5 .~ L--_~:..:.-~-V ~ 0 ~ ~ -0.5 E-< r r - _. r \_-----::z,="] -0.8 -0.6 -0.4 -0.2 0 0.2 0.. 0.6 0.8 1 3: 257 b) Dislribulion of Ihe cost C Figure 1: a) The lower figure shows the input distribution. The upper figure shows the true function f generated from a Gaussian prior with covariance matrix V (dot• ted line), the optimal function f = Ab (solid line) and the suboptimal solution m f (dashed line). b )The distribution of the cost function obtained by generating functions from the posterior Gaussian with covariance matrix A and calculating the cost according to equation 14. The mean and one standard deviation calculated analytically and numerically are shown by the lower and upper error bars respecti vely. Taking P (XIX(N)) to be Gaussian (the maximum likelihood estimate would be reasonable) and p, r, and q uniform, the first four integrals are straightforward. The latter two involve the model J, and were evaluated numerically in the work reported below. 5 Numerical results We present one numerical example to illustrate the formalism, and another to illustrate its application. For the first illustration, let the input and output variables be one dimensional real numbers. Let the input distribution P (x) be a Gaussian with mean I':e = 0 and standard deviation (1:e = 0.2. Nearly all inputs then fall within the range [-1,1]' which we uniformly quantise into 41 bins. The true function f is generated from a Gaussian distribution with 1', = 0 and 41 x 41 covariance matrix V with elements V:e:e' = e-1:e-:e'I . 50 training inputs x were generated from the input distribution and assigned corresponding outputs y = f:e + (, where ( is Gaussian noise with zero mean and standard deviation Jq:el/3 = 0.01. The cost weight r:e = 1. The inputs were thermometer coded3 over 256 bits, from which 100 subsets of 30 bits were randomly selected. Each of the 100 x 230 patterns formed over these bits defines a RAMnet feature which evaluates to 1 when that pattern is present in the input x. (Only those features which actually appear in the data need to be tabulated.) The 50 training data points were used in this way to train an n-tuple 3The first 256(x + 1)/2 bits are set to 1, and the remaining bits to O. 258 .. 2 0:: o .~ 1.5 0:: " ... , " 0:: .. 0.5 " ., 0:: 0. . ~ ... (U " ..0:: Eo< ... , ... 2 a) Neal's regression problem ... , [IJ t -~ o Dala 0. :z R. Rohwer and M. Morciniec b)Mean cosl (CIX(N)'Y(N» as a funclion of 0'1 and a 0..06'---~-r---r~-~~-"'--~-"'--~-' 0..045 0..04 0..036 ~ 0..025 ...!!. CJ 0..<>2 0.015 0' = O.b ........................... ... ... ~ ...... ~ ...... ----~ ... ~ ...... ~ ...... ~ ...... ~ ... ... ~ ...... ~ ... -~ ... 0.010 0.02 O.CM 0.08 0.01 0.1 0.12 0.1. 0.18 0.1. 0.2 a Figure 2: a) Neal's Regression problem. The true function f is indicated by a dotted • m line, the optimal function f is denoted by a solid line and the suboptimal solution f is indicated by a dashed line. Circles indicate the training data. b) Dependence of the cost prediction on the values of parameters a and O'f. The cost evaluated from the test set is plotted as a dashed line, predicted cost is shown as a solid line with one standard deviation indicated by a dotted line. m • regression network. The input distribution and functions f, f , f are plotted III figure 1a. • A Gaussian distribution with mean f and posterior covariance matrix A was then used to generate 104 functions. For each such function fp, the generalisation cost (14) x was computed. A histogram of these costs appears in figure 1b, together with the theoretical and numerically computed average generalisation cost and its variance. Good agreement is evident. Another one-dimensional problem illustrates the use of this formalism for predicting the generalisation performance of a RAMnet when the prior over functions can only be guessed. The true function, taken from (Neal, 1995) is given by fx = 0.3 + OAx + 0.5 sin(2.7x) + 1.1/(1 + x2) + ( (15) where the Gaussian noise variable ( has mean I'f = 0 and standard deviation Jq:c//3 == 0.1. The cost weight Tx == 1. The training and test set each comprised 100 data-points. The inputs were generated by the standard normal distribution (I'x = 0, O':c = 1) and converted into the binary strings using a thermometer code. The input range [-3,3] was quantised into 61 uniform bins. m • The training set and the functions f , f , f are shown on figure 2a for a = 0.1. The function space covariance matrix was defined to have the Gaussian form V xx' _1 (., _ .,')2 e 2 ,,; where O'f = 1.0. The Generalisation Cost of RAMnets 259 (Jf is the correlation length of the functions, which is of order 1, judging from figure 2a. The overall scale of variation is 1/ va, which appears to be about 3, so Q' should be about 1/9. Figure 2b shows the expected cost as a function of Q' for various choices of (Jf, with error bars on the (Jf = 1.0 curve. The actual cost m computed from the test set according to C = t E(yi -fx? is plotted with a dashed i line. There is good agreement around the sensible values of Q' and (Jf. 6 Conclusions This paper demonstrates that unusual models, such as the ultra-fast RAMnets which are not trained by directly optimising a cost function, can be analysed in a Bayesian framework to determine their generalisation cost. Because the formalism is constructed in terms of distributions over function space rather than distributions over model parameters, it can be used for model comparison, and in particular to select RAMnet parameters. The main drawback with this technique, as it stands, is the need to numerically integrate two expressions which involve the model. This difficulty intensifies rapidly as the input dimension increases. Therefore, it is now a research priority to search for RAMnet feature sets which allow these integrals to be performed analytically. It would also be interesting to average the expected costs over the training data, producing an expected generalisation cost for an algorithm. The Y(N) integral is straightforward, but the X(N) integral is difficult. However, similar integrals have been carried out in the thermodynamic limit (high input dimension) (Sollich, 1994), so the investigation of these techniques in the current setting is another promising research direction. 7 Acknowledgements We would like to thank the Aston Neural Computing Group, and especially Huaiyu Zhu, Chris Williams, and David Saad for helpful discussions. References Allinson, N.M., & Kolcz, A. 1995. N-tuple Regression Network. to be published in Neural Networks. Neal, R. 1995. Introductory} documentation for software implementing Bayesian learning for neural networks using Markov chain Monte Carlo techniques. Tech. rept. Dept of Computer Science, University of Toronto. Rohwer, R., & Morciniec, M. 1996. A theoretical and experimental account of the n-tuple classifier performance. Neural Computation, 8(3), 657-670. Sollich, Peter. 1994. Finite-size effects in learning and generalization in linear perceptrons. J. Phys. A, 27, 7771-7784. Zhu, H., & Rohwer, R. to appear 1996. Bayesian regression filters and the issue of priors. Neural Computing and Applications. ftp://cs.aston.ac.uk/neural/zhuh/reg~il-prior.ps.Z.	1996	105
1,157	Multilayer neural networks: one or two hidden layers? G. Brightwell Dept of Mathematics LSE, Houghton Street London WC2A 2AE, U.K. c. Kenyon, H. Paugam-Moisy LIP, URA 1398 CNRS ENS Lyon, 46 alIee d'Italie F69364 Lyon cedex, FRANCE Abstract We study the number of hidden layers required by a multilayer neural network with threshold units to compute a function f from n d to {O, I}. In dimension d = 2, Gibson characterized the functions computable with just one hidden layer, under the assumption that there is no "multiple intersection point" and that f is only defined on a compact set. We consider the restriction of f to the neighborhood of a multiple intersection point or of infinity, and give necessary and sufficient conditions for it to be locally computable with one hidden layer. We show that adding these conditions to Gibson's assumptions is not sufficient to ensure global computability with one hidden layer, by exhibiting a new non-local configuration, the "critical cycle", which implies that f is not computable with one hidden layer. 1 INTRODUCTION The number of hidden layers is a crucial parameter for the architecture of multilayer neural networks. Early research, in the 60's, addressed the problem of exactly realizing Boolean functions with binary networks or binary multilayer networks. On the one hand, more recent work focused on approximately realizing real functions with multilayer neural networks with one hidden layer [6, 7, 11] or with two hidden units [2]. On the other hand, some authors [1, 12] were interested in finding bounds on the architecture of multilayer networks for exact realization of a finite set of points. Another approach is to search the minimal architecture of multilayer networks for exactly realizing real functions, from nd to {O, I}. Our work, of the latter kind, is a continuation of the effort of [4, 5, 8, 9] towards characterizing the real dichotomies which can be exactly realized with a single hidden layer neural network composed of threshold units. Multilayer Neural Networks: One or Two Hidden Layers? 149 1.1 NOTATIONS AND BACKGROUND A finite set of hyperplanes {Hd1<i<h defines a partition of the d-dimensional space into convex polyhedral open regIons, the union of the Hi'S being neglected as a subset of measure zero. A polyhedral dichotomy is a function I : R d --t {O, I}, obtained by associating a class, equal to 0 or to 1, to each of those regions. Thus both 1-1 (0) and 1-1 (1) are unions of a finite number of convex polyhedral open regions. The h hyperplanes which define the regions are called the essential hyperplanes of I. A point P is an essential point if it is the intersection of some set of essential hyperplanes. In this paper, all multilayer networks are supposed to be feedforward neural networks of threshold units, fully interconnected from one layer to the next, without skipping interconnections. A network is said to realize a function I: Rd --t to, 1} if, for an input vector x, the network output is equal to I(x), almost everywhere in Rd. The functions realized by our multilayer networks are the polyhedral dichotomies. By definition of threshold units, each unit of the first hidden layer computes a binary function Yj of the real inputs (Xl, . .. ,Xd). Therefore, subsequent layers compute a Boolean function. Since any Boolean function can be written in DNF -form, two hidden layers are sufficient for a multilayer network to realize any polyhedral dichotomy. Two hidden layers are sometimes also necessary, e.g. for realizing the "four-quadrant" dichotomy which generalizes the XOR function [4]. For all j, the /h unit of the first hidden layer can be seen as separating the space by the hyperplane Hj : 2::=1 WijXi = OJ. Hence the first hidden layer necessarily contains at least one hidden unit for each essential hyperplane of I. Thus each region R can be labelled by a binary number Y = (Y1 , ... ,Yh) (see [5]). The /h digit Yj will be denoted by Hj(R). Usually there are fewer than 2h regions and not all possible labels actually exist. The Boolean family BJ of a polyhedral dichotomy I is defined to be the set of all Boolean functions on h variables which are equal to I on all the existing labels. 1.2 PREVIOUS RESULTS It is straightforward that all polyhedral dichotomies which have at least one linearly separable function in their Boolean family can be realized by a one-hidden-Iayer network. However the converse is far from true. A counter-example was produced in [5]: adding extra hyperplanes (i.e. extra units on the first hidden layer) can eliminate the need for a second hidden layer. Hence the problem of finding a minimal architecture for realizing dichotomies cannot be reduced to the neural computation of Boolean functions. Finding a generic description of all the polyhedral dichotomies which can be realized exactly by a one-hidden-Iayer network is still an open problem. This paper is a new step towards its resolution. One approach consists of finding geometric configurations which imply that a function is not realizable with a single hidden layer. There are three known such geometric configurations: the XOR-situation, the XOR-bow-tie and the XOR-at-infinity (see Figure 1). A polyhedral dichotomy is said to be in an XOR-situation iff one of its essential hyperplanes H is inconsistent, i.e. if there are four regions B, B', W, W' such that Band B' are in class 1, W and W' are in class 0, Band W' are on one side of H, B' and Ware on the other side of H, and Band Ware adjacent along H, as well as B' and W'. 150 G. Brightwell, C. Kenyon and H. Paugam-Moisy Given a point P, two regions containing P in their closure are called opposite with respect to P if they are in different halfspaces w.r.t. all essential hyperplanes going through P. A polyhedral dichotomy is said to be in an XOR-bow-tie iff there exist four distinct regions B,B', W, W', such that Band B', both in class 1 (resp. W and W', both in class 0), are opposite with respect to point P. The third configuration is the XOR-at-infinity, which is analogous to the XOR-bowtie at a point 00 added to n d. There exist four distinct unbounded regions B, B' (in class 1), W, W' (in class 0) such that, for every essential hyperplane H, either all of them are on the same side of H (e.g. the horizontal line), or Band B' are on opposite sides of H, and Wand W' are on opposite sides of H (see [3]) . B B' Figure 1: Geometrical representation of XOR-situation, XOR-bow-tie and XOR-atinfinity in the plane (black regions are in class 1, grey regions are in class 0). Theorem 1 If a polyhedral dichotomy I, from nd to {O, I}, can be realized by a one-hidden-layer network, then it cannot be in an XOR-situation, nor in an XORbow-tie, nor in an XOR-at-infinity. The proof can be found in [5] for the XOR-situation, in [13] for the XOR-bow-tie, and in [5] for the XOR-at-infinity. Another research direction, implying a function is realizable by a single hidden layer network, is based on the universal approximator property of one-hidden-Iayer networks, applied to intermediate functions obtained constructively adding extra hyperplanes to the essential hyperplanes of f. This direction was explored by Gibson [9], but there are virtually no results known beyond two dimensions. Gibson's result can be reformulated as follows: Theorem 2 II a polyhedral dichotomy I is defined on a compact subset of n 2 , if I is not in an XOR-situation, and if no three essential hyperplanes (lines) intersect, then f is realizable with a single hidden layer network. Unfortunately Gibson's proof is not constructive, and extending it to remove some of the assumptions or to go to higher dimensions seems challenging. Both XORbow-tie and XOR-at-infinity are excluded by his assumptions of compactness and no multiple intersections. In the next section, we explore the two cases which are excluded by Gibson's assumptions. We prove that, in n2 , the XOR-bow-tie and the XOR-at-infinity are the only restrictions to local realizability. 2 LOCAL REALIZATION IN 1(,2 2.1 MULTIPLE INTERSECTION Theorem 3 Let I be a polyhedral dichotomy on n2 and let P be a point of multiple intersection. Let Cp be a neighborhood of P which does not intersect any essential hyperplane other than those going through P . The restriction of I to Cp is realizable by a one-hidden-layer network iff I is not in an XOR-bow-tie at P. Multilayer Neural Networks: One or 1Wo Hidden Layers? 151 The proof is in three steps: first, we reorder the hyperplanes in the neighborhood of P, so as to get a nice looking system of inequalities; second, we apply Farkas' lemma; third, we show how an XOR-bow-tie can be deduced. Proof: Let P be the intersection of k 2': 3 essential hyperplanes of f. All the hyperplanes which intersect at P can be renumbered and re-oriented so that the intersecting hyperplanes are totally ordered. Thus the label of the regions which have the point P in their closure is very regular. If one drops all the digits corresponding to the essential hyperplanes of f which do not contain P, the remaining part of the region labels are exactly like those of Figure 2. fl 0 0 -fl f2 f2 0 0 -f2 fA:-l 0 H, A= fA: fA: fA: -fA: 0 fA:+! fA:+! -fA:+l 0 H. 0 fk+2 -fA:+2 H7 f2A:-l -f21:-1 0 0 -f2" H5 Figure 2: Labels of the regions in the neighborhood of P, and matrix A. The problem of finding a one-hidden-Iayer network which realizes f can be rewritten as a system of inequalities. The unknown variables are the weights Wi and threshold () of the output unit. Let (S) denote the subsystem of inequalities obtained from the 2k regions which have the point P in their closure. The regular numbering of these 2k regions allows us to write the system as follows I l<i<k [ 2:~=1 Wm < () if class 0 (S) 2:~=1 Wm > () if class 1 [ I:r=H+1 Wm < () if class 0 k + 1 ~ i ~ 2k 2:m=i-k+l Wm > () if class 1 The system (S) can be rewritten in the matrix form Ax ~ b, where x T = [Wl,W2, ... ,Wk,()] and bT = [b1,b2, ... ,bk,bk+1, ... ,b2k] where bi = -f, for all i, and f is an arbitrary small positive number. Matrix A can be seen in figure 2, where fj = +1 or -1 depending on whether region j is in class 0 or 1. The next step is to apply Farkas lemma, or an equivalent version [10], which gives a necessary and sufficient condition for finding a solution of Ax ~ b. Lemma 1 (Farkas lemma) There exists a vector x E nn such that Ax ~ b iff there does not exist a vector Y E nm such that y T A = 0, y 2': 0 and y T b < O. Assume that Ax ~ b is not solvable. Then, by Lemma 1 for n = k + 1 and m = 2k, a vector y can be found such that y 2': O. Since in addition y T b = -f 2:~~1 Yj, the condition y T b < 0 implies (3jt) Y31 > O. But y T A = 0 is equivalent to the system 152 G. Brightwell. C. Kenyon and H. Paugam-Moisy (t:) of k + 1 equations (t:) { ~::; i ::; k z=k+1 "i+k-l i..Jm=i Ym/clau 0 ,,2k i..Jm=l Ym/clau 0 "i+k-l i..Jm=i Ym/clau 1 ,,2k L..m=l Ym/clau 1 Since (3jt) Yil > 0, the last equation (Ek+l) of system (t:) implies that (3h / class (region jt) ::/= class(region h)) Yh > O. Without loss of generality, assume that it and h are less than k and that region it is in class 0 and region h is in class 1. Comparing two successive equations of (t:), for i < k, we can write (VA E {O, 1}) L(E.+d Ym/clau >. = L(E.) Ym/clau >. - Yi/clau >. + Yi+k/clau >. Since Yit > 0 and region it is in class 0, the transition from Ejl to EiI+1 implies that Yil +k = Yit > 0 and region it + k, which is opposite to region it, is also in class O. Similarly, the transition from Eh to Eh +1 implies that both opposite regions h + k and h are in class 1. These conditions are necessary for the system (t:) to have a non-negative solution and they correspond exactly to the definition of an XOR-bow-tie at point P. The converse comes from theorem 1. • 2.2 UNBOUNDED REGIONS If no two essential hyperplanes are parallel, the case of unbounded regions is exactly the same as a multiple intersection. All the unbounded regions can be labelled as on figure 2. The same argument holds for proving that, if the local system (S) Ax ::; b is not solvable, then there exists an XOR-at-infinity. The case of parallel hyperplanes is more intricate because matrix A is more complex. The proof requires a heavy case-by-case analysis and cannot be given in full in this paper (see [3]) . Theorem 4 Let f be a polyhedral dichotomy on 'R,2 . Let Coo be the complementary region of the convex hull of the essential points of f· The restriction of f to Coo is realizable by a one-hidden-layer network iff f is not in an XOR-at-in/inity. From theorems 3 and 4 we can deduce that a polyhedral dichotomy is locally realizable in 'R,2 by a one-hidden-Iayer network iff f has no XOR-bow-tie and no XORat-infinity. Unfortunately this result cannot be extended to the global realization of f in 'R, 2 because more intricate distant configurations can involve contradictions in the complete system of inequalities. The object of the next section is to point out such a situation by producing a new geometric configuration, called a critical cycle, which implies that f cannot be realized with one hidden layer. 3 CRITICAL CYCLES In contrast to section 2, the results of this section hold for any dimension d 2:: 2. We first need some definitions. Consider a pair of regions {T, T'} in the same class and which both contain an essential point P in their closure. This pair is called critical with respect to P and H if there is an essential hyperplane H going through P such that T' is adjacent along H to the region opposite to T . Note that T and T' are both on the same side of H. We define a graph G whose nodes correspond to the critical pairs of regions of f. There is a red edge between {T, T'} and {U, U'} if the pairs, in different classes, are both critical with respect to the same point (e.g., {Bp, Bp} and {Wp, Wi>} in figure 3). There is a green edge between {T, T'} and {U, U'} if the pairs are both critical with respect to the same hyperplane H, and either the two pairs are on the Multilayer Neural Networks: One or Two Hidden Layers? 153 same side of H, but in different classes (e.g., {W p, Wp} and {BQ, BQ})' or they are on different sides of H, but in the same class (e.g., {Bp,Bp} and {BR, Bk})· Definition 1 A critical cycle is a cycle in graph G, with alternating colors. -i B B' }--.. -.------- {Y Y'} f P. P .... ; P, P I ~ I ; I {B Q, B 'Q~- ·-·-{Y Q ,Y'Q}', I I I I " { B R, B'R}l---{Y R ,Y'R}" red edge green edge Figure 3: Geometrical configuration and graph of a critical cycle, in the plane. Note that one can augment the figure in such a way that there is no XOR-situation, no XOR-bow-tie, and no XOR-at-infinity. Theorem 5 If a polyhedral dichotomy I, from n-d to {O, I}, can be realized by a one-hidden-layer network, then it cannot have a critical cycle. Proof: For the sake of simplicity, we will restrict ourselves to doing the proof for a case similar to the example figure 3, with notation as given in that figure, but without any restriction on the dimension d of I. Assume, for a contradiction, that I has a critical cycle and can be realized by a one-hidden-Iayer network. Consider the sets of regions {Bp, Bp , BQ , BQ, BR, Bk} and {Wp, Wp, WQ , WQ, WR , WR}. Consider the regions defined by all the hyperplanes associated to the hidden layer units (in general, these hyperplanes are a large superset of the essential hyperplanes). There is a region b p ~ B p, whose border contains P and a (d - 1 )-dimensional subset of H 1. Similarly we can define bp, .. . ,bR, Wp , . . . ,wR' Let B be the set of such regions which are in class 1 and W be the set of such regions in class 0. Let H be the hyperplane associated to one of the hidden units. For T a region, let H(T) be the digit label of T w.r.t. H, i.e. H(T) = 1 or ° according to whether T is above or below H (cf. section 1.1). We do a case-by-case analysis. If H does not go through P, then H(bp) = H(b'p) = H(wp) = H(wp); similar equalities hold for lines not going through Q or R. If H goes through P but is not equal to H1 or to H2 , then, from the viewpoint of H, things are as if b'p was opposite to bp, and w'p was opposite to Wp, so the two regions of each pair are on different sides of H, and so H (bp )+H(b'p) = H( wp )+H(w'p) = 1; similar equalities hold for hyperplanes going through Q or R. If H = H 1, then we use the fact that there is a green edge between {W p, Wp} and {BQ, BQ}, meaning in the case of the figure that all four regions are on the same side of H 1 but in different classes. Then H(bp) +H(b'p) +H(bQ) +H(b'q) = H(wp) +H(wp)+ H(wQ) +H(w'q). In fact, this equality would also hold in the other case, as can easily be checked. Thus for all H, we have L,bEB H(b) = L,wEW H(w). But such an equality is impossible: since each b is in class 1 and each w is in class 0, this implies a contradiction in the system of inequalities and I cannot be realized by a one-hidden-Iayer network. Obviously there can exist cycles of length longer than 3, but the extension of the proof is straightforward. _ 154 G. Brightwell, C. Kenyon and H. Paugam-Moisy 4 CONCLUSION AND PERSPECTIVES This paper makes partial progress towards characterizing functions which can be realized by a one-hidden-Iayer network, with a particular focus on dimension 2. Higher dimensions are more challenging, and it is difficult to even propose a conjecture: new cases of inconsistency emerge in subspaces of intermediate dimension. Gibson gives an example of an inconsistent line (dimension 1) resulting of its intersection with two hyperplanes (dimension 2) which are not inconsistent in n3. The principle of using Farkas lemma for proving local realizability still holds but the matrix A becomes more and more complex. In nd , even for d = 3, the labelling of the regions, for instance around a point P of multiple intersection, can become very complex. In conclusion, it seems that neither the topological method of Gibson, nor our algebraic point of view, can easily be extended to higher dimensions. Nevertheless, we conjecture that in dimension 2, a function can be realized by a one-hiddenlayer network iff it does not have any of the four forbidden types of configurations: XOR-situation, XOR-bow-tie, XOR-at-infinity, and critical cycle. Acknowledgements This work was supported by European Esprit III Project nO 8556, NeuroCOLT. References [1] E. B. Baum. On the capabilities of multilayer perceptrons. Journal of Complexity, 4:193-215, 1988. [2] E. K. Blum and L. K. Li. Approximation theory and feedforward networks. Neural Networks, 4(4):511-516, 1991. [3] G. Brightwell, C. Kenyon, and H. Paugam-Moisy. Multilayer neural networks: one or two hidden layers? Research Report 96-37, LIP, ENS Lyon, 1996. [4] M. Cosnard, P. Koiran, and H. Paugam-Moisy. Complexity issues in neural network computations. In I. Simon, editor, Proc. of LATIN'92, volume 583 of LNCS, pages 530-544. Springer Verlag, 1992. [5] M. Cosnard, P. Koiran, and H. Paugam-Moisy. A step towards the frontier between one-hidden-Iayer and two-hidden layer neural networks. In I. Simon, editor, Proc. of IJCNN'99-Nagoya, volume 3, pages 2292-2295. Springer Verlag, 1993. [6] G. Cybenko. Approximation by superpositions of a sigmoidal function. Math. Control, Signal Systems, 2:303-314, October 1988. [7] K. F\mahashi. On the approximate realization of continuous mappings by neural networks. Neural Networks, 2(3):183-192, 1989. [8] G. J. Gibson. A combinatorial approach to understanding perceptron decision regions. IEEE Trans. Neural Networks, 4:989--992, 1993. [9] G. J. Gibson. Exact classification with two-layer neural nets. Journal of Computer and System Science, 52(2):349-356, 1996. [10] M. Grotschel, 1. Lovsz, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, Berlin, Heidelberg, 1988. [11] K. Hornik, M. Stinchcombe, and H. White. Multilayer feedforward networks are universal approximators. Neural Networks, 2(5):359-366, 1989. [12] S.-C. Huang and Y.-F. Huang. Bounds on the number of hidden neurones in multilayer perceptrons. IEEE Trans. Neural Networks, 2:47-55, 1991. [13] P. J. Zweitering. The complexity of multi-layered perceptrons. PhD thesis, Technische Universiteit Eindhoven, 1994.	1996	106
1,158	Improving the Accuracy and Speed of Support Vector Machines Chris J.C. Burges Bell Laboratories Lucent Technologies, Room 3G429 101 Crawford's Corner Road Holmdel, NJ 07733-3030 burges@bell-Iabs.com Abstract Bernhard Scholkopf" Max-Planck-Institut fur biologische Kybernetik, Spemannstr. 38 72076 Tubingen, Germany bs@mpik-tueb.mpg.de Support Vector Learning Machines (SVM) are finding application in pattern recognition, regression estimation, and operator inversion for ill-posed problems. Against this very general backdrop, any methods for improving the generalization performance, or for improving the speed in test phase, of SVMs are of increasing interest. In this paper we combine two such techniques on a pattern recognition problem. The method for improving generalization performance (the "virtual support vector" method) does so by incorporating known invariances of the problem. This method achieves a drop in the error rate on 10,000 NIST test digit images of 1.4% to 1.0%. The method for improving the speed (the "reduced set" method) does so by approximating the support vector decision surface. We apply this method to achieve a factor of fifty speedup in test phase over the virtual support vector machine. The combined approach yields a machine which is both 22 times faster than the original machine, and which has better generalization performance, achieving 1.1 % error. The virtual support vector method is applicable to any SVM problem with known invariances. The reduced set method is applicable to any support vector machine. 1 INTRODUCTION Support Vector Machines are known to give good results on pattern recognition problems despite the fact that they do not incorporate problem domain knowledge. ·Part of this work was done while B.S. was with AT&T Research, Holmdel, NJ. 376 C. 1. Burges and B. SchOlkopf However, they exhibit classification speeds which are substantially slower than those of neural networks (LeCun et al., 1995). The present study is motivated by the above two observations. First, we shall improve accuracy by incorporating knowledge about invariances of the problem at hand. Second, we shall increase classification speed by reducing the complexity of the decision function representation. This paper thus brings together two threads explored by us during the last year (Scholkopf, Burges & Vapnik, 1996; Burges, 1996). The method for incorporating invariances is applicable to any problem for which the data is expected to have known symmetries. The method for improving the speed is applicable to any support vector machine. Thus we expect these methods to be widely applicable to problems beyond pattern recognition (for example, to the regression estimation problem (Vapnik, Golowich & Smola, 1996)). After a brief overview of Support Vector Machines in Section 2, we describe how problem domain knowledge was used to improve generalization performance in Section 3. Section 4 contains an overview of a general method for improving the classification speed of Support Vector Machines. Results are collected in Section 5. We conclude with a discussion. 2 SUPPORT VECTOR LEARNING MACHINES This Section summarizes those properties of Support Vector Machines (SVM) which are relevant to the discussion below. For details on the basic SVM approach, the reader is referred to (Boser, Guyon & Vapnik, 1992; Cortes & Vapnik, 1995; Vapnik, 1995). We end by noting a physical analogy. Let the training data be elements Xi E C, C = R d, i = 1, ... ,f, with corresponding class labels Yi E {±1}. An SVM performs a mapping 4> : C ---+ 1i, x t-+ X into a high (possibly infinite) dimensional Hilbert space 1i. In the following, vectors in 1i will be denoted with a bar. In 1i, the SVM decision rule is simply a separating hyperplane: the algorithm constructs a decision surface with normal ~ E 1i which separates the Xi into two classes: ~·xi+b ~ kO-c'i, Yi=+1 (1) 1j, . Xi + b < kl + c'i, Yi = -1 (2) where the c'j are positive slack variables, introduced to handle the non-separable case (Cortes & Vapnik...l 1995), and where ko and kl are typically defined to be +1 and -1, respectively. W is computed by minimizing the objective function l W·W ""' -2- + C(L..J C,i)P i=l (3) subject to (1), (2), where C is a constant, and we choose p = 2. In the separable case, the SVM algorithm constructs that separating hyperplane for which the margin between the positive and negative examples in 1i is maximized. A test vector x E C is then assigned a class label {+ 1, -' 1} depending on whether 1j, . 4>( x) + b is greater or less than (ko + kt)/2. Support vectors Sj E C are defined as training samples for which one of Equations (1) or (2) is an equality. (We name the suppo!:t vectors S to distinguish them from the rest of the training data) . The solution W may be expressed Ns 1j, = I: O'jYj4>(Sj) (4) j=1 Improving the Accuracy and Speed of Support Vector Machines 377 where Cl:j ~ ° are the positive weights, determined during training, Yj E {±1} the class labels of the Sj , and N s the number of support vectors. Thus in order to classify a test point x one must compute Ns Ns N s q, . X = 2:' Cl:jYj Sj . x = 2: Cl:jYj4>(Sj) . 4>(x) = 2: Cl:jYj J«Sj, x). (5) j=l i=l j=l One of the key properties of support vector machines ,is the use of the kernel J< to compute dot products in 1-l without having to explicitly compute the mapping 4>. It is interesting to note that the solution has a simple physical interpretation in the high dimensional space 1-l. If we assume that each support vector Sj exerts a perpendicular force of size Cl:j and sign Yj on a solid plane sheet lying along the hyperplane ~ . x + b = (ko + kd/2 , then the solution satisfies the requirements of mechanical stability. At the solution, the Cl:j can be shown to satisfy 2:7';1 Cl:iYj = 0, which translates into the forces on the sheet summing to zero; and Equation (4) implies that the torques also sum to zero. 3 IMPROVING ACCURACY This section follows the reasoning of (Scholkopf, Burges, & Vapnik, 1996). Problem domain knowledge can be incorporated in two different ways: the knowledge can be directly built into the algorithm, or it can be used to generate artificial training examples ("virtual examples"). The latter significantly slows down training times, due to both correlations in the artificial data and to the increased training set size (Simard et aI. , 1992); however it has the advantage of being readily implemented for any learning machine and for any invariances. For instance, if instead of Lie groups of symmetry transformations one is dealing with discrete symmetries, such as the bilateral symmetries of Vetter, Poggio, & Biilthoff (1994) , then derivative-based methods (e.g. Simard et aI., 1992) are not applicable. For support vector machines, an intermediate method which combines the advantages of both approaches is possible. The support vectors characterize the solution to the problem in the following sense: If all the other training data were removed, and the system retrained, then the solution would be unchanged. Furthermore, those support vectors Si which are not errors are close to the decision boundary in 1-l, in the sense that they either lie exactly on the margin (ei = 0) or close to it (ei < 1). Finally, different types of SVM, built using different kernels, tend to produce the same set of support vectors (Scholkopf, Burges, & Vapnik, 1995). This suggests the following algorithm: first, train an SVM to generate a set of support vectors {Sl, .. . , SN. }; then, generate the artificial examples (virtual support vectors) by applying the desired invariance transformations to {Sl , ... , SN.} ; finally, train another SVM on the new set. To build a ten-class classifier, this procedure is carried out separately for ten binary classifiers. Apart from the increase in overall training time (by a factor of two, in our experiments), this technique has the disadvantage that many of the virtual support vectors become support vectors for the second machine, increasing the number of summands in Equation (5) and hence decreasing classification speed. However, the latter problem can be solved with the reduced set method, which we describe next. 378 C. J. Burges and B. SchOlkopf 4 IMPROVING CLASSIFICATION SPEED The discussion in this Section follows that of (Burges, 1996). Consider a set of vectors Zk E C, k = 1, ... , Nz and corresponding weights rk E R for which Nz ~I == L rk4>(Zk) (6) k=l minimizes (for fixed N z) the Euclidean distance to the original solution: p = II~ ~/II· (7) Note that p, expressed here in terms of vectors in 1i, can be expressed entirely in terms of functions (using the kernel K) of vectors in the input space C. The {( rk, Zk) I k = 1, ... , N z} is called the reduced set. To classify a test point x, the expansion in Equation (5) is replaced by the approximation Nz Nz ~/·X = 2:rkZk·X = LrkK(Zk'X). (8) k=l k=l The goal is then to choose the smallest N z ~ N s, and corresponding reduced set, such that any resulting loss in generalization performance remains acceptable. Clearly, by allowing N z = N s, P can be made zero. Interestingly, there are nontrivial cases where Nz < Ns and p = 0, in which case the reduced set leads to an increase in classification speed with no loss in generalization performance. Note that reduced set vectors are not support vectors, in that they do not necessarily lie on the separating margin and, unlike support vectors, are not training samples. While the reduced set can be found exactly in some cases, in general an unconstrained conjugate gradient method is used to find the Zk (while the corresponding optimal rk can be found exactly, for all k). The method for finding the reduced set is computationally very expensive (the final phase constitutes a conjugate gradient descent in a space of (d + 1) . N z variables, which in our case is typically of order 50,000). 5 EXPERIMENTAL RESULTS In this Section, by "accuracy" we mean generalization performance, and by "speed" we mean classification speed. In our experiments, we used the MNIST database of 60000+ 10000 handwritten digits, which was used in the comparison investigation of LeCun et al (1995). In that study, the error rate record of 0.7% is held by a boosted convolutional neural network ("LeNet4"). We start by summarizing the results of the virtual support vector method. We trained ten binary classifiers using C = 10 in Equation (3). We used a polynomial kernel K(x, y) = (x· y)5. Combining classifiers then gave 1.4% error on the 10,000 test set; this system is referred to as ORIG below. We then generated new training data by translating the resulting support vectors by one pixel in each of four directions, and trained a new machine (using the same parameters). This machine, which is referred to as VSV below, achieved 1.0% error on the test set. The results for each digit are given in Table 1. Note that the improvement in accuracy comes at a cost in speed of approximately a factor of 2. Furthermore, the speed of ORIG was comparatively slow to start with (LeCun et al., 1995), requiring approximately 14 million multiply adds for one Improving the Accuracy and Speed of Support Vector Machines 379 Table 1: Generalization Performance Improvement by Incorporating Invariances. N E and N sv are the number of errors and number of support vectors respectively; "ORIG" refers to the original support vector machine, "vsv" to the machine trained on virtual support vectors. Digit NE ORIG NE VSV Nsv ORIG Nsv VSV 0 17 15 1206 2938 1 15 13 757 1887 2 34 23 2183 5015 3 32 21 2506 4764 4 30 30 1784 3983 5 29 23 2255 5235 6 30 18 1347 3328 7 43 39 1712 3968 8 47 35 3053 6978 9 56 40 2720 6348 Table 2: Dependence of Performance of Reduced Set System on Threshold. The numbers in parentheses give the corresponding number of errors on the test set. Note that Thrsh Test gives a lower bound for these numbers. Digit Thrsh VSV Thrsh Bayes Thrsh Test 0 1.39606 (9) 1.48648 (8) 1.54696 (7) 1 3.98722 (24) 4.43154 (12) 4.32039 (10) 2 1.27175 (31) 1.33081 (30) 1.26466 (29) 3 1.26518 (29) 1.42589 (27) 1.33822 (26) 4 2.18764 (37) 2.3727 (35) 2.30899 (33) 5 2.05222 (33) 2.21349 (27) 2.27403 (24) 6 0.95086 (25) 1.06629 (24) 0.790952 (20) 7 3.0969 (59) 3.34772 (57) 3.27419 (54) 8 -1.06981 (39) -1.19615 (40) -1.26365 (37) 9 1.10586 (40) 1.10074 (40) 1.13754 (39) classification (this can be reduced by caching results of repeated support vectors (Burges, 1996)). In order to become competitive with systems with comparable accuracy, we will need approximately a factor of fifty improvement in speed. We therefore approximated VSV with a reduced set system RS with a factor of fifty fewer vectors than the number of support vectors in VSV. Since the reduced set method computes an approximation to the decision surface in the high dimensional space, it is likely that the accuracy of RS could be improved by choosing a different threshold b in Equations (1) and (2). We computed that threshold which gave the empirical Bayes error for the RS system, measured on the training set. This can be done easily by finding the maximum of the difference between the two un-normalized cumulative distributions of the values of the dot products q, . Xi, where the Xi are the original training data. Note that the effects of bias are reduced by the fact that VSV (and hence RS) was trained only on shifted data, and not on any of the original data. Thus, in the absence of a validation set, the original training data provides a reasonable means of estimating the Bayes threshold. This is a serendipitous bonus of the VSV approach. Table 2 compares results obtained using the threshold generated by the training procedure for the VSV system; the estimated Bayes threshold for the RS system; and, for comparison 380 C. 1. Burges and B. SchOlkopf Table 3: Speed Improvement Using the Reduced Set method. The second through fourth columns give numbers of errors on the test set for the original system, the virtual support vector system, and the reduced set system. The last three columns give, for each system, the number of vectors whose dot product must be computed in test phase. Digit ORIG Err VSV Err RS Err ORIG # SV VSV # SV #RSV 0 17 15 18 1206 2938 59 1 15 13 12 757 1887 38 2 34 23 30 2183 5015 100 3 32 21 27 2506 4764 95 4 30 30 35 1784 3983 80 5 29 23 27 2255 5235 105 6 30 18 24 1347 3328 67 7 43 39 57 1712 3968 79 8 47 35 40 3053 6978 140 9 56 40 40 2720 6348 127 purposes only (to see the maximum possible effect of varying the threshold) , the Bayes error computed on the test set. Table 3 compares results on the test set for the three systems, where the Bayes threshold (computed with the training set) was used for RS. The results for all ten digits combined are 1.4% error for ORIG, 1.0% for VSV (with roughly twice as many multiply adds) and 1.1% for RS (with a factor of 22 fewer multiply adds than ORIG). The reduced set conjugate gradient algorithm does not reduce the objective function p2 (Equation (7)) to zero. For example, for the first 5 digits, p2 is only reduced on average by a factor of 2.4 (the algorithm is stopped when progress becomes too slow). It is striking that nevertheless, good results are achieved. 6 DISCUSSION The only systems in LeCun et al (1995) with better than 1.1% error are LeNet5 (0.9% error, with approximately 350K multiply-adds) and boosted LeNet4 (0.7% error, approximately 450K mUltiply-adds). Clearly SVMs are not in this league yet (the RS system described here requires approximately 650K multiply-adds). However, SVMs present clear opportunities for further improvement. (In fact, we have since trained a VSV system with 0.8% error, by choosing a different kernel). More invariances (for example, for the pattern recognition case, small rotations, or varying ink thickness) could be added to the virtual support vector approach. Further, one might use only those virtual support vectors which provide new information about the decision boundary, or use a measure of such information to keep only the most important vectors. Known invariances could also be built directly into the SVM objective function. Viewed' as an approach to function approximation, the reduced set method is currently restricted in that it assumes a decision function with the same functional form as the original SVM. In the case of quadratic kernels, the reduced set can be computed both analytically and efficiently (Burges, 1996). However, the conjugate gradient descent computation for the general kernel is very inefficient. Perhaps reImproving the Accuracy and Speed of Support Vector Machines 381 laxing the above restriction could lead to analytical methods which would apply to more complex kernels also. Acknowledgements We wish to thank V. Vapnik, A. Smola and H. Drucker for discussions. C. Burges was supported by ARPA contract N00014-94-C-0186. B. Sch6lkopf was supported by the Studienstiftung des deutschen Volkes. References [1] Boser, B. E., Guyon, I. M., Vapnik, V., A Training Algorithm for Optimal Margin Classifiers, Fifth Annual Workshop on Computational Learning Theory, Pittsburgh ACM (1992) 144-152. [2] Bottou, 1., Cortes, C., Denker, J. S., Drucker, H., Guyon, I., Jackel, L. D., Le Cun, Y., Muller, U. A., Sackinger, E., Simard, P., Vapnik, V., Comparison of Classifier Methods: a Case Study in Handwritten Digit Recognition, Proceedings of the 12th International Conference on Pattern Recognition and Neural Networks, Jerusalem (1994) [3] Burges, C. J. C., Simplified Support Vector Decision Rules, 13th International Conference on Machine Learning (1996), pp. 71 - 77. [4] Cortes, C., Vapnik, V., Support Vector Networks, Machine Learning 20 (1995) pp. 273 - 297 [5] LeCun, Y., Jackel, 1., Bottou, L., Brunot, A., Cortes, C., Denker, J., Drucker, H., Guyon, I., Muller, U., Sackinger, E., Simard, P., and Vapnik, V., Comparison of Learning Algorithms for Handwritten Digit Recognition, International Conference on Artificial Neural Networks, Ed. F. Fogelman, P. Gallinari, pp. 53-60, 1995. [6] Sch6lkopf, B., Burges, C.J.C., Vapnik, V., Extracting Support Data for a Given Task, in Fayyad, U. M., U thurusamy, R. (eds.), Proceedings, First International Conference on Knowledge Discovery & Data Mining, AAAI Press, Menlo Park, CA (1995) [7] Sch6lkopf, B., Burges, C.J.C., Vapnik, V., Incorporating Invariances in Support Vector Learning Machines, in Proceedings ICANN'96 International Conference on Artificial Neural Networks. Springer Verlag, Berlin, (1996) [8] Simard, P., Victorri, B., Le Cun, Y., Denker, J., Tangent Prop a Formalism for Specifying Selected Invariances in an Adaptive Network, in Moody, J. E., Hanson, S. J:, Lippmann, R. P., Advances in Neural Information Processing Systems 4, Morgan Kaufmann, San Mateo, CA (1992) [9] Vapnik, V., Estimation of Dependences Based on Empirical Data, [in Russian] Nauka, Moscow (1979); English translation: Springer Verlag, New York (1982) [10] Vapnik, V., The Nature of Statistical Learning Theory, Springer Verlag, New York (1995) [11] Vapnik, V., Golowich, S., and Smola, A., Support Vector Method for Function Approximation, Regression Estimation, and Signal Processing, Submitted to Advances in Neural Information Processing Systems, 1996 [12] Vetter, T., Poggio, T., and Bulthoff, H., The Importance of Symmetry and Virtual Views in Three-Dimensional Object Recognition, Current Biology 4 (1994) 18-23	1996	107
1,159	Adaptive Access Control Applied to Ethernet Data Timothy X Brown Dept. of Electrical and Computer Engineering University of Colorado, Boulder, CO 80309-0530 timxb@colorado.edu Abstract This paper presents a method that decides which combinations of traffic can be accepted on a packet data link, so that quality of service (QoS) constraints can be met. The method uses samples of QoS results at different load conditions to build a neural network decision function. Previous similar approaches to the problem have a significant bias. This bias is likely to occur in any real system and results in accepting loads that miss QoS targets by orders of magnitude. Preprocessing the data to either remove the bias or provide a confidence level, the method was applied to sources based on difficult-to-analyze ethernet data traces. With this data, the method produces an accurate access control function that dramatically outperforms analytic alternatives. Interestingly, the results depend on throwing away more than 99% of the data. 1 INTRODUCTION In a communication network in which traffic sources can be dynamically added or removed, an access controller must decide when to accept or reject a new traffic source based on whether, if added, acceptable service would be given to all carried sources. Unlike best-effort services such as the internet, we consider the case where traffic sources are given quality of service (QoS) guarantees such as maximum delay, delay variation, or loss rate. The goal of the controller is to accept the maximal number of users while guaranteeing QoS. To accommodate diverse sources such as constant bit rate voice, variablerate video, and bursty computer data, packet-based protocols are used. We consider QOS in terms of lost packets (Le. packets discarded due to resource overloads). This is broadly applicable (e.g. packets which violate delay guarantees can be considered lost) although some QoS measures can not fit this model. The access control task requires a classification function-analytically or empirically derived-that specifies what conditions will result in QoS not being met. Analytic functions have been successful only on simple traffic models [Gue91], or they are so conservative that they grossly under utilize the network. This paper describes a neural network method that adapts an access control function based on historical data on what conditions packets have and have not been successfully carried. Neural based solutions have been previously applied to the access control problem [Hir90][Tra92] [Est94], but these Adaptive Access Control Applied to Ethernet Data 933 approaches have a distinct bias that under real-world conditions leads to accepting combinations of calls that miss QoS targets by orders of magnitude. Incorporating preprocessing methods to eliminate this bias is critical and two methods from earlier work will be described. The combined data preprocessing and neural methods are applied to difficultto-model ethernet traffic. 2 THE PROBLEM Since the decision to accept a multilink connection can be decomposed into decisions on the individual links, we consider only a single link. A link can accept loads from different source types. The loads consist of packets modeled as discrete events. Arriving packets are placed in a buffer and serviced in turn. If the buffer is full, excess packets are discarded and treated as lost. The precise event timing is not critical as the concern is with the number of lost packets relative to the total number of packets received in a large sample of events, the so-called loss rate. The goal is to only accept load combinations which have a loss rate below the QoS target denoted by p. Load combinations are described by a feature vector, $, consisting of load types and possibly other information such as time of day. Each feature vector, $, has an associated loss rate, p($), which can not be measured directly. Therefore, the goal is to have a classifier function, C($), such that C($) >, <, = 0 if p($) <, >, = p. Since analytic C($) are not in general available, we look to statistical classification methods. This requires training samples, a desired output for each sample, and a significance or weight for each sample. Loads can be dynamically added or removed. Training samples are generated at load transitions, with information since the last transition containing the number of packet arrivals, T, the number of lost packets, s, and the feature vector, $. A sample ($i' si' Ti), requires a desired classification, d($i> si' Ti) E {+1, -1}, and a weight, W($i' s;. Ti) E (0,00). Given a data set {($i' si' Ti)}, a classifier, C, is then chosen that minimizes the weighted sum squared error E = 2:j[w(~j, Sj, Tj)(C(~i) -d(~i' S;, T j»2]. A classifier, with enough degrees of freedom will set C($i) = d($i' si' 1j) if all the $i are different. With multiple samples at the same $ then we see that the error is minimized when C(~) = (2: _ _ [w(~j, Si' Tj)d(~j, Sj, T;)])/(2: _ _ W(~i' Sj, T;». (1) {iI~, "'~} {il~i "'~} Thus, the optimal C($) is the weighted average of the d($i' si' Ti) at $. If the classifier has fewer degrees of freedom (e.g. a low dimension linear classifier), C($) will be the average of the d($i' si' 1j) in the neighborhood of $, where the neighborhood is, in general, an unspecified function of the classifier. A more direct form of averaging would be to choose a specific neighborhood around $ and average over samples in this neighborhood. This suffers from having to store all the samples in the decision mechanism, and incurs a significant computational burden. More significant is how to decide the size of the neighborhood. If it is fixed, in sparse regions no samples may be in the neighborhood. In dense regions near decision boundaries, it may average over too wide a range for accurate estimates. Dynamically setting the neighborhood so that it always contains the k nearest neighbors solves this problem, but does not account for the size of the samples. We will return to this in Section 4. 3 THE SMALL SAMPLE PROBLEM Neural networks have previously been a£plied to the access control proElem [Hir91] [Tra92][Est94]. In [Hir90] and [Tra92], d(<I>i' si' Ti) = +1 when s;lTi < p, d(<I>i' si' 1j) =-1 otherwise, and the weighting is a uniform w($i' si' Ti) = 1 for all i. This desired out and 934 T. X. Brown uniform weighting we call the normal method. For a given load combination, lP, assume an idealized system where packets enter and with probability p(<P) independent of earlier or later packets, the packet is labeled as lost. In a sample of T such Bernoulli trials with S the number packets lost, let PB = P{s/T> p}. Since with the normal method d(lP, s, 1) = -1 if sIT> p, PB = P{d(lP, s, 1) = -I}. From (1), with uniform weighting the decision boundary is where PB = 0.5. If the samples are small (i.e. T < (In 2)/p < IIp), d(lP, s, 1) =-1 for all s > O. In this case PB = 1 - (1 -p(lP)l Solving for p(lP) at PB = 0.5 using In(1 - x) "" -x, the decision boundary is at p(lP) "" (In 2)ff > p. So, for small sample sizes, the normal method boundary is biased to greater than p* and can be made orders of magnitude larger as T becomes smaller. For larger T, e.g. Tp* > 10, this bias will be seen to be negligible. One obvious solution is to have large samples. This is complicated by three effects. The first is that desired loss rates in data systems are often small; typically in the range 1O-ti_1O-12. This implies that to be large, samples must be at least 107_1013 packets. For the latter, even at Gbps rates, short packets, and full loading this translates into samples of several hours of traffic. Even for the first at typical rates, this can translate into minutes of traffic. The second, related problem is that in dynamic data networks, while individual connections may last for significant periods, on the aggregate a given combination of loads may not exist for the requisite period. The third more subtle problem is that in any queueing system even with uncorrelated arrival traffic the buffering introduces memory in the system. A typical sample with losses may contain 100 losses, but a loss trace would show that all of the losses occurred in a single short overload interval. Thus the number of independent trials can be several orders of magnitude smaller than indicated by the raw sample size indicating that the loads must be stable for hours, days, or even years to get samples that lead to unbiased classification. An alternative approach used in [Hir95] sets d(lP, s, 1) = sIT and models p(lP) directly. The probabilities can vary over orders of magnitude making accurate estimates difficult. Estimating the less variable 10g(p(lP» with d = 10g(s/1) is complicated by the logarithm being undefined for small samples where most samples have no losses so that s = o. 4 METHODS FOR TREATING BIAS AND VARIANCE We present without proof two preprocessing methods derived and analyzed in [Br096]. The first eliminates the sample bias by choosing an appropriate d and w that directly solves (1) s.t. c(lP) >, <, = 0 if and only if p(lP) <, >, = p* i.e. it is an unbiased estimate as to whether the loss rate is above and below p. This is the weighting method shown in Table 1. The relative weighting of samples with loss rates above and below the critical loss rate is plotted in Figure 1. For large T, as expected, it reduces to the normal method. The second preprocessing method assigns uniform weighting, but classifies d(lP, s, 1) = 1 only if a certain confidence level, L, is met that the sample represents a combination where p(lP) < p. Such a confidence was derived in [Bro96]: Table 1: Summary of Methods. Sample Class Weighting, w(<!>j, Sj, TD, when Method d(<Pj, Sj, Ti) = + 1 if d(lPi, Sj, Tj ) = + 1 (i.e. w +) d(lPj, si, Tj) = -1 (i.e. w-) Normal si::;;LpTJ 1 1 Weighting si::;;LpT J TL (~)Pi(l_ p{-i TL (~)pi(l_ p{-i i>LpT J I!>LpTJ Aggregate Table 2 1 1 Adaptive Access Control Applied to Ethernet Data 935 S i T ·~(Tp) P{p(<I»>pls,T}=e- P £..J-'-'l. (2) i =0 For small T (e.g. T < IIp* and L > 1 - lie), even if s = 0 (no losses), this level is not met. But, a neighborhood of samples with similar load combinations may all have no losses indicating that this sample can be classified as having p( $) < p. Choosing a neighborhood requires a metric, m, between feature vectors, $. In this paper we simply use Euclidean distance. Using the above and solving for T when s = 0, the smallest meaningful neighborhood size is the smallest k such that the aggregate sample is greater than a critical size, 1' = -In(1- L)/p. From (2), this guarantees that if no packets in the aggregate sample are lost we can classify it as having p«(j») < p within our confidence level. For larger samples, or where samples are more plentiful and k can afford to be large, (2) can be used directly. Table 2 summarizes this aggregate method. The above preprocessing methods assume that the training samples consist of independent samples of Bernoulli trials. Because of memory introduced by the buffer and possible correlations in the arrivals, this is decidedly not true. The methods can still be applied, if samples can be subsampled at every Ith trial where I is large enough so that the samples are pseudo-independent, i.e. the dependency is not significant for our application. A simple graphical method for determining I is as follows. Observing Figure I, if the number of trials is artificially increased, for small samples the weighting method will tend to under weight the trials with errors, so that its decision boundary will be at erroneously high loss rates. This is the case with correlated samples. The sample size, T, overstates the number of independent trials. As the subsample factor is increased, the subsample size becomes smaller, the trials become increasingly independent, the weighting becomes more appropriate, and the decision boundary moves closer to the true decision boundary. At some point, the samples are sufficiently independent so that sparser subsampling does not change the decision boundary. By plotting the decision boundary of the classifier as a function of I, the point where the boundary is independent of the subsample factor indicates a suitable choice for I. In summary, the procedure consists of collecting traffic samples at different combinations of traffic loads that do and do not meet quality of service. These are then subsampled with a factor I determined as above. Then one of the sample preprocessing methods, summarized in Table I, are applied to the data. These preprocessed samples are then used in any neural network or classification scheme. Analysis in [Br096] derives the expected bias (shown in Figure 2) of the methods when used with an ideal classifier. The normal method can be arbitrarily biased, the weighting method is unbiased, and the aggregate method chooses a conservative boundary. Simulation experiments in [Br096] applying it to a well characterized MIMII queueing system to determine acceptable loads showed that the weighting method was able to produce unbiased threshold estimates over a range of valTable 2: Aggregate Classification Algorithm 1. Given Sample (<I>i' si, Ti) E {(<I>i' si,1j)}, metric, m, and confidence level, L. 2. Calculate T* = -InC 1 - L)/ p. 3. Find nearest neighbors no, nl' .. , where no = i and m($nj, $i) ~ m($nj+I' $i) for j ~ O. k k 4. Choose smallest k S.t. T' = LTnj ~ T. Let S' = L sn/ j=O j=O . _ {+1 ifP{p($»pls',T}«I-L) 5. Usmg (2), d(<I>;, s;, T;) = 1 . o.w. 936 le+02 ~ 8 ~ le+Ol o .f3 '" letOO le-OI L........_~_~_~_~_~--' 0.001 0.01 0.1 1 Tp· \0 100 1000 Figure 1: Plot of Relative Weighting of Samples with Losses Below (w-) and Above (w +) the Critical Loss Rate. T. X. Brown le+03 r---~'---~'---~'---~,--~'-------"--'" Nonnal Weighting ------Aggregate · . ~ ,5'le+02 J g le+Ol ~ IletOO " '" .................... . --_' .. n.!.: .. ! ... !<, .. : le-OI L........_~_~_'--''--~_~--' 0.001 0.01 0.1 1 Tp· \0 100 1000 Figure 2: Expected Decision Normalized by p. The nominal boundary is P/P* = 1. The aggregate method uses L = 0.95. ues; and the aggregate method produced conservative estimates that were always below the desired threshold, although in terms of traffic load were only 5% smaller. Even in this simple system where the input traffic is uncorrelated (but the losses become correlated due the memory in the queue), the subsample factor was 12, meaning that good results required more than 90% of the data be thrown out. 5 EXPERIMENTS WITH ETHERNET TRAFFIC DATA This paper set out to solve the problem of access control for real world data. We consider a system where the call combinations consist of individual computer data users trunked onto a single output link. This is modeled as a discrete-time single-server queueing model where in each time slot one packet can be processed and zero or more packets can arrive from the different users. The server has a buffer of fixed length 1000. To generate a realistic arrival process, we use ethernet data traces. The bandwidth of the link was chosen at from 100100Mbps. With 48 byte packets, the queue packet service rate was the bandwidth divided by 384. All arrival rates are normalized by the service rate. 5.1 THEDATA We used ethemet data described in [Le193] as the August 89 busy hour containing traffic ranging from busy file-servers/routers to users with just a handful of packets. The detailed data set records every packet's arrival time (to the nearest l00llsec), size, plus source and destination tags. From this, 108 "data traffic" sources were generated, one for each computer that generated traffic on the ethernet link. To produce uniform size packets, each ethernet packet (which ranged from 64 to 1518 bytes) was split into 2 to 32 48-byte packets (partial packets were padded to 48 bytes). Each ethernet packet arrival time was mapped into a particular time slot in the queueing model. All the packets arriving in a times lot are immediately added to the buffer, any buffer overflows would be discarded (counted as lost), and if the buffer was non-empty at the start of the timeslot, one packet sent. Ethernet contains a collision protocol so that only one of the sources is sending packets at anyone time onto a lOMbps connection. Decorrelating the sources via random starting offsets, produced independent data sources with the potential for overloads. Multiple copies at different offsets produced sufficient loads even for bandwidths greater than 10Mbps. The peak data rate with this data is fixed, while the load (the average rate over the one hour trace normalized by the peak rate) ranges over five orders of magnitude. Also troubling, analysis of this data [LeI93] shows that the aggregate traffic exhibits chaotic self-similar properties and suggests that it may be due to the sources' distribution of packet interarrival times following an extremely heavy tailed distribution with infinite higher order moments. No tractable closed form solution exists for such data to predict whether a particular load will result in an overload. Thus, we apply adaptive access control. Adaptive Access Control Applied to Ethernet Data 937 5.2 EXPERIMENT AND RESULTS We divided the data into two roughly similar groups of 54 sources each; one for training and one for testing. To create sample combinations we assign a distribution over the different training sources, choose a source combination from this distribution, and choose a random, uniform (over the period of the trace) starting time for each source. Simulations that reach the end of a trace wrap around to the beginning of the trace. The sources are described by a single feature corresponding to the average load of the source over the one hour data trace. A group of sources is described by the sum of the average loads. The source distribution was a uniformly chosen O-M copies of each of the 54 training samples. M was dynamically chosen so that the link would be sufficiently loaded to cause losses. Each sample combination was processed for 3x107 time slots, recording the load combination, the number of packets serviced correctly, and the number blocked. The experiment was repeated for a range of bandwidths. The bandwidths and number of samples at each bandwidth are shown in Table 3 We applied the three methods of Table 1 based on p* = 10-6 (L = 95% for the aggregate method) and used the resulting data in a linear classifier. Since the feature is the load and larger loads will always cause more blocking,p(<I» is a one variable monotonic function. A linear classifier is sufficient for this case and its output is simply a threshold on the load. To create pseudo-independent trials necessary for the aggregate and weighting methods, we subsampled every lth packet. Using the graphical method of Section 4, the resulting I are shown in column 4 of Table 3. A typical subsample factor is 200. The sample sizes ranged from 105 to 107 trials, But, after subsampling by a factor of 200, even for the largest samples, pT < 0.05 « 1. . The thresholds found by each method are shown in Table 3. To get loss rate estimates at these thresholds, the average loss rate of the 20% of source combinations below each method's threshold is computed. Since accepted loads would be below the threshold this is a typical loss rate. The normal scheme is clearly flawed with losses 10 times higher than p, the weighting scheme's loss rate is apparently unbiased with results around p, while the aggregate scheme develops a conservative boundary below p. To test the boundaries, we repeated the experiment generating source combination samples using the 54 sources not used in the training. Table 3 also shows the losses on this test set and indicates that the training set boundaries produce similar results on the test data. The boundaries are compared with those of more conventional, model-based techniques. One proposed technique for detecting overloads appears in [Gue91]. This paper assumes the sources are based on a Markov On/Off model. Applying the method to this ethernet data (treating each packet arrival as an On period and calculating necessary parameters from there), all but the very qighest loads in the training sets are classified as acceptable indicating that the loss rate would be orders of magnitude higher than p. A conservative technique is to accept calls only as long as the sum of the peak source transmission rates is less than the link bandwidth. For the lOMbps link, since this equals the original ethemet Table 3: Results from Experiments at Different Link Bandwidth. BandNumber of SubThreshold Found & Loss Rate width Samples sample at Threshold on (train/test) Set (Mbps) Train Test Factor Normal Weighting Aggregate 10 1569 1080 230 0.232 (1e-5/4e-6) 0.139 (8e-7/le-6) 0.105 (I e-7/8e-8) 17.5 2447 3724 180 0.415 (2e-5/3e-5) 0.268 (5e-7/ge-7) 0.215 (3e-9/4e-7) 30 6696 4219 230 0.508 (7e-6/4e-5) 0.333 (4e-6/5e-8) 0.286 (3e-7I2e-8) 100 1862 N.A. 180 0.688 (le-51N.A.) 0.566 (5e-71N.A.) 0.494 (Oe-OIN.A.) 938 T. X. Brown data rate, this peak rate method will accept exactly one source. Averaging over all sources, the average load would be 0.0014 and would not increase with increasing bandwidth. The neural method takes advantage of better trunking at increasing bandwidths, and carries two orders of magnitude more traffic. 6 CONCLUSION Access control depends on a classification function that decides if a given set of load conditions will violate quality of service constraints. In this paper quality of service was in terms of a maximum packet loss rate, p. Given that analytic methods are inadequate when given realistic traffic sources, a neural network classification method based on samples of traffic results at different load conditions is a practical alternative. With previous neural network approaches, the synthetic nature of the experiments obscured a significant bias that exists with more realistic data. This bias, due to the small sample sizes relative to l/p, is likely to occur in any real system and results in accepting loads with losses that are orders of magnitude greater than p. Preprocessing the data to either remove the bias or provide a confidence level, the neural network was applied to sources based on difficult-to-analyze ethernet data traces. A group of sources was characterized by its total load so that the goal was to simply choose a threshold on how much load the link would accept. The neural network was shown to produce accurate estimates of the correct threshold. Interestingly these good results depend on creating traffic samples representing independent packet transmissions. This requires more than 99% of the data to be thrown away indicating that for good performance an easy-to-implement sparse sampling of the packet fates is sufficient. It also indicates that unless the total number of packets that is observed is orders of magnitude larger than l/p*, the samples are actually small and preprocessing methods such as in this paper must be applied for accurate loss rate classification. In comparison to analytic techniques, all of the methods, are more accurate at identifying overloads. In comparison to the best safe alternative that works even on this ethernet data, the neural network method was able to carry two orders of magnitude more traffic. The techniques in this paper apply to a range of network problems from routing, to bandwidth allocation, network design, as well as access control. References [Br096] Brown, TX, "Classifying Loss Rates with Small Samples," Submitted to IEEE Tran. on Comm., April 1996. [Est94] Estrella, AD., Jurado, A, Sandoval, F., "New Training Pattern Selection Method for ATM Call Admission Neural Control," Elec. Let., Vol. 30, No.7, pp. 577-579, Mar. 1994. [Gue91] Guerin, R., Ahmadi, H., Naghshineh, M., "Equivalent Capacity and its Application to Bandwidth Allocation in High-Speed Networks," IEEE JSAC, vol. 9, no. 7,pp.968-981,1991. [Hir90] Hiramatsu, A., "ATM Communications Network Control by Neural Networks," IEEE Trans. on Neural Networks, vol. 1, no. 1, pp. 122-130, 1990. [Hir95] Hiramatsu, A, "Training Techniques for Neural Network Applications in ATM," IEEE Comm. Mag., October, pp. 58-67,1995. [LeI93] Leland, W.E., Taqqu, M.S., Willinger, W., Wilson, D.V., "On the Self-Similar Nature of Ethernet Traffic," in Proc. of ACM SIGCOMM 1993. pp. 183-193. [Tra92] Tran-Gia, P., Gropp, 0., "Performance of a Neural Net used as Admission Controller in ATM Systems," Pmc. Globecom 92, Orlando, FL, pp. 1303-1309.	1996	108
1,160	An Adaptive WTA using Floating Gate Technology w. Fritz Kruger, Paul Hasler, Bradley A. Minch, and Christ of Koch California Institute of Technology Pasadena, CA 91125 (818) 395 - 2812 stretch@klab.caltech.edu Abstract We have designed, fabricated, and tested an adaptive WinnerTake-All (WTA) circuit based upon the classic WTA of Lazzaro, et al [IJ. We have added a time dimension (adaptation) to this circuit to make the input derivative an important factor in winner selection. To accomplish this, we have modified the classic WTA circuit by adding floating gate transistors which slowly null their inputs over time. We present a simplified analysis and experimental data of this adaptive WTA fabricated in a standard CMOS 2f.tm process. 1 Winner-Take-All Circuits In a WTA network, each cell has one input and one output. For any set of inputs, the outputs will all be at zero except for the one which is from the cell with the maximum input. One way to accomplish this is by a global nonlinear inhibition coupled with a self-excitation term [2J. Each cell inhibits all others while exciting itself; thus a cell with even a slightly greater input than the others will excite itself up to its maximal state and inhibit the others down to their minimal states. The WTA function is important for many classical neural nets that involve competitive learning, vector quantization and feature mapping. The classic WTA network characterized by Lazzaro et. al. [IJ is an elegant, simple circuit that shares just one common line among all cells of the network to propagate the inhibition. Our motivation to add adaptation comes from the idea of saliency maps. Picture a saliency map as a large number of cells each of which encodes an analog value An Adaptive wrA using Floating Gate Technology 721 Vtun01 Vtun02 Vdd ± Vb1 ~C1 M4 , V i C2 V1 Vfg1 JLV~ 1--r---'-c2 -A ~5 M2 Figure 1: The circuit diagram of a two input winner-take-all circuit. reflecting some measure of the importance (saliency) of its input. We would like to pay attention to the most salient cell, so we employ a WTA function to tell us where to look. But if the input doesn't change, we never look away from that one cell. We would like to introduce some concept of fatigue and refraction to each cell such that after winning for some time, it tires, allowing other cells to win, and then it must wait some time before it can win again. We call this circuit an adaptive WTA. In this paper, we present an adaptive WTA based upon the classic WTA; Figure 1 shows a two-input, adaptive WTA circuit. The difference between the classic and adaptive WTA is that M4 and Ms are pFET single transistor synapses. A single transistor synapse [3] is either an nFET or pFET transistor with a floating gate and a tunneling junction. This enhancement results in the ability of each transistor to adapt to its input bias current. The adaptation is a result of the electron tunneling and hot-electron injection modifying the charge on the floating gate; equilibrium is established when the tunneling current equals the injection current. The circuit is devised in such a way that these are negative feedback mechanisms, consequently the output voltage will always return to the same steady state voltage determined by its bias current regardless of the DC input level. Like the autozeroing amplifier [4], the adaptive WTA is an example of a circuit where the adaptation occurs as a natural part of the circuit operation. 2 pFET hot-electron injection and electron tunneling Before considering the behavior of the adaptive WTA, we will review the processes of electron tunneling and hot-electron injection in pFETs. In subthreshold operation, we can describe the channel current of a pFET (Ip) for a differential change in gate voltage, ~ Vg, around a fixed bias current Iso, as Ip = Iso exp ( ,,~:g ) where Kp is the amount by which ~ Vg affects the surface potential of the pFET, and UT is ki. We will assume for this paper that all transistors are identical. First, we consider electron tunneling. We start with the classic model of electron 722 Ec~ ••• Ev ---. '- ... -....".~ Source Channel (a) W. F. Kruger, P. Hasler, B. A. Minch and C. Koch Drain L J ,,' ,0· -1 ... 2QOnA I .... ~ -1 •• 1nA -La.1inA e &.5 t IS 10 10.6 11 "'" (b) Figure 2: pFET Hot Electron Injection. (a) Band diagram of a subthreshold pFET transistor for favorable conditions for hot-electron injection. (b) Measured data of pFET injection efficiency versus the drain to channel voltage for four source currents. Injection efficiency is the ratio of injection current to source current. At cI>dc equal to 8.2V, the injection efficiency increases a factor of e for an increase cI>dc of 250mV. tunneling through a silicon - Si02 system [5]. As in the autozeroing amplifier [4], the tunneling current will be only a weak function for the voltage swing on the floating gate voltage through the region of subthreshold currents; therefore we will approximate the tunneling junction as a current source supplying I tunO current to the floating gate. Second, we derive a simple model of pFET hot-electron injection. Figure 2a shows the band diagram of a pFET operating at bias conditions which are favorable for hot-electron injection. Hot-hole impact ionization creates electrons at the drain edge of the depletion region. These secondary electrons travel back into the channel region gaining energy as they go. When their energy exceeds that of the Si02 barrier, they can be injected through the oxide to the floating gate. The hole impact ionization current is proportional to the source current, and is an exponential function of the voltage drop from channel to drain (c)de). The injection current is proportional to the hole impact ionization current and is an exponential function of the voltage drop from channel to drain. We will neglect the dependence of the floating-gate voltage for a given source current and c)de as we did in [4]. Figure 2b shows measured injection efficiency for several source currents, where injection efficiency is the ratio of the injection current to source current. The injection efficiency is independent of source current and is approximately linear over a 1 - 2V swing in c)de; therefore we model the injection efficiency as proportional to exp ( - t~~c ) within that 1 to 2V swing, where Vinj is a measured device parameter which for our process is 250mV at a bias c)de = 8.2V, and 6,c)de is the change in c) de from the bias level. An increasing voltage input will increase the pFET surface potential by capacitive coupling to the floating gate. Increasing the pFET surface potential will increase the source current thereby decreasing c) de for a fixed output voltage and lowering the injection efficiency. An Adaptive WTA using Floating Gate Technology ~~\ , \ 1.55 Culftlnt steP I nput ,I \ 10,77nA · 14.12nA - lO.11M ~,. / \ t / \ V .. n . 43.3SV ~ ! \ I \ ius ! \ J \ ! \ 1.4 j "j " I ~~+~ 1 .35 O~--;:20::----!:40'---:!:60:--:::'80-----:'::::OO--;'=:-20 ----:-, 40=-~' 60:::---:-:'80::--::!200 1111"18 (5) (a) 723 ,o'r-----~----~---___, 1000~------7.50:-------:'::::OO------'!'SO Input CuTent Step (% of bas cumtnt) (b) Figure 3: Illustration of the dynamics for the winning and losing input voltages. (a) Measured Vi verses time due to an upgoing and a downgoing input current step. The initial input voltage change due to the input step is much smaller than the voltage change due to the adaptation. (b) Adaptation time of a losing input voltage for several tunneling voltages. The adaptation time is the time from the start of the input current step to the time the input voltage is within 10% of its steady state voltage. A larger tunneling current decreases the adaptation time by increasing the tunneling current supplied to the floating gate. 3 Two input Adaptive WTA We will outline the general procedure to derive the general equations to describe the two input WTA shown in Fig. 1. We first observe that transistors M 1 , M 2 , and Ma make up a differential pair. Regardless of any adaptation, the middle V node and output currents are set by the input voltages (Vl and V2) , which are set by the input currents, as in the classic WTA [1]. The dynamics for high frequency operation are also similar to the classic WTA circuit. Next, we can write the two Kirchhoff Current Law (KCL) equations at Vl and V2 , which relate the change in ~ and V2 as a function of the two input currents and the floating gate voltages. Finally, we can write the two KCL equations at the two floating gates VJgl and VJ g2 , which relates the changes in the floating gate voltages in terms of Vl and V2. This procedure is directly extendable to multiple inputs. A full analysis of these equations is very difficult and will be described in another paper. For this discussion, we present a simplified analysis to develop the intuition of the circuit operation. At sufficiently high frequencies, the tunneling and injection currents do not adapt the floating gate voltages sufficiently fast to keep the input voltages at their steady state levels. At these frequencies, the adaptive WTA acts like the classic WTA circuit with one small difference. A change in the input voltages, Vl or V2 is linearly related to V by the capacitive coupling (~Vl = - §; ~ V), where this relationship is exponential in the classic WTA. There is always some capacitance C2 , even if not explicitly drawn due to the overlap capacitance from the floating gate to drain. This property gives the designer the added freedom to modify the gain. We will assume the circuit operates in its intended operating regime where the floating gate transistors settle sufficiently fast such that their channel 724 W. F. Kruger, P. Hasler, B. A. Minch and C. Koch 35 . ' . ,- .; " . .,.V ....... f ~25 L > 1 .. 10'· 10" 10" c~ ..... t2(A.. (a) '0' ~1fIp.ll12(A) (b) Figure 4: Measured change in steady state input voltages as a function of bias current. (a) Change in the two steady state output voltages as a function of the bias current of the second input. The bias current of the first input was held fixed at 8.14nA. (b) Change in the RMS noise of the two output voltages as a function of the bias current of the second input. The RMS noise is much higher for the losing input than for the winning input. Note that where the two bias currents crOSS roughly corresponds to the location where the RMS noise on the two input voltages is equal. current equals the input currents J. - I (_ K6,V/9i ) dIi _ -J.~ dV/gi ,80 exp UT -+ dt , UT dt (1) for all inputs indexed by i, but not necessarily fast enough for the floating gates to settle to their final steady state levels. To develop some initial intuition, we shall begin by considering one half of the two input WTA: transistors M 1 , M2 and M4 of Figure 1. First, we notice that Ioutl is equal to Ib (the current through transistor Mt}; note that this is not true for the multiple input case. By equating these two currents we get an equation for V as V = KV1 - KVb, where we will assume that Vb is a fixed bias voltage. Assuming the input current equals the current through M 4 , VI obeys the equation (KG1 + G2)- = ---- + ItunO exp( ---) -1 dVI GTUT dII ( II 6, VI ) dt KIt dt 180 Vinj (2) where CT is the total capacitance connected to the floating gate. The steady state of (2) is sv; = KVinj I (~) 'n U n I T 80 (3) which is exactly the same expression for each input in a multiple input WTA. We get a linear differential equation by making the substitution X = exp( D..v..Vl) [4], and we "'1 get similar solutions to the behavior of the autozeroing amplifier. Figure 3a shows measured data for an upgoing and a downgoing current step. The input current change results in an initial fast change in the input voltage, and the input voltage then adapts to its steady state voltage which is a much greater voltage change. From the voltage difference between the steady states, we get that Vinj is roughly 500mV. An Adaptive WTA using Floating Gate Technology 725 o 10 15 20 25 30 35 .a 45 50 o 5 10 ,5 20 25 30 35 .a 45 50 l1me(a) 11me(.) (a) (b) Figure 5: Experimental time traces measurements of the output current and voltage for small differential input current steps. (a) Time traces for small differential current steps around nearly identical bias currents of 8.6nA. (b) Time traces for small differential current steps around two different bias currents of 8.7nA and O.88nA. In the classic WTA, the output currents would show no response to the input current steps. Returning to the two input case, we get two floating gate equations by assuming that the currents through M4 and M5 are equal to their respective input currents and writing the KCL equations at each floating gate. If VI and V2 do not cross each other in the circuit operation, then one can easily solve these KCL equations. Assume without loss of generality that VI is the winning voltage; which implies that ~ V = K~ Vl . The initial input voltage change before the floating gate adaptation due to a step in the two input currents of II ~ It and 12 ~ It is ~VI = GT In (It) ~V2 ~ GT In (II It) KGl II' G2 It 12 (4) for G2 much less than KGl . In this case, Vl moves on the order of the floating gate voltage change, but V2 moves on the order of the floating gate change amplified up by .g;.. The response of ~ VI is governed by an identical equation to (2) ofthe earlier half-analysis, and therefore results in a small change in VI. Also, any perturbation of V is only slightly amplified at Vl due to the feedback; therefore any noise at V will only be slightly amplified into VI. The restoration of V2 is much quicker than the Vl node if G2 is much less than KGl ; therefore after the initial input step, one can safely assume that V is nearly constant. The voltage at V is amplified by - ~ at 112; therefore any noise at V is amplified at the losing voltage, but not at the winning voltage as the data in Fig. 4b shows. The losing dynamics are identical to the step response of an autozeroing amplifier [4]. Figure 3b shows the variation . of the adaptation time verses the percent input current change for several values of tunneling voltages. The main difficulty in exactly solving these KCL equations is the point in the dynamics where Vi crosses V2 , since the behavior changes when the signals move 726 W. F. Kruger, P. Hasler, B. A. Minch and C. Koch through the crossover point. If we get more than a sufficient Vi decrease to reach the starting V2 equilibrium, then the rest of the input change is manifested by an increase in V2 • If the voltage V2 crosses the voltage Vi, then V will be set by the new steady state, and Vi is governed by losing dynamics until Vi :::::l V2 • At this point Vi is nearly constant and V2 is governed by losing dynamics. This analysis is directly extendible to arbitrary number of inputs. Figure 5 shows some characteristic traces from the two-input circuit. Recall that the winning node is that with the lowest voltage, which is reflected in its corresponding high output current. In Fig. 5a, we see that as an input step is applied, the output current jumps and then begins to adapt to a steady state value. When the inputs are nearly equal, the steady state outputs are nearly equal; but when the inputs are different, the steady state output is greater for the cell with the lesser input. In general, the input current change that is the largest after reaching the previous equilibrium becomes the new equilibrium. This additional decrease in Vi would lead to an amplified increase in the other voltage since the losing stage roughly looks like an autozeroing amplifier with the common node as the input terminal. The extent to which the inputs do not equal this largest input is manifested as a proportionally larger input voltage. The other voltage would return to equilibrium by slowly, linearly decreasing in voltage due to the tunneling current. This process will continue until Vi equals V2. Note in general that the inputs with lower bias currents have a slight starting advantage over the inputs with higher bias currents. Figure 5b illustrates the advantage of the adaptive WTA over the classic WTA. In the classic WTA, the output voltage and current would not change throughout the experiment, but the adaptive WTA responds to changes in the input. The second input step does not evoke a response because there was not enough time to adapt to steady state after the previous step; but the next step immediately causes it to win. Also note in both of these traces that the noise is very large in the loosing node and small in the winner because of the gain differences (see Figure 4b). References [1] J. Lazzaro, S. Ryckebusch, M.A. Mahowald, and C.A. Mead "Winner-TakeAll Networks of O(N) Complexity" , NIPS 1 Morgan Kaufmann Publishers, San Mateo, CA, 1989, pp 703 - 711. [2] Grossberg S. "Adaptive Pattern Classification and Universal Recoding: I. Parallel Development and Coding of Neural Feature Detectors." Biological Cybernetics vol. 23, 121-134, 1988. [3] P. Hasler, C. Diorio, B. A. Minch, and C. Mead, "Single 'fransistor Learning Synapses", NIPS 7, MIT Press, 1995, 817-824. Also at http://www.pcmp.caitech.edu/ anaprose/paul. [4] P. Hasler, B. A. Minch, C. Diorio, and C. Mead, "An autozeroing amplifier using pFET Hot-Electron Injection", ISCAS, Atlanta, 1996, III-325 - III-328. Also at http://www.pcmp.caitech.edu/anaprose/paul. [5] M. Lenzlinger and E. H. Snow (1969), "Fowler-Nordheim tunneling into thermally grown Si02 ," J. Appl. Phys., vol. 40, pp. 278-283, 1969.	1996	109
1,161	Continuous sigmoidal belief networks trained using slice sampling Brendan J. Frey Department of Computer Science, University of Toronto 6 King's College Road, Toronto, Canada M5S 1A4 Abstract Real-valued random hidden variables can be useful for modelling latent structure that explains correlations among observed variables. I propose a simple unit that adds zero-mean Gaussian noise to its input before passing it through a sigmoidal squashing function. Such units can produce a variety of useful behaviors, ranging from deterministic to binary stochastic to continuous stochastic. I show how "slice sampling" can be used for inference and learning in top-down networks of these units and demonstrate learning on two simple problems. 1 Introduction A variety of unsupervised connectionist models containing discrete-valued hidden units have been developed. These include Boltzmann machines (Hinton and Sejnowski 1986), binary sigmoidal belief networks (Neal 1992) and Helmholtz machines (Hinton et al. 1995; Dayan et al. 1995). However, some hidden variables, such as translation or scaling in images of shapes, are best represented using continuous values. Continuous-valued Boltzmann machines have been developed (Movellan and McClelland 1993), but these suffer from long simulation settling times and the requirement of a "negative phase" during learning. Tibshirani (1992) and Bishop et al. (1996) consider learning mappings from a continuous latent variable space to a higher-dimensional input space. MacKay (1995) has developed "density networks" that can model both continuous and categorical latent spaces using stochasticity at the top-most network layer. In this paper I consider a new hierarchical top-down connectionist model that has stochastic hidden variables at all layers; moreover, these variables can adapt to be continuous or categorical. The proposed top-down model can be viewed as a continuous-valued belief network, which can be simulated by performing a quick top-down pass (Pearl 1988). Work done on continuous-valued belief networks has focussed mainly on Gaussian random variables that are linked linearly such that the joint distribution over all Continuous Sigmoidal Belief Networks Trained using Slice Sampling 453 (a) Zero-mean Gaussian noi~e with2 vaflance (T i Yi == ()(Xi) (b) ~~ -4 x 4 0 y 1 (d)~r1 -4 x 4 0 y 1 (c) rv-- p(y)A ()<:J1\!W ~ L -4 x 4 0 Y 1 (e)~()~d -400 x 400 0 y 1 Figure 1: (a) shows the inner workings of the proposed unit. (b) to (e) illustrate four quite different modes of behavior: (b) deterministic mode; (c) stochastic linear mode; (d) stochastic nonlinear mode; and (e) stochastic binary mode (note the different horizontal scale). For the sake of graphical clarity, the density functions are normalized to have equal maxima and the subscripts are left off the variables. variables is also Gaussian (Pearl 1988; Heckerman and Geiger 1995). Lauritzen et al. (1990) have included discrete random variables within the linear Gaussian framework. These approaches infer the distribution over unobserved unit activities given observed ones by "probability propagation" (Pearl 1988). However, this procedure is highly suboptimal for the richly connected networks that I am interested in. Also, these approaches tend to assume that all the conditional Gaussian distributions represented by the belief network can be easily derived using information elicited from experts. Hofmann and Tresp (1996) consider the case of inference and learning in continuous belief networks that may be richly connected. They use mixture models and Parzen windows to implement conditional densities. My main contribution is a simple, but versatile, continuous random unit that can operate in several different modes ranging from deterministic to binary stochastic to continuous stochastic. This spectrum of behaviors is controlled by only two parameters. Whereas the above approaches assume a particular mode for each unit (Gaussian or discrete), the proposed units are capable of adapting in order to operate in whatever mode is most appropriate. 2 Description of the unit The proposed unit is shown in Figure 1a. It is similar to the deterministic sigmoidal unit used in multilayer perceptrons, except that Gaussian noise is added to the total input, I-'i, before the sigmoidal squashing function is applied.1 The probability density over presigmoid activity Xi for unit i is p(xill-'i, u;) == exp[-(xi -l-'i)2 /2u;1/ yi27rUf, (1) where I-'i and U[ are the mean and variance for unit i. A postsigmoid activity, Yi, is obtained by passing the presigmoid activity through a sigmoidal squashing function: Yi == cI>(Xi). (2) Including the transformation Jacobian, the post sigmoid distribution for unit i is ( .1 . u~) = exp[-(cI>-1(Yi) -l-'i)2 /2ulj (3) P Yt 1-'" , cI>'(cI>-1(Yi))yi27ru;' IGeoffrey Hinton suggested this unit as a way to make factor analysis nonlinear. 454 B. 1. Frey I use the cumulative Gaussian squashing function: <)(x) == J~ooe-z2j2 /...tFi dz <)'(x) = </>(x) == e-z2j2/...tFi. (4) Both <)0 and <)-10 are nonanalytic, so I use the C-library erfO function to implement <)0 and table lookup with quadratic interpolation to implement <)-10. Networks of these units can represent a broad range of structures, including deterministic multilayer perceptrons, binary sigmoidal belief networks (aka. stochastic multilayer perceptrons), mixture models, mixture of expert models, hierarchical mixture of expert models, and factor analysis models. This versatility is brought about by a range of significantly different modes of behavior available to each unit. Figures 1b to Ie illustrate these modes. Deterministic mode: If the noise variance of a unit is very small, the postsigmoid activity will be a practically deterministic sigmoidal function of the mean. This mode is useful for representing deterministic nonlinear mappings such as those found in deterministic multilayer perceptrons and mixture of expert models. Stochastic linear mode: For a given mean, if the squashing function is approximately linear over the span of the added noise, the postsigmoid distribution will be approximately Gaussian with the mean and standard deviation linearly transformed. This mode is useful for representing Gaussian noise effects such as those found in mixture models, the outputs of mixture of expert models, and factor analysis models. Stochastic nonlinear mode: If the variance of a unit in the stochastic linear mode is increased so that the squashing function is used in its nonlinear region, a variety of distributions are producible that range from skewed Gaussian to uniform to bimodal. Stochastic binary mode: This is an extreme case of the stochastic nonlinear mode. If the variance of a unit is very large, then nearly all of the probability mass will lie near the ends of the interval (0,1) (see figure Ie). Using the cumulative Gaussian squashing function and a standard deviation of 150, less than 1% of the mass lies in (0.1,0.9). In this mode, the postsigmoid activity of unit i appears to be binary with probability of being "on" (ie., Yi > 0.5 or, equivalently, Xi > 0): (. I. 2) -100 eXP[-(X-JLi)2/2o-;]d -11'; exp[-x2/2CT;]d - ..T..(JLi) (5) P 't on JL~, CTi M:::7i X M:::7i x 'J.' • o V 27rCTi -00 V 27rCTi CTi This sort of stochastic activation is found in binary sigmoidal belief networks (Jaakkola et al. 1996) and in the decision-making components of mixture of expert models and hierarchical mixture of expert models. 3 Continuous sigmoidal belief networks If the mean of each unit depends on the activities of other units and there are feedback connections, it is difficult to relate the density in equation 3 to a joint distribution over all unit activities, and simulating the model would require a great deal of computational effort. However, when a top-down topology is imposed on the network (making it a directed acyclic graph), the densities given in equations 1 and 3 can be interpreted as conditional distributions and the joint distribution over all units can be expressed as p({Xi}) = n~lP(Xil{xjh<i) or p({Yi}) = n~lp(Yil{Yjh<i), (6) where N is the number of units. p(xil{xj}j<i) and p(Yil{yj}j<i) are the presigmoid and postsigmoid densities of unit i conditioned on the activities of units with lower Continuous Sigmoidal Belief Networks Trained using Slice Sampling 455 indices. This ordered arrangement is the foundation of belief networks (Pearl, 1988). I let the mean of each unit be determined by a linear combination of the postsigmoid activities of preceding units: J1.i = L,j<iWijYj, (7) where Yo == 1 is used to implement biases. The variance for each unit is independent of unit activities. A single sample from the joint distribution can be obtained by using the bias as the mean for unit 1, randomly picking a noise value for unit 1, applying the squashing function, computing the mean for unit 2, picking a noise value for unit 2, and so on in a simple top-down pass. Inference by slice sampling Given the activities of a set of visible (observed) units, V, inferring the distribution over the remaining set of hidden (unobserved) units, H, is in general a difficult task. The brute force procedure proceeds by obtaining the posterior density using Bayes theorem: p( {YihEHI{YihEV) = p( {Yi}iEH ,{yiliEV )/J{Yi}iEHP( {YihEH,{Yi}iEV )TIiEHdYi. (8) However, computing the integral in the denominator exactly is computationally intractable for any more than a few hidden units. The combinatorial explosion encountered in the corresponding sum for discrete-valued belief networks pales in comparison to this integral; not only is it combinatorial, but it is a continuous integral with a multimodal integrand whose peaks may be broad in some dimensions but narrow in others, depending on what modes the units are in. An alternative to explicit integration is to sample from the posterior distribution using Markov chain Monte Carlo. Given a set of observed activities, this procedure d { }(o) {}(l) {}(t) h· d pro uces a state sequence, Yi iEH' Yi iEH' ... , Yi iEH' ... , t at IS guarantee to converge to the posterior distribution. Each successive state is randomly selected based on knowledge of only the previous state. To simplify these random choices, I consider changing only one unit at a time when making a state transition. Ideally, the new activity of unit i would be drawn from the conditional distribution p(Yil{Yj}ji=i) (Gibbs sampling). However, it is difficult to sample from this distribution because it may have many peaks that range from broad to narrow. I use a new Markov chain Monte Carlo method called "slice sampling" (Neal 1997) to pick a new activity for each unit. Consider the problem of drawing a value Y from a univariate distribution P(y) in this application, P(y) is the conditional distribution p(Yil{yj}ji=i). Slice sampling does not directly produce values distributed according to P(y), but instead produces a sequence of values that is guaranteed to converge to P(y). At each step in the sequence, the old value Yold is used as a guide for where to pick the new value Ynew. To perform slice sampling, all that is needed is an efficient way to evaluate a function !(y) that is proportional to P(y) in this application, the easily computed value p(Yi' {Yj}ji=i) suffices, since p(Yi' {Yj}ji=i) <X p(Yil{Yj}ji=i). Figure 2a shows an example of a univariate distribution, P(y). The version of slice sampling that I use requires that all of the density lies within a bounded intenJal as shown. To obtain Ynew from Yold, !(Yold) is first computed and then a uniform random value is drawn from [0, !(Yold)]. The distribution is then horizontally "sliced" at this value, as shown in figure 2a. Any Y for which !(y) is greater than this value is considered to be part of the slice, as indicated by the bold line segments in the picture shown at the top of figure 2b. Ideally, Ynew would now be drawn uniformly from the slice. However, determining the line segments that comprise the slice is not easy, for although it is easy to determine whether a particular Y is in the slice, 456 B. 1. Frey (a) (b) • (c) • )( • . )( • • I )( • ~~ • I • Ynew I H. • Yold o Yold 1 Ynew X randomly drawn point Figure 2: After obtaining a random slice from the density (a), random values are drawn until one is accepted. (b) and (c) show two such sequences. it is much more difficult to determine the line segment boundaries, especially if the distribution is multimodal. Instead, a uniform value is drawn from the original interval as shown in the second picture of figure 2b. If this value is in the slice it is accepted as Ynew (note that this decision requires an evaluation of fey)). Otherwise either the left or the right interval boundary is moved to this new value, while keeping Yold in the interval. This procedure is repeated until a value is accepted. For the sequence in figure 2b, the new value is in the same mode as the old one, whereas for the sequence in figure 2c, the new value is in the other mode. Once Ynew is obtained, it is used as Yold for the next step. If the top-down influence causes there to be two very narrow peaks in p(Yil{Yi}j#i) (corresponding to a unit in the stochastic binary mode) the slices will almost always consist of two very short line segments and it will be very difficult for the above procedure to switch from one mode to another. To fix this problem, slice sampling is performed in a new domain, Zi = ~ ( {Xi - J.ti} / U i). In this domain the top-down distribution p(zil{Yi}j<i) is uniform on (0,1), so p(zil{Yj}j#i) = p(zil{Yj}j>i) and I use the following function for slice sampling: f(zi) = exp [ E~=i+l {Xk - J.t;i - Wki~(Ui~-l(Zi) + J.ti)} 2 /2u~], (9) where J.t;i = Ej<k,#i WkjYj. Since Xi, Yi and Zi are all deterministically related, sampling from the distribution of Zi will give appropriately distributed values for the other two. Many slice sampling steps could be performed to obtain a reliable sample from the conditional distribution for unit i, before moving on to the next unit. Instead, only one slice sampling step is performed for unit i before moving on. The latter procedure often converges to the correct distribution more quickly than the former. The Markov chain Monte Carlo procedure I use in the simulations below thus consists of sweeping a prespecified number of times through the set of hidden units, while updating each unit using slice sampling. Learning Given training examples indexed by T, I use on-line stochastic gradient ascent to perform maximum likelihood learning ie., maximize TIT p( {xihev). This consists of sweeping through the training set and for each case T following the gradient oflnp({xi}), while sampling hidden unit values fromp({xiheHI{xihev) using the sampling algorithm described above. From equations 1,6 and 7, flWjk == '" 8Inp({xi} )/8Wjk = ",(Xj - EI<jWjIYI)Yk/U], (10) fllnu] == ",81np({xi})/8Inu; = "'[(Xj - EI<jWjIYI)2 /U] -1]/2, (11) where", is the learning rate. Continuous Sigmoidal Belief Networks Trained using Slice Sampling (b) (1, .97) (1, .96) Yvis2 (1 , .69) (1, .55) (1, .19) 457 (0, .12)(0, .14) (0, .75)(0, .97) Yvisl Figure 3: For each experiment (a) and (b), contours show the distribution of the 2-dimensional training cases. The inferred postsigmoid activities of the two hidden units after learning are shown in braces for several training cases, marked by x. 4 Experiments I designed two experiments meant to elicit the four modes of operation described in section 2. Both experiments were based on a simple network with one hidden layer containing two units and one visible layer containing two units. Thaining data was obtained by carefully selecting model parameters so as to induce various modes of operation and then generating 10,000 two-dimensional examples. Before training, the weights and biases were initialized to uniformly random values between -0.1 and 0.1; log-variances were initialized to 10.0. Thaining consisted of 100 epochs using a learning rate of 0.001 and 20 sweeps of slice sampling to complete each training case. Each task required roughly five minutes on a 200 MHz MIPS R4400 processor. The distribution of the training cases in visible unit space (Yvisl Yvis2) for the first experiment is shown by the contours in figure 3a. After training the network, I ran the inference algorithm for each of ten representative training cases. The postsigmoid activities of the two hidden units are shown beside the cases in figure 3a; clearly, the network has identified four classes that it labels (0,0) ... (1,1). Based on a 30x30 histogram, the Kullback-Leibler divergence between the training set and data generated from the trained network is 0.02 bits. Figure 3b shows a similar picture for the second experiment, using different training data. In this case, the network has identified two categories that it labels using the first postsigmoid activity. The second postsigmoid activity indicates how far along the respective "ridge" the data point lies. The Kullback-Leibler divergence in this case is 0.04 bits. 5 Discussion The proposed continuous-valued nonlinear random unit is meant to be a useful atomic element for continuous belief networks in much the same way as the sigmoidal deterministic unit is a useful atomic element for multi-layer perceptrons. Four operational modes available to each unit allows small networks of these units to exhibit complex stochastic behaviors. The new "slice sampling" method that I employ for inference and learning in these networks uses easily computed local information. The above experiments illustrate how the same network can be used to model two quite different types of data. In contrast, a Gaussian mixture model would require many more components for the second task as compared to the first. Although the methods due to Tibshirani and Bishop et al. would nicely model each submanifold in the second task, they would not properly distinguish between categories of data in either task. MacKay's method may be capable of extracting both the sub manifolds and the categories, but I am not aware of any results on such a dual problem. 458 B. 1. Frey It is not difficult to conceive of models for which naive Markov chain Monte Carlo procedures will become fruitlessly slow. In particular, if two units have very highly correlated activities, the procedure of changing one activity at a time will converge extremely slowly. Also, the Markov chain method may be prohibitive for larger networks. One approach to avoiding these problems is to use the Helmholtz machine (Hinton et al. 1995) or mean field methods (Jaakkola et al. 1996). Other variations on the theme presented in this paper include the use of other types of distributions for the hidden units (e.g., Poisson variables may be more biologically plausible) and different ways of parameterizing the modes of behavior. Acknowledgements I thank Radford Neal and Geoffrey Hinton for several essential suggestions and I also thank Peter Dayan and Tommi Jaakkola for helpful discussions. This research was supported by grants from ITRC, IRIS, and NSERC. References Bishop, C. M, Svensen, M., and Williams, C.K.L 1996. EM optimization of latent-variable density models. In D. Touretzky, M. Mozer, and M. Hasselmo (editors), Advances in Neural Information Processing Systems 8, MIT Press, Cambridge, MA. Dayan, P., Hinton, G. E., Neal, R. M., and Zemel, R. S. 1995. The Helmholtz machine. Neural Computation 7, 889-904. Heckerman, D., and Geiger, D. 1994. Learning Bayesian networks: a unification for discrete and Gaussian domains. In P. Besnard and S. Hanks (editors), Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence, Morgan Kaufmann, San Francisco, CA, 274-284. Hinton, G. E., Dayan, P., Frey, B. J ., and Neal, R. M. 1995. The wake-sleep algorithm for unsupervised neural networks. Science 268, 1158-1161. Hinton, G. E., and Sejnowski, T. J. 1986. Learning and relearning in Boltzmann machines. In D. E. Rumelhart and J. L. McClelland (editors), Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Volume 1: Foundations . MIT Press, Cambridge, MA. Hofmann, R., and Tresp, V. 1996. Discovering structure in continuous variables using Bayesian networks. In D. Touretzky, M. Mozer, and M. Hasselmo (editors), Advances in Neural Information Processing Systems 8, MIT Press, Cambridge, MA. Jaakkola, T., Saul, L. K., and Jordan, M. 1. 1996. Fast learning by bounding likelihoods in sigmoid type belief networks. In D. Touretzky, M. Mozer and M. Hasselmo (editors), Advances in Neural Information Processing Systems 8, MIT Press, Cambridge, MA. Lauritzen, S. L., Dawid, A. P., Larsen, B. N., and Leimer, H. G. 1990. Independence properties of directed Markov Fields. Networks 20,491-505. MacKay, D. J. C. 1995. Bayesian neural networks and density networks. Nuclear Instruments and Methods in Physics Research, A 354, 73-80. Movellan, J. R., and McClelland, J . L. 1992. Learning continuous probability distributions with symmetric diffusion networks. Cognitive Science 17, 463-496. Neal, R. M. 1992. Connectionist learning of belief networks. Artificial Intelligence 56, 71-113. Neal, R. M. 1997. Markov chain Monte Carlo methods based on "sliCing" the density function. In preparation. Pearl, J. 1988. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo, CA. Tibshirani, R. (1992). Principal curves revisited. Statistics and Computing 2, 183-190.	1996	11
1,162	Representation and Induction of Finite State Machines using Time-Delay Neural Networks Daniel S. Clouse Computer Science & Engineering Dept. C. Lee Giles NEC Research Institute 4 Independence Way Princeton, NJ 08540 giles@research.nj .nec.com University of California, San Diego La Jolla, CA 92093-0114 dclouse@ucsd.edu Bill G. Horne NEC Research Institute 4 Independence Way Princeton, N J 08540 horne@research.nj.nec.com Garrison W. Cottrell Computer Science & Engineering Dept. University of California, San Diego La Jolla, CA 92093-0114 gcottrell@ucsd.edu Abstract This work investigates the representational and inductive capabilities of time-delay neural networks (TDNNs) in general, and of two subclasses of TDNN, those with delays only on the inputs (IDNN), and those which include delays on hidden units (HDNN). Both architectures are capable of representing the same class of languages, the definite memory machine (DMM) languages, but the delays on the hidden units in the HDNN helps it outperform the IDNN on problems composed of repeated features over short time windows. 1 Introduction In this paper we consider the representational and inductive capabilities of timedelay neural networks (TDNN) [Waibel et al., 1989] [Lang et al., 1990], also known as NNFIR [Wan, 1993]. A TDNN is a feed-forward network in which the set of inputs to any node i may include the output from previous layers not only in the current time step t, but from d earlier time steps as well. The activation function 404 D. S. Clouse, C. L Giles, B. G. Home and G. W. Cottrell for node i at time t in such a network is given by equation 1: i-l d y! = h(I: I: yJ-kWijk) (1) j=lk=O where y: is the activation of node i at time t, Wijk is the connection strength from node j to node i at delay k, and h is the squashing function. TDNNs have been used in speech recognition [Waibel et al., 1989], and time series prediction [Wan, 1993]. In this paper we concentrate on the language induction problem. A training set of variable-length strings taken from a discrete alphabet {O, 1} is generated. Each string is labeled as to whether it is in some language L or not. The network must learn to discriminate strings which are in the language from those which are not, not only for the training set strings, but for strings the network has never seen before. The language induction problem provides a simple, familiar domain in which to gain insight into the capabilities of different network archi tect ures. Specifically, in this paper, we will look at the representational and inductive capabilities of the general class of TDNNs versus a subclass of TDNNs, the input-delay neural networks (IDNNs). An IDNN is a TDNN in which delays are limited to the network inputs. In section 2, we will show that the classes of functions representable by general TDNNs and IDNNs are equivalent. In section 3, we will show that the class of languages representable by the TDNNs, are the definite memory machine (DMM) languages. In section 4, we will demonstrate the inductive capability of the TDNNs in a simulation in which a large DMM is learned using a small percentage of the possible, short training examples. In section 5, a second set of simulations will show the difference between representational and inductive bias, and will demonstrate the utility of internal delays in a TDNN network. 2 TDNN sand IDNN s Are Functionally Equivalent Since every IDNN is also a TDNN, the set of functions computable by any TDNN includes all those computable by the IDNNs. [Wan, 1993] also shows that the IDNNs can compute any function computable by the TDNNs making these two classes of network architectures functionally equivalent. For completeness, here we include a description of how to construct from a TDNN, an equivalent IDNN. Figure 1a shows a TDNN with a single input U at the current time (Ut), and at four earlier time steps (Ut-l ... Ut-4). The inputs to node R consist of the outputs of nodes P and Q at the current time step along with one or two previous time steps. At time t, node P computes !p(Ut, . . . Ut-4), a function of the current input and four delays. At time t -1, node P computes !P(Ut-l, ... Ut-s). This serves as one of the delayed inputs to node R. This value could also be computed by sliding node P over one step in the input tap-delay line along with its incoming weights as shown in figure lb. Using this construction, all the internal delays can be removed, and replaced by copies of the original nodes P and Q, along with their incoming weights. This method can be applied recursively to remove any internal delay in any TDNN network. Thus, for any function computable by a TDNN, we can construct an IDNN which computes the same function. 3 TDNNs Can Represent the DMM Languages In this section, we show that the set of languages which are representable by some TDNN are exactly those languages representable by the definite memory machines Representation and Induction of Finite State Machines using TDNNs a) Generu TDNN fp (u1 •. ··• ul-4 ) fp(u 1.!.,,·. u1•5 ) fp (u 1.2 •. ". u1•6 ) Ut ut.! · · ut·6 b) Equivalent IDNN Figure 1: Constructing an IDNN equivalent to a given TDNN 405 (DMMs). According to Kohavi (1978) a DMM of order d is a finite state machine (FSM) whose present state can always be determined uniquely from the knowledge of the most recent d inputs. We equivalently define a DMM of order d as an FSM whose accepting/rejecting behavior is a function of only the most recent d inputs. To fit TDNNs and IDNNs into the language induction framework, we consider only networks with a single 0/1 input. Since any boolean function can be represented by a feed-forward network with enough hidden units [Horne and Hush, 1994], an IDNN exists which can perform the mapping from d most recent inputs to any accepting/rejecting behavior. Therefore, any DMM language can be represented by some IDNN. Since every IDNN computes a function of its most recent d inputs, by the definition of DMM, there is no boolean output IDNN which represents a non-DMM language. Therefore, the IDNNs represent exactly the DMM languages. Since the TDNN and IDNN classes are functionally equivalent, TDNNs implement exactly the DMM languages as well. The shift register behavior of the input tap-delay line in an IDNN completely determines the state transition behavior of any machine represented by the network. This state transition behavior is fixed by the architecture. For example, figure 2a shows the state transition diagram for any machine representable by an IDNN with two input delays. The mapping from the current state to "accept" or "reject" is all that can be changed with training. Clouse et al. (1994) describes the conditions under which such a mapping results in a minimal FSM. All mappings used in the subsequent simulations are minimal FSM mappings. 4 Simulation 1: Large DMM To demonstrate the close relationship between TDNNs and DMMs, here we present the results of a simulation in which we trained an IDNN to reproduce the behavior of a DMM of order 11. The mapping function for the DMM is given in equation 2. Figure 2b shows the minimal 2048 state transition diagram required to represent the DMM. The symbol ~ in equation 2 represents the if-and-only-iffunction. The overbar notation, Uk, represents the negation of Uk, the input at time k. Yk is the network output at time k. Yk > 0.5 is interpreted as "accept the string seen so far." Yk ~ 0.5 means "reject." Yk = Uk-IO ~ (Uk U k-IU k-2 + Uk-2Uk-3 + Uk-l Uk-2) (2) To create training and test sets, we randomly split in two the set of all 4094 406 D. S. Clouse, C. L. Giles, B. G. Home and G. W Cottrell a) DMM of order 3 b) DMM of order 11 Figure 2: Transition diagrams for two DMMs. 0.4 Is S ! I ~ 0.2 J II 0.0 ............... !.i. ....•..... l ....•.....•....•....•....•....•....•....•....•....•.. 10 20 30 Percent of total samples (4094) used in training Figure 3: Generalization error on 2048 state DMM. strings of length 11 or less. We will report results using various percentages of possible strings for the training set. The IDNN had 10 input tap-delays, and seven hidden units. All tap-delays were cleared to 0 before introduction of a new input string. Weights were trained using online back propagation with learning rate 0.25, and momentum 0.25. To speed up the algorithm, weights were updated only if the absolute error on an example was greater than 0.2. Training was stopped when weight updates were required for no examples in the training set. This generally required 200 epochs or fewer, though there were trials which required almost 4000 epochs. Each point in figure 3 represents the mean classification error on the test set across 20 trials. Error bars indicate one standard deviation on each side of the mean. Each trial consists of a different randomly-chosen training set. The graph plots error at various training set sizes. Note that with training sets as small as 12 percent of possible strings the network generalizes perfectly to the remaining 88 percent. This kind of performance is possible because of the close match between the representational bias of the IDNN and this specific problem. 5 Simulation 2: Inductive biases of IDNNs and HDNNs In section 2, we showed that the IDNNs and general TDNNs can represent the same class offunctions. It does not follow that these two architectures are equally capable of learning the same functions. In this section, we show that the inductive biases are Representation and Induction of Finite State Machines using TDNNs 407 indeed different. We will present our intuitions about the kinds of problems each architecture is well suited to learning, then back up our intuitions with supporting simulations. In the following simulations, we compare two specific networks. The network representing the general TDNNs includes delays on hidden layer outputs. We'll refer to this as the hidden delay neural network or HDNN. All delays in the second network are confined to the network inputs, and so we call this the IDNN. We have been careful to design the two networks to be comparable in size. Each of the networks contains two hidden layers. The first hidden layer of the IDNN has four units, and the second five. The IDNN has eight input delays. Each of the two hidden layers of the HDNN has three units. The HDNN has three input delays, and five delays on the output of each node of the first hidden layer. Note that in each network the longest path from input to output requires eight delays. The number of weights, including bias weights, are also similar - 76 for the HDNN, and 79 for the IDNN. In order for the size of the two networks to be similar, the HDNN must have fewer delays on the network inputs. If we think of each unit in the first hidden layer as a feature detector, the feature detectors in the HDNN will span a smaller time window than the IDNN. On the other hand, the HDNN has a second set of delays which saves the output of the feature detectors over several time steps. If some narrow feature repeats over time, this second set of delays should help the HDNN to pick up this regularity. The IDNN, lacking the internal delays, should find it more difficult to detect this kind of repeated regularity. To test these ideas, we generated four DMM problems. We call equation 3 the narrow-repeated problem because it contains a number of identical terms shifted in time, and because each of these terms is narrow enough to fit in the time window of the HDNN first layer feature detectors. Yk = Uk-8 +-+ (Uk Uk-2U k-3 + Uk-1Uk-3U k-4 + Uk-3Uk-SU k-6 + Uk-4 Uk-6U k-7) (3) The wide-repeated problem, represented by equation 4, is identical to the narrowrepeated problem except that each term has been stretched so that it will no longer fit in the HDNN feature detector time window. Yk = Uk-8 +-+ (Uk Uk-2U k-4 + Uk-1 Uk-3U k-S + Uk-2 Uk-4Uk-6 + Uk-3Uk-SU k-7) (4) The narrow-unrepeated problem, represented by equation 5, is composed of narrow terms, but none of these terms is simply a shifted reproduction of another. Yk = Uk-8 +-+ (Uk Uk-2U k-3 + Uk-l Uk-3Uk-4 + Uk-3U k-SUk-6 + Uk-4U k-6U k-7) (5) Lastly, the wide-unrepeated problem of equation 6 contains wide terms which do not repeat. Yk = Uk-8 +-+ (Uk Uk-3U k-4 + Uk-1Uk-4Uk-S + Uk-2U k-SUk-6 + Uk-3U k-6U k-7) (6) Each problem in this section requires a minimum of 512 states to represent. Similar to the simulation of section 3, we trained both networks on subsets of all possible strings of length 9 or less. Since these problems were more difficult than that of section 3, often the networks were unable to find a solution which performed perfectly on the training set. In this case, training was stopped after 8000 epochs. The results reported later include these trials as well as trials in which training ended because of perfect performance on the training set. Training for the HDNN 408 D. S. Clouse, C. L. Giles, B. G. Home and G. W Cottrell 0.4 I I 04 I I ~ r:s o IDNN architecture j -HDNN iKhitectun: J 02 f f 02 , IIII!LI\J e ! " IJ1!lJLU 0.0 0.0 -._ ..... p_orlalal_ _iIIlralII/Ioi 20 40 60 20 40 60 (a) Narrow·Repealed (b) Narrow·Unrepealed .. / " I i I .. 'I ., iii 11\111\ 111111 . .... . .. .f 0.0 • •. . • .•.... • ..•••.. _ ..• 0.0 .. .........•.... ... 20 40 60 20 40 60 (e) Wide·Repeated (d) Wide-U"",peated Figure 4: Generalization of a HDNN and an IDNN on four DMM problems was identical to that of the IDNN except that error was propagated back across the internal delays as in Wan (1993). Figure 4 plots generalization error versus percentage of possible strings used in training for the two networks for each of the four DMM problems. If our intuitions were correct we would expect to see evidence here that the effect of wider terms, and lack of repetition would have a stronger adverse effect on the HDNN network than on the IDNN. This is exactly what we see. The position of the curve for the IDNN network is stable compared to that of the HDNN when changes are made to the width and repetition factors. Statistical analysis supports this conclusion. We ran an ANOVA test [Rice, 1988] with four factors (which network, term width, term repetition, and training set size) on the data summarized by the graphs of figure 4. The test detected a significant interaction between the network and width factors (M Snetxwid = 0.3430, F(l, 1824) = 234.4), and between the network and repetition factors (MSnetxrep = 0.1181, F(l, 1824) = 80.694). These two interactions are significant at p < 0.001, agreeing with our conclusion that width and repetition each has a stronger effect on the performance of the HDNN network. Further planned tests reveal that the effects of width and repetition are strong enough to change which network generalizes better. We ran a one-way ANOVA test on each problem individually to see which network performs better across the entire curve. The tests reveal that the HDNN performs with significantly less error than the IDNN in the narrow-repeated problem (M Serror = 0.0015, M Snet 0.5400, F(1,1824) = 369.0), and in the narrow-unrepeated problem (M Snet = 0.0683, F(1 , 1824) = 46.7). Performance of the IDNN is significantly better in the wide-unrepeated problem (M Snet = 0.0378, F(l, 1824) = 25.83). All of these comparisons are significant at p < 0.001. The test on the wide-repeated problem finds no significant difference in performance of the two networks (M Snet = 0.0004, Representation and Induction of Finite State Machines using TDNNs 409 F(l, 1824) = 0.273, p > 0.05). In addition to confirming our intuitions about the kinds of problems that internal delays should be helpful in solving, this set of simulations demonstrates the difference between representational and inductive bias. For all DMM problems except for the wide-unrepeated one, we were able to find, for each network, at least one set of weights which solve the problem perfectly. Despite the fact that the two networks are both capable of representing the problems, the differing way in which they respond to the width and repetition factors demonstrates a difference in their learning biases. 6 Conclusions This paper presents a number of interesting ideas concerning TDNNs using both theoretical and empirical techniques. On the theoretical side, we have precisely defined the subclass of FSMs which can be represented by TDNNs, the DMM languages. It is interesting to note that this network architecture which has no recurrent connections is capable of representing languages whose transition diagrams require loops. Other ideas were demonstrated using empirical techniques. First, we have shown that the number of states required to represent an FSM may be a poor predictor of how difficult the language is to learn. We were able to learn a 2048-state FSM using a small percentage of the possible training examples. This is possible because of the close match between the representational bias of the network, and the language learned. Second, we presented a set of simulations which demonstrated the utility of internal delays in a TDNN. These delays were shown to improve generalization on problems composed of features over short time intervals which reappear repeatedly. Third, that same set of simulations highlights the difference between representational bias, and inductive bias. Though these two terms are sometimes used interchangeably in the theoretical literature, this work shows that the two concepts are, in fact, separable. References [Clouse et al., 1994] Clouse, D. S., Giles, C. 1., Horne, B. G., and Cottrell, G. W. (1994). Learning large debruijn automata with feed-forward neural networks. Technical Report CS94-398, University of California, San Diego, Computer Science and Engineering Dept. [Horne and Hush, 1994] Horne, B. G. and Hush, D. R. (1994). On the node complexity of neural networks. Neural Networks, 7(9):1413-1426. [Kohavi, 1978] Kohavi, Z. (1978). Switching and Finite Automata Theory. McGraw-Hill, Inc., New York, NY, second edition. [Lang et al., 1990] Lang, K, Waibel, A., and Hinton, G. (1990). A time-delay neural network architecture for isolated word recognition. Neural Networks, 3(1):23-44. [Rice, 1988] Rice, J. A. (1988). Mathematical Statistics and Data Analysis. Brooks/Cole Publishing Company, Monterey, California. [Waibel et al., 1989] Waibel, A., Hanazawa, T., Hinton, G., Shikano, K, and Lang, K (1989). Phoneme recognition using time-delay neural networks. IEEE Transactions on Acoustics, Speech and Signal Processing, 37(3):328-339. [Wan, 1993] Wan, E. A. (1993). Time series prediction by using a connectionist network with internal delay lines. In Weigend, A. S. and Gershenfeld, N. A., editors, Time Series Prediction: Forecasting the Future and Understanding the Past. Addison Wesley.	1996	110
1,163	Probabilistic Interpretation of Population Codes Richard S. Zemel Peter Dayan zemeleu.arizona.edu dayaneai.mit.edu Abstract Alexandre Pouget alexesalk.edu We present a theoretical framework for population codes which generalizes naturally to the important case where the population provides information about a whole probability distribution over an underlying quantity rather than just a single value. We use the framework to analyze two existing models, and to suggest and evaluate a third model for encoding such probability distributions. 1 Introduction Population codes, where information is represented in the activities of whole populations of units, are ubiquitous in the brain. There has been substantial work on how animals should and/or actually do extract information about the underlying encoded quantity. 5,3,11,9,12 With the exception of Anderson, l this work has concentrated on the case of extracting a single value for this quantity. We study ways of characterizing the joint activity of a population as coding a whole probability distribution over the underlying quantity. Two examples motivate this paper: place cells in the hippocampus of freely moving rats that fire when the animal is at a particular part of an environment,S and cells in area MT of monkeys firing to a random moving dot stimulus.7 Treating the activity of such populations of cells as reporting a single value of their underlying variables is inadequate if there is (a) insufficient information to be sure (eg if a rat can be uncertain as to whether it is in place XA or XB then perhaps place cells for both locations should fire; or (b) if multiple values underlie the input, as in the whole distribution of moving random dots in the motion display. Our aim is to capture the computational power of representing a probability distribution over the underlying parameters.6 RSZ is at University of Arizona, Tucson, AZ 85721; PD is at MIT, Cambridge, MA 02139; AP is at Georgetown University, Washington, DC 20007. This work was funded by McDonnell-Pew, NIH, AFOSR and startup funds from all three institutions. Probabilistic Interpretation of Population Codes 677 In this paper, we provide a general statistical framework for population codes, use it to understand existing methods for coding probability distributions and also to generate a novel method. We evaluate the methods on some example tasks. 2 Population Code Interpretations The starting point for almost all work on neural population codes is the neurophysiological finding that many neurons respond to particular variable( s) underlying a stimulus according to a unimodal tuning function such as a Gaussian. This characterizes cells near the sensory periphery and also cells that report the results of more complex processing, including receiving information from groups of cells that themselves have these tuning properties (in MT, for instance). Following Zemel & Hinton's13 analysis, we distinguish two spaces: the explicit space which consists of the activities r = {rd of the cells in the population, and a (typically low dimensional) implicit space which contains the underlying information X that the population encodes in which they are tuned. All processing on the basis of the activities r has to be referred to the implicit space, but it itself plays no explicit role in determining activities. Figure 1 illustrates our framework. At the top is the measured activities of a population of cells. There are two key operations. Encoding: What is the relationship between the activities r of the cells and the underlying quantity in the world X that is represented? Decoding: What information about the quantity X can be extracted from the activities? Since neurons are generally noisy, it is often convenient to characterize encoding (operations A and B) in a probabilistic way, by specifying P[rIX]. The simplest models make a further assumption of conditional independence of the different units given the underlying quantity P[rIX] = I1i P[riIX] although others characterize the degree of correlation between the units. If the encoding model is true, then a Bayesian decoding model specifies that the information r carries about X can be characterized precisely as: P[Xlr] ex P[rIX]P[X], where P[ X] is the prior distribution about X and the constant of proportionality is set so that Ix P[Xlr]dX = 1. Note that starting with a deterministic quantity X in the world, encoding in the firing rates r, and decoding it (operation C) results in a probability distribution over X. This uncertainty arises from the stochasticity represented by P[rIX]. Given a loss function, we could then go on to extract a single value from this distribution (operation D). We attack the common assumption that X is a single value of some variable x, eg the single position of a rat in an environment, or the single coherent direction of motion of a set of dots in a direction discrimination task. This does not capture the subtleties of certain experiments, such as those in which rats can be made to be uncertain about their position, or in which one direction of motion predominates yet there are several simultaneous motion directions.7 Here, the natural characterization of X is actually a whole probability distribution P[xlw] over the value of the variable x (perhaps plus extra information about the number of dots), where w represents all the available information. We can now cast two existing classes of proposals for population codes in terms of this framework. The Poisson Model Under the Poisson encoding model, the quantity X encoded is indeed one particular value which we will call x, and the activities of the individual units are independent, 678 R. S. Zemel, P. Dayan and A. Pouget enmde 1'1''''11 ,--_0_ +_ 0 _,_t_f_t_o _ ..... x' x i iD f " x Figure 1: Left: encoding maps X from the world through tuning functions (A) into mean activities (B), leading to Top: observed activities r. We assume complete knowledge of the variables governing systematic changes to the activities of the cells. Here X is a single value x· in the space of underlying variables. Right: decoding extracts 1'[Xlr) (C)j a Single value can be picked (D) from this distribution given a loss function. with the terms P[rilx] = e-h(x) (h(x)t' jriL The activity ri could, for example, be the number of spikes the cell emits in a fixed time interval following the stimulus onset. A typical form for the tuning function h(x) is Gaussian h(x) <X e-(X-Xi)2/20'2 about a preferred value Xi for cell i. The Poisson decoding model is: 3, 11, 9, 12 (1) where K is a constant with respect to x. Although simple, the Poisson model makes the the assumption criticized above, that X is just a single value x. We argued for a characterization of the quantity X in the world that the activities of the cells encode as now P[xlw]. We describe below a method of encoding that takes exactly this definition of X. However, wouldn't P[xlr] from Equation 1 be good enough? Not if h(x) are Gaussian, since logP[xlr] = K' _ ~ (L:i ri) (X _ L:i riXi)2, 2 a2 L:i ri completing the square, implying that P[xlr] is Gaussian, and therefore inevitably unimodal. Worse, the width of this distribution goes down with L:i ri, making it, in most practical cases, a close approximation to a delta function. The KDE Model Anderson1,2 set out to represent whole probability distributions over X rather than just single values. Activities r represent distribution pr(x) through a linear combination of basis functions tPi(X), ie pr(x) = L:i r~tPi(x) where r~ are normalized such that pr(x) is a probability distribution. The kernel functions tPi(X) are not Probabilistic Interpretation of Population Codes 679 the tuning functions Ji(x) of the cells that would commonly be measured in an experiment. They need have no neural instantiation; instead, they form part of the interpretive structure for the population code. If the tPi(X) are probability distributions, and so are positive, then the range of spatial frequencies in P[xlw] that they can reproduce in pr(x) is likely to be severely limited. In terms of our framework, the KDE model specifies the method of decoding, and makes encoding its corollary. Evaluating KDE requires some choice of encoding representing P[xlw] by pr(x) through appropriate r. One way to encode is to use the Kullback-Leibler divergence as a measure of the discrepancy between P[xlw] and Ei r~tPi(x) and use the expectation-maximization (EM) algorithm to fit the ira, treating them as mixing proportions in a mixture mode1.4 This relies on {tPi(X)} being probability distributions themselves. The projection methodl is a one-shot linear filtering based alternative using the £2 distance. ri are computed as a projection of P[xlw] onto tuning functions Ji(x) that are calculated from tPj(x). ri = Ix P[xlw]Ji(x)dx fi(X) = L Aij1tPj(x) j Aij = Ix tPi (x)tPj (x)dx (2) Ji(x) are likely to need regularizing, 1 particularly if the tPi(X) overlap substantially. 3 The Extended Poisson Model The KDE model is likely to have difficulty capturing in pr(x) probability distributions P[xlw] that include high frequencies, such as delta functions. Conversely, the standard Poisson model decodes almost any pattern of activities r into something that rapidly approaches a delta function as the activities increase. Is there any middle ground? We extend the standard Poisson encoding model to allow the recorded activities r to depend on general P[xlw], having Poisson statistics with mean: (ri) = Ix P[xlw]Ji(x)dx. (3) This equation is identical to that for the KDE model (Equation 2), except that variability is built into the Poisson statistics, and decoding is now required to be the Bayesian inverse of encoding. Note that since ri depends stochastically on P[xlw], the full Bayesian inverse will specify a distribution P[P[xlw]lr] over possible distributions. We summarize this by an approximation to its most likely memberwe perform an approximate form of maximum likelihood, not in the value of x, but in distributions over x. We approximate P[xlw] as a piece-wise constant histogram which takes the value ¢>j in (xj, Xj+l], and Ji(x) by a piece-wise constant histogram that take the values Jij in (xj, xj+d. Generally, the maximum a posteriori estimate for {¢>j} can be shown to be derived by maximizing: (4) where € is the variance of a smoothness prior. We use a form of EM to maximize the likelihood and adopt the crude ·approximation of averaging neighboring values 680 R. S. Zemel, P. Dayan and A. Pouget Operation Extended Poisson KDE (Projection) KDE(EM) Encode (r.) = h [I" P(xlw]f.(x)dxj (r.) = h [R.n •• I" P[xlw]f.(x)dx] (r.) = h [Rm .. r:J (r.) f;(x) = R.n .. N(x •• u) f.(x) = L:J Aijl.pj(x) ri to max. L A.j = I" .p.(x).pj(x)dx Decode pr(x) to max. L pr(x) = L:. ri.p.(x) pr(x) = L:. r:.p.(x) pr(x) ri = I% pr(x)f.(x)dx::::: L:j tPilij r: = r./ L:J rj Likelihood L = log P [{tPi}l{ri}] ::::: L:.r;logf. L = I" P[xIwJlogpr(x)dx Error G = L:. ri log(r;!f.) E = I" [pr(x) - P[xlwJ] 2 dx G = I" P[xlw] log ~~X)JdX Table 1: A summary of the key operations with respect to the framework of the interpretation methods compared here. hO is a rounding operator to ensure integer firing rates, and 'l/Ji(X) = N(xi, 0') are the kernel functions for the KDE method. of ~j on successive iterations. By comparison with the linear decoding of the KDE method, Equation 4 offers a non-linear way of combining a set of activities {rd to give a probability distribution pr(x) over the underlying variable x. The computational complexities of Equation 4 are irrelevant, since decoding is only an implicit operation that the system need never actually perform. 4 Comparing the Models We illustrate the various models by showing the faithfulness with which they can represent two bimodal distributions. We used 0' = 0.3 for the kernel functions (KDE) and the tuning functions (extended Poisson model) and used 50 units whose Xi were spaced evenly in the range x = [-10,10]. Table 1 summarizes the three methods. Figure 2a shows the decoded version of a mixture of two broad Gaussians 1/2N[-2, 1] + 1/2N[2,1]. Figure 2b shows the same for a mixture of two narrow Gaussians tN[-2, .2] + tN[2, .2]. All the models work well for representing the broad Gaussians; both forms of the KDE model have difficulty with the narrow Gaussians. The EM version of KDE puts all its weight on the nearest kernel functions, and so is too broad; the projection version 'rings' in its attempt to represent the narrow components of the distributions. The extended Poisson model reconstructs with greater fidelity. 5 Discussion Informally, we have examined the consequences of the seemingly obvious step of saying that if a rat, for instance, is uncertain about whether it is at one of two places, then place cells representing both places could be activated. The complications Probabilistic Interpretation of Population Codes 01 A A OIl 01 OJ O. .. 01 ... ...,IM~_01 tIl 01 OJ 11 OJ u u A A ... ...,IMIBI) 681 01 Oil 01 O~--~--~--~--~ ·10 01 OJ 'T D.I OJ Ol 02 01 ... ...,~Figure 2: a) (upper) All three methods provide a good fit to the bimodal Gaussian distribution when its variance is sufficiently large (7 = 1.0). b) (lower) The KDE model has difficulty when 7 = 0.2. come because the structure of the interpretation changes - for instance, one can no longer think of maximum likelihood methods to extract a single value from the code directly. One main fruit of our resulting framework is a method for encoding and decoding probability distributions that is the natural extension of the (provably inadequate) standard Poisson model for encoding and decoding single values. Cells have Poisson statistics about a mean determined by the integral of the whole probability distribution, weighted by the tuning function of the cell. We suggested a particular decoding model, based on an approximation to maximum likelihood decoding to a discretized version of the whole probability distribution, and showed that it reconstructs broad, narrow and multimodal distributions more accurately than either the standard Poisson model or the kernel density model. Stochasticity is built into our method, since the units are supposed to have Poisson statistics, and it is therefore also quite robust to noise. The decoding method is not biologically plausible, but provides a quantitative lower bound to the faithfulness with which a set of activities can code a distribution. Stages of processing subsequent to a population code might either extract a single value from it to control behavior, or integrate it with information represented in other population codes to form a combined population code. Both operations must be performed through standard neural operations such as taking non-linear weighted sums and possibly products of the activities. We are interested in how much information is preserved by such operations, as measured against the non-biological 682 R. S. Zemel. P. Dayan and A. Pouget standard of our decoding method. Modeling extraction requires modeling the loss function - there is some empirical evidence about this from a motion experiment in which electrical stimulation of MT cells was pitted against input from a moving stimulus.lO However, much works remains to be done. Integrating two or more population codes to generate the output in the form of another population code was stressed by Hinton,6 who noted that it directly relates to the notion of generalized Hough transforms. We are presently studying how a system can learn to perform this combination, using the EM-based decoder to generate targets. One special concern for combination is how to understand noise. For instance, the visual system can be behaviorally extraordinarily sensitive - detecting just a handful of photons. However, the outputs of real cells at various stages in the system are apparently quite noisy, with Poisson statistics. If noise is added at every stage of processing and combination, then the final population code will not be very faithful to the input. There is much current research on the issue of the creation and elimination of noise in cortical synapses and neurons. A last issue that we have not treated here is certainty or magnitude. Hinton's6 idea of using the sum total activity of a population to code the certainty in the existence of the quantity they represent is attractive, provided that there is some independent way of knowing what the scale is for this total. We have used this scaling idea in both the KDE and the extended Poisson models. In fact, we can go one stage further, and interpret greater activity still as representing information about the existence of multiple objects or multiple motions. However, this treatment seems less appropriate for the place cell system the rat is presumably always certain that it is somewhere. There it is plausible that the absolute level of activity could be coding something different, such as the familiarity of a location. An entire collection of cells is a terrible thing to waste on representing just a single value of some quantity. Representing a whole probability distribution, at least with some fidelity, is not more difficult, provided that the interpretation of the encoding and decoding are clear. We suggest some steps in this direction. References [1) Anderson, CH (1994). International Journal of Modern Physics C, 5, 135-137. [2) Anderson, CH & Van Essen, DC (1994). In Computational Intelligence Imitating Life, 213-222. New York: IEEE Press. [3) Baldi, P & Heiligenberg, W (1988). Biological Cybernetics, 59, 313-318. (4) Dempster, AP, Laird, NM & Rubin, DB (1997). Proceedings of the Royal Statistical Society, B 39, 1-38. [5) [6) (7) (8) [9) (10) [11) Georgopoulos, AP, Schwartz, AB & Kettner, RE (1986). Science, 243, 1416-1419. Hinton, GE (1992). Scientific American, 267(3) 105-109. Newsome, WT, Britten, KH & Movshon, JA (1989). Nature, 341, 52-54. O'Keefe, J & Dostrovsky, J (1971). Brain Research, 34,171-175. Salinas, E & Abbott, LF (1994). Journal of Computational Neuroscience, 1, 89-107. Salzman, CD & Newsome, WT (1994). Science, 264, 231-237. Seung, HS & SompoJinsky, H (1993). Proceedings of the National Academy of Sciences, USA, 90, 10749-10753. (12) Snippe, HP (1996). Neural Computation, 8, 29-37. (13) Zemel, RS & Hinton, GE (1995). Neural Computation, 7, 549-564. PART V IMPLEMENTATION	1996	111
1,164	Analog VLSI Circuits for Attention-Based, Visual Tracking Timothy K. Horiuchi Computation and Neural Systems California Institute of Technology Pasadena, CA 91125 timmer@klab.caltech.edu Christof Koch Computation and Neural Systems California Institute of Technology Pasadena, CA 91125 Tonia G. Morris Electrical and Computer Engineering Georgia Institute of Technology Atlanta, GA, 30332-0250 tmorris@eecom.gatech.edu Stephen P. DeWeerth Electrical and Computer Engineering Georgia Institute of Technology Atlanta, GA, 30332-0250 Abstract A one-dimensional visual tracking chip has been implemented using neuromorphic, analog VLSI techniques to model selective visual attention in the control of saccadic and smooth pursuit eye movements. The chip incorporates focal-plane processing to compute image saliency and a winner-take-all circuit to select a feature for tracking. The target position and direction of motion are reported as the target moves across the array. We demonstrate its functionality in a closed-loop system which performs saccadic and smooth pursuit tracking movements using a one-dimensional mechanical eye. 1 Introduction Tracking a moving object on a cluttered background is a difficult task. When more than one target is in the field of view, a decision must be made to determine which target to track and what its movement characteristics are. If motion information is being computed in parallel across the visual field, as is believed to occur in the middle temporal area (MT) of primates, some mechanism must exist to preferentially extract the activity of the neurons associated with the target at the appropriate Analog VLSI Circuits for Attention-Based, VISual Tracking Photoreceptors Temporal Derivative Spatial Derivative Direction of Motion Hysteretic winner-take-all ST Target ......... ----......... ----......... --- Position ---~~---~----~-. Sa~ade Trigger 707 Figure 1: System Block Diagram: P = adaptive photoreceptor circuit, TD = temporal derivative circuit, SD = spatial derivative, DM = direction of motion, HYS WTA = hysteretic winner-take-all, P2V = position to voltage, ST = saccade trigger. The TD and SD are summed to form the saliency map from which the WTA finds the maximum. The output of the WTA steers the direction-of-motion information onto a common output line. Saccades are triggered when the selected pixel is outside a specified window located at the center of the array. time. Selective visual attention is believed to be this mechanism. In recent years, many studies have indicated that selective visual attention is involved in the generation of saccadic [10] [7] [12] [15] and smooth pursuit eye movements [9] [6] [16]. These studies have shown that attentional enhancement occurs at the target location just before a saccade as well as at the target location during smooth pursuit. In the case of saccades, attempts to dissociate attention from the target location has been shown to disrupt the accuracy or latency. Koch and Ullman [11] have proposed a model for attentional selection based on the formation of a saliency map by combining the activity of elementary feature maps in a topographic manner. The most salient locations are where activity from many different feature maps coincide or at locations where activity from a preferentiallyweighted feature map, such as temporal change, occurs. A winner-take-all (WTA) mechanism, acting as the center of the attentional "spotlight," selects the location with the highest saliency. Previous work on analog VLSI-based, neuromorphic, hardware simulation of visual tracking include a one-dimensional, saccadic eye movement system triggered by temporal change [8] and a two-dimensional, smooth pursuit system driven by visual motion detectors [5]. Neither system has a mechanism for figure-ground discrimination of the target. In addition to this overt form of attentional shifting, covert 708 g 0 .t::. C. 10.5 V > E ~ 6 0 C/) E CD t: ::J 0 0 f0E CD t: ::J 0 ,t it •• ~ 0 F? ' T. Horiuchi, T. G. Morris, C. Koch and S. P. DeWeerth "~....-r, .•. ,,..,..,. Vphoto I Spatial I Derivative I Temporal I Derivative Direction of Motion 4 7 10 13 16 19 22 Pixel Position Figure 2: Example stimulus - Traces from top to bottom: Photoreceptor voltage, absolute value of the spatial derivative, absolute value of the temporal derivative, and direction-of-motion. The stimulus is a high-contrast, expanding bar, which provides two edges moving in opposite directions. The signed, temporal and spatial derivative signals are used to compute the direction-of-motion shown in the bottom trace. attentional shifts have been modeled using analog VLSI circuits [4] [14], based on the Koch and Ullman model. These circuits demonstrate the use of delayed, transient inhibition at the selected location to model covert attentional scanning. In this paper we describe an analog VLSI implementation of an attention-based, visual tracking architecture which combines much of this previous work. Using a hardware model of the primate oculomotor system [8], we then demonstrate the use of the tracking chip for both saccadic and smooth pursuit eye movements. 2 System Description The computational goal of this chip is the selection of a target, based on a given measure of saliency, and the extraction of its retinal position and direction of motion. Figure 1 shows a block diagram of the computation. The first few stages of processing compute simple feature maps which drive the WTA-based selection of a target to track. The circuits at the selected location signal their position and the computed direction-of-motion. This information is used by an external saccadic and smooth pursuit eye movement system to drive the eye. The saccadic system uses the position information to foveate the target and the smooth pursuit system uses the motion information to match the speed of the target. Adaptive photoreceptors [2] (at the top of Figure 1) transduce the incoming pattern of light into an array of voltages. The temporal (TD) and spatial (SD) derivatives are computed from these voltages and are used to generate the saliency map and direction of motion. Figure 2 shows an example stimulus and the computed features. The saliency map is formed by summing the absolute-value of each derivative (ITDI + ISD I) and the direction-of-motion (DM) signal is a normalized product of the two Analog VLSI Circuits for Attention-Based, VISual Tracking 3.5 3.0 2.5 2.0 1.5 Target Position 1 .0+---r-~---+---r--~--+---r-~---+---r--~ -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Time (seconds) 709 Figure 3: Extracting the target's direction of motion: The WTA output voltage is used to switch the DM current onto a common current sensing line. The output of this signal is seen in the top trace. The zero-motion level is indicated by the flat line shown at 2.9 volts. The lower trace shows the target's position from the position-to-voltage encoding circuits. The target's position and direction of motion are used to drive saccades and smooth pursuit eye movements during tracking. d ·· TD·SD erlvatIves. ITDI+lsDI In the saliency map, the temporal and spatial derivatives can be differentially weighted to emphasize moving targets over stationary targets. The saliency map provides the input to a winner-take-all (WTA) computation which finds the maximum in this map. Spatially-distributed hysteresis is incorporated in this winnertake-all computation [4] by adding a fixed current to the winner's input node and its neighbors. This distributed hysteresis is motivated by the following two ideas: 1) once a target has been selected it should continue to be tracked even if another equally interesting target comes along, and 2) targets will typically move continuously across the array. Hysteresis reduces oscillation of the winning status in the case where two or more inputs are very close to the winning input level and the local distribution of hysteresis allows the winning status to freely shift to neighboring pixels rather than to another location further away. The WTA output signal is used to drive three different circuits: the position-tovoltage (P2V) circuit [3], the DM-current-steering circuit (see Figure 3), and the saccadic triggering (ST) circuit. The only circuits that are active are those at the winning pixel locations. The P2V circuit drives the common position output line to a voltage representing it's position in the array, the DM-steering circuit puts the local DM circuit's current onto the common motion output line, and the ST circuit drives a position-specific current onto a common line to be compared against an externally-set threshold value. By creating a "V" shaped profile of ST currents centered on the array, winning pixels away from the center will exceed the threshold 710 "2 o 2.7 E (/) o Q. a; .~ Q. 2.6 -; 2.5 N '0 > 2.4 T. Horiuch~ T. G. Mo"is, C. Koch and S. P. DeWeerth 2.1+----+----+---~----~--_4----~--~~--~ o 5 10 15 20 25 30 35 40 Time (msec) Figure 4: Position vs. time traces for the passage of a strong edge across the array at five different speeds. The speeds shown correspond to 327, 548, 1042, 1578,2294 pixels/sec. and send saccade requests off-chip. Figure 3 shows the DM and P2V outputs for an oscillating target. To test the speed of the tracking circuit, a single edge was passed in front of the array at varying speeds. Figure 4 shows some of these results. The power consumption of the chip (23 pixels and support circuits, not including the pads) varies between 0.35 m Wand 0.60 m W at a supply voltage of 5 volts. This measurement was taken with no clock signal driving the scanners since this is not essential to the operation of the circuit. 3 System Integration The tracking chip has been mounted on a neuromorphic, hardware model of the primate oculomotor system [8] and is being used to track moving visual targets. The visual target is mounted to a swinging apparatus to generate an oscillating motion. Figure 5 shows the behavior of the system when the retinal target position is used to drive re-centering saccades and the target direction of motion is used drive smooth pursuit. Saccades are triggered when the selected pixel is outside a specified window centered on the array and the input to the smooth pursuit system is suppressed during saccades. The smooth pursuit system mathematically integrates retinal motion to match the eye velocity to the target velocity. 4 Acknowledgements T. H. is supported by an Office of Naval Research AASERT grant and by the NSF Center for Neuromorphic Systems Engineering at Caltech. T . M. is supported by the Georgia Tech Research Institute. Analog VLSI Circuits for Attention-Based, VISual Tracking 711 40 Analog VLSI Human Subject ·200 ~ __ --~--~~~ __ --~--~~oo From Collewijn and Tamminga, 1984 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.S Time (seconds) Figure 5: Saccades and Smooth Pursuit: In this example, a swinging target is tracked over a few cycles. Re-centering saccades are triggered when the target leaves a specified window centered on the array. For comparison, on the right, we show human data for the same task [1]. References [1] H. Collewijn and E. Tamminga, "Human smooth and saccadic eye movements during voluntary pursuit of different target motions on different backgrounds" J. Physiol., VoL 351, pp. 217-250. (1984) [2] T. Delbriick, Ph.D. Thesis, Computation and Neural Systems Program California Institute of Technology (1993) [3] S. P. DeWeerth, "Analog VLSI Circuits for Stimulus Localization and Centroid Computation" Inti. 1. Compo Vis. 8(3), pp. 191-202. (1992) [4] S. P. DeWeerth and T. G. Morris, "CMOS Current Mode Winner-Take-All with Distributed Hysteresis" Electronics Letters, Vol. 31, No. 13, pp. 1051-1053. (1995) [5] R. Etienne-Cummings, J. Van der Spiegel, and P. Mueller "A Visual Smooth Pursuit Tracking Chip" Advances in Neural Information Processing Systems 8 (1996) [6] V. Ferrara and S. Lisberger, "Attention and Target Selection for Smooth Pursuit Eye Movements" J. Neurosci., 15(11), pp. 7472-7484, (1995) [7] J. Hoffman and B. Subramaniam, "The Role of Visual Attention in Saccadic Eye Movements" Perception and Psychophysics, 57(6), pp. 787-795, (1995) [8] T . Horiuchi, B. Bishofberger, and C. Koch, "An Analog VLSI Saccadic System" Advances in Neural Information Processing Systems 6, Morgan Kaufmann, pp. 582-589, (1994) [9] B. Khurana, and E. Kowler, "Shared Attentional Control of Smooth Eye Movement and Perception" Vision Research, 27(9), pp. 1603-1618, (1987) 712 T. Horiuchi, T. G. Morris, C. Koch and S. P. DeWeerth 10 Analog VLSI Monkey 5 Target Position .............. 0 Ci) Cl Q) -5 :s Q) C> c: -10 « Eye Position -15 100 msec saccadic delay added -20 -I---+--+---+-----1--+--_+_---.,r-----+---+----i 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 From Lisberger et aI., 1987 nme (seconds) Figure 6: Step-Ramp Experiment: In this experiment, the target jumps from the fixation point to a new location and begins moving with constant velocity. On the left, the analog VLSI system tracks the target. For comparison, on the right, we show data from a monkey performing the same task [13]. [10] E. Kowler, E. Anderson, B. Dosher, E. Blaser, "The Role of Attention in the Programming of Saccades" Vision Research, 35(13), pp. 1897-1916, (1995) [11] C. Koch and S. Ullman, "Shifts in selective visual attention: towards the underlying neural circuitry" Human Neurobiology, 4:219-227, (1985) [12] R. Rafal, P. Calabresi, C. Brennan, and T. Scioltio, "Saccade Preparation Inhibits Reorienting to Recently Attended Locations" 1. Exp. Psych: Hum. Percep. and Perf., 15, pp. 673-685, (1989) [13] S. G. Lisberger, E. J. Morris, and L. Tychsen, "Visual motion processing and sensory-motor integration for smooth pursuit eye movements." In Ann. Rev. Neurosci., Cowan et al., editors. Vol. 10, pp. 97-129, (1987) [14] T. G. Morris and S. P. DeWeerth, "Analog VLSI Circuits for Covert Attentional Shifts" Proc. 5th Inti. Conf. on Microelectronics for Neural Networks and Fuzzy Systems - MicroNeur096, Feb 12-14, 1996. Lausanne, Switzerland, IEEE Computer Society Press, Los Alamitos, CA, pp. 30-37, (1996) [15] S. Shimojo, Y. Tanaka, O. Hikosaka, and S. Miyauchi, "Vision, Attention, and Action - inhibition and facilitation in sensory motor links revealed by the reaction time and the line-motion." In Attention and Performance XVI, T. Inui & J. L. McClelland, editors. MIT Press, (1995) [16] W. J. Tam and H. Ono, "Fixation Disengagement and Eye-Movement Latency" Perception and Psychophysics, 56(3) pp. 251-260, (1994)	1996	112
1,165	Online learning from finite training sets: An analytical case study Peter Sollich* Department of Physics University of Edinburgh Edinburgh EH9 3JZ, U.K. P.SollichOed.ac.uk David Barbert Neural Computing Research Group Department of Applied Mathematics Aston University Birmingham B4 7ET, U.K. D.BarberOaston.ac.uk Abstract We analyse online learning from finite training sets at noninfinitesimal learning rates TJ. By an extension of statistical mechanics methods, we obtain exact results for the time-dependent generalization error of a linear network with a large number of weights N. We find, for example, that for small training sets of size p ~ N, larger learning rates can be used without compromising asymptotic generalization performance or convergence speed. Encouragingly, for optimal settings of TJ (and, less importantly, weight decay ,\) at given final learning time, the generalization performance of online learning is essentially as good as that of offline learning. 1 INTRODUCTION The analysis of online (gradient descent) learning, which is one of the most common approaches to supervised learning found in the neural networks community, has recently been the focus of much attention [1]. The characteristic feature of online learning is that the weights of a network ('student') are updated each time a new training example is presented, such that the error on this example is reduced. In offline learning, on the other hand, the total error on all examples in the training set is accumulated before a gradient descent weight update is made. Online and * Royal Society Dorothy Hodgkin Research Fellow t Supported by EPSRC grant GR/J75425: Novel Developments in Learning Theory for Neural Networks Online Leamingfrom Finite Training Sets: An Analytical Case Study 275 offline learning are equivalent only in the limiting case where the learning rate T) --* 0 (see, e.g., [2]). The main quantity of interest is normally the evolution of the generalization error: How well does the student approximate the input-output mapping ('teacher') underlying the training examples after a given number of weight updates? Most analytical treatments of online learning assume either that the size of the training set is infinite, or that the learning rate T) is vanishingly small. Both of these restrictions are undesirable: In practice, most training sets are finite, and noninfinitesimal values of T) are needed to ensure that the learning process converges after a reasonable number of updates. General results have been derived for the difference between online and offline learning to first order in T), which apply to training sets of any size (see, e. g., [2]). These results, however, do not directly address the question of generalization performance. The most explicit analysis of the time evolution of the generalization error for finite training sets was provided by Krogh and Hertz [3] for a scenario very similar to the one we consider below. Their T) --* 0 (i.e., offline) calculation will serve as a baseline for our work. For finite T), progress has been made in particular for so-called soft committee machine network architectures [4, 5], but only for the case of infinite training sets. Our aim in this paper is to analyse a simple model system in order to assess how the combination of non-infinitesimal learning rates T) and finite training sets (containing a examples per weight) affects online learning. In particular, we will consider the dependence of the asymptotic generalization error on T) and a, the effect of finite a on both the critical learning rate and the learning rate yielding optimal convergence speed, and optimal values of T) and weight decay A. We also compare the performance of online and offline learning and discuss the extent to which infinite training set analyses are -applicable for finite a. 2 MODEL AND OUTLINE OF CALCULATION We consider online training of a linear student network with input-output relation Here x is an N-dimensional vector of real-valued inputs, y the single real output and w the wei~t vector of the network. ,T, denotes the transpose of a vector and the factor 1/VN is introduced for convenience. Whenever a training example (x, y) is presented to the network, its weight vector is updated along the gradient of the squared error on this example, i. e., where T) is the learning rate. We are interested in online learning from finite training sets, where for each update an example is randomly chosen from a given set {(xll,yll),j.l = l. .. p} ofp training examples. (The case of cyclical presentation of examples [6] is left for future study.) If example J.l is chosen for update n, the weight vector is changed to (1) Here we have also included a weight decay 'Y. We will normally parameterize the strength of the weight decay in terms of A = 'YO' (where a = p / N is the number 276 P. Sollich and D. Barber of examples per weight), which plays the same role as the weight decay commonly used in offline learning [3]. For simplicity, all student weights are assumed to be initially zero, i.e., Wn=o = o. The main quantity of interest is the evolution of the generalization error of the student. We assume that the training examples are generated by a linear 'teacher', i.e., yJJ = W. T x JJ IVN +e, where eJJ is zero mean additive noise of variance (72. The teacher weight vector is taken to be normalized to w. 2 = N for simplicity, and the input vectors are assumed to be sampled randomly from an isotropic distribution over the hypersphere x2 = N. The generalization error, defined as the average of the squared error between student and teacher outputs for random inputs, is then where Vn = Wn - W •. In order to make the scenario analytically tractable, we focus on the limit N -+ 00 of a large number of input components and weights, taken at constant number of examples per weight a = piN and updates per weight ('learning time') t = niN. In this limit, the generalization error fg(t) becomes self-averaging and can be calculated by averaging both over the random selection of examples from a given training set and over all training sets. Our results can be straightforwardly extended to the case of percept ron teachers with a nonlinear transfer function, as in [7]. The usual statistical mechanical approach to the online learning problem expresses the generalization error in terms of 'order parameters' like R = ~wJw. whose (self-averaging) time evolution is determined from appropriately averaged update equations. This method works because for infinite training sets, the average order parameter updates can again be expressed in terms of the order parameters alone. For finite training sets, on the other hand, the updates involve new order parameters such as Rl = ~wJ Aw., where A is the correlation matrix of the training inputs, A = ~L-P =lxJJ(xJJ)T. Their time evolution is in turn determined by order parameters involving higher powers of A, yielding an infinite hierarchy of order parameters. We solve this problem by considering instead order parameter (generating) junctions [8] such as a generalized form of the generalization error f(t;h) = 2~vJexp(hA)vn . This allows powers of A to be obtained by differentiation with respect to h, reSUlting in a closed system of (partial differential) equations for f(t; h) and R(t; h) = ~ wJ exp(hA)w •. The resulting equations and details of their solution will be given in a future publication. The final solution is most easily expressed in terms of the Laplace transform of the generalization error fg(Z) = '!!.. fdt fg(t)e-z(f//a)t = fdz) + T}f2(Z) + T}2f3(Z) (2) a ~ 1 T}f4(Z) The functions fi (z) (i = 1 ... 4) can be expressed in closed form in terms of a, (72 and A (and, of course, z). The Laplace transform (2) yields directly the asymptotic value of the generalization error, foo = fg(t -+ (0) = limz--+o zig{z) , which can be calculated analytically. For finite learning times t, fg(t) is obtained by numerical inversion of the Laplace transform. 3 RESULTS AND DISCUSSION We now discuss the consequences of our main result (2), focusing first on the asymptotic generalization error foo, then the convergence speed for large learning times, Online Learningfrom Finite Training Sets: An Analytical Case Study 277 a=O.s a=i <1=2 Figure 1: Asymptotic generalization error (00 vs 1] and A. a as shown, (1"2 = 0.1. and finally the behaviour at small t. For numerical evaluations, we generally take (1"2 = 0.1, corresponding to a sizable noise-to-signal ratio of JQ.I ~ 0.32. The asymptotic generalization error (00 is shown in Fig. 1 as a function of 1] and A for a = 0.5, 1, 2. We observe that it is minimal for A = (1"2 and 1] = 0, as expected from corresponding resul ts for offline learning [3]1. We also read off that for fixed A, (00 is an increasing function of 1]: The larger 1], the more the weight updates tend to overshoot the minimum of the (total, i.e., offline) training error. This causes a diffusive motion of the weights around their average asymptotic values [2] which increases (00. In the absence of weight decay (A = 0) and for a < 1, however, (00 is independent of 1]. In this case the training data can be fitted perfectly; every term in the total sum-of-squares training error is then zero and online learning does not lead to weight diffusion because all individual updates vanish. In general, the relative increase (00(1])/(00(1] = 0) - 1 due to nonzero 1] depends significantly on a. For 1] = 1 and a = 0.5, for example, this increase is smaller than 6% for all A (at (1"2 = 0.1), and for a = 1 it is at most 13%. This means that in cases where training data is limited (p ~ N), 1] can be chosen fairly large in order to optimize learning speed, without seriously affecting the asymptotic generalization error. In the large a limit, on the other hand, one finds (00 = ((1"2/2)[1/a + 1]/(2 - 1])]. The relative increase over the value at 1] = a therefore grows linearly with a; already for a = 2, increases of around 50% can occur for 1] = 1. Fig. 1 also shows that (00 diverges as 1] approaches a critical learning rate 1]e: As 1] -+ 1]e, the 'overshoot' of the weight update steps becomes so large that the weights eventually diverge. From the Laplace transform (2), one finds that 1]e is determined by 1]e(4(Z = 0) = 1; it is a function of a and A only. As shown in Fig. 2b-d, 1]e increases with A. This is reasonable, as the weight decay reduces the length of the weight vector at each update, counteracting potential weight divergences. In the small and large a limit, one has 1]e = 2( 1 + A) and 1]e = 2( 1 + A/a), respectively. For constant A, 1]e therefore decreases2 with a (Fig. 2b-d) . We now turn to the large t behaviour of the generalization error (g(t). For small 1], the most slowly decaying contribution (or 'mode') to (g(t) varies as exp( -ct), its 1 The optimal value of the unscaledweight decay decreases with a as 'Y = (1"2 ja, because for large training sets there is less need to counteract noise in the training data by using a large weight decay. 2Conversely, for constant 'Y, f"/e increases with a from 2(1 + 'Ya) to 2(1 + 'Y): For large a , the weight decay is applied more often between repeat presentations of a training example that would otherwise cause the weights to diverge. 278 P. Sollich and D. Barber decay constant c = 71['\ + (va - 1 )2]/ a scaling linearly with 71, the size of the weight updates, as expected (Fig. 2a). For small a, the condition ct » 1 for fg(t) to have reached its asymptotic value foo is 71(1 + ,\)(t/a) » 1 and scales with tla, which is the number of times each training example has been used. For large a, on the other hand, the condition becomes 71t » 1: The size of the training set drops out since convergence occurs before repetitions of training examples become significant. For larger 71, the picture changes due to a new 'slow mode' (arising from the denominator of (2)). Interestingly, this mode exists only for 71 above a finite threshold 71min = 2/(a1/ 2 + a- 1/ 2 -1). For finite a, it could therefore not have been predicted from a small 71 expansion of (g(t). Its decay constant Cslow decreases to zero as 71 -t 71e, and crosses that of the normal mode at 71x(a,'\) (Fig. 2a). For 71 > 71x, the slow mode therefore determines the convergence speed for large t, and fastest convergence is obtained for 71 = 71x. However, it may still be advantageous to use lower values of 71 in order to lower the asymptotic generalization error (see below); values of 71 > 71x would deteriorate both convergence speed and asymptotic performance. Fig. 2b-d shows the dependence of 71min, 71x and 71e on a and'\. For ,\ not too large, 71x has a maximum at a ~ 1 (where 71x ~ 71e), while decaying as 71x = 1+2a- 1/ 2 ~ ~71e for larger a. This is because for a ~ 1 the (total training) error surface is very anisotropic around its minimum in weight space [9]. The steepest directions determine 71e and convergence along them would be fastest for 71 = ~71e (as in the isotropic case). However, the overall convergence speed is determined by the shallow directions, which require maximal 71 ~ 71e for fastest convergence. Consider now the small t behaviour of fg(t). Fig. 3 illustrates the dependence of fg(t) on 71; comparison with simulation results for N = 50 clearly confirms our calculations and demonstrates that finite N effects are not significant even for such fairly small N. For a = 0.7 (Fig. 3a), we see that nonzero 71 acts as effective update noise, eliminating the minimum in fg(t) which corresponds to over-training [3]. foo is also seen to be essentially independent of 71 as predicted for the small value of ,\ = 10-4 chosen. For a = 5, Fig. 3b clearly shows the increase of foo with 71. It also illustrates how convergence first speeds up as 71 is increased from zero and then slows down again as 71e ~ 2 is approached. Above, we discussed optimal settings of 71 and ,\ for minimal asymptotic generalization error foo. Fig. 4 shows what happens if we minimize fg(t) instead for a given final learning time t, corresponding to a fixed amount of computational effort for training the network. As t increases, the optimal 71 decreases towards zero as required by the tradeoff between asymptotic performance and convergence 1..=0 1..=0.1 1..=1 4,-------, 4,---------, (a) (b) (c) c 11m in o o J 2a 3 4 S o J 2 a 3 4 S Figure 2: Definitions of71min, 71x and 71e, and their dependence on a (for'\ as shown). Online Learning /rom Finite Training Sets: An Analytical Case Study 279 O.511---~--~--~---., (a) a = 0.7 (b) a = 5 O.20'--~5--1~O-~15--2~O-~25--3~o--'t Figure 3: fg vs t for different TJ. Simulations for N = 50 are shown by symbols (standard errors less than symbol sizes). A=1O- 4 , 0-2=0.1, a as shown. The learning rate TJ increases from below (at large t) over the range (a) 0.5 .. . 1.95, (b) 0.5 ... 1. 75. 0.8 0.6 0.4 0.2 (a) " " ------..... -... '-----0.0 L-L--'---~_'___'______'_~___'___'__' o 10 20 30 40 50 t (b) / / (c) 0.08 0.25 0.06 , 0.04 0.02 O. 00 '---'--'----'-..t.......'---'-~...J........_'___' 10 20 30 40 50 o 10 20 30 40 50 Figure 4: Optimal TJ and A vs given final learning time t, and resulting (g. Solid/dashed lines: a = 1 / a =2; bold/thin lines: online/offline learning. 0-2 =0.1. Dotted lines in (a): Fits of form TJ = (a + bIn t)/t to optimal TJ for online learning. speed. Minimizing (g(t) ::::: (00+ const . exp( -ct) ~ Cl + TJC2 + C3 exp( -C4TJt) leads to TJopt = (a + bIn t)/t (with some constants a, b, Cl...4). Although derived for small TJ, this functional form (dotted lines in Fig. 4a) also provides a good description down to fairly small t , where TJopt becomes large. The optimal weight decay A increases3 with t towards the limiting value 0-2 . However, optimizing A is much less important than choosing the right TJ: Minimizing (g(t) for fixed A yields almost the same generalization error as optimizing both TJ and A (we omit detailed results here4 ). It is encouraging to see from Fig. 4c that after as few as t = 10 updates per weight with optimal TJ, the generalization error is almost indistinguishable from its optimal value for t --t 00 (this also holds if A is kept fixed). Optimization of the learning rate should therefore be worthwhile in most practical scenarios. In Fig. 4c, we also compare the performance of online learning to that of offline learning (calculated from the appropriate discrete time version of [3]), again with 30ne might have expected the opposite effect of having larger>. at low t in order to 'contain' potential divergences from the larger optimal learning rates tJ. However, smaller >. tends to make the asymptotic value foo less sensitive to large values of tJ as we saw above, and we conclude that this effect dominates. 4Por fixed>. < u 2 , where fg(t) has an over-training minimum (see Pig. 3a), the asymptotic behaviour of tJopt changes to tJopt <X C 1 (without the In t factor), corresponding to a fixed effective learning time tJt required to reach this minimum. 280 P. SalJich and D. Barber optimized values of TJ and A for given t. The performance loss from using online instead of offline learning is seen to be negligible. This may seem surprising given the effective noise on weight updates implied by online learning, in particular for small t. However, comparing the respective optimal learning rates (Fig. 4a), we see that online learning makes up for this deficiency by allowing larger values of TJ to be used (for large a, for example, TJc(offline) = 2/0' « TJc(online) = 2). Finally, we compare our finite a results with those for the limiting case a -+ 00. Good agreement exists for any learning time t if the asymptotic generalization error (00 (a < 00) is dominated by the contribution from the nonzero learning rate TJ (as is the case for a -+ 00). In practice, however, one wants TJ to be small enough to make only a negligible contribution to (00(0' < 00); in this regime, the a -+ 00 results are essentially useless. 4 CONCLUSIONS The main theoretical contribution of this paper is the extension of the statistical mechanics method of order parameter dynamics to the dynamics of order parameter (generating) functions. The results that we have obtained for a simple linear model system are also of practical relevance. For example, the calculated dependence on TJ of the asymptotic generalization error (00 and the convergence speed shows that, in general, sizable values of TJ can be used for training sets of limited size (a ~ 1), while for larger a it is important to keep learning rates small. We also found a simple functional form for the dependence of the optimal TJ on a given final learning time t. This could be used, for example, to estimate the optimal TJ for large t from test runs with only a small number of weight updates. Finally, we found that for optimized TJ online learning performs essentially as well as offline learning, whether or not the weight decay A is optimized as well. This is encouraging, since online learning effectively induces noisy weight updates. This allows it to cope better than offline learning with the problem of local (training error) minima in realistic neural networks. Online learning has the further advantage that the critical learning rates are not significantly lowered by input distributions with nonzero mean, whereas for offline learning they are significantly reduced [10]. In the future, we hope to extend our approach to dynamic (t-dependent) optimization of TJ (although performance improvements over optimal fixed TJ may be small [6]), and to more complicated network architectures in which the crucial question of local minima can be addressed. References [1] See for example: The dynamics of online learning. Workshop at NIPS '95. [2] T. Heskes and B. Kappen. Phys. Ret). A, 44:2718, 1991. [3] A. Krogh and J. A. Hertz. J. Phys. A, 25:1135, 1992. [4] D. Saad and S. Solla. Phys. Ret). E, 52:4225, 1995; also in NIPS-8. [5] M. Biehl and H. Schwarze. J. Phys. A, 28:643-656, 1995. [6] Z.-Q. Luo. Neur. Comp., 3:226, 1991; T. Heskes and W. Wiegerinck. IEEE Trans. Neur. Netw., 7:919, 1996. [7] P. Sollich. J. Phys. A, 28:6125, 1995. [8] 1. L. Bonilla, F. G. Padilla, G. Parisi and F. Ritort. Europhys. Lett., 34:159, 1996; Phys. Ret). B, 54:4170, 1996. [9] J. A. Hertz, A. Krogh and G. I. Thorbergsson. J. Phys. A, 22:2133, 1989. [10] T. L. H. Watkin, A. Rau and M. Biehl. Ret). Modern Phys., 65:499, 1993.	1996	113
1,166	Spatiotemporal Coupling and Scaling of Natural Images and Human Visual Sensitivities Dawei W. Dong California Institute of Technology Mail Code 139-74 Pasadena, CA 91125 dawei@hope.caltech.edu Abstract We study the spatiotemporal correlation in natural time-varying images and explore the hypothesis that the visual system is concerned with the optimal coding of visual representation through spatiotemporal decorrelation of the input signal. Based on the measured spatiotemporal power spectrum, the transform needed to decorrelate input signal is derived analytically and then compared with the actual processing observed in psychophysical experiments. 1 Introduction The visual system is concerned with the perception of objects in a dynamic world. A significant fact about natural time-varying images is that they do not change randomly over space-time; instead image intensities at different times and/or spatial positions are highly correlated. We measured the spatiotemporal correlation function - equivalently the power spectrum - of natural images and we find that it is non-separable, i.e., coupled in space and time, and exhibits a very interesting scaling behaviour. When expressed as a function of an appropriately scaled frequency variable, the spatiotemporal power spectrum is given by a simple power-law. We point out that the same kind of spatiotemporal coupling and scaling exists in human visual sensitivity measured in psychophysical experiments. This poses the intriguing question of whether there is a quantitative relationship between the power spectrum of natural images and visual sensitivity. We answer this question by showing that the latter can be predicted from measurements of the power spectrum. 860 D. W. Dong 2 Spatiotemporal Coupling and Scaling Interest in properties of time-varying images dates back to the early days of development of the television [1]. But systematic studies have not been possible previously primarily due to technical obstacles, and our knowledge of the regularities of timevarying images has so far been very limited. . ~ ",,'- ... } , ~ , r <. )~ 't --=,.1 Figure 1: Natural time-varying images are highly correlated in space and time. Shown on the top are two frames of a motion scene separated by thirty three milliseconds. These two frames are highly repetitive, in fact the light intensities of most corresponding pixels are similar. Shown on the bottom are light increase (on the left) and light decrease (on the right) between the above two snapshots indicated by greyscale of pixels (white means no change). One can immediately see that only a small portion of the image changes significantly over this time scale. Our methods have been described previously [3J. To summerize, more than one thousand segments of videos on 8mm video tape (NTSC format RGB) are digitized to 8 bits greyscale using a Silicon Graphics Video board with default factory settings. Two types of segments are analyzed. The first are segments from movies on video tapes (e.g. "Raiders of the Lost Ark", "Uncommon Valor"). The second type of segments that we analyzed are videos made by the authors. The scene of the moving egret shown here is taken at Central Park in New York City. We have systematically measured the two point correlation matrix or covariance matrix of lOoxlOox2s (horizontalxverticalxtemporal digitized to 64x64x64) segments of natural time-varying images by averaging over 1049 movie segments. An example of two consecutive frames from a typical segment is given in Figure 1. The Fourier transform of the correlation matrix, or the power spectrum, turns out to be a non-separable function of spatial and temporal frequencies and exhibits an interesting scaling behaviour. From our measurements (see Figure 2) we find R(j,w) = R(jw) where 1w is a scaled frequency which is simply the spatial frequency 1 scaled by G(wl1), a function of the ratio of temporal and spatial frequencies, i.e., 1w = G(wl1)1. This behaviour is revealed most clearly by plotting the power spectrum as a function of 1 for fixed wi 1 ratio: the curves for different wi 1 ratios are just a horizontal shift from each other. Spatiotemporal Coupling/Scaling of Natural Images & VISual Sensitivity 861 A ~ .....:; ~ B 10- 1 W = 0.9 Hz 10-2 10- 3 10-4 0.1 1 Spatial Frequency I (cycle/degree) 10-1 10-3 10-4 0.1 w/I = 7°/s w/I=2.3°/s 1 Spatial Frequency I (cycle/degree) Figure 2: Spatiotemporal power spectra of natural time-varying images. (A) plotted as a function of spatial frequency for three temporal frequencies (0.9, 3, 10) Hz; (:8) plotted for three velocities ratios of temporal and spatial frequencies (0.8, 2.3, 7) degree/second. There are some important conclusions that can be drawn from this measurement. First, it is obvious that the power spectrum cannot be separated into pure spatial and pure temporal parts; space and time are coupled in a non-trivial way. The power spectrum at low temporal frequency decreases more rapidly with increasing spatial frequency. Second, underlying this data is an interesting scaling behaviour which can be easily seen from the curves for constant w / I ratios: each curve is simply shifted horizontally from each other in the log-log plot. Thus curves for constant w/ I ratio overlap with each other when shifted by an amount of G(w/J), Le., when plotted against a scaled frequency Iw = G(w/f)I. The similar spatio-temporal coupling and scaling for hunam visual sensitivity is shown in Figure 3. Interestingly, the human visual system seems to be designed to take advantage of such regularity in natural images. The spatiotemporal contrast sensitivity of human K(f, w), i.e., the visual responses to a sinewave grating of spatial frequency f modulated at temporal frequency w, exhibits the same kind of spatiotemporal coupling and scaling (see Figure 3), K(f, w) = K(fw). Again, when the contrast sensitivity curves are plotted as a function of f for fixed wi f ratios, the curves have the same shape and are only shifted from each other [2]. A ~ ;3 ::5 ~ B w =2 Hz 100 ~ 100 .....:; ~ 10 10 0.1 1 10 0.1 1 10 Spatial Frequency I (cycle/degree) Spatial Frequency I (cycle/degree) Figure 3: Spatiotemporal contrast sensitivities of human vision. (A) plotted as a function of spatial frequency for two temporal frequencies (2, 13) Hz; (B) plotted for two w/ I ratios (0.15, 3) degree/second. The solid lines in both A and B are the empirical fits. The experimental data points and empirical fitting curves are from reference [2]. First, it can be seen that the human visual sensitivity curve is band-pass filter at low temporal frequency and approaches low-pass filter for higher temporal frequency. The space and time are coupled. Second, it is clear that the curves for different w / I ratios have the same shape and are only shifted horizontally from each other in the log-log plot. Again, curves for constant w/I ratio overlap with each other when shifted by an amount of G(w/f), i.e., when plotted against a scaled frequency Iw = G(w/f)I. The similar behaviour of spatiotemporal coupling and scaling for the power spectra of natural images is shown in Figure 2. 862 D. W. Dong 3 Relative Motion of Visual Scene Why does the human visual sensitivity have the same spatiotemporal coupling and scaling as natural images? The intuition underlying the spatiotemporal coupling and scaling of natural images is that when viewing a real visual scene the natural eye and/or body movements translate the entire scene across the retina and every spatial Fourier component of the scene moves at the same velocity. Thus it is reasonable to assume that for constant velocity, Le., wi 1 ratio, the power spectrum show the same universal behaviour. This assumption is tested quantitatively in the following. Our measurements reveal that the spatiotemporal power spectrum has a simple form R(fw) '" 1;;;3 which is shown in Figure 6A. This behaviour can be accounted for if the dominant component in the temporal signal comes from motion of objects with static power spectra of Rs(f)'" 1-2 • The static power spectra for the same collection of images is measured by treating frames as snapshots (Figure 4A); the measurement confirmed the above assumption and is in agreement with earlier works on the statistical properties of static natural images [5, 6, 7]. It is easy to derive that for a rotationally symmetric static spectrum Rs (f) = KIP (K is a constant), the spatiotemporal power spectrum of moving images is K w R(f,w) = pP(j)' (1) where P( 7) is the function of velocity distribution, which is shown as the solid curve in Figure 4B (measured independently from the optical flows between frames). A 10-2 0.1 1 Spatial Frequency f (cycle/degree) B :::c: 10-1 ........ ., ...... ..-.. ~ ....:; ~ 10-3 10-5~----~----------------~ 1 10 v, w / f (degree/second) Figure 4: Spatial power spectrum and velocity distribution. (A) the measured spatial power spectrum of snap shot images, which shows that Rs(f) rv K/ P is a good approximation to the spectrum; (B) the measured velocity distribution P(v) (solid curve), in which the data of Figure 2 for the power spectrum were replotted as a function of w / f after multiplication by j3 all the data points fall on the P( v) curve. In summary, the measured spatiotemporal power spectrum is dominated by images of spatial power spectrum'" 1/12 moving with a velocity distribution P(v) '" 1/(v + vO)2 (similar velocity distribution has been proposed earlier [8, 3] . Thus R(f, w) = KI 13(wl 1 + VO)2 and G(wl f) '" (wi 1 + VO)2/3. Spatiotemporal Coupling/Scaling of Natural Images & Visual Sensitivity 863 Based on the assumption that the visual system is optimized to transmit information from natural scenes, we have derived and pointed out in references [3, 4] that the spatiotemporal contrast sensitivity K is a function of the power spectrum R, and thus the spatiotemporal coupling and scaling of R of natural images translates directly to the spatiotemporal coupling and scaling of K of visual sensitivity i.e., R is a function of f w only, so is K. 4 Spatiotemporal Decorrelation The theory of spatiotemporal decorrelation is based on ideas of optimal coding from information theory: decorrelation of inputs to make statistically independent representations when signal is strong and smoothing where noise is significant. The end result is that by chosing the correct degree of decorrelation the signal is compressed by elimination of what is irrelevant without significant loss of information. The following relationship can be derived for the visual sensitivity K and the power spectrum R in the presence of noise power N: The figure below illustrates the predicted filter for the case of white noise (constant N). 0.1 1 10 Figure 5: Predicted optimal filter (curve I): in the low noise regime, it is given by whitening filter R- 1/ 2 (curve II), which achieves spatiotemporal decorrelationj while at high noise regime it asymptotes the low-pass filter (curve III) which suppresses noise. As shown in Figure 6, the relation between the contrast sensitivity and the power spectrum predicts ( fw )~ K(fw)""" 1 + Nf~ in which N is the power of the white noise. This prediction is compared with psychophysical data in Figure 6B where we have used the scaling function G(w/ f) = (w/ f + VO)2/3 which has the same asymptotic behaviour as we have shown for the natural time-varying images [3]. We find that for Vo = 1 degree/second, the human 864 D. W. Dong contrast sensitivity curves for w/ f from 0.1 to 4 degree/second, measured in reference [2], overlap very well with the theoretical prediction from the power spectrum of our measurements. A 10- 1 10-3 10-4 ~~----------~~----~--~ 0.1 1 Scaled Frequency 1 w B ~ ...:; "-' ~ 100 10 0.1 1 10 Scaled Frequency 1 w Figure 6: Relation between the power spectrum of natural images and the human visual sensitivities. (A) the measured spatiotemporal power spectrum (Figure 2B) replotted as a function of the scaled frequency can be fit very well by R '" 1;;3 (solid line); (B) the spatiotemporal contrast sensitivities of human vision (Figure 3B) replotted as a function of the scaled frequency can be fit very well by our theoretical prediction (solid line). Our theory on the relation between the visual sensitivity K and the power spectrum of natural time-varyin~ images R in the presence of noise power N has been described in detail in reference [4] . To summarize, the visual sensitivity in Fourier space is simply K = R-1/2(1 + N/ R)- 3/2. In a linear system, this is proportional to the visual response to a sinewave of spatial frequency 1 modulated at temporal frequency w, Le., the contrast sensitivity curves shown in Figure 3. In the case of white noise, Le., N is independent of 1 and w, K depends on 1 and w through the power spectrum R . Since R is a function of the scaled frequency Iw only, so is K . From our measurement R '" I~ , thus K '" I:,P(1 + N/~)-3/2 . This curve is plotted in the figure as the solid line with N = 0.01. The agreement is very good. 5 Conclusions and Discussions A simple relati9nship is revealed between the statistical structure of natural timevarying images and the spatiotemporal sensitivity of human vision. The existence of this relationship supports the hypothesis that visual processing is optimized to compress as much information as possible about the outside world into the limited dynamic range of the visual channels. We should point out that this scaling behaviour is expected to break down for very high temporal and spatial frequency where the effect of the temporal and spatial modulation function of the eye [9, 10] cannot be ignored. Finally while our predictions show that, in general, the human visual sensitivity is strongly space-time coupled, we do predict a regime where decoupling is a good approximation. This is based on the fact that in the regime of relatively high temporal frequency and relatively low spatial frequency we find that the power spectrum of natural images is separable into spatial and temporal parts [3]. In a previous work we have used this decoupling to model response properties of cat LGN cells where we have shown that these can be accounted for by the theoretical prediction based on the power spectrum in that regime [4]. Acknowledgements The author gratefully acknowledges the discussions with Dr. Joseph Atick. Spatiotemporal Coupling/Scaling of Natural Images & Visual Sensitivity 865 References [1] Kretzmer ER, 1952. Statistics of television signals. The bell system technical journal. 751-763. [2] Kelly DR, 1979 Motion and vision. II. Stabilized spatio-temporal threshold surface. J. Opt. Soc. Am. 69, 1340-1349. [3] Dong DW, Atick JJ, 1995 Statistics of natural time-varying images. Network: Computation in Neural Systems, 6, 345-358. [4] Dong DW, Atick JJ, 1995 Temporal decorrelation: a theory of lagged and nonlagged responses in the lateral geniculate nucleus. Network: Computation in Neural Systems, 6, 159-178. [5] Burton GJ, Moorhead JR, 1987. Color and spatial structure in natural scenes. Applied Optics. 26(1): 157-170. [6] Field DJ, 1987. Relations between the statistics of natural images and the response properties of cortical cells .. J. Opt. Soc. Am. A 4: 2379-2394. [7] Ruderman DL, Bialek W , 1994. Statistics of natural images: scaling in the woods. Phy. Rev. Let. 73(6): 814-817. [8] Van Rateren JR, 1993. Spatiotemporal Contrast sensitivity of early vision. Vision Res. 33(2): 257-267. [9] Campbell FW, Gubisch RW, 1966. Optical quality of the human eye. J. Physiol. 186: 558-578. [10] Schnapf JL, Baylor DA, 1987. Row photoreceptor cells respond to light. Scientific American 256(4): 40-47.	1996	114
1,167	U sing Curvature Information for Fast Stochastic Search Genevieve B. Orr Dept of Computer Science Willamette University 900 State Street Salem, OR 97301 gorr@willamette.edu Todd K. Leen Dept of Computer Science and Engineering Oregon Graduate Institute of Science and Technology P.O.Box 91000, Portland, Oregon 97291-1000 tleen@cse.ogi.edu Abstract We present an algorithm for fast stochastic gradient descent that uses a nonlinear adaptive momentum scheme to optimize the late time convergence rate. The algorithm makes effective use of curvature information, requires only O(n) storage and computation, and delivers convergence rates close to the theoretical optimum. We demonstrate the technique on linear and large nonlinear backprop networks. Improving Stochastic Search Learning algorithms that perform gradient descent on a cost function can be formulated in either stochastic (on-line) or batch form. The stochastic version takes the form Wt+l = Wt + J1.t G( Wt, Xt ) (1) where Wt is the current weight estimate, J1.t is the learning rate, G is minus the instantaneous gradient estimate, and Xt is the input at time t i . One obtains the corresponding batch mode learning rule by taking J1. constant and averaging Gover all x. Stochastic learning provides several advantages over batch learning. For large datasets the batch average is expensive to compute. Stochastic learning eliminates the averaging. The stochastic update can be regarded as a noisy estimate of the batch update, and this intrinsic noise can reduce the likelihood of becoming trapped in poor local optima [1, 2J. 1 We assume that the inputs are i.i.d. This is achieved by random sampling with replacement from the training data. Using Curvature Informationfor Fast Stochastic Search 607 The noise must be reduced late in the training to allow weights to converge. After settling within the basin of a local optimum W., learning rate annealing allows convergence of the weight error v == W - w •. It is well-known that the expected squared weight error, E[lv12] decays at its maximal rate ex: l/t with the annealing schedule flo/to FUrthermore to achieve this rate one must have flo > flcnt = 1/(2Amin) where Amin is the smallest eigenvalue of the Hessian at w. [3, 4, 5, and references therein]. Finally the optimal flo, which gives the lowest possible value of E[lv12] is flo = 1/ A. In multiple dimensions the optimal learning rate matrix is fl(t) = (l/t) 1-£-1 ,where 1-£ is the Hessian at the local optimum. Incorporating this curvature information into stochastic learning is difficult for two reasons. First, the Hessian is not available since the point of stochastic learning is not to perform averages over the training data. Second, even if the Hessian were available, optimal learning requires its inverse - which is prohibitively expensive to compute 2. The primary result of this paper is that one can achieve an algorithm that behaves optimally, i.e. as if one had incorporated the inverse of the full Hessian, without the storage or computational burden. The algorithm, which requires only V(n) storage and computation (n = number of weights in the network), uses an adaptive momentum parameter, extending our earlier work [7] to fully non-linear problems. We demonstrate the performance on several large back-prop networks trained with large datasets. Implementations of stochastic learning typically use a constant learning rate during the early part of training (what Darken and Moody [4] call the search phase) to obtain exponential convergence towards a local optimum, and then switch to annealed learning (called the converge phase). We use Darken and Moody's adaptive search then converge (ASTC) algorithm to determine the point at which to switch to l/t annealing. ASTC was originally conceived as a means to insure flo > flcnt during the annealed phase, and we compare its performance with adaptive momentum as well. We also provide a comparison with conjugate gradient optimization. 1 Momentum in Stochastic Gradient Descent The adaptive momentum algorithm we propose was suggested by earlier work on convergence rates for annealed learning with constant momentum. In this section we summarize the relevant results of that work. Extending (1) to include momentum leaves the learning rule wt+ 1 = Wt + flt G ( Wt, x t) + f3 ( Wt Wt -1 ) (2) where f3 is the momentum parameter constrained so that 0 < f3 < 1. Analysis of the dynamics of the expected squared weight error E[ Ivl2 ] with flt = flo/t learning rate annealing [7, 8] shows that at late times, learning proceeds as for the algorithm without momentum, but with a scaled or effective learning rate _ flo ( ) fleff = 1 _ f3 . 3 This result is consistent with earlier work on momentum learning with small, constant fl, where the same result holds [9, 10, 11] 2Venter [6] proposed a I-D algorithm for optimizing the convergence rate that estimates the Hessian by time averaging finite differences of the gradient and scalin~ the learning rate by the inverse. Its extension to multiple dimensions would require O(n ) storage and O(n3) time for inversion. Both are prohibitive for large models. 608 G. B. Orr and T. K. Leen If we allow the effective learning rate to be a matrix, then, following our comments in the introduction, the lowest value of the misadjustment is achieved when /leff = ti- 1 [7, 8]. Combining this result with (3) suggests that we adopt the heuristic3 /3opt = I - /loti. (4) where /3opt is a matrix of momentum parameters, I is the identity matrix, and /lo is a scalar. We started with a scalar momentum parameter constrained by 0 < /3 < 1. The equivalent constraint for our matrix /3opt is that its eigenvalues lie between 0 and 1. Thus we require /lo < 1/ Amoz where Amoz is the largest eigenvalue of ti. A scalar annealed learning rate /loft combined with the momentum parameter /3opt ought to provide an effective learning rate asymptotically equal to the optimal learning rate ti- 1. This rate 1) is achieved without ever performing a matrix inversion on ti and 2) is independent of the choice of /lo, subject to the restriction in the previous paragraph. We have dispensed with the need to invert the Hessian, and we next dispense with the need to store it. First notice that, unlike its inverse, stochastic estimates of ti are readily available, so we use a stochastic estimate in (4). Secondly according to (2) we do not require the matrix /3opt, but rather /3opt times the last weight update. For both linear and non-linear networks this dispenses with the O( n 2 ) storage requirements. This algorithm, which we refer to as adaptive momentum, does not require explicit knowledge or inversion of the Hessian, and can be implemented very efficiently as is shown in the next section. 2 Implementation The algorithm we propose is Wt+! = Wt + /It G( Wt, Xt) + (I - /lo iit ) ~Wt (5) where ~Wt = Wt - Wt-l and iit is a stochastic estimate of the Hessian at time t. We first consider a single layer feedforward linear network. Since the weights connecting the inputs to different outputs are independent of each other we need only discuss the case for one output node. Each output node is then treated identically. For one output node and N inputs, the Hessian is ti = (xxT}z E n NxN where 0:1: indicates expectation over the inputs x and where xT is the transpose of x. The single-step estimate of the hessian is then just iit = xtxi. The momentum term becomes ~ T T (I - /lotit) ~Wt = (I - /lo(XtXt ))~Wt = ~Wt - /loXt(Xt ~Wt). (6) Written in this way, we note that there is no matrix multiplication, just the vector dot product xi ~Wt and vector addition that are both O(n). For M output nodes, the algorithm is then O(Nw ) where Nw = NM is the total number weights in the network. For nonlinear networks the problem is somewhat more complicated. To compute iit~Wt we use the algorithm developed by Pearlmutter [12] for computing the product of the hessian times an arbitrary vector.4 The equivalent of one forward-back 3We refer to (4) as a heuristic since we have no theoretical results on the dynamics of the squared weight error for learning with this matrix of momentum parameters. ·We actually use a slight modification that calculates the linearized Hessian times a vector: D f @D f ~Wt where D f is the Jacobian of the network output (vector) with respect to the weights, and @ indicates a tensor product. Using Curvature Information for Fast Stochastic Search Log( E[ Ivl2 1 ) I ~o=O·1 I ·1 ·2 ·3 B=adaptlve 2 3 5 a) Log(t) b) Log( E[ Iv12]) '--------""_--..flo=O.1 ·1 ·2L.------flo=O·01 ·3 2 3 Log(t) 609 I B=adaptlve I Figure 1: 2·D LMS Simulations: Behavior of log(E[lvI 2]) over an ensemble of 1000 networks with Al = .4 and Al = 4, (J'~ = 1. a) 1-'0 = 0.1 with various 13. Dashed curve corresponds to adaptive momentum. b) 13 adaptive for various 1-'0. propagation is required for this calculation. Thus, to compute the entire weight update requires two forward-backward propagations, one for the gradient calculation and one for computing iltllWt. The only constraint on JJo is that JJo < 1/ Amax. We use the on-line algorithm developed by LeCun, Simard, and Pearlmutter [13] to find the largest eigenvalue prior to the start of training. 3 Examples In the following two subsections we examine the behavior of annealed learning with adaptive momentum on networks previously trained to a point close to an optimum, where the noise dominates. We look at very simple linear nets, large linear nets, and a large nonlinear net. In section 3.3 we couple adaptive momentum with automatic switching from constant to annealed learning. 3.1 Linear Networks We begin with a simple 2-D LMS network. Inputs Xt are gaussian distributed with zero mean and the targets d at each timestep t are dt = W,!, Xt + Et where Et is zero mean gaussian noise, and W* is the optimal weight vector. The weight error at time t is just v == Wt w*. Figure 1 displays results for both constant and adaptive momentum with averages computed over an ensemble of 1000 networks. Figure (la) shows the decay of E[lv1 2 ] for JJo = 0.1 and various values of f3. As momentum is increased, the convergence rate increases. The optimal scalar momentum parameter is f3 == (1- JJOAmin) = .96. Adaptive momentum achieves essentially the same rate of convergence without prior knowledge of the Hessian. Figure 1b shows the behavior of E[lvI 2 ] for various JJo when adaptive momentum is used. One can see that after a few hundred iterations the value of E[lv1 2] is independent of JJo (in all cases JJo < l/Amax < JJcrit ). Figure 2 shows the behavior of the misadjustment (mean squared error in excess of the optimum~ for a 4-D LMS problem with a large condition number P == Amax/Arr;in = 10 . We compare 3 cases:. 1) the opt~mal learning rate matrix JJo = 1iwIthout momentum, 2) JJo = .5 wIth the optzmal constant momentum matrix f3 = I - JJo 1i, and 3) JJo = .5 with the adaptive momentum. All three cases show similar behavior, showing the efficacy with which the matrix momentum 610 10. 0.1 0.001 Figure 2: 4-D LMS with p = 105 : Plot displays misadjustment. Annealing starts at t = 10. For {3adapt and {3 = I 1-'01i, we use 1-'0 = .5. Each curve is an average of 10 runs. G. B. Orr and T. K. Leen 10. 100. 1000. 10000. 5 6 I 10 10 Figure 3: Linear Prediction: 1-'0 = 0.26. Curves show constant learning rate, annealing started at t = 50 without momentum, and with adaptive momentum. mocks up the optimal learning rate matrix J1.0 = 1£ -1, and lending credence to the stochastic estimate of the Hessian used in adaptive momentum. We next consider a large linear prediction problem (128 inputs, 16 outputs and eigenvalues ranging from 1.06 x 10-5 to 19.98 - condition number p = 1.9 X 106)5. Figure 3 displays the misadjustment for 1) annealed learning with f3 = f3adapt, 2) annealed learning with f3 = 0, and 3) constant learning rate (for comparison purposes). As before, we have first trained (not shown completely) at constant learning rate J1.0 = .026 until the MSE and the weight error have leveled out. As can be seen f3adapt does much better than annealing without momentum. 3.2 Phoneme Classification We next use phoneme classification as an example of a large nonlinear problem. The database consists of 9000 phoneme vectors taken from 48 50-second speech monologues. Each input vector consists of 70 PLP coefficients. There are 39 target classes. The architecture was a standard fully connected feedforward network with 71 (includes bias) input nodes, 70 hidden nodes, and 39 output nodes for a total of 7700 weights. We first trained the network with constant learning rate until the MSE flattened out. At that point we either annealed without momentum, annealed with adaptive momentum, or used ASTC (which attempts to adjust J1.0 to be above J1.crit - see next section). When annealing was used without momentum, we found that the noise went away, but the percent of correctly classified phonemes did not improve. Both the adaptive momentum and ASTC resulted in significant increases in the percent correct, however, adaptive momentum was significantly better than ASTC. In the next section, we examine this problem in more detail. 3.3 Switching on Annealing A complete algorithm must choose an appropriate point to change from constant J1. search to annealed learning. We use Moody and Darken's ASTC algorithm [4, 14] to accomplish this. ASTC measures the roughness of trajectories, switching to 1ft annealing when the trajectories become very rough - an indication that the noise in the updates is dominating the algorithm's behavior. In an attempt to satisfy 5Prediction of a 4 X 4 block of image pixels from the surrounding 8 blocks. Using Curvature Information for Fast Stochastic Search 611 50 50 40 40 ~30 0 ~30 0 0 (.)20 (.)20 ;,I! ~ 0 0 10 10 100000 qo 20 50 100 a) b) epoch Figure 4: Phoneme Classification: Percent Correct a) ASTC without momentum (bottom curve) and adaptive momentum (top) as function of the number of input presentations. b) Conjugate Gradient Descent - one epoch equals one pass through the data, i.e. 9000 input presentations. J.lo > J.lcrit, ASTC can also switch back to constant learning when trajectories become too smooth. We return to the phoneme problem using three different training methods: 1) ASTC without momentum (with switching back and forth between annealed and constant learning), 2) adaptive momentum with annealing turned on when ASTC first suggests the transition (but no subsequent return to constant learning rate), and 3) standard conjugate gradient descent. Figure 4a compares ASTC (no momentum) with adaptive momentum (using ASTC to turn on annealing). After annealing is turned on, the classification accuracy improves far more quickly with adaptive momentum. Figure 4b displays the classification performance as a function of epoch using conjugate gradient descent (CGD). After 100 passes through the 9000 example dataset (900,000 presentations), the classification accuracy is 39.6%, or 7% below adaptive momentum's performance at 100,000 presentations. Note also that adaptive momentum is continuing to improve the optimization, while the ASTC and conjugate gradient descent curves have flattened out. The cpu time used for the optimization was about the same for the CGD and adaptive momentum algorithms. It thus appears that our implementation of adaptive momentum costs about 9 times as much per pattern as CGD. We believe that the performance can be improved. Our complexity analysis [8] predicts a 3:1 cost ratio, rather than 9:1, and optimization comparable to that applied to the CGD code6 should enhance the run-time performance of CGD. For this problem, the performance of the two algorityms on the test set (no shown on graph) is not much different (31.7% for CGD versus 33.4% for adaptive momentum. Howver we are concerned here with the efficiency of the optimization, not generalization performance. The latter depends on dataset size and regularization techniques, which can easily be combined with any optimizer. 4 Summary We have presented an efficient O( n) stochastic algorithm with few adjustable parameters that achieves fast convergence during the converge phase for both linear and nonlinear problems. It does this by incorporating curvature information without 6CGD was performed using nopt written by Etienne Barnard and made available through the Center for Spoken Language Understanding at the Oregon Graduate Institute. 612 G. B. Orr and T. K. Leen explicit computation of the Hessian. We also combined it with a method (ASTC) for detecting when to make the transition between search and converge regimes. Acknowledgments The authors thank Yann LeCun for his helpful critique. This work was supported by EPRl under grant RPB015-2 and AFOSR under grant FF4962-93-1-0253. References [1] Genevieve B. Orr and Todd K. Leen. Weight space probability densities in stochastic learning: II. Transients and basin hopping times. In Giles, Hanson, and Cowan, editors, Advances in Neural Information Processing Systems, vol. 5, San Mateo, CA, 1993. Morgan Kaufmann. [2] William Finnoff. Diffusion approximations for the constant learning rate backpropagation algorithm and resistence to local minima. In Giles, Hanson, and Cowan, editors, Advances in Neural Information Processing Systems, vol. 5, San Mateo, CA, 1993. Morgan Kaufmann. [3] Larry Goldstein. Mean square optimality in the continuous time Robbins Monro procedure. Technical Report DRB-306, Dept. of Mathematics, University of Southern California, LA, 1987. [4] Christian Darken and John Moody. Towards faster stochastic gradient search. In J.E. Moody, S.J. Hanson, and R.P. Lipmann, editors, Advances in Neural Information Processing Systems 4. Morgan Kaufmann Publishers, San Mateo, CA, 1992. [5] Halbert White. Learning in artificial neural networks: A statistical perspective. Neural Computation, 1:425-464, 1989. [6] J. H. Venter. An extension of the robbins-monro procedure. Annals of Mathematical Statistics, 38:117-127, 1967. [7] Todd K. Leen and Genevieve B. Orr. Optimal stochastic search and adaptive momentum. In J.D. Cowan, G. Tesauro, and J. Alspector, editors, Advances in Neural Information Processing Systems 6, San Francisco, CA., 1994. Morgan Kaufmann Publishers. [8] Genevieve B. Orr. Dynamics and Algorithms for Stochastic Search. PhD thesis, Oregon Graduate Institute, 1996. [9] Mehmet Ali Tugay and Yalcin Tanik. Properties of the momentum LMS algorithm. Signal Processing, 18:117-127, 1989. [10] John J. Shynk and Sumit Roy. Analysis of the momentum LMS algorithm. IEEE Transactions on Acoustics, Speech, and Signal Processing, 38(12):2088-2098, 1990. [11] W. Wiegerinck, A. Komoda, and T. Heskes. Stochastic dynamics of learning with momentum in neural networks. Journal of Physics A, 27:4425-4437, 1994. [12] Barak A. Pearlmutter. Fast exact multiplication by the hessian. Neural Computation, 6:147-160, 1994. [13] Yann LeCun, Patrice Y. Simard, and Barak Pearlmutter. Automatic learning rate maximization by on-line estimation of the hessian's eigenvectors. In Giles, Hanson, and Cowan, editors, Advances in Neural Information Processing Systems, vol. 5, San Mateo, CA, 1993. Morgan Kaufmann. [14J Christian Darken. Learning Rate Schedules for Stochastic Gradient Algorithms. PhD thesis, Yale University, 1993.	1996	115
1,168	ARC-LH: A New Adaptive Resampling Algorithm for Improving ANN Classifiers Friedrich Leisch Friedrich.Leisch@ci.tuwien.ac.at Kurt Hornik Kurt.Hornik@ci.tuwien.ac.at Institut fiir Statistik und Wahrscheinlichkeitstheorie Technische UniversWit Wien A-I040 Wien, Austria Abstract We introduce arc-Ih, a new algorithm for improvement of ANN classifier performance, which measures the importance of patterns by aggregated network output errors. On several artificial benchmark problems, this algorithm compares favorably with other resample and combine techniques. 1 Introduction The training of artificial neural networks (ANNs) is usually a stochastic and unstable process. As the weights of the network are initialized at random and training patterns are presented in random order, ANNs trained on the same data will typically be different in value and performance. In addition, small changes in the training set can lead to two completely different trained networks with different performance even if the nets had the same initial weights. Roughly speaking, ANNs have a low bias because of their approximation capabilities, but a rather high variance because of the instability. Recently, several resample and combine techniques for improving ANN performance have been proposed. In this paper we introduce an new arcing ("~aptive resample and £ombine") method called arc-Ih. Contrary to the arc-fs method by Freund & Schapire (1995), which uses misclassification rates for adapting the resampling probabilities, arc-Ih uses the aggregated network output error. The performance of arc-Ih is compared with other techniques on several popular artificial benchmark problems. ARC-Uf: A New Adaptive Resampling Algorithm/or ANN Classifiers 523 2 Bias-Variance Decomposition of 0-1 Loss Consider the task of classifying a random vector e taking values in X into one of c classes G1 , ... , Ge , and let g(.) be a classification function mapping the input space on the finite set {I, ... , c}. The classification task is to find an optimal function g minimizing the risk Rg = IELg(e) = l Lg(x) dF(x) (1) where F denotes the (typically unknown) distribution function of e, and L is a loss function. In this paper, we consider 0-1 loss only, i.e., the loss is 1 for all misclassified patterns and zero otherwise. It is well known that the optimal classifier, i.e., the classifier with minimum risk, is the Bayes classifier g* assigning to each input x the class with maximum posterior probability IP(Gnlx) . These posterior probabilities are typically unknown, hence the Bayes classifier cannot be used directly. Note that Rg* = 0 for disjoint classes and Rg* > 0 otherwise. Let X N = {xt, ... ,xN} be a set of independent input vectors for which the true class is known, available for training the classifier. Further, let g X N ( .) denote a classifier trained using set X N. The risk Rg x N ~ Rg* of classifier g x N is a random variable depending on the training sample X N. In the case of ANN classifiers it also depends on the network training, i.e., even for fixed X N the performance of a trained ANN is a random variable depending on the initialization of weights and the (often random) presentation of the patterns [xnl during training. Following Breiman (1996a) we decompose the risk of a classifier into the (minimum possible) Bayes error, a systematic bias term of the model class and the variance of the classifier within its model class. We call a classifier model unbiased for input x if, over replications of all possible training sets X N of size N, network initializations and pattern presentations, g picks the correct class more often than any other class. Let U = U(g) denote the set of all x E X where g is unbiased; and B = B(g) = X\U the set of all points where g is biased. The risk of classifier g can be decomposed as Rg = Rg* + Bias(g) + Var(g) where Rg* is the risk of the Bayes classifier, Bias(g) Var(g) Rag - Rag* Rug - Rug* (2) and Ra and Ru denote the risk on set Band U, respectively, i.e., the integration in Equation 1 is over B or U instead of X, repectively. A simpler bias-variance decomposition has been proposed by Kong & Dietterich (1995): Bias(g) Var(g) IP{B} Rg - Bias(g) 524 F. LeischandK. Hornik The size of the bias set is seen as the bias of the model (i.e., the error the model class "typically" makes) . The variance is simply the difference between the actual risk and this bias term. This decompostion yields negative variance if the current classifier performs better than the average classifier. In both decompositions, the bias gives the systematic risk of the model, whereas the variance measures how good the current realization is compared to the best possible realization of the model. Neural networks are very powerful but rather unstable approximators, hence their bias should be low, but the variance may be high. 3 Resample and Combine Suppose we had k independent training sets X N1 , .. . , X Nk and corresponding classifiers 91' . .. , 9k trained using these sets, respectively. We can then combine these single classifiers into ajoint voting classifier 9~ by assigning to each input x the class the majority of the 9j votes for. If the 9j have low bias, then 9~ should have low bias, too. If the model is unbiased for an input x, then the variance of 9~ vanishes as k -+ 00 , and 9 v = limk --+ oo 9k is optimal for x. Hence, by resampling training sets from the original training set and combining the resulting classifiers into a voting classifier it might be possible to reduce the high variance of unstable classification algorithms. Training sets ANN classifiers X N1 t::91 ~ ~ . - ." ...c:,.~ .... '&.'" ........ XN'J t::92 X N . ~~-.... -.. -;/ 9k • t::• ... -:::1; .. ··-· .. • ... • .-.-.... -.. X Nk I> 9k resample combine adapt 3.1 Bagging Breiman (1994, 1996a) introduced a procedure called bagging ("Qootstrap aggregating") for tree classifiers that may also be used for ANNs. The bagging algorithm starts with a training set X N of size N. Several bootstrap replica X J.." . .. ,X7v are constructed and a neural network is trained on each. These networks are finally combined by majority voting. The bootstrap sets X1 consist of N patterns drawn with replacement from the original training set (see Efron & Tibshirani (1993) for more information on the bootstrap). ARC-ill: A New Adaptive Resampling Algorithm/or ANN Classifiers 525 3.2 Arcing 3.2.1 Arcing Based on Misclassification Rates Arcing, which is a more sophisticated version of bagging, was first introduced by Freund & Schapire (1995) and called boosting. The new training sets are not constructed by uniformly sampling from the empirical distribution of the training set XN" but from a distribution over XN that includes information about previous misclassifications. Let P~ denote the probability that pattern xn is included into the i-th training set X},y and initialize with P~ = 1/ N. Freund and Schapire's arcing algorithm, called arc-fs as in Breiman (1996a), works as follows: 1. Construct a pattern set Xiv by sampling with replacement with probabilities P~ from X N and train a classifier 9i using set xiv. 2. Set dn = 1 for all patterns that are misclassified by 9i and zero otherwise. With fi = L~=lp~dn and!3i = (1- fi)/fi update the probabilities by i /3dn HI _ Pn i Pn N 'd Ln=l P~(3i n 3. Set i := i + 1 and repeat. After k steps, 91' . . . ,gk are combined with weighted voting were each 9j'S vote has weight log!3i. Breiman (1996a) and Quinlan (1996) compare bagging and arcing for CART and C4.5 classifiers, respectively. Both bagging and arc-fs are very effective in reducing the high variance component of tree classifiers, with adaptive resampling being a bit better than simple bagging. 3.2.2 Arcing Based on Network Error Independently from the arcing and bagging procedures described above, adaptive resampling has been introduced for active pattern selection in leave-k-out crossvalidation CV / APS (Leisch & Jain, 1996; Leisch et al., 1995). Whereas arc-fs (or Breiman's arc-x4) uses only the information whether a pattern is misclassified or not, in CV / APS the fact that MLPs approximate the posterior probabilities of the classes (Kanaya & Miyake, 1991) is utilized, too. We introduce a simple new arcing method based on the main idea of CV / APS that the "importance" of a pattern for the learning process can be measured by the aggregated output error of an MLP for the pattern over several training runs. Let the classifier 9 be an ANN using l-of-c coding, i.e., one output node per class, the target t(x) for each input x is one at the node corresponding to the class of x and zero at the remaining output nodes. Let e(x) = It(x) - 9(x))12 be the squared error ofthe network for input x. Patterns that 'repeatedly have high output errors are somewhat harder to learn for the network and therefore their resampling probabilities are increased proportionally to the error. Error-dependent resampling 526 F. Leisch and K. Hornik introduces a "grey-scale" of pattern-importance as opposed to the "black and white" paradigm of misclassification dependent resampling. Again let p~ denote the probability that pattern xn is included into the i-th training set Xiv and initialize with p; = 1/ N. Our new arcing algorithm, called arc-Ih, works as follows: 1. Construct a pattern set xiv by sampling with replacement with probabilities p~ from X N and train a classifier gj using set xiv. 2. Add the network output error of each pattern to the resampling probabilities: 3. Set i := i + 1 and repeat. After k steps, g1' ... ,gk are combined by majority voting. 3.3 Jittering In our experiments, we also compare the above resample and combine methods with jittering, which resamples the training set by contaminating the inputs by artificial noise. No voting is done, but the size of the training set is increased by creation of artificial inputs "around" the original inputs, see Koistinen & Holmstrom (1992). 4 Experiments We demonstrate the effects of bagging and arcing on several well known artificial benchmark problems. For all problems, i - h - c single hidden layer perceptrons (SHLPs) with i input, h hidden and c output nodes were used. The number of hidden nodes h was chosen in a way that the corresponding networks have reasonably low bias. 2 Spirals with noise: 2-dimensional input, 2 classes. Inputs with uniform noise around two spirals. N = 300. Rg* = 0%. 2-14-2 SHLP. Continuous XOR: 2-dimensional input, 2 classes. Uniform inputs on the 2dimensional square -1 :::; x, y :::; 1 classified in the two classes x * y ~ 0 and x * y < O. N = 300. Rg* = 0%. 2-4-2 SHLP. Ringnorm: 20-dimensional input, 2 classes. Class 1 is normal wit mean zero and covariance 4 times the identity matrix. Class 2 is a unit normal with mean (a, a, ... , a). a = 2/.../20. N = 300. Rg* = 1.2%. 20-4-2 SHLP. The first two problems are standard benchmark problems (note however that we use a noisy variant of the standard spirals problem); the last one is, e.g., used in Breiman (1994, 1996a). ARC-LH: A New Adaptive Resampling Algorithm/or ANN Classifiers 527 All experiments were replicated 50 times, in each bagging and arcing replication 10 classifiers were combined to build a voting classifier. Generalization errors were computed using Monte Carlo techniques on test sets of size 10000. Table 1 gives the average risk over the 50 replications for a standard single SHLP, an SHLP trained on a jittered training set and for voting classifiers using ten votes constructed with bagging, arc-Ih and arc-fs, respectively. The Bayes risk ofthe spiral and xor example is zero, hence the risk of a network equals the sum of its bias and variance. The Bayes risk of the ringnorm example is 1.2%. Breiman Kong & Dietterich Rg Bias(g) Var(g) Bias(g) Var(g) 2 Spirals standard 7.75 0.32 7.43 0.82 6.93 jitter 6.53 0.26 6.27 0.52 6.02 bagging 4.39 0.35 4.04 0.68 3.71 arc-fs 4.31 0.35 3.96 0.60 3.71 arc-Ih 4.32 0.31 4.01 0.72 3.60 XOR standard 6.54 0.53 6.01 1.32 5.22 jitter 6.29 0.37 5.92 1.08 5.21 bagging 3.69 0.59 3.09 1.22 2.47 arc-fs 3.73 0.58 3.15 1.12 2.61 arc-Ih 3.58 0.50 3.08 1.20 2.38 Ringnorm standard 18.64 9.19 8.26 13.84 4.80 jitter 18.56 9.03 8.34 13.72 4.84 bagging 15.72 9.61 4.91 13.54 2.18 arc-fs 15.71 9.70 4.81 13.58 2.13 arc-Ih 15.63 9.30 5.13 13.20 2.43 Table 1: Bias-variance decompositions. The variance part was drastically reduced by the res ample & combine methods, with only a negligible change in bias. Note the low bias in the spiral and xor problems. ANNs obviously can solve these classification tasks (one could create appropriate nets by hand), but of course training cannot find the exact boundaries between the classes. Averaging over several nets helps to overcome this problem. The bias in the ringnorm example is rather high, indicating that a change of network topology (bigger net, etc.) or training algorithm (learning rate, etc.) may lower the overall risk. 5 Summary Comparison of of the resample and combine algorithms shows slight advantages for adaptive resampling, but no algorithm dominates the other two. Further im528 F. Leisch and K. Hornik provements should be possible based on a better understanding of the theoretical properties of resample and combine techniques. These issues are currently being investigated. References Breiman, L. (1994). Bagging predictors. Tech. Rep. 421, Department of Statistics, University of California, Berkeley, California, USA. Breiman, 1. (1996a). Bias, variance, and arcing classifiers. Tech. Rep. 460, Statistics Department, University of California, Berkeley, CA, USA. Breiman, L. (1996b). Stacked regressions. Machine Learning, 24,49. Drucker, H. & Cortes, C. (1996). Boosting decision trees. In Touretzky, S., Mozer, M. C., & Hasselmo, M. E. (eds.), Advances in Neural Information Processing Systems, vol. 8. MIT Press. Efron, B. & Tibshira...u, R. J. (1993). An introduction to the bootstrap. Monographs on Statistics and Applied Probability. New York: Chapman & Hall. Freund, Y. & Schapire, R. E. (1995). A decision-theoretic generalization of on-line learning and an application to boosting. Tech. rep., AT&T Bell Laboratories, 600 Mountain Ave, Murray Hill, NJ, USA. Kanaya, F. & Miyake, S. (1991). Bayes statistical behavior and valid generalization of pattern classifying neural networks. IEEE Transactions on Neural Networks, 2(4), 471475. Kohavi, R. & Wolpert, D. H. (1996). Bias plus variance decomposition for zero-one loss. In Machine Learning: Proceedings of the 19th International Conference. Koistinen, P. & Holmstrom, L. (1992). Kernel regression and backpropagation training with noise. In Moody, J. E., Hanson, S. J., & Lippmann, R. P. (eds.), Advances in Neural Information Processing Systems, vol. 4, pp. 1033-1039. Morgan Kaufmann Publishers, Inc. Kong, E. B. & Dietterich, T. G. (1995). Error-correcting output coding corrects bias and variance. In Machine Learning: Proceedings of the 12th International Conference, pp. 313-321. Morgan-Kaufmann. Leisch, F. & Jain, 1. C. (1996). Cross-validation with active pattern selection for neural network classifiers. Submitted to IEEE Transactions on Neural Networks, in Review. Leisch, F., Jain, 1. C., & Hornik, K. (1995). NN classifiers: Reducing the computational cost of cross-validation by active pattern selection. In Artificial Neural Networks and Expert Systems, vol. 2. Los Alamitos, CA, USA: IEEE Computer Society Press. Quinlan, J. R. (1996). Bagging, boosting and C4.5. University of Sydney, Australia. Ripley, B. D. (1996). Pattern recognition and neural networks. Cambridge, UK: Cambridge University Press. Tibshirani, R. (1996a). Bias, variance and prediction error for classification rules. University of Toronto, Canada. Tibshirani, R. (1996b). A comparison of some error estimates for neural network models. Neural Computation, 8(1), 152-163.	1996	116
1,169	Estimating Equivalent Kernels For Neural Networks: A Data Perturbation Approach A. Neil Burgess Department of Decision Science London Business School London, NW1 4SA, UK (N.Burgess@lbs.lon.ac.uk) ABSTRACT We describe the notion of "equivalent kernels" and suggest that this provides a framework for comparing different classes of regression models, including neural networks and both parametric and non-parametric statistical techniques. Unfortunately, standard techniques break down when faced with models, such as neural networks, in which there is more than one "layer" of adjustable parameters. We propose an algorithm which overcomes this limitation, estimating the equivalent kernels for neural network models using a data perturbation approach. Experimental results indicate that the networks do not use the maximum possible number of degrees of freedom, that these can be controlled using regularisation techniques and that the equivalent kernels learnt by the network vary both in "size" and in "shape" in different regions of the input space. 1 INTRODUCTION The dominant approaches within the statistical community, such as multiple linear regression but even extending to advanced techniques such as generalised additive models (Hastie and Tibshirani, 1990), projection pursuit regression (Friedman and Stuetzle, 1981), and classification and regreSSion trees (Breiman et al., 1984), tend to err, when they do, on the high-bias side due to restrictive assumptions regarding either the functional form of the response to individual variables and/or the limited nature of the interaction effects which can be accommodated. Other classes of models, such as multi-variate adaptive regression spline models of high-order (Friedman, 1991), interaction splines (Wahba, 1990) and especially non-parametric regression techniques (HardIe, 1990) are capable of relaxing some or all of these restrictive assumptions, but run the converse risk of suffering high-variance, or "over fitting". A large literature of experimental results suggests that, under the right conditions, the flexibility of neural networks allows them to out-perform other techniques. Where the current understanding is limited, however, is in analysing trained neural networks to understand how the degrees of freedom have been allocated, in a way which allows meaningful comparisons with other classes of models. We propose that the notion of Estimating Equivalent Kernels: A Data Perturbation Approach 383 "equivalent kernels" [ego (Hastie and Tibshirani, 1990)] can provide a unifying framework for neural networks and other classes of regression model, as well as providing important information about the neural network itself. We describe an algorithm for estimating equivalent kernels for neural networks which overcomes the limitations of existing analytical methods. In the following section we describe the concept of equivalent kernels. In Section 3 we describe an algorithm which estimates how the response function learned by the neural network would change if the training data were modified slightly, from which we derive the equivalent kernels for the network. Section 4 provides simulation results for two controlled experiments. Section 5 contains a brief discussion of some of the implications of this work, and highlights a number of interesting directions for further research. A summary of the main points of the paper is presented in Section 6. 2 EQUIVALENT KERNELS Non-parametric regression techniques, such as kernel smoothing, local regression and nearest neighbour regression, can all be expressed in the form: <Xl y(z) = f ((J(z,x).J(x).t(x) dx (1) X=·<Xl where y(z) is the response at the query point z, <p(z. x) is the weighting, or kernel, which is "centred" at z, f(x) is the input density and t(x) is the target function. In finite samples, this is approximated by: n y(xJ = L<f>(x;,xj).tj (2) j=1 and the response at point Xj is a weighted average of the sampled target values across the entire dataset. Furthermore, the response can be viewed as a least squares estimate for y(Xj) because we can write it as a solution to the minimization problem: ~ (r.CjJ(xj,Xj).tj- y(xi)Y (3) ];1 ) We can combine the kernel functions to define the smoother matrix S, given by: <P(Xl,Xl) <P(Xl,X2) s= <P(X2 ,Xl) <P(X2 ,X2) (4) From which we obtain: y=S.t (5) 384 A. N. Burgess Where y = (y(XI), y(X2), ... ,y(xn) )T, and t = (t\, h, ... , tJT is the vector of target values. From the smoother matrix S, we can derive many kinds of important infonnation. The model is represented in tenns of the influence of each observation on the response at each sample point, allowing us to quantify the effect of outliers for instance. It is also possible to calculate the model bias and variance at each sample point [see (Hardie, 1990) for details]. One important measure which we will return to below is the number of degrees of freedom which are absorbed by the model; a number of definitions can be motivated, but in the case of least squares estimators they turn out to be equivalent [see pp 52-55 of (Hastie and Tibshirani, 1990)], perhaps the most intuitive is: dofs = trace( S ) (6) thus a model which is a look up table, i.e. y(Xj) = tj, absorbs all 'n' degrees of freedom, whereas the sample mean, y(Xj) = lin L tj , absorbs only one degree of freedom. The degrees of freedom can be taken as a natural measure of model complexity, which fonnulated with respect to the data itself, rather than to the number of parameters. The discussion above relates only to models which can be expressed in the fonn given by equation (2), i.e. where the "kernel functions" can be computed. Fortunately, many types of parametric models can be "inverted" in this manner, providing what are known as "equivalent kernels". Consider a model of the fonn: (7) i.e. a weighted function of some arbitrary transfonnations of the input variables. In the case of fitting using a least squares approach, then the optimal weights w = ( WI, W2, ... , Wn)T are given by: (8) where <1>+ is the pseudo-inverse of the transfonned data matrix <1>. The network output can then be expressed as: (9) = ~k <P(Xj, Xk) .~ and the cp(Xj, Xk) are then the "equivalent kernels" of the original model which is now in the same fonn as equation (2). Examples of equivalent kernels for different classes of parametric and non-parametric models are given by (Hastie and Tibshirani, 1990) whilst a treatment for Radial Basis Function (RBF) networks is presented in (Lowe, 1995). 3 EQUIVALENT KERNELS FOR NEURAL NETWORKS The analytic approach described above relies on the ability to calculate the optimal weights using the pseudo-inverse of the data matrix. This is only possible if the transfonnations, ~(x) , are fixed functions, as is typically the case in parametric models or single-layer neural networks. However, for a neural network with more than one layer of Estimating Equivalent Kernels: A Data Perturbation Approach 385 adjustable weights, the basis functions are parametrised rather than fixed and are thus themselves a function of the training data. Consequently the equivalent kernels are also dependent on the data, and the problem of finding the equivalent kernels becomes nonlinear. We adopt a solution to this problem which is based on the following observation. In the case where the equivalent kernels are independent of the observed values tj, we notice from equation (2): By; ) = <p(x;,Xj (10) Btj i.e. the basis function <p(Xj, x) is equal to the sensitivity of the response y(Xj) to a small change in the observed value tj. This suggests that we approximate the equivalent kernels by turning the above expression around: (11) where E is a small perturbation of the training data and <p(Xj) is the response of the reoptimised network: If/(X j ) = <p·(x;,x).(tj +e)+ L<p·(x;,Xk)·tk (12) k~j The notation <p. indicates that the new kernel functions derive from the network fitted to perturbed data. Note that this takes into account all of the adjustable parameters in the network. Whereas treating the basis functions as fixed would give simply the number of additive terms in the final layer of the network. Calculating the equivalent kernels in this fashion is a computationally intensive procedure, with the network needing to be retrained after perturbing each point in tum. Note that regularisation techniques such as weight decay should be incorporated within this procedure as with initial training and are thus correctly accounted for by the algorithm. The retraining step is facilitated by using the optimised weights from the unperturbed data, causing the network to re-train from weights which are initially almost optimal (especially if the perturbation is small). 4 SIMULATION RESULTS In order to investigate the practical viability of estimating equivalent kernels using the perturbation approach, we performed a controlled experiment on simulated data. The target function used was the first two periods of a sine-wave, sampled at 41 points evenly spaced between 0 and 47t. This function was estimated using a neural network with a single layer of four sigmoid units, a shortcut connection from input to output, and a linear output unit, trained using standard backpropogation. From the trained network we then estimated the equivalent kernels using the perturbation method described in the previous section. The resulting kernels for points 0, 7t, and 27t are shown in figure 2, below. 386 A. N. Burgess Figure 2: Equivalent Kernels for sine-wave problem As discussed in the previous section, we can combine the estimated kernels to construct a linear smoother. The correlation coefficient between the function reconstructed from the approximated smoother matrix and the original neural network is found to be 0.995. From equation (6) we find that the network contains approx. 8.2 degrees of freedom; this compares to the 10 potential degrees of freedom, and also to the 6 degrees of freedom which we would expect for an equivalent model with fixed transfer functions. Clearly, to some degreee, perturbations in the training data are accommodated by adjustments to the sigmoid functions. Using this approach we can also investigate the effects of weight decay on (a) the ability of the network to reproduce the target function, (b) the number of degrees of freedom absorbed by the network, and (c) the kernel functions themselves. We use a standard quadratic weight decay, leading to a cost function of the form: C = (y - f(x)i + y.LW2 (13) The effect of gradually increaSing the weight decay factor, y, on both network performance and capacity is shown in figure 3(b), below: 15 .15 9 .... -.......-~·. · ~· ..... ;,;.·h················- -·- ---· --- ·-- ----·---·lilJ'11, ... • .... •....... .•••. ••••. ..•.... ...... .,., .........•.•.••...•..••••• .• OM : ::::::::::::::::::::::::::::::::::::: .. :: : :'~:':"'~:: ::::: ::: : :::::: : : , ........ ...... ....... ........ .. ....... . . . .... ........ \ ................. OM " "'" , ............... .. ...... ...... ... . ........... .............. \ ........... .. : :::::1 ::=(=:...~ .... I !::::::::::::: .. ~~:~:~ :: 1 --- ••• • - -- ----- ----------.--.-- • •• • •• -------- - -- --- - -- -- ---.- - . ---.'--. 1(M o ~ Q ~ • m m ~ ~ ~ ~ ~ ~ ! I ~ i ~ ~ ~ ~ 'I m Iii ;:j t _o.aw_ Figure 3: (a) Comparison of network and reconstructed functions with target, and (b) effect of weight decay Looking at figure 3(b) we note that the two curves follow each other very closely. As the weight decay factor is increased, the effective capacity of the network is reduced and the performance drops off accordingly. In one dimension, the main flexibility for the equivalent kernels is one of scale: narrow, concentrated kernels which rely heavily on nearby observations versus broad, diffuse kernels in which the response is conditioned on a larger number of observations. In higher dimensions, however, the real power of neural networks as function estimators lies in the fact that the sensitivity of the estimated network function is itself a flexible Estimating Equivalent Kernels: A Data Perturbation Approach 387 function of the input vector. Viewed from the perspective of equivalent kernels, this property might be expected to manifest itself in a change in the shape of the kernels in different regions of the input space. In order to investigate this effect we applied the perturbation approach in estimating equivalent kernels for a network trained to reproduce a two-dimensional function; the function chosen was a "ring" defined by: z = II ( 1 + 30.( x2 + y2 - 0.5)2) (14) For ease of visualisation the input points were chosen on a regular 15 by 15 grid running between plus and minus one. This function was approximated using a 2(+ 1 )-8-1 network with sigmoidal hidden units and a linear output unit. Selected kernel functions, estimated from this network, are shown in figure 4, below: Figure 4: Equivalent Kernels: approximated using the perturbation method This result clearly shows the changing shape of the kernel functions in different parts of the input space. The function reconstructed from the estimated smoother matrix has a correlation coefficient of 0.987 with the original network function. 5. Discussion The ability to transform neural network regression models into an equivalent kernel representation raises the possibility of harnessing the whole battery of statistical methods which have been developed for non-parametric techniques: model selection procedures, prediction interval estimation, calculation of degrees of freedom, and statistical significance testing amongst others. The algorithm described in this paper raises the possibility of applying these techniques to more-powerful networks with two or more layers of adaptable weights, be they based on sigmoids, radial functions, splines or whatever, albeit at the price of significant computational effort. Another opportunity is in the area of model combination where the added value from combining models in an ensemble is related to the degree of correlation between the different models (Krogh and Vedelsby, 1995). Typically the pointwise correlation between two models will be related to the similarity between their equivalent kernels and so the equivalent kernel approach opens new possibilities for conditionally modifying the ensemble weights without a need for an additional level of learning. The influence-based method for estimating the number of degrees of freedom absorbed by a neural network model, focuses attention on uncertainty in the data itself, rather than taking the indirect route based on uncertainty in the model parameters; in future work 388 A. N. Burgess we propose to investigate the similarities and differences between our approach and those based on the "effective number of parameters" (Moody, 1992) and Bayesian methods (MacKay, 1992). 6. Summary We suggest that equivalent kernels provide an important tool for understanding what neural networks do and how they go about doing it; in particular a large battery of existing statistical tools use information derived from the smoother matrix. The perturbation method which we have presented overcomes the limitations of standard approaches, which are only appropriate for models with a single layer of adjustable weights, albeit at considerable computational expense. It has the added bonus of automatically taking into account the effect of regularisation techniques such as weight decay. The experimental results illustrate the application of the technique to two simple problems. As expected the number of degrees of freedom in the models is found to be related to the amount of weight decay used during training. The equivalent kernels are found to vary significantly in different regions of input space and the functions reconstructed from the estimated smoother matrices closely match the origna! networks. 7. References Breiman, 1., Friedman, J. H., Olshen, R. A, and Stone C. 1., 1984, Classification and Regression Trees, Wadsworth and Brooks/Cole, Monterey. Friedman, J.H. and Stuetzle, W., 1981. Projection pursuit regression. Journal of the American Statistical Association. Vol. 76, pp. 817-823. Friedman, J.H., 1991. Multivariate Adaptive Regression Splines (with discussion). Annals of Statistics. Vol 19, num. 1, pp. 1-141. HardIe, W., 1990. Applied non parametric regression. Cambridge University Press. Hastie, T.J. and Tibshirani, R.J., 1990. Generalised Additive Models. Chapman and Hall, London. Krogh, A, and Vedelsby, 1., New-al network ensembles, cross-validation and active learning, NIPS 7, pp231-238. Lowe, D., 1995, On the use of nonlocal and non positive definite basis functions in radial basis function networks, Proceedings of the Fourth lEE Conference on ArtifiCial Neural Networks, pp. 206-211. MacKay, D. J. C., 1992, A practical Bayesian framework for backprop networks, Neural Computation, 4,448-472. Moody, J. E., 1992, The effective number of parameters: an analysis of generalisation and regularization in nonlinear learning systems, NIPS 4, 847-54, Morgan Kaufmann, San Mateo Wahba, G., 1990, Spline Models for Observational Data. Society for Industrial and Applied Mathematics, Philadelphia.	1996	117
1,170	Monotonicity Hints Joseph Sill Computation and Neural Systems program California Institute of Technology email: joe@cs.caltech.edu Abstract Yaser S. Abu-Mostafa EE and CS Deptartments California Institute of Technology email: yaser@cs.caltech.edu A hint is any piece of side information about the target function to be learned. We consider the monotonicity hint, which states that the function to be learned is monotonic in some or all of the input variables. The application of mono tonicity hints is demonstrated on two real-world problems- a credit card application task, and a problem in medical diagnosis. A measure of the monotonicity error of a candidate function is defined and an objective function for the enforcement of monotonicity is derived from Bayesian principles. We report experimental results which show that using monotonicity hints leads to a statistically significant improvement in performance on both problems. 1 Introduction Researchers in pattern recognition, statistics, and machine learning often draw a contrast between linear models and nonlinear models such as neural networks. Linear models make very strong assumptions about the function to be modelled, whereas neural networks are said to make no such assumptions and can in principle approximate any smooth function given enough hidden units. Between these two extremes, there exists a frequently neglected middle ground of nonlinear models which incorporate strong prior information and obey powerful constraints. A monotonic model is one example which might occupy this middle area. Monotonic models would be more flexible than linear models but still highly constrained. Many applications arise in which there is good reason to believe the target function is monotonic in some or all input variables. In screening credit card applicants, for instance, one would expect that the probability of default decreases monotonically Monotonicity Hints 635 with the applicant's salary. It would be very useful, therefore, to be able to constrain a nonlinear model to obey monotonicity. The general framework for incorporating prior information into learning is well established and is known as learning from hints[l]. A hint is any piece of information about the target function beyond the available input-output examples. Hints can improve the performance oflearning models by reducing capacity without sacrificing approximation ability [2]. Invariances in character recognition [3] and symmetries in financial-market forecasting [4] are some of the hints which have proven beneficial in real-world learning applications. This paper describes the first practical applications of monotonicity hints. The method is tested on two noisy real-world problems: a classification task concerned with credit card applications and a regression problem in medical diagnosis. Section II derives, from Bayesian principles, an appropriate objective function for simultaneously enforcing monotonicity and fitting the data. Section III describes the details and results of the experiments. Section IV analyzes the results and discusses possible future work. 2 Bayesian Interpretation of Objective Function Let x be a vector drawn from the input distribution and Xl be such that \.I • ../... I VJ T 1, Xj = Xj (1) (2) The statement that ! is monotonically increasing in input variable Xi means that for all such x, x' defined as above !(x/) ~ !(x) (3) Decreasing monotonicity is defined similarly. We wish to define a single scalar measure of the degree to which a particular candidate function y obeys monotonicity in a set of input variables. One such natural measure, the one used in the experiments in Section IV, is defined in the following way: Let x be an input vector drawn from the input distribution. Let i be the index of an input variable randomly chosen from a uniform distribution over those variables for which monotonicity holds. Define a perturbation distribution, e.g., U[O,l], and draw ,sXi from this distribution. Define x' such that \.I • ../... I VJ T 1, Xj = Xj (4) X~ = Xi + sgn( i),sXi (5) 636 J. Sill and Y. S. Abu-Mosta/a where sgn( i) = 1 or -1 depending on whether f is monotonically increasing or decreasing in variable i. We will call Eh the monotonicity error of y on the input pair (x, x'). {o Eh -(y(x) - Y(X'))2 y(x') ;::: y(x) y(x') < y(x) (6) Our measure of y's violation of monotonicity is £[Eh], where the expectation is taken with respect to random variables x, i and 8Xi . We believe that the best possible approximation to f given the architecture used is probably approximately monotonic. This belief may be quantified in a prior distribution over the candidate functions implementable by the architecture: (7) This distribution represents the a priori probability density, or likelihood, assigned to a candidate function with a given level of monotonicity error. The probability that a function is the best possible approximation to f decreases exponentially with the increase in monotonicity error. ). is a positive constant which indicates how strong our bias is towards monotonic functions. In addition to obeying prior information, the model should fit the data well. For classification problems, we take the network output y to represent the probability of class c = 1 conditioned on the observation of the input vector (the two possible classes are denoted by 0 and 1). We wish to pick the most probable model given the data. Equivalently, we may choose to maximize log(P(modelldata)). Using Bayes' Theorem, log(P(modelldata)) ex log(P(datalmodel) + log(P(model)) (8) M = L: cmlog(Ym) + (1 - cm)log(l - Ym) - ).£[Eh] (9) m=l For continuous-output regression problems, we interpret y as the conditional mean of the observed output t given the observation of x . If we assume constant-variance gaussian noise, then by the same reasoning as in the classification case, the objective function to be maximized is : M - L (Ym - tm)2 - )'£[Eh] (10) m=l The Bayesian prior leads to a familiar form of objective function, with the first term reflecting the desire to fit the data and a second term penalizing deviation from mono tonicity. Monotonicity Hints 637 3 Experimental Results Both databases were obtained via FTP from the machine learning database repository maintained by UC-Irvine 1. The credit card task is to predict whether or not an applicant will default. For each of 690 applicant case histories, the database contains 15 features describing the applicant plus the class label indicating whether or not a default ultimately occurred. The meaning of the features is confidential for proprietary reasons. Only the 6 continuous features were used in the experiments reported here. 24 of the case histories had at least one feature missing. These examples were omitted, leaving 666 which were used in the experiments. The two classes occur with almost equal frequency; the split is 55%-45%. Intuition suggests that the classification should be monotonic in the features. Although the specific meanings of the continuous features are not known, we assume here that they represent various quantities such as salary, assets, debt, number of years at current job, etc. Common sense dictates that the higher the salary or the lower the debt, the less likely a default is, all else being equal. Monotonicity in all features was therefore asserted. The motivation in the medical diagnosis problem is to determine the extent to which various blood tests are sensitive to disorders related to excessive drinking. Specifically, the task is to predict the number of drinks a particular patient consumes per day given the results of 5 blood tests. 345 patient histories were collected, each consisting of the 5 test results and the daily number of drinks. The "number of drinks" variable was normalized to have variance 1. This normalization makes the results easier to interpret, since a trivial mean-squared-error performance of 1.0 may be obtained by simply predicting for mean number of drinks for each patient, irrespective of the blood tests. The justification for mono tonicity in this case is based on the idea that an abnormal result for each test is indicative of excessive drinking, where abnormal means either abnormally high or abnormally low. In all experiments, batch-mode backpropagation with a simple adaptive learning rate scheme was used 2. Several methods were tested. The performance of a linear perceptron was observed for benchmark purposes. For the experiments using nonlinear methods, a single hidden layer neural network with 6 hidden units and direct input-output connections was used on the credit data; 3 hidden units and direct input-output connections were used for the liver task. The most basic method tested was simply to train the network on all the training data and optimize the objective function as much as possible. Another technique tried was to use a validation set to avoid overfitting. Training for all of the above models was performed by maximizing only the first term in the objective function, i.e., by maximizing the log-likelihood of the data (minimizing training error). Finally, training the networks with the monotonicity constraints was performed, using an approximation to (9) lThey may be obtained as follows: ftp ics.uci.edu. cd pub/machine-Iearning-databases. The credit data is in the subdirectory /credit-screening, while the liver data is in the subdirectory /liver-disorders. 2If the previous iteration resulted in a increase in likelihood, the learning rate was increased by 3%. If the likelihood decreased, the learning rate was cut in half 638 1. Sill and Y. S. Abu-Mostafa and (10). A leave-k-out procedure was used in order to get statistically significant comparisons of the difference in performance. For each method, the data was randomly partitioned 200 different ways (The split was 550 training, 116 test for the credit data; 270 training and 75 test for the liver data). The results shown in Table 1 are averages over the 200 different partitions. In the early stopping experiments, the training set was further subdivided into a set (450 for the credit data, 200 for the liver data) used for direct training and a second validation set (100 for the credit data, 70 for the liver data). The classification error on the validation set was monitored over the entire course of training, and the values of the network weights at the point of lowest validation error were chosen as the final values. The process of training the networks with the monotonicity hints was divided into two stages. Since the meanings of the features were unaccessible, the directions of mono tonicity were not known a priori. These directions were determined by training a linear percept ron on the training data for 300 iterations and observing the resulting weights. A positive weight was taken to imply increasing monotonicity, while a negative weight meant decreasing monotonicity. Once the directions of mono tonicity were determined, the networks were trained with the monotonicity hints. For the credit problem, an approximation to the theoretical objective function (10) was maximized: (13) For the liver problem, objective function (12) was approximated by (14) Eh,n represents the network's monotonicityerror on a particular pair of input vectors x, x'. Each pair was generated according to the method described in Section II. The input distribution was modelled as a joint gaussian with a covariance matrix estimated from the training data. For each input variable, 500 pairs of vectors representing monotonicity in that variable were generated. This yielded a total of N=3000 hint example pairs for the credit problem and N=2500 pairs for the liver problem. A was chosen to be 5000. No optimization of A was attempted; 5000 was chosen somewhat arbitrarily as simply a high value which would greatly penalize non-monotonicity. Hint generalization, i.e. monotonicity test error, was measured by using 100 pairs of vectors for each variable which were not trained on but whose mono tonicity error was calculated. For contrast, monotonicity test error was also monitored for the two-layer networks trained only on the input-output examples. Figure 1 shows test error and monotonicity error vs. training time for the credit data for the networks trained only on the training data (i.e, no hints), averaged over the 200 different data splits. Monotonicity Hints 639 .. o Test Error and Monotonicity Error vs. Iteration Number 0 . 3 r---~-----r----~----r---~-----r----~----r---~----~ 0.25 0 . 2 "testcurve.data" 0 ·'hintcurve.data" + t ~~~--------------~ ~ .. 0.15 ~ 0 . 1 0.05 500 1000 1500 2000 2500 3000 Iteration Number 3500 4000 4500 5000 Figure 1: The violation of monotonicity tracks the overfitting occurring during training The monotonicity error is multiplied by a factor of 10 in the figure to make it more easily visible. The figure indicates a substantial correlation between overfitting and monotonicity error during the course of training. The curves for the liver data look similar but are omitted due to space considerations. Method training error test error hint test error Linear 22.7%± 0.1% 23.7%±0.2% 6-6-1 net 15.2%± 0.1% 24.6% ± 0.3% .005115 6-6-1 net, w/val. 18.8%± 0.2% 23.4% ± 0.3% 6-6-1 net, w /hint 18.7%±0.1% 21.8% ± 0.2% .000020 Table 1: Performance of methods on credit problem The performance of each method is shown in tables 1 and 2. Without early stopping, the two-layer network overfits and performs worse than a linear model. Even with early stopping, the performance of the linear model and the two-layer network are almost the same; the difference is not statistically significant. This similarity in performance is consistent with the thesis of a monotonic target function. A monotonic classifier may be thought of as a mildly nonlinear generalization of a linear classifier. The two-layer network does have the advantage of being able to implement some of this nonlinearity. However, this advantage is cancelled out (and in other cases could be outweighed) by the overfitting resulting from excessive and unnecessary degrees of freedom. When monotonicity hints are introduced, much of this unnecessary freedom is eliminated, although the network is still allowed to implement monotonic nonlinearities. Accordingly, a modest but clearly statistically significant improvement on the credit problem (nearly 2%) results from the introduction of 640 J. Sill and Y. S. Abu-Mosta/a Method training error test error hint test error Linear .802 ± .005 .873 ± .013 5-3-1 net .640 ± .003 .920 ± .014 .004967 5-3-1 net, w/val. .758 ± .008 .871 ± .013 5-3-1 net, w/hint .758± .003 .830 ± .013 .000002 Table 2: Performance of methods on liver problem monotonicity hints. Such an improvement could translate into a substantial increase in profit for a bank. Monotonicity hints also significantly improve test error on the liver problem; 4% more of the target variance is explained. 4 Conclusion This paper has shown that monotonicity hints can significantly improve the performance of a neural network on two noisy real-world tasks. It is worthwhile to note that the beneficial effect of imposing monotonicity does not necessarily imply that the target function is entirely monotonic. If there exist some nonmonotonicities in the target function, then monotonicity hints may result in some decrease in the model's ability to implement this function. It may be, though, that this penalty is outweighed by the improved estimation of model parameters due to the decrease in model complexity. Therefore, the use of monotonicity hints probably should be considered in cases where the target function is thought to be at least roughly monotonic and the training examples are limited in number and noisy. Future work may include the application of monotonicity hints to other real world problems and further investigations into techniques for enforcing the hints. Aclmowledgements The authors thank Eric Bax, Zehra Cataltepe, Malik Magdon-Ismail, and Xubo Song for many useful discussions. References [1] Y. Abu-Mostafa (1990). Learning from Hints in Neural Networks Journal of Complexity 6, 192-198. [2] Y. Abu-Mostafa (1993) Hints and the VC Dimension Neural Computation 4, 278-288 [3] P. Simard, Y. LeCun & J Denker (1993) Efficient Pattern Recognition Using a New Transformation Distance NIPS5, 50-58. [4] Y. Abu-Mostafa (1995) Financial Market Applications of Learning from Hints Neural Networks in the Capital Markets, A. Refenes, ed., 221-232. Wiley, London, UK.	1996	118
1,171	A Convergence Proof for the Softassign Quadratic Assignment Algorithm Anand Rangarajan Department of Diagnostic Radiology Yale University School of Medicine New Haven, CT 06520-8042 e-mail: anand<Onoodle. med. yale. edu Alan Yuille Smith-Kettlewell Eye Institute 2232 Webster Street San Francisco, CA 94115 e-mail: yuille<oskivs.ski. org Eric Mjolsness Dept. of Compo Sci. and Engg. Steven Gold CuraGen Corporation 322 East Main Street Branford, CT 06405 e-mail: gold-steven<ocs. yale. edu Univ. of California San Diego (UCSD) La Jolla, CA 92093-0114 e-mail: emj <Ocs . ucsd. edu Abstract The softassign quadratic assignment algorithm has recently emerged as an effective strategy for a variety of optimization problems in pattern recognition and combinatorial optimization. While the effectiveness of the algorithm was demonstrated in thousands of simulations, there was no known proof of convergence. Here, we provide a proof of convergence for the most general form of the algorithm. 1 Introduction Recently, a new neural optimization algorithm has emerged for solving quadratic assignment like problems [4, 2]. Quadratic assignment problems (QAP) are characterized by quadratic objectives with the variables obeying permutation matrix constraints. Problems that roughly fall into this class are TSP, graph partitioning (GP) and graph matching. The new algorithm is based on the softassign procedure which guarantees the satisfaction of the doubly stochastic matrix constraints (resulting from a "neural" style relaxation of the permutation matrix constraints). While the effectiveness of the softassign procedure has been demonstrated via thousands A Convergence Proof for the Sojtassign Quadratic Assignment Algorithm 621 of simulations, no proof of convergence was ever shown. Here, we show a proof of convergence for the soft assign quadratic assignment algorithm. The proof is based on algebraic transformations of the original objective and on the non-negativity of the Kullback-Leibler measure. A central requirement of the proof is that the softassign procedure always returns a doubly stochastic matrix. After providing a general criterion for convergence, we separately analyze the cases of TSP and graph matching. 2 Convergence proof The deterministic annealing quadratic assignment objective function is written as [4, 5]: ')'~ 2 1~ -- ~Mai + -(3 ~MailogMai (1) 2 . . a~ a~ Here M is the desired N x N permutation matrix. This form of the energy function has a self-amplification term with a parameter,)" two Lagrange parameters J-L and l/ for constraint satisfaction, an x log x barrier function which ensures positivity of Mai and a deterministic annealing control parameter (3. The QAP benefit matrix Cai;bj is preset based on the chosen problem, for example, graph matching or TSP. In the following deterministic annealing pseudocode (30 and (3, are the initial and final values of (3, (3r is the rate at which (3 is increased, IE is an iteration cap and ~ is an N x N matrix of small positive-valued random numbers. Initialize (3 to (30, Mai to ~ + ~ai Begin A: Deterministic annealing. Do A until (3 ~ (3, Begin B: Relaxation. Do B until all Mai converge or number of iterations> IE Qai +- 'Ebj Cai;bjMbj + ')'Mai Begin Softassign: Mai +- exp ((3Qai) Begin C: Sinkhorn. Do C until all Mai converge Update Mai by normalizing the rows: Mai +- 2:Mt- . . 'u . Update Mai by normalizing the columns: Mai +- 2::J:ta, End C End Soft assign End B End A 622 A. Rangarajan, A. Yuille, S. Gold and E. Mjolsness The softassign is used for constraint satisfaction. The softassign is based on Sinkhorn's theorem [4] but can be independently derived as coordinate ascent on the Lagrange parameters 11 and 1/. Sinkhorn's theorem ensures that we obtain a doubly stochastic matrix by the simple process of alternating row and column normalizations. The QAP algorithm above was developed using the graduated assignment heuristic [1] with no proof of convergence until now. We simplify the objective function in (1) by collecting together all terms quadratic in M ai . This is achieved by defining (2) Then we use an algebraic transformation [3] to transform the quadratic form into a more manageable linear form: -- --t min -X(J + _(J2 X2 ( 1) 2 u 2 (3) Application of the algebraic transformation (in a vectorized form) to the quadratic term in (1) yields: Eqap(M, (J, 11, 1/) = L ciI;~jMai(Jbj + ~ L Cinj(Jai(Jbj aibj aibj 1 + L l1a(~ Mai - 1) + ~ l/i(L Mai - 1) + fJ L Mai log Mai (4) a t t a at Extremizing (4) w.r.t. (J, we get '" c(-y) Mb' - '" c(-y) (Jb ' ->.. (J . M . ~ ai;bj J ~ ai;bj J -at at (5) bj bj is a minimum, provided certain conditions hold which we specify below. In the first part of the proof, we show that setting (Jai = Mai is guaranteed to decrease the energy function. Restated, we require that M . ( '" C(-y) M 1 '" C(-y) ) (Jai = ai - argmin - ~ ai;bj aiCJbj + 2 ~ ai;bj(Jai(Jbj aibj aibj (6) If C~l~j is positive definite in the subspace spanned by M, then (Jai = Mai is a minimum of the energy function - :Eaibj C~I~jMaiCJbj + ! :Eaibj Cil~j(JaiCJbj . At this juncture, we make a crucial assumption that considerably Simplifies the proof. Since this assumption is central, we formally state it here: "M is always constrained to be a doubly stochastic matrix." In other words, for our proof of convergence, we require the soft assign algorithm to return a doubly stochastic matrix (as Sinkhorn's theorem guarantees that it will) instead of a matrix which is merely close to being doubly stochastic (based on some reasonable metric). We also require the variable (J to be a doubly stochastic matrix. Since M is always constrained to be a doubly stochastic matrix, cilL is required to be positive definite in the linear subspace of rows and columns of M summing to A Convergence Proof for the Softassign Quadratic Assignment Algorithm 623 one. The value of "f should be set high enough such that ciJ;~j does not have any negative eigenvalues in the subspace spanned by the row and column constraints. This is the same requirement imposed in [5] to ensure that we obtain a permutation matrix at zero temperature. To derive a more explicit criterion for "f, we first define a matrix r in the following manner: def 1, r == IN - -ee N (7) where IN is the N x N identity matrix, e is the vector of all ones and the "prime" indicates a transpose operation. The matrix r has the property that any vector rs with s arbitrary will sum to zero. We would like to extend such a property to cover matrices whose row and column sums stay fixed. To achieve this, take the Kronecker product of r with itself: R def 10. =r'6Jr (8) R has the property that it will annihilate all row and column sums. Form a vector m by concatenating all the columns of the matrix M together into a single column [m = vec(M)]. Then the vector Rm has the equivalent property of the "rows" and "columns" summing to zero. Hence the matrix RC(-Y) R (where C("'() is the matrix equivalent of ciJ;~j) satisfies the criterion of annihilated row and column sums in any quadratic form; m'RC(-Y)Rm = (Rm)'C("'()(Rm). The parameter "f is chosen such that all eigenvalues of RC(-Y) R are positive: "f = - min >'(RCR) + € A (9) where € > 0 is a small quantity. Note that C is the original QAP benefit matrix whereas C("'() is the augmented matrix of (2). We cannot always efficiently compute the largest negative eigenvalue of the matrix RCR. Since the original Cai;bj is four dimensional, the dimensions of RC Rare N 2 x N 2 where N is the number of elements in one set. Fortunately, as we show later, for specific problems it's possible to break up RC R into its constituents thereby making the calculation of the largest negative eigenvalue of RCR more efficient. We return to this point in Section 3. The second part of the proof involves demonstrating that the softassign operation also decreases the objective in (4). (Note that the two Lagrange parameters J-t and 1/ are specified by the softassign algorithm [4]). M = Softassign(Q,,B) where Qai = 2: Ci7;~jabj bj (10) Recall that the step immediately preceding the softassign operation sets a ai Mai. We are therefore justified in referring to aai as the "old" value of M ai . For convergence, we have to show that Eqap(a, a) 2: Eqap(M, a) in (4). Minimizing (4) w.r.t. Mai, we get (11) 624 A. Rangarajan, A. Yuille, S. Gold and E. Mjolsness From (11), we see that ~ 2;: Mai log Mai = ~ C~IitjMai(Jbj - 2: /la 2: Mai - 2: IIi 2: Mai - ~ 2: Mai at atbJ a t t a at (12) and ~ 2;: (Jai log Mai = ~ C~J;tj(JaWbj - 2: /la ~ (Jai - ~ IIi 2: (Jai - ~ ~ (Jai (13) at atbJ a t t a at From (12) and (13), we get (after some algebraic manipulations) Eqap((J,(J) Eqap(M,(J) = - 2: C~Iitj(JaWbj 1 1 + ~ ~ (Jai log (Jai ~ ~ Mai log Mai aibj at at by the non-negativity of the Kullback-Leibler measure. We have shown that the change in energy after (J has been initialized with the "old" value of M is nonnegative. We require that (J and M are always doubly stochastic via the action of the softassign operation. Consequently, the terms involving the Lagrange parameters /l and II can be eliminated from the energy function (4). Setting (J = M followed by the softassign operation decreases the objective in (4) after excluding the terms involving the Lagrange parameters. We summarize the essence of the proof to bring out the salient points. At each temperature, the quadratic assignment algorithm executes the following steps until convergence is established. Step 1: (Jai +- M ai . Step 2: Step 2a: Qai +- L:bj C~Iiti(Jbj. Step 2b: M +- Softassign(Q,,8). Return to Step 1 until convergence. Our proof is based on demonstrating that an appropriately designed energy function decreases in both Step 1 and Step 2 (at fixed temperature). This energy function is Equation (4) after excluding the Lagrange parameter terms. Step 1: Energy decreases due to the positive definiteness of C~Iitj in the linear subspace spanned by the row and column constraints. 1 has to be set high enough for this statement to be true. Step 2: Energy decreases due to the non-negativity of the Kullback-Leibler measure and due to the restriction that M (and (J) are doubly stochastic. A Convergence Prooffor the Softassign Quadratic Assignment Algorithm 625 3 Applications 3.1 Quadratic Assignment The QAP benefit matrix is chosen such that the softassign algorithm will not converge without adding the, term in (1). To achieve this, we randomly picked a unit vector v of dimension N2. The benefit matrix C is set to -vv'. Since C has only one negative eigenvalue, the softassign algorithm cannot possibly converge. We ran the softassign algorithm with f30 = 1, (3r = 0.9 and, = O. The energy difference plot on the left in Figure 1 shows the energy never decreasing with increasing iteration number. Next, we followed the recipe for setting, exactly as in Section 2. After projecting C into the subspace of the row and column constraints, we calculated the largest negative eigenvalue of the matrix RCR which turned out to be -0.8152. We set, to 0.8162 (€ = 0.001) and reran the softassign algorithm. The energy difference plot shows (Figure 1) that the energy never increases. We have shown that a proper choice of , leads to a convergent algorithm. g-O.5 ~ -1 :s:: U ~-1 .5 2! Ql -2 -2.5'-----~--~--......J o 10 20 30 iterations 0.8 gO.6 !!! Ql ;g 0.4 >~ Ql Iii 0.2 0 0 20 40 60 80 iterations Figure 1: Energy difference plot. Left: , = 0 and Right: , = 0.8162. While the change in energy is always negative when, = 0, it is always non-negative when , = 0.8162. The negative energy difference (on the left) implies that the energy function increases whereas the non-negative energy difference (on the right) implies that the energy function never increases. 3.2 TSP The TSP objective function is written as follows: Given N cities, Etsp(M) = L dijMaiM(aEIH)j = trace(DM'T M) aij (14) where the symbol EB is used to indicate that the summation in (14) is taken modulo N, dij (D) is the inter-city distance matrix and M is the desired permutation matrix. T is a matrix whose (i,j)th entry is 6(i$1)j (6ij is the Kronecker delta function). Equation (14) is transformed into the m'Cm form: Etsp(m) = trace(m'(D (9 T)m) (15) where m = vec(M). We identify our general matrix C with -2D (9 T. 626 A. Rangarajan, A. Yuille, S. Gold and E. Mjolsness For convergence, we require the largest eigenvalue of -RCR = 2(r 0 r)(D 0 T)(r 0 r) = 2(rDr) 0 (rTr) = 2(rDr) 0 (rT) (16) The eigenvalues of rT are bounded by unity. The eigenvalues of r Dr will depend on the form of D. Even in Euclidean TSP the values will depend on whether the Euclidean distance or the distance squared between the cities is used. 3.3 Graph Matching The graph matching objective function is written as follows: Given Nl and N2 node graphs with adjacency matrices G and 9 respectively, 1 Egm(M) = -2 L CaiibjMai Mbj (17) aibj where Caiibj = 1 - 31Gab gijl is the compatibility matrix [1]. The matching constraints are somewhat different from TSP due to the presence of slack variables [1]. This makes no difference however to our projection operators. We add an extra row and column of zeros to 9 and G in order to handle the slack variable case. Now Gis (Nl + 1) X (Nl + 1) and 9 is (N2 + 1) X (N2 + 1). Equation (17) can be readily transformed into the m'Cm form. Our projection apparatus remains unchanged. For convergence, we require the largest negative eigenvalue of RC R. 4 Conclusion We have derived a convergence proof for the softassign quadratic assignment algorithm and specialized to the cases of TSP and graph matching. An extension to graph partitioning follows along the same lines as graph matching. Central to our proof is the requirement that the QAP matrix M is always doubly stochastic. As a by-product, the convergence proof yields a criterion by which the free self-amplification parameter I is set. We believe that the combination of good theoretical properties and experimental success of the softassign algorithm make it the technique of choice for quadratic assignment neural optimization. References [1] S. Gold and A. Rangarajan. A graduated assignment algorithm for graph matching. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18(4):377~388, 1996. [2] S. Gold and A. Rangarajan. Softassign versus softmax: Benchmarks in combinatorial optimization. In Advances in Neuml Information Processing Systems 8, pages 626-632. MIT Press, 1996. [3] E. Mjolsness and C. Garrett. Algebraic transformations of objective functions. Neuml Networks, 3:651-669, 1990. [4] A. Rangarajan, S. Gold, and E. Mjolsness. A novel optimizing network architecture with applications. Neuml Computation, 8(5):1041~1060, 1996. [5] A. L. Yuille and J. J. Kosowsky. Statistical physics algorithms that converge. Neuml Computation, 6(3):341~356, May 1994.	1996	119
1,172	Compositionality, MDL Priors, and Object Recognition Elie Bienenstock (elie@dam.brown.edu) Stuart Geman (geman@dam.brown.edu) Daniel Potter (dfp@dam.brown.edu) Division of Applied Mathematics, Brown University, Providence, RI 02912 USA Abstract Images are ambiguous at each of many levels of a contextual hierarchy. Nevertheless, the high-level interpretation of most scenes is unambiguous, as evidenced by the superior performance of humans. This observation argues for global vision models, such as deformable templates. Unfortunately, such models are computationally intractable for unconstrained problems. We propose a compositional model in which primitives are recursively composed, subject to syntactic restrictions, to form tree-structured objects and object groupings. Ambiguity is propagated up the hierarchy in the form of multiple interpretations, which are later resolved by a Bayesian, equivalently minimum-description-Iength, cost functional. 1 Bayesian decision theory and compositionaiity In his Essay on Probability, Laplace (1812) devotes a short chapter-his "Sixth Principle" -to what we call today the Bayesian decision rule. Laplace observes that we interpret a "regular combination," e.g., an arrangement of objects that displays some particular symmetry, as having resulted from a "regular cause" rather than arisen by chance. It is not, he argues, that a symmetric configuration is less likely to happen by chance than another arrangement. Rather, it is that among all possible combinations, which are equally favored by chance, there are very few of the regular type: "On a table we see letters arranged in this order, Constantinople, and we judge that this arrangement is not the result of chance, not because it is less possible than the others, for if this word were not employed in any language Compositionality, MDL Priors, and Object Recognition 839 we should not suspect it came from any particular cause, but this word being in use amongst us, it is incomparably more probable that some person has thus arranged the aforesaid letters than that this arrangement is due to chance." In this example, regularity is not a mathematical symmetry. Rather, it is a convention shared among language users, whereby Constantinople is a word, whereas Jpctneolnosant, a string containing the same letters but arranged in a random order, is not. Central in Laplace's argument is the observation that the number of words in the language is smaller, indeed "incomparably" smaller, than the number of possible arrangements of letters. Indeed, if the collection of 14-letter words in a language made up, say, half of all 14-letter strings- a rich language indeed-we would, upon seeing the string Constantinople on the table, be far less inclined to deem it a word, and far more inclined to accept it as a possible coincidence. The sparseness of allowed combinations can be observed at all linguistic articulations (phonetic-syllabic, syllabic-lexical, lexical-syntactic, syntactic-pragmatic, to use broadly defined levels), and may be viewed as a form of redundancy-by analogy to error-correcting codes. This redundancy was likely devised by evolution to ensure efficient communication in spite of the ambiguity of elementary speech signals. The hierarchical compositional structure of natural visual scenes can also be thought of as redundant: the rules that govern the composition of edge elements into object boundaries, of intensities into surfaces etc., all the way to the assembly of 2-D projections of named objects, amount to a collection of drastic combinatorial restrictions. Arguably, this is why in all but a few-generally hand-crafted-cases, natural images have a unique high-level interpretation in spite of pervasive low-level ambiguity-this being amply demonstrated by the performances of our brains. In sum, compositionality appears to be a fundamental aspect of cognition (see also von der Malsburg 1981, 1987; Fodor and Pylyshyn 1988; Bienenstock, 1991, 1994, 1996; Bienenstock and Geman 1995). We propose here to account for mental computation in general and scene interpretation in particular in terms of elementary composition operations, and describe a mathematical framework that we have developed to this effect. The present description is a cursory one, and some notions are illustrated on two simple examples rather than formally defined-for a detailed account, see Geman et al. (1996), Potter (1997). The binary-image example refers to an N x N array of binary-valued pixels, while the Laplace-Table example refers to a one-dimensional array of length N, where each position can be filled with one of the 26 letters of the alphabet or remain blank. 2 Labels and composition rules The objects operated upon are denoted Wi, i = 1,2, ... , k. Each composite object W carries a label, I = L(w), and the list of its constituents, (Wt,W2,·· .). These uniquely determine w, so we write W = I (WI, W2, .•. ) . A scene S is a collection of primitive objects. In the binary-image case, a scene S consists of a collection of black pixels in the N x N array. All these primitives carry the same label, L(w) = p (for "Point"), and a parameter 7r(w) which is the position in the image. In Laplace's Table, a scene S consists of an arrangement of characters on the table. There are 26 primitive labels, "A", "B" , ... , "Z" , and the parameter of a primitive W is its position 1 ~ 7r(w) ~ N (all primitives in such a scene must have different positions). An example of a composite W in the binary-image case is an arrangement composed 840 E. Bienenstock. S. Geman and D. Potter of a black pixel at any position except on the rightmost column and another black pixel to the immediate right of the first one. The label is "Horizontal Linelet," denoted L(w) = hl, and there are N(N - 1) possible horizontallinelets. Another non-primitive label, "Vertical Linelet," or vl, is defined analogously. An example of a composite W for Laplace's Table is an arrangement of 14 neighboring primitives carrying the labels "G", "0", "N", "S", ... , "E" in that order, wherever that arrangement will fit. We then have L(w) = Ganstantinople, and there are N - 13 possible Constantinople objects. The composition rule for label type 1 consists of a binding junction, B" and a set of allowed binding-function values, or binding support, S,: denoting by 0 the set of all objects in the model, we have, for any WI, ' .. ,Wk E 0, B, (WI. ... ,Wk) E s, ¢:} l(WI"" ,Wk) E O. In the binary-image example, Bhl(WI,W2) = Bv,(WI,W2) = (L(WI),L(W2),7I'(W2) -7I'(WI)), Sh' = {(P,p,(I,O))} and Sv' = {(p,p,(O,I))} define the hl- and vl-composition rules, p+p -+ hl and p+p -+ vl. In Laplace's Table, G + 0+· .. + E -+ Ganstantinpole is an example of a 14-ary composition rule, where we must check the label and position of each constituent. One way to define the binding function and support for this rule is: B(WI, ' " ,WI4) = (L(WI),' " ,L(WI4), 71'(W2) 71'(Wt} , 71'(W3) - 71'(WI),"', 71'(W14) - 71'(WI)) and S = (G,"', E, 1,2"",13). We now introduce recursive labels and composition rules: the label of the composite object is identical to the label of one or more of its constituents, and the rule may be applied an arbitrary number of times, to yield objects of arbitrary complexity. In the binary-image case, we use a recursive label c, for Curve, and an associated binding function which creates objects of the form hl + p -+ c, vl + p -+ c, c + p -+ c, p + hl -+ c, p + vl -+ c, p + c -+ c, and c + c -+ c. The reader may easily fill in the details, i.e., define a binding function and binding support which result in "c" -objects being precisely curves in the image, where a curve is of length at least 3 and may be self-intersecting. In the previous examples, primitives were composed into compositions; here compositions are further composed into more complex compositions. In general, an object W is a labeled tree, where each vertex carries the name of an object, and each leaf is associated with a primitive (the association is not necessarily one-to-one, as in the case of a self-intersecting curve). Let M be a model-Le., a collection of labels with their binding functions and binding supports-and 0 the set of all objects in M . We say that object W E o covers S if S is precisely the set of primitives that make up w's leaves. An interpretation I of S is any finite collection of objects in 0 such that the union of the sets of primitives they cover is S. We use the convention that, for all M and S, 10 denotes the trivial interpretation, defined as the collection of (unbound) primitives in S. In most cases of interest, a model M will allow many interpretations for a scene S . For instance, given a long curve in the binary-image model, there will be many ways to recursively construct a "c"-labeled tree that covers exactly that curve. 3 The MDL formulation In Laplace's Table, a scene consisting of the string Constantinople admits, in addition to 10 , the interpretation II = {WI}, where WI is a "G anstantinople" object. We wish to define a probability distribution D on interpretations such that D(I1 ) » D(Io), in order to realize Laplace's "incomparably more probable". Our Compositionality, MDL Priors, and Object Recognition 841 definition of D will be motivated by the following use of the Minimum Description Length (MDL) principle (Rissanen 1989). Consider a scene S and pretend we want to transmit S as quickly as possible through a noiseless channel, hence we seek to encode it as efficiently as possible, i.e., with the shortest possible binary code c. We can always use the trivial interpretation 10: the codeword c(Io) is a mere list of n locations in S. We need not specify labels, since there is only one primitive label in this example. The length, or cost, of this code for S is Ic(Io)1 = nlog2(N 2 ). Now however we want to take advantage of regularities, in the sense of Laplace, that we expect to be present in S. We are specifically interested in compositional regularities, where some arrangements that occur more frequently than by chance can be interpreted advantageously using an appropriate compositional model M. Interpretation I is advantageous if Ic(I)1 < Ic(Io)l. An example in the binary-image case is a linelet scene S. The trivial encoding of this scene costs us Ic(Io)1 = 2[log2 3+ log2(N2)] bits, whereas the cost of the compositional interpretation II = {wI} is Ic(Idl = log2 3+log2(N(N -1)), where WI is an hI or vl object, as the case may be. The first log23 bits encode the label L(WI) E {p, hi, vi}, and the rest encodes the position in the image. The compositional {p, hl, vl} model is therefore advantageous for a linelet scene, since It affords us a gain in encoding cost of about 2log2 N bits. In general, the gain realized by encoding {w} = {I (WI, W2)} instead of {WI, W2} may be viewed as a binding energy, measuring the affinity that WI and W2 exhibit for each other as they assemble into w. This binding energy is c, = IC(WI)I + IC(W2)1 I c( I (WI, W2) ) I, and an efficient M is one that contains judiciously chosen cost-saving composition rules. In effect, if, say, linelets were very rare, we would be better off with the trivial model. The inclusion of non-primitive labels would force us to add at least one bit to the code of every object-to specify its label-and this would increase the average encoding cost, since the infrequent use of non-primitive labels would not balance the extra small cost incurred on primitives. In practical applications, the construction of a sound M is no trivial issue. Note however the simple rationale for including a rule such as p + p --7 hl. Giving ourselves the label hi renders redundant the independent encoding of the positions of horizontally adjacent pixels. In general, a good model should allow one to hierarchically compose with each other frequently occurring arrangements of objects. This use of MDL leads in a straightforward way to an equivalent Bayesian formulation. Setting P'(w) = 2- lc(w)lj L:w'EO 2-lc(w')I yields a probability distribution P' on n for which c is approximately a Shannon code (Cover and Thomas 1991). With this definition, the decision to include the label hl-or the label Con8tantinoplewould be viewed, in principle, as a statement about the prior probability of finding horizontal linelets-or Constantinople strings-in the scene to be interpreted. 4 The observable-measure formulation The MDL formulation however has a number of shortcomings; in particular, computing the binding energy for composite objects can be problematic. We outline now an alternative approach (Geman et al. 1996, Potter 1997), where a probability distribution P(w) on n is defined through a family of observable measures Q,. These measures assign probabilities to each possible binding-function value, s E S" and also to the primitives. We require L:'EM L:sEsr Q,(8) = 1, where the notion of binding function has been extended to primitives via Bprim (w) = 7r(w) for primitive 842 E. Bienenstoc/c, S. Geman and D. Potter w. The probabilities induced on 0 by Q, are given by P(w) = Qprim(Bprim(w)) for a primitive w, and P(w) = Q,(B,(WI,W2))P2(WI,W2IB,(WI,W2)) for a composite object w = l(wI, W2).1 Here p 2 = P X P is the product probability, i.e., the free, or not-bound, distribution for the pair (WI, W2) E 0 2. For instance, with C + ... + E -? Canstantinople, p 14(WI,W2,'" ,w14IBcons ... (W1, ... ,W14) = (C, 0,···,13)) is the conditional probability of observing a particular string Constantinople, under the free distribution, given that (WI, ... , W14) constitutes such a string. With the reasonable assumption that, under Q, primitives are uniformly distributed over the table, this conditional probability is simply the inverse of the number of possible Constantinople strings, Le., 1/(N - 13). The binding energy, defined, by analogy to the MDL approach, as [, = log2(P(w)/(P(wdP(w2))), now becomes [, = log2(Q,(B,(wI,w2))) - log2(P x P(B'(Wl,W2)))' Finally, if I is the collection of all finite interpretations / c 0, we define the probability of / E I as D(/) = IIwElP(w)/Z, with Z = L:I'EI IIwEl'P(w), Thus, the probability of an interpretation containing several free objects is obtained by assuming that these objects occurred in the scene independently of each other. Given a scene S, recognition is formulated as the task of maximizing D over all the l's in I that are interpretations of S. We now illustrate the use of D on our two examples. In the binary-image example with model M = {p, hi, vi}, we use a parameter q, 0 ~ q ~ 1, to adjust the prior probability of linelets. Thus, Qprim(Bprim(W)) = (1 - q)/N2 for primitives, and Qh'«P,p,O, 1)) = Qv'«P,p, 1,0)) = q/2 for linelets. It is easily seen that regardless of the normalizing constant Z, the binding energy of two adjacent pixels into a linelet is [h' = [v, = log2(q/2) - log2[{lNf N(N - 1)]. Interestingly, as long as q =1= 0 and q =1= 1, the binding energy, for large N, is approximately 2log2 N, which is independent of q. Thus, the linelet interpretation is "incomparably" more likely than the independent occurrence of two primitives at neighboring positions. We leave it to the reader to construct a prior P for the model {p, hl, vI, c}, e.g. by distributing the Q-mass evenly between all composition rules. Finally, in Laplace's Table, if there are M equally likely non-primitive labels-say city names-and q is their total mass, the binding energy for Constantinople is [Cons ... = log2 M(r! -13) log2[~~.&]14, and the "regular" cause is again "incomparably" more likely. There are several advantages to this reformulation from codewords into probabilities using the Q-parameters. First, the Q-parameters can in principle be adjusted to better account for a particular world of images. Second, we get an explicit formula for the binding energy, (namely log2 (Q / P x P)). Of course, we need to evaluate the product probability P x P, and this can be highly non-trivial-one approach is through sampling, as demonstrated in Potter (1997). Fi~ally, this formulation is well-suited for parameter estimation: the Q's, which are the parameters of the distribution P, are indeed observables, Le., directly available empirically. 5 Concluding remarks The approach described here was applied by X. Xing to the recognition of "online" handwritten characters, using a binary-image-type model as above, enriched IThis is actually an implicit definition. Under reasonable conditions, it is well definedsee Geman et al. (1996). Compositionality, MDL Priors, and Object Recognition 843 with higher-level labels including curved lines, straight lines, angles, crossings, Tjunctions, L-junctions {right angles}, and the 26 letters of the alphabet. In such a model, the search for an optimal solution cannot be done exhaustively. We experimented with a number of strategies, including a two-step algorithm which first generates all possible objects in the scene, and then selects the "best" objects, Le., the objects with highest total binding energy, using a greedy method, to yield a final scene interpretation. (The total binding energy of W is the sum of the binding energies £, over all the composition rules I used in the composition of w. Equivalently, the total binding energy is the log-likelihood ratio log2{P{w}/IIi P{Wi)), where the product is taken over all the primitives Wi covered by w.} The first step of the algorithm typically results in high-level objects partly overlapping on the set of primitives they cover, i.e., competing for the interpretation of shared primitives. Ambiguity is thus propagated in a "bottom-up" fashion. The ambiguity is resolved in the second "top-down" pass, when high-level composition rules are used to select the best compositions, at all levels including the lower ones. A detailed account of our experiments will be given elsewhere. We found the results quite encouraging, particularly in view of the potential scope of the approach. In effect, we believe that this approach is in principle capable of addressing unrestricted vision problems, where images are typically very ambiguous at lower levels for a variety of reasons-including occlusion and mutual overlap of objects-hence purely bottom-up segmentation is impractical. Turning now to biological implications, note that dynamic binding in the nervous system has been a subject of intensive research and debate in the last decade. Most interesting in the present context is the suggestion, first clearly articulated by von der Malsburg {1981}, that composition may be performed thanks to a dual mechanism of accurate synchronization of spiking activity-not necessarily relying on periodic firing-and fast reversible synaptic plasticity. If there is some neurobiological truth to the model described in the present paper, the binding mechanism proposed by von der Malsburg would appear to be an attractive implementation. In effect, the use of fine temporal structure of neural activity opens up a large realm of possible high-order codes in networks of neurons. In the present model, constituents always bind in the service of a new object, an operation one may refer to as triangular binding. Composite objects can engage in further composition, thus giving rise to arbitrarily deep tree-structured constructs. Physiological evidence of triangular binding in the visual system can be found in Sillito et al. {1994}; Damasio {1989} describes an approach derived from neuroanatomical data and lesion studies that is largely consistent with the formalism described here. An important requirement for the neural representation of the tree-structured objects used in our model is that the doing and undoing of links operating on some constituents, say Wi and W2, while affecting in some useful way the high-order patterns that represent these objects, leaves these patterns, as representations of Wi and W2, intact. A family of biologically plausible patterns that would appear to satisfy this requirement is provided by synfire patterns {Abeles 1991}. We hypothesized elsewhere {Bienenstock 1991, 1994, 1996} that synfire chains could be dynamically bound via weak synaptic couplings; such couplings would synchronize the wave-like activities of two synfire chains, in much the same way as coupled oscillators lock 844 E. Bienenstock, S. Geman and D. Potter their phases. Recursiveness of compositionality could, in principle, arise from the further binding of these composite structures. Acknow ledgements Supported by the Army Research Office (DAAL03-92-G-0115), the National Science Foundation (DMS-9217655), and the Office of Naval Research (N00014-96-1-0647). References Abeles, M. (1991) Corticonics: Neuronal circuits of the cerebral cortex, Cambridge University Press. Bienenstock, E. (1991) Notes on the growth of a composition machine, in Proceedings of the Royaumont Interdisciplinary Workshop on Compositionality in Cognition and Neural Networks-I, D. Andler, E. Bienenstock, and B. Laks, Eds., pp. 25--43. (1994) A Model of Neocortex. Network: Computation in Neural Systems, 6:179-224. (1996) Composition, In Brain Theory: Biological Basis and Computational Principles, A. Aertsen and V. Braitenberg eds., Elsevier, pp 269-300. Bienenstock, E., and Geman, S. (1995) Compositionality in Neural Systems, In The Handbook of Brain Theory and Neural Networks, M.A. Arbib ed., M.I.T./Bradford Press, pp 223-226. Cover, T.M., and Thomas, J.A. (1991) Elements of Information Theory, Wiley and Sons, New York. Damasio, A. R. (1989) Time-locked multiregional retroactivation: a systems-level proposal for the neural substrates of recall and recognition, Cognition, 33:25-62. Fodor, J.A., and Pylyshyn, Z.W. (1988) Connectionism and cognitive architecture: a critical analysis, Cognition, 28:3-71. Geman, S., Potter, D., and Chi, Z. (1996) Compositional Systems, Technical Report, Division of Applied Mathematics, Brown University. Laplace, P.S. (1812) Esssai philosophique sur les probabiliUs. Translation of Truscott and Emory, New York, 1902. Potter, D. (1997) Compositional Pattern Recognition, PhD Thesis, Division of Applied Mathematics, Brown University, In preparation. Rissanen, J. (1989) Stochastic Complexity in Statistical Inquiry World Scientific Co, Singapore. Sillito, A.M., Jones, H.E, Gerstein, G.L., and West, D.C. (1994) Feature-linked synchronization of thalamic relay cell firing induced by feedback from the visual cortex, Nature, 369: 479-482 von der Malsburg, C. (1981) The correlation theory of brain function. Internal report 81-2, Max-Planck Institute for Biophysical Chemistry, Dept. of Neurobiology, Gottingen, Germany. (1987) Synaptic plasticity as a basis of brain organization, in The Neural and Molecular Bases of Learning (J.P. Changeux and M. Konishi, Eds.), John Wiley and Sons, pp. 411--432.	1996	12
1,173	Removing Noise in On-Line Search using Adaptive Batch Sizes Genevieve B. Orr Department of Computer Science Willamette University 900 State Street Salem, Oregon 97301 gorr@willamette.ed-u Abstract Stochastic (on-line) learning can be faster than batch learning. However, at late times, the learning rate must be annealed to remove the noise present in the stochastic weight updates. In this annealing phase, the convergence rate (in mean square) is at best proportional to l/T where T is the number of input presentations. An alternative is to increase the batch size to remove the noise. In this paper we explore convergence for LMS using 1) small but fixed batch sizes and 2) an adaptive batch size. We show that the best adaptive batch schedule is exponential and has a rate of convergence which is the same as for annealing, Le., at best proportional to l/T. 1 Introduction Stochastic (on-line) learning can speed learning over its batch training particularly ,,,,hen data sets are large and contain redundant information [M0l93J. However, at late times in learning, noise present in the weight updates prevents complete convergence from taking place. To reduce the noise, the learning rate is slowly decreased (annealed{ at late times. The optimal annealing schedule is asymptotically proportional to T where t is the iteration [GoI87, L093, Orr95J. This results in a rate of convergence (in mean square) that is also proportional to t. An alternative method of reducing the noise is to simply switch to (noiseless) batch mode when the noise regime is reached. However, since batch mode can be slow, a better idea is to slowly increase the batch size starting with 1 (pure stochastic) and slowly increasing it only "as needed" until it reaches the training set size (pure batch). In this paper we 1) investigate the convergence behavior of LMS when Removing Noise in On-Line Search using Adaptive Batch Sizes 233 using small fixed batch sizes, 2) determine the best schedule when using an adaptive batch size at each iteration, 3) analyze the convergence behavior of the adaptive batch algorithm, and 4) compare this convergence rate to the alternative method of annealing the learning rate. Other authors have approached the problem of redundant data by also proposing techniques for training on subsets of the data. For example, Pluto [PW93] uses active data selection to choose a concise subset for training. This subset is slowly added to over time as needed. Moller [1'10193] proposes combining scaled conjugate gradient descent (SCG) with \ .... hat he refers to as blocksize updating. His algorithm uses an iterative approach and assumes that the block size does not vary rapidly during training. In this paper, we take the simpler approach of just choosing exemplars at mndom at each iteration. Given this, we then analyze in detail the convergence behavior. Our results are more of theoretical than practical interest since the equations we derive are complex functions of quantities such as the Hessian that are impractical to compute. 2 Fixed Batch Size In this section we examine the convergence behavior for LMS using a fixed batch siL~e. vVe assume that we are given a large but finite sized training set T == {Zj == (Xi, dd }~1 where Xi E nm is the ith input and dj En is the corresponding target. \Ve further assume that the targets are generated using a signal plus noise model so that we can write dj = w; Xj + fj (1) where w. E nm is the optimal weight vector and the €j is zero mean noise. Since the training set is assumed to be large we take the average of fj and Xj€j over the training set to be approximately zero. Note that we consider only the problem of optimization of the w over the training set and do not address the issue of obtaining good generalization over the distribution from which the training set was drawn. At each iteration, we assume that exactly n samples are randomly drawn without Teplacement from T where 1 S n S N. vVe denote this batch of size n drawn at time t by Bn(t) == {ZkJ~l' When n = 1 we have pure on-line training and when /I = lV we have pure batch. \Ve choose to sample without replacement so that as the batch size is increased, we have a smooth transition from on-line to batch. FoI' L1'1S, the squared error at iteration t /OT' a batch of size n is (2) and where Wt E nm is the current weight in the network. The update weight equation is then Wt+l = Wt /.L &~B" where /.L is the fixed learning rate. Rewriting UWt this in terms of the weight error v == W w. and defining gBn,t == 8£Bn(t)/8vt we obtain (3) Convergence (in mean square) to (,v'. can be characterized by the rate of change of the average squared norm of the weight error E[v2] where v2 == vT v. From (3) we obtain an expression for V;+l in terms of Vt, , ,2 v2 2 ,Tg + 2 g2 Vt+1 = t /.LVt B,,,t /JBn,t· (4) 234 G. B. Orr To compute the expected value of Vi+l conditioned on Vt we can average the right side of (4) over all possible ways that the batch Bn (t) can be chosen from the N training examples. In appendix A, we show that (gB,,,t)B (gi ,t)N (5) 2 N-n 2 (n-1)N 2 (gBn,t)B = n(N -1) (gi,t)N + (N -1)n (gi,t}N (6) where (. ) N denotes average over all examples in the training set, (.) B denotes average over the possible batches drawn at time t, and gi,t == O£(Zi)/OVt. The averages over the entire training set are N N ~ L O£(Zi) = - L fiXi - vT XiXi = RVt N i=l OVt i=l = (7) N ~ L(fiXi - vT XiXj)T (fjXj - vT XjXj) = u;('I'r R) + vT SVt i=l = (8) where R == (xxT)N' S == (xxTxxT)N 1 , u; == (f2) and (Tr R) is the trace of R. These equations together with (5) and (6) in (4) gives the expected value of Vt+l conditioned on Vt . 2 T { 2 (N(n - 1) 2 N - n )} 1J2 U;(Tr R)(N - n) (Vt+tlVt) = Vt J- 2IJR + IJ (N _ 1)n R + (N _ 1)n S Vt + n(N -1) . (9) Note that this reduces to the standard stochastic and batch update equations when n = 1 and n = N, respectively. 2.0.1 Special Cases: 1-D solution and Spherically Symmetric In I-dimension we can average over Vt in (9) to give (Vi+l) = a(vi) + (3 where 2 (N(n - 1) 2 N - n ) a = 1 - 211R + I-l (N _ l)n R + (N _ 1)n S , (10) and where Rand S simplifY to R = (x2)N' S = (x 4 ) N. This is a difference equation which can be solved exactly to give 1- at-to (vi) = a t - tO (v5) + 1- a (3 (12) where (v5) is the expected squared weight error at the initial time to. Figure la compares equation (12) with simulations of 1-D LMS with gaussian inputs for N = 1000 and batch sizes n = 10, 100, and 500. As can be seen, the agreement is good. Note that (v 2) decreases exponentially until flattening out. The equilibrium value can be computed from (12) by setting t = 00 (assuming lal < 1) to give ( 2) (3 WT;R(N - n) (13) v 00 = 1- (~ = 2Rn(N - 1) -1J(N(n - 1)R2 + (N - n)S)· Note that (v 2 )00 decreases as n increases and is zero only if n = N. 1 For real zero mean gaussian inputs, we can apply the gaussian moment factoring theorem [Hay91] which states that (XiXjXkXI}N = RijRkl + RikRjl + RilRjk where the subscripts on x denote components of x. From this, we find that S = (Trace R)R+ 2R2. Removing Noise in On-Line Search using Adaptive Batch Sizes E[ v"2] 1 n=10 O.OOl!-----:":"iil'~-----E[ v"2) 1 0.1 0.01 235 n=10 n,.l00 0.001 0 't (a) 0.0001 n,.500 (b) 0 Figure 1: Simulations(solid) vs Theoretical (dashed) predictions of the squared weight error of 1-D LMS a) as function of the number of t, the batch updates (iterations), and b) as function of the number of input presentations, T. Training set size is N = 1000 and batch sizes are n =10, 100, and 500. Inputs were gaussian with R = 1, 0'( = 1 and p. = .1. Simulations used 10000 networks Equation (9) can also be solved exactly in multiple dimensions in the rather restrictive case where we assume that the inputs are spherically symmetric gaussians with R = a1 where a is a constant, 1 is the identity matrix, and m is the dimension. The update equation and solution are the same as (10) and (12), respectively, but where a and {J are now 2 2 (N(n - 1) N - n ) p.20';ma(N - n) C\' = 1 - 2p.a + p. a (N _ 1)n + (N _ 1)n (m + 2) , f3 = n(N _ 1) . (14) 3 Adaptive Batch Size The time it takes to compute the weight update in one iteration is roughly proportional to the number of input presentations, i.e the batch size. To make the comparison of convergence rates for different batch sizes meaningful, we must compute the change in squared weight error as a function of the number of input presentations, T, rather than iteration number t. For fixed batch size, T = nt. Figure 1b displays our 1-D LMS simulations plotted as a function of T. As can be seen, training \vith a large batch size is slow but results in a lower equilibrium value than obtained with a small batch size .. This suggests that we could obtain the fastest decrease of (v 2 ) overall by varying the batch size at each iteration. The batch size to choose for the current (v2 ) would be the smallest n that has yet to reach equilibrium, i.e. for which (v 2) > {v 2)oo. To determine the best batch size, we take the greedy approach by demanding that at each itemtion the batch size is chosen so as to reduce the weight error at the next iteration by the greatest amount per input presentation. This is equivalent to asking what value of n maximizes h == ((v;) (V~+l))/n? Once we determine n we then express it as a function of T. We treat the 1-D case, although the analysis would be similar for the spherically symmetric case. From (10) we have h = ~ (((~ - 1)(v;) + {J). Differentiating h with respect to n and solving yields the batch size that decreases the weight error the most to be . ( 2J.1.N((S-R2)(v;)+(1~R) ) nt = mm N, (2R(N -1) + J.I.(S _ NR2))(v;) + J.I.(1~R . (15) We have nt exactly equal to N when the current value of (v;) satisfies ( 2) _ J.I.(f~ R vt < Ie = 2R(N _ 1) - J.I.(R2(N - 2) - S) (nt = N). (16) 236 G. B. Orr E[ v"2J 1 0.1 0.01 0.001 Adaptive Batch theory - - - Simulation 100. 50. 10. 5 n Batch Size 1~~?=~~~~~~~~~ (a) 0 10 20 30 40 50 (b) 0 10 20 30 40 50 60 70 Figure 2: 1-D LMS: a) Comparison of the simulated and theoretically predicted (equation (18)) squared weight error as a function of t with N = 1000, R = 1, u. = 1, p. = .1, and 10000 networks. b) Corresponding batch sizes used in the simulations. Thus, after (v;) has decreased to Ie, training will proceed as pure batch. When (vn > Ie, we have nt < N and we can put (15) into (10) to obtain ( 2 ) _ ( 2 (NR2 -S)) (2) jJ2u;R Vt+l 1- I-'-R + I-'2(N _ 1) Vt - 2(N -1)' (17) Solving (17) we get 1 - nt-to (v;) = ni- tO (v6) + 1- :~1 {31 (18) where (tl, and (31 are constants 2 u;R (31 = -j.t 2(N - 1)" (19) Figure 2a compares equation (18) with 1-D LMS simulations. The adaptive batch size was chosen by rounding (15) to the nearest integer. Early in training, the predicted nt is always smaller than 1 but the simulation always rounds up to 1 (can't have n=O). Figure 2b displa:ys the batch sizes that were used in the simulations. A logarithmic scale is used to show that the batch size increases exponentially in t. \Ve next examine the batch size as a function of T. 3.1 Convergence Rate per Input Presentation \Vhen we use (15) to choose the batch size, the number of input presentations will vary at each iteration. Thus, T is not simply a mUltiple of t. Instead, we have t T(t) = TO + L nj i=to (20) where TO is the number of inputs that have been presented by to. This can be evaluated when N is very large. In this case, equations (18) and (15) reduce to t 1 2 2 (-un (V6) Ct3 -to where n3 == 1 - I-'-R + "21-'- R (21) 21-'-«S- R2)(v;) + u;R) 21-'-(S - R2) 2j.tu; (2R - I-'-R2) (v;) = 2R - I-'-R2 + (2 _ jJR)(v6)(};~-to' (22) Putting (22) into (20), and summing gives gives ~ 2j.t(S - R2) ~ 21-'-u; Ct3~t (};3 T(t) = (2 _ I-'-R)R t + (2 - I-'-R)(V6) 1 Ct3 (23) Removing Noise in On-Line Search using Adaptive Batch Sizes E[v"2J 1f':-----____ ~n-~-1~OO~ ____ __ n=1 \ . o.~·g~ n:::~~iivE!-'-___ '_" __ _ n=10 (a) ~~--------L-~5-0~10-0~15~0-2~OO--2~5~0-3~OO~3~50~400~ E[v"2J 1 0.01 annealed ••• 10;f····· .. I adaptive batch: 0.001 theory (solid) simulation (dashed) (b) ~~~~~~~~~~~~.~ 10. 20. 50. 100. 200. 500. 10002000. 237 Figure 3: I-D LMS: a) Simulations of the squared weight error as a function of T , the number of input presentations for N = 1000, R = 1, u. = 1, 11 = .1, and 10000 networks. Batch sizes are n =1, 10, 100, and nt (see (15)). b) Comparison of simulation (dashed) and theory (see (24)) using adaptive batch size. Simulation (long dash) using an annealed learning rate with n = 1 and 11 = R- 1 is also shown. where D.t == t - to and D.T == T - TO. Assuming that la31 < 1, the term with a;-At will dominate at late times. Dropping the other terms and solving for a~ gives 40-2 (v~) = (vi)a~t ~ (2 -pR)2 R(T _ TO)' (24) Thus, when using an adaptive batch size, (v 2 ) converges at late times as ~. Figure 3a compares simulations of (v 2 ) with adaptive and constant batch sizes. As can be seen, the adaptive n curve follows the n = 1 curve until just before the n = 1 curve starts to flatten out. Figure 3b compares (24) with the simulation. Curves are plotted on a log-log plot to illustrate the l/T relationship at late times (straight line with slope of -1). 4 Learning Rate Annealing vs Increasing Batch Size \Vith online learning (n = 1), we can reduce the noise at late times by annealing the learning rate using a p/t schedule. During this phase, (v 2 ) decreases at a rate of l/T if J.1 > R-l /2 [L093] and slower otherwise. In this paper, we have presented an alternative method for reducing the noise by increasing the batch size exponentially in t. Here, (v2 ) also decreases at rate of l/T so that, from this perspective, an adaptive batch size is equivalent to annealing the learning rate. This is confirmed in Figure 3b which compares using an adaptive batch size with annealing. An advantage of the adaptive batch size comes when n reaches N. At this point n remains constant so that (v 2 ) decreases exponentially in T. However, with annealing, the convergence rate of (v 2 ) always remains proportional to l/T. A disadvantage, though, occurs in multiple dimensions with nonspherical R where the best choice of Ht would likely be different along different directions in the weight space. Though it is possible to have a different learning rate along different directions, it is not possible to have different batch sizes. 5 Appendix A In this appendix we use simple counting arguments to derive the two results in equations (5) and (6). We first note that there are M == ( ~ ) ways of choosing n examples out of a total of N examples. Thus, (5) can be rewritten as 1 M 1 M 1 (gB".t)B= MLgB~i),t= ML;; L gj,t. (25) i=l i=l ZjEB~i) 238 G. B. Orr where B~i) is the ith batch (i = 1, ... ,M), and gj,t == a£(Zj )jaVt for j = 1, ... ,N. If we were to expand (25) we would find that there are exactly nlvl terms. From symmetry and since there are only N unique gj,t, we conclude that each gj,t occurs exactly n;: times. The above expression can then be written as (26) Thus, we have equation (5). The second equation (6) is By the same argument to derive (5), the first term on the right (ljn)(gr,t)N' In the second term, there are a total n(n -1)M terms in the sum, of which only N(N -1) are unique. Thus, a given gJ.tgk,t occurs exactly n(n - I)Mj(N(N - 1)) times so that 1 ~" 1 n(n-l)M ~ n2M ~ ~ gj,t gk,t = n2M N(N _ 1) . ~. gj,t gk,t .=1 Zj ,z"EB~') ,j~k J,k=l,J~k N(n - 1) (1 ( N N 2) 1 (1 N 2)) n(N - 1) N2' . L. gj,tgk,t + ~gj,t - N N ~gj,t J,k=l,J~k J=l J=l = = N(n - 1) 2 (n - 1) 2 n(N - 1) (gi,t)N - n(N -1) (9i,t)N. (28) Putting the simplified first term together and (28) both into (27) we obtain our second result in equation (6). References [Go187] Larry Goldstein. Mean square optimality in the continuous time Robbins Monro procedure. Technical Report DRB-306, Dept. of Mathematics, University of Southern California, LA, 1987. [Hay91] Simon Haykin. Adaptive Filter Theory. Prentice Hall, New Jersey, 1991. [L093] Todd K. Leen and Genevieve B. Orr. Momentum and optimal stochastic search. In Advances in Neural Information Processing Systems, vol. 6, 1993. to appear. [M!2I193] Martin M!2Iller. Supervised learning on large redundant training sets. International Journal of Neural Systems, 4{1}:15-25, 1993. [Orr95] Genevieve B. Orr. Dynamics and Algorithms for Stochastic learning. PhD thesis, Oregon Graduate Institute, 1995. [PW93] Mark Plutowski and Halbert White. Selecting concise training sets from clean data. IEEE Transactions on Neural Networks, 4:305-318, 1993.	1996	120
1,174	Size of multilayer networks for exact learning: analytic approach Andre Elisseefl' D~pt Mathematiques et Informatique Ecole Normale Superieure de Lyon 46 allee d'Italie F69364 Lyon cedex 07, FRANCE Helene Paugam-Moisy LIP, URA 1398 CNRS Ecole Normale Superieure de Lyon 46 allee d'Italie F69364 Lyon cedex 07, FRANCE Abstract This article presents a new result about the size of a multilayer neural network computing real outputs for exact learning of a finite set of real samples. The architecture of the network is feedforward, with one hidden layer and several outputs. Starting from a fixed training set, we consider the network as a function of its weights. We derive, for a wide family of transfer functions, a lower and an upper bound on the number of hidden units for exact learning, given the size of the dataset and the dimensions of the input and output spaces. 1 RELATED WORKS The context of our work is rather similar to the well-known results of Baum et al. [1, 2,3,5, 10], but we consider both real inputs and outputs, instead ofthe dichotomies usually addressed. We are interested in learning exactly all the examples of a fixed database, hence our work is different from stating that multilayer networks are universal approximators [6, 8, 9]. Since we consider real outputs and not only dichotomies, it is not straightforward to compare our results to the recent works about the VC-dimension of multilayer networks [11, 12, 13]. Our study is more closely related to several works of Sontag [14, 15], but with different hypotheses on the transfer functions of the units. Finally, our approach is based on geometrical considerations and is close to the model of Coetzee and Stonick [4]. First we define the model of network and the notations and second we develop our analytic approach and prove the fundamental theorem. In the last section, we discuss our point of view and propose some practical consequences of the result. Size of Multilayer Networks for Exact Learning: Analytic Approach 163 2 THE NETWORK AS A FUNCTION OF ITS WEIGHTS General concepts on neural networks are presented in matrix and vector notations, in a geometrical perspective. All vectors are written in bold and considered as column vectors, whereas matrices are denoted with upper-case script. 2.1 THE NETWORK ARCHITECTURE AND NOTATIONS Consider a multilayer network with N/ input units, N H hidden units and N s output units. The inputs and outputs are real-valued. The hidden units compute a nonlinear function f which will be specified later on. The output units are assumed to be linear. A learning set of Np examples is given and fixed. For allp E {1..Np }, the pth example is defined by its input vector dp E iRNI and the corresponding desired output vector tp E iRNs. The learning set can be represented as an input matrix, with both row and column notations, as follows Similarly, the target matrix is T = [ti, ... ,ttp (, with independent row vectors. 2.2 THE NETWORK AS A FUNCTION g OF ITS WEIGHTS For all h E {1..N H }, w; = (w;I' ... ,WkNI? E iRNI is the vector of the weights between all the input units and the hth hidden unit. The input weight matrix WI is defined as WI = [wi, . .. ,wJvH ]. Similarly, a vector w~ = (w;I' ... ,W~NHf E iRNH represents the weights between all the hidden units and the sth output unit, for all s E {1..N s}. Thus the output weight matrix W 2 is defined as W 2 = [w~, ... ,wJ.,.s]' For an input matrix V, the network computes an output matrix where each output vector z(dp ) must be equal to the target tp for exact learning. The network computation can be detailed as follows, for all s E {1..N s} NH NI L w~h.f(L dpi.wt) h=1 i=1 NH L w;h.f(d;.w;) h=1 Hence, for the whole learning set, the sth output component is NH 2 [f(di .. wu ] L W 8h' : h=l f(d~p.w;) NH (1) L W;h·F(V.W;) h=l 164 A. Elisseeff and H. Paugam-Moisy In equation (1), F is a vector operator which transforms a n vector v into a n vector F(v) according to the relation [F(V)]i = f([v]d, i E {1..n}. The same notation F will be used for the matrix operator. Finally, the expression of the output matrix can be deduced from equation (1) as follows 2(V) [F(V.wt), ... ,F(V.WhH )] : [wi, . .. ,w~s] (2) 2(V) = F(V.Wl).W2 From equation (2), the network output matrix appears as a simple function of the input matrix and the network weights. Unlike Coetzee and Stonick, we will consider that the input matrix V is not a variable of the problem. Thus we express the network output matrix 2(V) as a function of its weights. Let 9 be this function 9 : n.N[xNH+NHxNs --t n.NpxNs W = (Wl, W2) --t F(V.Wl).W2 The 9 function clearly depends on the input matrix and could have be denoted by g'D but this index will be dropped for clarity. 3 FUNDAMENTAL RESULT 3.1 PROPERTY OF FUNCTION 9 Learning is said to be exact on V if and only if there exists a network such that its output matrix 2(V) is equal to the target matrix T. If 9 is a diffeomorphic function from RN[xNH+NHXNS onto RNpxNs then the network can learn any target in RNpxNs exactly. We prove that it is sufficient for the network function 9 to be a local diffeomorphism. Suppose there exist a set of weights X, an open subset U C n.N[NH+NHNS including X and an open subset V C n.NpNs including g(X) such that 9 is diffeomorphic from U to V. Since V is an open neighborhood of g(X), there exist a real ..\ and a point y in V such that T = ..\(y - g(X)) . Since 9 is diffeomorphic from U to V, there exists a set of weights Y in U such that y = g(Y), hence T = ..\(g(Y) - g(X)). The output units of the network compute a linear transfer function, hence the linear combination of g(X) and g(Y) can be integrated in the output weights and a network with twice N/ N H + N H N s weights can learn (V, T) exactly (see Figure 1). g(Y) (J;) ~)---z 'T=A.(g(Y)-g(X)) Figure 1: A network for exact learning of a target T (unique output for clarity) For 9 a local diffeomorphism, it is sufficient to find a set of weights X such that the Jacobian of 9 in X is non-zero and to apply the theorem of local inversion. This analysis is developed in next sections and requires some assumptions on the transfer function f of the hidden units. A function which verifies such an hypothesis 11. will be called a ll-function and is defined below. Size of Multilayer Networks for Exact Learning,' Analytic Approach 165 3.2 DEFINITION AND THEOREM Definition 1 Consider a function f : 'R ~ 'R which is C1 ('R) (i.e. with continuous derivative) and which has finite limits in -00 and +00. Such a function is called a 1l-function iff it verifies the following property (1l) (Va E'RI I a I> 1) lim I ff'~(ax)) 1= 0 x--+±oo x From this hypothesis on the transfer function of all the hidden units, the fundamental result can be stated as follows Theorem 1 Exact learning of a set of Np examples, in general position, from'RNr to 'RNs , can be realized by a network with linear output units and a transfer function which is a 1l-function, if the size N H of its hidden layer verifies the following bounds Lower Bound N H = r !:r~ 1 hidden units are necessary Upper Bound NH = 2 r N~'Ns 1 Ns hidden units are sufficient The proof of the lower bound is straightforward, since a condition for g to be diffeomorphic from RNrxNH+NHXNs onto RNpxNs is the equality of its input and output space dimensions NJNH + NHNS = NpNs . 3.3 SKETCH OF THE PROOF FOR THE UPPER BOUND The 9 function is an expression of the network as a function of its weights, for a given input matrix: g(W1, W2) = F(V.W1 ).W2 and 9 can be decomposed according to its vectorial components on the learning set (which are themselves vectors of size Ns) . For all p E {1..Np} The derivatives of 9 w.r.t. the input weight matrix WI are, for all i E {1..NJ}, for all h E {l..NH} :!L = [W~h !,(d~.wl)dpi"" ,WJvsh f'(d~ .wl)dpi]T For the output weight matrix W2, the derivatives of 9 are, for all h E {1..NH}, for all s E {l..Ns} 88g~ = [ 0, ... ,O,f(d~ .w~), 0, .. . , 0 y W 8h '--" '--" 8-1 NS-8 The Jacobian matrix MJ(g) of g, the size of which is NJNH + NHNS columns and NsNp rows, is thus composed of a block-diagonal part (derivatives w.r.t. W2) and several other blocks (derivatives w.r.t. WI). Hence the Jacobian J(g) can be rewritten J(g) =1 J1, h,·· . ,JNH I, after permutations of rows and columns, and using the Hadamard and Kronecker product notations, each J h being equal to (3) Jh = [F(v.wl) ® INs, [F'(v.wl) 061 " .F'(v.wl) 06Nr ] 0 [W~h" ,WJvsh]] where INs is for the identity matrix in dimension Ns. 166 A. Elisseeff and H. Paugam-Moisy Our purpose is to prove that there exists a point X = (Wi, W2) such that the Jacobian J(g) is non-zero at X, i.e. such that the column vectors of the Jacobian matrix MJ(g) are linearly independent at X. The proof can be divided in two steps. First we address the case of a single output unit. Afterwards, this proof can be used to extend the result to several output units. Since the complete development of both proofs require a lot of calculations, we only present their sketchs below. More details can be found in [7]. 3.3.1 Case of a single output unit The proof is based on a linear arrangement of the projections of the column vectors of Jh onto a subspace. This subspace is orthogonal to all the Ji for i < h. We build a vector wi and a scalar w~h such that the projected column vectors are an independent family, hence they are independent with the Ji for i < h. Such a construction is recursively applied until h = N H. We derive then vectors wi, .. . ,wkrH and wi such that J(g) is non-zero. The assumption on 1l-fonctions is essential for proving that the projected column vectors of Jh are independent. 3.3.2 Case of multiple output units In order to extend the result from a single output to s output units, the usual idea consists in considering as many subnetworks as the number of output units. From this point of view, the bound on the hidden units would be N H = 2 'f.!;~f which differs from the result stated in theorem 1. A new direct proof can be developed (see [7]) and get a better bound: the denominator is increased to N/ + N s . 4 DISCUSSION The definition of a 1l-function includes both sigmoids and gaussian functions which are commonly used for multilayer perceptrons and RBF networks, but is not valid for threshold functions. Figure 2 shows the difference between a sigmoid, which is a 1l-function, and a saturation which is not a 1l-function. Figures (a) and (b) represent the span of the output space by the network when the weights are varying, i.e. the image of g. For clarity, the network is reduced to 1 hidden unit, 1 input unit, 1 output unit and 2 input patterns. For a 1l-function, a ball can be extracted from the output space 'R}, onto which the 9 function is a diffeomorphism. For the saturation, the image of 9 is reduced to two lines , hence 9 cannot be onto on a ball of R2. The assumption of the activation function is thus necessary to prove that the jacobian is non-zero. Our bound on the number of hidden units is very similar to Baum's results for dichotomies and functions from real inputs to binary outputs [1] . Hence the present result can be seen as an extension of Baum's results to the case of real outputs, and for a wide family of transfer functions, different from the threshold functions addressed by Baum and Haussler in [2]. An early result on sigmoid networks has been stated by Sontag [14]: for a single output and at least two input units, the number of examples must be twice the number of hidden units. Our upper bound on the number of hidden units is strictly lower than that (as soon as the number of input units is more than two). A counterpart of considering real data is that our results bear little relation to the VC-dimension point of view. Size of Multilayer Networksfor Exact Learning: Analytic Approach 167 D~ 0.5 -.!, t .5 -1 ~J.5 0 05 1 1.5 2 2.5 :I 1.5 0.5 o •••••• -----~I__-----0.5 -1 -1.5 -~2~---:'-1.5=----~1 --:-07.5 -~---'0-:-:.5----7---:':1 .5:---!. (a) : A saturation function (b) : A sigmoid function Figure 2: Positions of output vectors, for given data, when varying network weights 5 CONCLUSION In this paper, we show that a number of hidden units N H = 2 r N p N s / (Nr + N s) 1 is sufficient for a network ofll-functions to exactly learn a given set of Np examples in general position. We now discuss some of the practical consequences of this result. According to this formula, the size of the hidden layer required for exact learning may grow very high if the size of the learning set is large. However, without a priori knowledge on the degree of redundancy in the learning set, exact learning is not the right goal in practical cases. Exact learning usually implies overfitting, especially if the examples are very noisy. Nevertheless, a right point of view could be to previously reduce the dimension and the size of the learning set by feature extraction or data analysis as pre-processing. Afterwards, our theoretical result could be a precious indication for scaling a network to perform exact learning on this representative learning set, with a good compromise between, bias and variance. Our bound is more optimistic than the rule-of-thumb N p = lOw derived from the theory of PAC-learning. In our architecture, the number of weights is w = 2NpNs. However the proof is not constructive enough to be derived as a learning algorithm, especially the existence of g(Y) in the neighborhood of g(X) where 9 is a local diffeomorphism (cf. figure 1). From this construction we can only conclude that NH = r NpNs/(Nr+Ns)l is necessary and NH = 2 fNpNs/(Nr+Ns)l is sufficient to realize exact learning of Np examples, from nNr to nNs. 168 A. Elisseeff and H. Paugam-Moisy The opportunity of using multilayer networks as auto-associative networks and for data compression can be discussed at the light of this results. Assume that N s = NJ and the expression of the number of hidden units is reduced to N H = N p or at least NH = Np /2. Since N p ~ NJ + Ns, the number of hidden units must verify N H ~ NJ. Therefore, an architecture of "diabolo" network seems to be precluded for exact learning of auto-associations. A consequence may be that exact retrieval from data compression is hopeless by using internal representations of a hidden layer smaller than the data dimension. Acknowledgements This work was supported by European Esprit III Project nO 8556, NeuroCOLT Working Group. We thank C.S. Poon and J.V. Shah for fruitful discussions. References [1] E. B. Baum. On the capabilities of multilayer perceptrons. J. of Complexity, 4:193-215, 1988. [2] E. B. Baum and D. Haussler. What size net gives valid generalization? Neural Computation, 1:151- 160, 1989. [3] E. K. Blum and L. K. Li. Approximation theory and feedforward networks. Neural Networks, 4(4):511-516, 1991. [4] F. M. Coetzee and V. L. Stonick. Topology and geometry of single hidden layer network, least squares weight solutions. Neural Computation, 7:672-705, 1995. [5] M. Cosnard, P. Koiran, and H. Paugam-Moisy. Bounds on the number of units for computing arbitrary dichotomies by multilayer perceptrons. J. of Complexity, 10:57-63, 1994. [6] G. Cybenko. Approximation by superpositions of a sigmoidal function. Math. Control, Signal Systems, 2:303-314, October 1988. [7] A. Elisseeff and H. Paugam-Moisy. Size of multilayer networks for exact learning: analytic approach. Rapport de recherche 96-16, LIP, July 1996. [8] K. Funahashi. On the approximate realization of continuous mappings by neural networks. Neural Networks, 2(3):183- 192, 1989. [9] K. Hornik, M. Stinchcombe, and H. White. Multilayer feedforward networks are universal approximators. Neural Networks, 2(5):359-366, 1989. [10] S.-C. Huang and Y.-F. Huang. Bounds on the number of hidden neurones in multilayer perceptrons. IEEE Trans. Neural Networks, 2:47- 55, 1991. [11] M. Karpinski and A. Macintyre. Polynomial bounds for vc dimension of sigmoidal neural networks. In 27th ACM Symposium on Theory of Computing, pages 200-208, 1995. [12] P. Koiran and E. D. Sontag. Neural networks with quadratic vc dimension. In Neural Information Processing Systems (NIPS *95), 1995. to appear. [13] W. Maass. Bounds for the computational power and learning complexity of analog neural networks. In 25th ACM Symposium on Theory of Computing, pages 335-344, 1993. [14] E. D. Sontag. Feedforward nets for interpolation and classification. J. Compo Syst. Sci., 45:20-48, 1992. [15] E. D. Sontag. Shattering all sets of k points in "general position" requires (k1)/2 parameters. Technical Report Report 96-01, Rutgers Center for Systems and Control (SYCON), February 1996.	1996	121
1,175	Support Vector Regression Machines Harris Drucker· Chris J.C. Burges" Linda Kaufman" Alex Smola·· Vladimir Vapoik + Bell Labs and Monmouth University Department of Electronic Engineering West Long Branch. NJ 07764 BellLabs + AT&T Labs Abstract A new regression technique based on Vapnik's concept of support vectors is introduced. We compare support vector regression (SVR) with a committee regression technique (bagging) based on regression trees and ridge regression done in feature space. On the basis of these experiments, it is expected that SVR will have advantages in high dimensionality space because SVR optimization does not depend on the dimensionality of the input space. 1. Introduction In the following, lower case bold characters represent vectors and upper case bold characters represent matrices. Superscript "t" represents the transpose of a vector. y represents either a vector (in bold) or a single observance of the dependent variable in the presence of noise. yCp) indicates a predicted value due to the input vector x(P) not seen in the training set. Suppose we have an unknown function G(x) (the "truth") which is a function of a vector x (termed input space). The vector xt = [.XI,X2, ... ,Xd] has d components where d is termed the dimensionality of the input space. F(x, w) is a family of functions parameterized by w. w is that value of w that minimizes a measure of error between G(x) and F(x, w). Our objective is to estimate w with w by observing the N training instances Vj, j=l, .. ·,N. We will develop two apprOximations for the truth G(x). The first one is F 1 (x, w) which we term a feature space representation. One (of many) such feature vectors is: Zt_[x2 ... X2d X X ... x·x· ... Xd X·-J X ... x 1] I, , , 1 2, , I l' ' -I u' .. , d, which is a quadratic function of the input space components. Using the feature space representation, then F 1 (x, w) = z tw , that is, F 1 (x, w) is linear in feature space although 156 H. Drucker, C. J. Burges, L. Kaufman. A. Smola and V. Vapnik it is quadratic in input space. In general, for a p'th order polynomial and d'th dimensional input space, the feature dimensionality / of w is p+d-l . / = L CJ-l ;=d-l h Cn n! were k = k !(n-k)! The second representation is a support vector regression (SVR) representation that was developed by Vladimir Vapnik (1995): N F2(x,w)=L(at-a;)(v~x+1)P + b ;=1 F 2 is an expansion explicitly using the training examples. The rationale for calling it a support vector representation will be clear later as will the necessity for having both an a and an a rather than just one multiplicative constant. In this case we must choose the 2N + 1 values of a; at and b. If we expand the term raised to the p'th power, we find/ coefficients that multiply the various powers and cross product terms of the components of x. So, in this sense Fl looks very similar to F2 in that they have the same number of terms. However F} has/free coefficients while F2 has 2N+1 coefficients that must be determined from the N training vectors. We let a represent the 2N values of aj and at. The optimum values for the components of w or a depend on our definition of the loss function and the objective function. Here the primal objective function is: N ULL[Yj-F(vj, w)]+11 w 112 j=l where L is a general loss function (to be defined later) and F could be F 1 or F 2, Yj is the observation of G(x) in the presence of noise, and the last term is a regularizer. The regularization constant is U which in typical developments multiplies the regularizer but is placed in front of the first term for reasons discussed later. If the loss function is quadratic, i.e., we L[·]=[·J2, and we let F=F 1, i.e., the feature space representation, the objective function may be minimized by using linear algebra techniques since the feature space representation is linear in that space. This is termed ridge regression (Miller, 1990). In particular let V be a matrix whose i'th row is the i'th training vector represented in feature space (including the constant term "1" which represents a bias). V is a matrix where the number of rows is the number of examples (N) and the number of columns is the dimensionality of feature space f Let E be the tx/ diagonal matrix whose elements are 11U. y is the Nxl column vector of observations of the dependent variable. We then solve the following matrix formulation for w using a linear technique (Strang, 1986) with a linear algebra package (e.g., MA TLAB): Vly = [VtV +E] w The rationale for the regularization term is to trade off mean square error (the first term) in the objective function against the size of the w vector. If U is large, then essentially we are minimizing the mean square error on the training set which may give poor generalization to a test sel. We find a good value of U by varying U to find the best performance on a validation set and then applying that U to the test set. Support Vector Regression Machines 157 Let us now define a different type of loss function termed an E-insensitive loss (Vapnik, 1995): L _ { 0 if I Yj-F2(X;,w) 1< E I Yj-F 2(Xj, w) I - E otherwise This defines an E tube (Figure 1) so that if the predicted value is within the tube the loss is zero, while if the predicted pOint is outside the tube, the loss is the magnitude of the difference between the predicted value and the radius E of the tube. Specifically, we minimize: N N 1 U(~~ • + ~~) + "2(w tw) where ~j or ~. is zero if the sample point is inside the tube. If the observed point is "above" the tube, l;; is the positive difference between the observed value and E and aj will be nonzero. Similary, ~j. will be nonzero if the observed point is below the tube and in this case a7 will be nonzero. Since an observed point can not be simultaneously on both sides of the tube, either aj or a7 will be nonzero, unless the point is within the tube, in which case, both constants will be zero. If U is large, more emphasis is placed on the error while if U is small, more emphasis is placed on the norm of the weights leading to (hopefully) a better generalization. The constraints are: (for all i, i=1,N) Yi-(wtVi)--b~~; (wtvi)+b-yj~~; l;; ~ ~~ The corresponding Lagrangian is: 1 N N N L=-(wtw) + U(L~j + L~i) La;[yi-{wtvi)-b+E~;] 2 i=1 i=1 i=1 N N - Lai[(wtvi)+b-Yi+E~i] L(17~7+Y;~i) i=1 i=1 where the 1i and aj are Lagrange multipliers. We find a saddle point of L (Vapnik. 1995) by differentiating with respect to Wi , b, and ~ which results in the equivalent maximization of the (dual space) objective function: N N 1 N W(a,a*) = -E~(a7 +Clj)+ ~yj(a~ -Clj) "2.~ (a7-Clj)(a;-ai)(v~vj + 11 1=1 1=1 I.J=I with the constraints: ~Clj~U ~aj·~U i=1, ... ,N N N La; = Lai i=1 i=1 We must find N Largrange multiplier pairs (ai, (7). We can also prove that the product of Cl; and a; is zero which means that at least one of these two terms is zero. A Vi corresponding to a non-zero Clj or a; is termed a support vector. There can be at most N support vectors. Suppose now, we have a new vector x(P), then the corresponding 158 H. Drucker. C. J Burges, L Kaufman, A. Smola and V. Vapnik prediction of y(P) is: N y(P) = L(a: - ai)(vfx(p) + 1)P+b i=1 Maximizing W is a quadratic programming problem but the above expression for W is not in standard fonn for use in quadratic programming packages (which usually does minimization). If we let then we minimize: subject to the constraints where A . = a~ 1-'1 I N '1N ~i+N = Clj i=1, ... ,N L~i = L~; and 05~i~U i=1,··· ,2N ;=1 N+l C'=[E-YloE-Y2, ... ,E-YN,E+Yl ,E+Y2, ... ,E+YN] Q= [! :] djj = (vfVj + 1)P i,j = 1, ... ,N We use an active set method (Bunch and Kaufman, 1980) to solve this quadratic programming problem. 2. Nonlinear Experiments We tried three artificial functions from (Friedman, 1991) and a problem (Boston Housing) from the UCI database. Because the first three problems are artificial, we know both the observed values and the truths. Boston Housing has 506 cases with the dependent variable being the median price of housing in the Boston area. There are twelve continuous predictor variables. This data was obtaining from the UCI database (anonymous ftp at ftp.ics.ucLedu in directory Ipub/machine-learning-databases) In this case, we have no "truth", only the observations. In addition to the input space representation and the SVR representation, we also tried bagging. Bagging is a technique that combines regressors, in this case regression trees (Breiman, 1994). We used this technique because we had a local version available. In the case of regression trees, the validation set was used to prune the trees. Suppose we have test points with input vectors xfP) i=1,M and make a prediction yr) using any procedure discussed here. Suppose Yi is the actually observed value, which is the truth G(x) plus noise. We define the prediction error (PE) and the modeling error (ME): 1 M ME=-L(YrLG(Xj»2 M ;=1 1 M PE=-L(YfPL y;)2 M i=1 For the three Friedman functions we calculated both the prediction error and modeling Support Vector Regression Machines 159 error. For Boston Housing, since the "truth" was not known, we calculated the prediction error only. For the three Friedman functions, we generated (for each experiment) 200 training set examples and 40 validation set examples. The validation set examples were used to find the optimum regularization constant in the feature space representation. The following procedure was followed. Train on the 200 members of the training set with a choice of regularization constant and obtain the prediction error on the validation set. Now repeat with a different regularization constant until a minimum of prediction error occurs on the validation set. Now, use that regularizer constant that minimizes the validation set prediction error and test on a 1000 example test set. This experiment was repeated for 100 different training sets of size 200 and validation sets of size 40 but one test set of size 1000. Different size polynomials were tried (maximum power 3). Second order polynomials fared best. For Friedman function #1, the dimensionality of feature space is 66 while for the last two problems, the dimensionality of feature space was 15 (for d=2). Thus the size of the feature space is smaller than that of the number of examples and we would expect that a feature space representation should do well. A similar procedure was followed for the SVR representation except the regularizer constant U, £ and power p were varied to find the minimum validation prediction error. In the majority of cases p=2 was the optimum choice of power. For the Boston Housing data, we picked randomly from the 506 cases using a training set of size 401, a validation set of size 80 and a test set of size 25. This was repeated 100 times. The optimum power as picked by the validations set varied between p=4 and p=5. 3. Results of experiments The first experiments we tried were bagging regression trees versus support regression (Table I). 1#1 1#2 1#3 Table I. Modeling error and prediction error on the three Friedman problems (100 trials). bagging SVR bagging SVR 1# trials ME ME PE PE better 2.26 .67 3.36 1.75 100 10,185 4,944 66,077 60,424 92 .0302 .0261 .0677 .0692 46 Rather than report the standard error, we did a comparison for each training set. That is, for the first experiment we tried both SVR and bagging on the same training, validation, and test set. If SVR had a better modeling error on the test set, it counted as a win. Thus for Friedman 1#1. SVR was always better than bagging on the 100 trials. There is no clear winner for Friedman function #3. Subsequent to our comparison of bagging to SVR, we attempted working directly in feature space. That is. we used F 1 as our approximating function with square loss and a second degree polynomial. The results of this ridge regression (Table TI) are better than SVR. In retrospect, this is not surprising since the dimensionality of feature space is small (/=66 for Friedman #1 and.t=15 for the two remaining functions) in relation to the number of training examples (200). This was due to the fact that the best approximating polynomial is second order. The other advantages of the feature space representation in 160 H. Drucker; C. J. Burges, L Kaufman, A. Smola and V. Vapnik this particular case are that both PE and ME are mean squared error and the loss function is mean squared error also. Table ll. Modeling error for SVR and feature space polynomial approximation. SVR feature space #1 .67 .61 #2 4,944 3051 #3 .0261 .0176 We now ask the question whether U and E are important in SVR by comparing the results in Table I with the results obtaining by setting E to zero and U to 100,000 making the regularizer insignificant (Table DI). On Friedman #2 (and less so on Friedman #3), the proper choice of E and U are important. Table m. Comparing the results above with those obtained by setting E to zero and U to 100,000 (labeled suboptimum). optimum suboptimum ME ME #1 .67 .70 #2 4,944 34,506 #3 .0261 .0395 For the case of Boston Housing, the prediction error using bagging was 12.4 while for SVR we obtained 7.2 and SVR was better than bagging on 71 out of 100 trials. The optimum power seems to be about five. We never were able to get the feature representation to work well because the number of coefficients to be determined (6885) was much larger than the number of training examples (401). 4 Conclusions Support vector regression was compared to bagging and a feature space representation on four nonlinear problems. On three of these problems a feature space representation was best, bagging was worst, and SVR came in second. On the fourth problem, Boston Housing, SVR was best and we were unable to construct a feature space representation because of the high dimensionality required of the feature space. On linear problems, forward subset selection seems to be the method of choice for the two linear problems we tried at varying signal to noise ratios. In retrospect, the problems we decided to test on were too simple. SVR probably has greatest use when the dimensionality of the input space and the order of the approximation creates a dimensionality of a feature space representation much larger than that of the number of examples. This was not the case for the problems we considered. We thus need real life examples that fulfill these requirements. 5. Acknowledgements This project was supported by ARPA contract number NOOOl4-94-C-1 086. Support Vector Regression Machines 161 6. References Leo Breiman, "Bagging Predictors", Technical Report 421, September 1994, Department of Statistics, University of California Berkeley, CA Also at anonymous ftp site: ftp.stat.berkeley.edulpub/tech-reports/421.ps.Z. Jame R. Bunch and Linda C. Kaufman, " A Computational Method of the Indefinite Quadratic Programming Problem", Linear Algebra and Its Applications, Elsevier-North Holland, 1980. Jerry Friedman, "Multivariate Adaptive Regression Splines", Annal Of Statistics, vol 19, No.1, pp. 1-141 Alan J. Miller, Subset Selection in Regression, Chapman and Hall, 1990. Gilbert Strang, Introduction to Applied Mathematics, Wellesley Cambridge Press, 1986. VladimirN. Vapnik, The Nature of Statistical Learning Theory, Springer, 1995. Figure 1: The p ... .meters for the support vector regression.	1996	122
1,176	An Hierarchical Model of Visual Rivalry Peter Dayan Department of Brain and Cognitive Sciences E25-21O Massachusetts Institute of Technology Cambridge, MA 02139 dayan@psyche.mit.edu1 Abstract Binocular rivalry is the alternating percept that can result when the two eyes see different scenes. Recent psychophysical evidence supports an account for one component of binocular rivalry similar to that for other bistable percepts. We test the hypothesisl9, 16, 18 that alternation can be generated by competition between topdown cortical explanations for the inputs, rather than by direct competition between the inputs. Recent neurophysiological evidence shows that some binocular neurons are modulated with the changing percept; others are not, even if they are selective between the stimuli presented to the eyes. We extend our model to a hierarchy to address these effects. 1 Introduction Although binocular rivalry leads to distinct perceptual distress, it is revealing about the mechanisms of visual information processing. The first accounts for rivalry argued on the basis of phenomena such as increases in thresholds for test stimuli presented in the suppressed eye24 , 8, 3 that there was a early competitive process, the outcome of which meant that the system would just ignore input from one eye in favour of the other. Various experiments have suggested that simple input competition cannot be the whole story. For instance, in a case in which rivalry is between a vertical grating in the left eye and a horizontal one in the right, and in which a vertical grating is presented prior to rivalry to cause adaptation, the relative suppression of vertical during rivalry is independent of 1 I am very grateful to Bart Anderson, Adam Elga, Geoff Goodhill, Geoff Hinton, David Leopold, Earl Miller, Read Montague, Bruno Olshausen, Pawan Sinha, Rich Zemel, and particularly Zhaoping Li and Tommi Jaakkola for their comments on earlier drafts and discussions. This work was supported by the NIH. A Hierarchical Model of Visual Rivalry 49 the eye of origin of the adapting grating.4 Even more compelling, if the rivalrous stimuli in the two eyes are switched rapidly, the percept switches only slowly competition is more between coherent percepts than merely inputs. Rivalry is an attractive paradigm for studying models of cortex like the Helmholtz machine12, 7 that construct coherent percepts, and in particular for studying hierarchical models, because of electrophysiological data on the behaviour during rivalry of cells at different levels of the visual processing hierarchy.16 Leopold & Logothetis16 trained monkeys to report their percepts during rivalrous and non-rivalrous stimuli whilst recording from neurons VI/2 and V4. Important findings are that striate monocular neurons are unaffected by rivalry; some striate binocular neurons that are selective between the stimuli modulate their activities during rivalry; others do not; some fire more when their preferred stimuli are suppressed; others still are only selective during rivalry. In this paper we consider one form of analysis-by-synthesis model of cortical processing7 and show how it can exhibit rivalry between explanations in the case that the eyes receive different input. This model can provide an account for many of the behaviours described above. 2 The Model Figure Ia shows the full generative model. Units in layers y (modeling VI) and x and w (modeling early and late extra-striate areas) are all binocular and jointly explain successively more complex features in the input z according to a top-down generative model. Apart from the half bars in y, the model is similar to that learned by the Helmholtz machine12, 7 for which increasing complexity in higher layers rather than the increasing input scale is key. In this case, for instance, w2 specifies the occurrence of vertical bars anywhere in the 8 x 8 input grids; X16 specifies the rightmost vertical bar; and Y31 and Y32 the top and bottom half of this vertical bar. These specifications are provided by a top-down generative model in which the activations of units are specified by probabilities such as P[Yi = Ilx] = a (by + Lk xkJ!~) where the sum k is over all the units in the x layer, and 0'0 is a robust normal distribution function. We model the percept in terms of the activation in the w layer. We model differing input contrasts by representing the input to Zi by di , where P[Zi = 1] = O'(di ) and all the Zi are independent. Recognition is formally the statistical inverse to generation, and should produce distribution P[w, x, yld] over all the choices of the hidden activations. We use a mean field inversion method,13 using a factorised approximation Q[w,x, y; Il,~,~] = Q[w; Il]Q][x; ~]Q[y; ~], with Q[w; Il] = TIi O'(lli)Wi (1 - O'(lli))l-Wi, etc, and fitting the parameters Il,~, ~ to minimise the approximation cost: [ ] "'P[ d] '" Q[ C .1']1 Q[w,x,y;Il,~,~] :F Il,~, ~ = L.J z; L.J w, x, y; Il, ,-, 'f/ og P[w x Iz] . z wxy , ,y , , We report the mean activities of the units in the graphs and use a modified gradient descent method to find appropriate parameters. Figure Ib shows the resulting activities of units in response to binocular horizontal (i) and vertical (ii) bars, and also the two equally likely explanations for rivalrous input (iii and iv). For rivalry, 50 LAYER wi y % (a) ••••••• b • • _ • • • • ~ YII •••••• ........ •••••• L R lib (b) wi Ia:::::QJ "II~I yl·1_ L R (i) wi !I:::::iJI "II~I yllil ·1_ P. Dayan wl~ ,,11l1lil1li1111 II ylllllill ·111111111 L R (ii) wl~ "llmBEl yl·1_ L R L R (iii) (iv) Figure 1: a) Hierarchical generative model for 8 x 8 bar patterns across the two eyes. Units are depicted by their net projective (generative) fields, and characteristic weights are shown. Even though the net projective field of Xl is the top horizontal bar in both eyes, note that it generates this by increasing the probability that units YI and Y9 in the y layer will be active, not by having direct connections to the input z. Unit WI connects to Xl, X2, .•• Xs through Jwx = 0.8; XI6 connects to Y31 , Y32 through JXY = 1.0 and Y32 connects to the bottom right half vertical bar through Jyz = 5.8. Biases are bw = -0.75,bx = -1.5, by = -2.7 and bz = -3.3. b) Recognition activity in the network for four different input patterns. The units are arranged in the same order as (a), and white and black squares imply activities for the units whose means are less than and greater than 0.5. (i) and (ii) represent normal binocular stimulation; (iii) and (iv) show the two alternative stable states during rivalrous stimulation, without the fatigue process. there is direct competition in the top left hand quadrant of z, which is reflected in the competition between YI, Y3 and Y17, Y21. However, the input regions (top right of L and bottom left of R) for which there is no competition, require the constant activity of explanations Y9, Yu ,Y18 and Y22. Under the generative model, the coactivation of YI and Y9 without Xl is quite unlikely (P[XI = OIYI = 1, Y3 = 1] = 0.1), which is why XI, X3 and also WI become active with YI and Y3. Given just gradient descent for the rivalrous stimulus, the network would just find one of the two equally good (or rather bad) solutions in figure 1b(iii,iv). Alternation ensues when descent is augmented by a fatigue process: = 'l/JI(t) +<5(-\7I/JIF[1L,~,'l/J] + a ({3'l/JI (t)) -'l/J~(t)) 'l/J~ (t) + <5('l/JI (t) {3'l/J~ (t)), where {3 is a decay term. In all the simulations, a = 0.5, {3 = 0.1 and <5 = 0.01. We adopted various heuristics to simplify the process of using this rather cumbersome mean field model. First, fatigue is only implemented for the units in the y A Hierarchical Model of Visual Rivalry 51 layer, and the 'I.jJ follow the equivalent of the dynamical equations above. Although adaptation processes can clearly occur at many levels in the system, and indeed have been used to try to diagnose the mechanisms of rivalry,15 their exact form is not clear. Bialek & DeWeesel argue that the rate of a switching process should be adaptive to the expected rate of change of the associated signal on the basis of prior observations. This is clearly faster nearer to the input. The second heuristic is that rather than perform gradient descent for the nonfatiguing units, the optimal values of f.1. and ~ are calculated on each iteration by solving numerically equations such as The dearth of connections in the network of figure la allows f.1. and ~ to be calculated locally at each unit in an efficient manner. Whether this is reasonable depends on the time constants of settling in the mean field model with respect to the dynamics of switching, and, more particularly on the way that this deterministic model is made appropriately stochastic. Figure 2a shows the resulting activities during rivalry of units at various levels of the hierarchy including the fatigue process. Broadly, the competing explanations in figure Ib(iii;iv), ie horizontal and vertical percepts, alternate, and units without competing inputs, such as Y9, are much less modulated than the others, such as Yl. The activity of Y9 is slightly elevated when horizontal bars are dominant, based on top-down connections. The activities of the units higher up, such as Xl and WI, do not decrease to 0 during the suppression period for horizontal bars, leaving weak activity during suppression. Many of the modulating cells in monkeys were not completely silent during their periods of less activity.16 Figure 2b shows that the hierarchical version of the model also behaves in accordance with experimental results on the effects of varying the input contrast,17, 10, 22, 16 which suggest that increasing the contrast in both eyes decreases the period of the oscillation (ie increases the frequency), and increasing the contrast in just one eye decreases the suppression period for that eye much more than it increases its dominance period. 3 Discussion Following Logothetis and his colleaguesl9, 16, 18 (see also Grossbergll ) we have suggested an account of rivalry based on competing top-down hierarchical explanations, and have shown how it models various experimental observations on rivalry. Neurons explain inputs in virtue of being capable of generating their activities through a top-down statistical generative model. Competition arises between higher-level explanations of overlapping active regions (ie those involving contrast changes) of the input rather than between inputs themselves. Note that alternating the input between the two eyes would have no effect on this behaviour of the model, since explanations are competing rather than inputs. Of course, the model is greatly simplified - for instance, it only has units that are not modulating with the percept in the earliest binocular layer (layer y), whereas in the monkeys, more than half the cells in V4 were unmodulated during rivalry.I6 The model's accounts of the neurophysiological findings described in the introduction are: i) monocular cells will generally not be modulated if they are involved in 52 a) Iterations ------+ o ::r 0.0 :: x, ~ 00 Q) ~ ~ 3 Q ~::n, 1 I 11.1 11-" I ' .0~ 0.5~-Y. 0,0 O'-----,:-:':OOO:::-----::2000:::-:-----:::3000~---::.OOO P. Dayan b) Contrast Dependence 600 r---------------------, - - equal contrast horizontal dominance (1=1 .25) ---- vertical dominance (r) -----------500 1.0 12 lA 1~ Test vertical 'contrast' (r) Figure 2: a) Mean activities of units at three levels of the hierarchy in response to rivalrous stimuli with input strengths I = r = 1.75. b) Contrast dependence of the oscillation periods for equal input strengths, and when I = 1.25 and r is varied. explaining local correlations in the input from a single eye. This model does not demonstrate this explicitly, but would if, for instance, each of the inputs Zi actually consisted of two units, which are always on or off together. In this case one could get a compact explanation of the joint activities with a set of monocular units which would then not be modulated. ii) Units such as Y9 in the hierarchical model are binocular, are selective between the binocular version of the stimuli, and are barely modulated with the percept. iii) Units such as YI, Xl and WI are binocular, are selective between the stimuli, and are significantly modulated with the percept. The final neurophysiological finding is to do with cells that fire when their preferred stimuli are suppressed, or fire selectively between the stimuli only during rivalry. There are no units in this model that are selective between the stimuli and are preferentially activated during suppression of their preferred stimuli. However, in a model with more complicated stimulus contingencies, they would emerge to account for the parts of the stimulus in the suppressed eye that are not accounted for by the explanation of the overlying parts of the dominant explanation, at least provided that this residual between the true monocular stimulus and the current explanation is sufficiently complex as to require explaining itself. We would expect to find two sets of cells that are activated during the suppressed period by this residual, some of which will form part of the representation of the stimulus when presented binocularly and some of which will not. Those that do not (class A) will only even appear to be selective between the stimuli during rivalry, and will represent parts of the residual that are themselves explained by more overarching explanations for parts of the complete (binocularly presented) stimulus. This suggests the experimental test of presenting binocularly a putative form of the residual (eg dotted lines for competing horizontal and vertical gratings). We predict that these cells should be activated. If there are cells that do participate in the binocular representation, then they will be selective, but will preferentially fire during suppression (class B). Certainly, the A Hierarchical Model of Visual Rivalry 53 residual will have a high correlation with the full suppressed pattern, and so a cell that is selective for part of the residual could have appropriate properties. However, why should such a cell not fire when the full, but currently suppressed, pattern is dominant? In monkeys,16 there are fewer class B than class A cells (0 versus 3 of 33 cells in Vl/2; 6 versus 8 of 68 cells in V4). Under the model, we account for these cells based on a competition between units that represent the residual and those that represent overlapping parts of the complete pattern. In binocular viewing, explanations are generally stronger than during rivalry. So even if both such units participate in representing a binocular stimulus, the cells representing the residual might not reach threshold during the dominance period. However, during suppression, they no longer suffer from competition, and so will be activated. The model's explanation for class B cells seems far less natural than that for class A cells. One experimental test would be to present the preferred pattern binocularly, reduce the contrast, and see if these cells are suppressed more strongly. The overall model mechanistically has much in common with models which place the competition in rivalry at the level of binocular oriented cells rather than between monocular cells.11,2 Indeed, the model is based on an explanation-driven account for normal binocular processing, so this is to be expected. The advantage of couching rivalry in terms of explanations is that this provides a natural way of accounting for top-down influences. In fact, one can hope to study top-down control through studying its effects on the behaviour of cells during rivalry. The model suffers from various lacunce. Foremost, it is necessary to model the stochasticity of switching between explanations.9,17 The distributions of dominance times for both humans and monkeys is well characterised by a r distribution (Lehky14 argues that this is descriptive rather than normative), with strong independence between successive dominance periods. Our mean field recognition process is deterministic. The stochastic analogue would be some form of Markov chain Monte-Carlo method such as Gibbs sampling. However, it is not obvious how to incorporate the equivalent of fatigue in a computationally reasonable way. In any case, the nature of neuronal randomness is subject to significant debate at present. Note that the recognition model of a stochastic Helmholtz machine7,6 would be unsuitable, since it is purely feedforward and does not integrate bottomup and top-down information. We have adopted a very simple mean field approach to recognition, giving up neurobiological plausibility for convenience. The determinism of the mean field model in any case rules it out as a complete explanation, but it does at least show clearly the nature of competition between explanations. The architecture of the model is also incomplete. The cortex is replete with what we would model as lateral connections between units within a single layer. We have constructed generative models in which there are no such direct connections, because they significantly complicate the mean field recognition method. It could be that these connections are important for the recognition process,6 but modeling their effect would require representing them explicitly. This would also allow modeling of the apparent diffusive process by which patches of dominance spread and alter. In a complete model, it would also be necessary to account for competition between eyes in addition to competition between explanations. 24,8,3 54 P. Dayan Another gap is some form of contrast gain control.s The model is quite sensitive to input contrast. This is obviously important for the effects shown in figures 2, however the range of contrasts over which it works should be larger. It would be particularly revealing to explore the effects of changing the contrast in some parts of images and examine the consequent effects on the spreading of dominance. References [1] Bialek, W & DeWeese, M (1995). Random switching and optimal processing in the perception of ambiguous Signals. Physical Review Letters, 74, 3077-3080. [2] Blake, R (1989). A neural theory of binocular rivalry. Psyclwlogical Review, 96, 145-167. [3] Blake, R & Fox, R (1974). Binocular rivalry suppression: Insensitive to spatial frequency and orientation change. Vision Research, 14, 687-692. [4] Blake, R, Westendorf, DH & Overton, R (1980). What is suppressed during binocular rivalry? Perception, 9, 223-231. [5] Carandini, M & Heeger, DJ (1994). Summation and division by neurons in primate visual cortex. Science, 264, 1333-1336. [6] Dayan, P & Hinton, GE (1996). Varieties of Helmholtz machine. Neural Networks, 9, 1385-1403. [7] Dayan, P, Hinton, GE, Neal, RM & Zemel, RS (1995). The Helmholtz machine. Neural Computation, 7,889-904. [8] Fox, R & Check, R (1972). Independence between binocular rivalry suppression duration and magnitude of suppression. Journal of Experimental Psyclwlogy, 93, 283-289. [9] Fox, R & Herrmann, J (1%7). Stochastic properties of binocular rivalry alternations. Perception and Psychophysics, 2, 432-436. [10] Fox, R & Rasche, F (1969). Binocular rivalry and reciprocal inhibition. Perception and Psychophyics, 5,215-217. [11] Grossberg, S (1987). Cortical dynamiCS of three-dimensional form, color and brightness perception: 2. Binocular theory. Perception & Psychphysics, 41, 117-158. [12] Hinton, GE, Dayan, P, Frey, BJ & Neal, RM (1995). The wake-sleep algorithm for unsupervised neural networks. Science, 268,1158-1160. [13] Jaakkola, T, Saul, LK & Jordan, MI (1996). Fast learning by bounding likelihoods in sigmoid type belief networks. Advances in Neural Information Processing Systems, 8, forthcoming. [14] Lehky, SR (1988). An astable multivibrator model of binocular rivalry. Perception, 17, 215-228. [15] Lehky, SR & Blake, R (1991). Organization of binocular pathways: Modeling and data related to rivalry. Neural Computation, 3,44-53. [16] Leopold, DA & Logothetis, NK (1996). Activity changes in early visual cortex reflect monkeys' percepts during binocular rivalry. Nature, 379, 549-554. [17] Levelt, WJM (1968). On Binocular Rivalry. The Hague, Paris: Mouton. [18] Logothetis, NK, Leopold, DA & Sheinberg, DL (1996). What is rivalling during binocular rivalry. Nature, 380, 621-624. [19] Logothetis, NK & Schall, JD (1989). Neuronal correlates of subjective visual perception. Science" 245,761-763. [20] Matsuoka, K (1984). The dynamic model of binocular rivalry. Biological Cybernetics, 49, 201-208. [21] Mueller, 11 (1990). A physiological model of binocular rivalry. Visual Neuroscience, 4, 63-73. [22] Mueller, 11 & Blake, R (1989). A fresh look at the temporal dynamiCS of binocular rivalry. Biological Cybernetics, 61, 223-232. [23] Pearl, J (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. San Mateo, CA: Morgan Kaufmann. [24] Wales, R & Fox, R (1970). Increment detection thresholds during binocular rivalry suppression. Perception and PsychophysiCS, 8, 90-94. [25] Wheatstone, C (1838). Contributions to the theory of vision. I: On some remarkable and hitherto unobserved phenomena of binocular vision. Philosophical Transactions of the Royal Society of London, 128,371-394. [26] Wolfe, JM (1986). Stereopsis and binocular rivalry. PsycholOgical Review, 93,269-282.	1996	123
1,177	Dynamically Adaptable CMOS Winner-Take-AII Neural Network Kunihiko Iizuka, Masayuki Miyamoto and Hirofumi Matsui Information Technology Research Laboratories Sharp Tenri, Nara, lAP AN Abstract The major problem that has prevented practical application of analog neuro-LSIs has been poor accuracy due to fluctuating analog device characteristics inherent in each device as a result of manufacturing. This paper proposes a dynamic control architecture that allows analog silicon neural networks to compensate for the fluctuating device characteristics and adapt to a change in input DC level. We have applied this architecture to compensate for input offset voltages of an analog CMOS WTA (Winner-Take-AlI) chip that we have fabricated. Experimental data show the effectiveness of the architecture. 1 INTRODUCTION Analog VLSI implementation of neural networks, such as silicon retinas and adaptive filters, has been the focus of much active research. Since it utilizes physical laws that electric devices obey for neural operation, circuit scale can be much smaller than that of a digital counterpart and massively parallel implementation is possible. The major problem that has prevented practical applications of these LSIs has been fluctuating analog device characteristics inherent in each device as a result of manufacturing. Historically, this has been the main reason most analog devices have been superseded by digital devices. Analog neuro VLSI is expected to conquer this problem by making use of its adaptability. This optimistic view comes from the fact that in spite of the unevenness of their components, biological neural networks show excellent competence. This paper proposes a CMOS circuit architecture that dynamically compensates for fluctuating component characteristics and at the same time adapts device state to incoming signal levels. There are some engineering techniques available to compensate 714 K. lizuka, M. Miyamoto and H. Matsui for MOS threshold fluctuation, e.g., the chopper comparator, but they need a periodical change of mode to achieve the desired effect. This is because there are two modes one for the adaptation and one for the signal processing. This is quite inconvenient because extra clock signals are needed and a break of signal processing takes place. Incoming signals usually consist of a rapidly changing foreground component and a slowly varying background component. To process these signals incessantly, biological neural networks make use of multiple channels having different temporaVspatial scales. While a relatively slow/large channel is used to suppress background floating, a faster/smaller channel is devoted to process the foreground signal. The proposed method inspired by this biological consideration utilizes different frequency bands for adaptation and signal processing (Figure 1), where negative feedback is applied through a low pass filter so that the feedback will not affect the foreground signal processing. COMI'ARATOR Input Signal Output SignAl (a) Gain LOW PASS FILTER BACKGROUND nAND (b) FOREGROUND BAND Frequency Figure 1: Dynamic adaptation by frequency divided control. (a) model diagram, (b) frequency division. In the first part of this paper, a working analog CMOS WTA chip that we have test fabricated is introduced. Then, dynamical adaptation for this WT A chip is described and experimental results are presented. 2 ANALOG CMOS WTA CHIP 2.1 ARCHITECTURE AND SPECIFICATION v," I 2nd LAYER Figure 2: Analog CMOS WT A chip architecture • CM FEEDBACK CONTROLLER Dynamically Adaptable CMOS Wmner-Take-All Neural Network 715 Vdd M5 M4 Vhl--f I--Vb2 eM M In l)U~ Olltput VII\~ I MI I I 1142 V!!IS (ll) (b) Figure 3: Circuit diagrams for (a) the competitive cell and (b) the feedback controller. As a basic building block to construct neuro-chips, analog Wf A circuits have been investigated by researchers such as [Lazzaro, 1989] and [Pedroni, 1994]. All CMOS analog WfA circuits are based on voltage follower circuits [Pedroni, 1995] to realize competition through inhibitory interaction, and they use feedback mechanisms to enhance resolution gain. The architecture of the chip that we have fabricated is shown in Figure 2 and the circuit diagram is in Figure 3. This Wf A chip indicates the lowest input voltage by making the output voltage corresponds to the lowest input voltage near Vss (winner), and others nearly the power supply voltage Vdd (loser). The circuit is similar to [Sheu, 1993], but represents two advances. 1. The steering current that the feedback controller absorbs from the line CM is enlarged, allowing the winner cell can compete with others in the region where resolution gain is the largest. 2 The feedback controller originally placed after the second competitive layer is removed in order to guarantee the existence of at least one output node whose voltage is nearly zero. Table 1 shows the specifications of the fabricated chip. Table 1: Specifications of the fabricated WfA chip 2.2 INPUT OFFSET VOLTAGE Input offset voltages of a Wf A chip may greatly deteriorate chip performance. Examples of input offset voltage distribution of the fabricated chips are shown in Figure 4. Each input offset voltage is measured relative to the first input node. The input offset voltage 716 K. lizuJea, M. Miyamoto and H. Matsui ~Vj of the j-th input node is defined as ~Vj = Vinj - Vin1 when the voltages of output nodes Outj and Out1 are equal; Vin1 is fixed to a certain voltage and the voltage of other input nodes are fixed at a relatively high voltage. Figure 4: Examples of measured input offset voltage distribution. The primary factor of the input offset voltage is considered to be fluctuation of MOS transistor threshold voltages in the first layer competitive cell. Then, the input offset voltage ~ Vj of this cell yielded by the small fluctuation ~ Vthi of Vthi is calculated as follows: _ -~Vtht + gd1 + gd2 + gm2 (~Vth2 _ ~Vth3) + gm4(gd1 + gd2 + gm2) ~Vth4 gmt gm1gm3 where gmi and gdl are the transconductance and the drain conductance of MOS Mi, respectively. Using design and process parameters, we can estimate the input offset voltage to be AVj. -AVthl + (AVth2 -AVth3)+O.l5AVth4 , Based on our experiences, the maximum fluctuation of Vthi in a chip is usually smaller than 20 mY, and it is reasonable to consider that the difference I~Vth2 - ~VtJrI is even smaller; perhaps less than 5 m V, because M2 and M3 compose a current mirror and are closely placed. This implies that the maximum of ~Vj is about 28 mY, which is in rough agreement with the measured data. 3 DYNAMICAL ADAPTATION ARCmTECTURE In Figure 5, we show circuit implementation of the dynamically adaptable wr A function. In each feedback channel, the difference between each output and the reference Vref is fed back to the input node through a low pass filter consisting of Rand C. The charge stored in capacitor C is controlled by this feedback signal. Let the linear approximation of the wr A chip DC characteristic be Vouti = A ( Vin, - VOJ, where Vini and Voutt are the voltages at the nodes In/ and Out, respectively, andA and VOL are functions of Vinj (j '" i ). The input offset voltage relative to the node In] is considered to be the difference between VOL and V01. On the other hand, the DC characteristic of the i-th feedback path can be approximated as Dynamically Adaptable CMOS Winner-Take-All Neural Network 717 Inl' In2' In32 , Cl.. Cl.. cl.. R • • • R' Inl In2 • • • WTA Chip • • • Out l Out2 • • • Vref Vref • • • Outl Out2 Outn Figure 5: wrA chip equipped with adaptation circuit where R=10MQ and C=0.33JAF. Yin; = B (Yout; - Vref). It follows from the above two equations that AB B Vin· ---- Vo· --- Vrer• Vo · I 1- AB I 1- AB '.J I The last term is derived using the assumptions A » 1 and B « -1. This means that the voltage difference between the DC level of the input and YO; is clamped on the capacitor C. This in turn implies that the input offset voltage will be successfully compensated for. The role of the low pass filters is twofold. 1. They guarantee stable dynamics of the feedback loop; we can make the cutoff frequency of the low pass filters small enough so that the gain of the feedback path is attenuated before the phase of the feedback signal is delayed by more than 1800 • 2. They prevent the feed-forward wr A operation from being affected, as shown in Figure 1, the adaptive control is carried out on a different, non-overlapped frequency band than wr A operation. 4 EXPERIMENTAL RESULTS Experiments concerning the adaptable wr A function were carried out by applying pulses of 90% duty to the input nodes In', and In'l, while other input nodes were fixed to a certain voltage. In Figures 6 (a) and 6 (b), the output waveforms of Outl , Out2, Out] and the waveform of the pulse applied to the node In', are shown. Figure 6(a) shows the result when the same pulse was applied to both In', and In'z. Figure 6(b) shows the result when the amplitude of the pulse to In', was greater than that of the pulse to In'z by 10 mY. The schematic explanation of this behavior is in Figure 7. The outputs remained at the same levels for a while after the inputs were shut off, since there was no strong inducement. As a result of adaptation, the winning frequencies of every output nodes become equal in a long time scale. This explains the unstable output during the period of quiescent inputs. 718 K. Iizuka. M. Miyamoto and H. Matsui The chip used in this measurement had a relative input offset voltage of 15 mV between nodes In1 and In2• We can see in Figure 6 (a) that this offset voltage was completely compensated for because the output waveforms of corresponding nodes were the same. (a) (b) Figure 6: The output waveforms ot the dynamically adaptable CMOS WT A neural network. Pulse waves were applied to nodes In'] and In'2; other nodes voltages were fixed. When the amplitude of each pulse was the same (a), the corresponding output waveforms were the same. When the amplitude of the pulse fed to In'I was greater than that to In'2 by 10 mV (b), the output voltage at Out] was low (winner) and that at Out2 was high (loser) during the period the pulse was low (on). • • • Inputs quiescent Outputs Out 1 i;i Q, L.." mii~iii~~iii~!iim Out2 ~ Out3 iiii .......... Lo- s-er~mmmmmm~mm • • • roser . ~mmmmm~mmm ~'---v---' Hysteresis Unstable Figure 7: The schematic explanation of the dynamically adaptable WT A behavior. 5 CONCLUSION We have proposed a dynamic adaptation architecture that uses frequency divided control and applied this to a CMOS WT A chip that we have fabricated. Experimental results show that the architecture successfully compensated for input offset voltages of the WT A Dynamically Adaptable CMOS Winner-Take-All Neural Network 719 chip due to inherent device characteristic fluctuations. Moreover, this architecture gives analog neuro-chips the ability to adapt to incoming signal background levels. This adaptability has a lot of applications. For example, in vision chips, the adaptation may be used to compensate for the fluctuation of photo sensor characteristics, to adapt the gain of photo sensors to background illumination level and to automatically control color balance. As another application, Figure 8 describes an analog neuron with weighted synapses, where the time constant RC is much larger than the time constant of input signals. Inputs c 11 11 ••• Output Figure 8: Analog neuron with weighted synapses where the time constant RC is much larger than that of input signals. The key to this architecture is use of non-overlapping frequency bands for adaptation to background and foreground signal processing. For neuro-VLSIs, this requires implementing circuits with completely different time scale constants. In modern VLSI technology, however, this is not a difficult problem because processes for very high resistances, i.e., teraohms, are available. Acknowledgment The authors would like to thank Morio Osaka for his help in chip fabrication and Kazuo Hashiguchi for his support in experimental work. References Choi, J. & Sheu, B.J. (1993) A high-precision VLSI winner-take-all circuit for selforganizing neural networks. IEEE J. Solid-State Circuits, vo1.28, no.5, pp.576-584. Lazzaro, J., Ryckebush, S., Mahowald, M.A., & Mead, C. (1989) Winner-take-all networks of O(N) complexity. In D.S. Touretzky (eds.), Advances in Neural Information Processing Systems 1, pp. 703-711. Cambridge, MA: MIT Press. Pedroni, V.A. (1994) Neural n-port voltage comparator network, Electron. Lett., vo1.30, no.21, pp1774-1775. Pedroni, V.A. (1995) Inhibitory Mechanism Analysis of Complexity O(N) MOS WinnerTake-All Networks. IEEE Trans. Circuits Syst. I, vo1.42, no.3, pp.172-175.	1996	124
1,178	Ensemble Methods for Phoneme Classification Steve Waterhouse Gary Cook Cambridge University Engineering Department Cambridge CB2 IPZ, England, Tel: [+44] 1223 332754 Email: srwl00l@eng.cam.ac.uk.gdc@eng.cam.ac.uk Abstract This paper investigates a number of ensemble methods for improving the performance of phoneme classification for use in a speech recognition system. Two ensemble methods are described; boosting and mixtures of experts, both in isolation and in combination. Results are presented on two speech recognition databases: an isolated word database and a large vocabulary continuous speech database. These results show that principled ensemble methods such as boosting and mixtures provide superior performance to more naive ensemble methods such as averaging. INTRODUCTION There is now considerable interest in using ensembles or committees of learning machines to improve the performance of the system over that of a single learning machine. In most neural network ensembles, the ensemble members are trained on either the same data (Hansen & Salamon 1990) or different subsets of the data (Perrone & Cooper 1993). The ensemble members typically have different initial conditions and/or different architectures. The subsets of the data may be chosen at random, with prior knowledge or by some principled approach e.g. clustering. Additionally, the outputs of the networks may be combined by any function which results in an output that is consistent with the form of the problem. The expectation of ensemble methods is that the member networks pick out different properties present in the data, thus improving the performance when their outputs are combined. The two techniques described here, boosting (Drucker, Schapire & Simard 1993) and mixtures of experts (Jacobs, Jordan, Nowlan & Hinton 1991), differ from simple ensemble methods. Ensemble Methods/or Phoneme Classification 801 In boosting, each member of the ensemble is trained on patterns that have been filtered by previously trained members of the ensemble. In mixtures, the members of the ensemble, or "experts", are trained on data that is stochastically selected by a gate which additionally learns how to best combine the outputs of the experts. The aim of the work presented here is twofold and inspired from two differing but complimentary directions. Firstly, how does one select which data to train the ensemble members on and secondly, given these members how does one combine them to achieve the optimal result? The rest of the paper describes how a combination of boosting and mixtures may be used to improve phoneme error rates. PHONEME CLASSIFICATION Speech Figure 1: The ABBOT hybrid connectionist-HMM speech recognition system with an MLP ensemble acoustic model The Cambridge University Engineering Department connectionist speech recognition system (ABBOT) uses a hybrid connectionist - hidden Markov model (HMM) approach. This is shown in figure 1. A connectionist acoustic model is used to map each frame of acoustic data to posterior phone probabilities. These estimated phone probabilities are then used as estimates of the observation probabilities in an HMM framework. Given new acoustic data and the connectionist-HMM framework, the maximum a posteriori word sequence is then extracted using a single pass, start synchronous decoder. A more complete description of the system can be found in (Hochberg, Renals & Robinson 1994). Previous work has shown how a novel boosting procedure based on utterance selection can be used to increase the performance of the recurrent network acoustic model (Cook & Robinson 1996). In this work a combined boosting and mixturesof-experts approach is used to improve the performance of MLP acoustic models. Results are presented for two speech recognition tasks. The first is phonetic classification on a small isolated digit database. The second is a large vocabulary continuous speech recognition task from the Wall Street Journal corpus. ENSEMBLE METHODS Most ensemble methods can be divided into two separate methods; network selection and network combination. Network selection addresses the question of how to 802 S. Waterhouse and G. Cook choose the data each network is trained on. Network combination addresses the question of what is the best way to combine the outputs of these trained networks. The simplest method for network selection is to train separate networks on separate regions of the data, chosen either randomly, with prior knowledge or according to some other criteria, e.g. clustering. The simplest method of combining the outputs of several networks is to form an average, or simple additive merge: y(t) = k L~=l Yk(t), where Yk(t) is the output of the kth network at time t. Boosting Boosting is a procedure which results in an ensemble of networks. The networks in a boosting ensemble are trained sequentially on data that has been filtered by the previously trained networks in the ensemble. This has the advantage that only data that is likely to result in improved generalization performance is used for training. The first practical application of a boosting procedure was for the optical character recognition task (Drucker et al. 1993). An ensemble offeedforward neural networks was trained using supervised learning. Using boosting the authors reported a reduction in error rate on ZIP codes from the United States Postal Service of 28% compared to a single network. The boosting procedure is as follows: train a network on a randomly chosen subset of the available training data. This network is then used to filter the remaining training data to produce a training set for a second network with an even distribution of cases which the first network classifies correctly and incorrectly. After training the second network the first and second networks are used to produce a training set for a third network. This training set is produced from cases in the remaining training data that the first two networks disagree on. The boosted networks are combined using either a voting scheme or a simple add as described in the previous section. The voting scheme works as follows: if the first two networks agree, take their answer as the output, if they disagree, use the third network's answer as the output. Mixtures of Experts The mixture of experts (Jacobs et al. 1991) is a different type of ensemble to the two considered so far. The ensemble members or experts are trained with data which is stochastically selected by a 9ate. The gate in turn learns how to best combine the experts given the data. The training of the experts, which are typically single or multi-layer networks, proceeds as for standard networks, with an additional weighting of the output error terms by the posterior probabilty hi (t) of selecting expert i given the current data point at time (t): hi(t) = 9i(t).Pj(t) /Lj 9j(t).Pj(t) , where 9j(t) is the output of the gate for expert i, and Pj(t) is the probability of obtaining the correct output given expert i. In the case of classification, considered here, the experts use softmax output units. The gate, which is typically a single or multi-layered network with softmax output units is trained using the posterior probabilities as targets. The overall output y(t) ofthe mixture of experts is given by the weighted combination of the gate and expert outputs: y(t) = L~=l 9k(t).Yk(t), where Yk(t) is the output of the kth expert, and 9k(t) is the output of the gate for Ensemble Methods/or Phoneme Classification 803 expert k at time t. The mixture of experts is based on the principle of divide and conquer, in which a relatively hard problem is broken up into a series of smaller easier to solve problems. By using the posterior probabilities to weight the experts and provide targets for the gate, the effective data sets used to train each expert may overlap. SPEECH RECOGNITION RESULTS This section describes the results of experiments on two speech databases: the Bellcore isolated digits database and the Wall Street Journal Corpus (Paul & Baker 1992). The inputs to the networks consist of 9 frames of acoustic feature vectors; the frame on which the network is currently performing classification, plus 4 frames of left context and 4 frames of right context. The context frames allow the network to take account of the dynamical nature of speech. Each acoustic feature vector consists of 8th order PLP plus log energy coefficients along with the dynamic delta coeficients of these coefficients, computed with an analysis window of 25ms, every 12.5 ms at a sampling rate of 8kHz. The speech is labelled with 54 phonemes according to the standard ABBOT phone set. Bellcore Digits The Bellcore digits database consists of 150 speakers saying the words "zero" through "nine", "oh", "no" and "yes". The database was divided into a training set of 122 speakers, a cross validation set of 13 speakers and a test set of 15 speakers. Each method was evaluated over 10 partitions of the data into different training, cross validation and test sets. In all the experiments on the Bellcore digits multi-layer perceptrons with 200 hidden units were used as the basic network members in the ensembles. The gates in the mixtures were also multi-layer perceptrons with 20 hidden units. Ensemble Combination Phone Error Rate Method Average (J' Simple ensemble cheat 14.7 % 0.9 Simple ensemble vote 20.3 % 1.2 Simple ensemble average 19.3 % 1.2 Simple ensemble soft gated 20.9 % 1.2 Simple ensemble hard gated 19.3 % 1.0 Simple ensemble mixed 17.1 % 1.3 Boosted ensemble cheat 11.9 % 1.0 Boosted ensemble vote 17.8 % 1.1 Boosted ensemble average 17.4 % 1.1 Boosted ensemble soft gated 17.8 % 1.0 Boosted ensemble hard gated 17.4 % 1.2 Boosted ensemble mixed 16.4 % 1.0 Table 1: Comparison of phone error rates using different ensemble methods on the Bellcore isolated digits task. 804 S. Waterhouse and G. Cook Table 1 summarises the results obtained on the Bellcore digits database. The meaning of the entries are as follows. Two types of ensemble were trained: Simple Ensemble: consisting of 3 networks each trained on 1/3 of the training data each (corresponding to 40 speakers used for training and 5 for cross validation for each network), Boosted Ensemble: consisting of 3 networks trained according to the boosting algorithm of the previous section. Due to the relatively small size of the data set, it was necessary to ensure that the distributions of the randomly chosen data were consistent with the overall training data distribution. Given each set of ensemble networks, 6 combination methods were evaluated: cheat: The cheat scheme uses the best ensemble member for each example in the data set. The best ensemble member is determined by looking at the correct label in the labelled test set (hence cheating). This method is included as a lower bound on the error. Since the tests are performed on unseen data, this bound can only be approached by learning an appropriate combination function of the ensemble member outputs. average: The ensemble members' outputs are combined using a simple average. vote: The voting scheme outlined in the previous section. gated: In the gated combination method, the ensemble networks were kept fixed whilst the gate was trained. Two types of gating were evaluated, standard or sojtgating, and hard or winner take all (WTA) training. In WTA training the targets for the gate are binary, with a target of l.0 for the output corresponding to the expert whose probability of generating the current data point correctly is greatest, and 0.0 for the other outputs. mixed: In contrast to the gated method, the mixed combination method both trains a gate and retrains the ensemble members using the mixture of experts framework. From these results it can be concluded that boosting provides a significant improvement in performance over a simple ensemble. In addition, by training a gate to combine the boosted networks performance can be further enhanced. As might be expected, re-training both the boosted networks and the gate provides the biggest improvement, as shown by the result for the mixed boosted networks. Wall Street Journal Corpus The training data used in these experiments is the short term speakers from the Wall Street Journal corpus. This consists of approximately 36,400 sentences from 284 different speakers (SI284). The first network is trained on l.5 million frames randomly selected from the available training data (15 million frames). This is then used to filter the unseen training data to select frames for training the second network. The first and second networks are then used to select data for the third network as described previously. The performance of the boosted MLP ensemble Ensemble Methods/or Phoneme Classification 805 Test Language Word Error Rate Set Model Lexicon Single MLP Boosted Gated Mixed eLh2_93 trigram 20k 16.0% 12.9% 12.9% 11.2 % dt....s5_93 bigram 5k 20.4% 16.5% 16.5% 15.1% Table 2: Evaluation of the performance of boosting MLP acoustic models was evaluated on a number of ARPA benchmark tests. The results are summarised in Table 2. Initial experiments use the November 1993 Hub 2 evaluation test set (eLh2-93) . This is a 5,000 word closed vocabulary, non-verbalised punctuation test. It consists of 200 utterances, 20 from each of 10 different speakers, and is recorded using a Sennheiser HMD 410 microphone. The prompting texts are from the Wall Street Journal. Results are reported for a system using the standard ARPA 5k bigram language model. The Spoke 5 test (dt....s5_93) is designed for evaluation of unsupervised channel adaptation algorithms. It consists of a total of 216 utterances from 10 different speakers. Each speaker's data was recorded with a different microphone. In all cases simultaneous recordings were made using a Sennheiser microphone. The task is a 5,000 word, closed vocabulary, non-verbalised punctuation test. Results are only reported for the data recorded using the Sennheiser microphone. This is a matched test since the same microphone is used to record the training data. The standard ARPA 5k bigram language model was used for the tests. Further details of the November 1993 spoke 5 and hub 2 tests, can be found in (Pallett, Fiscus, Fisher, Garofolo, Lund & Pryzbocki 1994). Four techniques were evaluated on the WSJ corpus; a single network with 500 hidden units, a boosted ensemble with 3 networks with 500 hidden units each, a gated ensemble of the boosted networks and a mixture trained from boosted ensembles. As can be seen from the table, boosting has resulted in significant improvements in performance for both the test sets over a single model. In addition, in common with the results on the Bellcore digits, whilst the gating combination method does not give an improvement over simple averaging, the retraining of the whole ensemble using the mixed combination method gives an average improvement of a further 8% over the averaging method. CONCLUSION This paper has described a number of ensemble methods for use with neural network acoustic models. It has been shown that through the use of principled methods such as boosting and mixtures the performance of these models may be improved over standard ensemble techniques. In addition, by combining the techniques via boot-strapping mixtures using the boosted networks the performance of the models can be improved further. Previous work, which focused on boosting at the word level showed improvements for a recurrent network:HMM hybrid at the word level over the baseline system (Cook & Robinson 1996). This paper has shown how the performance of a static MLP system can also be improved by boosting at the frame level. 806 S. Waterhouse and G. Cook Acknowledgements Many thanks to Bellcore for providing the digits data set to our partners, ICSI; Nikki Mirghafori for help with datasets; David Johnson for providing the starting point for our code development; and Dan Kershaw for his invaluable advice. References Cook, G. & Robinson, A. (1996), Boosting the performance of connectionist large -vocabulary speech recognition, in 'International Conference on Spoken Language Processing'. Drucker, H., Schapire, R. & Simard, P. (1993), Improving Performance in Neural Networks Using a Boosting Algorithm, in S. Hanson, J. Cowan & C. Giles, eds, 'Advances in Neural Information Processing Systems 5', Morgan Kauffmann, pp.42-49. Hansen, L. & Salamon, P. (1990), 'Neural Network Ensembles', IEEE Transactions on Pattern Analysis and Machine Intelligence 12, 993-1001. Hochberg, M., Renals, S. & Robinson, A. (1994), 'ABBOT: The CUED hybrid connectionist-HMM large-vocabulary recognition system', Proc. of Spoken Language Systems Technology Worshop, ARPA. Jacobs, R. A., Jordan, M. I., Nowlan, S. J . & Hinton, G. E. (1991), 'Adaptive mixtures oflocal experts' , Neural Computation 3 (1), 79-87. Pallett, D., Fiscus, J., Fisher, W., Garofolo, J., Lund, B. & Pryzbocki, M. (1994), '1993 Benchmark Tests for the ARPA Spoken Language Program', ARPA Workshop on Human Language Technology pp. 51-73. Merrill Lynch Conference Center, Plainsboro, NJ. Paul, D. & Baker, J . (1992), The Design for the Wall Street Journal-based CSR Corpus, in 'DARPA Speech and Natural Language Workshop', Morgan Kaufman Publishers, Inc., pp. 357-62. Perrone, M. P. & Cooper, L. N. (1993), When networks disagree: Ensemble methods for hybird neural networks, in 'Neural Networks for Speech and Image Processing', Chapman-Hall.	1996	125
1,179	Maximum Likelihood Blind Source Separation: A Context-Sensitive Generalization of ICA Barak A. Pearlmutter Computer Science Dept, FEC 313 University of New Mexico Albuquerque, NM 87131 bap@cs.unm.edu Abstract Lucas C. Parra Siemens Corporate Research 755 College Road East Princeton, N J 08540-6632 lucas@scr.siemens.com In the square linear blind source separation problem, one must find a linear unmixing operator which can detangle the result Xi(t) of mixing n unknown independent sources 8i(t) through an unknown n x n mixing matrix A( t) of causal linear filters: Xi = E j aij * 8 j . We cast the problem as one of maximum likelihood density estimation, and in that framework introduce an algorithm that searches for independent components using both temporal and spatial cues. We call the resulting algorithm "Contextual ICA," after the (Bell and Sejnowski 1995) Infomax algorithm, which we show to be a special case of cICA. Because cICA can make use of the temporal structure of its input, it is able separate in a number of situations where standard methods cannot, including sources with low kurtosis, colored Gaussian sources, and sources which have Gaussian histograms. 1 The Blind Source Separation Problem Consider a set of n indepent sources 81 (t), . .. ,8n (t). We are given n linearly distorted sensor reading which combine these sources, Xi = E j aij8j, where aij is a filter between source j and sensor i, as shown in figure 1a. This can be expressed as 00 Xi(t) = 2: 2: aji(r)8j(t - r) = 2: aji * 8j j r=O j 614 B. A. Pearlmutter and L. C. Parra IftY/(t )IY/(t-l) • ... ;,,(1» f-h-..... ~,,---------..,------~ 11"f, ~Y~~-+~-+-~ x. Figure 1: The left diagram shows a generative model of data production for blind source separation problem. The cICA algorithm fits the reparametrized generative model on the right to the data. Since (unless the mixing process is singular) both diagrams give linear maps between the sources and the sensors, they are mathematicallyequivalent. However, (a) makes the transformation from s to x explicit, while (b) makes the transformation from x to y, the estimated sources, explicit. or, in matrix notation, x(t) = L~=o A(T)S(t - T) = A * s. The square linear blind source separation problem is to recover S from x. There is an inherent ambiguity in this, for if we define a new set of sources s' by s~ = bi * Si where bi ( T) is some invertable filter, then the various s~ are independent, and constitute just as good a solution to the problem as the true Si, since Xi = Lj(aij * bjl) * sj. Similarly the sources could be arbitrarily permuted. Surprisingly, up to permutation of the sources and linear filtering of the individual sources, the problem is well posed-assuming that the sources Sj are not Gaussian. The reason for this is that only with a correct separation are the recovered sources truly statistically independent, and this fact serves as a sufficient constraint. Under the assumptions we have made, I and further assuming that the linear transformation A is invertible, we will speak of recovering Yi(t) = Lj Wji * Xj where these Yi are a filtered and permuted version of the original unknown Si. For clarity of exposition, will often refer to "the" solution and refer to the Yi as "the" recovered sources, rather than refering to an point in the manifold of solutions and a set of consistent recovered sources. 2 Maximum likelihood density estimation Following Pham, Garrat, and Jutten (1992) and Belouchrani and Cardoso (1995), we cast the BSS problem as one of maximum likelihood density estimation. In the MLE framework, one begins with a probabilistic model of the data production process. This probabilistic model is parametrized by a vector of modifiable parameters w, and it therefore assigns a w-dependent probability density p( Xo, Xl, ... ; w) to a each possible dataset xo, Xl, .... The task is then to find a w which maximizes this probability. There are a number of approaches to performing this maximization. Here we apply lWithout these assumptions, for instance in the presence of noise, even a linear mixing process leads to an optimal un mixing process that is highly nonlinear. Maximum Likelihood Blind Source Separation: ContextuallCA 615 the stochastic gradient method, in which a single stochastic sample x is chosen from the dataset and -dlogp(x; w)/dw is used as a stochastic estimate of the gradient of the negative likelihood 2:t -dlogp(x(t); w)/dw. 2.1 The likelihood of the data The model of data production we consider is shown in figure 1a. In that model, the sensor readings x are an explicit linear function of the underlying sources s. In this model of the data production, there are two stages. In the first stage, the sources independently produce signals. These signals are time-dependent, and the probability density of source i producing value Sj(t) at time t is f;(Sj(t)lsj(t 1), Sj(t - 2), ... ). Although this source model could be of almost any differentiable form, we used a generalized autoregressive model described in appendix A. For expository purposes, we can consider using a simple AR model, so we model Sj(t) = bj(l)sj(t -1) + bj(2)sj(t - 2) + ... + bj(T)sj(t - T) + Tj, where Tj is an iid random variable, perhaps with a complicated density. It is important to distinguish two different, although related, linear filters. When the source models are simple AR models, there are two types of linear convolutions being performed. The first is in the way each source produces its signal: as a linear function of its recent history plus a white driving term, which could be expressed as a moving average model, a convolution with a white driving term, Sj = bj * Tj. The second is in the way the sources are mixed: linear functions of the output of each source are added, Xi = 2:j aji * Sj = 2: j (aji * bj) *Tj. Thus, with AR sources, the source convolution could be folded into the convolutions of the linear mixing process. If we were to estimate values for the free parameters of this model, i.e. to estimate the filters, then the task of recovering the estimated sources from the sensor output would require inverting the linear A = (aij), as well as some technique to guarantee its non-singularity. Such a model is shown in figure 1a. Instead, we parameterize the model by W = A-I, an estimated unmixing matrix, as shown in figure lb. In this indirect representation, s is an explicit linear function of x, and therefore x is only an implicit linear function of s. This parameterization of the model is equally convenient for assigning probabilities to samples x, and is therefore suitable for MLE. Its advantage is that because the transformation from sensors to sources is estimated explicitly, the sources can be recovered directly from the data and the estimated model, without invertion. Note that in this inverse parameterization, the estimated mixture process is stored in inverse form. The source-specific models Ii are kept in forward form. Each source-specific model i has a vector of parameters, which we denote w(i). We are now in a position to calculate the likelihood of the data. For simplicity we use a matrix W of real numbers rather than FIR filters. Generalizing this derivation to a matrix of filters is straightforward, following the same techniques used by Lambert (1996), Torkkola (1996), A. Bell (1997), but space precludes a derivation here. The individual generative source models give p(y(t)ly(t - 1), y(t - 2), ... ) = II Ii(Yi(t)IYi(t - 1), Yi(t - 2), ... ) (1) 616 B. A. Pear/mutter and L. C. Parra where the probability densities h are each parameterized by vectors w(i). Using these equations, we would like to express the likelihood of x(t) in closed form, given the history x(t - 1), x(t - 2), .... Since the history is known, we therefore also know the history of the recovered sources, y(t - 1),y(t - 2), .... This means that we can calculate the density p(y(t)ly(t - 1), . .. ). Using this, we can express the density of x(t) and expand G = logp(x; w) = log IWI + 2:j log fj(Yj(t)IYj(t 1), Yj(t - 2), ... ; wU») There are two sorts of parameters which we must take the derivative with respect to: the matrix W and the source parameters wU). The source parameters do not influence our recovered sources, and therefore have a simple form dG dfJ(Yj;wj)/dwj dWj fj(Yj; Wj) However, a change to the matrix W changes y, which introduces a few extra terms. Note that dlog IWI/dW = W- T , the transpose inverse. Next, since y = Wx, we see that dYj/dW = (OlxIO)T, a matrix of zeros except for the vector x in row j . Now we note that dfJO/dW term has two logical components: the first from the effect of changing W upon Yj(t), and the second from the effect of changing W upon Yj(t -1), Yj(t - 2), .... (This second is called the "recurrent term", and such terms are frequently dropped for convenience. As shown in figure 3, dropping this term here is not a reasonable approximation.) dfJ(Yj(t)IYj(t-1), ... ;wj) = afj dYj(t) + 2: afJ dYj(t-T) dW aYj(t) dW aYj(t - T) dW T Note that the expression dYij:;T) is the only matrix, and it is zero except for the jth row, which is x(t - T). The expression afJ/aYj(t) we shall denote fjO, and the expression afjaYj(t - T) we shall denote f(T}(.). We then have ! = _W-T (f~(:)) x(tf - f (ft}:·)) x(t - Tf (2) fJ() j T=l fJ() j where (expr(j))j denotes the column vector whose elements are expr(1), . .. , expr(n). 2.2 The natural gradient Following Amari, Cichocki, and Yang (1996), we follow a pseudogradient. Instead of using equation 2, we post-multiply this quantity by WTW. Since this is a positivedefinite matrix, it does not affect the stochastic gradient convergence criteria, and the resulting quantity simplifies in a fashion that neatly eliminates the costly matrix inversion otherwise required. Convergence is also accelerated. 3 Experiments We conducted a number of experiments to test the efficacy of the cICA algorithm. The first, shown in figure 2, was a toy problem involving a set of processed deliberately constructed to be difficult for conventional source separation algorithms. In the second experiment, shown in figure 3, ten real sources were digitally mixed with an instantaneous matrix and separation performance was measured as a funciton of varying model complexity parameters. These sources have are available for benchmarking purposes in http://www.cs.unm.edu;-bap/demos.html. Maximum Likelihood Blind Source Separation: ContextuallCA 617 Figure 2: cICA using a history of one time step and a mixture of five logistic densities for each source was applied to 5,000 samples of a mixture of two one-dimensional uniform distributions each filtered by convolution with a decaying exponential of time constant of 99.5. Shown is a scatterplot of the data input to the algorithm, along with the true source axes (left), the estimated residual probability density (center), and a scatterplot of the residuals of the data transformed into the estimated source space coordinates (right). The product of the true mixing matrix and the estimated unmixing matrix deviates from a scaling and permutation matrix by about 3%. o Truncated Gradient Full Gradient Noise Model 5 10 t5 20 o number 01 AR filter taps 5 10 15 20 number 01 AR filter taps 100 ·8 II ~ 10 number oIlogistica 2 Figure 3: The performance of cICA as a function of model complexity and gradient accuracy. In all simulations, ten five-second clips taken digitally from ten audio CD were digitally mixed through a random ten-by-ten instantanious mixing matrix. The signal to noise ratio of each original source as expressed in the recovered sources is plotted. In (a) and (b), AR source models with a logistic noise term were used, and the number of taps of the AR model was varied. (This reduces to Bell-Sejnowski infomax when the number of taps is zero.) Is (a), the recurrent term of the gradient was left out, while in (b) the recurrent term was included. Clearly the recurrent term is important. In (c), a degenerate AR model with zero taps was used, but the noise term was a mixture of logistics, and the number of logistics was varied. 4 Discussion The Infomax algorithm (Baram and Roth 1994) used for source separation (Bell and Sejnowski 1995) is a special case of the above algorithm in which (a) the mixing is not convolutional, so W(l) = W(2) = ... = 0, and (b) the sources are assumed to be iid, and therefore the distributions fi(y(t)) are not history sensitive. Further, the form of the Ii is restricted to a very special distribution: the logistic density, 618 B. A. Pearlmuner and L. C. Parra the derivative of the sigmoidal function 1/{1 + exp -{). Although ICA has enjoyed a variety of applications (Makeig et al. 1996; Bell and Sejnowski 1996b; Baram and Roth 1995; Bell and Sejnowski 1996a), there are a number of sources which it cannot separate. These include all sources with Gaussian histograms (e.g. colored gaussian sources, or even speech to run through the right sort of slight nonlinearity), and sources with low kurtosis. As shown in the experiments above, these are of more than theoretical interest. If we simplify our model to use ordinary AR models for the sources, with gaussian noise terms of fixed variance, it is possible to derive a closed-form expression for W (Hagai Attias, personal communication). It may be that for many sources of practical interest, trading away this model accuracy for speed will be fruitful. 4.1 Weakened assumptions It seems clear that, in general, separating when there are fewer microphones than sources requires a strong bayesian prior, and even given perfect knowledge of the mixture process and perfect source models, inverting the mixing process will be computationally burdensome. However, when there are more microphones than sources, there is an opportunity to improve the performance of the system in the presence of noise. This seems straightforward to integrate into our framework. Similarly, fast-timescale microphone nonlinearities are easily incorporated into this maximum likelihood approach. The structure of this problem would seem to lend itself to EM. Certainly the individual source models can be easily optimized using EM, assuming that they themselves are of suitable form. References A. Bell, T.-W. L. (1997). Blind separation of delayed and convolved sources. In Advances in Neural Information Processing Systems 9. MIT Press. In this volume. Amari, S., Cichocki, A., and Yang, H. H. (1996). A new learning algorithm for blind signal separation. In Advances in Neural Information Processing Systems 8. MIT Press. Baram, Y. and Roth, Z. (1994). Density Shaping by Neural Networks with Application to Classification, Estimation and Forecasting. Tech. rep. CIS-9420, Center for Intelligent Systems, Technion, Israel Institute for Technology, Haifa. Baram, Y. and Roth, Z. (1995). Forecasting by Density Shaping Using Neural Networks. In Computational Intelligence for Financial Engineering New York City. IEEE Press. Bell, A. J. and Sejnowski, T. J. (1995). An Information-Maximization Approach to Blind Separation and Blind Deconvolution. Neural Computation, 7(6), 1129-1159. Bell, A. J. and Sejnowski, T. J. (1996a). The Independent Components of Natural Scenes. Vision Research. Submitted. Maximum Likelihood Blind Source Separation: ContextuallCA 619 Bell, A. J. and Sejnowski, T. J. (1996b). Learning the higher-order structure of a natural sound. Network: Computation in Neural Systems. In press. Belouchrani, A. and Cardoso, J.-F. (1995). Maximum likelihood source separation by the expectation-maximization technique: Deterministic and stochastic implementation. In Proceedings of 1995 International Symposium on Non-Linear Theory and Applications, pp. 49- 53 Las Vegas, NV. In press. Lambert, R. H. (1996). Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures. Ph.D. thesis, USC. Makeig, S., Anllo-Vento, L., Jung, T.-P., Bell, A. J., Sejnowski, T. J., and Hillyard, S. A. (1996). Independent component analysis of event-related potentials during selective attention. Society for Neuroscience Abstracts, 22. Pearlmutter, B. A. and Parra, L. C. (1996). A Context-Sensitive Generalization of ICA. In International Conference on Neural Information Processing Hong Kong. Springer-Verlag. Url ftp:/ /ftp.cnl.salk.edu/pub/bap/iconip-96cica.ps.gz. Pham, D., Garrat, P., and Jutten, C. (1992). Separation of a mixture of independent sources through a maximum likelihood approach. In European Signal Processing Conference, pp. 771-774. Torkkola, K. (1996). Blind separation of convolved sources based on information maximization. In Neural Networks for Signal Processing VI Kyoto, Japan. IEEE Press. In press. A Fixed mixture AR models The fj{uj; Wj) we used were a mixture AR processes driven by logistic noise terms, as in Pearlmutter and Parra (1996). Each source model was fj{Uj{t)IUj{t -1), Uj{t - 2), ... ; Wj) = I: mjk h{{u){t) - Ujk)/Ujk)/Ujk (3) k where Ujk is a scale parameter for logistic density k of source j and is an element of Wj, and the mixing coefficients mjk are elements of Wj and are constrained by 'Ek mjk = 1. The component means Ujk are taken to be linear functions of the recent values of that source, Ujk = L ajk(r) Uj{t - r) + bjk (4) T=l where the linear prediction coefficients ajk{r) and bias bjk are elements of Wj' The derivatives of these are straightforward; see Pearlmutter and Parra (1996) for details. One complication is to note that, after each weight update, the mixing coefficients must be normalized, mjk t- mjk/ 'Ekl mjk' .	1996	126
1,180	Temporal Low-Order Statistics of Natural Sounds H. Attias· and C.E. Schreinert Sloan Center for Theoretical Neurobiology and W.M. Keck Foundation Center for Integrative Neuroscience University of California at San Francisco San Francisco, CA 94143-0444 Abstract In order to process incoming sounds efficiently, it is advantageous for the auditory system to be adapted to the statistical structure of natural auditory scenes. As a first step in investigating the relation between the system and its inputs, we study low-order statistical properties in several sound ensembles using a filter bank analysis. Focusing on the amplitude and phase in different frequency bands, we find simple parametric descriptions for their distribution and power spectrum that are valid for very different types of sounds. In particular, the amplitude distribution has an exponential tail and its power spectrum exhibits a modified power-law behavior, which is manifested by self-similarity and long-range temporal correlations. Furthermore, the statistics for different bands within a given ensemble are virtually identical, suggesting translation invariance along the cochlear axis. These results show that natural sounds are highly redundant, and have possible implications to the neural code used by the auditory system. 1 Introduction The capacity of the auditory system to represent the auditory scene is restricted by the finite number of cells and by intrinsic noise. This fact limits the ability of the organism to discriminate between different sounds with similar spectro-temporal ·Corresponding author. E-mail: hagai@phy.ucsf.edu. tE-mail: chris@phy.ucsf.edu. 28 H. Attias and C. E. Schreiner characteristics. However, it is possible to enhance the discrimination ability by a suitable choice of the encoding procedure used by the system, namely of the transformation of sounds reaching the cochlea to neural spike trains generated in successive processing stages in response to these sounds. In general, the choice of a good encoding procedure requires knowledge of the statistical structure of the sound ensemble. For the visual system, several investigations of the statistical properties of image ensembles and their relations to neuronal response properties have recently been performed (Field 1987, Atick and Redlich 1990, Ruderman and Bialek 1994). In particular, receptive fields of retinal ganglion and LG N cells were found to be consistent with an optimal-code prediction formulated within information theory (Atick 1992, Dong and Atick 1995), suggesting that the visual periphery may be designed as to take advantage of simple statistical properties of visual scenes. In order to investigate whether the auditory system is similarly adapted to the statistical structure of its own inputs, a good characterization of auditory scenes is necessary. In this paper we take a first step in this direction by studying low-order statistical properties of several sound ensembles. The quantities we focus on are the spectro-temporal amplitude and phase defined as follows. For the sound s(t), let SII(t) denote its components at the set of frequencies v, obtained by filtering it through a bandpass filter bank centered at those frequencies. Then SII(t) = XII (t)cos (vt + rPlI(t)) (1) where xlI(t) ~ 0 and rPlI(t) are the spectro-temporal amplitude (STA) and phase (STP), respectively. A complete characterization of a sound ensemble with respect to a given filter bank must be given by the joint distribution of amplitudes and phases at all times, P (XlIl (tl), rPlII (tD, ... , XII" (tn ), rPlI" (t~)). In this paper, however, we restrict ourselves to second-order statistics in the time domain and examine the distribution and power spectrum of the stochastic processes xlI(t) and rPlI(t). Note that the STA and STP are quantities directly relevant to auditory processing. The different stages of the auditory system are organized in topographic frequency maps, so that cells tuned to the same sound frequency v are organized in stripes perpendicular to the direction of frequency progression (see, e.g., Pickles 1988). The neuronal responses are thus determined by XII and rPlI' and by XII alone when phase-locking disappears above 4-5KHz. 2 Methods Since it is difficult to obtain a reliable sample of an animal's auditory scene over a sufficiently long time, we chose instead to analyze several different sound ensembles, each consisting of a 15min sound of a certain type. We used cat vocalizations, bird songs, wolf cries, environmental sounds, symphonic music, jazz, pop music, and speech. The sounds were obtained from commercially available compact discs and from recordings of animal vocalizations in two laboratories. No attempt has been made to manipulate the recorded sounds in any way (e.g., by removing noise). Each sound ensemble was loaded into the computer by 30sec segments at a sampling rate of Is = 44.1KHz. After decimating to Is/2, we performed the following frequency-band analysis. Each segment was passed through a bandpass filTemporal Low-Order Statistics of Natural Sounds 29 Symphonic music Speech Or---~------~----~--~ Or---~------~--------~ -0.5 _ -1 as 0:: 0-1.5 ~ g> _3L---~------~----~~~ -2 o 2 Cat vocalizations -0.5 _ -1 as 0:: 0-1.5 ~ g> -2 -2 o 2 Environmental sounds Or---~------~----~--~ O~--~------~----~--~ -0.5 _ -1 -5: 0-1 .5 8> -2 -2.5 -0.5 _ -1 ~ CI.. 0-1.5 ~ E -2 _3ll-----------~----~--~ _3L-----------~----~--~ -2 0 2 -2 0 2 a=log10(x) a=log10(x) Figure 1: Amplitude probability distribution in different frequency bands for four sound ensembles. ter bank with impulse responses hv{t) to get the narrow-band component signals sv{t) = s(t) * hv{t). We used square, non-overlapping filters with center frequencies II logarithmically spaced within the range of 100 - 11025Hz. The filters were usually 1/8-octave wide, but we experimented with larger bandwidths as well. The amplitude and phase in band II were then obtained via the Hilbert transform i J s{t') H [sv{t)] = sv{t) + :; dt' t _ t' = xv{t)ei(vHtPv(t» . (2) The frequency content of Xv is bounded by 0 and by the bandwidth of hv (Flanagan 1980), so keeping the latter below II guarantees that the low frequencies in sv are all contained in Xv, confirming its interpretation as the amplitude modulator of the carrier cos lit suggested by (1). The phase ,pv, being time-dependent, produces frequency modulation. For a given II the results were averaged over all segments. 3 Amplitude Distribution We first examined the STA distribution in different frequency bands II. Fig. 1 presents historgrams of P{IOglO xv) on a logarithmic scale for four different sound ensembles. In order to facilitate a comparison among different bands and ensembles, we normalized the variable to have zero mean and unit variance, (loglO xv(t)) = 0, ((IOglO x v (t))2) = 1, corresponding to a linear gain control. 30 Symphonic music o.---~------------~---. -0.5 _ -1 :§: c.. 0'-1 .5 g -2 -2.5 H. Attias and C. E. Schreiner Speech o.---~------------~---. -0.5 = -1 S c.. 0'-1.5 'Ol .Q -2 -2.5 -3~--~------~----~--~ _3~--~------~----~-L~ -2 o 2 -2 0 2 a=10910(x) a=10910(x) Figure 2: n-point averaged amplitude distributions for v = 800Hz in two sound ensembles, using n = 1,20,50,100,200. The speech ensemble is different from the one used in Fig. 1. As shown in the figure, within a given ensemble, the histograms corresponding to different bands lie atop one another. Furthermore, although curves from different ensembles are not identical, we found that they could all be fitted accurately to the same parametric functional form, given by e-'")'Z", p(x,,) ex (b5 + X~){J/2 (3) with parameter values roughly in the range of 0.1 ~ 'Y ~ 1, 0 ~ f3 ~ 2.5, and 0.1 ~ bo ~ 0.6. In some cases, a mixture of two distributions of the form (3) was necessary, suggesting the presence of two types of sound sources; see, e.g., the slight bimodality in the lower parts of Fig. 1. Details of the fitting procedure will be given in a longer paper. We found the form (3) to be preserved as the filter bandwidths increased. Whereas this distribution decays exponentially fast at high amplitudes (p ex e-'")'z", /xe), it does not vanish at low amplitudes, indicating a finite probability for the occurence of arbitrarily soft sounds. In contrast, the STA of a Gaussian noise signal can be shown to be distributed according to p ex x"e-'\z~, which vanishes at x" = 0 and decays faster than (3) at large x". Hence, the origin of the large dynamic range usually associated with audio signals can be traced to the abundance of soft sounds rather than of loud ones. 4 Amplitude Self-Similarity An interesting probe of the STA temporal correlations is the property of scale invariance (also called statistical self-similarity). The process x,,(t) is scale-invariant when any statistical quantity on a given scale (e.g., at a given temporal resolution, determined by the sampling rate) does not change as that scale is varied. To observe this property we examined the STA distribution p(x,,) at different temporal resolutions, by defining the n-point averaged amplitude 1 n-l x~n)(t) = - L x,,(t + k6.) n k=O (4) Temporal Low-Order Statistics of Natural Sounds Symphonic music or-__ ~~--~----------~ -0.5 -1 12 - 1.5 en 0" -2 ~-2.5 -3 -3.5 -4~~------~----~----~ -1 o 1 2 Cat vocalizations Or--.~----~----------~ -1 6: en '0-2 g. -3 -4~~------~----~----~ -1 o 1 log10(f) 31 Speech 0 -1 6: en '0-2 ~ -3 -4 -1 0 1 2 Environmental sounds Or-~------~-----------. -1 ~ '0-2 i -3 -4~~------~----~----~ -1 o 1 2 log10(f) Figure 3: Amplitude power spectrum in different frequency bands for four sound ensembles. (A = 1/ is) and computing its distribution. Fig. 2 displays the histograms of P(IOglO x~n) for the II = 800Hz frequency band in two sound ensembles on a logarithmic scale, using n = 1,20,50, 100, 200 which correspond to a temporal resolution range of 0.75 - 150msec. Remarkably, the histogram remains unmodified even for n = 200. Had the xlI(t + kA) been statistically independent variables, the central limit theorem would have predicted a Gaussian p(x~n) for large n. The fact that this non-Gaussian distribution preserves its form as n increases implies the presence of temporal STA correlations over long periods. Notice the analogy between the invariance of p(xII) under a change in filter bandwidth, reported in the previous section, and under a change in temporal resolution. An XII with a broad bandwidth is essentially an average over the XII'S with narrow bandwidth within the same band, thus bandwidth invariance is a manifestation of STA correlations across frequency bands. 5 Amplitude Power Spectrum In order to study the temporal amplitude correlations directly, we computed the STA power spectrum BII(w) = (I XII(W) 12) in different bands II, where xII(w) is the Fourier transform of the log-amplitude loglO xlI(t) obtained by a 512-point FFT. As is well-known, the spectrum BII(w) is the Fourier transform of the logamplitude auto-correlation function clI(r) = (IOglO xlI(t) loglO xlI(t + r»). We used 32 H. Attias and C. E. Schreiner the zero-mean, unit-variance normalization of IOglO Xv, which implies the normalization J dJ..J8v (w) = const. of the spectra. Fig. 3 presents 8v as a function of the modulation frequency j = w /21r on a logarithmic scale for four different sound ensembles. Notice that, as in the case of the STA distribution, the different curves corresponding to different frequency bands within a given ensemble lie atop one another, including individual peaks; and whereas spectra in different ensembles are not identical, we found a simple parametric description valid for all ensembles which is given by (5) with parameter values roughly in the range of 1 ::; a ::; 2.5 and 10-4 ::; Wo ::; 1. This is a modified power-law form (note that 8v -+ C / wQ at large w), implying longrangle temporal correlations in the amplitude: these correlations decrease slowly (as a power law in t) on a time scale of l/wo, beyond which they decay exponentially fast. Larger Wo contributes more to the flattening of the spectrum at low frequencies (see especially the speech spectra) and corresponds to a shorter correlation time. Again, in some cases a sum of two such forms was necessary, corresponding to a mixture STA distribution as mentioned above; see, e.g., the environmental sound spectra (lower right part of Fig. 3 and Fig. 1). The form (5) persisted as the filter bandwidth increased. In the limit of allpass filter (not shown) we still observed this form, a fact related to the report of (Voss and Clarke 1975) on 1/ j-like power spectra of sound 'loudness' S(t)2 found in several speech and music ensembles. 6 Phase Distribution and Power Spectrum Whereas the STA is a non-stationary process which is locally stationary and can thus be studied on the appropriate time scale using our methods, the STP is nonstationary even locally. A more suitable quantity to examine is its rate of change d¢v / dt, called the instantaneous frequency. We studied the statistics of I d¢v / dt I in different ensembles, and found its distribution to be described accurately by the parametric form (3) with 'Y = 0, whereas its power spectrum could be well fitted by the form (5). In addition, those quantities were virtually identical in different bands within a given ensemble. More details on this work will be provided in a longer paper. 7 Implications for Auditory Processing We have shown that auditory scenes have several robust low-order statistical properties. The STA power spectrum has a modified power-law behavior, which is manifested in self-similarity and temporal correlations over a few hundred milliseconds. The distribution has an exponential tail and features a finite probability for arbitrarily soft sounds. Both the phase and amplitude statistics can be described by simple parametrized functional forms which are valid for very different types of sounds. These results lead to the conclusion that natural sounds are highly redundant, i.e., they occupy a very small subspace in the space of all possible sounds. It would therefore be beneficial for the auditory system to adapt its sound representation to these statistics, thus improving the animal discrimination ability. Whether Temporal Low-Order Statistics of Natural Sounds 33 the auditory system actually follows this design principle is an empirical question which can be attacked by suitable experiments. Furthermore, since different frequency bands correspond to different spatial locations on the basal membrane (Pickles 1988), the fact that the distributions and spectra in different bands within a given ansemble are identical suggests the existence of translation invariance along the cochlear axis, i.e., all the locations in the cochlea 'see' the same statistics. This is analogous to the translation invariance found in natural images. Finally, a recent theory for peripheral visual processing (Dong and Atick 1995) proposes that, in order to maximize information transmission into cortex, the LGN performs temporal correlation of retinal images. Within an analogous auditory model, the decorrelation time for sound ensembles reported here implies that the auditory system should process incoming sounds by a few hundred msec-Iong segments. The ability of cortical neurons to follow in their response modulation rates near and below 10Hz but usually not higher (Schreiner and Urbas 1988) may reflect such a process. Acknowledgements We thank B. Bonham, K. Miller, S. Nagarajan, and especially W. Bialek for helpful discussions and suggestions. We also thank F. Theunissen for making his bird song recordings available to us. Supported by The Office of Naval Research (NOOOI4-941-0547). H.A. was supported by a Sloan Foundation grant for the Sloan Center for Theoretical Neurobiology. References J.J. Atick and N. Redlich (1990), Towards a theory of early visual processing. Neural Comput. 2, 308-320. J.J. Atick (1992), Could information theory provide an ecological theory of sensory processing. Network 3, 213-25l. D.W. Dong and J.J. Atick (1995), Temporal decorrelation: a theory of lagged and non-lagged responses in the lateral geniculate nucleus. Network 6, 159-178. D.J. Field (1987), Relations between the statistics of natural images and the response properties of cortical cells. J. Opt. Soc. Am. 4, 2379-2394. J.L. Flanagan (1980), Parametric coding of speech spectra. J. Acoust. Soc. Am. 68, 412-419. J.O. Pickles (1988), An introduction to the physiology of hearing (2nd Ed.). San Diego, CA: Academic Press. D.L. Ruderman and W. Bialek (1994), Statistics of natural images: scaling in the woods. Phys. Rev. Lett. 73, 814-817. C.E. Schreiner and J.V. Urbas, Representation of amplitude modulation in the auditory cortex of the cat. II. Comparison between cortical fields. Hear. Res. 32, 49-63. R.F. Voss and J. Clarke (1975), 1/ f noise in music and speech. Nature 258,317-318.	1996	127
1,181	Limitations of self-organizing maps for vector quantization and multidimensional scaling Arthur Flexer The Austrian Research Institute for Artificial Intelligence Schottengasse 3, A-lOlO Vienna, Austria and Department of Psychology, University of Vienna Liebiggasse 5, A-lOlO Vienna, Austria arthur~ai.univie.ac.at Abstract The limitations of using self-organizing maps (SaM) for either clustering/vector quantization (VQ) or multidimensional scaling (MDS) are being discussed by reviewing recent empirical findings and the relevant theory. SaM's remaining ability of doing both VQ and MDS at the same time is challenged by a new combined technique of online K-means clustering plus Sammon mapping of the cluster centroids. SaM are shown to perform significantly worse in terms of quantization error, in recovering the structure of the clusters and in preserving the topology in a comprehensive empirical study using a series of multivariate normal clustering problems. 1 Introduction Self-organizing maps (SaM) introduced by [Kohonen 84] are a very popular tool used for visualization of high dimensional data spaces. SaM can be said to do clustering/vector quantization (VQ) and at the same time to preserve the spatial ordering of the input data reflected by an ordering of the code book vectors (cluster centroids) in a one or two dimensional output space, where the latter property is closely related to multidimensional scaling (MDS) in statistics. Although the level of activity and research around the SaM algorithm is quite large (a recent overview by [Kohonen 95] contains more than 1000 citations), only little comparison among the numerous existing variants of the basic approach and also to more traditional statistical techniques of the larger frameworks of VQ and MDS is available. Additionally, there is only little advice in the literature about how to properly use 446 A. Flexer SOM in order to get optimal results in terms of either vector quantization (VQ) or multidimensional scaling or maybe even both of them. To make the notion of SOM being a tool for "data visualization" more precise, the following question has to be answered: Should SOM be used for doing VQ, MDS, both at the same time or none of them? Two recent comprehensive studies comparing SOM either to traditional VQ or MDS techniques separately seem to indicate that SOM is not competitive when used for either VQ or MDS: [Balakrishnan et al. 94J compare SOM to K-means clustering on 108 multivariate normal clustering problems with known clustering solutions and show that SOM performs significantly worse in terms of data points misclassified1 , especially with higher numbers of clusters in the data sets. [Bezdek & Nikhil 95J compare SOM to principal component analysis and the MDS-technique Sammon mapping on seven artificial data sets with different numbers of points and dimensionality and different shapes of input distributions. The degree of preservation of the spatial ordering of the input data is measured via a Spearman rank correlation between the distances of points in the input space and the distances of their projections in the two dimensional output space. The traditional MDS-techniques preserve the distances much more effectively than SOM, the performance of which decreases rapidly with increasing dimensionality of the input data. Despite these strong empirical findings that speak against the use of SOM for either VQ or MDS there remains the appealing ability ofSOM to do both VQ and MDS at the same time. It is the aim of this work to find out, whether a combined technique of traditional vector quantization (clustering) plus MDS on the code book vectors (cluster centroids) can perform better than Kohonen's SOM on a series of multivariate normal clustering problems in terms of quantization error (mean squared error), recovering the cluster structure (Rand index) and preserving the topology (Pearson correlation). All the experiments were done in a rigoruos statistical design using multiple analysis of variance for evaluation of the results. 2 SOM and vector quantization/clustering A vector quantizer (VQ) is a mapping, q, that assigns to each input vector x a reproduction (code book) vector x = q( x) drawn from a finite reproduction alphabet A = {Xi, i = 1, ... , N}. The quantizer q is completely described by the reproduction alphabet (or codebook) A together with the partition S = {Si , i = 1, .. . , N}, of the input vector space into the sets Si = {x : q(x) = xd of input vectors mapping into the ith reproduction vector (or code word) [Linde et al. 80J. To be compareable to SO M, our VQ assigns to each of the input vectors x = (xO, xl, . .. , x k- l ) a socalled code book vector x = (xO, xl, ... , xk -1) of the same dimensionality k. For reasons of data compression, the number of code book vectors N ~ n, where n is the number of input vectors. Demanded is a VQ that produces a mapping q for which the expected distortion caused by reproducing the input vectors x by code book vectors q( x) is at least locally minimal. The expected distortion is usually esimated by using the average distortion D, where the most common distortion measure is the squared-error 1 Although SOM is an unsupervised technique not built for classification, the number of points missclassified to a wrong cluster center is an appropriate and commonly used performance measure for cluster procedures if the true cluster structure is known. Limitations of Self-organizing Maps 447 distortion d: k-l d(x, x) = L 1 Xi - Xi 12 (2) i=O The classical vector quantization technique to achieve such a mapping is the LBGalgorithm [Linde et al. 80], where a given quantizer is iteratively improved. Already [Linde et al. 80] noted that their proposed algorithm is almost similar to the k-means approach developed in the cluster analysis literature starting from [MacQueen 67]. Closely related to SOM is online K-means clustering (oKMC) consisting of the following steps: 1. Initialization: Given N = number of code book vectors, k = dimensionality of the vectors, n = number of input vectors, a training sequence {Xj; j = 0, ... , n -I}, an initial set Ao of N code book vectors x and a discrete-time coordinate t = 0 ... , n - 1. 2. Given At = {Xi ; i = 1, .. . , N}, find the minimum distortion partition peAt) = {Si; i = 1, ... , N}. Compute d(Xt, Xi) for i = 1, .. . , N. If d(Xt, Xi) ~ (Xt, XI) for alII, then Xt E Si. 3. Update the code book vector with the minimum distortion X(t)(Si) = x(t-1)(S;) + O'[X(t) - X(t-l)(Si)] (3) where 0' is a learning parameter to be defined by the user. Define At+1 = x(P(At», replace t by t + 1, ift = n -1, halt. Else go to step 2. The main difference between the SOM-algorithm and oKMC is the fact that the code book vectors are ordered either on a line or on a planar grid (i.e. in a one or two dimensional output space). The iterative procedure is the same as with oKMC where formula (3) is replaced by X(t)(S;) = X(t-1)(Si) + h[x(t) - X(t-l)(Si)] (4) and this update is not only computed for the Xi that gives minimum distortion, but also for all the code book vectors which are in the neighbourhood of this Xi on the line or planar grid. The degree of neighbourhood and amount of code book vectors which are updated together with the Xi that gives minimum distortion is expressed by h, a function that decreases both with distance on the line or planar grid and with time and that also includes an additional learning parameter 0' . If the degree of neighbourhood is decreased to zero, the SOM-algorithm becomes equal to the oKMC-algorithm. Whereas local convergence is guaranteed for oKMC (at least for decreasing 0', [Bot.t.ou & Bengio 95]), no general proof for the convergence of SOM with nonzero neighbourhood is known. [Kohonen 95, p.128] notes that the last. steps of the SOM algorithm should be computed with zero neighbourhood in order to guarantee "the most. accurate density approximation of the input samples" . 3 SOM and multidimensional scaling Formally, a topology preserving algorithm is a t.ransformation <1l : Rk ....... RP, that either preserves similarities or just. similarity orderings of the points in the input space Rk when they are mapped into the outputspace R? For most algorithms it is the case t.hat both the number of input vectors 1 x E Rk 1 and the number of output 448 A. Flexer vectors I x E RP I are equal to n. A transformation !l> : x = !l>( x), that preserves similarities poses the strongest possible constraint since d( Xi, Xj) = cf( Xi, X j) for all Xi, X JERk, all Xi, X j E RP, i, j = 1, .. . , n - 1 and d (cf) being a measure of distance in Rk (RP). Such a transformation is said to produce an isometric image. Techniques for finding such transformations !l> are, among others, various forms of multidimensional scalinl (MDS) like metric MDS [Torgerson 52], nonmetric MDS [Shepard 62] or Sammon mapping [Sammon 69], but also principal component analysis (PCA) (see e.g. [Jolliffe 86]) or SOM. Sammon mapping is doing MDS by minimizing the following via steepest descent: Since the SOM has been designed heuristically and not to find an extremum for a certain cost or energy function3 and the theoretical connection to the other MDS algorithms remains unclear. It should be noted that for SOM the number of output vectors I x E RP I is limited to N, the number of cluster centroids x and that the x are further restricted to lie on a planar grid. This restriction entails a discretization of the outputspace RP . 4 Online [(-means clustering plus Sammon mapping of the cl uster centroids Our new combined approach consists of simply finding the set of A = {Xi, i = 1, ... , N} code book vectors that give the minimum distortion partition P(A) = {8i; i = 1, . .. , N} via the oKMC algorithm and then using the Xi as input vectors to Sammon mapping and thereby obtaining a two dimensional representation of the Xi via minimizing formula (5). Contrary to SOM, this two dimensional representation is not restricted to any fixed form and the distances between the N mapped Xi directly correspond to those in the original higher dimension. This combined algorithm is abbreviated oKMC+. 5 Empirical comparison The empirical comparison was done using a 3 factorial experimental design with 3 dependent variables. The multivariate normal distributions were generated using the procedure by [Milligan & Cooper 85], which since has been used for several comparisons of cluster algorithms (see e.g. [Balakrishnan et al. 94]). The marginal normal distributions gave internal cohesion of the clusters by warranting that more than 99% of the data lie within 3 standard deviations (IT). External isolation was defined as having the first dimension nonoverlapping by truncating the normal distributions in the first dimension to ±2IT and defining the cluster centroids to be 4.5IT apart. In all other dimensions the clusters were allowed to overlap by setting the distance per dimension between two centroids randomly to lie between ±6IT. The data was normalized to zero mean and unit variance in all dimensions. 2Note that for MDS not the actual coordinates of the points in the input space but only their distances or the ordering of the latter are needed. 3[Erwin et al. 92] even showed that such an objective function cannot exist for SOM. Limitations of Self-organizing Maps 449 algorithm no. clusters dimension msqe Rand corr. SOM 4 4 0.53 1.00 0.64 6 1.53 0.91 0.72 8 1.15 O.YY 0.74 9 4 0.33 0.97 0.48 6 0.54 0.97 0.66 8 0.81 0.96 0.74 mean SOM 0 .81 0.97 0.67 oKMC+ 4 4 0.53 0.99 0.87 6 1.06 0.99 0.87 8 1.17 1.00 O.Yl 9 4 0.29 0.98 0.89 6 0.47 0.99 0.87 8 0.56 0.98 0.86 mean oKMC+ 0 .68 0.99 0.88 Factor 1, Type of algorithm: The number of code book vectors of both the SOM and the oKMC+ were set equal to the number of clusters known to be in the data. The SOMs were planar grids consisting of 2 x 2 (3 x 3) code book vectors. During the first phase (1000 code book updates) a was set to 0.05 and the radius of the neighbourhood to 2 (5). During the second phase (10000 code book updates) a was set to 0.02 and the radius ofthe neighbourhood to 0 to guarantee the most accurate vector quantization [Kohonen 95, p.128]. The oKMC+ algorithm had the parameter a fixed to 0.02 and was trained using each data set 20 times, the minimization of formula (5) was stopped after 100 iterations. Both SOM and oKMC+ were run 10 times on each data set and only the best solutions, in terms of mean squared error, were used for further analysis. Factor 2, Number of clusters was set to 4 and 9. Factor 3, Number of dimensions was set to 4,6, or8. Dependent variable 1: mean squared error was computed using formula (1). Dependent variable 2, Rand index (see [Hubert & Arabie 85]) is a measure of agreement between the true, known partition structure and the obtained clusters. Both the numerator and the denominator of the index reflect frequency counts. The numerator is the number of times a pair of data is either in the same or in different clusters in both known and obtained clusterings for all possible comparisons of data points. Since the denominator is the total number of all possible pairwise comparisons, an index value of 1.0 indicates an exact match of the clusterings. Dependent variable 3, correlation is a measure of the topology preserving abilities of the algorithms. The Pearson correlation of the distances d( Xl, X2) in the input space and the distances d( Xi, X j) in the output space for all possible pairwise comparisons of data points is computed. Note that for SOM the coordinates of the code book vectors on the planar grid were used to compute the d. An algorithm that preserves all dist.ances in every neighbourhood would produce an isometric image and yield a value of 1.0 (see [Bezdek & Nikhil 95] for a discussion of measures of topolgy preservation) . For each cell in the full-factorial 2 x 2 x 3 design 3 data sets with 25 points for each cluster were generated resulting in a total of 36 data sets. A multiple analysis of variance (MANOVA) yielded the following significant effects at the .05 error level: The mean squared error is lower for oKMC+ than for SOM, it is lower for the 9cluster problem than for the 4-cluster problem and is higher for higher dimensional 450 A. Flexer data. There is also a combined effect of the number of clusters and dimensions on the mean squared error. The Rand index is higher for oKMC+ than for SOM, there is also a combined effect of the number of clusters and dimensions. The correlation index is higher for oKMC+ than for SOM. Since the main interest of this study is the effect of the type of algorithm on the dependent variables, the mean performances for SOM and oKMC+ are printed in bold letters in the table. Note that the overall differences in the performances of the two algorithms are blurred by the significant effects of the other factors and that therefore the differences of the grand means across the type of algorithms appear rather small. Only by applying a MANOVA, effects of the factor 'type of algorithms' that are masked by additional effects of the other two factors 'number of clusters' and 'number of dimensions' could still be detected. 6 Discussion and Conclusion From the theoretical comparison of SOM to oKMC it should be clear that in terms of quantization error, SOM should only be possible to perform as good as oKMC if SOM's neighbourhood is set to zero. Additional experiments, not reported here in detail for brevity, with nonzero neighbourhood till the end of SOM training gave even worse results since the neighbourhood tends to pull the obtained cluster centroids away from the true ones. The Rand index is only slightly better for oKMC+. The high values indicate that both algorithms were able to recover the known cluster structure. Topology preserving is where SOM performs worst compared to oKMC+. This is a direct implication of the restriction to planar grids which allows only 2::=2 i,(&~2) different distances in an s x s planar grid instead of N(~ -1) different distances for N = s x s cluster centroids mapped via Sammon mapping in the case of oKMC+. Using a nonzero neighbourhood at the end of SOM training did not warrant any significant improvements. An argument that could be brought forward against our approach towards comparing SOM and oKMC+ is that it would be unfair or not correct to set the number of SOM's code book vectors equal to the number of clusters known to be in the data. In fact it seems to be common practice to apply SOM with numbers of code book vectors that are a multiple of the input vectors available for training (see e.g. [Kohonen 95, pp.113]). Two things have to be said against such an argumentation: First if one uses more or even only the same amount of code book vectors than input vectors during vector quantization, each code book vector will become identical to one of the input vectors in the limit of learning. So every Xi is replaced with an identical Xi, which does not make any sense and runs counter to every notion of vector quantization. This means that SOMs employing numbers of code book vectors t.hat are a multiple of the input vectors available can be used for MDS only. But even such big SOMs do MDS in a very crude way: We computed SOMs consisting of either 20 x 20 (for data sets consisting of 4 clusters and 100 points) or 30 x 30 (for data sets consisting of 9 clusters and 225 points) code book vectors for all 36 data sets which gave an average correlation of 0.77 between the distances di and di . This is significantly worse at the .05 error level compared to the average correlation of 0.95 achieved by Sammon mapping applied to the input data directly. Our data sets consisted of iid multivariate normal distributions which therefore have spherical shape. All VQ algorithms using squared distances as a distortion measure, including our versions of oKMC as well as SOM, are inherently designed for such distributions. Therefore, the clustering problems in this study, being also perfectly seperable in one dimension, were very simple and should be solveable with little or no error by any clustering or MDS algorithm. Limitations of Self-organizing Maps 451 In this work we examined the vague concept of using SOM as a "data visualization tool" both from a theoretical and empirical point of view. SOM cannot outperform traditional VQ techniques in terms of quantization error and should therefore not be used for doing VQ. From [Bezdek & Nikhil 95] as well as from our discussion of SOM's restriction to planar grids in the output space which allows only a restricted number of different distances to be represented, it should be evident that SOM is also a rather crude way of doing MDS. Our own empirical results show that if one wants to have an algorithm that does both VQ and MDS at the same time, there exists a very simple combination oftraditional techniques (our oKMC+) with wellknown and established properties that clearly outperforms SOM. Whether it is a good idea to combine clustering or vector quantization and multidimensional scaling at all and whether more principled approaches (see e.g. [Bishop et al. this volume], also for pointers to further related work) can yield even better results than our oKMC+ and last but not least what self-organizing maps shmtld be used for under this new light remain questions to be answered by future investigations. Acknowledgements: Thanks are due to James Pardey, University of Oxford, for the Sammon code. The SOM_PAK, Helsinki University of Technology, was used for all computations of self-organizing maps. This work has been started within the framework of the BIOMED-1 concerted action ANNDEE, sponsored by the European Commission, DG XII, and the Austrian Federal Ministry of Science, Transport, and the Arts, which is also supporting the Austrian Research Institute for Artificial Intelligence. The author is supported by a doctoral grant of the Austrian Academy of Sciences. References [Balakrishnan et al. 94] Balakrishnan P.V., Cooper M.C., Jacob V.S., Lewis P.A.: A study of the classification capabilities of neural networks using unsupervised learning: a comparison with k-means clustering, Psychometrika, Vol. 59, No.4, 509-525, 1994. [Bezdek & Nikhil 95] Bezdek J.C. , Nikhil R.P.: An index of topological preservation for feature extraction, Pattern Recognition, Vol. 28, No.3, pp.381-391, 1995. [Bishop et al. this volume] Bishop C.M., Svensen M., Williams C.K.I.: GTM: A Principled Alternative to the Self-Organizing Map, this volume. [Bottou & Bengio 95] Bottou 1., Bengio Y.: Convergence Properties of the K-Means Algorithms, in Tesauro G., et al.(eds.), Advances in Neural Information Processing System 7, MIT Press, Cambridge, MA, pp.585-592, 1995. [Erwin et al. 92] Erwin E., Obermayer K., Schulten K.: Self-organizing maps: ordering, convergence properties and energy functions, Biological Cybernetics, 67,47- 55, 1992. [Hubert & Arabie 85] Hubert L.J., Arabie P.: Comparing partitions, J. of Classification, 2, 63-76, 1985. [Jolliffe 86] Jolliffe I.T.: Principal Component Analysis, Springer, 1986. [Kohonen 84] Kohonen T.: Self-Organization and Associative Memory, Springer, 1984. [Kohonen 95] Kohonen T.: Self-organizing maps, Springer, Berlin, 1995. [Linde et al. 80] Linde Y., Buzo A., Gray R.M.: An Algorithm for Vector Quantizer Design, IEEE Transactions on Communications, Vol. COM-28, No.1, January, 1980. [MacQueen 67] MacQueen J.: Some Methods for Classification and Analysis of Multivariate Observations, Proc. of the Fifth Berkeley Symposium on Math., Stat. and Prob., Vol. 1, pp. 281-296, 1967. [Milligan & Cooper 85] Milligan G.W., Cooper M.C.: An examination of procedures for determining the number of clusters in a data set, Psychometrika 50(2), 159-179, 1985. [Sammon 69] Sammon J .W.: A Nonlinear Mapping for Data Structure Analysis, IEEE Transactions on Comp., Vol. C-18, No.5, p.401-409, 1969. [Shepard 62] Shepard R.N.: The analysis of proximities: multidimensional scaling with an unknown distance function. I., Psychometrika, Vol. 27, No. 2, p.125-140, 1962. [Torgerson 52] Torgerson W .S.: Multidimensional Scaling, I: theory and method, Psychometrika, 17, 401-419, 1952.	1996	128
1,182	A spike based learning neuron in analog VLSI Philipp Hiifliger Institute of Neuroinformatics ETHZjUNIZ Gloriastrasse 32 CH-8006 Zurich Switzerland e-mail: haftiger@neuroinf.ethz.ch tel: ++41 1 257 26 84 Misha Mahowald Institute of Neuroinformatics ETHZjUNIZ Gloriastrasse 32 CH-8006 Zurich Switzerland e-mail: misha@neuroinf.ethz.ch tel: ++41 1 257 26 84 Lloyd Watts Arithmos, Inc. 2730 San Tomas Expressway, Suite 210 Santa Clara, CA 95051-0952 USA e-mail: lloyd@arithmos.com tel: 408 982 4490, x219 Abstract Many popular learning rules are formulated in terms of continuous, analog inputs and outputs. Biological systems, however, use action potentials, which are digital-amplitude events that encode analog information in the inter-event interval. Action-potential representations are now being used to advantage in neuromorphic VLSI systems as well. We report on a simple learning rule, based on the Riccati equation described by Kohonen [1], modified for action-potential neuronal outputs. We demonstrate this learning rule in an analog VLSI chip that uses volatile capacitive storage for synaptic weights. We show that our time-dependent learning rule is sufficient to achieve approximate weight normalization and can detect temporal correlations in spike trains. A Spike Based Learning Neuron in Analog VLSI 693 1 INTRODUCTION It is an ongoing debate how information in the nervous system is encoded and carried between neurons. In many subsystems of the brain it is now believed that it is done by the exact timing of spikes. Furthermore spike signals on VLSI chips allow the use of address-event busses to solve the problem of the large connectivity in neural networks [3, 4]. For these reasons our artificial neuron and others [2] use spike signals to communicate. Additionally the weight updates at the synapses are determined by the relative timing of presynaptic and postsynaptic spikes, a mechanism that has recently been discovered to operate in cortical synapses [5, 7, 6]. Weight normalization is a useful property of learning rules. In order to perform the normalization, some information about the whole weight vector must be available at every synapse. We use the neuron's output spikes (The neuron's output is the product of the weight and the input vector), which retrogradely propagate through the dendrites to the synapses (as has been observed in biological neurons [5]). In our model approximate normalization is an implicit property of the learning rule. 2 THE LEARNING RULE presynaptic spikes 7 .~ ____ L-~ ____ ~ ______ ~_ correlation signat 3 ynaptic weight postsynaptic spikes Figure 1: A snapshot of the simulation variables involved at one synapse. With r = 0.838 The core of the learning rule is a local 'correlation signal' c at every synapse. It records the 'history' of presynaptic spikes. It is incremented by 1 with every presynaptic spike and decays in time with time constant r: c(tm,O) = 0 ttn,n- t 171.,n-l c(tm,n) = eT c(tm,n-d + 1 n>O (1) tm,o is the time of the m'th postsynaptic spike and tm,n (n > 0) is the time of the n'th presynaptic spike after the m'th postsynaptic spike. The weight changes when the cell fires an action potential: 694 P. Hiijliger, M. Mahowald and L Watts tTn o-tTTl-l • w(tm,o) = W(tm-l,O) + ae. T • c(tm-1,s) - ,8w(tm- 1,o) s = max{v : tm-l,v :s; tm,o} (2) where w is the weight at this synapse. tm-l,s means the last event (presynaptic or postsynaptic spike) before the m'th postsynaptic spike. a and ,8 are parameters influencing learning speed and weight vector normalization (see (5)). Our learning rule is designed to react to temporal correlations between spikes in the input signals. However, to show the normalizing of the weights we analyze its behavior by making some simplifying assumptions on the input and output signals; e.g. the intervals of the presynaptic and the postsynaptic spike train are Poisson distributed and there is no correlation between single spikes. Therefore we can represent the signals by their instantaneous average frequencies 0 and i. Now the simplified learning rule can be written as: :t w = al(O)f - ,8wO (3) (4) l(O) represents the average percentage to which the correlation signal is reduced between weight updates (output spikes). So when the neuron's average firing rate fulfills 0 » ~, one can approximate l(O) ~ 1. (3) is thus reduced to the Riccati equation described by Kohonen [1]. This rule would not be Hebbian, but normalizes the weight vector (see (5)). Note that if the correlation signal does not decay, then our rule matches exactly the Riccati equation. We will further refer to it as the Modified Riccati Rule (MRR). Whereas if 0 « ~ then l(O) ~ Or, which is a Hebbian learning rule also described in [1]. If we assume that the spiking mechanism preserves 0 = wT f and insert it in (3), it follows for the equilibrium state: IIwll = Jl(O)~ (5) Since l(O) < 1 the weight vector will never be longer than Ii' This property also holds when the simplifying assumptions are removed. The vector will always be smaller, as it is with no decay of the correlation signals, since the decay only affects the incrementing part of the rule. Matters get much more complicated with the removal of the assumption of the pre- and postsynaptic trains being independently Poisson distributed. With an integrate-and-fire neuron for instance, or if there exist correlations between spikes of the input trains, it is no longer possible to express what happens in terms of rate A Spike Based Learning Neuron in Analog VLSI 695 coding only (with f and 0). (3) is still valid as an approximation but temporal relationships between pre- and postsynaptic spikes become important. Presynaptic spikes immediately followed by an action potential will have the strongest increasing effect on the synapse's weight. 3 IMPLEMENTATION IN ANALOG VLSI We have implemented a learning rule in a neuron circuit fabricated in a 2.0f..Lm CMOS process. This neuron is a preliminary design that conforms only approximately to the MRR. The neuron uses an integrate-and-fire mechanism to generate action potentials (Figure 2). Figure 2: Integrate-and-fire neuron. The sarna capacitor holds the somatic membrane voltage. This voltage is compared to a threshold thresh with a differential pair. When it crosses this threshold it gets pulled up through the mirrored current from the differential pair. This same current gets also mirrored to the right and starts to pull up a second leaky capacitor (setback) through a small W / L transistor, so this voltage rises slowly. This capacitor voltage finally opens a transistor that pulls sarna back to ground where it restarts integrating the incoming current. The parameters tonic+ and tonic- are used to add or subtract a constant current to the soma capacitor. tre! allows the spike-width to be changed. Not shown, but also part of the neuron, are two non-learning synapses: one excitatory and one inhibitory. Each of three learning synapses contains a storage capacitor for the synaptic weight and for the correlation signal (Figure 3). The correlation signal c is simplified to a binary variable in this implementation. When an input spike occurs, the correlation signal is set to 1. It is set to 0 whenever the neuron produces an output-spike or after a fixed time-period (T in (7)) if there is no other input spike: c(tm,o) = 0 c(tm,n) = 1 n > 0 , tm,n::; tm+1,O (6) This approximation unfortunately tends to eliminate differences between highly active inputs and weaker inputs. Nevertheless the weight changes with every output spike: 696 P. Hiijiiger, M. Mahowald and L. Watts Figure 3: The CMOS learning-synapse incorporates the learning mechanism. The weight capacitor holds the weight, the carr capacitor stores the correlation signal representation. The magnitude of the weight increment and decrement are computed by a differential pair (upper left w50). These currents are mirrored to the synaptic weight and gated by digital switches encoding the state of the correlation signal and of the somatic action potential. The correlation signal reset is mediated by a leakage transistor, decayin, which has a tonic value, but is increased dramatically when the output neuron fires. if C{tm-l s) = 1 and tm 0 - tm-l s < T , " otherwise (7) w is the weight on one synapse, c is the correlation signal of that synapse, and a is a parameter that controls how fast the weight changes. (See in the previous section for a description of tm,n.) The weight, W50, is the equilibrium value of the synaptic weight when the occurrence of an input spike is fifty percent correlated with the occurrence of an output spike. This implementation differs from the Riccati rule in that either the weight increment or the weight decrement, but not both, are executed upon each output spike. Also, the weight increment is a function of the synaptic weight. The circuit was implemented this way to try and achieve an equilibrium value for the synaptic weight equal to the fraction of the time that the input neuron fired relative to the times the output neuron fired. This is the correct equilibrium value for the synaptic weight in the Riccati rule. The evolution of a synaptic weight is depicted in Figure 4. The synaptic weight vector normalization in this implementation is accurate only when the assumptions of the design are met. These assumptions are that there is one or fewer input spikes per synapse for every output spike. This assumption is easier to meet when there are many synapses formed with the neuron, so that spikes from multiple inputs combine to drive the cell to threshold. Since we have only three synapses, this approximation is usually violated. Nevertheless, the weights compete with one another and therefore the length of the weight vector is limited. Competition between synaptic weights occurs because if one weight is stronger, it causes the output neuron to spike and this suppresses the other input that has not A Spike Based Learning Neuron in Analog VLSI fired. Future revision of the chip will conform more closely to the MRR. 20 18 8.S 9 tlma 9.S 10 X 10-3 697 Figure 4: A snapshot of the learning behavior of a single VLSI synapse: The top trace is the neuron output (IV/division), the upper middle trace is the synaptic weight (lower voltage means a stronger synaptic weight) (25mV /division), the lower middle trace is a representation of the correlation signal (1 V /division)(it has inverted sense too) and the bottom trace is the presynaptic activity (1 V / division). The weight changes only when an output spike occurs. The timeout of the correlation signal is realized with a decay and a threshold. If the correlation signal is above threshold, the weight is strengthened. If the signal has decayed below threshold at the time of an output spike, the weight is weakened. The magnitude of the change of the weight is a function of the absolute magnitude of the weight. This weight was weaker than W50, so the increments are bigger than the decrements. 4 TEMPORAL CORRELATION IN INPUT SPIKE TRAINS Figure 5 illustrates the ability of our learning rule to detect temporal correlations in spike trains. A simulated neuron strengthens those two synapses that receive 40% coincident spikes, although all four synapses get the same average spike frequencies. 5 DISCUSSION Learning rules that make use of temporal correlations in their spike inputs/outputs provide biologically relevant mechanisms of synapse modification [5, 7, 6]. Analog VLSI implementations allow such models to operate in real time. We plan to develop such analog VLSI neurons using floating gates for weight storage and an addressevent bus for interneuronal connections. These could then be used in realtime applications in adaptive 'neuromorphic' systems. Acknowledgments We thank the following organizations for their support: SPP Neuroinformatik des Schweizerischen Nationalfonds, Centre Swiss d'Electronique et de Microtechnique, U.S. Office of Naval Research and the Gatsby Charitable Foundation. 698 P. Hiijiiger, M. Mahowald and L. Watts synapse 2 0.4 0.36 0.3 015 0.1 O·060l..----:-O:,00--='200":----,300~~400.,----='"="--":-----=-'700 timers) Figure 5: In this simulation we use a neuron with four synapses. All of them get input trains of the same average frequency (20Hz). Two of those input trains are the result of independent Poisson processes (synapses 3 and 4), the other two are the combination oftwo Poisson processes (synapses 1 and 2): One that is independent of any other (12Hz) and one that is shared by the two with slightly different time delays (8Hz): Synapse 1 gets those coincident spikes 0.01 seconds earlier than synapse 2. Synapse 2 gets stronger because when it together with synapse 1 triggered an action potential, it was the last synapse being active before the postsynaptic spike. The parameters were: Q = 0.004, {3 = 0.02, T = 11ms References [1] Thevo Kohonen. Self-Organization and Associative Memory. Springer, Berlin, 1984. [2] D.K. Ferry L.A. Akers and R.O. Grondin. Synthetic neural systems in the 1990s. An introduction to neural and electronic networks, Academic Press (Zernetzer, Davis, Lau, McKenna), pages 359-387, 1995. [3] J. Lazzaro, J. Wawrzynek, M. Mahowald, M. Sivilotti, and D. Gillespie. Silicon auditory processors as computer peripherals. IEEE Trans. Neural Networks, 4:523-528, 1993. [4] A. Mortara and E. A. Vittoz. A communication architecture tailored for analog VLSI artificial neural networks: intrinsic performance and limitations. IEEE Translation on Neural Networks, 5:459-466, 1994. [5] G. J. Stuart and B. Sakmann. Active propagation of somatic action potentials into neocortical pyramidal cell dendrites. Nature, 367:600, 1994. [6] M. V. Tsodyks and H. Markram. Redistribution of synaptic efficacy between neocortical pyramidal neurons. Nature, 382:807-810, 1996. [7] R. Yuste and W. Denk. Dendritic spines as basic functional units of neuronal integration. Nature, 375:682-684, 1995.	1996	129
1,183	A mean field algorithm for Bayes learning in large feed-forward neural networks Manfred Opper Institut fur Theoretische Physik Julius-Maximilians-Universitat, Am Hubland D-97074 Wurzburg, Germany opperOphysik.Uni-Wuerzburg.de Abstract Ole Winther CONNECT The Niels Bohr Institute Blegdamsvej 17 2100 Copenhagen, Denmark wintherGconnect.nbi.dk We present an algorithm which is expected to realise Bayes optimal predictions in large feed-forward networks. It is based on mean field methods developed within statistical mechanics of disordered systems. We give a derivation for the single layer perceptron and show that the algorithm also provides a leave-one-out cross-validation test of the predictions. Simulations show excellent agreement with theoretical results of statistical mechanics. 1 INTRODUCTION Bayes methods have become popular as a consistent framework for regularization and model selection in the field of neural networks (see e.g. [MacKay,1992]). In the Bayes approach to statistical inference [Berger, 1985] one assumes that the prior uncertainty about parameters of an unknown data generating mechanism can be encoded in a probability distribution, the so called prior. Using the prior and the likelihood of the data given the parameters, the posterior distribution of the parameters can be derived from Bayes rule. From this posterior, various estimates for functions ofthe parameter, like predictions about unseen data, can be calculated. However, in general, those predictions cannot be realised by specific parameter values, but only by an ensemble average over parameters according to the posterior probability. Hence, exact implementations of Bayes method for neural networks require averages over network parameters which in general can be performed by time consuming 226 M. Opper and O. Winther Monte Carlo procedures. There are however useful approximate approaches for calculating posterior averages which are based on the assumption of a Gaussian form of the posterior distribution [MacKay,1992]. Under regularity conditions on the likelihood, this approximation becomes asymptotically exact when the number of data is large compared to the number of parameters. This Gaussian ansatz for the posterior may not be justified when the number of examples is small or comparable to the number of network weights. A second cause for its failure would be a situation where discrete classification labels are produced from a probability distribution which is a nonsmooth function of the parameters. This would include the case of a network with threshold units learning a noise free binary classification problem. In this contribution we present an alternative approximate realization of Bayes method for neural networks, which is not based on asymptotic posterior normality. The posterior averages are performed using mean field techniques known from the statistical mechanics of disordered systems. Those are expected to become exact in the limit of a large number of network parameters under additional assumptions on the statistics of the input data. Our analysis follows the approach of [Thouless, Anderson& Palmer,1977] (TAP) as adapted to the simple percept ron by [Mezard,1989]. The basic set up of the Bayes method is as follows: We have a training set consisting of m input-output pairs Dm = {(sll,ull),m = 1, ... ,/J}, where the outputs are generated independently from a conditional probability distribution P( ull Iw, sll). This probability is assumed to describe the output ull to an input sll of a neural network with weights w subject to a suitable noise process. If we assume that the unknown parameters w are randomly distributed with a prior distribution p(w), then according to Bayes theorem our knowledge about w after seeing m examples is expressed through the posterior distribution m p(wIDm) = Z-lp(w) II P(ulllw,sll) ( 1) 11=1 where Z = J dwp(w) n;=l P(ulllw, sll) is called the partition function in statistical mechanics and the evidence in Bayesian terminology. Taking the average with respect to the posterior eq. (1), which in the following will be denoted by angle brackets, gives Bayes estimates for various quantities. For example the optimal predictive probability for an output u to a new input s is given by pBayes(uls) = (P(ulw, s». In section 2 exact equations for the posterior averaged weights (w) are derived for arbitrary networks. In 3 we specialize these equations to a perceptron and develop a mean field ansatz in section 4. The resulting system of mean field equations equations is presented in section 5. In section 6 we consider Bayes optimal predictions and a leave-one-out estimator for the generalization error. We conclude in section 7 with a discussion of our results. 2 A RESULT FOR POSTERIOR AVERAGES FROM GAUSSIAN PRIORS In this section we will derive an interesting equation for the posterior mean of the weights for arbitrary networks when the prior is Gaussian. This average of the Mean Field Algorithm/or Bayes Learning 227 weights can be calculated for the distribution (1) by using the following simple and well known result for averages over Gaussian distributions. Let v be a Gaussian random variable with zero means. Then for any function f(v), we have 2 df(v) (vf(v»a = (v )a· (~)a . (2) Here ( .. . )a denotes the average over the Gaussian distribution of v. The relation is easily proved from an integration by parts. In the following we will specialize to an isotropic Gaussian prior p(w) = ~Ne-!w.w. In [Opper & Winter,1996] anisotropic priors are treated as well. v 21r Applying (2) to each component of wand the function n;=l P(o-Illw,sll), we get the following equations (w) = Z-l J dw wp(w) Ii P(o-"Iw, s") ,,=1 = Z-l t J dwp(w) Ii P(o-"Iw, s")\7w P(o-lllw, sll) 1l=1 "icll (3) J dWp(w) ... n P(a"lw ,s") Here ( . . . ) Il = J n "1t' is a reduced average over a posterior where dwp(w) "~t' P(a"lw ,s") the Jl-th example is kept out of the training set and \7 w denotes the gradient with respect to w. 3 THE PERCEPTRON In the following, we will utilize the fact that for neural networks, the probability (1) depends only on the so called internal fields 8 = JNw . s. A simple but nontrivial example is the perceptron with N dimensional input vector s and output 0-( W, s) = sign( 8). We will generalize the noise free model by considering label noise in which the output is flipped, i.e. 0-8 < 0 with a probability (1 +e.B)-l. (For simplicity, we will assume that f3 is known such that no prior on f3 is needed.) The conditional probability may thus be written as e-.B9( -at' At') P(0-1l81l) = P(o-Illw sll) (4) ,1 + e-.B ' where 9(x) = 1 for x > 0 and 0 otherwise. Obviously, this a nonsmooth function of the weights w, for which the posterior will not become Gaussian asymptotically. For this case (3) reads (w) = _1_ t (P'(0-1l81l»1l o-Ilsll = .jN 1l=1 (P(0-1l81l»1l _1_ f J d8fll (8)P'(0-1l8) o-Ilsll .jN 1l=1 J d8fll(A)P(0-1l8) (5) 228 M. Opper and O. Winther IIJ (~) is the density of -dNw . glJ, when the weights ware randomly drawn from a posterior, where example (glJ , (TIJ) was kept out of the training set. This result states that the weights are linear combinations of the input vectors. It gives an example of the ability of Bayes method to regularize a network model: the effective number of parameters will never exceed the number of data points. 4 MEAN FIELD APPROXIMATION Sofar, no approximations have been made to obtain eqs. (3,5). In general IIJ(~) depends on the entire set of data Dm and can not be calculated easily. Hence, we look for a useful approximation to these densities. We split the internal field into its average and fluctuating parts, i.e. we set ~IJ = (~IJ)IJ + v lJ , with vlJ = IN(w - (w)lJ)glJ. Our mean field approximation is based on the assumption of a central limit theorem for the fluctuating part of the internal field, vlJ which enters in the reduced average of eq. (5). This means, we assume that the non-Gaussian fluctuations of Wi around (Wi)IJ' when mulitplied by sr will sum up to make vlJ a Gaussian random variable. The important point is here that for the reduced average, the Wi are not correlated to the sr! 1 We expect that this Gaussian approximation is reasonable, when N, the number of network weights is sufficiently large.Following ideas of [Mezard, Parisi & Virasoro,1987] and [Mezard,1989]' who obtained mean field equations for a variety of disordered systems in statistical mechanics, one can argue that in many cases this assumption may be exactly fulfilled in the 'thermodynamic limit' m, N ~ 00 with a = ~ fixed. According to this ansatz, we get in terms of the second moment of vlJ AIJ := ~ 2:i,j srsj (WiWj)1J - (Wi)IJ(Wj)IJ). To evaluate (5) we need to calculate the mean (~IJ)IJ and the variance AIJ. The first problem is treated easily within the Gaussian approximation. (6) In the third line (2) has been used again for the Gaussian random variable vlJ . Sofar, the calculation of the variance AIJ for general inputs is an open problem. However, we can make a further reasonable ansatz, when the distribution of the inputs is known. The following approximation for AIJ is expected to become exact in the thermodynamic limit if the inputs of the training set are drawn independently 1 Note that the fluctuations of the internal field with respect to the full posterior mean (which depends on the input si-') is non Gaussian, because the different terms in the sum become slightly correlated. Mean Field Algorithm/or Bayes Learning 229 from a distribution, where all components Si are uncorrelated and normalized i.e. Si = 0 and Si Sj = dij. The bars denote expectation over the distribution of inputs. For the generalisation to a correlated input distribution see [Opper& Winther,1996]. Our basic mean field assumption is that the fluctuations of the All with the data set can be neglected so that we can replace them by their averages All. Since the reduced posterior averages are not correlated with the data sf, we obtain All ~ tr 2:i(wl}1l - (Wi)!). Finally, we replace the reduced average by the expectation over the full posterior, neglecting terms of order liN. Using 2:i(wl) = N, which follows from our choice of the Gaussian prior, we get All ~ A = 1 - tr 2:i (Wi)2. This depends only on known quantities. 5 MEAN FIELD EQUATIONS FOR THE PERCEPTRON (5) and (6) give a selfconsistent set of equations for the variable xll == \~{::::N: . We finally get (7) (8) with (9) These mean field equations can be solved by iteration. It is useful to start with a small number of data and then to increase the number of data in steps of 1 - 10. Numerical work show that the algorithm works well even for small systems sizes, N ~ 15. 6 BAYES PREDICTIONS AND LEAVE-ONE-OUT After solving the mean field equations we can make optimal Bayesian classifications for new data s by chosing the output label with the largest predictive probability. In case of output noise this reduces to uBayes(s) = sign(u(w, s» Since the posterior distribution is independent of the new input vector we can apply the Gaussian assumption again to the internal field, d. and obtain uBayes(s) = u( (w), s), i.e for the simple perceptron the averaged weights implement the Bayesian prediction. This will not be the case for multi-layer neural networks. We can also get an estimate for the generalization error which occurs on the prediction of new data. The generalization error for the Bayes prediction is defined by {Bayes = (8 (-u(s)(u(w,s»»s, where u(s) is the true output and ( ... )s denotes average over the input distribution. To obtain the leave-one-out estimator of { one 230 M. Opper and O. Winther 0.50 0.40 .""-. >0.30 ... 0.20 0.10 I J 1- I I o. 00 '---"'----'-.J'-----' __ --'---_ ---L _ _ -'--_~ _ _ _L_ _ ___"_ _ _ L__ _ -'--_--'--_ ----' o 2 4 Figure 1: Error vs. a = mj N for the simple percept ron with output noise f3 = 0.5 and N = 50 averaged over 200 runs. The full lines are the simulation results (upper curve shows prediction error and the lower curve shows training error). The dashed line is the theoretical result for N -+ 00 obtained from statistical mechanics [Opper & Haussler, 1991] . The dotted line with larger error bars is the moving control estimate. removes the p-th example from the training set and trains the network using only the remaining m - 1 examples. The p'th example is used for testing. Repeating this procedure for all p an unbiased estimate for the Bayes generalization error with m-1 training data is obtained as the mean value f~~r8 = ! EI' e (-ul'(O'(w, 81'»1') which is exactly the type of reduced averages which are calculated within our approach. Figure 1 shows a result of simulations of our algorithm when the inputs are uncorrelated and the outputs are generated from a teacher percept ron with fixed noise rate f3. 7 CONCLUSION In this paper we have presented a mean field algorithm which is expected to implement a Bayesian optimal classification well in the limit of large networks. We have explained the method for the single layer perceptron. An extension to a simple multilayer network, the so called committee machine with a tree architecture is discussed in [Opper& Winther,1996]. The algorithm is based on a Gaussian assumption for the distribution of the internal fields, which seems reasonable for large networks. The main problem sofar is the restriction to ideal situations such as a known distri6 Mean FieLd Algorithm/or Bayes Learning 231 bution of inputs which is not a realistic assumption for real world data. However, this assumption only entered in the calculation of the variance of the Gaussian field. More theoretical work is necessary to find an approximation to the variance which is valid in more general cases. A promising approach is a derivation of the mean field equations directly from an approximation to the free energy -In(Z). Besides a deeper understanding this would also give us the possibility to use the method with the so called evidence framework, where the partition function (evidence) can be used to estimate unknown (hyper-) parameters of the model class [Berger, 1985]. It will further be important to extend the algorithm to fully connected architectures. In that case it might be necessary to make further approximations in the mean field method. ACKNOWLEDGMENTS This research is supported by a Heisenberg fellowship of the Deutsche Forschrmgsgemeinschaft and by the Danish Research Councils for the Natural and Technical Sciences through the Danish Computational Neural Network Center (CONNECT) . REFERENCES Berger, J. O. (1985) Statistical Decision theory and Bayesian Analysis, SpringerVerlag, New York. MacKay, D. J. (1992) A practical Bayesian framework for backpropagation networks, Neural Compo 4 448. Mezard, M., Parisi G. & Virasoro M. A. (1987) Spin Glass Theory and Beyond, Lecture Notes in Physics, 9, World Scientific, . Mezard, M. (1989) The space of interactions in neural networks: Gardner's calculation with the cavity method J. Phys. A 22, 2181 . Opper, M. & Haussler, D. (1991) in IVth Annual Workshop on Computational Learning Theory (COLT91), Morgan Kaufmann. Opper M. & Winther 0 (1996) A mean field approach to Bayes learning in feedforward neural networks, Phys. Rev. Lett. 76 1964. Thouless, D.J ., Anderson, P. W . & Palmer, R.G. (1977), Solution of 'Solvable model of a spin glass' Phil. Mag. 35, 593.	1996	13
1,184	One-unit Learning Rules for Independent Component Analysis Aapo Hyvarinen and Erkki Oja Helsinki University of Technology Laboratory of Computer and Information Science Rakentajanaukio 2 C, FIN-02150 Espoo, Finland email: {Aapo.Hyvarinen.Erkki.Oja}(Qhut.fi Abstract Neural one-unit learning rules for the problem of Independent Component Analysis (ICA) and blind source separation are introduced. In these new algorithms, every ICA neuron develops into a separator that finds one of the independent components. The learning rules use very simple constrained Hebbianjanti-Hebbian learning in which decorrelating feedback may be added. To speed up the convergence of these stochastic gradient descent rules, a novel computationally efficient fixed-point algorithm is introduced. 1 Introduction Independent Component Analysis (ICA) (Comon, 1994; Jutten and Herault, 1991) is a signal processing technique whose goal is to express a set of random variables as linear combinations of statistically independent component variables. The main applications of ICA are in blind source separation, feature extraction, and blind deconvolution. In the simplest form of ICA (Comon, 1994), we observe m scalar random variables Xl, ... , Xm which are assumed to be linear combinations of n unknown components 81, ... 8 n that are zero-mean and mutually statistically independent. In addition, we must assume n ~ m. If we arrange the observed variables Xi into a vector x = (Xl,X2, ... ,xm)T and the component variables 8j into a vector s, the linear relationship can be expressed as x=As (1) Here, A is an unknown m x n matrix of full rank, called the mixing matrix. Noise may also be added to the model, but it is omitted here for simplicity. The basic One-unit Learning Rules for Independent Component Analysis 481 problem of ICA is then to estimate (separate) the realizations of the original independent components Sj, or a subset of them, using only the mixtures Xi. This is roughly equivalent to estimating the rows, or a subset of the rows, of the pseudoinverse of the mixing matrix A. The fundamental restriction of the model is that we can only estimate non-Gaussian independent components, or ICs (except if just one of the ICs is Gaussian). Moreover, the ICs and the columns of A can only be estimated up to a multiplicative constant, because any constant multiplying an IC in eq. (1) could be cancelled by dividing the corresponding column of the mixing matrix A by the same constant. For mathematical convenience, we define here that the ICs Sj have unit variance. This makes the (non-Gaussian) ICs unique, up to their signs. Note the assumption of zero mean of the ICs is in fact no restriction, as this can always be accomplished by subtracting the mean from the random vector x. Note also that no order is defined between the lCs. In blind source separation (Jutten and Herault, 1991), the observed values of x correspond to a realization of an m-dimensional discrete-time signal x(t), t = 1,2, .... Then the components Sj(t) are called source signals. The source signals are usually original, uncorrupted signals or noise sources. Another application of ICA is feature extraction (Bell and Sejnowski, 1996; Hurri et al., 1996), where the columns of the mixing matrix A define features, and the Sj signal the presence and the amplitude of a feature. A closely related problem is blind deconvolution, in which a convolved version x(t) of a scalar LLd. signal s(t) is observed. The goal is then to recover the original signal s(t) without knowing the convolution kernel (Donoho, 1981). This problem can be represented in a way similar to eq. (1), replacing the matrix A by a filter. The current neural algorithms for Independent Component Analysis, e.g. (Bell and Sejnowski, 1995; Cardoso and Laheld, 1996; Jutten and Herault, 1991; Karhunen et al., 1997; Oja, 1995) try to estimate simultaneously all the components. This is often not necessary, nor feasible, and it is often desired to estimate only a subset of the ICs. This is the starting point of our paper. We introduce learning rules for a single neuron, by which the neuron learns to estimate one of the ICs. A network of several such neurons can then estimate several (1 to n) ICs. Both learning rules for the 'raw' data (Section 3) and for whitened data (Section 4) are introduced. If the data is whitened, the convergence is speeded up, and some interesting simplifications and approximations are made possible. Feedback mechanisms (Section 5) are also mentioned. Finally, we introduce a novel approach for performing the computations needed in the ICA learning rules, which uses a very simple, yet highly efficient, fixedpoint iteration scheme (Section 6). An important generalization of our learning rules is discussed in Section 7, and an illustrative experiment is shown in Section 8. 2 Using Kurtosis for leA Estimation We begin by introducing the basic mathematical framework of ICA. Most suggested solutions for ICA use the fourth-order cumulant or kurtosis of the signals, defined for a zero-mean random variable vas kurt(v) = E{v4 } - 3(E{V2})2. For a Gaussian random variable, kurtosis is zero. Therefore, random variables of positive kurtosis are sometimes called super-Gaussian, and variables of negative kurtosis sub-Gaussian. Note that for two independent random variables VI and V2 and for a scalar 0:, it holds kurt(vi + V2) = kurt(vJ) + kurt(v2) and kurt(o:vd = 0:4 kurt(vd· 482 A. Hyviirinen and E. Oja Let us search for a linear combination of the observations Xi, say, w T x, such that it has maximal or minimal kurtosis. Obviously, this is meaningful only if w is somehow bounded; let us assume that the variance of the linear combination is constant: E{(wTx)2} = 1. Using the mixing matrix A in eq. (1), let us define z = ATw. Then also IIzl12 = w T A ATw = w T E{xxT}w = E{(WTx)2} = 1. Using eq. (1) and the properties of the kurtosis, we have n kurt(wT x) = kurt(wT As) = kurt(zT s) = L zJ kurt(sj) (2) j=1 Under the constraint E{(wT x)2} = IIzll2 = 1, the function in (2) has a number of local minima and maxima. To make the argument clearer, let us assume for the moment that the mixture contains at least one Ie whose kurtosis is negative, and at least one whose kurtosis is positive. Then, as may be obvious, and was rigorously proven by Delfosse and Loubaton (1995), the extremal points of (2) are obtained when all the components Zj of z are zero except one component which equals ±1. In particular, the function in (2) is maximized (resp. minimized) exactly when the linear combination w T x = zT S equals, up to the sign, one of the les Sj of positive (resp. negative) kurtosis. Thus, finding the extrema of kurtosis of w T x enables estimation of the independent components. Equation (2) also shows that Gaussian components, or other components whose kurtosis is zero, cannot be estimated by this method. To actually minimize or maximize kurt(wT x), a neural algorithm based on gradient descent or ascent can be used. Then w is interpreted as the weight vector of a neuron with input vector x and linear output w T x. The objective function can be simplified because of the constraint E{ (wT X)2} = 1: it holds kurt(wT x) = E{ (wT x)4} - 3. The constraint E{(wT x)2} = 1 itself can be taken into account by a penalty term. The final objective function is then of the form (3) where a, (3 > 0 are arbitrary scaling constants, and F is a suitable penalty function. Our basic leA learning rules are stochastic gradient descents or ascents for an objective function of this form. In the next two sections, we present learning rules resulting from adequate choices of the penalty function F . Preprocessing of the data (whitening) is also used to simplify J in Section 4. An alternative method for finding the extrema of kurtosis is the fixed-point algorithm; see Section 6. 3 Basic One-Unit leA Learning Rules In this section, we introduce learning rules for a single neural unit. These basic learning rules require no preprocessing of the data, except that the data must be made zero-mean. Our learning rules are divided into two categories. As explained in Section 2, the learning rules either minimize the kurtosis of the output to separate les of negative kurtosis, or maximize it for components of positive kurtosis. Let us assume that we observe a sample sequence x(t) of a vector x that is a linear combination of independent components 81, ... , 8 n according to eq. (1). For separating one of the les of negative kurtosis, we use the following learning rule for One-unit Learning Rules for Independent Component Analysis 483 the weight vector w of a neuron: Aw(t) <X x(t)g-(w(t? x(t)) (4) Here, the non-linear learning function g- is a simple polynomial: g-(u) = au - bu3 with arbitrary scaling constants a, b > O. This learning rule is clearly a stochastic gradient descent for a function of the form (3), with F(u) = -u. To separate an IC of positive kurtosis, we use the following learning rule: Aw(t) <X x(t)g!(t) (w(t? x(t)) (5) where the learning function g!(t) is defined as follows: g~(u) -au(w(t)TCw(t))2 + bu3 where C is the covariance matrix of x(t), i.e. C E{x(t)X(t)T}, and a, b > a are arbitrary constants. This learning rule is a stochastic gradient ascent for a function of the form (3), with F(u) = -u2. Note that w(t)TCw(t) in g+ might also be replaced by (E{(w(t)Tx(t))2})2 or by IIw(t)114 to enable a simpler implementation. It can be proven (Hyvarinen and Oja, 1996b) that using the learning rules (4) and (5), the linear output converges to CSj(t) where Sj(t) is one of the ICs, and C is a scalar constant. This multiplication of the source signal by the constant c is in fact not a restriction, as the variance and the sign of the sources cannot be estimated. The only condition for convergence is that one of the ICs must be of negative (resp. positive) kurtosis, when learning rule (4) (resp. learning rule (5)) is used. Thus we can say that the neuron learns to separate (estimate) one of the independent components. It is also possible to combine these two learning rules into a single rule that separates an IC of any kurtosis; see (Hyvarinen and Oja, 1996b). 4 One-Unit ICA Learning Rules for Whitened Data Whitening, also called sphering, is a very useful preprocessing technique. It speeds up the convergence considerably, makes the learning more stable numerically, and allows some interesting modifications of the learning rules. Whitening means that the observed vector x is linearly transformed to a vector v = Ux such that its elements Vi are mutually uncorrelated and all have unit variance (Comon, 1994). Thus the correlation matrix of v equals unity: E{ vvT} = I. This transformation is always possible and can be accomplished by classical Principal Component Analysis. At the same time, the dimensionality of the data should be reduced so that the dimension of the transformed data vector v equals n, the number of independent components. This also has the effect of reducing noise. Let us thus suppose that the observed signal vet) is whitened (sphered). Then, in order to separate one of the components of negative kurtosis, we can modify the learning rule (4) so as to get the following learning rule for the weight vector w: Aw(t) <X v(t)g- (w(t? vet)) - wet) (6) Here, the function g- is the same polynomial as above: g-(u) = au - bu3 with a > 1 and b > O. This modification is valid because we now have Ev(wT v) = w and thus we can add +w(t) in the linear part of g- and subtract wet) explicitly afterwards. The modification is useful because it allows us to approximate g- with 484 A. Hyviirinen and E. Oja the 'tanh' function, as w(t)T vet) then stays in the range where this approximation is valid. Thus we get what is perhaps the simplest possible stable Hebbian learning rule for a nonlinear Perceptron. To separate one of the components of positive kurtosis, rule (5) simplifies to: dw(t) <X bv(t) (w(t)T v(t))3 - allw(t)114w (t). (7) 5 Multi-Unit ICA Learning Rules If estimation of several independent components is desired, it is possible to construct a neural network by combining N (1 ~ N ~ n) neurons that learn according to the learning rules given above, and adding a feedback term to each of those learning rules. A discussion of such networks can be found in (Hyv~rinen and Oja, 1996b). 6 Fixed-Point Algorithm for ICA The advantage of neural on-line learning rules like those introduced above is that the inputs vet) can be used in the algorithm at once, thus enabling faster adaptation in a non-stationary environment. A resulting trade-off, however, is that the convergence is slow, and depends on a good choice of the learning rate sequence, i.e. the step size at each iteration. A bad choice of the learning rate can, in practice, destroy convergence. Therefore, some ways to make the learning radically faster and more reliable may be needed. The fixed-point iteration algorithms are such an alternative. Based on the learning rules introduced above, we introduce here a fixed-point algorithm, whose convergence is proven and analyzed in detail in (Hyv~rinen and Oja, 1997). For simplicity, we only consider the case of whitened data here. Consider the general neural learning rule trying to find the extrema of kurtosis. In a fixed point of such a learning rule, the sum of the gradient of kurtosis and the penalty term must equal zero: E{v(wT v)3} - 311wll2w + f(lIwI1 2)w = 0. The solutions of this equation must satisfy (8) Actually, because the norm of w is irrelevant, it is the direction of the right hand side that is important. Therefore the scalar in eq. (8) is not significant and its effect can be replaced by explicit normalization. Assume now that we have collected a sample of the random vector v , which is a whitened (or sphered) version of the vector x in eq. (1). Using (8), we obtain the following fixed-point algorithm for leA: 1. Take a random initial vector w(o) of norm 1. Let k = 1. 2. Let w(k) = E{v(w(k - I)T v)3} - 3w(k - 1). The expectation can be estimated using a large sample of v vectors (say, 1,000 points). 3. Divide w(k) by its norm. 4. If IW(k)Tw(k - 1)1 is not close enough to 1, let k = k + 1 and go back to step 2. Otherwise, output the vector w(k). One-unit Learning Rules for Independent Component Analysis 485 The final vector w* = limk w(k) given by the algorithm separates one of the nonGaussian les in the sense that wT v equals one of the les Sj. No distinction between components of positive or negative kurtosis is needed here. A remarkable property of our algorithm is that a very small number of iterations, usually 5-10, seems to be enough to obtain the maximal accuracy allowed by the sample data. This is due to the fact that the convergence of the fixed point algorithm is in fact cubic, as shown in (Hyv:trinen and Oja, 1997). To estimate N les, we run this algorithm N times. To ensure that we estimate each time a different Ie, we only need to add a simple projection inside the loop, which forces the solution vector w(k) to be orthogonal to the previously found solutions. This is possible because the desired weight vectors are orthonormal for whitened data (Hyv:trinen and Oja, 1996bj Karhunen et al., 1997). Symmetric methods of orthogonalization may also be used (Hyv:trinen, 1997). This fixed-point algorithm has several advantages when compared to other suggested leA methods. First, the convergence of our algorithm is cubic. This means very fast convergence and is rather unique among the leA algorithms. Second, contrary to gradient-based algorithms, there is no learning rate or other adjustable parameters in the algorithm, which makes it easy to use and more reliable. Third, components of both positive and negative kurtosis can be directly estimated by the same fixedpoint algorithm. 7 Generalizations of Kurtosis In the learning rules introduced above, we used kurtosis as an optimization criterion for leA estimation. This approach can be generalized to a large class of such optimizaton criteria, called contrast functions. For the case of on-line learning rules, this approach is developed in (Hyv:trinen and Oja, 1996a), in which it is shown that the function 9 in the learning rules in section 4 can be, in fact, replaced by practically any non-linear function (provided that w is normalized properly). Whether one must use Hebbian or anti-Hebbian learning is then determined by the sign of certain 'non-polynomial cumulants'. The utility of such a generalization is that one can then choose the non-linearity according to some statistical optimality criteria, such as robustness against outliers. The fixed-point algorithm may also be generalized for an arbitrary non-linearity, say g. Step 2 in the fixed-point algorithm then becomes (for whitened data) (Hyv:trinen, 1997): w(k) = E{vg(w(k -l)Tv)} - E{g'(w(k -l)Tv)}w(k -1). 8 Experiments A visually appealing way of demonstrating how leA algorithms work is to use them to separate images from their linear mixtures. On the left in Fig. 1, four superimposed mixtures of 4 unknown images are depicted. Defining the j-th Ie Sj to be the gray-level value of the j-th image in a given position, and scanning the 4 images simultaneously pixel by pixel, we can use the leA model and recover the original images. For example, we ran the fixed-point algorithm four times, estimating the four images shown on the right in Fig. 1. The algorithm needed on the average 7 iterations for each Ie. 486 A. Hyviirinen and E. Oja Figure 1: Three photographs of natural scenes and a noise image were linearly mixed to illustrate our algorithms. The mixtures are depicted on the left. On the right, the images recovered by the fixed-point algorithm are shown. References Bell, A. and Sejnowski, T. (1995). An information-maximization approach to blind separation and blind deconvolution. Neural Computation, 7:1129-1159. Bell, A. and Sejnowski, T. J. (1996). Edges are the independent components of natural scenes. In NIPS 96, Denver, Colorado. Cardoso, J.-F. and Laheld, B. H. (1996). Equivariant adaptive source separation. IEEE Trans. on Signal Processing. 44(12). Comon, P. (1994). Independent component analysis - a new concept? Signal Processing, 36:287-314. Delfosse, N. and Loubaton, P. (1995). Adaptive blind separation of independent sources: a deflation approach. Signal Processing, 45:59- 83. Donoho, D. (1981). On minimum entropy deconvolution. In Applied Time Series Analysis II. Academic Press. Hurri, J., Hyv:irinen, A., Karhunen, J., and Oja, E. (1996). Image feature extraction using independent component analysis. In Proc. NORSIG'96, Espoo, Finland. Hyv:irinen, A. (1997). A family of fixed-point algorithms for independent component analysis. In Pmc. ICASSP'9'1, Munich, Germany. Hyv:irinen, A. and Oja, E. (1996a). Independent component analysis by general nonlinear hebbian-like learning rules. Technical Report A41, Helsinki University of Technology, Laboratory of Computer and Information Science. Hyv:irinen, A. and Oja, E. (1996b). Simple neuron models for independent component analysis. Technical Report A37, Helsinki University of Technology, Laboratory of Computer and Information Science. Hyv:irinen, A. and Oja, E. (1997). A fast fixed-point algorithm for independent component analysis. Neural Computation. To appear. Jutten, C. and Herault, J. (1991). Blind separation of sources, part I: An adaptive algorithm based on neuromimetic architecture. Signal Processing, 24:1-10. Karhunen, J., Oja, E., Wang, L., Vigario, R., and Joutsensalo, J. (1997). A class of neural networks for independent component analysis. IEEE Trans. on Neural Networks. To appear. Oja, E. (1995). The nonlinear PCA learning rule and signal separation - mathematical analysis. Technical Report A 26, Helsinki University of Technology, Laboratory of Computer and Information Science. Submitted to a journal.	1996	130
1,185	Analysis of Temporal-Difference Learning with Function Approximation John N. Tsitsiklis and Benjamin Van Roy Laboratory for Information and Decision Systems Massachusetts Institute of Technology Cambridge, MA 02139 e-mail: jnt@mit.edu, bvr@mit.edu Abstract We present new results about the temporal-difference learning algorithm, as applied to approximating the cost-to-go function of a Markov chain using linear function approximators. The algorithm we analyze performs on-line updating of a parameter vector during a single endless trajectory of an aperiodic irreducible finite state Markov chain. Results include convergence (with probability 1), a characterization of the limit of convergence, and a bound on the resulting approximation error. In addition to establishing new and stronger results than those previously available, our analysis is based on a new line of reasoning that provides new intuition about the dynamics of temporal-difference learning. Furthermore, we discuss the implications of two counter-examples with regards to the Significance of on-line updating and linearly parameterized function approximators. 1 INTRODUCTION The problem of predicting the expected long-term future cost (or reward) of a stochastic dynamic system manifests itself in both time-series prediction and control. An example in time-series prediction is that of estimating the net present value of a corporation, as a discounted sum of its future cash flows, based on the current state of its operations. In control, the ability to predict long-term future cost as a function of state enables the ranking of alternative states in order to guide decision-making. Indeed, such predictions constitute the cost-to-go function that is central to dynamic programming and optimal control (Bertsekas, 1995). Temporal-difference learning, originally proposed by Sutton (1988), is a method for approximating long-term future cost as a function of current state. The algorithm 1076 1. N. Tsitsiklis and B. Van Roy is recursive, efficient, and simple to implement. Linear combinations of fixed basis functions are used to approximate the mapping from state to future cost. The weights of the linear combination are updated upon each observation of a state transition and the associated cost. The objective is to improve approximations of long-term future cost as more and more state transitions are observed. The trajectory of states and costs can be generated either by a physical system or a simulated model. In either case, we view the system as a Markov chain. Adopting terminology from dynamic programming, we will refer to the function mapping states of the Markov chain to expected long-term cost as the cost-to-go function. In this paper, we introduce a new line of analysis for temporal-difference learning. In addition to providing new intuition about the dynamics of the algorithm, this approach leads to a stronger convergence result than previously available, as well as an interpretation of the limit of convergence and bounds on the resulting approximation error, neither of which have been available in the past. Aside from the statement of results, we maintain the discussion at an informal level, and make no attempt to present a complete or rigorous proof. The formal and more general analysis based on our line of reasoning can found in (Tsitsiklis and Van Roy, 1996), which also discusses the relationship between our results and other work involving tem poral-difference learning. The convergence results assume the use of both on-line updating and linearly parameterized function approximators. To clarify the relevance of these requirements, we discuss the implications of two counter-examples that are presented in (Tsitsiklis and Van Roy, 1996). These counter-examples demonstrate that temporal-difference learning can diverge in the presence of either nonlinearly parameterized function approximators or arbitrary (instead of on-line) sampling distributions. 2 DEFINITION OF TD(A) In this section, we define precisely the nature of temporal-difference learning, as applied to approximation of the cost-to-go function for an infinite-horizon discounted Markov chain. While the method as well as our subsequent results are applicable to Markov chains with fairly general state spaces, including continuous and unbounded spaces, we restrict our attention in this paper to the case where the state space is finite. Discounted Markov chains with more general state spaces are addressed in (Tsitsiklis and Van Roy, 1996). Application of this line of analysis to the context of undiscounted absorbing Markov chains can be found in (Bertsekas and Tsitsiklis, 1996) and has also been carried out by Gurvits (personal communication). We consider an aperiodic irreducible Markov chain with a state space S = {I, ... , n}, a transition probability matrix P whose (i, j)th entry is denoted by Pij, transition costs g(i,j) associated with each transition from a state i to a state j, and a discount factor Q E (0,1). The sequence of states visited by the Markov chain is denoted by {it I t = 0,1, ... }. The cost-to-go function J* : S t-+ ~ associated with this Markov chain is defined by J(i) ~ E [f: olg(it, it+d I io = ij. t=o Since the number of dimensions is finite, it is convenient to view J as a vector instead of a function. We consider approximations of J* using a function of the form J(i, r) = (<I>r)(i). Analysis ofTemporal-Diflference Learning with Function Approximation 1077 Here, r = (r(l), ... ,r(K)) is a parameter vector and cI> is a n x K. We denote the ith row of cI> as a (column) vector </J(i). Suppose that we observe a sequence of states it generated according to the transition probability matrix P and that at time t the parameter vector r has been set to some value rt. We define the temporal difference dt corresponding to the transition from it to it+l by dt = g(it, it+1) + aJ(it+1' rt) - J(it, rt). We define a sequence of eligibility vectors Zt (of dimension K) by t Zt = 2)aA)t-k</J(ik). k=O The TD(A) updates are then given by rt+l = rt + "Itdtzt, where ro is initialized to some arbitrary vector, "It is a sequence of scalar step sizes, and A is a parameter in [0,1]. Since temporal-difference learning is actually a continuum of algorithms, parameterized by A, it is often referred to as TD(A). Note that the eligibility vectors can be updated recursively according to Zt+1 aAzt + </J(it+d, initialized with Z-l = O. 3 ANALYSIS OF TD("\) Temporal-difference learning originated in the field of reinforcement learning. A view commonly adopted in the original setting is that the algorithm involves "looking back in time and correcting previous predictions." In this context, the eligibility vector keeps track of how the parameter vector should be adjusted in order to appropriately modify prior predictions when a temporal-difference is observed. Here, we take a different view which involves examining the "steady-state" behavior of the algorithm and arguing that this characterizes the long-term evolution of the parameter vector. In the remainder ofthis section, we introduce this view of TD(A) and provide an overview of the analysis that it leads to. Our goal in this section is to convey some intuition about how the algorithm works, and in this spirit, we maintain the discussion at an informal level, omitting technical assumptions and other details required to formally prove the statements we make. These technicalities are addressed in (Tsitsiklis and Van Roy, 1996), where formal proofs are presented. We begin by introducing some notation that will make our discussion here more concise. Let 71"(1), .. . , 7I"(n) denote the steady-state probabilities for the process it. We assume that 7I"(i) > 0 for all i E S. We define an n x n diagonal matrix D with diagonal entries 71"(1), ... , 7I"(n). We define a weighted norm II ·IID by IIJIID = L 7I"(i)J2(i). iES We define a "projection matrix" II by IIJ = arg !llin IIJ - JIID. J=tf>r It is easy to show that II = cI>(cI>' DcI»-lcI>' D. We define an operator T(>") : ~n I-t ~n, indexed by a parameter A E [0,1) by (T(» J)(i) = (1 - ~) %;. ~m E [t, o/g(i" it+1) + "m+l J(im+l) I io = i) . 1078 1. N. Tsitsiklis and B. Van Roy For A = 1 we define (T(l)J)(i) = J(i), so that lim>.tl(T(>')J)(i) = (T(l)J)(i). To interpret this operator in a meaningful manner, note that, for each m, the term E [f cig(it, it+d + am+! J(im+d I io = i] t=o is the expected cost to be incurred over m transitions plus an approximation to the remaining cost to be incurred, based on J. This sum is sometimes called the "m-stage truncated cost-to-go." Intuitively, if J is an approximation to the costto-go function, the m-stage truncated cost-to-go can be viewed as an improved approximation. Since T(>') J is a weighted average over the m-stage truncated costto-go values, T(>') J can also be viewed as an improved approximation to J. A property of T(>') that is instrumental in our proof of convergence is that T(>') is a contraction of the norm II·IID. It follows from this fact that the composition IIT(>') is also a contraction with respect to the same norm, and has a fixed point of the form cf>r* for some parameter vector r* . To clarify the fundamental structure of TD(A), we construct a process X t = (it, it+!, Zt)· It is easy to see that Xt is a Markov process. In particular, Zt+l and it+! are deterministic functions of X t and the distribution of it+2 only depends on it+l. Note that at each time t, the random vector X t , together with the current parameter vector rt, provides all necessary information for computing rt+l. By defining a function s with s(r, X) = (g(i,j)+aJ(j, r) -J(i, r))z, where X = (i,j, z), we can rewrite the TD(A) algorithm as rt+1 = rt + Its(rt, Xd· For any r, s(r,Xt) has a "steady-state" expectation, which we denote by Eo[s(r, X t)]. Intuitively, once X t reaches steady-state, the TD(A) algorithm, in an "average" sense, behaves like the following deterministic algorithm: TT+l = TT + ITEO[S(TT' X t)]. Under some technical assumptions, a theorem from (Benveniste, et al., 1990) can be used to deduce convergence TD(A) from that of the deterministic counterpart. Our study centers on an analysis of this deterministic algorithm. A theorem from (Benveniste, et aI, 1990) is used to formally deduce convergence of the stochastic algorithm. It turns out that Eo[s(r,Xt )] = cf>'D(T(>')(cf>r) - cf>r). Using the contraction property of T(>'), (r - r)'Eo[s(r,Xt)] = (cf>r - cf>r)'D(IIT(>')(cf>r) - cf>r* + (cf>r* - cf>r)) < lIcf>r - cf>rIID . IlIIT(>') (cf>r) - cf>rIID -11cf>r* - cf>r1l1 < (0: -1)IIcf>r - cf>r1I1. Since a < 1, this inequality shows that the steady state expectation Eo[s(r, Xd] generally moves the parameter vector towards r, the fixed point of IIT(>'), where "closeness" is measured in terms of the norm II . liD. This provides the main line of reasoning behind the proof of convergence provided in (Tsitsiklis and Van Roy, 1996). Some illuminating interpretations of this deterministic algorithm, which are useful in developing an intuitive understanding of temporal difference learning, are also discussed in (Tsitsiklis and Van Roy, 1996). Analysis ofTemporal-DiflJerence Learning with Function Approximation 1079 4 CONVERGENCE RESULT We now present our main result concerning temporal-difference learning. A formal proof is provided in (Tsitsiklis and Van Roy, 1996). Theorem 1 Let the following conditions hold: (a) The Markov chain it has a unique invariant distribution 71" that satisfies 71"' P = 71"', with 71"( i) > 0 for all i. (b) The matrix 4> has full column rank; that is, the "basis functions" {¢k I k = 1, ... ,K} are linearly independent. (c) The step sizes 'Yt are positive, nonincreasing, and predetermined. Furthermore, they satisfy 2::0 'Yt = 00, and 2::0 'Yt < 00. We then have: (a) For any A E [0,1]' the TD(A) algorithm, as defined in Section 2, converges with probability 1. (b) The limit of convergence r* is the unique solution of the equation IIT(>') (4)r) = 4>r. (c) Furthermore, r* satisfies l14>r* - J* liD :S 1 - Aa IlIIJ* - J* liD. I-a Part (b) of the theorem leads to an interesting interpretation of the limit of convergence. In particular, if we apply the TD (A) operator to the final approximation 4>r, and then project the resulting function back into the span of the basis functions, we get the same function 4>r. Furthermore, since the composition IIT(>') is a contraction, repeated application of this composition to any function would generate a sequence of functions converging to 4>r. Part (c) of the theorem establishes that a certain desirable property is satisfied by the limit of convergence. In particular, if there exists a vector r such that 4>r = J, then this vector will be the limit of convergence of TD(A), for any A E [0, 1]. On the other hand, if no such parameter vector exists, the distance between the limit of convergence 4>r* and J* is bounded by a multiple of the distance between the projection IIJ* and J*. This latter distance is amplified by a factor of (1 - Aa)/(1 - a), which becomes larger as A becomes smaller. 5 COUNTER-EXAMPLES Sutton (1995) has suggested that on-line updating and the use of linear function approximators are both important factors that make temporal-difference learning converge properly. These requirements also appear as assumptions in the convergence result of the previous section. To formalize the fact that these assumptions are relevant, two counter-examples were presented in (Tsitsiklis and Van Roy, 1996). The first counter-example involves the use of a variant of TD(O) that does not sample states based on trajectories. Instead, the states it are sampled independently from a distribution q(.) over S, and successor states jt are generated by sampling according to Pr[jt = jlit] = Pid. Each iteration of the algorithm takes on the form rt+I = rt + 'Yt¢(it) (g(it,jt) + a¢'(jt)rt - ¢'(it)rt). We refer to this algorithm as q-sampled TD(O). Note that this algorithm is closely related to the original TD(A) algorithm as defined in Section 2. In particular, if it is 1080 J. N. Tsitsiklis and B. Van Roy generated by the Markov chain and jt = it+! , we are back to the original algorithm. It is easy to show, using a subset of the arguments required to prove Theorem 1, that this algorithm converges when q(i) = 7r(i) for all i, and the Assumptions of Theorem 1 are satisfied. However, results can be very different when q(.) is arbitrary. In particular, the counter-example presented in (Tsitsiklis an Van Roy, 1996) shows that for any sampling distribution q(.) that is different from 7r(-) there exists a Markov chain with steady-state probabilities 7r(-) and a linearly parameterized function approximator for which q-sampled TD(O) diverges. A counter-example with similar implications has also been presented by Baird (1995). A generalization of temporal difference learning is commonly used in conjunction with nonlinear function approximators. This generalization involves replacing each vector </J( it) that is used to construct the eligibility vector with the vector of derivatives of J(it, .), evaluated at the current parameter vector rt. A second counterexample in (Tsitsiklis and Van Roy, 1996), shows that there exists a Markov chain and a nonlinearly parameterized function approximator such that both the parameter vector and the approximated cost-to-go function diverge when such a variant of TD(O) is applied. This nonlinear function approximator is "regular" in the sense that it is infinitely differentiable with respect to the parameter vector. However, it is still somewhat contrived, and the question of whether such a counter-example exists in the context of more standard function approximators such as neural networks remains open. 6 CONCLUSION Theorem 1 establishes convergence with probability 1, characterizes the limit of convergence, and provides error bounds, for temporal-difference learning. It is interesting to note that the margins allowed by the error bounds are inversely proportional to >.. Although this is only a bound, it strongly suggests that higher values of >. are likely to produce more accurate approximations. This is consistent with the examples that have been constructed by Bertsekas (1994). The sensitivity of the error bound to >. raises the question of whether or not it ever makes sense to set >. to values less than 1. Many reports of experimental results, dating back to Sutton (1988), suggest that setting>. to values less than one can often lead to significant gains in the rate of convergence. A full understanding of how>. influences the rate of convergence is yet to be found, though some insight in the case of look-up table representations is provided by Dayan and Singh (1996). This is an interesting direction for future research. Acknowledgments We thank Rich Sutton for originally making us aware of the relevance of on-line state sampling, and also for pointing out a simplification in the expression for the error bound of Theorem l. This research was supported by the NSF under grant DMI-9625489 and the ARO under grant DAAL-03-92-G-01l5. References Baird, L. C. (1995). "Residual Algorithms: Reinforcement Learning with Function Approximation," in Prieditis & Russell, eds. Machine Learning: Proceedings of the Twelfth International Conference, 9-12 July, Morgan Kaufman Publishers, San Francisco, CA. Bertsekas, D. P. (1994) "A Counter-Example to Temporal-Difference Learning," Analysis ofTemporal-Diffference Learning with Function Approximation 1081 Neural Computation, vol. 7, pp. 270-279. Bertsekas, D. P. (1995) Dynamic Programming and Optimal Control, Athena Scientific, Belmont, MA. Bertsekas, D. P. & Tsitsiklis, J. N. (1996) Neuro-Dynamic Programming, Athena Scientific, Belmont, MA. Benveniste, A., Metivier, M., & Priouret, P., (1990) Adaptive Algorithms and Stochastic Approximations, Springer-Verlag, Berlin. Dayan, P. D. & Singh, S. P (1996) "Mean Squared Error Curves in Temporal Difference Learning," preprint. Gurvits, L. (1996) personal communication. Sutton, R. S., (1988) "Learning to Predict by the Method of Temporal Differences," Machine Learning, vol. 3, pp. 9-44. Sutton, R.S. (1995) "On the Virtues of Linear Learning and Trajectory Distributions," Proceedings of the Workshop on Value Function Approximation, Machine Learning Conference 1995, Boyan, Moore, and Sutton, Eds., p. 85. Technical Report CMU-CS-95-206, Carnegie Mellon University, Pittsburgh, PA 15213. Tsitsiklis, J. N. & Van Roy, B. (1996) "An Analysis of Temporal-Difference Learning with Function Approximation," to appear in the IEEE Transactions on Automatic Control.	1996	131
1,186	Fast Network Pruning and Feature Extraction Using the Unit-OBS Algorithm Achim Stahlberger and Martin Riedmiller Institut fur Logik, Komplexitiit und Deduktionssysteme Universitiit Karlsruhe, 76128 Karlsruhe, Germany email: stahlb@ira.uka.de. riedml@ira.uka.de Abstract The algorithm described in this article is based on the OBS algorithm by Hassibi, Stork and Wolff ([1] and [2]). The main disadvantage of OBS is its high complexity. OBS needs to calculate the inverse Hessian to delete only one weight (thus needing much time to prune a big net) . A better algorithm should use this matrix to remove more than only one weight, because calculating the inverse Hessian takes the most time in the OBS algorithm. The algorithm, called Unit- OBS, described in this article is a method to overcome this disadvantage. This algorithm only needs to calculate the inverse Hessian once to remove one whole unit thus drastically reducing the time to prune big nets. A further advantage of Unit- OBS is that it can be used to do a feature extraction on the input data. This can be helpful on the understanding of unknown problems. 1 Introduction This article is based on the technical report [3] about speeding up the OBS algorithm. The main target of this work was to reduce the high complexity O(n2p) of the OBS algorithm in order to use it for big nets in a reasonable time. Two "exact" algorithms were developed which lead to exactly the same results as OBS but using less time. The first with time O( n1.8p) makes use of Strassens' fast matrix multiplication algorithm. The second algorithm uses algebraic transformations to speed up calculation and needs time O(np2). This algorithm is faster than OBS in the special case of p < n. 656 A. Stahlberger and M. Riedmiller To get a much higher speedup than these exact algorithms can do, an improved OBS algorithm was developed which reduces the runtime needed to prune a big network drastically. The basic idea is to use the inverse Hessian to remove a group of weights instead of only one, because the calculation of this matrix takes the most time in the OBS algorithm. This idea leads to an algorithm called Unit- OBS that is able to remove whole units. Unit-OBS has two main advantages: First it is a fast algorithm to prune big nets, because whole units are removed in every step instead of slow pruning weight by weight. On the other side it can be used to do a feature extraction on the input data by removing unimportant input units. This is helpful for the understanding of unknown problems. 2 Optimal Brain Surgeon This section gives a small summary of the OBS algorithm described by Hassibi, Stork and Wolff in [1] and [2] . As they showed the increase in error (when changing weights by ~ w) is (1) where H is the Hessian matrix. The goal is to eliminate weight Wq and minimize the increase in error given by Eq. 1. Eliminating Wq can be expressed by Wq + ~Wq = 0 which is equivalent to (w + ~ W f eq = 0 where eq is the unit vector corresponding to weight Wq (wT eq = wq ) . Solving this extremum problem with side condition using Lagrange's method leads to the solution 2 ~E= Wq 2H-1 qq Wq -1 ~W = - H-1 H eq qq (2) (3) H -1 qq denotes the element (q, q) of matrix H -1 . For every weight Wq the minimal increase in error ~E(wq) is calculated and the weight which leads to overall minimum will be removed and all other weights be adapted referring to Eq. 3. Hassibi, Stork and Wolff also showed how to calculate H- 1 using time O(n2p) where n is the number of weights and p the number of pattern. The main disadvantage of the OBS algorithm is that it needs time O(n2p) to remove only one weight thus needing much time to prune big nets. The basic idea to soften this disadvantage is to use H- 1 to remove more than only one weight! This generalized OBS algorithm is described in the next section. 3 Generalized OBS (G-OBS) This section shows a generalized OBS algorithm (G-OBS) which can be used to delete m weights in one step with minimal increase in error. Like in the OBS algorithm the increase in error is given by ~E = ~~wT H ~w . But the condition Wq + ~Wq = 0 is replaced by the generalized condition (4) Fast Network Pruning by using the Unit-OBS Algoritiun 657 where M is the selection matrix (selecting the weights to be removed) and ql, q2, . .. , qm are the indices of the weights that will be removed. Solving this extremum problem with side condition using Lagrange's method leads to the solution jj.E = !wT M(MT H- 1 M)-l MT w 2 jj.w = _H- 1 M(MT H- 1 M)-l MT w (5) (6) Choosing M = eq Eq. 5 and 6 reduce to Eq. 2 and 3. This shows that OBS is (as expected) a special case of G-OBS. The problem of calculating H- 1 was already solved by Hassibi, Stork and Wolff ([1] and [2]) . 4 Analysis of G-OBS Hassibi, Stork and Wolff ([1] and [2]) showed that the time to calculate H-l is in O(n2p). The calculation of jj.E referring to Eq. 5 needs time O(m3)t where m is the number of weights to be removed. The calculation of jj.w (Eq. 6) needs time O(nm + m 3 ). The problem within this solution consists of not knowing which weights should be deleted and thus jj.E has to be calculated for all possible combinations to find the global minimum in error increase. Choosing m weights out of n can be done with (;:J possible combinations. This takes time (~)O(m3) to find the minimum. Therefore the total runtime of the generalized OBS algorithm to remove m weights (with minimal increase in error) is The problem is that for m > 3 the term C:Jm3 dominates and TG-OBS is in O(n4 ). In other words G-OBS can be used only to remove a maximum of three weights in one step. But this means little advantage over OBS. To overcome this problem the set of possible combinations has to be restricted to a small subset of combinations that seem to be "good" combinations. This reduces the term (~)m3 to a reasonable amount. One way to do this is that a good combination exists of all outgoing connections of a unit. This reduces the number of combinations to the number of units! The basic idea for that subset is: If all outgoing connections of a unit can be removed then the whole unit can be deleted because it can not influence the net output anymore. Therefore choosing this subset leads to an algorithm called Unit- OBS that is able to remove whole units without the need to recalculate H- 1 . 5 Special Case of G-OBS: Unit-OBS With the results of the last sections we can now describe an algorithm called UnitOBS to remove whole units. 1. Train a network to minimum error. t M is a matrix of special type and thus the calculation of (MT H- J M) needs only O(m2 ) operations! 658 A. Stahlbergerand M. Riedmiller 2. Compute H- 1 . 3. For each unit u (a) Compute the indices Ql, Q2 , .. . ,Qm(u) of the outgoing connections of unit u where m(u) is the number of outgoing connections of unit u. (b) M := (eq1 eq2 ... eqm(u») (c) D..E(u) := ~wT M(MT H- 1 M)-l MT w 4. Find the Uo that gives the smallest increase in error D..E(uo). 5. M := M(uo) (refer to steps 3.(a) and 3.(b)) 6. D..w := _H- 1 M(MT H- 1 M)-l MT w 7. Remove unit Uo and use D..w to update all weights. 8. Repeat steps 2 to 7 until a break criteria is reached. Following the analysis of G-OBS the time to remove one unit is TUnit-OBS = O(n2p + um3 ) (7) where u is the number of units in the network and m is the maximum number of outgoing connections. If m is much smaller than n we can neglect the term um3 and the main problem is to calculate H- 1. Therefore, if m is small, we can say that Unit-OBS needs the same time to remove a whole unit as OBS needs to remove a single weight. The speedup when removing units with an average of s outgoing connections should then be s. 6 Simulation results 6.1 The Monk-1 benchmark Unit- OBS was applied to the MONK's problems because the underlying logical rules are well known and it is easy to say which input units are important to the problem and which input units can be removed. The simulations showed that in no case Unit-OBS removed a wrong unit and that it has the ability to remove all unimportant input units. Figure 1 shows a MONK-I- net pruned with Unit-OBS. This net is the minimal network that can be found by Unit-OBS. Table 1 shows the speedup of Unit-OBS compared to OBS to find an equal-size network for the MONK-I problem. The network shown in Fig. 1 is only minimal in the number of units but not minimal with respect to the number of weights. Hassibi, Stork and Wolff ([1] and [2]) found a network with only 14 weights by applying OBS (Fig. 3). In the framework of UnitOBS, OBS can be used to do further pruning on the network after all possible units have been pruned. The advantage lies in the fact that now the time consuming OBS- algorithm is applied to a much smaller network (22 weights instead of 58). The result of this combination of Unit-OBS and OBS is a network with only 14 weights (Fig. 2) which has also 100 % accuracy like the minimal net found by OBS (see Table 1). . Fast Network Pruning by using the Unit-OBS Algorithm 659 Atuibute 1 Attribute 2 Attribute 3 Attribute 4 Attribute 5 Atuibute 6 Figure 1: MONK-I- net pruned with Unit-OBS, 22 weights. All unimportant units are removed and this net needs less units than the minimal network found by OBS! Atuibute 1 Attribute 2 Attribute 3 Attribute 4 Atuibute 5 Atuibute 6 Figure 2: Minimal network (14 weights) for the MONK-I problem found by the combination of Unit-OBS with OBS. The logical rule for the MONK- I problem is more evident in this network than in the minimal network found by OBS (comp. Fig. 3) . Atuibute 1 Attribute 2 Attribute 3 Attribute 4 Attribute 5 Attribute 6 Figure 3: Minimal network (14 weights) for the MONK-I problem found by OBS (see [1] and [2]) . 660 A. Stahlberger and M. Riedmiller algorithm # weights topology speedup+ perf. perf. train test no prumng 58 17-3-1 100% 100% OBS 14 6-3-1 1.0 100% 100% Unit-OBS 22 5-3-1 2.8 100% 100% Unit-OBS + OBS 14 5-3-1 2.6 100% 100% Table 1: The Monk- l problem For the initial Monk-l network the maximum number of outgoing connections (m in Eq. 7) is 3 and this is much smaller than the number of weights. The average number of outgoing connections of the removed units is 3 and therefore we expect a speedup by factor 3 (compare Table 1). By comparing the two minimal nets found by Unit-OBSjOBS (Fig. 2) and OBS (Fig. 3) it can be seen that the underlying logical rule (out=1 ¢:} AttribuLl=AttribuL2 or AttribuL5=1) is more evident in the network found by UnitOBSjOBS. The other advantage of Unit-OBS is that it needs only 38 % of the time OBS needs to find this minimal network. This advantage makes it possible to apply Unit-OBS to big nets for which OBS is not useful because of its long computation time. 6.2 The Thyroid Benchmark The following describes the application of pruning on a medical classification problem. The task is to classify measured data values of patients into three categories. The output of the three layered feed forward network therefore consists of three neurons indicating the corresponding class. The input consists of 21 both continuos and binary signals. The task was first described in [4]. The results obtained there are shown in the first row of Table 2. The initially used network has 21 input neurons, 10 hidden and 3 output neurons, which are fully connected using shortcut connections. When applying OBS to prune the network weights, more than 90 % of the weights can be pruned. However, over 8 hours of cpu-time on a sparc workstation are used to do so (row 2 in Table 2). The solution finally found by OBS uses only 8 of the originally 21 input features. The pruned network shows a slightly improved classification rate on the test set. Unit-OBS finds a solution with 41 weights in only 76 minutes of cpu-time. In comparison to the original OBS algorithm, Unit-OBS is about 8 times as fast when deleting the same number of weights. Also another important fact can be seen from the result: The Unit-OBS network considers only 7 of the originally 21 inputs, 1 less than the weight-focused OBS- algorithm. The number of hidden units is reduced to 2 units, 5 units less than the OBS network uses. When further looking for an absolute minimum in the number of used weights, the Unit-OBS network can be additionally pruned using OBS. This finally leeds to an optimized network with only 24 weights. The classification performance of this very tCompared to OBS deleting the same number of weights. Fast Network Pruning by using the Unit-DBS Algorithm 661 small network is 98.5 % which is even slightly better than obtained by the much bigger initial net. algorithm # weights topology speedup I cpu-time perf. test no prunmg 316 21-10-3 98.4% OBS 28 8-7-3 1.0 511 min . 98.5% Unit-OBS 41 7-2-3 7.8 76 min. 98.4% Unit-OBS + OBS 24 7-2-3 137 min. 98.5% Table 2: The thyroid benchmark 7 Conclusion The article describes an improvement of the OBS-algorithm introduced in [1] called Generalized OBS (G-OBS). The underlying idea is to exploit second order information to delete mutliple weights at once. The aim to reduce the number of different weight groups leads to the formulation of the Unit-OBS algorithm, which considers the outgoing weights of one unit as a group of candidate weights: When all the weights of a unit can be deleted, the unit itself can be pruned. The new Unit-OBS algorithm has two major advantages: First, it considerably accelerates pruning by a speedup factor which lies in the range of the average number of outgoing weights of each unit. Second, deleting complete units is especially interesting to determine the input features which really contribute to the computation of the output. This information can be used to get more insight in the underlying problem structure, e.g. to facilitate the process of rule extraction. References [1] B. Hassibi, D. G. Storck: Second Order Derivatives for Network Pruning: Optimal Brain Surgeon . Advances in Neural Information Processing Systems 5, Morgan Kaufmann, 1993, pages 164- 171. [2] B. Hassibi, D. G. Stork, G. J. Wolff: Optimal Brain Surgeon and general Network Pruning. IEEE International Conference on Neural Networks, 1993 Volume 1, pages 293-299. [3] A. Stahlberger: OBS - Verbesserungen und neue Ansatze. Diplomarbeit, Universitat Karlsruhe, Institut fur Logik, Komplexitat und Deduktionssysteme, 1996. [4] W . Schiffmann, M. Joost, R. Werner: Optimization of the Backpropagation Algorithm for Training Multilayer Perceptrons. Technical Report, University of Koblenz, Institute of Physics, 1993.	1996	132
1,187	Learning with Noise and Regularizers Multilayer Neural Networks David Saad Dept. of Compo Sci. & App. Math. Aston University Birmingham B4 7ET, UK D .Saad@aston.ac.uk Abstract Sara A. Solla AT &T Research Labs Holmdel, NJ 07733, USA solla@research .at t .com We study the effect of noise and regularization in an on-line gradient-descent learning scenario for a general two-layer student network with an arbitrary number of hidden units. Training examples are randomly drawn input vectors labeled by a two-layer teacher network with an arbitrary number of hidden units; the examples are corrupted by Gaussian noise affecting either the output or the model itself. We examine the effect of both types of noise and that of weight-decay regularization on the dynamical evolution of the order parameters and the generalization error in various phases of the learning process. 1 Introduction • In One of the most powerful and commonly used methods for training large layered neural networks is that of on-line learning, whereby the internal network parameters {J} are modified after the presentation of each training example so as to minimize the corresponding error. The goal is to bring the map fJ implemented by the network as close as possible to a desired map j that generates the examples. Here we focus on the learning of continuous maps via gradient descent on a differentiable error function. Recent work [1]-[4] has provided a powerful tool for the analysis of gradient-descent learning in a very general learning scenario [5]: that of a student network with N input units, I< hidden units, and a single linear output unit, trained to implement a continuous map from an N-dimensional input space e onto a scalar (. Examples of the target task j are in the form of input-output pairs (e', (1'). The output labels (JAto independently drawn inputs e' are provided by a teacher network of similar Learning with Noise and Regularizers in Multilayer Neural Networks 261 architecture, except that its number M of hidden units is not necessarily equal to K . Here we consider the possibility of a noise process pI-' that corrupts the teacher output. Learning from corrupt examples is a realistic and frequently encountered scenario. Previous analysis of this case have been based on various approaches: Bayesian [6], equilibrium statistical physics [7], and nonequilibrium techniques for analyzing learning dynamics [8]. Here we adapt our previously formulated techniques [2] to investigate the effect of different noise mechanisms on the dynamical evolution of the learning process and the resulting generalization ability. 2 The model We focus on a soft committee machine [1], for which all hidden-to-output weights are positive and of unit strength. Consider the student network: hidden unit i receives information from input unit r through the weight Jir, and its activation under presentation of an input pattern e = (6,· .. , ~N) is Xi = J i . e, with J i = (Jil , . .. , JiN ) defined as the vector of incoming weights onto the i-th hidden unit. The output of the student network is O"(J, e) = 2:~1 g (Ji . e), where g is the activation function of the hidden units, taken here to be the error function g(x) == erf(x/v'2), and J == {Jdl~i~K is the set of input-to-hidden adaptive weights. The components of the input vectors el-' are uncorrelated random variables with zero mean and unit variance. Output labels (I-' are provided by a teacher network of similar architecture: hidden unit n in the teacher network receives input information through the weight vector Bn = (Bn 1, . . . , BnN), and its activation under presentation of the input pattern e is Y~ = Bn . e. In the noiseless case the teacher output is given by (t = 2:~=1 g (Bn . e). Here we concentrate on the architecturally matched case M = K, and consider two types of Gaussian noise: additive output noise that results in (I-' = pI-' + 2:~=1 g (Bn . e), and model noise introduced as fluctuations in the activations Y~ of the hidden units, (I-' = 'E~=1 g (p~ + Bn . e). The random variables pI-' and p~ are taken to be Gaussian with zero mean and variance (J'2 . The error made by a student with weights J on a given input e is given by the quadratic deviation (1) measured with respect to the noiseless teacher (it is also possible to measure performance as deviations with respect to the actual output ( provided by the noisy teacher) . Performance on a typical input defines the generalization error Eg(J) == < E(J,e) >{O' through an average over all possible input vectors e to be performed implicitly through averages over the activations x = (Xl, ... , X K) and Y = (Yl, . . . , YK) . These averages can be performed analytically [2] and result in a compact expression for Eg in terms of order parameters: Qik == Ji ·Jk, Rin == Ji· B n , and Tnm == Bn . B m , which represent student-student, student-teacher, and teacherteacher overlaps, respectively. The parameters Tnm are characteristic of the task to be learned and remain fixed during training, while the overlaps Qik among student hidden units and R in between a student and a teacher hidden units are determined by the student weights J and evolve during training. A gradient descent rule on the error made with respect to the actual output provided 262 D. Saad and S. A. SolLa by the noisy teacher results in Jr+1 = Jf + N 8f e for the update of the student weights, where the learning rate 1] has been scaled with the input size N, and 8f depends on the type of noise. The time evolution of the overlaps Rin and Qik can be written in terms of similar difference equations. We consider the large N limit, and introduce a normalized number of examples Q' = III N to be interpreted as a continuous time variable in the N -+ 00 limit. The time evolution of Rin and Qik is thus described in terms of first-order differential equations. 3 Output noise The resulting equations of motion for the student-teacher and student-student overlaps are given in this case by: (2) where each term is to be averaged over all possible ways in which an example e could be chosen at a given time step. These averages have been performed using the techniques developed for the investigation of the noiseless case [2]; the only difference due to the presence of additive output noise is the need to evaluate the fourth term in the equation of motion for Qik, proportional to both 1]2 and 0'2. We focus on isotropic un correlated teacher vectors: Tnm = T 8nm , and choose T = 1 in our numerical examples. The time evolution of the overlaps Rin and Qik follows from integrating the equations of motion (2) from initial conditions determined by a random initialization of the student vectors {Jd1<i<K. Random initial norms Qii for the student vectors are taken here from a uniform distribution in the [0,0.5] interval. Overlaps Qik between independently chosen student vectors J i and J k, or Rin between Ji and an unknown teacher vector B n , are small numbers of order 1/-..iN for N » K, and taken here from a uniform distribution in the [0,10- 12] interval. We show in Figures l.a and 1. b the evolution of the overlaps for a noise variance 0'2 = 0.3 and learning rate 1] = 0.2. The example corresponds to M = K = 3. The qualitative behavior is similar to the one observed for M = K in the noiseless case extensively analyzed in [2]. A very short transient is followed by a long plateau characterized by lack of differentiation among student vectors: all student vectors have the same norm Qii = Q, the overlap between any two different student vectors takes a unique value Qik = C for i :j:. k, and the overlap Rin between an arbitrary student vector i and a teacher vector n is independent of i (as student vectors are indistinguishable in this regime) and of n (as the teacher is isotropic), resulting in Rin = R. This phase is characterized by an unstable symmetric solution; the perturbation introduced through the nonsymmetric initialization of the norms Qii and overlaps Rin eventually takes over in a transition that signals the onset of specialization. This process is driven by a breaking of the uniform symmetry of the matrix of student-teacher overlaps: each student vector acquires an increasingly dominant overlap R with a specific teacher vector which it begins to imitate, and a gradually decreasing secondary overlap S with the remaining teacher vectors. In the example of Figure l.b the assignment corresponds to i = 1 -+ n = 1, i = 2 -+ n = 3, and i = 3 -+ n = 2. A relabeling of the student hidden units allows us to identify R with the diagonal elements and S with the off-diagonal elements of the matrix of student-teacher overlaps. Learning with Noise and Regularizers in Multilayer Neural Networks 263 1.5 1.0 ~ CI 0.5 0.0 0.03 (a) QII -- QI2 ----- QI3 r --- Qu .. -.... -. Q" -----. Q" I,f --=-~~--------j:/ r--:--... ~ 0 2000 4000 6000 (c) ----------------------~ 0.025 - -.- - - - -- - - - - - - - - .... til 0.02 W 0.015 0.0\ 0.005 -u~o 1 l 17.02 l 17.0 a , , , , , \ " \ , \ \ \ , \ " \ , \ \ \0,,:::.. __ - =::"_-:.=.:"-::..=.:" O.o-+==::::;:::===r~-.--'-=:::::::::=F====? o 500 1000 1500 2000 2500 3000 (b) 1.2-.-~'--------------, 1.0 0.8 r:l 0.6 0.4 0.2 0.0 o (d) r-0.030.025til 0.02 W 0.015 0.01 0.005\ RII -- RI2 -.---- RI3 R" ------ Ru ... ---- R" R" -.- - - R" -----. R" 2000 4000 6000 Zoo Zos -- Z007 -------- Zos ~~:.:.:.:.:::---:.::::.:.:.:--------: --.: - ------::: -0.0..L-.-1.::..:.::::=:::::::::~~1~~~~~1~~;d 410' 610' 8103 Figure 1: Dependence of the overlaps and the generalization error on the normalized number of examples a for a three-node student learning corrupted examples generated by an isotropic three-node teacher. (a) student-student overlaps Qik and (b) student-teacher overlaps Rin for 172 = 0.3. The generalization error is shown in (c) for different values of the noise variance 172 , and in (d) for different powers of the polynomial learning rate decay, focusing on a > 0'0 ( asymptotic regime). Asymptotically the secondary overlaps S decay to zero, while Rin -+ -ICJii indicates full alignment for Tnn = L As specialization proceeds, the student weight vectors grow in length and become increasingly uncorrelated. It is interesting to observe that in the presence of noise the student vectors grow asymptotically longer than the teacher vectors: Qii -+ Qoo > 1, and acquire a small negative correlation with each other. Another detectable difference in the presence of noise is a larger gap between the values of Q and C in the symmetric phase. Larger norms for the student vectors result in larger generalization errors: as shown in Figure I.c, the generalization error increases monotonically with increasing noise level, both in the symmetric and asymptotic regimes. For an isotropic teacher, the teacher-student and student-student overlaps can thus be fully characterized by four parameters: Qik = QCik +C(I- Cik) and R;n = RCin + S(I-Cin). In the symmetric phase the additional constraint R = S reflects the lack of differentiation among student vectors and reduces the number of parameters to three. The symmetric phase is characterized by a fixed point solution to the equations 264 D. Saad and S. A. Solfa of motion (2) whose coordinates can be obtained analytically in the small noise approximation: R = I/JK(2K -1) + 1/ 0'2 r8 , Q* = 1/(2K -1) + 1/ 0'2 q8 , and C* = 1/(2K -1) + 1/ 0'2 C8, with r 8, q8, and C8 given by relatively simple functions of K. The generalization error in this regime is given by: * K (7r , . ( 1 )) 0'21/ (2K - 1 ?/2 fg = -; 6' - Ii arCSIn 2K + 27r (2K + 1)1/2 ; (3) note its increase over the corresponding noiseless value, recovered for 0'2 = O. The asymptotic phase is characterized by a fixed point solution with R* =j:. S. The coordinates of the asymptotic fixed point can also be obtained analytically in the small noise approximation: R = 1 + 1/ 0'2 r a , S* = -1/ 0'2 Sa, Q* = 1 + 1/ 0'2 qa, and C* = -1/ 0'2 Ca , with r a, Sa, qa, and Ca given by rational functions of K with corrections of order 1/. The asymptotic generalization error is given by * .J3 2 T.' (4) f g = 67r 1/ 0' .Ii . Explicit expressions for the coefficients r 8, q8' C8 , r a, Sa, qa, and Ca will not be given here for lack of space; suffice it to say that the fixed point coordinates predicted on the basis of the small noise approximation are found to be in excellent agreement with the values obtained from the numerical integration of the equations of motion for 0'2 ~ 0.3. It is worth noting in Figure I.c that in the small noise regime the length of the symmetric plateau decreases with increasing noise. This effect can be investigated analytically by linearizing the equations of motion around the symmetric fixed point and identifying the positive eigenvalue responsible for the escape from the symmetric phase. This calculation has been carried out in the small noise approximation, to obtain A = (2/7r)K(2K - 1)-1/2(2K + 1)-3/2 + Au0'21/, where Au is positive and increases monotonically with K for K > 1. A faster escape from the symmetric plateau is explained by this increase of the positive eigenvalue. The calculation is valid for 0'21/ ~ 1; we observe experimentally that the trend is reversed as 0'2 increases. A small level of noise assists in the process of differentiation among student vectors, while larger levels of noise tend to keep student vectors equally ignorant about the task to be learned. The asymptotic value (4) for the generalization error indicates that learning at finite 1/ will result in asymptotically suboptimal performance for 0'2 > O. A monotonic decrease ofthe learning rate is necessary to achieve optimal asymptotic performance with f; = O. Learning at small 1/ results in long trapping times in the symmetric phase; we therefore suggest starting the training process with a relatively large value of 1/ and switching to a decaying learning rate at 0' = 0'0, after specialization begins. We propose 1/ = 1/0 for 0' ~ 0'0 and 1/ = 1/0/(0' - O'oy for 0' > 0'0 . Convergence to the asymptotic solution requires z ~ 1. The value z = 1 corresponds to the fastest decay for 1/(0'); the question of interest is to determine the value of z which results in fastest decay for fg(O'). Results shown in Figure l.d for 0' > 0'0 = 4000 correspond to M = K = 3, 1/0 = 0.7, and 0'2 = 0.1. Our numerical results indicate optimal decay of fg(O') for z = 1/2. A rigorous justification of this result remains to be found. 4 Model noise The resulting equations of motion for the student-teacher and student-student overlaps can also be obtained analytically in this case; they exhibit a structure remarkLearning with Noise and Regularizers in Multilayer Neural Networks 0.06, ------- -- ---.-.---·,1 .~, 1-------.. :' tIGo.04 ~~ W ~ 0.02 -u~o 5 u~o 1 --- u'JJ 9 ~ -._.- ----.-- ---_.- -- - - --. '._---_._-..... _----_ ... _-------_._ .. _------0.0-+------.--------.----' 010' 510' 110' 0.04 ~0.03 100.02 0.01 \0 20 30 40 50 60 70 80 90 K 265 Figure 2: Left - The generalization error for different values of the noise variance (72; training examples are corrupted by model noise. Right - 7max as a function of K. ably similar to those for the noiseless case reported in [2], except for some changes in the relevant covariance matrices. A numerical investigation of the dynamical evolution of the overlaps and generalization error reveals qualitative and quantitative differences with the case of additive output noise: 1) The sensitivity to noise is much higher for model noise than for output noise. 2) The application of independent noise to the individual teacher hidden units results in an effective anisotropic teacher and causes fluctuations in the symmetric phase; the various student hidden units acquire some degree of differentiation and the symmetric phase can no longer be fully characterized by unique values of Q and C. 3) The noise level does not affect the length of the symmetric phase. The effect of model noise on the generalization error is illustrated in Figure 2 for M = K = 3, 'rJ = 0.2, and various noise levels. The generalization error increases monotonically with increasing noise level, both in the symmetric and asymptotic regimes, but there is no modification in the length of the symmetric phase. The dynamical evolution of the overlaps, not shown here for the case of model noise, exhibits qualitative features quite similar to those discussed in the case of additive output noise: we observe again a noise-induced widening of the gap between Q and C in the symmetric phase, while the asymptotic phase exhibits an enhancement of the norm of the student vectors and a small degree of negative correlation between them. Approximate analytic expressions based on a small noise expansion have been obtained for the coordinates of the fixed point solutions which describe the symmetric and asymptotic phases. In the case of model noise the expansions for the symmetric solution are independent of 'rJ and depend only on (72 and K. The coordinates of the asymptotic fixed point can be expressed as: R = 1 + (72 r a, S* = _(72 Sa, Q* = 1 + (72 qa, C* = _(72 Ca , with coefficients ra , Sa, qa, and Ca given by rational functions of K with corrections of order 'rJ. The important difference with the output noise case is that the asymptotic fixed point is shifted from its noiseless position even for 'rJ = O. It is therefore not possible to achieve optimal asymptotic performance even if a decaying learning rate is utilized. The asymptotic generalization error is given by * y'3 2}-' 2K' (}., ) (g=--(7 \+'rJ(7 (a \,'rJ . 1271" (5) 266 D. Saad and S. A. Solla Note that the asymptotic generalization error remains finite even as TJ O. 5 Regularlzers A method frequently used in real world training scenarios to overcome the effects of noise and parameter redundancy (1< > M) is the use of regularizers such as weight decay (for a review see [6]). Weight-decay regularization is easily incorporated within the framework of on-line learning; it leads to a rule for the update of the student weights of the form Jf+l = Jf + 11 6r e - 1:r Jf· The corresponding equations of motion for the dynamical evolution of the teacher-student and student-student overlaps can again be obtained analytically and integrated numerically from random initial conditions. The picture that emerges is basically similar to that described for the noisy case: the dynamical evolution of the learning process goes through the same stages, although specific values for the order parameters and generalization error at the symmetric phase and in the asymptotic regime are changed as a consequence of the modification in the dynamics. Our numerical investigations have revealed no scenario, either when training from noisy data or in the presence of redundant parameters, where weight decay improves the system performance or speeds up the training process. This lack of effect is probably a generic feature of on-line learning, due to the absence of an additive, stationary error surface defined over a finite and fixed training set. In off-line (batch) learning, regularization leads to improved performance through the modification of such error surface. These observations are consistent with the absence of 'overfitting' phenomena in on-line learning. One of the effects that arises when weight-decay regularization is introduced in on-line learning is a prolongation of the symmetric phase, due to a decrease in the positive eingenvalue that controls the onset of specialization. This positive eigenvalue, which signals the instability of the symmetric fixed point, decreases monotonically with increasing regularization strength 'Y, and crosses zero at 'Ymax = TJ 7max. The dependence of 7max on 1< is shown in Figure 2; for 'Y > 'Ymax the symmetric fixed point is stable and the system remains trapped there for ever. The work reported here focuses on an architecturally matched scenario, with M = 1<. Over-realizable cases with 1< > M show a rich behavior that is rather less amenable to generic analysis. It will be of interest to examine the effects of different types of noise and regularizers in this regime. Acknowledgement: D.S. acknowledges support from EPSRC grant GRjLl9232. References [1] M. Biehl and H. Schwarze, J. Phys. A 28, 643 (1995). [2] D. Saad and S.A. Solla, Phys. Rev. E 52, 4225 (1995). [3] D. Saad and S.A. Solla, preprint (1996). [4] P. Riegler and M. Biehl, J. Phys. A 28, L507 (1995). [5] G. Cybenko, Math. Control Signals and Systems 2, 303 (1989). [6] C.M. Bishop, Neural networks for pattern recognition, (Oxford University Press, Oxford, 1995). [7] T.L.H. Watkin, A. Rau, and M. Biehl, Rev. Mod. Phys. 65, 499 (1993). [8] K.R. Muller, M. Finke, N. Murata, K. Schulten, and S. Amari, Neural Computation 8, 1085 (1996).	1996	133
1,188	The Learning Dynamics of a Universal Approximator Ansgar H. L. West1,2 A.H.L.West~aston.ac.uk David Saad1 D.Saad~aston.ac.uk Ian T. N abneyl I.T.Nabney~aston.ac.uk 1 Neural Computing Research Group, University of Aston Birmingham B4 7ET, U.K. http://www.ncrg.aston.ac.uk/ 2Department of Physics, University of Edinburgh Edinburgh EH9 3JZ, U.K. Abstract The learning properties of a universal approximator, a normalized committee machine with adjustable biases, are studied for on-line back-propagation learning. Within a statistical mechanics framework, numerical studies show that this model has features which do not exist in previously studied two-layer network models without adjustable biases, e.g., attractive suboptimal symmetric phases even for realizable cases and noiseless data. 1 INTRODUCTION Recently there has been much interest in the theoretical breakthrough in the understanding of the on-line learning dynamics of multi-layer feedforward perceptrons (MLPs) using a statistical mechanics framework. In the seminal paper (Saad & Solla, 1995), a two-layer network with an arbitrary number of hidden units was studied, allowing insight into the learning behaviour of neural network models whose complexity is of the same order as those used in real world applications. The model studied, a soft committee machine (Biehl & Schwarze, 1995), consists of a single hidden layer with adjustable input-hidden, but fixed hidden-output weights. The average learning dynamics of these networks are studied in the thermodynamic limit of infinite input dimensions in a student-teacher scenario, where a stu.dent network is presented serially with training examples (elS , (IS) labelled by a teacher network of the same architecture but possibly different number of hidden units. The student updates its parameters on-line, i.e., after the presentation of each example, along the gradient of the squared error on that example, an algorithm usually referred to as back-propagation. Although the above model is already quite similar to real world networks, the approach suffers from several drawbacks. First, the analysis of the mean learning dynamics employs the thermodynamic limit of infinite input dimension a problem which has been addressed in (Barber et al., 1996), where finite size effects have been studied and it was shown that the thermodynamic limit is relevant in most The Learning Dynamcis of a UniversalApproximator 289 cases. Second, the hidden-output weights are kept fixed, a constraint which has been removed in (Riegler & Biehl, 1995), where it was shown that the learning dynamics are usually dominated by the input-hidden weights. Third, the biases of the hidden units were fixed to zero, a constraint which is actually more severe than fixing the hidden-output weights. We show in Appendix A that soft committee machines are universal approximators provided one allows for adjustable biases in the hidden layer. In this paper, we therefore study the model of a normalized soft committee machine with variable biases following the framework set out in (Saad & Solla, 1995). We present numerical studies of a variety of learning scenarios which lead to remarkable effects not present for the model with fixed biases. 2 DERIVATION OF THE DYNAMICAL EQUATIONS The student network we consider is a normalized soft committee machine of K hidden units with adjustable biases. Each hidden unit i consists of a bias (Ji and a weight vector lVi which is connected to the N-dimensional inputs e. All hidden units are connected to a linear output unit with arbitrary but fixed gain 'Y by couplings of fixed strength. The activation of any unit is normalized by the inverse square root of the number of weight connections into the unit, which allows all weights to be of 0(1) magnitude, independent of the input dimension or the number of hidden units. The implemented mapping is therefore /w(e) = (-Y/VK) L:~1 g(Ui - (Ji), where Ui = lVi ·e/.,fJii and g(.) is a sigmoidal transfer function. The teacher network to be learned is of the same architecture except for a possible difference in the number of hidden units M and is defined by the weight vectors En and biases Pn (n = 1, ... , M). Training examples are of the form (e, (1-'), where the input vectors el-' are drawn form the normal distribution and the outputs are (I-' = (-Y/.JiJ) L:~1 g(v~ - Pn), where v~ = Bn ·el-' /.,fJii. The weights and biases are updated in response to the presentation of an example (el-', (1-'), along the gradient of the squared error measure € = ![(I-' - /w(el-')F Wol-'+! - Wol-' = 1/ 61!' el-' and (J.I-'+! (J .I-' = - 1/0 61!' (1) t I Wt.,fJii I I Nt with 6f == [(I-' - /w(el-')]g'(uf - (Ji). The two learning rates are 1/w for the weights and 1/0 for the biases. In order to analyse the mean learning dynamics resulting from the above update equations, we follow the statistical mechanics framework in (Saad & Solla, 1995). Here we will only outline the main ideas and concentrate on the results of the calculation. As we are interested in the typical behaviour of our training algorithm we average over all possible instances of the examples e. We rewrite the update equations (1) in lVi as equations in the order parameters describing the overlaps between pairs of student nodes Qij = lVi·W;/N, student and teacher nodes Rin = lVi·En/N, and teacher nodes Tnm = Bn ·Bm/N. The generalization error €g, measuring the typical performance, can be expressed solely in these variables and the biases (Ji and Pn. The order parameters Qij, Rin and the biases (Ji are the dynamical variables. These quantities need to be self-averaging with respect to the randomness in the training data in the thermodynamic limit (N ~ 00), which enforces two necessary constraints on our calculation. First, the number of hidden units K « N, whereas one needs K", O(N) for the universal approximation proof to hold. Second, one can show that the updates of the biases have to be of 0(1/N), i.e., the bias learning rate has to be scaled by 1/ N, in order to make the biases self-averaging quantities, a fact that is confirmed by simulations [see Fig. 1]. If we interpret the normalized 290 A. H. L. West, D. Saad and I. T. Nabney example number 0 = piN as a continuous time variable, the update equations for the order parameters and the biases become first order coupled differential equations dQij do dRin do TJw (8iuj + 8j U i}e + TJ!. (8i8j }e· dOi TJw (8ivn }e ' and do = -TJo (8i }e . (2) Choosing g(x) = erf(xlV2) as the sigmoidal transfer, most integrations in Eqs. ~2) can be performed analytically, but for single Gaussian integrals remaining for TJw terms and the generalization error. The exact form of the resulting dynamical equations is quite complicated and will be presented elsewhere. Here we only remark, that the gain "/ of the linear output unit, which determines the output scale, merely rescales the learning rates with ,,/2 and can therefore be set to one without loss of generality. Due to the numerical integrations required, the differential equations can only be solved accurately in moderate times for smaller student networks (K ~ 5) but any teacher size M. 3 ANALYSIS OF THE DYNAMICAL EQUATIONS The dynamical evolution of the overlaps Qij, Rin and the biases Oi follows from integrating the equations of motion (2) from initial conditions determined by the (random) initialization of the student weights Wi and biases Oi. For random initialization the resulting norms Qii of the student vector will be order 0(1), while the overlaps Qij between different student vectors, and student-teacher vectors Rin will be only order CJ(I/VN). A random initialization of the weights and biases can therefore be simulated by initializing the norms Qii, the biases Oi and the normalized overlaps Qij = Qij I JQiiQjj and Rin = Rinl JQiiTnn from uniform distributions in the [0,1]' [-1,1], and [_10- 12,10- 12] intervals respectively. We find that the results of the numerical integration are sensitive to these random initial values, which has not been the case to this extent for fixed biases. Furthermore, the dynamical behaviour can become very complex even for realizable cases (K = M) and networks with three or four hidden units. For sake of simplicity, we will therefore restrict our presentation to networks with two hidden units (K = M = 2) and uncorrelated isotropic teachers, defined by Tnm = 8nm, although larger networks and graded teacher scenarios were investigated extensively as well. We have further limited our scope by investigating a common learning rate (TJo = TJo = TJw) for biases and weights. To study the effect of different weight initialization, we have fixed the initial values of the student-student overlaps Qij and biases Oi, as these can be manipulated freely in any learning scenario. Only the initial student-teacher overlaps Rin are randomized as suggested above. In Fig. 1 we compare the evolution of the overlaps, the biases and the generalization error for the soft committee machine with and without adjustable bias learning a similar realizable teacher task. The student denoted by * lacks biases, Le., Oi = 0, and learns to imitate an isotropic teacher with zero biases (Pn = 0). The other student features adjustable biases, trained from an isotropic teacher with small biases (Pl,2 = =FO.I). For both scenarios, the learning rate and the initial conditions were judiciously chosen to be TJo = 2.0, Qll = 0.1, Q22 = 0.2, Rin = Q12 = U[ _10- 12,10-12] with 01 = 0.0 and O2 = 0.5 for the student with adjustable biases. In both cases, the student weight vectors (Fig. Ia) are drawn quickly from their initial values into a suboptimal symmetric phase, characterized by the lack of specialization of the student hidden units on a particular teacher hidden unit, as can be depicted from the similar values of ~n in Fig. 1 b. This symmetry is broken 1.0 0.8 0.6 Qij 0.4 0.2 0.0 0.3 0.2 0.1 (}i 0.0 -0.1 -0.2 -0.3 The Learning Dynamcis of a UniversalApproximator ,0 Q (N 0) Q11 ,Q22 o 11 =1 :} "Q11 (N=100) Qil-/1 A Q12 (N=10) Q22··········· o Q12 (N=100) Qi2-'-'Q11-----· ~n 'Q* Q* Q22--III 22 _.-.- -_.-. __ ._._. '-. Q12----· ~-.-.-.-.-.-.-.-.-:~.'::,'"------. o 100 200 300 400 500 600 700 ex 1.0 0.8 0.6 0.4 0.2 0.0 291 o 100 200 300 400 500 600 700 ex 0.02-r-------------, (d) .. , , -..---==:0-..... -,.:.<--- '1< ....... ,. <:: , 0 '. ,". ' ,... .... 100 200 300 400 500 600 700 ex 0.015 0.005 fg(O.Ol) - -fg(O.l) _ ... - . fg(0.5)fg(l) -fg(O) --fg(O) .......... . f g(10-S) ----_. fg(10-4) - .- .N=200 0 N=500" o 100 200 300 400 500 600 700 ex Figure 1: The dynamical evolution of the student-student overlaps Qij (a), and the student-teacher overlaps Rin (b) as a function of the normalized example number 0 is compared for two student-teacher scenarios: One student (denoted by ) has fixed zero biases, the other has adjustable biases. The influence of the symmetry in the initialization of the biases on the dynamics is shown for the student biases (Ji (c), and the generalization error fg (d): (Jl = 0 is kept for all runs, but the initial value of (J2 varies and is given in brackets in the legends. Finite size simulations for input dimensions N = 10 ... 500 show that the dynamical variables are self-averaging. almost immediately in the learning scenario with adjustable biases and the student converges quickly to the optimal solution, characterized by the evolution of the overlap matrices Q, R and biases (Ji (see Fig. 1c) to their optimal values T and Pn (up to the permutation symmetry due to the arbitrary labeling of the student nodes). Likewise, the generalization error fg decays to zero in Fig. 1d. The student with fixed biases is trapped for most of its training time in the symmetric phase before it eventually converges. Extensive simulations for input dimensions N = 10 ... 500 confirm that the dynamic variables are self-averaging and show that variances decrease with liN. The mean trajectories are in good agreement with the theoretical predictions even for very small input dimensions (N = 10) and are virtually indistinguishable for N = 500. The length of the symmetric phase for the isotropic teacher scenario is dominated by the learning ratel , hut also exhibits a logarithmic dependence on the typical 1The length of the symmetric phase is linearly dependent on 110 for small learning rates. 292 I \ \ (}i O.O-t----· -' '-' -' '-' -' '-'-' -' ' -,;--'~; -~--i ..... ..... .................... " I " I -0.2- .,;:.:-=:::~.::::.------_ " I r'" .:::--:..:-~-..:.;::.-- .. ' i -0.4- ...... O2 -- --. (h .... :::- '~.-j (0.25) (05) .... .......... 01 -----. Ol -0.6 I I o 400 800 a I 1200 Ja) I 1600 3200 2800 2400 a c 2000 1600 1200 A. H. L. West, D. Saad and I. T. Nabney 2-0 !I 1 I --710!I I 1 I ........... 710=0.01!1 I 1 I -----·710=0.1 ;1 I I -I I I - .- .- 710=0.5 II r I r r - - - 710=1 il j I iI I -"'- 710=1.5 I I il I I --- 710=2 I I il i I I I ---. ''10=3 !I i I , , !I , i , !I I , , , --------------}/ I I / ,. , /: / . ,/ :77~.:7"''=:- .~;''- ' ~ :-,." 0.0 0.2 0.4 0.6 0.8 1.0 (}2 Figure 2: (a) The dynamical evolution of the biases Oi for a student imitating an isotropic teacher with zero biases. reveals symmetric dynamics for 01 and O2 • The student was randomly initialized identically for the different runs, but for a change in the range of the random initialization of the biases (U[-b,b]), with the value of b given in the legend. Above a critical value of b the student remains stuck in a suboptimal phase. (b) The normalized convergence time ~ == TJoQc is shown as a function of the initialization of O2 for varios learning rates TJo (see legend, TJ5 = 0 symbolizes the dynamics neglecting TJ5 terms.). differences in the initial student-teacher overlaps Rin (Biehl et al., 1996) which are typically of order O(I/..fN) and cannot be influenced in real scenarios without a priori knowledge. The initialization of the biases, however, can be controlled by the user and its influence on the learning dynamics is shown in Figs. lc and Id for the biases and the generalization error respectively. For initially identical biases (01 = O2 = 0), the evolution of the order parameters and hence the generalization error is almost indistinguishable from the fixed biases case. A breaking of this symmetry leads to a decrease of the symmetric phase linear in log(IOl - ( 2 1) until it has all but disappeared. The dynamics are again slowed down for very large initialization of the biases (see Id), where the biases have to travel a long way to their optimal values. This suggests that for a given learning rate the biases have a dominant effect in the learning process and strongly break existent symmetries in weight space. This is argueably due to a steep minimum in the generalization error surface along the direction of the biases. To confirm this, we have studied a range of other learning scenarios including larger networks and non-isotropic teachers, e.g., graded teachers with Tnm = n6nm . Even when the norms of the teacher weight vectors are strongly graded, which also breaks the weight symmetry and reduces the symmetric phase significantly in the case of fixed biases, we have found that the biases usually have the stronger symmetry breaking effect: the trajectories of the biases never cross, provided that they were not initialized too symmetrically. This would seem to promote initializing the biases of the student hidden units evenly across the input domain, which has been suggested previously on a heuristic basis (Nguyen & Widrow, 1990). However, this can lead to the student being stuck in a suboptimal configuration. In Fig. 2a, we show the dynamics of the student biases Oi when the teacher biases are symmetric (Pn = 0). We find that the student progress is inversely related to the magnitude of the bias initialization and finally fails to converge at all. It remains in a suboptimal phase characterized by biases of the same large magnitude but opposite sign and highly correlated weight vectors. In effect, the outputs of the two student nodes cancel out over most of the input domain. In The Learning Dynamcis of a Universal Approximator 293 Fig. 2b, the influence of the learning rate in combination with the bias initialization in determining convergence is illustrated. The convergence time Qc, defined as the example number at which the generalization error has decayed to a small value, here judiciously chosen to be 10-8 , is shown as a function of the initial value of ()2 for various learning rates 'TJo. For convenience, we have normalized the convergence time with 1/""0. The initialization of the other order parameters is identical to Fig. 1a. One finds that the convergence time diverges for all learning rates, above a critical initial value of (h. For increasing learning rates, this transition becomes sharper and occurs at smaller ()2, i.e., the dynamics become more sensitive to the bias initialization. 4 SUMMARY AND DISCUSSION This research has been motivated by recent progress in the theoretical study of on-line learning in realistic two-layer neural network models the soft-committee machine, trained with back-propagation (Saad & Solla, 1995). The studies so far have excluded biases to the hidden layers, a constraint which has been removed in this paper, which makes the model a universal approximator. The dynamics of the extended model turn out to be very rich and more complex than the original model. In this paper, we have concentrated on the effect of initialization of student weights and biases. We have further restricted our presentation for simplicity to realizable cases and small networks with two hidden units, although larger networks were studied for comparison. Even in these simple learning scenarios, we find surprising dynamical effects due to the adjustable biases. In the case where the teacher network exhibits distinct biases, unsymmetric initial values of the student biases break the node symmetry in weight space effectively and can speed up the learning process considerably, suggesting that student biases should in practice be initially spread evenly across the input domain if there is no a priori knowledge of the function to be learned. For degenerate teacher biases however such a scheme can be counterproductive as different initial student bias values slow down the learning dynamics and can even lead to the student being stuck in suboptimal fixed points, characterized by student biases being grouped symmetrically around the degenerate teacher biases and strong correlations between the associated weight vectors. In fact, these attractive suboptimal fixed points exist even for non-degenerate teacher biases, but the range of initial conditions attracted to these suboptimal network configurations decreases in size. Furthermore, this domain is shifted to very large initial student biases as the difference in the values of the teacher biases is increased. We have found these effects also for larger network sizes, where the dynamics and number of attractive suboptimal fixed points with different internal symmetries increases. Although attractive suboptimal fixed points were also found in the original model (Biehl et al., 1996), the basins of attraction of initial values are in general very small and are therefore only of academic interest. However, our numerical work suggests that a simple rule of thumb to avoid being attracted to suboptimal fixed points is to always initialize the squared norm of a weight vector larger than the magnitude of the corresponding bias. This scheme will still support spreading of the biases across the main input domain in order to encourage node symmetry breaking. This is somewhat similar to previous findings (Nguyen & Widrow, 1990; Kim & Ra, 1991), the former suggesting spreading the biases across the input domain, the latter relating the minimal initial size of each weight with the learning rate. This work provides a more theoretical motivation for these results and also distinguishes between the different roles of biases and weights. In this paper we have addressed mainly one important issue for theoreticians and 294 A. H. L West, D. Saad and l. T. Nabney practitioners alike: the initialization of the student network weights and biases. Other important issues, notably the question of optimal and maximal learning rates for different network sizes during convergence, will be reported elsewhere. A THEOREM Let S9 denote the class of neural networks defined by sums of the form L~l nig(ui - (h) where K is arbitrary (representing an arbitrary number of hidden units), (h E lR and ni E Z (i.e. integer weights). Let 'I/J(x) == ag(x)/ax and let 1>", denote the class of networks defined by sums of the form L~l Wi'I/J(Ui -0;) where W; E lR. If 9 is continuously differentiable and if the class 1>", are universal approximators, then S9 is a class of universal approximatorsj that is, such functions are dense in the space of continuous functions with the Loo norm. As a corollary, the normalized soft committee machine forms a class of universal approximators with both sigmoid and error transfer functions [since radial basis function networks are universal (Park & Sandberg, 1993) and we need consider only the one-dimensional input case as noted in the proof below). Note that some restriction on 9 is necessary: if 9 is the step function, then with arbitrary hidden-output weights, the network is a universal approximator, while with fixed hidden-output weights it is not. A.! Proof By the arguments of (Hornik et al., 1990) which use the properties of trigonometric polynomials, it is sufficient to consider the case of one-dimensional input and output spaces. Let I denote a compact interval in lR and let f be a continuous function defined on I. Because 1>", is universal, given any E > 0 we can find weights Wi and biases Oi such that K f- LW;'I/J(u-Oi) ;=1 00 E <-2 (i) Because the rationals are dense in the reals, without loss of generality we can assume that the weights Wi E Q. Since 'I/J(x) is continuous and I is compact, the convergence of [g(x + h) - g(x)J1h to ag(x)/ax is uniform and hence for all n> n (21;Wi) the following i~.lity hblds: (ii) I Also note that for suitable ni > n (2~Wi)' rn. = now; E Z, as Wi is a rational number. Thus, by the triangle inequality, K K L .rn; [g(u+ ~i -0;) -g(u-Oi)] - LWi'I/J(u-Oi) .=1 i=l 00 The result now follows from equations (i) and (iii) and the triangle inequality. References Barber, D., Saad, D., & Sollich, P. 1996. Europhys. Lett., 34, 151-156. Biehl, M., & Schwarze, H. 1995. J. Phys. A, 28, 643-656. (iii) Biehl, M., Riegler, P., & Wohler, C. 1996. University of Wiirzburg Preprint WUE-ITP96-003. Hornik, K., Stinchcombe, M., & White, H. 1990. Neural Networks, 3, 551-560. Kim, Y. K., & Ra, J. ,B. 1991. Pages 2396-2401 of: International Joint Conference on Neural Networks 91. Nguyen, D., & Widrow, B. 1990. Pages C21-C26 of: IJCNN International Conference on Neural Networks 90. Park, J., & Sandberg, 1. W. 1993. Neural Computation, 5, 305-316. Riegler, P., & Biehl, M. 1995. J. Phys. A, 28, L507-L513. Saad, D., & SoHa, S. A. 1995. Phys. Rev. E, 52, 4225-4243.	1996	134
1,189	Gaussian Processes for Bayesian Classification via Hybrid Monte Carlo David Barber and Christopher K. I. Williams Neural Computing Research Group Department of Computer Science and Applied Mathematics Aston University, Birmingham B4 7ET, UK d.barber~aston.ac.uk c.k.i.williams~aston.ac.uk Abstract The full Bayesian method for applying neural networks to a prediction problem is to set up the prior/hyperprior structure for the net and then perform the necessary integrals. However, these integrals are not tractable analytically, and Markov Chain Monte Carlo (MCMC) methods are slow, especially if the parameter space is high-dimensional. Using Gaussian processes we can approximate the weight space integral analytically, so that only a small number of hyperparameters need be integrated over by MCMC methods. We have applied this idea to classification problems, obtaining excellent results on the real-world problems investigated so far. 1 INTRODUCTION To make predictions based on a set of training data, fundamentally we need to combine our prior beliefs about possible predictive functions with the data at hand. In the Bayesian approach to neural networks a prior on the weights in the net induces a prior distribution over functions. This leads naturally to the idea of specifying our beliefs about functions more directly. Gaussian Processes (GPs) achieve just that, being examples of stochastic process priors over functions that allow the efficient computation of predictions. It is also possible to show that a large class of neural network models converge to GPs in the limit of an infinite number of hidden units (Neal, 1996). In previous work (Williams and Rasmussen, 1996) we have applied GP priors over functions to the problem of predicting a real-valued output, and found that the method has comparable performance to other state-of-the-art methods. This paper extends the use of GP priors to classification problems. The G Ps we use have a number of adjustable hyperparameters that specify quantities like the length scale over which smoothing should take place. Rather than Gaussian Processes/or Bayesian Classification via Hybrid Monte Carlo 341 optimizing these parameters (e.g. by maximum likelihood or cross-validation methods) we place priors over them and use a Markov Chain Monte Carlo (MCMC) method to obtain a sample from the posterior which is then used for making predictions. An important advantage of using GPs rather than neural networks arises from the fact that the GPs are characterized by a few (say ten or twenty) hyperparameters, while the networks have a similar number of hyperparameters but many (e.g. hundreds) of weights as well, so that MCMC integrations for the networks are much more difficult . We first briefly review the regression framework as our strategy will be to transform the classification problem into a corresponding regression problem by dealing with the input values to the logistic transfer function. In section 2.1 we show how to use Gaussian processes for classification when the hyperparameters are fixed , and then describe the integration over hyperparameters in section 2.3. Results of our method as applied to some well known classification problems are given in section 3, followed by a brief discussion and directions for future research. 1.1 Gaussian Processes for regression We outline the GP method as applied to the prediction of a real valued output y* = y(x) for a new input value x, given a set of training data V = {(Xi, ti), i = 1. .. n} Given a set of inputs X,X1, ... Xn, a GP allows us to specify how correlated we expect their corresponding outputs y = (y(xt), y(X2), ... , y(xn)) to be. We denote this prior over functions as P(y), and similarly, P(y, y) for the joint distribution including y. If we also specify P( tly), the probability of observing the particular values t = (t1, .. . tn)T given actual values y (i.e. a noise model) then P(ylt) = J P(y,ylt)dy = P~t) J P(y,y)P(tly)dy (1) Hence the predictive distribution for y. is found from the marginalization of the product of the prior and the noise model. If P(tly) and P(y, y) are Gaussian then P(Y. It) is a Gaussian whose mean and variance can be calculated using matrix computations involving matrices of size n x n. Specifying P(y, y) to be a multidimensional Gaussian (for all values of n and placements of the points X, Xl , . .. Xn) means that the prior over functions is a G P. More formally, a stochastic process is a collection ofrandom variables {Y(x )Ix E X} indexed by a set X. In our case X will be the input space with dimension d, the number of inputs. A GP is a stochastic process which can be fully specified by its mean function J.l(x) = E[Y(x)] and its covariance function C(x,x') = E[(Y(x) - J.l(x))(Y(x') - J.l(x'))]; any finite set of Y -variables will have a joint multivariate Gaussian distribution. Below we consider GPs which have J.l(x) == o. 2 GAUSSIAN PROCESSES FOR CLASSIFICATION For simplicity of exposition, we will present our method as applied to two class problems as the extension to multiple classes is straightforward. By using the logistic transfer function u to produce an output which can be interpreted as 11"(x), the probability of the input X belonging to class 1, the job of specifying a prior over functions 11" can be transformed into that of specifying a prior over the input to the transfer function. We call the input to the transfer function the activation, and denote it by y, with 11"(x) = u(y(x)). For input Xi, we will denote the corresponding probability and activation by 11"i and Yi respectively. 342 D. Barber and C. K. l Williams To make predictions when using fixed hyperparameters we would like to compute 11-. = !7r.P(7r.lt) d7r., which requires us to find P(7r.lt) = P(7r(z.)lt) for a new input z •. This can be done by finding the distribution P(y. It) (Y. is the activation of 7r.) and then using the appropriate Jacobian to transform the distribution. Formally the equations for obtaining P(y. It) are identical to equation 1. However, even if we use a GP prior so that P(Y., y) is Gaussian, the usual expression for P(tly) = ni 7r;' (1 - 7rd 1- t , for classification data (where the t's take on values of 0 or 1), means that the marginalization to obtain P(Y. It) is no longer analytically tractable. We will employ Laplace's approximation, i.e. we shall approximate the integrand P(Y., ylt, 8) by a Gaussian distribution centred at a maximum of this function with respect to Y., Y with an inverse covariance matrix given by - v"v log P(Y., ylt, 8). The necessary integrations (marginalization) can then be carried out analytically (see, e.g. Green and Silverman (1994) §5.3) and we provide a derivation in the following section. 2.1 Maximizing P(y.,ylt) Let y+ denote (Y. , y), the complete set of activations. By Bayes' theorem log P(y+ It) = log P(tly)+log P(y+)-log P(t), and let 'It+ = log P(tly)+log P(y+) . As P(t) does not depend on y+ (it is just a normalizing factor), the maximum of P(y+ It) is found by maximizing 'It + with respect to y+. We define 'It similarly in relation to P(ylt). Using log P(tdyd = tiYi -log(1 + eY'), we obtain T ~ 1 T -1 1 T n + 1 'It + t y-~log(1+eY')-2y+J{+ y+-210glli.+I--2-log27r (2) i=1 (3) where J{+ is the covariance matrix of the GP evaluated at Z1, . .. Zn,Z •. J{+ can be partitioned in terms of an n x n matrix J{, a n x 1 vector k and a scalar k., viz. ~ ) (4) As y. only enters into equation 2 in the quadratic prior term and has no data point associated with it, maximizing 'It + with respect to y+ can be achieved by first maximizing 'It with respect to y and then doing the further quadratic optimization to determine the posterior mean y •. To find a maximum of 'It we use the NewtonRaphson (or Fisher scoring) iteration ynew = y - ('V'V'It)-1'V'It. Differentiating equation 3 with respect to y we find (t - 1r) - J{-1 y _J{-1 - W where W = diag( 7r1 (1 - 7r1), .. , 7rn (1 - 7rn )), which gives the iterative equation1, (5) (6) (7) IThe complexity of calculating each iteration using standard matrix methods is O( n3 ). In our implementation, however, we use conjugate gradient methods to avoid explicitly inverting matrices. In addition, by using the previous iterate y as an initial guess for the conjugate gradient solution to equation 7, the iterates are computed an order of magnitude faster than using standard algorithms. Gaussian Processes for Bayesian Classification via Hybrid Monte Carlo 343 Given a converged solution y for Y, fl. can easily be found using y. = kT f{-ly = kT(t -i'). var(y.) is given by (f{+l + W+)(n1+l)(n+l)' where W+ is the W matrix with a zero appended in the (n + l)th diagonal position. Given the (Gaussian) distribution of y. we then wish to find the mean of the distribution of P(11".lt) which is found from 71-. = J u(y.)P(y.lt). We calculate this by approximating the sigmoid by a set of five cumulative normal densities (erf) that interpolate the sigmoid at chosen points. This leads to a very fast and accurate analytic approximation for the mean class prediction. The justification of Laplace's approximation in our case is somewhat different from the argument usually put forward, e.g. for asymptotic normality of the maximum likelihood estimator for a model with a finite number of parameters. This is because the dimension of the problem grows with the number of data points. However, if we consider the "infill asymptotics" , where the number of data points in a bounded region increases, then a local average of the training data at any point x will provide a tightly localized estimate for 11"( x) and hence y( x), so we would expect the distribution P(y) to become more Gaussian with increasing data. 2.2 Parameterizing the covariance function There are many reasonable choices for the covariance function . Formally, we are required to specify functions which will generate a non-negative definite covariance matrix for any set of points (Xl, . .. , Xk). From a modelling point of view we wish to specify covariances so that points with nearby inputs will give rise to similar predictions. We find that the following covariance function works well: C(x , x') = Vaexp {-~ t WI(XI - XD2} 1=1 (8) where XI is the Ith component of x and 8 = log(va, W1, .. . , Wd) plays the role of hyperparameters2. We define the hyperparameters to be the log of the variables in equation 8 since these are positive scale-parameters. This covariance function has been studied by Sacks et al (1989) and can be obtained from a network of Gaussian radial basis functions in the limit of an infinite number of hidden units (Williams, 1996). The WI parameters in equation 8 allow a different length scale on each input dimension. For irrelevant inputs, the corresponding WI will become small, and the model will ignore that input. This is closely related to the Automatic Relevance Determination (ARD) idea of MacKay and Neal (Neal, 1996). The Va variable gives the overall scale of the prior; in the classification case, this specifies if the 11" values will typically be pushed to 0 or 1, or will hover around 0.5. 2.3 Integration over the hyperparameters Given that the GP contains adjustable hyperparameters, how should they be adapted given the data? Maximum likelihood or (generalized) cross-validation methods are often used, but we will prefer a Bayesian solution. A prior distribution over the hyperparameters P( 8) is modified using the training data to obtain 'the posterior distribution P(8It) ex P(tI8)P(8). To make predictions we integrate 2We call f) the hyperparameters rather than parameters as they correspond closely to hyperparameters in neural networks. 344 D. Barber and C. K. I. Williams the predicted probabilities over the posterior; for example, the mean value 7f(:I:) for test input :1:* is given by 7f(:I:.) = J 1i-(:I:. 19)P(9It )d9, (9) where 1i-(:I:* 19) is the mean prediction for a fixed value of the hyperparameters, as given in section 2. For the regression problem P(tI9) can be calculated exactly using P(tI9) = J P(tly)P(yI9)dy , but this integral is not analytically tractable for the classification problem. Again we use Laplace's approximation and obtain3 logP(tI9) c:= w(y) + ~IJ{-l + WI + i log27r (10) where y is the converged iterate of equation 7. We denote the right-hand side of equation 10 by log Pa(tI9) (where a stands for approximate). The integration over 9-space also cannot be done analytically, and we employ a Markov Chain Monte Carlo method. We have used the Hybrid Monte Carlo (HMC) method of Duane et al (1987), with broad Gaussian hyperpriors on the parameters. HMC works by creating a fictitious dynamical system in which the hyperparameters are regarded as position variables, and augmenting these with momentum variables p. The purpose of the dynamical system is to give the hyperparameters "inertia" so that random-walk behaviour in 9-space can be avoided. The total energy, 1l, of the system is the sum of the kinetic energy, K = pT pj2 and the potential energy, £ . The potential energy is defined such that p( 91D) <X exp( -£), i.e. £ = -log P( tI9)logP(9). In practice logPa(tI9) is used instead of log P(tI9). We sample from the joint distribution for 9 and p given by P( 9, p) <X exp( -£ - K); the marginal of this distribution for 9 is the required posterior. A sample of hyperparameters from the posterior can therefore be obtained by simply ignoring the momenta. Sampling from the joint distribution is achieved by two steps: (i) finding new points in phase space with near-identical energies 1l by simulating the dynamical system using a discretised approximation to Hamiltonian dynamics, and (ii) changing the energy 1l by Gibbs sampling the momentum variables. Hamilton's first order differential equations for 1l are approximated using the leapfrog method which requires the derivatives of £ with respect to 9. Given a Gaussian prior on 9, log P(9) is straightforward to differentiate. The derivative of log Pa(9) is also straightforward, although implicit dependencies of y (and hence ir) on 9 must be taken into account by using equation 5 at the maximum point to obtain ayjae = (I + J{W)-l (aKjae)(t - 7r). The computation of the energy can be quite expensive as for each new 9, we need to perform the maximization required for Laplace's approximation, equation 10. The Newton-Raphson iteration was initialized each time with 7r = 0.5, and continued until the mean relative difference of the elements of W between consecutive iterations was less than 10-4 . The same step size [ is used for all hyperparameters, and should be as large as possible while keeping the rejection rate low. We have used a trajectory made up of L = 20 leapfrog steps, which gave a low correlation between successive states4 . This proposed state is then accepted or rejected using the Metropolis rule depending on 3This requires O( n 3 ) computation. 4In our experiments, where () is only 7 or 8 dimensional, we found the trajectory length needed is much shorter than that for neural network HMC implementations. Gaussian Processes/or Bayesian Classification via Hybrid Monte Carlo 345 the final energy 1{* (which is not necessarily equal to the initial energy 1{ because of the discretization of Hamilton's equations). The priors over hyperparameters were set to be Gaussian with a mean of -3 and a standard deviation of 3. In all our simulations a step size € = 0.1 produced a very low rejection rate « 5%). The hyperparameters corresponding to the WI'S were initialized to -2 and that for Va to O. The sampling procedure was run for 200 iterations, and the first third of the run was discarded; this "burn-in" is intended to give the hyperparameters time to come close to their equilibrium distribution. 3 RESULTS We have tested our method on two well known two-class classification problems, the Leptograpsus crabs and Pima Indian diabetes datasets and the multiclass Forensic Glass dataset5 . We first rescale the inputs so that they have mean zero and unit variance on the training set. Our Matlab implementations for the HMC simulations for both tasks each take several hours on a SGI Challenge machine (R10000), although good results can be obtained in less time. We also tried a standard Metropolis MCMC algorithm for the Crabs problem, and found similar results, although the sampling by this method is slower than that for HMC. Comparisons with other methods are taken from Ripley (1994) and Ripley (1996). Our results for the two-class problems are presented in Table 1: In the Leptograpsus crabs problem we attempt to classify the sex of crabs on the basis of five anatomical attributes. There are 100 examples available for crabs of each sex, making a total of 200 labelled examples. These are split into a training set of 40 crabs of each sex, making 80 training examples, with the other 120 examples used as the test set. The performance of the G P is equal to the best of the other methods reported in Ripley (1994), namely a 2 hidden unit neural network with direct input to output connections and a logistic output unit which was trained with maximum likelihood (Network(l) in Table 1). For the Pima Indians diabetes problem we have used the data as made available by Prof. Ripley, with his training/test split of 200 and 332 examples respectively (Ripley, 1996). The baseline error obtained by simply classifying each record as coming from a diabetic gives rise to an error of 33%. Again, the GP method is comparable with the best alternative performance, with an error of around 20%. Table 1 Pima Crabs Neural Network(l) 3 Neural Network(2) 3 Neural Network( 3 ) 75+ Linea.r Discrimina.nt 67 8 Logistic regression 66 4 MARS (degree _ 1) 75 4 PP (4 ridge functions) 75 6 2 Ga.ussia.n Mixture 64 . Gaussian Process (HMC) 68 3 Table 2 Forenslc Gla.ss Neural Network (4HU) 23.8% Linea.r Discriminant 36% MARS (degree 1) 32 .2'70 PP (5 ridge funct ions ) 35% Ga.u ssia.n Mixture 30.8% Decision Tree 32.2% Gaussian Process (MAP) 23 .3'70 Gaussian Process (MAP) 69 3 Table 1: Number of test errors for the Pima Indian diabetes and Leptograpsus crabs tasks. Network(2) used two hidden units and the predictive approach (Ripley, 1994), which uses Laplace's approximation to weight each network local minimum. Network(3) had one hidden unit and was trained with maximum likelihood; the results were worse for nets with two or more hidden units (Ripley, 1996). Table 2: Percentage classification error on the Forensic Glass task. 5 All available from http://markov .stats. ox. ac. uk/pub/PRNN. 346 D. Barber and C. K I Williams Our method is readily extendable to multiple class problems by using the softmax function . The details of this work which will be presented elsewhere, and we simply report here our initial findings on the Forensic Glass problem (Table 2). This is a 6 class problem, consisting of 214 examples containing 9 attributes. The performance is estimated using 10 fold cross validation. Computing the MAP estimate took ~ 24 hours and gave a classification error of 23.3%, comparable with the best of the other presented methods. 4 DISCUSSION We have extended the work of Williams and Rasmussen (1996) to classification problems, and have demonstrated that it performs well on the datasets we have tried so far. One of the main advantages of this approach is that the number of parameters used in specifying the covariance function is typically much smaller than the number of weights and hyperparameters that are used in a neural network, and this greatly facilitates the implementation of Monte Carlo methods. Furthermore, because the Gaussian Process is a prior on function space (albeit in the activation function space), we are able to interpret our prior more readily than for a model in which the priors are on the parametrization of the function space, as in neural network models. Some of the elegance that is present using Gaussian Processes for regression is lost due to the inability to perform the required marginalisation exactly. Nevertheless, our simulation results suggest that Laplace's approximation is accurate enough to yield good results in practice. As methods based on GPs require the inversion of n x n matrices, where n is the number of training examples, we are looking into methods such as query selection for large dataset problems. Other future research directions include the investigation of different covariance functions and improvements on the approximations employed. We hope to make our MATLAB code available from http://www.ncrg.aston.ac.uk/ Acknowledgements We thank Prof. B. Ripley for making available the Leptograpsus crabs and Pima Indian diabetes datasets. This work was partially supported by EPSRC grant GRj J75425, "Novel Developments in Learning Theory for Neural Networks" . References Duane, S., A. D. Kennedy, B. J. Pendleton, and D. Roweth (1987). Hybrid Monte Carlo. Physics Letters B 195, 216-222. Green, P. J.and Silverman, B. W . (1994). Nonparametric regression and generalized linear models. Chapman and Hall. Neal, R. M. (1996). Bayesian Learning for Neural Networks. Springer. Lecture Notes in Statistics 118. Ripley, B. (1996). Pattern Recognition and Neural Networks. Cambridge. Ripley, B. D. (1994). Flexible Non-linear Approaches to Classification. In V. Cherkassy, J. H. Friedman, and H. Wechsler (Eds.), From Statistics to Neural Networks, pp. 105-126. Springer. Sacks, J., W. J. Welch, T. J. Mitchell, and H. P. Wynn (1989). Design and analysis of computer experiments. Statistical Science 4(4), 409- 435. Williams, C. K. 1. Computing with infinite networks. This volume. Williams, C. K. I. and C. E. Rasmussen (1996). Gaussian processes for regression. In D. S. Touretzky, M. C. Mozer, and M. E. Hasselmo (Eds.), Advances in Neural Information Processing Systems 8, pp. 514- 520. MIT Press.	1996	135
1,190	Genetic Algorithms and Explicit Search Statistics Shumeet 8a1uja baluja@cs.cmu.edu Justsystem Pittsburgh Research Center & School of Computer Science, Carnegie Mellon University Abstract The genetic algorithm (GA) is a heuristic search procedure based on mechanisms abstracted from population genetics. In a previous paper [Baluja & Caruana, 1995], we showed that much simpler algorithms, such as hillcIimbing and PopulationBased Incremental Learning (PBIL), perform comparably to GAs on an optimization problem custom designed to benefit from the GA's operators. This paper extends these results in two directions. First, in a large-scale empirical comparison of problems that have been reported in GA literature, we show that on many problems, simpler algorithms can perform significantly better than GAs. Second, we describe when crossover is useful, and show how it can be incorporated into PBIL. 1 IMPLICIT VS. EXPLICIT SEARCH STATISTICS Although there has recently been controversy in the genetic algorithm (GA) community as to whether GAs should be used for static function optimization, a large amount of research has been, and continues to be, conducted in this direction [De Jong, 1992]. Since much of GA research focuses on optimization (most often in static environments), this study examines the performance of GAs in these domains. In the standard GA, candidate solutions are encoded as fixed length binary vectors. The initial group of potential solutions is chosen randomly. At each generation, the fitness of each solution is calculated; this is a measure of how well the solution optimizes the objective function. The subsequent generation is created through a process of selection, recombination, and mutation. Recombination operators merge the information contained within pairs of selected "parents" by placing random subsets of the information from both parents into their respective positions in a member of the subsequent generation. The fitness proportional selection works as selective pressure; higher fitness solution strings have a higher probability of being selected for recombination. Mutations are used to help preserve diversity in the population by introducing random changes into the solution strings. The GA uses the population to implicitly maintain statistics about the search space. The selection, crossover, and mutation operators can be viewed as mechanisms of extracting the implicit statistics from the population to choose the next set of points to sample. Details of GAs can be found in [Goldberg, 1989] [Holland, 1975]. Population-based incremental learning (PBIL) is a combination of genetic algorithms and competitive learning [Baluja, 1994]. The PBIL algorithm attempts to explicitly maintain statistics about the search space to decide where to sample next. The object of the algorithm is to create a real valued probability vector which, when sampled, reveals high quality solution vectors with high probability. For example, if a good solution can be encoded as a string of alternating O's and l's, a suitable final probability vector would be 0.01, 0.99, 0.01, 0.99, etc. The PBIL algorithm and parameters are shown in Figure 1. Initially, the values of the probability vector are initialized to 0.5. Sampling from this vector yields random solution vectors because the probability of generating a I or 0 is equal. As search progresses, the values in the probability vector gradually shift to represent high 320 •••••• Initialize Probability Vector ••••• for i :=1 to LENGTH do P[i] = 0.5; while (NOT tennination condition) ..... Generate Samples ..... for i := 1 to SAMPLES do S. Baluja sample_vectors[i] := generate_sample_vector_according_to-probabilities (P); evaluations[I] := evaluate(sample_vectors[i)); besLvector:= find_vectocwith_besLevaluation (sample_vectors, evaluations); worsLvector := find_vectocwith_worsLevaluation (sample_vectors, evaluations); ••••• Update Probability Vector Towards Best Solution ••••• for i :=1 to LENGTH do P[i] := P[I] • (1.0 • LA) + besLvector[i] • (LA); PBIL: USER DEFINED CONSTANTS (Values Used In this Study): SAMPLES: the number of vectors generated before update of the probability vector (100). LA: the leaming rate, how fast to exploit the search perfonned (0.1). NEGATIVE_LA: negative leaming rate, how much to leam from negative examples (PBIL 1=0.0, PBIL2= 0.075). LENGTH: the number of bits In a generated vBCtor (problem specific). Figure 1: PBILIIPBIL2 algorithm for a binary alphabet. PBIL2 includes shaded region. Mutations not shown. evaluation solution vectors through the following process. A number of solution vectors are generated based upon the probabilities specified in the probability vector. The probability vector is pushed towards the generated solution vector with the highest evaluation. After the probability vector is updated, a new set of solution vectors is produced by sampling from the updated probability vector, and the cycle is continued. As the search progresses, entries in the probability vector move away from their initial settings of 0.5 towards either 0.0 or 1.0. One key feature of the early generations of genetic optimization is the parallelism in the search; many diverse points are represented in the population of points during the early generations. When the population is diverse, crossover can be an effective means of search, since it provides a method to explore novel solutions by combining different members of the population. Because PBIL uses a single probability vector, it may seem to have less expressive power than a GA using a full population, since a GA can represent a large number of points simultaneously. A traditional single population GA, however, would not be able to maintain a large number of points. Because of sampling errors, the population will converge around a single point. This phenomenon is summarized below: " ... the theorem [Fundamental Theorem of Genetic Algorithms [Goldberg, 1989]], assumes an infinitely large population size. In a finite size population, even when there is no selective advantage for either of two competing alternatives ... the population will converge to one alternative or the other in finite time (De Jong, 1975; [Goldberg & Segrest, 1987]). This problem of finite populations is so important that geneticists have given it a special name, genetic drift. Stochastic errors tend to accumulate, ultimately causing the population to converge to one alternative or another" [Goldberg & Richardson, 1987]. Diversity in the population is crucial for GAs. By maintaining a population of solutions, the GA is able-in theory at least-to maintain samples in many different regions. Crossover is used to merge these different solutions. A necessary (although not sufficient) condition for crossover to work well is diversity in the popUlation. When diversity is lost, crossover begins to behave like a mutation operator that is sensitive to the convergence of the value of each bit [Eshelman, 1991]. If all individuals in the population converge at Genetic Algorithms and Explicit Search Statistics 321 some bit position, crossover leaves those bits unaltered. At bit positions where individuals have not converged, crossover will effectively mutate values in those positions. Therefore, crossover creates new individuals that differ from the individuals it combines only at the bit positions where the mated individuals disagree. This is analogous to PBIL which creates new trials that differ mainly in positions where prior good performers have disagreed. As an example of how the PBIL algorithm works, we can examine the values in the probability vector through multiple generations. Consider the following maximization problem: 1.0/1(366503875925.0 - X)I, 0 ~ X < 240. Note that 366503875925 is represented in binary as a string of 20 pairs of alternating '01'. The evolution of the probability vector is shown in Figure 2. Note that the most significant bits are pinned to either 0 or 1 very quickly, while the least significant bits are pinned last. This is because during the early portions of the search, the most significant bits yield more information about high evaluation regions of the search space than the least significant bits. o 5 '0 ~"'5 "0 Sl 20 6: g 25 30 35 Generation Figure 2: Evolution of the probability vector over successive generations. White represents a high probability of generating a 1. black represents a high probability of generating a O. Intennediate grey represent probabilities close to 0.5 - equal chances of generating a 0 or 1. Bit 0 is the most significant. bit 40 the least. 2 AN EMPIRICAL COMPARISON This section provides a summary of the results obtained from a large scale empirical comparison of seven iterative and evolution-based optimization heuristics. Thirty-four static optimization problems, spanning six sets of problem classes which are commonly explored in the genetic algorithm literature, are examined. The search spaces in these problems range from 2128 to 22040. The results indicate that, on many problems, using standard GAs for optimizing static functions does not yield a benefit, in terms of the final answer obtained, over simple hillclimbing or PBIL. Recently, there have been other studies which have examined the perfonnance of GAs in comparison to hillclimbing on a few problems; they have shown similar results [Davis, 1991][Juels & Wattenberg, 1996]. Three variants of Multiple-Restart Stochastic Hillclimbing (MRS H) are explored in this paper. The first version, MRSH-l, maintains a list of the position of the bit flips which were attempted without improvement. These bit flips are not attempted again until a better solution is found. When a better solution is found, the list is emptied. If the list becomes as large as the solution encoding, MRSH-l is restarted at a random solution with an empty list. MRSH-2 and MRSH-3 allow moves to regions of higher and equal evaluation. In MRSH-2, the number of evaluations before restart depends upon the length of the encoded solution. MRSH-2 allows 1O(length of solution) evaluations without improvement before search is restarted. When a solution with a higher evaluation is found, the count is reset. In MRSH-3, after the total number of iterations is specified, restart is forced 5 times during search, at equally spaced intervals. Two variants of the standard GA are tested in this study. The first, tenned SGA, has the following parameters: Two-Point crossover, with a crossover rate of 100% (% of times crossover occurs, otherwise the individuals are copied without crossover), mutation probability of 0.001 per bit, population size of 100, and elitist selection (the best solution in 322 s. Haluja generation N replaces the worst solution in generation N+ 1). The second GA used, termed GA-Scale, uses the same parameters except: uniform crossover with a crossover rate of 80% and the fitness of the worst member in a generation is subtracted from the fitnesses of each member of the generation before the probabilities of selection are determined. Two variants of PBIL are tested. Both move the probability vector towards the best example in each generated population. PBIL2 also moves the probability vector away from the worst example in each generation. Both variants are shown in Figure 1. A small mutation, analogous to the mutation used in genetic algorithms, is also used in both PBILs. The mutation is directly applied to the probability vector. The results obtained in this study should not be considered to be state-of-the-art. The problem encodings were chosen to be easily reproducible and to allow easy comparison with other studies. Alternate encodings may yield superior results. In addition, no problem-specific information was used for any of the algorithms. Problem-specific information, when available, could help all of the algorithms examined. All of the variables in the problems were encoded in binary, either with standard Graycode or base-2 representation. The variables were represented in non-overlapping, contiguous regions within the solution encoding. The results reported are the best evaluations found through the search of each algorithm, averaged over at least 20 independent runs per algorithm per problem; the results for GA-SCALE and PBIL2 algorithms are the average of at least 50 runs. All algorithms were given 200,000 evaluations per run. In each run, the GA and PBIL algorithms were given 2000 generations, with 100 function evaluations per generation. In each run, the MRSH algorithms were restarted in random locations as many times as needed until 200,000 evaluations were performed. The best answer found in the 200,000 evaluations was returned as the answer found in the run. Brief notes about the encodings are given below. Since the numerical results are not useful without the exact problems, relative results are provided in Table I. For most of the problems, exact results and encodings are in [Baluja, 1995). To measure the significance of the difference between the results obtained by PBIL2 and GA-SCALE, the Mann-Whitney test is used. This is a non-parametric equivalent to the standard two-sample pooled t-tests. • TSP: 128,200 & 255 city problems were tried. The "sort" encoding [Syswerda, 1989] was used. The last problem was tried with the encoding in binary and Gray-Code. • Jobshop: Two standard JS problems were tried with two encodings. The first encoding is described in [Fang et. ai, 1993]. The second encoding is described in [Baluja, 1995]. An additional, randomly generated, problem was also tried with the second encoding. • Knapsack: Problem 1&2: a unique element is represented by each bit. Problem 3&4: there are 8 and 32 copies of each element respectively. The encoding specified the number of copies of each element to include. Each element is assigned a "value" and "weight". Object: maximize value while staying under pre-specified weight. • Bin-Packing/EquaI Piles: The solution is encoded in a bit vector of length M log2N (N bins, M elem.). Each element is assigned a substring of length log2N, which specifies a bin. Object: pack the given bins as tightly as possible. Because of the large variation in results which is found by varying the number of bins and elements, the results from 8 problems are reported. • Neural-Network Weight Optimization: Problem 1&2: identify the parity of7 inputs. Problem 3&4: determine whether a point falls within the middle of 3 concentric squares. For problems 3&4, 5 extra inputs, which contained noise, were used. The networks had 8 inputs (including bias), 5 hidden units, and 1 output. The network was fully connected between sequential layers. • Numerical Function Optimization (FI-FJ): Problems 1&2: the variables in the first portions of the solution string have a large influence on the quality of the rest of the solution. In the third problem, each variable can be set independently. See [Baluja, 1995] for details. • Graph Coloring: Select 1 of 4 colors for nodes of a partially connected graph such that connected nodes are not the same color. The graphs used were not necessarily planar. Genetic Algorithms and Explicit Search Statistics 323 Table I: Summary of Empirical Results - Relative Ranks (l=best, 7=worst). 3 EXPLICITL Y PRESERVING DIVERSITY Although the results in the previous section showed that PBIL often outperformed GAs and hillclimbing, PBIL may not surpass GAs at all population sizes. As the population size increases, the observed behavior of a GA more closely approximates the ideal behavior predicted by theory [Holland, 1975]. The population may contain sufficient samples from distinct regions for crossover to effectively combine "building blocks" from multiple solutions. However, the desire to minimize the total number of function evaluations often prohibits the use of large enough populations to make crossover behave ideally. One method of avoiding the cost of using a very large population is to use a parallel GA (pGA). Many studies have found pGAs to be very effective for preserving diversity for function optimization [Cohoon et al., 1988][Whitley et ai., 1990]. In the pGA, a collection of independent GAs, each maintaining separate populations, communicate with each other 324 S. Baluja via infrequent inter-population (as opposed to intra-population) matings. pGAs suffer less from premature convergence than single population GAs. Although the individual populations typically converge, different populations converge to different solutions, thus preserving diversity across the populations. Inter-population mating permits crossover to combine solutions found in different regions of the search space. We would expect that employing mUltiple PBIL evolutions, parallel PBIL (pPBIL), has the potential to yield performance improvements similar to those achieved in pGAs. Multiple PBIL evolutions are simulated by using multiple probability vectors to generate solutions. To keep the evolutions independent, each probability vector is only updated with solutions which are generated by sampling it. The benefit of parallel populations (beyond just multiple runs) is in using crossover to combine dissimilar solutions. There are many ways of introducing crossover into PBIL. The method which is used here is to sample two probability vectors for the creation of each solution vector, see Figure 3. The figure shows the algorithm with uniform crossover; nonetheless, many other crossover operators can be used. The randomized nature of crossover often yields unproductive results. If crossover is to be used, it is important to simulate the crossover operation many times. Therefore, crossover is used to create each member of the population (this is in contrast to crossing over the probability vectors once, and generating the entire population from the newly created probability vector). More details on integrating crossover and PBIL, and its use in combinatorial problems in robotic surgery can be found in [Baluja & Simon, 1996]. Results with using pPBIL in comparison to PBIL, GA, and pGA are shown in Table II. For many of the problems explored here, parallel versions of GAs and PBIL work better than the sequential versions, and the parallel PBIL models work better than the parallel GA models. In each of these experiments, the parameters were hand-tuned for each algorithms. In every case, the GA was given at least twice as many function evaluations as PBIL. The crossover operator was chosen by trying several operators on the GA, and selecting the best one. The same crossover operator was then used for PBIL. For the pGA and pPBIL experiments, 10 subpopulations were always used . ..... Generate Samples With Two Probability Vectors ..... for i :=1 to SAMPLES do vector I := generate_sample_vector_with_probabilities (PI); vector2 := generate_sample_vector_with_probabilities (P2); for j := I to LENGTH_do if (random (2) = 0) sample_ vector{i]lil := vector I [j] else sample_ vector{i][j] := vector2[j] evaluations[i] := Evaluate_Solution (sample[i]); besevector := best_evaluation (sample_vectors. evaluations); ..... Update Both Probability Vectors Towards Best Solution ..... for i :=1 to LENGTH do PI[i] := Pl[i] • (1.0 - LR) + best_vector[i] • (LR); P2[i] := P2[i] * (1 .0 - LR) + besevector[i] • (LR); Figure 3: Generating samples based on two probability vectors. Shown with uniform crossover [Syswerda, 1989] (50% chance of using probability vector 1 or vector 2 for each bit position). Every 100 generations, each population makes a local copy of another population's probability vector (to replace vector2). In these experiments, there are a total of 10 subpopulations. Table IT: Sequential & Parallel, GA & PBIL, Avg. 25 runs - 200 city (minimize tour length) "'lIIont:lI' Optim. Highly Correlated Parameters - Base-2 Code (max) Optim. Highly Correlated Parameters - Gray Code (max) Optim. Independent Parameters - Base-2 Code (max) ... lIlmo'",,,,,. Optim. Independent Parameters - Gray Code (max) (Problem with many maxima, see [Baluja, 1994]) (max) Genetic Algorithms and Explicit Search Statistics 325 4 SUMMARY & CONCLUSIONS PBIL was examined on a very large set of problems drawn from the GA literature. The effectiveness of PBIL for finding good solutions for static optimization functions was compared with a variety of GA and hillclimbing techniques. Second, Parallel-PBIL was introduced. pPBIL is designed to explicitly preserve diversity by using multiple parallel evolutions. Methods for reintroducing crossover into pPBIL were given. With regard to the empirical results, it should be noted that it is incorrect to say that one procedure will always perform better than another. The results do not indicate that PBIL will always outperform a GA. For example, we have presented problems on which GAs work better. Further, on problems such as binpacking, the relative results can change drastically depending upon the number of bins and elements. The conclusion which should be reached from these results is that algorithms, like PBIL and MRSH, which are much simpler than GAs, can outperform standard GAs on many problems of interest. The PBIL algorithm presented here is very simple and should serve as a prototype for future study. Three directions for future study are presented here. First, the most obvious extension to PBIL is to track more detailed statistics, such as pair-wise covariances of bit positions in high-evaluation vectors. ~eliminary work in this area has been conducted, and the results are very promising. Second, another extension is to quickly determine which probability vectors, in the pPBIL model, are unlikely to yield promising answers; methods such as Hoeffding Races may be adapted here [Maron & Moore, 1994]. Third, the manner in which the updates to the probability vector occur is similar to the weight update rules used in Learning Vector Quantization (LVQ). Many of the heuristics used in L VQ can be incorporated into the PBIL algorithm. Perhaps the most important contribution of the PBIL algorithm is a novel way of examining GAs. In many previous studies of the GA, the GA was examined at a micro-level, analyzing the preservation of building blocks and frequency of sampling hyperplanes. In this study, the statistics at the population level were examined. In the standard GA, the population serves to implicitly maintain statistics about the search space. The selection and crossover mechanisms are ways of extracting these statistics from the population. PBIL's population does not maintain the information that is carried from one generation to the next. The statistics of the search are explicitly kept in the probability vector. References Baluja, S. (1995) "An Empirical Comparison of Seven Iterative and Evolutionary Function Optimization Heuristics," CMU-CS95-193. Available via. http://www.cs.cmu.edul-baluja. Baluja, S. (1994) "Population-Based Incremental Learning". Carnegie MeUon University. Technical Repon. CMU-CS-94-163. Baluja, S. & Caruana, R. (1995) "Removing the Genetics from the Standard Genetic Algorithm", Imer.Con! Mach. uarning-12. Baluja, S. & Simon, D. (1996) "Evolution-Based Methods for Selecting Point Data for Object Localization: Applications to Computer Assisted Surgery". CMU·CS·96 -183. Cohoon, J., Hedge, S., Martin, W., Richards, D., (1988) "Distributed Genetic Algorithms for the Floor Plan Design Problem," School of Engineering and Applied Science, Computer Science Dept., University of Virginia, TR-88-12. Davis, L.1. (1991) "Bit-Climbing, Representational Bias and Test Suite Design".lntemational Con! on Genetic Algorilhms 4. De Jong, K. (1975) An Analysis of the Behavior of a Class of Genetic Adaptive Systems. Ph.D. Dissenation. De Jong, K. (1993) "Genetic Algorithms are NOT Function Optimizers". In Whitley (ed.) Foundations of GAs-2. 5-17. Eshelman, L.J. (1991) "The CHC Adaptive Search Algorithm," in Rawlings (ed.) Foundations of GAs-I. 265-283. Fang, H.L, Ross, P., Come, D. (1993) "A Promising Genetic Algorithm Approach to Job-Shop Scheduling, Rescheduling, and Open- Shop Scheduling Problems". In Forrest, S. Imernational Conference on Genetic Algorithms 5. GOldberg, D.E. (1989) Genetic Algorithms in Search, Optimization, and Machine uarning. Addison-Wesley. Goldberg & Richardson (1987) "Genetic Algorithms with Sharing for Multimodal Function Optimization" - Proceedings of the Second International Conference on Genetic Algorithms. HoUand, J. H. (1975) Adaptation in Natural and Ani/icial Systems. Ann Arbor: The University of Michigan Press. Juels, A. & Wattenberg, M. (1994) "Stochastic Hillclimbing as a Baseline Method for Evaluating Genetic Algorithms" NIPS 8. Maron, O. & Moore, A.(1994) "Hoeffding Races:Accelerating Model Selection for Classification and Function Approx." NIPS 6 Mitchell, M., Holland, 1. & Forrest, S. (1994) "When will a Genetic Algorithm Outperform Hill Climbing" NIPS 6. Syswerda, G. (1989) "Uniform Crossover in Genetic Algorithms," International Conference on Genetic Algorithms 3.2-9. Whitley, D., & Starkweather, T. "Genitor II: A Distributed Genetic Algorithm". }ETAl2: 189-214.	1996	136
1,191	.. Learning Exact Patterns of Quasi-synchronization among Spiking Neurons from Data on Multi-unit Recordings Laura Martignon Max Planck Institute for Psychological Research Adaptive Behavior and Cognition 80802 Munich, Germany laura@mpipf-muenchen.mpg.de Gustavo Deco Siemens AG Central Research Otto Hahn Ring 6 81730 Munich gustavo.deco@zfe.siemens.de Abstract Kathryn Laskey Dept. of Systems Engineering and the Krasnow Institute George Mason University Fairfax, Va. 22030 klaskey@gmu.edu Eilon Vaadia Dept. of Physiology Hadassah Medical School Hebrew University of Jerusalem Jerusalem 91010, Israel eilon@hbf.huji.ac.il This paper develops arguments for a family of temporal log-linear models to represent spatio-temporal correlations among the spiking events in a group of neurons. The models can represent not just pairwise correlations but also correlations of higher order. Methods are discussed for inferring the existence or absence of correlations and estimating their strength. A frequentist and a Bayesian approach to correlation detection are compared. The frequentist method is based on G 2 statistic with estimates obtained via the Max-Ent principle. In the Bayesian approach a Markov Chain Monte Carlo Model Composition (MC3) algorithm is applied to search over connectivity structures and Laplace's method is used to approximate their posterior probability. Performance of the methods was tested on synthetic data. The methods were applied to experimental data obtained by the fourth author by means of measurements carried out on behaving Rhesus monkeys at the Hadassah Medical School of the Hebrew University. As conjectured, neural connectivity structures need not be neither hierarchical nor decomposable. Learning Quasi-synchronization Patterns among Spiking Neurons 77 1 INTRODUCTION Hebb conjectured that information processing in the brain is achieved through the collective action of groups of neurons, which he called cell assemblies (Hebb, 1949). His followers were left with a twofold challenge: • to define cell assemblies in an unambiguous way. • to conceive and carry out the experiments that demonstrate their existence. Cell assemblies have been defined in various sometimes conflicting ways, both in terms of anatomy and of shared function. One persistent approach characterizes the cell assembly by near-simultaneity or some other specific timing relation in the firing of the involved neurons. If two neurons converge on a third one, their synaptic influence is much larger for near-coincident firing, due to the spatio-temporal summation in the dendrite (Abeles, 1991; Abeles et al. 1993). Thus syn-jiring is directly available to the brain as a potential code. The second challenge has led physiologists to develop methods to observe the simultaneous activity of individual neurons to seek evidence for spatio-temporal patterns. It is now possible to obtain multi-unit recordings of up to 100 neurons in awake behaving animals. In the data we analyze, the spiking events (in the 1 msec range) are encoded as sequences of O's and 1 's, and the activity of the whole group is described as a sequence of binary configurations. This paper presents a statistical model in which the parameters represent spatio-temporal firing patterns. We discuss methods for estimating these pararameters and drawing inferences about which interactions are present. 2 PARAMETERS FOR SPATIO-TEMPORAL FIRING PATTERNS The term spatial correlation has been used to denote synchronous firing of a group of neurons, while the term temporal correlation has been used to indicate chains of firing events at specific temporal intervals. Terms like "couple" or "triplet" have been used to denote spatio-temporal patterns of two or three neurons (Abeles et al., 1993; GrOn, 1996) frring simultaneously or in sequence. Establishing the presence of such patterns is not straightforward. For example, three neurons may fire together more often than expected by chancel without exhibiting an authentic third order interaction. This phenomenon may be due, for instance, to synchronous frring of two couples out of the three neurons. Authentic triplets, and, in general, authentic n-th order correlations, must therefore be distinguished from correlations that can be explained in terms of lower order interactions. In what follows, we present a parameterized model that represents a spatio-temporal correlation by a parameter that depends on the involved neurons and on a set of time intervals, where synchronization is characterized by all time intervals being zero. Assume that the sequence of configurations !:t = ( x (W'· .. , x ( N.lJ ) of N neurons forms a Markov chain of order r. Let 8 be the time step, and denote the conditional distribution for :!t given previous configurations by p(:!t I :!(t-oJ' :!(t-2oJ , ,,·':!(t-roJ ). We assume that all transition probabilities are strictly positive and expand the logarithm of the conditional distribution as: I that is to say, more often than predicted by the null hypothesis of independence. 78 L. Martignon, K. Laskey, G. Deco and E. Vaadia p(!t I !(t-O) '!(t-2o) ,···'!(t-ro}) = czp{ (}o + L (J A X A) (1) Ae=: where each A is a subset of pairs of subscripts of the form (i, t - sO) that includes at least one pair of the form (i, t). Here X A = II x(i t-m 1» denotes the event that all l$J$k i ' J neurons in A are active. The set ::: c 2 A of all subsets for which () A is non-zero is called the interaction structure for the distribution p. The effect () A is called the interaction strength for the interaction on subset A. Clearly, () A = 0 is equivalent to A e::: and is taken to indicate absence of an order-I A I interaction among neurons in A . We denote the structure-specific vector of non-zero interaction strengths by (J s. Consider a set A of N binary neurons and denote by p the probability distribution on the binary configurations of A. DEFINITION 1: We say that neurons (i1,i2 , .. ..ik ) exhibit a spatio-temporal pattern if there is a set of time intervals mI8,m28, ... ,mk8 with at least one mi = 0, such that () A "# 0 in Equation (1), where A = {( i1,t- m18J,..J ik ,t- mk8)). DEFINITION 2: A subset (i1.i2 , ... , ik ) of neurons exhibits a synchronization or spatial correlation if (J A * a for A = {( iI' 0 J, ... , ( ik, 0)) . In the case of absence of any temporal dependencies the configurations are independent and we drop the time index: p(!) = czp{(}o + I.(}AXA) (2) where A is any nonempty subset of A and X A = n Xi . ieA Of course (2) is unrealistic. Temporal correlation of some kind is always present, one such example being the refractory period after firing. Nevertheless, (2) may be adequate in cases of weak temporal correlation. Although the models (1) and (2) are statistical not physiological, it is an established conjecture that synaptic connection between two neurons will manifest as a non-zero (J A for the corresponding set A in the temporal model (1). Another example leading to non-zero (J A will be simultaneous activation cf the neurons in A due to a common input, as illustrated in Figure 1 below. Such a (J A will appear in model (1) with time intervals equal to O. An attractive feature of our models is that it is capable of distinguishing between cases a. and b. of Figure 1. This can be seen by extending the model (2) to include the external neurons (H in case a., H,K in case b.) and then marginalizing. An information-theoretic argument supports the choice of (J A * a as a natural indicator of an order-I A I interaction among the neurons in A. Assume that we are in the case of no temporal correlation. The absence cf interaction of order I A I Learning Quasi-synchronization Patterns among Spiking Neurons 79 H a. Figure 1 b. among neurons in A should be taken to mean that the distribution is determined by the marginal distributions on proper subsets of A. A well established criterion for selecting a distribution among those matching the lower order marginals fIxed by proper subsets <f A, is Max-Ent. According to the Max-Ent principle the distribution that maximizes entropy is the one which is maximally non-committal with regard to missing information. The probability distribution p * that maximizes entropy among distributions with the same marginals as the distribution p on proper subsets of A has a log-linear expansion in which only OB ' B c A , B '* A can possibly be non-zero.2 3 THE FREQUENTIST APPROACH We treat here the case of no temporal dependencies. The general case is treated in Martignon-Deco,1997; Deco-Martignon,1997. We also assume that our data are stationary. We test the presence of synchronization of neurons in A by the following procedure: we condition on silence of neurons in the complement of A in A and call the reSUlting frequency distribution p . We construct the Max-Ent model determined by the marginals of p on proper subsets of A. The well-known method for constructing this type of Max-Ent models is the I.P.F.P. Algorithm (Bishop et al.,1975). We propose here another simpler and quicker procedure: If B is a subset of A, denote by X B the confIguration that has a component 1 for every index in B and a elsewhere. Defme p ( X B) = p( X B) + ( _l)IBI ~, where ~ is to be determined by solving for o ~ == 0, where 0 A is the coefficient corresponding to A in the log-expansion of p . As can be shown (Martignon et ai, 1995), 0 A can be written as 2 This was observed by J. Good in 1963 (Bishop et al. 1~75). It is interesting to note that P* minimizes the Kullback-Leibler distance from P in the manifold of distributions with a loglinear expansion in which only 0 B' Be A, B '* A can possibly be non-zero. 80 L. Martignon, K. Laskey, G. Deco and E. Vaadia e A = L ( -1) IA-BI In P ( X B ). The distribution p * maximizes entropy among those BcA with the same marginals of p on proper subsets of A. 3 We use p * as estimate of p for tests by means of C2statistic (Bishop et aI., 1975). 4 THE BAYESIAN APPROACH We treat here the case of no temporal dependencies. The general case is treated in LaskeyMartignon, 1997. Information about p ( X) prior to observing any data is represented by a joint probability distribution called the prior distribution over.3 and the e,s. Observations are used to update this probability distribution to obtain a posterior distribution over structures and parameters. The posterior probability of a cluster A can be interpreted as the probability that the r nodes in cluster A exhibit a degree-r interaction. The posterior distribution for () A represents structure-specific information about the magnitude of the interaction. The mean or mode of the posterior distribution can be used as a point estimate of the interaction strength; the standard deviation of the posterior distribution reflects remaining uncertainty about the interaction strength. We exhibit a family of log-linear models capable of capturing interactions of all orders. An algorithm is presented for learning both structure and parameters in a unified Bayesian framework. Each model structure specifies a set of clusters of nodes, and structure-specific parameters represent the directions and strengths of interactions among them. The Bayesian learning algorithm gives high posterior probability to models that are consistent with the data. Results include a probability, given the observations, that a set of neurons fires simultaneously, and a posterior probability distribution for the strength of the interaction, conditional on its occurrence. The prior distribution we used has two components. The first component assigns a prior probability to each structure. In our model, interactions are independent of each other and each interaction has a probability of .1. This reflects the prior expectation that not many interactions are expected to be present. The second component of the prior distribution is the conditional distribution of interaction strengths given the structure. If an interaction is not in the structure, the corresponding strength parameter e A is taken to be identicalIy zero given structure':::'. All interactions belonging to .3 are taken to be independent and normally distributed with mean zero and standard deviation 2. This reflects the prior expectation that interaction strength magnitudes are rarely larger than 4 in absolute value. Computing the posterior probability of a structure .3 requires integrating out of the joint mass-density function of the structure S, the interaction strength e A' and the data X. The solution to this integral cannot be obtained in closed form. We use Laplace's method (Kass-Raftery, 1995; Tierney-Kadane,1986) to estimate the posterior probability of structures. The posterior distribution of e A given frequency data also 3 This is due to the fact that there is a unique distribution with the same marginals of pon proper subsets of A such that the coefficient corresponding to A in its log-expansion is zero. Learning Quasi-synchronization Patterns among Spiking Neurons 81 cannot be obtained in closed form. We use the mode of the posterior distribution as a point estimate of () A . The standard deviation of () A' which indicates how precisely () A can be estimated from the given data, is estimated using a normal approximation to the posterior distribution (Laskey-Martignon, 1997). The covariance matrix of the () A is estimated as the inverse Fisher information matrix evaluated at the mode of the posterior distribution. The posterior probability of an interaction () A is the sum over the posterior probabilities of all structures containing A. We used a Markov chain Monte Carlo Model Composition algorithm (MC3) to search over structures. This stochastic algorithm converges to a stationary distribution in which structure .3 is visited with probability equal to its posterior probability. We ran the Me3 algorithm for 15,000 runs and estimated the posterior probability of a structure as its frequency of occurrence over the 15,000 runs. We estimated interaction strength parameters and standard deviations using only the 100 highest-probability structures. Although the number of possible structures is astronomical, typically most of the posterior probability is contained in relatively few structures. We found this to be the case, which justifies using only the most probable structures to estimate interaction strength parameters. 5 RESULTS We applied our models to data from an experiment in which spiking events among groups of neurons were analyzed through multi-unit recordings of 6-16 units in the frontal cortex of Rhesus monkeys. The monkeys were trained to localize a source of light and, after a delay, to touch the target from which the light blink was presented. At the beginning of each trial the monkeys touched a "ready-key", then the central ready light was turned on. Later, a visual cue was given in the form of a 200-ms light blink coming from either the left or the right. Then, after a delay of 1 to 32 seconds, the color of the ready light changed from red to orange and the monkeys had to release the ready key and touch the target from which the cue was given. The spiking events (in the 1 millisecond range) of each neuron were encoded as a sequence of zeros and ones, and the activity of the group was described as a sequence of configurations of these binary states. The fourth author provided data corresponding to piecewise stationary segments of the trials, which presented weak temporal correlation, corresponding to intervals of 2000 milliseconds around the ready-signal. He adjoined these 94 segments and formed a data-set of 188,000 msec. The data were then binned in time windows of 40 milliseconds. The criterion we used to fix the binwidth was robustness with regards to variations of the offsets. We selected a subset of eight of the neurons for which data were recorded. We analyzed recordings prior to the ready-signal separately from data recorded after the ready-signal. Each of these data sets is assumed to consist of independent trials from a model of the form (2). Cluster Postenor prob. Posterior prob. MAP esttmate of Standard SIgnifIcance A of A of A (}A deviation of (frequency) (best (). IOOmodels) 6,8 .Y .89 0.47 u.lI 4.0853 4,5,6,7 .30 0.32 2.3U U.64 No 2,3,6 .4U Q]8 2.30 0.64 2.35 1,3,4 close to pnor close to pnor 4.7 Table1: results for pre-ready signal data. Effects with posterior prob. > 0.1 82 L. Martignon, K. Laskey, G. Deco and E. Vaadia Cluster Posterior prob. Posterior prob. MAP estimate ot Standard Slgnincance A of A of A fJ, deviation of (frequency) (best 100 fJ, models) 5,6 .79 0.96 1.00 0.27 1.82 4,7 .246 0.18 0.93 0.34 2.68 1,4,5,6 0.18 0.13 1.06 0.36 No 1,3,4,6,7 0.24 0.17 2.69 0.13 No Table2:results for post-ready signal data. Effects with posterior prob >0.1 Another set of data from 5 simulated neurons was provided by the fourth author for a double-check of the methods. Only second order correlations had been simulated: a synapse lasting 2 msec, an inhibitory common input, and two excitatory common inputs. The Bayesian method was very accurate, detecting exactly the simulated interactions. The frequentist method made one mistake. Other data sets with temporal correlations have also been analyzed. By means of the frequentist approach on shifted data, temporal triplets have been detected and even fourth order correlations. Temporal correlograms are computed for shifts of up to 50 msec (Martignon-Deco, 1997). References Hebb, D. (1949) The Organization of Behavior. New York: Wiley, 1949. Abeles, M.(l991)Corticonics: Neural Circuits of the Cerebral Cortex. Cambridge: Cambridge University Press, 1991. Abeles, M., H. Bergman, E. Margalit, and E. Vaadia. (1993) "SpatiotemporaI Firing Patterns in the Frontal Cortex of Behaving Monkeys." Journal of Neurophysiology 70, 4:, 1629-1638. Griin S. (1996) Unitary Joint-Events in Multiple-Neuron Spiking Activity-Detection, Signijicance and Interpretation. Verlag Harry Deutsch, Frankfurt. Martignon L. and Deco G. (1997) "Neurostatistics of Spatio-Temporal Patterns of Neural Activation: the frequentist approach" Technical Report, MPI-ABC no.3. Deco G. and Martignon L. (1997) "Higher-order Phenomena among Spiking Events of Groups of Neurons" Preprint. Bishop, Y., S. Fienberg, and P. Holland (1975) Discrete Multivariate Analysis. Cambridge, MA: MIT Press. Martignon L,.v.Hasseln H. Griin S, Aertsen A, Palm G.(1995) "Detecting Higher Order Interactions among the Spiking Events of a Group of Neurons" Biol.Cyb. 73, 69-81 . Kass, . and Raftery A. (1995) "Bayes factors"Journal of the American Statistical Association 90, no. 430:, 773-795. Tierney, L., and J. B. Kadane (J 986) "Accurate Approximations for Posterior Moments and Marginal Densities." Journal of the American Statistical Association 81, 82-86 Laskey K., and Martignon L.( 1997) "Neurostatistics of Spatio-temporal Patterns of Neural Activation: the Bayesian Approach", in preparation Laskey K., and Martignon, L.(1996) "Bayesian Learning of Log-linear Models for Neural Connectivity" Proceedings of the XII Conference on Uncertainty in Artijiciallntelligence, Horvitz E. ed., Morgan-Kaufmann, San Mateo.	1996	137
1,192	Efficient Nonlinear Control with Actor-Tutor Architecture Kenji Doya* A.TR Human Information Processing Research Laboratories 2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02, Japan. Abstract A new reinforcement learning architecture for nonlinear control is proposed. A direct feedback controller, or the actor, is trained by a value-gradient based controller, or the tutor. This architecture enables both efficient use of the value function and simple computation for real-time implementation. Good performance was verified in multi-dimensional nonlinear control tasks using Gaussian softmax networks. 1 INTRODUCTION In the study of temporal difference (TD) learning in continuous time and space (Doya, 1996b), an optimal nonlinear feedback control law was derived using the gradient of the value function and the local linear model of the system dynamics. It was demonstrated in the simulation of a pendulum swing-up task that the value-gradient based control scheme requires much less learning trials than the conventional "actor-critic" control scheme (Barto et al., 1983). In the actor-critic scheme, the actor, a direct feedback controller, improves its control policy stochastically using the TD error as the effective reinforcement (Figure 1a). Despite its relatively slow learning, the actor-critic architecture has the virtue of simple computation in generating control command. In order to train a direct controller while making efficient use of the value function, we propose a new reinforcement learning scheme which we call the "actor-tutor" architecture (Figure 1b). ·Current address: Kawato Dynamic Brain Project, JSTC. 2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02, Japan. E-mail: doya@erato.atr.co.jp Efficient Nonlinear Control with Actor-Tutor Architecture 1013 In the actor-tutor scheme, the optimal control command based on the current estimate of the value function is used as the target output of the actor. With the use of supervised learning algorithms (e.g., LMSE), learning of the actor is expected to be faster than in the actor-critic scheme, which uses stochastic search algorithms (e.g., A RP )' The simulation result below confirms this prediction. This hybrid control architecture provides a model of functional integration of motor-related brain areas, especially the basal ganglia and the cerebellum (Doya, 1996a). 2 CONTINUOUS TD LEARNING First, we summarize the theory of TD learning in continuous time and space (Doya, 1996b), which is basic to the derivation of the proposed control scheme. 2.1 CONTINUOUS TD ERROR Let us consider a continuous-time, continuous-state dynamical system d~;t) = f(x(t), u(t» (I) where x E X C R n is the state and u E U C R m is the control input (or the action). The reinforcement is given as the function of the state and the control r(t) = r(x(t), u(t». (2) For a given control law (or a policy) u(t) = p(x(t», (3) we define the "value function" of the state x(t) as 1 00 1 .-j VJ'(x(t» = -e-r r(x(s), u(s»ds, t T (4) where x(s) and u(s) (t :5 s < 00) follow the system dynamics (I) and the control law (3). Our goal is to find an optimal control law p* that maximizes VJ'(x) for any state x EX. Note that T is the time constant of imminence-weighting, which is related to the discount factor 'Y of the discrete-time TD as 'Y = 1 _ ~t. By differentiating (4) by t, we have a local consistency condition for the value function (5) Let P(x(t» be the prediction of the value function VJ'(x(t» from x(t) by a neural network, or some function approximator that has enough capability of generalization. The prediction should be adjusted to minimize the inconsistency r(t) = r(t) - P(x(t» + T dP~~(t» , (6) which is a continuous version of the TD error. Because the boundary condition for the value function is given on the attractor set of the state space, correction of P(x(t» should be made backward into time. The correspondence between continuous-time TD algorithms and discrete-time TD(A) algorithms (Sutton, 1988) is shown in (Doya, 1996b). 1014 K. Doya Figure 1: (a) Actor-critic (b) Actor-tutor 2.2 OPTIMAL CONTROL BY VALUE GRADIENT According to the principle of dynamic programming (Bryson and Ho, 1975), the local constraint for the value function V· for the optimal control law p. is given by the Hamilton-Jacobi-Bellman equation V·(t) = max [r(x(t), u(t)) + T av·~x(t)) I(x(t), u(t))] . (7) u(1)EU x The optimal control p* is given by solving the maximization problem in the HJB equation, i.e., ar(x, u) aV·(x) al(x, u) _ 0 au +T ax au -. (8) When the cost for each control variable is given by a convex potential function Gj 0 r(x,u) = R(x) - L:Gj(Uj), j equation (8) can be solved using a monotonic function gj(x) = (Gj)-l(x) as Uj = gj (TaV;~X) a/~:~ u)) . (9) (10) If the system is linear with respect to the input, which is the case with many mechanical systems, al(x, u)/aUj is independent of u and the above equation gives a closed-form optimal feedback control law u = p·(x). In practice, the optimal value function is unknown and we replace V·(x) with the current estimate of the value function P(x) ( aPex) al(x, u)) u=g T~ au . (11) While the system evolves with the above control law, the value function P(x) is updated to minimize the TD error (6). In (11), the vector aP(x)/ax represents the desired motion direction in the state space and the matrix al(x, u)/au transforms it into the action space. The function g, which is specified by the control cost, determines the amplitude of control output. For example, if the control cost G is quadratic, then (11) reduces to a linear feedback control. A practically important case is when 9 is a sigmoid, because this gives a feedback control law for a system with limited control amplitude, as in the examples below. Efficient Nonlinear Control with Actor-Tutor Architecture 1015 3 ACTOR-TUTOR ARCHITECTURE It was shown in a task of a pendulum swing-up with limited torque (Doya, 1996b) that the above value-gradient based control scheme (11 can learn the task in much less trials than the actor-critic scheme. However, computation of the feedback command by (11) requires an on-line calculation of the gradient of the value function oP(x)/ox and its multiplication with the local linear model of the system dynamics a lex, u)/ou, which can be too demanding for real-time implementation. One solution to this problem is to use a simple direct controller network, as in the case of the actor-critic architecture. The training of the direct controller, or the actor, can be performed by supervised learning instead of trial-and-error learning because the target output of the controller is explicitly given by (11). Although computation of the target output may involve a processing time that is not acceptable for immediate feedback control, it is still possible to use its output for training the direct controller provided that there is some mechanism of short-term memory (e.g., eligibility trace in the connection weights). Figure l(b) is a schematic diagram of this "actor-tutor" architecture. The critic monitors the performance of the actor and estimates the value function. The "tutor" is a cascade of the critic, its gradient estimator, the local linear model ofthe system, and the differential model of control cost. The actor is trained to minimize the difference between its output and the tutor's output. 4 SIMULATION We tested the performance of the actor-tutor architecture in two nonlinear control tasks; a pendulum swing-up task (Doya, 1996b) and the global version of a cart-pole balancing task (Barto et al., 1983). The network architecture we used for both the actor and the critic was a Gaussian soft-max network. The output of the network is given by K Y = I: Wkbk(X), k=l b ( ) exp[- L:~=1 (~)2] k X =",K [_",n (X.-Cli)2]' ul=l exp ui=l 3/0 where (CkI' ... , Ckn) and (Ski, ... , Skn) are the center and the size of the k-th basis function. It is in general possible to adjust the centers and sizes of the basis function, but in order to assure predictable transient behaviors, we fixed them in a grid. In this case, computation can be drastically reduced by factorizing the activation of basis functions in each input dimension. 4.1 PENDULUM SWING-UP TASK The first task was to swing up a pendulum with a limited torque ITI ~ Tmax , which was about one fifth of the torque that was required to statically bring the pendulum up (Figure 2 (a)). This is a nonlinear control task in which the controller has to swing the pendulum several times at the bottom to build up enough momentum. 1016 K. Doya triat. (a) Pendulum (b) Value gradient trial_ ( c) Actor-Critic ( d) Actor-Tutor Figure 2: Pendulum swing-up task. The dynamics of the pendulum (a) is given by mle = -ti; + mglsin{} + T. The parameters were m = I = 1, g = 9.8, Jl. = 0.01, and Tmax = 2.0. The learning curves for value-gradient based optimal control (b), actor-critic (c), and actor-tutor (d); t_up is time during which I{}I < 45°. The state space for the pendulum x = ({},w) was 2D and we used 12 x 12 basis functions to cover the range I{} I ~ 180° and Iw I ~ 180° / s. The reinforcement for the state was given by the height of the tip of the pendulum, i.e., R(x) = cos {} and the cost for control G and the corresponding output sigmoid function g were selected to match the maximal output torque ymax. Figures 2 (b), (c), and (d) show the learning curves for the value-gradient based control (11), actor critic, and actor-tutor control schemes, respectively. As we expected, the learning of the actor-tutor was much faster than that of the actorcritic and was comparable to the value-gradient based optimal control schemes. 4.2 CART-POLE SWING-UP TASK Next we tested the learning scheme in a higher-dimensional nonlinear control task, namely, a cart-pole swing-up task (Figure 3). In the pioneering work of , the actorcritic system successfully learned the task of balancing the pole within ± 12° of the upright position while avoiding collision with the end of the cart track. The task we chose was to swing up the pole from an arbitrary angle and to balance it upright. The physical parameters of the cart-pole were the same as in (Barto et al., 1983) except that the length of the track was doubled to provide enough room for swinging. Efficient Nonlinear Control with Actor-Tutor Architecture J017 (a) (b) (c) Figure 3: Cart-pole swing-up task. (a) An example of a swing-up trajectory. (b) Value function learned by the critic. (c) Feedback force learned by the actor. Each square in the plot shows a slice of the 4D state space parallel to the (0, w) plane. Figure 3 (a) shows an example of a successful swing up after 1500 learning trials with the actor-tutor architecture. We could not achieve a comparable performance with the actor-critic scheme within 3000 learning trials. Figures 3 (b) and (c) show the value function and the feedback force field, respectively, in the 4D state space x = (x, v, 0, w), which were implemented in 6 x 6 x 12 x 12 Gaussian soft-max networks. We imposed symmetric constraints on both actor and critic networks to facilitate generalization. It can be seen that the paths to the upright position in the center of the track are represented as ridges in the value function. 5 DISCUSSION The biggest problem in applying TD or DP to real-world control tasks is the curse of dimensionality, which makes both the computation for each data point and the numbers of data points necessary for training very high. The actor-tutor architecture provides a partial solution to the former problem in real-time implementation. The grid-based Gaussian soft-max basis function network was successfully used in a 4D state space. However, a more flexible algorithm that allocates basis functions only in the relevant parts of the state space may be necessary for dealing with higher-dimension systems (Schaal and Atkeson, 1996). In the above simulations, we assumed that the local linear model of the system dynamics fJf(x,u)/fJu was available. In preliminary experiments, it was verified that the critic, the system model, and the actor can be trained simultaneously. 1018 K. Doya The actor-tutor architecture resembles "feedback error learning" (Kawato et al., 1987) in the sense that a nonlinear controller is trained by the output of anther controller. However, the actor-tutor scheme can be applied to a highly nonlinear control task to which it is difficult to prepare a simple linear feedback controller. Motivated by the performance of the actor-tutor architecture and the recent physiological and fMRI experiments on the brain activity during the course of motor learning (Hikosaka et al., 1996; Imamizu et al., 1996), we proposed a framework of functional integration of the basal ganglia, the cerebellum, and cerebral motor areas (Doya, 1996a). In this framework, the basal ganglia learns the value function P(x) (Houk et al., 1994) and generates the desired motion direction based on its gradient oP(x)/ox. This is transformed into a motor command by the "transpose model" of the motor system (of (x, u)/ouf in the lateral cerebellum (cerebrocerebellum). In early stages of learning, this output is used for control, albeit its feedback latency is long. As the subject repeats the same task, a direct controller is constructed in the medial and intermediate cerebellum (spinocerebellum) with the above motor command as the teacher. The direct controller enables quick, near-automatic performance with less cognitive load in other parts of the brain. 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 Systems, Man, and Cybernetics, 13:834-846. Bryson, Jr., A. E .. and Ho, Y -C. (1975). Applied Optimal Control. Hemisphere Publishing, New York, 2nd edition. Doya, K. (1996a). An integrated model of basal ganglia and cerebellum in sequential control tasks. Society for Neuroscience Abstracts, 22:2029. Doya, K. (1996b). Temporal difference learning in continuous time and space. In Touretzky, D. S., Mozer, M. C., and Hasselmo, M. E., editors, Advances in Neural Information Processing Systems 8, pages 1073-1079. MIT Press, Cambridge, MA. Hikosaka, 0., Miyachi, S., Miyashita, K., and Rand, M. K. (1996). Procedural learning in monkeys Possible roles of the basal ganglia. In Ono, T., McNaughton, B. 1., Molotchnikoff, S., Rolls, E. T ., and Nishijo, H., editors, Perception, Memory and Emotion: Frontiers in Neuroscience, pages 403-420. Pergamon, Oxford. Houk, J . C., Adams, J. L., and Barto, A. G. (1994). A model of how ,the basal ganglia generate and use neural signals that predict reinforcement. In Houk, J . C., Davis, J. L., and Beiser, D. G., editors, Models of Information Processing in the Basal Ganglia, pages 249-270. MIT Press, Cambrigde, MA. Imamizu, H., Miyauchi, S., Sasaki, Y, Takino, R., Putz, B., and Kawato, M. (1996). A functional MRI study on internal models of dynamic transformations during learning a visuomotor task. Society for Neuroscience Abstracts, 22:898. Kawato, M., Furukawa, K., and Suzuki, R. (1987). A hierarchical neural network model for control and learning of voluntary movement. Biological Cybernetics, 57:169-185. Schaal, S. and Atkeson, C. C. (1996). From isolation to cooperation: An alternative view of a system of experts. In Touretzky, D. S., Mozer, M. C., and Hasselmo, ,M. E., editors, Advances in Neural Information Processing Systems 8, pages 605-611. MIT Press, Cambridge, MA, USA. Sutton, R. S. (1988). Learning to predict by the methods of temporal difference. Machine Learning, 3:9-44.	1996	138
1,193	Interpolating Earth-science Data using RBF Networks and Mixtures of Experts E.VVan D.Bone Division of Infonnation Technology Canberra Laboratory, CSIRO GPO Box 664, Canberra, ACT, 2601, Australia {ernest, don} @cbr.dit.csiro.au Abstract We present a mixture of experts (ME) approach to interpolate sparse, spatially correlated earth-science data. Kriging is an interpolation method which uses a global covariation model estimated from the data to take account of the spatial dependence in the data. Based on the close relationship between kriging and the radial basis function (RBF) network (Wan & Bone, 1996), we use a mixture of generalized RBF networks to partition the input space into statistically correlated regions and learn the local covariation model of the data in each region. Applying the ME approach to simulated and real-world data, we show that it is able to achieve good partitioning of the input space, learn the local covariation models and improve generalization. 1. INTRODUCTION Kriging is an interpolation method widely used in the earth sciences, which models the surface to be interpolated as a stationary random field (RF) and employs a linear model. The value at an unsampled location is evaluated as a weighted sum of the sparse, spatially correlated data points. The weights take account of the spatial correlation between the available data points and between the unknown points and the available data points. The spatial dependence is specified in the form of a global covariation model. Assuming global stationarity, the kriging predictor is the best unbiased linear predictor of the un sampled value when the true covariation model is used, in the sense that it minimizes the squared error variance under the unbiasedness constraint. However, in practice, the covariation of the data is unknown and has to be estimated from the data by an initial spatial data analysis. The analysis fits a covariation model to a covariation measure of the data such as the sample variogram or the sample covariogram, either graphically or by means of various least squares (LS) and maximum likelihood (ML) approaches. Valid covariation models are all radial basis functions. Optimal prediction is achieved when the true covariation model of the data is used. In general, prediction (or generalization) improves as the covariation model used more Interpolating Earth-science Data using RBFN and Mixtures of Experts 989 closely matches the true covariation of the data. Nevertheless, estimating the covariation model from earth-science data has proved to be difficult in practice due to the sparseness of data samples. Furthermore for many data sets the global stationarity assumption is not valid. To address this, data sets are commonly manually partitioned into smaller regions within which the stationarity assumption is valid or approximately so. In a previous paper, we showed that there is a close, formal relationship between kriging and RBF networks (Wan & Bone, 1996). In the equivalent RBF network formulation of kriging, the input vector is a coordinate and the output is a scalar physical quantity of interest. We pointed out that, under the stationarity assumption, the radial basis function used in an RBF network can be viewed as a covariation model of the data. We showed that an RBF network whose RBF units share an adaptive norm weighting matrix, can be used to estimate the parameters of the postulated covariation model, outperforming more conventional methods. In the rest of this paper we will refer to such a generalization of the RBF network as a generalized RBF (GRBF) network. In this paper, we discuss how a mixture of GRBF networks can be used to partition the input space into statistically correlated regions and learn the local covariation model of each region. We demonstrate the effectiveness of the ME approach with a simulated data set and an aero-magnetic data set. Comparisons are also made of prediction accuracy of a single GRBF network and other more traditional RBF networks. 2 MIXTURE OF GRBF EXPERTS Mixture of experts (Jacobs et al , 1991) is a modular neural network architecture in which a number of expert networks augmented by a gating network compete to learn the data. The gating network learns to assign probability to the experts according to their performance over various parts of the input space, and combines the outputs of the experts accordingly. During training, each expert is made to focus on modelling the local mapping it performs best, improving its performance further. Competition among the experts achieves a soft partitioning of the input space into regions with each expert network learning a separate local mapping. An hierarchical generalization of ME, the hierarchical mixture of experts (HME), in which each expert is allowed to expand into a gating network and a set of sub-experts, has also been proposed (Jordan & Jacobs, 1994). Under the global stationarity assumption, training a GRBF network by minimizing the mean squared prediction error involves adjusting its norm weighting matrix. This can be interpreted as an attempt to match the RBF to the covariation of the data. It then seems natural to use a mixture of GRBF networks when only local stationarity can be assumed. After training, the gating network soft partitions the input space into statistically correlated regions and each GRBF network provides a model of the covariation of the data for a local region. Instead of an ME architecture, an HME architecture can be used. However, to simplify the discussion we restrict ourselves to the ME architecture. Each expert in the mixture is a GRBF network. The output of expert i is given by: ... Yi(X;Oi) = L Wijq,(x;cij~Mi)+ WiD j =\ (2.1) where ni is the number of RBF units, 0i = {{wi);~o,{cij}i=\,Md are the parameters of the expert and q,(x;c,M)=qX:II x-c II M). Assuming zero-mean Gaussian error and common variance a/, the conditional probability of y given x and ~ is given by: (2.3) 990 E. Wan and D. Bone Since the radial basis functions we used bave compact support and eacb expert only learns a local covariation model, small GRBF networks spanning overlapping regions can be used to reduce computation at the expense of some resolution in locating the boundaries of the regions. Also, only the subset of data within and around the region spanned by a GRBF network is needed to train it, further reducing computational effort. With m experts, the ilb output of the gating network gives the probability of selecting the expert i and is given by the normalized function: g, (x~'U) = P(ilx, '0) = Il, exp(q(x~'UJ)/ ~lllj exp{q(x;'U J) (2.4) wbere'U = { raj::\, {'UJ::1}. Using q(x~ '0,) = 'U;[x T If and setting all a, 's to 1, the gating network implements the softmax function and partitions the input space into a smoothed planar tessellation. Alternatively, with q(x~1>i)=-IITi(X-u;)112 (wbere 1>i={u;,Td consists of a location vector and an affine transformation matrix) and restricting the a/s to be non-negative, the gating network divides the input space into packed anisotropic ellipsoids. These two partitionings are quite convenient and adequate for most earth-science applications wbere x is a 2D or 3D coordinate. The output of the experts are combined to give the overall output of the mixture: III III Y{x~a) = L P(ilx, '\»)9i (x;ai) = L g, (x; '0 )Yi (x;a,) (2.5) i=1 i=1 wbere a = {'U, {ai }::1} and the conditional probability of observing y given x and a is: III p(ylx,a) = L P(ilx, '0 )p(ylx,a,) . (2.6) ,=1 3 THE TRAINING ALGORITHM The Expectation-Maximization (EM) algorithm of Jordan and Jacobs is used to train the mixture of GRBF networks. Instead of computing the ML estimates, we extend the algorithm by including priors on the parameters of the experts and compute the maximum a posteriori (MAP) estimates. Since an expert may be focusing on a small subset of the data, the priors belp to prevent over-fitting and improve generalization. Jordan & Jacobs introduced a set of indicator random variables Z = {Z<t)}~1 as missing data to label the experts that generate the observable data D = ((x(t), y<t»} ~1. The log joint probability of the complete data Dc = {D, Z} and parameters a can be written as: wbere A. is a set of byperparameters. Assuming separable priors on the parameters of the model i.e. p(alA.) = p('UIAo)D p(ail~) with A. = {~}:o' (3.1) can be rewritten as: N III In p(Dc,alA.) = L L Zi(t) In P(ilx(t) , '0)+ In P('UIAo) /=1 ,=1 (3.2) Interpolating Earth-science Data using RBFN and Mixtures of Experts 991 Since the posterior probability of the model parameters is proportional to the joint probability, maximizing (3.2) is equivalent to maximizing the log posterior. In the Estep, the observed data and the current network parameters are used to compute the expected value of the complete-data log joint probability: N '" Q(OIO(k) ) = L L h;(k)(t)In P(ilx(I), '\))+ In p( '\)IAo) 1=1 1=1 (3.3) where (3.4) In the M-step, Q(OIO(k) is maximized with respect to e to obtain 0(1+1). As a result of the use of the indicator variables, the problem is decoupled into a separate set of interim MAP estimations: N '" '\)(k+1) = arg max L L hi(k)(t) In P(ilx(I), '\)) + In p( '\)IAo) 1) 1=1 i=1 (3.5) N O~HI) = arg lI}.ax L ~(1)(t)In P(l')lx(I),OJ+ In p(OP't) I 1=1 (3.6) We assume a flat prior for the gating network parameters and the prior II; II; P(Oi I~) = exp(-t ~ L L WiT Wi.rq,(CiT -Ci.r» I ZR(~) where ZR(A-.) is a normalization constant, for the experts. This smoothness prior is used on the GRBF networks because it can be derived from regularization theory (Girosi & Poggio, 1990) and at the same time is consistent with the interpretation of the radial basis function as a covariation model. Hence, maximizing e i with (3.6) is equivalent to minimizing the cost function: where A-.' = ~(ji2. The value of the effective regularization parameter, ~', can be set by generalized cross validation (GCV) (Orr, 1995) or by the 'evidence' method of (Mackay, 1991) using re-estimation formulas. However, in the simulations, for simplicity, we preset the value of the regularization parameter to a fixed value. 4 SIMULATION RESULTS Using the Cholesky decomposition method (Cressie, 1993), we generate four 2D data sets using the four different covariation models shown in Figure 1. The four data set are then joined together to form a single 64x64 data set. Figure 3a shows the original data set and the hard boundaries of the 4 statistically distinct regions. We randomly sample the data to obtain a 400 sample training set and use the rest of the data for validation. Two GRBF networks, with 64 and 144 adaptive anistropic spherical! units respectively, are used to learn the postulated global covariation model and the mapping. A 2-level I The spherical model is widely used in geostatistics and when used as a covariance function is defined as lI'(h;a) = 1- {7(~) - t<l!)3} for ~llhll~ and rp{b;a) = 0 for Ilhll>a. Spherical does NOT mean isotropic. 992 E. Wan and D. Bone HME with 4 GRBF network experts each with 36 spherical units are used to learn the local covariation models and the mapping. Softmax gating networks are used and each expert is somewhat 'localized' in each quadrant of the input space. The units of the experts are located at the same locations as the units of the 64-unit GRBF network with 24 overlapping units between any two of the experts. The design ensures that the HME does not have an advantage over the 64-unit GRBF network if the data is indeed globally stationary. Figure 2 shows the local covariation models learned by the HME with the smoothness priors and Figure 3b shows the interpolant generated and the partitioning. (a) NW (exponential) (b) NE (spherical) (a) NW (spherical) (b) NE (spherical) ~ lij" ~ ~"01 ~ [11"" ~ ~.01 -10 -10 -10 -10 . -20 -20 -20-20 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 (c) SW (spherical) (d) SE (spherical) (c) SW (spherical) (d) SE (spherical) ~~"~1 ~~"1" ~~~[I}. -10 ~ -10 • -10 -10 -20 -20 -20-20 -20-10 0 1020 -20-10 0 1020 -20-10 0 1020 -20-10 0 1020 Figure 1: The profile of the true local Figure 2: The profile of the local covariation models of the simulated data set. covariation models learned by the HME. Exponential and spherical models are used. (a) (b) (c) 60 60 60 40 40 40 20 20 20 20 40 60 20 40 60 20 40 60 Figure 3: (a) Simulated data set and true partitions. (b) Interpolant generated by the 144 spherical unit GRBFN. (c) The HME interpolant and the soft partitioning learned (0.5, 0.9 probability contours of the 4 experts shown in solid and dotted lines respectively) Table 1: Nonnalized mean squared prediction error for the simulated data set. Network RBF unit NMSE RBFN (isotropic RBF units with width set to the 64, Gaussian 0.761 distance to the nearest neighbor) 144, Gaussian 0.616 400, Gaussian 0.543 RBFN (identical isotropic RBF units with adaptive 64, Gaussian 0.477 width) 144, Gaussian 0.475 GRBFN (identical RBF units with adaptive norm 64, spherical 0.506 weiRhtinR matrix) 144, spherical 0.431 HME (2 levels, 4 GRBFN eXlJerts) without lJriors 4x36, spherical 0.938 HME (2 levels 4 GRBFN eXlJerts) with lJriors 4x36, spherical 0.433 kriging predictor (usinR true local models) 0.372 For comparison, a number of ordinary RBF networks are also used to learn the mapping. In all cases, the RBF units of networks of the same size share the same locations which Interpolating Earth-science Data using RBFN and Mixtures of Experts 993 are preset by a Kohonen map. Table 1 summarizes the normalized mean squared prediction error (NMSE)- the squared prediction error divided by the variance of the validation set - for each network. With the exception of HME, all results listed are obtained with a smoothness prior and a regularization parameter of 0.1. Ordinary weight decay is used for RBF networks with units of varying widths and the smoothness prior discussed in section 3 are used for the remaining networks. The NMSE of the kriging predictor that uses the true local models is also listed as a reference. Similar experiments are also conducted on a real aero-magnetic data set. The flight paths along which the data is collected are divided into a 740 data points training set and a 1690 points validation set. The NMSE for each network is summarized in Table 2, the local covariation models learned by the HME is shown in Figure 4, and the interpolant generated by the HME and the partitioning is shown in Figure 5b. (a) NW (spherical) (b) NE (spherical) (a) (b) -::.::~ 80 120 80 120 40 40 -100 0 100 -100 0 100 40 80 120 40 80 120 (c) SW (spherical) (d) SE (spherical) '~[I]'~~ -100 l00~ Figure 5: (a) Thin-plate interpolant of the entire aero-magnetic data set. (b) The HME interpolant and the soft partitioning (0.5, 0.9 probability contours of the 4 experts shown in solid and dotted lines respectively). -100 0 100 -100 0 100 Figure 4: The profile of the local covariation models of the aero-magnetic data set learned by the HME. Table 2: Normalized mean squared prediction error for the aero-magnetic data set. Network RBF units NMSE RBFN (isotropic RBF units with width set to the 49, Gaussian 1.158 distance to the nearest neighbor) 100, Gaussian 1.256 RBFN (isotropic RBF units with width set to the 49, Gaussian 0.723 mean distance to the 8 nearest neighbors) 100, Gaussian 0.699 RBFN (identical isotropic RBF units with adaptive 49, Gaussian 0.692 width) 100, Gaussian 0.614 GRBFN (identical RBF units with adaptive norm 49, spherical 0.684 weighting matrix) 100 spherical 0.612 HME (2 levels, 4 GRBFN experts) without priors 4x25, spherical 0.389 HME (2 levels, 4 GRBFN expertsl with priors 4x25, spherical 0.315 5 DISCUSSION The ordinary RBF networks perform worst with both the simulated data and the aeromagnetic data. As neither data set is globally stationary, the GRBF networks do not improve prediction accuracy over the corresponding RBF networks that use isotropic Gaussian units. In both cases, the hierarchical mixture of GRBF networks improves the prediction accuracy when the smoothness priors are used. Without the priors, the ML estimates of the HME parameters lead to improbably high and low predictions. 994 E. Wan and D. Bone The improvement in prediction accuracy is more significant for the aero-magnetic data set than for the simulated data set due to some apparent global covariation of the simulated data which only becomes evident when the directional variograms of the data are plotted. However, despite the similar NMSE, Figure 3 shows that the interpolant generated by the 144-unit GRBF network does not contain the structural information that is captured by the HME interpolant and is most evident in the north-east region. In the case of the simulated data set, the HME learns the local covariation models accurately despite the fact that the bottom level gating networks fail to partition the input space precisely along the north-south direction. The availability of more data and the straight east-west discontinuity allows the upper gating network to partition the input space precisely along the east-west direction. In the north-west region, although the class of function the expert used is different from that of the true model, the model learned still resembles the true model especially in the inner region where it matters most. In the case of the aero-magnetic data set, the RBF and GRBF networks perform poorly due to the considerable extrapolation that is required in the prediction and the absence of global stationarity. However, the HME whose units capture the local covariation of the data interpolates and extrapolates significantly better. The partitioning as well as the local covariation model learned by the HME seems to be reasonably accurate and leads to the construction of prominent ridge-like structures in the north-west and south-east which are only apparent in the thin-plate interpolant of the entire data set of Figure Sa. 6 CONCLUSIONS We show that a mixture of GRBF networks can be used to learn the local covariation of spatial data and improve prediction (or generalization) when the data is approximately locally stationary - a viable assumption in many earth-science applications. We believe that the improvement will be even more significant for data sets with larger spatial extent especially if the local regions are more statistically distinct. The estimation of the local covariation models of the data and the use of these models in producing the interpolant helps to capture the structural information in the data which, apart from accuracy of the prediction, is of critical importance to many earth-science applications. The ME approach allows the objective and automatic partitioning of the input space into statistically correlated regions. It also allows the use of a number of small local GRBF networks each trained on a subset of the data making it scaleable to large data sets. The mixture of GRBF networks approach is motivated by the statistical interpolation method of kriging. The approach therefore has a very sound physical interpretation and all the parameters of the network have clear statistical and/or physical meanings. References Cressie, N. A (1993). Statistics for Spatial Data. Wiley, New York. Jacobs, R. A, Jordan, M. I., Nowlan, S. J. & Hinton, G. E. (1991). Adaptive Mixtures of Local Experts. Neural Computation 3, pp. 79-87. Jordan, M. I. & Jacobs, R. A (1994). Hierarchical Mixtures of Experts and the EM Algorithm. Neural Computation 6, pp. 181-214. MacKay, D. J. (1992). Bayesian Interpolation. Neural Computation 4, pp. 415-447. Orr, M. J. (1995). Regularization in the Selection of Radial Basis Function Centers. Neural Computation 7, pp. 606-623. Poggio, T. & Girosi, F. (1990). Networks for Approximation and Learning. In Proceedings of the IEEE 78, pp. 1481-1497. Wan, E. & Bone, D. (1996). A Neural Network Approach to Covariation Model Fitting and the Interpolation of Sparse Earth-science Data. In Proceedings of the Seventh Australian Conference on Neural Networks, pp. 121-126.	1996	139
1,194	Ordered Classes and Incomplete Examples in Classification Mark Mathieson Department of Statistics, University of Oxford 1 South Parks Road, Oxford OXI 3TG, UK E-mail: mathies@stats.ox.ac.uk Abstract The classes in classification tasks often have a natural ordering, and the training and testing examples are often incomplete. We propose a nonlinear ordinal model for classification into ordered classes. Predictive, simulation-based approaches are used to learn from past and classify future incomplete examples. These techniques are illustrated by making prognoses for patients who have suffered severe head injuries. 1 Motivation Jennett et al. (1979) reported data on patients with severe head injuries. For each patient some of the information in Table 1 was available shortly after injury. The objective is to predict the degree of recovery attained within six months as measured by outcome. This problem exhibits two characteristics that are common in classification tasks: allocation qf examples into classes which have a natural ordering, and learning from past and classifying future incomplete examples. 2 A Flexible Model for Ordered Classes The Bayes decision rule (see, for example, Ripley, 1996) depends on the loss L(j, k) incurred in assigning to class k an object belonging to class j. When better information is unavailable, for unordered or nominal classes we treat every mis-classification as equally serious: LU, k) is 0 when j = k and 1 otherwise. For ordered classes, when the K classes are numbered from 1 to K in their natural order, a better default choice is LU, k) =1 j - k I. A class is then given support by its position in the ordering, and the Bayes rule will sometimes assign patterns to classes that do not have maximum posterior probability to avoid making a serious error. Ordered Classes and Incomplete Examples in Classification 551 Table 1: Definition of variables with proportion missing. Variable Definition Missing % age emv motor change eye pupils outcome Age in decades (1=0-9, 2=10-19, ... ,8=70+). Measure of eye, motor and verbal response to stimulation (1-7). Motor response patterns for all limbs (1-7). Change in neurological function over the first 24 hours (-1,0,+1). Eye indicant. 1 (bad), 2 (impaired), 3 (good). Pupil reaction to light. 1 (non-reacting), 2 (reacting). Recovery after six months based on Glasgow Outcome Scale. 1 (dead/vegetative), 2 (severe disability), 3 (moderate/good recovery). ° 41 33 78 65 30 ° If the classes in a classification problem are ordered the ordering should also be reflected in the probability model. Methods for nominal tasks can certainly be used for ordinal problems, but an ordinal model should have a simpler parameterization than comparable nominal models, and interpretation will be easier. Suppose that an example represented by a row vector X belongs to class C = C(X). To make the Bayes-optimal classification it is sufficient to know the posterior probabilities p(C = k I X = x). The ordinallogistic regression (OLR) model for K ordered classes models the cumulative posterior class probabilities p( C ~ k I X = x) by [ p( C ~ k I X = x) ] log 1 _ p(C ~ k I X = x) = ¢>k -1](x) k = 1, ... ,K -1, (1) for some function 1]. We impose the constraints ¢>1 ~ . . . ~ ¢>K-l on the cut-points to ensure thatp(C ~ k I X = x) increases with k. If ¢>o = -00 and ¢>K = 00 then (1) gives p(C = k I X = x) = a(¢>k -1](x)) - a(¢>k-l -1](x)) k=l, ... ,K where a(x) = 1/(1 + e- X ). McCullagh (1980) proposed linear OLR where 1](x) = x{3. The posterior probabilities depend on the patterns x only through 1], and high values of 1](x) correspond to higher predicted classes (Figure la). This can be useful for interpreting the fitted model. However, linear OLR is rather inflexible since the decision boundaries are always parallel hyperplanes. Departures from linearity can be accommodated by allowing 1] to be a non-linear function of the feature space. We extend OLR to non-linear ordinal logistic regression (NOLR) by letting 1](x) be the single linear output of a feed-forward neural network with input vector x, having skip-layer connections and sigmoid transfer functions in the hidden layer (Figure Ib). Then for weights Wij and biases bj we have 1](x) = 2: WioXCi) + 2: wjoa(bj + 2: WijXCi»), i-to j-to i-tj where :Li-tj denotes the sum over i such that node i is connected to node j, and node o is the single output node. The usual output-unit bias is incorporated in the cut-points. Observe that OLR is the special case of NOLR with no hidden nodes. Although the network component of NOLR is a universal approximator the NOLR model cannot approximate all probability densities arbitrarily well (unlike 'softmax', the most similar nominal method). The likelihood for the cut-points l/> = (¢>1, ... ,¢> K -1) and network parameters w given a training set T = {(Xi, Ci) Ii = 1, ... ,n} ofn correctly classified examples is n n £(w, l/» = IIp(Ci I Xi) = II [a(¢>Ci -1](Xi; w)) - a(¢>ci-l -1](Xi; w))] . (2) i=l i=l 552 q,------------------, co o '" o -10 -8 p( 11 eta) p(21 eta) p(31 eta) p(41 eta) p(SI eta) -6 -4 -2 Network output (eta) o 2 M. Mathieson o 20 40 60 age (years) Figure 1: (a) p(k 1 "I) plotted against "I for an OLR model with K = 5 classes and 4> = (-7, -6, -3, -1). (b) The network output TJ(x) from a NOLR model used to predict change given all other variables (except outcome) predicts that young patients with high emv score are likely to improve over first 24 hours. While age and emv are varied, other variables are fixed. Dark shading denotes low values ofTJ(x). The Bayes decision boundaries are shown for loss L(j, k) =1 j - k I. If we estimate the classifier by substituting the maximum likelihood estimates we must maximize (2) whilst constraining the cut-points to be increasing (Mathieson, 1996). To avoid over-fitting we regularize both by weight decay (which is equivalent to putting independent Gaussian priors on the network weights) and by imposing independent Gamma priors on the differences between adjacent cut-points. The minim and is now -log f(w, l/» + >..D(w) + E(l/>; t, 0:) with hyperparameters >.. > 0, t, 0: (to be chosen by cross-validation, for example, or averaged over under a Bayesian scheme) where D(w) = 2:i,j W;j and K-l E(l/» = L [t(<Pi - <pi-d + (1 - 0:) log(<pi - <Pi-d] . i=2 3 Classification and Incomplete Examples We now consider simulation-based methods for training diagnostic paradigm classifiers from incomplete examples, and classifying future incomplete examples. To avoid modelling the missing data we assume that the missing data mechanism is independent of the missing values given the observed values (missing at random) and that the missing data and data generation mechanisms are independent (ignorable) (Little & Rubin, 1987). This assumption is rarely true but is usually less damaging than adopting crude ad hoc approaches to missing values. 3.1 Learning from Incomplete Examples The training set is r = {(xi, Ci) I i = 1, ... ,n} where xi, xi are the observed and unobserved parts of the ith example, which belongs to class Ci. Define XO = {xi I i = 1, ... ,n} and Xu = {xi I i = 1, ... ,n}, and use C to denote all the classes, so r = (XO, C). We assume that C is fully observed. Under the diagnostic paradigm (which includes logistic regression and its non-linear and ordinal variants such as 'softmax' and Ordered Classes and Incomplete Examples in Classification 553 NOLR) we model p( C Ix) by p( C I Xj 8) giving the conditional likelihood n n n £(8) = IIp(ci I xf;8) = IIIEx:'lxfP(ci I xf,Xi;8) = IExulXo IIp(ci I xf,Xi;8) i=1 i=l i=l (3) when the examples are independent. The model for p( C Ix) contains no information about p(x) and so we construct a model for p(XU I XO) separately using T (Section 3.2). Once we can sample xfu' .. ,xfm from p(xf I xi, Ci) a Monte Carlo approximation for £(8) based on the last expression of (3) by averaging over repeated imputations of the missing values in the training set (Little & Rubin, 1987, and earlier): ( 1 m n ) log£(8) ;:::;; log m ~ 1jP(Ci I xf, xij; 8) . (4) Existing algorithms for finding maximum likelihood estimates for 8 allow maximization of the individual summands in (4) with respect to 8, but in general the software will require extensive modification in order to maximize the average. This problem can be avoided if we approximate the arithmetic average over the imputations by a geometric one so that ( ) 1/m £( 8);:::;; TIj TIi p( Ci I xi, xt; 8) . Now the log-posterior averages over the log of the likelihoods of the completed training sets, so standard estimation algorithms can be used on a training set formed by pooling all completions of the training set, giving each weight 11m. The approximation log ! I:j Pj ;:::;; ! I:j logpj has been made, where we define Pj (8) = TIi p( Ci I xi, xij; 8), although in fact log ! I:j Pj ~ ! I:j log Pj everywhere. Suppose that the Pj are well approximated by some function P for the region of interest in the parameter space. Then in this region I lL 1 Ll 1 L (Pi - p)2 1 L (Pi - p)(Pj - p) ogp._ogp·;:::;;---m J m J 2m P 2m2 p2 j j i i,j (5) and so the approximation will be reasonable when the imputed values have little effect on the likelihood of the completed training sets. Note that the approximation cannot be improved by increasing m; (5) does not tend to zero as m ---t 00. The relative effects on the likelihood of making this approximation and the Monte Carlo approximation (4) will be problem specific and dependent on m . The predictive approach (Ripley, 1996, for example) incorporates uncertainty in 8 by estimatingp(c I x) asp(c I x) = IEOITP(C I x;8). Changing the order of integration gives p(C I x) = J p(c I Xj 8)p(8 I T) d8 ex J p(c I x; 8)p(8) IT IExulxfP(Ci I xf, Xi; 8) d8 \=1 = IExulXo J p(c I x; 8)p(8) ITp(Ci I xi, Xi; 8) d8 (6) i=1 This justifies applying standard techniques for complete data to build a separate classifier using each completed training set, and then averaging the posterior class probabilities that they predict. The integral over 8 in (6) will usually require approximation; in particular we could average over plug-in estimates toobtainp(c I x);:::;;! I:~1P(C I x; OJ), where OJ is the MAP estimate of 8 based only on the jth imputed training set. A more subtle approach 554 M. Mathieson Table 2: Classifier performance on 301 complete test examples. See Section 4. Training set 40 complete training examples only 40 complete + 206 incomplete training examples: • Median imputation (In each variable, substitute the median for missing values whenever they occur.) • Averaging predicted probabilities over 1000 completions of T generated by: [> Unconditional imputation (Sample missing values from the empirical distribution of each variable in the training set.) [> Gibbs sampling from p(XU I xo,,,fJ) Pool the 1000 completions from the line above to form a single training set Test set loss 132 149 133 118 117 (Ripley, 1994) approximates each posterior by a mixture of Gaussians centred at the local maxima Oj1, ... ,0jRj of p( fJ 1 T, X}L) to give (7) where: N(·; j.£, E) is the Gaussian density function with mean j.£ and covariance matrix E, the Hessian Hjr = &()~~&() 10gp(fJ 1 T, XJ') is evaluated at Ojr and, using Laplace's approximation, Wjr = p(Ojr 1 T, Xl) 1 Hjr 1- 1/ 2 . We can average over the maxima to get p(c 1 x) ~ (m l:j,r Wjr )-ll:j,r P(c I x; Ojr), butthe full-blooded approach samples from the 'mixture of mixtures' approximation to p( fJ 1 T) and also uses importance sampling to compute the predictive estimates p. 3.2 The Imputation Model We need samples from p(xy I xi, Ci) for each i. When many patterns of missing values occur it is not practical to model p(XU 1 xo, c) for each pattern, but Markov chain Monte Carlo methods can be employed. The Gibbs sampler is convenient and in its most basic form requires models for the distribution of each element of x given the others, that is p(x(j) 1 x( -j), c) where x( -j) = (X(l), ... ,x(j-1), x(j+1) , ... ,x(p». We model these full conditionals parametrically as p( xU) 1 x( - j) , c; 'I/J) and assume here that the parameters for each of the full conditionals are disjoint, so p(x(j) I x( -j), C; 'I/J(j» where 'I/J = ('I/J(1), ... ,'I/J(p». When x(j} takes discrete values this is a classification task, and for continuous values a regression problem. Under certain conditions the chain of dependent samples of Xu converges in distribution to p( XU I xo, 'I/J) and the ergodic average of p(c I xo, XU) converges as required to the predictive estimate p(c I Xo). We usually take every wth sample to provide a cover of the space in fewer samples, reducing the computation required to learn the classifier. It is essential to check convergence of the Gibbs sampler although we do not give details here. If we have sufficient complete examples we might use them to estimate 'I/J to be -J; and Gibbs sample from p(XU 1 xo; -J;). Otherwise, in the Bayesian framework, incorporate 'I/J into the sampling scheme by Gibbs sampling from p( 'I/J, Xu I XO) (the solution suggested by Li, 1988). In the head injury example we report results using the former approach. (The latter was found to make little improvement and requires considerably more computation time.) Ordered Classes and Incomplete Examples in Classification 555 Table 3: Predictive approximations for a NOLR model fitted to a single completion T, Xu of the training set. The likelihood maxima at {h and {h account for over 0.99 of the posterior probability. Posterior probability -logp(Oi I T, XU} Test set loss: • using the plug-in classifier p( c I x; Oi} • averaging over 10,000 samples from Gaussian 3.3 Classifying Incomplete Examples 0.929 176.10 128 120 0.071 174.65 Predictive: 149 126 137 119 We could build a separate classifier for each pattern of missing data that occurs, but this can be computationally expensive, will lose information and the classifiers need not make consistent predictions. We know that p(c I XO) = IExulxop(c I xo, XU) so it seems better to classify Xo by averaging over repeated imputations of XU from the imputation model. 4 Prognosis After Head Injury We now return to the head injury prognosis example to learn a NOLR classifier from a training set containing 40 complete and 206 incomplete examples. The NOLR architecture (4 nodes, skip-layer connections and A = 0.01) was selected by cross-validation on a single imputation of the training set, and we use a predictive approximation. 1 Table 2 shows the performance of this classifier on a test set of 301 complete examples and loss L (j, k) = I j - k I for different strategies for dealing with the missing values. For imputation by Gibbs sampling we modelled each of the full conditionals using NOLR because all variables in this dataset are ordinal. Categorical inputs to models are put in as level indicators, so change corresponds to two indicators taking values (0,0), (1,0) and (1,1). Throughout this example we predict age, emv and motor as categorical variables but treat them as continuous inputs to models. Models were selected by cross-validation based on the complete training examples only and used the predictive approximation described above. Several full conditionals benefited from a non-linear model. We now classify 199 incomplete test examples using the classifier found in the last line of Table 2. Median imputation of missing values in the test set incurs loss 132 whereas unconditional imputation incurs loss 106. The Gibbs sampling imputation model incurs loss 91 and is predicting probabilities accurately (Figure 2). Michie et al. (1994) and references therein give alternative analyses of the head injury data. NOLR has provided an interpretable network model for ordered classes, the missing data strategy successfully learns from incomplete training examples and classifies incomplete future examples, and the predictive approach is beneficial. IFor each completion T, X jU of the training set we form a mixture approximation (7) to p(O I T, XjU }, sample from this 10,000 times and average the predicted probabilities. These predictions are averaged over completions. Maxima were found by running the optimizer 50 times from randomized starting weights. Up to 26 distinct maxima were found and approximately 5 generally accounted for over 95% of the posterior probability in most cases. Table 3 gives an example: averaging over maxima has greater effect than sampling around them, although both are useful. The cut-points cI> in the NOLR model must satisfy order constraints, so we rejected samples of () where these did not hold. However, the parameters were sufficiently well determined that this occurred in less than 0.5% of samples. 556 M. Mathieson severe disability good recovery Figure 2: (a) Test set calibration for median imputation (dashed) and conditional imputation (solid). For predictions by conditional imputation we average p( c 1 xo, XU) over 100 pseudo-independent samples from p(XU 1 Xo). Ticks on the lower (upper) axis denote predicted probabilities for the test examples using median (conditional) imputation. (b) In 100 pseudo-independent conditional imputations of the missing parts XU of a particular incomplete test example eight distinct values xf (i = 1, . . . ,8) occur. (Recall that all components of x are discrete.) For each distinct imputation we plot a circle with centre corresponding to (p(1 1 xO,xf),p(2 1 xO,xf),p(3 1 xO,xf)) and area proportional to the number of occurrences of xf in the 100 imputations. The prediction by median imputation is located by x; the average prediction over conditional imputations is located by • . Actual outcome is 'good recovery'. The conditional method correctly classifies the example and shows that the example is close to the Bayes decision boundary under loss L(j, k) =1 j - k 1 (dashed). Median imputation results in a confident and incorrect classification. Software: A software library for fitting NOLR models in S-Plus is available at URL http://www.stats.ox.ac.uk/-mathies Acknowledgements: The author thanks Brian Ripley for productive discussions of this work and Gordon Murray for permission to use the head injury dataset. This research was funded by the UK EPSRC and investment managers GMO Woolley Ltd. References Jennett, B., Teasdale, G., Braakman, R., Minderhoud, J., Heiden, J. & Kurze, T. (1979) Prognosis of patients with severe head injury. Neurosurgery, 4782-790. Li, K.-H. (1988) Imputation using Markov chains. Journal of Statistical Computation and Simulation, 3057-79. Little, R. & Rubin, D. B. (1987) Statistical Analysis with Missing Data. (Wiley, New York). Mathieson, M. J. (1996) Ordinal models for neural networks. In Neural Networks in Financial Engineering, eds A.-P. Refenes, Y. Abu-Mostafa, J. Moody and A. S. Weigend (World Scientific, Singapore) 523-536. McCullagh, P. (1980) Regression models for ordinal data. Journal of the Royal Statistical Society Series B, 42 109-142. Michie, D., Spiegelhalter, D. J. & Taylor, C. C. (eds) (1994) Machine Learning, Neural and Statistical Classification. (Ellis Horwood, New York). Ripley, B. D. (1994) Flexible non-linear approaches to classification. In From Statistics to Neural Networks. Theory and Pattern Recognition Applications, eds V. Cherkassky, J. H. Friedman and H. Wechsler (Springer Verlag, New York) 108-126. Ripley, B. D. (1996) Pattern Recognition and Neural Networks. (Cambridge University Press, Cambridge).	1996	14
1,195	Statistically Efficient Estimation Using Cortical Lateral Connections Alexandre Pouget alex@salk.edu Abstract Kechen Zhang zhang@salk.edu Coarse codes are widely used throughout the brain to encode sensory and motor variables. Methods designed to interpret these codes, such as population vector analysis, are either inefficient, i.e., the variance of the estimate is much larger than the smallest possible variance, or biologically implausible, like maximum likelihood. Moreover, these methods attempt to compute a scalar or vector estimate of the encoded variable. Neurons are faced with a similar estimation problem. They must read out the responses of the presynaptic neurons, but, by contrast, they typically encode the variable with a further population code rather than as a scalar. We show how a non-linear recurrent network can be used to perform these estimation in an optimal way while keeping the estimate in a coarse code format. This work suggests that lateral connections in the cortex may be involved in cleaning up uncorrelated noise among neurons representing similar variables. 1 Introduction Most sensory and motor variables in the brain are encoded with coarse codes, i.e., through the activity of large populations of neurons with broad tuning to the variables. For instance, direction of visual motion is believed to be encoded in visual area MT by the responses of a large number of cells with bell-shaped tuning, as illustrated in figure I-A. Neurophysiological recordings have shown that, in response to an object moving along a particular direction, the pattern of activity across such a population would look like a noisy hill of activity (figure I-B). On the basis of this activity, A, the best that can be done is to recover the conditional probability of the direction of motion, (), given the activity, p( (}IA). A slightly less ambitious goal is to come up with a good "guess", or estimate, 0, of the direction, (), given the activity. Because of the stochastic nature of the noise, the estimator is a random variable, i.e, for o AP is at the Institute for Computational and Cognitive Sciences, Georgetown University, Washington, DC 20007 and KZ is at The Salk Institute, La Jolla, CA 92037 . This work was funded by McDonnell-Pew and Howard Hughes Medical Institute. 98 A OL-----------------------J 100 200 300 Direction (deg) B 3 2.5 .~ 2 ."5 « 1.5 A. Pouget and K. Zhang i\ i \ 1 \ 100 200 300 Preferred Direction (deg) Figure 1: A- Tuning curves for 16 direction tuned neurons. B- Noisy pattern of activity (0) from 64 neurons when presented with a direction of 1800 • The ML estimate is found by moving an "expected" hill of activity (dotted line) until the squared distance with the data is minimized (solid line) the same image, B will vary from trial to trial. A good estimator should have the smallest possible variance across those trials because the variance determines how well two similar directions can be discriminated using this estimator. The Cramer-Rao bound provides an analytical lower bound for this variance given the noise in the system and the unit tuning curves [5] . Typically, computationally simple estimators, such as optimum linear estimator (OLE), are very inefficient; their variances are several times the bound. By contrast, Bayesian or maximum likelihood (ML) estimators (which are equivalent for the case under consideration in this paper) can reach this bound but require more complex calculations [5]. These decoding technics are valuable for a neurophysiologist interested in reading out the population code but they are not directly relevant for understanding how neural circuits perform estimation. In particular, they all provide the estimate in a format which is incompatible with what we know of sensory representations in the cortex. For example, cells in V 4 are estimating orientation from the noisy responses of orientation tuned VI cells, but, unlike ML or OLE which provide a scalar estimate, V4 neurons retain orientation in a coarse code format, as demonstrated by the fact that V4 cells are just as broadly tuned to orientation as VI neurons. Therefore, it seems that a theory of estimation in biological networks should have two critical characteristics: 1- it should preserve the estimate in a coarse code and 2- it should be efficient, i.e., the variance should be close to the Cramer-Rao bound. We explore in this paper various network architectures for performing estimations with coarse code using lateral connections. We start by briefly describing several classical estimators such as OLE or ML. Then, we consider linear and non-linear recurrent networks and compare their performances with the classical estimators. 2 Classical Methods The simplest estimators are linear of the form BOLE = WT A. Better performance can be obtained with a center of mass estimator (COM), BeoM = 2:i Biai/ 2:i ai; however, in the case of a periodic variable, such as direction of motion, the best one-shot method known is the complex estimator (CaMP), BeoMP = phase(z) where z = 2::=1 akei91c [5]. This estimator consists in fitting a cosine through the pattern of activity, like the one shown in figure I-B, and using the phase of Statistically Efficient Estimations Using CorticaL LateraL Connections A B 40 o Activity over Time 200 300 pret~~~d Direction (deg) 99 Figure 2: A- Circular network of 64 units. Only the connections originating from one unit are shown. B- Activity over time in the non-linear network when initialized with a random pattern at t = O. The activity of the units are plotted as a function of their position along the circle which is equivalent to their preferred direction of motion with appropriate choice of weights. the best cosine fit as the estimate of direction. This method is suboptimal if the data were not generated by cosine tuning functions as in the case illustrated in figure I-A. It is possible to obtain optimum performance by fitting the curve that was actually used to generate the data, i.e, the actual tuning curves of the units. A maximum likelihood estimate, defined as being the direction maximizing p(AIO), involves exactly this type of curve fitting, a process illustrated in figure 1-B [5]. The estimate is computed by finding first the "expected" hill- the hill that would be obtained in a noise free system- minimizing the distance with the data. In the case of gaussian noise, the appropriate distance measure to minimize is the euclidian squared distance. The final position of the peak of the hill corresponds to the maximum likelihood estimate, OML. 3 Recurrent Networks Consider a circular network of 64 units fully connected like the one depicted in figure 2-A. With an appropriate choice of weights and activation function, this network will develop a hill-shaped pattern of activity in response to a transient input as illustrated in figure 2-B. If we initialize this networks with activity patterns A = {ad corresponding to the responses of 64 direction tuned units (figure 1), we can use the final position of the hill across the neuronal array after relaxation as an estimate of the direction, O. The variance of this estimator will depend on the exact choice of activation function and weights. 3.1 Linear Network We first consider a network of 64 units whose dynamics is governed by the following difference equation: (1) The dynamics of such networks is well understood [3]. If each unit receives the same weight vector 'Iii, then the weight matrix W is symmetric. In this case, the 100 A. Pouget and K. Zhang network dynamics amplifies or suppresses the Fourier component of the initial input pattern, {ad, independently by a factors equal to the corresponding component of the Fourier transform, ;];, of w. For example, if the first component of;]; is more than one (resp. less than one) the first Fourier component of the initial pattern of activity will be amplified (resp. suppressed). Thus, we can choose W such that the network amplifies selectively the first Fourier component of the data while suppressing the others. The network would be unstable but if we stop after a large, yet fixed, number of iterations, the activity pattern would look like a cosine function of direction with a phase corresponding to the phase of the first Fourier components of the data. In other words, the network would end up fitting a cosine function in the data which is equivalent to the CaMP method described above. A network for orientation selectivity proposed by Ben-Yishai et al [1] is closely related to this linear network. Although this method keeps the estimate in a coarse code format, it suffers two problems: it is unclear how it could be extended to non periodic variables, such as disparity, and it is suboptimal since it is equivalent to the CaMP estimator. 3.2 Non-Linear Network We consider next a network of 64 units fully connected whose dynamics is governed by the following difference equations: Oi(t) = g( Ui(t» = 6.3 (log ( 1 + e5+1 0U,(t») ) 0.8 (2) u,( t Ht) = u, (t) Ht ( -u,( t) + t, W'jOj (t) ) (3) Zhang (1996) has demonstrated that with appropriate symmetric weights, {Wij}, this network develops a stable hill of activity in response to an arbitrary transient input pattern {Id(figure 2-B). The shape of the hill is fully specified by the weights and activation function whereas, by contrast, the final position of the hill on the neuronal array depends only on the initial input. Therefore, like ML, the network fits an "expected" function through the data. We first present a set of simulations in which we investigated whether ML and the network place the hill at the same location. Methods: The simulations consisted estimating the value of the direction of a moving bar based on the activity, A = {ad, of 64 input units with hill-shaped tuning to direction corrupted by noise. We used circular normal functions like the ones showed in figure I-A to model the mean activities, fiCO): fiCO) = 3exp(7(cos(O - Od - 1» + 0.3 (4) The value 0.3 corresponds to the mean spontaneous activity of each unit. The peak, OJ, of the circular normal functions were uniformly spread over the interval [0°,360°]. The activities, {ad, depended on the noise distribution. We used two types of noise, normally distributed with fixed variance, O'~ = 1 and Poisson distributed: 1 ( (a - f'(e»2) P(ai = ale) = exp ' , J27r0'2 20'2 n n 1.(O)k -f,(9) P(ai = kle) = J, :! (5) Our results compare the standard deviation offour estimators, OLE, COM, CaMP and ML to the non-linear recurrent network (RN) with transient inputs (the input patterns are shown on the first iteration only). In the case of ML, we used the Statistically Efficient Estimations Using Cortical Lateral Connections 101 Noise with Normal Distribution Noise with Poisson Distribution OLE COM COMP ML RN Figure 3: Histogram of the standard deviations of the estimate for all five methods Cramer-Rao bound to compute the standard deviation as described in Seung and Sompolinsky (1993). The weights in the recurrent network were chosen such that the final pattern of activity in the network have a profile very similar to the tuning function fi(O). Results: Since the preferred direction of two consecutive units in the network are more than 50 apart, we first wonder whether RN estimates exhibit a biasa difference between the mean estimate and the true direction- in particular for directions between the peaks of two consecutive units. Our simulations showed no significant bias for any of the orientations tested (not shown). Next, we compared standard deviations of the estimates for all five methods and for the two types of noise. The RN method was found to outperform the OLE, COM and COMP estimators in both cases and to match the Cramer-Rao bound for gaussian noise (figure 3) as suggested by our analysis. For noise with Poisson distribution, the standard deviation for RN was only 0.3440 above ML (figure 3). We also estimated numerically -lJORN j lJai 19=1700, the derivative of the RN estimate with respect to the initial activity of each of 64 units for an orientation of 1700 • This derivative in the case of ML matches closely the derivative of the cell tuning curve, /,(0). In other words, in ML, units contribute to the estimate according to the amplitude of the derivative of the tuning curve. As shown in figure the same is true for RN, -lJORN jlJai 19=1700 matches closely the derivative of the units tuning curves. In contrast, the same derivatives for the COMP estimate, (dotted line), or the COM estimate, (dash-dotted line), do not match the profile of /'(0). In particular, units with preferred direction far away from 1700 , i.e. units whose activity is just noise, end up contributing to the final estimate, hindering the performance of the estimator. We also looked at the standard deviation of the RN as a function of time, i.e., the number of iterations. Reaching a stable state can take up to several hundred iterations which could make the RN method too slow for any practical purpose. We found however that the standard deviation decreases very rapidly over the first 5-6 iterations and reaches asymptotic values after around 20 iterations (figure 4-B). Therefore, there is no need to wait for a perfectly stable pattern of activity to obtain minimum standard deviation. Analysis: One way to determine which factors control the final position of the hill is to find a function, called a Lyapunov function, which is minimized over time by the network dynamics. Cohen and Grossberg (1983) have shown that network characterized by the dynamical equation above and in which the input pattern {sIi} 102 CD > A 1 , ~ 0.5 'c CD o , . ' : , , RN COMP COM u 01----'-" 16 ~ E -0.5 o z , " ' . -1L-__ ~ __ ~_'_ " _ ' ~--J' 100 200 300 Preferred Direction (deg) B A. Pouget and K. Zhang 20.-~--~--~-~--, OL-~--~-~--~~ o 20 40 60 Time (# of iterations) Figure 4: A- Comparison of g'(B) (solid line), -oO/oai!9=1700 for RN, CaMP and COM. All functions have been normalized to one. B- Standard deviation as a function of the number of iterations for RN. is clamped, minimizes a Lyapunov function of the form : (6) The last term is the dot product between the input pattern, {sIi }, and the current activity pattern, {g( Ui)}, on the neuronal array. Here s is simply a scaling factor for the input pattern. The dynamics of the network will therefore tend to minimize - Li Iig( ud, or equivalently, to maximize the overlap between the stable pattern and the input pattern. The other terms however are also dependent on Ii because the shape of the final stable activity profile depends on the input pattern. Therefore the network will settle into a compromise between maximizing overlap and getting the right profile given the clamped input. We can show however that, for small input (i.e., as the scaling factor s -+ 0), the dominant term in the Lyapunov function is the dot product. To see this, we consider the Taylor expansion of Lyapunov function L with respect to s. First, let {Ui } denote the profile of the stable activity {ud in the limit of zero input (s -+ 0), and then write the corresponding value of the Lyapunov function at zero input as La. Now keeping only the first-order terms of s in the Taylor expansion, we obtain: (7) This means that the dot product is the only first order term of s, and disturbances to the shape of the final activity profile contribute only to higher order terms of s, which are negligible when s is small. Notice that in the limit of zero input, the shape of the activity profile {Ui} is fixed, and the only thing unknown is its peak position. Because La is a constant, the global minimum of the Lyapunov function here should correspond to a peak position which maximizes the dot product. The difference between Ui and Ui is negligible for sufficiently small input because, by definition, Ui -+ Ui as s -+ O. Consequently, for small input, the network will converge to a solution maximizing primarily Li Iig( Ui), which is mathematically equivalent to minimizing the square distance between the input and the output pattern. Therefore, if we use an activity pattern, A = {ad, as the input to this network, the stable hill should have its peak at a position very close to the direction correStatistically Efficient Estimations Using Cortical Lateral Connections 103 sponding to the maximum likelihood estimate (under the assumption of gaussian noise), provided the network is not attracted into a local minimum of the Liapunov function. This result is valid when using a small clamped input but our simulations show that a transient input is sufficient to reach the Cramer-Rao bound. 4 Discussion Our results demonstrate that it is possible to perform efficient unbiased estimation with coarse coding using a neurally plausible architecture. Our model relies on lateral connections to implement a prior expectation on the profile of the activity patterns. As a consequence, units determine their activation according to their own input and the activity of their neighbors. This approach shows that one of the advantages of coarse code is to provide a representation which simplifies the problem of cleaning up uncorrelated noise within a neuronal population. Unlike OLE, COM and CaMP, the RN estimate is not the result of a voting process in which units vote from their preferred direction, (Ji. Instead, units turn out to contribute according to the derivatives of their tuning curves, If( (J), as in the case of ML. This feature allows the network to ignore background noise, that is to say, responses due to other factors beside the variable of interest. This property also predicts that discrimination of directions around the vertical (90°) would be most affected by shutting off the units tuned at 60° and 120°. This prediction is consistent with psychophysical experiments showing that discrimination around the vertical in human is affected by prior adaptation to orientations displaced from the vertical by ±300 [4]. Our approach can be readily extended to any other periodic sensory or motor variables. For non periodic variables such as the disparity of a line in an image, our network needs to be adapted since it currently relies on circular symmetrical weights. Simply unfolding the network will be sufficient to deal with values around the center of the interval under consideration, but more work is needed to deal with boundary values. We can also generalize this approach to arbitrary mapping between two coarse codes for variables x and y where y is a function of x. Indeed, a coarse code for x provides a set of radial basis functions of x which can be subsequently used to approximate arbitrary functions. It is even conceivable to use a similar approach for one-to-many mappings, a common situation in vision or robotics, by adapting our network such that several hills can coexist simultaneously. References [1] R. Ben-Yishai, R. L. Bar-Or, and H. Sompolinsky. Proc. Natl. Acad. Sci. USA, 92:3844-3848, 1995. [2] M. Cohen and S. Grossberg. IEEE Trans. SMC, 13:815-826, 1983. [3] M. Hirsch and S. Smale. Differential equations, dynamical systems and linear algebra. Academic Press, New York, 1974. [4] D. M. Regan and K. 1. Beverley. J. Opt. Soc. Am., 2:147-155, 1985. [5] H. S. Seung and H. Sompolinsky. Proc. Natl. Acad. Sci. USA, 90:10749-10753, 1993.	1996	140
1,196	Recursive algorithms for approximating probabilities in graphical models Tommi S. Jaakkola and Michael I. Jordan {tommi,jordan}Opsyche.mit.edu Department of Brain and Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 Abstract We develop a recursive node-elimination formalism for efficiently approximating large probabilistic networks. No constraints are set on the network topologies. Yet the formalism can be straightforwardly integrated with exact methods whenever they are/become applicable. The approximations we use are controlled: they maintain consistently upper and lower bounds on the desired quantities at all times. We show that Boltzmann machines, sigmoid belief networks, or any combination (i.e., chain graphs) can be handled within the same framework. The accuracy of the methods is verified experimentally. 1 Introduction Graphical models (see, e.g., Lauritzen 1996) provide a medium for rigorously embedding domain knowledge into network models. The structure in these graphical models embodies the qualitative assumptions about the independence relationships in the domain while the probability model attached to the graph permits a consistent computation of belief (or uncertainty) about the values of the variables in the network. The feasibility of performing this computation determines the ability to make inferences or to learn on the basis of observations. The standard framework for carrying out this computation consists of exact probabilistic methods (Lauritzen 1996). Such methods are nevertheless restricted to fairly small or sparsely connected networks and the use of approximate techniques is likely to be the rule for highly interconnected graphs of the kind studied in the neural network literature. There are several desiderata for methods that calculate approximations to posterior probabilities on graphs. Besides having to be (1) reasonably accurate and fast to compute, such techniques should yield (2) rigorous estimates of confidence about 488 T. S. Jaakkola and M. I. Jordan the attained results; this is especially important in many real-world applications (e.g., in medicine). Furthermore, a considerable gain in accuracy could be obtained from (3) the ability to use the techniques in conjunction with exact calculations whenever feasible. These goals have been addressed in the literature with varying degrees of success. For inference and learning in Boltzmann machines, for example, classical mean field approximations (Peterson & Anderson, 1987) address only the first goal. In the case of sigmoid belief networks (Neal 1992), partial solutions have been provided to the first two goals (Dayan et al. 1995; Saul et al. 1996; Jaakkola & Jordan 1996). The goal of integrating approximations with exact techniques has been introduced in the context of Boltzmann machines (Saul & Jordan 1996) but nevertheless leaving the solution to our second goal unattained. In this paper, we develop a recursive node-elimination formalism that meets all three objectives for a powerful class of networks known as chain graphs (see, e.g., Lauritzen 1996); the chain graphs we consider are of a restricted type but they nevertheless include Boltzmann machines and sigmoid belief networks as special cases. We start by deriving the recursive formalism for Boltzmann machines. The results are then generalized to sigmoid belief networks and the chain graphs. 2 Boltzmann machines We begin by considering Boltzmann machines with binary (Ojl) variables. We assume the joint probability distribution for the variables S = {SI,' .. , Sn} to be given by 1 Pn(Slh, J) = Zn(h, J) Bn(Slh, J) (1) where hand J are the vector of biases and weights respectively, and the Boltzmann factor B has the form (2) The partition function Zn(h, J) = Z=s Bn(Slh, J) normalizes the distribution. The Boltzmann distribution defined in this manner is tractable insofar as we are able to compute the partition function; indeed, all marginal distributions can be reduced to ratios of partition functions in different settings. We now turn to methods for computing the partition function. In special cases (e.g., trees or chains) the structure of the weight matrix Jij may allow us to employ exact methods for calculating Z. Although exact methods are not feasible in more generic networks, selective approximations may nevertheless restore their utility. The recursive framework we develop provides a general and straightforward methodology for combining approximate and exact techniques. The crux of our approach lies in obtaining variational bounds that allow the creation of recursive node-elimination formulas of the form 1 : Zn(h,J) < > C(h, J) Zn-l(h, J) (3) Such formulas are attractive for three main reasons: (1) a variable (or many at the same time) can be eliminated by merely transforming the model parameters (h and J); (2) the approximations involved in the elimination are controlled, i.e., they 1 Related schemes in the physics literature (renormalization group) are unsuitable here as they generally don't provide strict upper/lower bounds. Recursive Bounds for Graphical Models 489 consistently yield upper or lower bounds at each stage of the recursion; (3) most importantly, if the remaining (simplified) partition function Zn-l(h, j) allow the use of exact methods, the corresponding model parameters hand j can simply be passed on to such routines. Next we will consider how to obtain the bounds and outline their implications. Note that since the quantities of interest are predominantly ratios of partition functions, it is the combination of upper and lower bounds that is necessary to rigorously bound the target quantities. This applies to parameter estimation as well even if only a lower bound on likelihood of examples is used; such likelihood bound relies on both upper and lower bounds on partition functions. 2.1 Simple recursive factorizations We start by developing a lower bound recursion. Consider eliminating the variable Si: Zn(h,J) LBn(Slh,J) = L LBn(Slh,J) (4) S S\S, S, L (1 + eh,+L::, J,jSj)Bn_l(S \ Silh, J) (5) S\Si > L el',(h,+L:: j J,jSj)+H(I'.) Bn-l(S \ Silh, J) (6) S\S, el',h,+H(I',) L Bn- 1(S \ Si Ih, J) (7) S\S, el',hi+H (l'i) Zn-l(h, J) (8) where hi = hi + l'iJii for j =j:. i, H(·) is the binary entropy function and I'i are free parameters that we will refer to as "variational parameters." The variational bound introduced in eq. (6) can be verified by a direct maximization which recovers the original expression. This lower bound recursion bears a connection to mean field approximation and in particular to the structured mean field approximation studied by Saul and Jordan (1996).2 Each recursive elimination translates into an additional bound and therefore the approximation (lower bound) deteriorates with the number of such iterations. It is necessary, however, to continue with the recursion only to the extent that the prevailing partition function remains unwieldy to exact methods. Consequently, the problem becomes that of finding the variables the elimination of which would render the rest of the graph tractable. Figure 1 illustrates this objective. Note that the simple recursion does not change the connection matrix J for the remaining variables; thus, graphically, the operation translates into merely removing the variable. The above recursive procedure maintains a lower bound on the partition function that results from the variational representation introduced in eq. (6). For rigorous 2Each lower bound recursion can be shown to be equivalent to a mean field approximation of the eliminated variable(s). The structured mean field approach of Saul and Jordan (1996) suggests using exact methods for tractable substructures while mean field for the variables mediating these structures. Translated into our framework this amounts to eliminating the mediating variables through the recursive lower bound formula with a subsequent appeal to exact methods. The connection is limited to the lower bound. 490 T. S. Jaakkola and M. I. Jordan Figure 1: Enforcing tractable networks. Each variable in the graph can be removed (in any order) by adding the appropriate biases for the existing adjacent variables. The elimination of the dotted nodes reveals a simplified graph underneath. bounds we need an upper bound as well. In order to preserve the graphical interpretation of the lower bound, the upper bound should also be factorized. With this in mind, the bound of eq. (6) can be replaced with (9) where (10) for j > 0, ho = f(h i ), f(x) = log(1 + eX), and qj are variational parameters such that Lj qj = 1. The derivation of this bound can be found in appendix A. 2.2 Refined recursive bound If the (sub )network is densely or fully connected the simple recursive methods presented earlier can hardly uncover any useful structure. Thus a large number of recursive steps are needed before relying on exact methods and the accuracy of the overall bound is compromised. To improve the accuracy, we introduce a more sophisticated variational (upper) bound to replace the one in eq. (6). By denoting Xi = hi + Lj Jij Sj we have: 1 + eX. ::; ex./2+>.(x.)X~-F(>'.Xi) (11) The derivation and the functional forms of A(Xi) and F(A, Xi) are presented in appendix B. We note here, however, that the bound is exact whenever Xi = Xi. In terms of the recursion we obtain Zn(h,J) < eh./2+>.(x.)h~-F(>'.Xi) Zn-l(h, J) (12) where h· 3 hj + 2hjA(Xi)Jij + Jij/2 + A(Xi)Ji} (13) Jjk Jjk + 2A(Xi)JjJik (14) for j 1= k 1= i. Importantly and as shown in figure 2a, this refined recursion imposes (qualitatively) the proper structural changes on the remaining network: the variables adjacent to the eliminated (or marginalized) variable become connected. In other words, if Jij 1= 0 and hk 1= 0 then Jjk 1= 0 after the recursion. To substantiate the claim of improved accuracy we tested the refined upper bound recursion against the factorized lower bound recursion in random fully connected networks with 8 variables3 . The weights in these networks were chosen uniformly in the range [-d, d) and all the initial biases were set to zero. Figure 3a plots the relative errors in the log-partition function estimates for the two recursions as a 3The small network size was chosen to facilitate comparisons with exact results. Recursive Bounds for Graphical Models 491 a) :'~"""""""""' \ : t .~ " -. -- --- - --- .b) ( ............ ~~\_ .. J Figure 2: a) The graphical changes in the network following the refined recursion match those of proper marginalization. b) Example of a chain graph. The dotted ovals indicate the undirected clusters. 0.04.---~----.----~-----, 0.03 0.025 0.02 0.02 refined o~~~~~ __ ~-~-~-~~-·~ ____ M-40.015 0.D1 -0.02 0.005 ...... a) -0·040 0.25 0.5 0.75 b) 00 0.5 0.75 Figure 3: a) The mean relative errors in the log-partition function as a function of the scale of the random weights (uniform in [-d, dJ). Solid line: factorized lower bound recursion; dashed line: refined upper bound. b) Mean relative difference between the upper and lower bound recursions as a function of dJn/8, where n is the network size. Solid: n = 8; dashed: n = 64; dotdashed: n = 128. function of the scale d. Figure 3b reveals how the relative difference between the two bounds is affected by the network size. In the illustrated scale the size has little effect on the difference. We note that the difference is mainly due to the factorized lower bound recursion as is evident from Figure 3a. 3 Chain graphs and sigmoid belief networks The recursive bounds presented earlier can be carried over to chain graphs4. An example of a chain graph is given in figure 2b. The joint distribution for a chain graph can written as a product of conditional distributions for clusters of variables: Pn(SIJ) = II p(Sk Ipa[k], hk, Jk) (15) k where Sk = {SdiECk is the set of variables in cluster k. In our case, the conditional probabilities for each cluster are conditional Boltzmann distributions given by p(Sk I [k] hk Jk) = B(Sk Ih~, Jk) pa " Z(h~,Jk) (16) where the added complexity beyond that of ordinary Boltzmann machines is that the Boltzmann factors now include also outside cluster biases: [h~]i = hf + L Ji~ · out Sj (17) j~Ck 4While Boltzmann machines are undirected networks (interactions defined through potentials), sigmoid networks are directed models (constructed from conditional probabilities). Chain graphs contain both directed and undirected interactions. 492 T. S. Jaakkola and M. I. Jordan where the index i stays within the kth cluster. We note that sigmoid belief networks correspond to the special case where there is only single binary variable in each cluster; Boltzmann machines, on the other hand, have only one cluster. We now show that the recursive formalism can be extended to chain graphs. This is achieved by rewriting or bounding the conditional probabilities in terms of variational Boltzmann factors. Consequently, the joint distribution - being a product of the conditionals - will also be a Boltzmann factor. Computing likelihoods (marginals) from such a joint distribution amounts to calculating the value of a particular partition function and therefore reduces to the case considered earlier. It suffices to find variational Boltzmann factors that bound (or rerepresent in some cases) the cluster partition functions in the conditional probabilities. We observe first that in the factorized lower bound or in the refined upper bound recursions, the initial biases will appear in the resulting expressions either linearly or quadratically in the exponent5 . Since the initial biases for the clusters are of the form of eq. (17), the resulting expressions must be Boltzmann factors with respect to the variables outside the cluster. Thus, applying the recursive approximations to each cluster partition function yields an upper/lower bound in the form of a Boltzmann factor. Combining such bounds from each cluster finally gives upper/lower bounds for the joint distribution in terms of variational Boltzmann factors. We note that for sigmoid belief networks the Boltzmann factors bounding the joint distribution are in fact exact variational translations of the true joint distribution. To see this, let us denote Xi = L: Jij Sj + hi and use the variational forms, for example, from eq. (6) and (11): O'(Xi) = (1 + e- X • )-1 < el1•Xi - H (l1i) > eXi/2->'(Xi)X~+F(>.,Xi) (18) (19) where the sigmoid function 0'(.) is the inverse cluster partition function in this case. Both the variational forms are Boltzmann factors (at most quadratic in Xi in the exponent) and are exact if minimized/maximized with respect to the variational parameters. In sum, we have shown how the joint distribution for chain graphs can be bounded by (translated into) Boltzmann factors to which the recursive approximation formalism is again applicable. 4 Conclusion To reap the benefits of probabilistic formulations of network architectures, approximate methods are often unavoidable in real-world problems. We have developed a recursive node-elimination formalism for rigorously approximating intractable networks. The formalism applies to a large class of networks known as chain graphs and can be straightforwardly integrated with exact probabilistic calculations whenever they are applicable. Furthermore, the formalism provides rigorous upper and lower bounds on any desired quantity (e.g., the variable means) which is crucial in high risk application domains such as medicine. bThis follows from the linearity of the propagation rules for the biases, and the fact that the emerging prefactors are either linear or quadratic in the exponent. Recursive Bounds for Graphical Models 493 References P. Dayan, G. Hinton, R. Neal, and R. Zemel (1995). The Helmholtz machine. Neural Computation 7: 889-904. S. L. Lauritzen (1996). Graphical Models. Oxford: Oxford University Press. T . Jaakkola and M. Jordan (1996). Computing upper and lower bounds on likelihoods in intractable networks. To appear in Proceedings of the twelfth Conference on Uncertainty in Artificial Intelligence. R. Neal. Connectionist learning of belief networks (1992). Artificial Intelligence 56: 71-113. C. Peterson and J. R. Anderson (1987). A mean field theory learning algorithm for neural networks. Complex Systems 1: 995-1019. L. K. Saul, T. Jaakkola, and M. I. Jordan (1996). Mean field theory for sigmoid belief networks. lAIR 4: 61-76. L. Saul and M. Jordan (1996). Exploiting tractable substructures in intractable networks. To appear in Advances of Neural Information Processing Systems 8. MIT Press. A Factorized upper bound The bound follows from the convexity of f( x) = loge 1 + eX) and from an application of Jensen's inequality. Let fk(x) = I(x + hk) and note that Ik(X) has the same convexity properties as I. For any convex function Ik then we have (by Jensen's inequality) Ik (L.ilkiSi) = Ik (L.i% lki~i) :::; ~qilk (lkj~i) q) i q) (20) By rewriting Ik (Jk;jSj) = Si [/A: ( ~) - Ik(O)] + Ida) we get the desired result. B Refined upper bound To derive the upper bound consider first 1 + eX = exj2 + log(e- x / 2 + eX/ 2 ) (21) Now, g(x) = log(e-x / 2 + ex / 2 ) is a symmetric function of x and also a concave function of x 2 . Any tangent line for a concave function always remains above the function and so it also serves as an upper bound. Therefore we may bound g(x) by the tangents of g(.,fY) (due to the concavity in x 2). Thus where log(e- x/ 2 + eX / 2 ) < ag~v:) (x 2 - y) + g(.JY) (22) A(y) F(A, y) A(Y)X2 - F(A, y) (23) a ayg( .JY) A(y) y - g(.JY) (24) (25) The desired result now follows the change of variables: y = xlNote that the tangent bound is exact whenever Xi = x (a tangent defined at that point).	1996	141
1,197	Clustering via Concave Minimization P. S. Bradley and O. L. Mangasarian Computer Sciences Department University of Wisconsin 1210 West Dayton Street Madison, WI 53706 email: paulb@es.wise.edu, olvi@es.wise.edu Abstract w. N. Street Computer Science Department Oklahoma State University 205 Mathematical Sciences Stillwater, OK 74078 email: nstreet@es. okstate. edu The problem of assigning m points in the n-dimensional real space Rn to k clusters is formulated as that of determining k centers in Rn such that the sum of distances of each point to the nearest center is minimized. If a polyhedral distance is used, the problem can be formulated as that of minimizing a piecewise-linear concave function on a polyhedral set which is shown to be equivalent to a bilinear program: minimizing a bilinear function on a polyhedral set. A fast finite k-Median Algorithm consisting of solving few linear programs in closed form leads to a stationary point of the bilinear program. Computational testing on a number of realworld databases was carried out. On the Wisconsin Diagnostic Breast Cancer (WDBC) database, k-Median training set correctness was comparable to that of the k-Mean Algorithm, however its testing set correctness was better. Additionally, on the Wisconsin Prognostic Breast Cancer (WPBC) database, distinct and clinically important survival curves were extracted by the k-Median Algorithm, whereas the k-Mean Algorithm failed to obtain such distinct survival curves for the same database. 1 Introduction The unsupervised assignment of elements of a given set to groups or clusters of like points, is the objective of cluster analysis. There are many approaches to this problem, including statistical [9], machine learning [7], integer and mathematical programming [18,1]. In this paper we concentrate on a simple concave minimization formulation of the problem that leads to a finite and fast algorithm. Our point of Clustering via Concave Minimization 369 departure is the following explicit description of the problem: given m points in the n-dimensional real space Rn , and a fixed number k of clusters, determine k centers in Rn such that the sum of "distances" of each point to the nearest center is minimized. If the I-norm is used, the problem can be formulated as the minimization of a piecewise-linear concave function on a polyhedral set. This is a hard problem to solve because a local minimum is not necessarily a global minimum. However, by converting this problem to a bilinear program, a fast successive-linearization kMedian Algorithm terminates after a few linear programs (each explicitly solvable in closed form) at a point satisfying the minimum principle necessary optimality condition for the problem. Although there is no guarantee that such a point is a global solution to our original problem, numerical tests on five real-world databases indicate that the k-Median Algorithm is comparable to or better than the k-Mean Algorithm [18, 9, 8]. This may be due to the fact that outliers have less influence on the k-Median Algorithm which utilizes the I-norm distance. In contrast the kMean Algorithm uses squares of 2-norm distances to generate cluster centers which may be inaccurate if outliers are present. We also note that clustering algorithms based on statistical assumptions that minimize some function of scatter matrices do not appear to have convergence proofs [8, pp. 508-515]' however convergence to a partial optimal solution is given in [18] for k-Mean type algorithms. We outline now the contents of the paper. In Section 2, we formulate the clustering problem for a fixed number of clusters, as that of minimizing the sum of the I-norm distances of each point to the nearest cluster center. This piecewise-linear concave function minimization on a polyhedral set turns out to be equivalent to a bilinear program [3]. We use an effective linearization of the bilinear program proposed in [3, Algorithm 2.1] to solve our problem by solving a few linear programs. Because of the simple structure, these linear programs can be explicitly solved in closed form, thus leading to the finite k-Median Algorithm 2.3 below. In Section 3 we give computational results on five real-world databases. Section 4 concludes the paper. A word about our notation now. All vectors are column vectors unless otherwise specified. For a vector x E Rn, Xi, i = 1, ... ,n, will denote its components. The norm II . lip will denote the p norm, 1 ~ p ~ 00, while A E RTnxn will signify a real m x n matrix. For such a matrix, AT will denote the transpose, and Ai will denote row i. A vector of ones in a real space of arbitrary dimension will be denoted bye. 2 Clustering as Bilinear Programming Given a set A of m points in Rn represented by the matrix A E RTnxn and a number k of desired clusters, we formulate the clustering problem as follows. Find cluster centers Gl, e = 1, ... , k, in Rn such that the sum of the minima over e E {I, ... , k} of the I-norm distance between each point Ai, i = 1, ... , m, and the cluster centers Gl , e = 1, ... , k, is minimized. More specifically we need to solve the following mathematical program: Tn minimize L min { e T Dil} C ,D i=l l=l , ... ,k (1) subject to -Dil ~ AT - Gl ~ Dil' i = 1, ... ,m, e = 1, ... k Here Dil E Rn, is a dummy variable that bounds the components of the difference 370 P. S. Bradley, O. L. Mangasarian and W. N. Street AT - Ct between point AT and center Ct, and e is a vector of ones in Rn. Hence eT Dit bounds the I-norm distance between Ai and Ct. We note immediately that since the objective function of (1) is the sum of minima of k linear (and hence concave) functions, it is a piecewise-linear concave function [13, Corollary 4.1.14]. If the 2-norm or p-norm, p oF 1,00, is used, the objective function will be neither concave nor convex. Nevertheless, minimizing a piecewise-linear concave function on a polyhedral set is NP-hard, because the general linear complementarity problem, which is NP-complete [4], can be reduced to such a problem [11, Lemma 1]. Given this fact we try to look for effective methods for processing this problem. We propose reformulation of problem (1) as a bilinear program. Such reformulations have been very effective in computationally solving NP-complete linear complementarity problems [14] as well as other difficult machine learning [12] and optimization problems with equilibrium constraints [12]. In order to carry out this reformulation we need the following simple lemma. Lemma 2.1 Let a E Rk. Then min {at} = min {t altl ttl = 1, tt ~ 0, f = 1, ... , k} (2) 1<t<k tERk l=l t=1 Proof This essentially obvious result follows immediately upon writing the dual of the linear program appearing on the right-hand side of (2) which is Tl;{hlh:::; at, f = 1, . .. k} (3) Obviously, the maximum of this dual problem is h = minl<t<k {at}. By linear programming duality theory, this maximum equals the minimum of the primal linear program in the right hand side of (2). This establishes the equality of (2). 0 By defining a~ = eT Dit, i = 1, ... , m, f = 1, ... , k, Lemma 2.1 can be used to reformulate the clustering problem (1) as a bilinear program as follows. Proposition 2.2 Clustering as a Bilinear Program The clustering problem (1) is equivalent to the following bilinear program: minimize CtERn,DttERn ,TilER subject to E:'l E;=1 eT DitTit - Dil :::; AT - Cl :::; Dil' i = 1 ... ,m, f = 1, ... , k (4) E;=l Til = 1 Til ~ 0, i = 1, ... ,m, f = 1, ... , k Note that the constraints of (4) are uncoupled in the variables (C, D) and the variable T. Hence the Uncoupled Bilinear Program Algorithm UBPA [3, Algorithm 2.1] is applicable. Simply stated, this algorithm alternates between solving a linear program in the variable T and a linear program in the variables (C, D). The algorithm terminates in a finite number of iterations at a stationary point satisfying the minimum principle necessary optimality condition for problem (4) [3, Theorem 2.1]. We note however, because of the simple structure the bilinear program (4), the two linear programs can be solved explicitly in closed form. This leads to the following algorithmic implementation. Algorithm 2.3 k-Median Algorithm Given cf, ... ,ct at iteration j, compute cf+! , ... ,ct+! by the following two steps: Clustering via Concave Minimization 371 (a) Cluster Assignment: For each AT, i = 1, ... m, determine £( i) such that C1(i) is closest to AT in the 1-norm. '+1 (b) Cluster Center Update: For £ = 1, ... ,k choose Cj as a median of all AT assigned to CI. Stop when cI+ 1 = cl, £ = 1, ... , k. Although the k-Median Algorithm is similar to the k-Mean Algorithm wherein the 2-norm distance is used [18, 8, 9], it differs from it computationally, and theoretically. In fact, the underlying problem (1) of the k-Median Algorithm is a concave minimization on a polyhedral set while the corresponding problem for the p-norm, p"# 1, is: minimize C,D subject to L min IIDillip , l=I"",k .=1 (5) -Dil ~ AT - Cl ~ Dil' i = 1 ... , m, £ = 1, ... , k. This is not a concave minimization on a polyhedral set, because the minimum of a set of convex functions is not in general concave. The concave minimization problem of [18] is not in the original space of the problem variables, that is, the cluster center variables, (C, D), but merely in the space of variables T that assign points to clusters. We also note that the k-Mean Algorithm finds a stationary point not of problem (5) with p = 2, but of the same problem except that IIDill12 is replaced by IIDilll~. Without this squared distance term, the subproblem of the k-Mean Algorithm becomes the considerably harder Weber problem [17, 5] which locates a center in Rn closest in sum of Euclidean distances (not their squares!) to a finite set of given points. The Weber problem has no closed form solution. However, using the mean as a cluster center of points assigned to the cluster, minimizes the sum of the squares of the distances from the cluster center to the points. It is precisely the mean that is used in the k-Mean Algorithm subproblem. Because there is no guaranteed way to ensure global optimality of the solution obtained by either the k-Median or k-Mean Algorithms, different starting points can be used to initiate the algorithm. Random starting cluster centers or some other heuristic can be used such as placing k initial centers along the coordinate axes at densest, second densest, ... , k densest intervals on the axes. 3 Computational Results An important computational issue is how to measure the correctness of the results obtained by the proposed algorithm. We decided on the following three ways. Remark 3.1 Training Set Correctness The k-Median algorithm (k = 2) is applied to a database with two known classes to obtain centers. Training correctness is measured by the ratio of the sum of the number examples of the majority class in each cluster to the total number of points in the database. The k-Median training set correctness is compared to that of the k-Mean Algorithm as well as the training correctness of a supervised learning method, a perceptron trained by robust linear programming [2l. Table 1 shows results averaged over ten random starts for the 372 P. S. Bradley, O. L. Mangasarian and W. N. Street publicly available Wisconsin Diagnostic Breast Cancer (WDBC) database as well as three others [15, 16). We note that for two of the databases k-Median outperformed k-Mean, and for the other two k-Mean was better. Algorithm .J.. Database -t WDBC Cleveland Votes Star / Galaxy-Bright Unsupervised k-Median 93.2% 80.6% 84.6% 87.6% Unsupervised k-Mean 91.1% 83.1% 85.5% 85.6% Supervised Robust LP 100% 86.5% 95.6% 99.7% Table 1 Training set correctness using the unsupervised k-Median and k-Mean Algorithms and the supervised Robust LP on four databases Remark 3.2 Testing Set Correctness The idea behind this approach T eoIing Set Correctness vo. T eoIing Set Size 94 is that supervised learning may be costly due to problem size, difficulty in obtaining true classification, etc., hence ~~------------92 the importance of good per- I formance of an unsupervised ~ 90 learning algorithm on a test<1 ~ ing subset of a database. The '" :j88 ' WDBC database [15} is split ~ into training and testing sub... sets of different proportions. The k-Median and k-Mean Algorithms (k = 2) are applied to 86 84 o I I I I 10 -- k-Median k-Meen Robust lP 15 20 25 30 35 40 45 50 Testing Set Size (% 01 Original) the training subset. The centers are given class labels determined by the majority class of training subset points assigned to the cluster. Class labels are assigned to the testing subset by the label of the closFigure 1: Correctness on variable-size test set of unsupervised k-Median & k-Mean Algorithms versus correctness of the supervised Robust LP on WDBC est center. Testing correctness is determined by the number of points in testing subset correctly classified by this assignment. This is compared to the correctness of a supervised learning method, a perceptron trained via robust linear programming [2}, using the leave-one-out strategy applied to the testing subset only. This comparison is then carried out for various sizes of the testing subset. Figure 1 shows the results averaged over 50 runs for each of 7 testing subset sizes. As expected, the performance of the supervised learning algorithm (Robust LP) improved as the size of the testing subset increases. The k-Median Algorithm test set correctness remained fairly constant in the range of 92.3% to 93.5%, while the k-Mean Algorithm test set correctness was lower and more varied in the range 88.0% to 91.3%. Remark 3.3 Separability of Survival Curves In mining medical databases, survival curves [10} are important prognostic tools. We applied the k-Median and k-Mean (k = 3) Algorithms, as knowledge discovery in database (KDD) tools [6}, to the Wisconsin Prognostic Breast Cancer Database (WPBC) [15} using only two features: tumor size and lymph node status. Survival curves were constructed for Clustering via Concave Minimization 373 :f\,' '.. I~ 10.7 ,, '-------, j08 \ ··' ...... i \ 1 :: it .................... .. 0." 08 ~0 . 7 JOB 1:: .. ......... -. 03 03 02 02 0.1 0.1 ° '40 ° ~L-~ro~~40~~ro---M~~,OO~~'ro~~ 2C 40 60 80 100 120 UO "0'''',. Moo.,. (a) k-Median (b) k-Mean Figure 2: Survival curves for the 3 clusters obtained by k-Median and k-Mean Algorithms each cluster, representing expected percent of surviving patients as a function of time, for patients in that cluster. Figure 2( a) depicts the survival curves from clusters obtained from the k-Median Algorithm, Figure 2(b) depicts curves for the k-Mean Algorithm. The key observation to make here is that curves in Figure 2(a) are well separated, and hence the clusters can be used as prognostic indicators. In contrast, the curves in Figure 2(b) are poorly separated, and hence are not useful for prognosis. 4 Conclusion We have proposed a new approach for assigning points to clusters based on a simple concave minimization model. Although a global solution to the problem cannot be guaranteed, a finite and simple k-Median Algorithm quickly locates a very useful stationary point. Utility of the proposed algorithm lies in its ability to handle large databases and hence would be a useful tool for data mining. Comparing it with the k-Mean Algorithm, we have exhibited instances where the k-Median Algorithm is superior, and hence preferable. Further research is needed to pinpoint types of problems for which the k-Median Algorithm is best. 5 Acknowledgements Our colleague Jude Shavlik suggested the testing set strategy used in Remark 3.2. This research is supported by National Science Foundation Grants CCR-9322479 and National Institutes of Health !NRSA Fellowship 1 F32 CA 68690-01. References [1] K. AI-Sultan. A Tabu search approach to the clustering problem. Pattern Recognition, 28(9):1443-1451, 1995. 374 p. S. Bradley, 0. L Mangasarian and W N. Street [2] K. P. Bennett and O. L. Mangasarian. Robust linear programming discrimination of two linearly inseparable sets. Optimization Methods and Software, 1:23-34, 1992. [3] K. P. Bennett and O. L. Mangasarian. Bilinear separation of two sets in nspace. Computational Optimization £3 Applications, 2:207-227, 1993. [4] S.-J. Chung. NP-completeness of the linear complementarity problem. Journal of Optimization Theory and Applications, 60:393-399, 1989. [5] F. Cordellier and J. Ch. Fiorot. On the Fermat-Weber problem with convex cost functionals. Mathematical Programming, 14:295-311, 1978. [6] U. Fayyad, G. Piatetsky-Shapiro, and P. Smyth. The KDD process for extracting useful knowledge from volumes of data. Communications of the ACM, 39:27-34, 1996. [7] D. Fisher. Knowledge acquisition via incremental conceptual clustering. Machine Learning, 2:139-172, 1987. [8] K. Fukunaga. Statistical Pattern Recognition. Academic Press, NY, 1990. [9] A. K. Jain and R. C. Dubes. Algorithms for Clustering Data. Prentice-Hall, Inc, Englewood Cliffs, NJ, 1988. [10] E. L. Kaplan and P. Meier. Nonparametric estimation from incomplete observations. J. Am. Stat. Assoc., 53:457-481, 1958. [11] O. L. Mangasarian. Characterization of linear complementarity problems as linear programs. Mathematical Programming Study, 7:74-87, 1978. [12] O. L. Mangasarian. Misclassification minimization. Journal of Global Optimization, 5:309-323, 1994. [13] O. L. Mangasarian. Nonlinear Programming. SIAM, Philadelphia, PA, 1994. [14J O. L. Mangasarian. The linear complementarity problem as a separable bilinear program. Journal of Global Optimization, 6:153-161, 1995. [15] P. M. Murphy and D. W. Aha. UCI repository of machine learning databases. Department of Information and Computer Science, University of California, Irvine, www.ics.uci.edu/AI/ML/MLDBRepository.html, 1992. [16] S. Odewahn, E. Stockwell, R. Pennington, R. Hummphreys, and W. Zumach. Automated star/galaxy discrimination with neural networks. Astronomical Journal, 103(1):318-331, 1992. [17] M. L. Overton. A quadratically convergent method for minimizing a sum of euclidean norms. Mathematical Programming, 27:34-63, 1983. [18J S. Z. Selim and M. A. Ismail. K-Means-Type algorithms: a generalized convergence theorem and characterization of local optimality. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-6:81-87, 1984.	1996	142
1,198	Balancing between bagging and bumping Tom Heskes RWCP Novel Functions SNN Laboratory; University of Nijmegen Geert Grooteplein 21, 6525 EZ Nijmegen, The Netherlands tom@mbfys.kun.nl Abstract We compare different methods to combine predictions from neural networks trained on different bootstrap samples of a regression problem. One of these methods, introduced in [6] and which we here call balancing, is based on the analysis of the ensemble generalization error into an ambiguity term and a term incorporating generalization performances of individual networks. We show how to estimate these individual errors from the residuals on validation patterns. Weighting factors for the different networks follow from a quadratic programming problem. On a real-world problem concerning the prediction of sales figures and on the well-known Boston housing data set, balancing clearly outperforms other recently proposed alternatives as bagging [1] and bumping [8]. 1 EARLY STOPPING AND BOOTSTRAPPING Stopped training is a popular strategy to prevent overfitting in neural networks. The complete data set is split up into a training and a validation set. Through learning the weights are adapted in order to minimize the error on the training data. Training is stopped when the error on the validation data starts increasing. The final network depends on the accidental subdivision in training and validation set, and often also on the, usually random, initial weight configuration and chosen minimization procedure. In other words, early stopped neural networks are highly unstable: small changes in the data or different initial conditions can produce large changes in the estimate. As argued in [1, 8], with unstable estimators it is advisable to resample, i.e., to apply the same procedure several times using different subdivisions in training and validation set and perhaps starting from different initial RWCP: Real World Computing Partnership; SNN: Foundation for Neural Networks. Balancing Between Bagging and Bumping 467 configurations. In the neural network literature resampling is often referred to as training ensembles of neural networks [3, 6]. In this paper, we will discuss methods for combining the outputs of networks obtained through such a repetitive procedure. First, however, we have to choose how to generate the subdivisions in training and validation sets. Options are, among others, k-fold cross-validation, subsampling and bootstrapping. In this paper we will consider bootstrapping [2] which is based on the idea that the available data set is nothing but a particular realization of some probability distribution. In principle, one would like to do inference on this "true" yet unknown probability distribution. A natural thing to do is then to define an empirical distribution. With so-called naive bootstrapping the empirical distribution is a sum of delta peaks on the available data points, each with probability content l/Pdata with Pdata the number of patterns. A bootstrap sample is a collection of Pdata patterns drawn with replacement from this empirical probability distribution. Some of the data points will occur once, some twice and some even more than twice in this bootstrap sample. The bootstrap sample is taken to be the training set, all patterns that do not occur in a particular bootstrap sample constitute the validation set. For large Pdata, the probability that a pattern becomes part of the validation set is (1 - l/Pdata)Pda.ta. ~ l/e ~ 0.368. An advantage of bootstrapping over other resampling techniques is that most statistical theory on resampling is nowadays based on the bootstrap. Using naive bootstrapping we generate nrun training and validation sets out of our complete data set of Pdata input-output combinations {iI', tl'}. In this paper we will restrict ourselves to regression problems with, for notational convenience, just one output variable. We keep track of a matrix with components q; indicating whether pattern p is part of the validation set for run i (q; = 1) or of the training set (qf = 0). On each subdivision we train and stop a neural network with one layer of nhidden hidden units. The output or of network i with weight vector w( i) on input il' reads o~ I + wo(i) , where we use the definition x~ == 1. The validation error for run i can be written 1 Pda.ta. Evalidation(i) == -:- L qrr; , PI /.'=1 with Pi == L:/.' qf ~ 0.368 Pdata, the number of validation patterns m run z, and r; == (or - ttl)2/2, the error of network i on pattern p. After training we are left with nrun networks, with, in practice, quite different performances on the complete data set. How should we combine all these outputs to get the best possible performance on new data? 2 COMBINING ESTIMATORS Several methods have been proposed to combine estimators (see e.g. (5) for a review). In this paper we will only consider estimators with the same architecture 468 T. Heskes but trained and stopped on different subdivisions of the data in training and validation sets. Recently, two such methods have been suggested for bootstrapped estimators: bagging [1], an acronym for bootstrap aggregating, and bumping [8], meaning bootstrap umbrella of model parameters. With bagging, the prediction on a newly arriving input vector is the average over all network predictions. Bagging completely disregards the performance of the individual networks on the data used for training and stopping. Bumping, on the other hand, throws away all networks except the one with the lowest error on the complete data set 1 • In the following we will describe an intermediate form due to [6], which we here call balancing. A theoretical analysis of the implications of this idea can be found in [7]. Suppose that after training we receive a new set of Ptest test patterns for which we do not know the true targets [II, but can calculate the network output OJ for each network i. We give each network a weighting factor aj and define the prediction of all networks on pattern 1/ as the weighted average nrun -II _ ~ -II m = L- ajOi . i=1 The goal is to find the weighting factors aj, subject to the constraints nrun L aj = 1 and aj ~ 0 Vj , j=1 yielding the smallest possible generalization error 1 Ptest E ~ ( - II t-II) 2 test = -- L- m . Ptest 11:1 (1) The problem, of course, is our ignorance about the targets [II. Bagging simply takes ai = l/nrun for all networks, whereas bumping implies aj = din. with 1 Pd .. t .. K. argmin -- L (or - t JJ )2 . j Pdata JJ=1 As in [6, 7] we write the generalization error in the form E test _1_ L L ajaj(or - [1I)(oj - [II) Ptest .. II I,) 2p1 L L ajaj [(or - [11)2 + (oj - ill)2 - (or - oj)2] test II j ,j L ajaj [Etest(i) + Etest(j) - ~ L(or - 5j )2]. (2) . . Ptest IJ II The last term depends only on the network outputs and can thus be calculated. This "ambiguity" term favors networks with conflicting outputs. The first part, lThe idea behind bumping is more general and involved than discussed here. The interested reader is referred to [8]. In this paper we will only consider its naive version. Balancing Between Bagging and Bumping 469 containing the generalization errors Etest(i) for individual networks, depends on the targets tV and is thus unknown. It favors networks that by themselves already have a low generalization error. In the next section we will find reasonable estimates for these generalization errors based on the network performances on validation data. Once we have obtained these estimates, finding the optimal weighting factors Cti under the constraints (1) is a straightforward quadratic programming problem. 3 ESTIMATING THE GENERALIZATION ERROR At first sight, a good estimate for the generalization error of network i could be the performance on the validation data not included during training. However, the validation error Evalidation (i) strongly depends on the accidental subdivision in training and validation set. For example, if there are a few outliers which, by pure coincidence, are part of the validation set, the validation error will be relatively large and the training error relatively small. To correct for this bias as a result of the random subdivision, we introduce the "expected" validation error for run i. First we define nil as the number of runs in which pattern J.l is part of the validation set and E~alidation as the error averaged over these runs: nrun 1 nrun nil == L qf and E~alidation == nil ?= qf rf , i=1 .=1 The expected validation error then follows from , 1 Pda.ta. Evalidation (i) == --:- L qf E~alidation . P. 11=1 The ratio between the observed and the expected validation error indicates whether the validation error for network i is relatively high or low. Our estimate for the generalization error of network i is this ratio multiplied by an overall scaling factor being the estimated average generalization error: E (.) 1 Pda.ta. E (.) ~ validation t __ '"" Ell. . test t , . ~ validation' Evalidation (t) Pdata 11=1 Note that we implicitly make the assumption that the bias introduced by stopping at the minimal error on the validation patterns is negligible, i.e., that the validation patterns used for stopping a network can be considered as new to this network as the completely independent test patterns. 4 SIMULATIONS We compare the following methods for combining neural network outputs. Individual: the average individual generalization error, i.e., the generalization error we will get on average when we decide to perform only one run. It serves as a reference with which the other methods will be compared. Bumping: the generalization of the network with the lowest error on the data available for training and stopping. 470 T. Heskes unfair unfair bumping bagging ambiguity balancing bumping balancing store 1 4% 9% 10% 17 % 17 % 24 % store 2 5% 15 % 22 % 23 % 23 % 34 % store 3 -7 % 11% 18 % 25 % 25 % 36 % store 4 6% 11% 17 % 26 % 26 % 31 % store 5 6% 10% 22 % 19 % 22 % 26 % store 6 1% 8% 14 % 19 % 16 % 26 % mean 3% ) 11% 17 % 22 % 22 % 30 % Table 1: Decrease in generalization error relative to the average individual generalization error as a result of several methods for combining neural networks trained to predict the sales figures for several stores. Bagging: the generalization error when we take the average of all n run network outputs as our prediction. Ambiguity: the generalization error when the weighting factors are chosen to maximize the ambiguity, i.e., taking identical estimates for the individual generalization errors of all networks in expression (2). Balancing: the generalization error when the weighting factors are chosen to minimize our estimate of the generalization error. Unfair bumping: the smallest generalization error for an individual error, i.e., the result of bumping if we had indeed chosen the network with the smallest generalization error. Unfair balancing: the lowest possible generalization error that we could obtain if we had perfect estimates of the individual generalization errors. The last two methods, unfair bumping and unfair balancing, only serve as some kind of reference and can never be used in practice. We applied these methods on a real-world problem concerning the prediction of sales figures for several department stores in the Netherlands. For each store, 100 networks with 4 hidden units were trained and stopped on bootstrap samples of about 500 patterns. The test set, on which the performances of the various methods for combination were measured, consists of about 100 patterns. Inputs include weather conditions, day of the week, previous sales figures, and season. The results are summarized in Table 1, where we give the decrease in the generalization error relative to the average individual generalization error. As can be seen in Table 1, bumping hardly improves the performance. The reason is that the error on the data used for training and stopping is a lousy predictor of the generalization error, since some amount of overfitting is inevitable. The generalization performance obtained through bagging, i.e., first averaging over all outputs, can be pro"en to be always better than the average individual generalization error. Balancing Between Bagging and Bumping 80r---~----~--~----~-' E Q) E 60 ~ E .§40 Q) C> ~ 20 > «I O~--~----~----~--~~ o 20 40 60 80 number of replicates 471 ~ 30 r-------..-------.-----~---~......, E ~ 25 -lIE E lI('lIE- ""*- __ • - - a. .s 20 20 40 60 number of replicates 80 Figure 1: Decrease of generalization error relative to the average individual generalization error as a function of the number of bootstrap replicates for different combination methods: bagging (dashdot, star), ambiguity (dotted, star), bumping (dashed, star), balancing (solid, star) , unfair bumping (dashed, circle), unfair balancing (solid, circle). Shown are the mean (left) and the standard deviation (right) of the decrease in percentages. Networks are trained and tested on the Boston housing database. On these data bagging is definitely better than bumping, but also worse than maximizing the ambiguity. In all cases, except for store 5 where maximization of the ambiguity is slightly better, balancing is a clear winner among the "fair" methods. The last column in Table 1 shows how much better we can get if we could find more accurate estimates for the generalization errors of individual networks. The method of balancing discards most of the networks, i.e., the solution to the quadratic programming problem (2) under constraints (1) yields just a few weighting factors different from zero (on average about 8 for this set of simulations). Balancing is thus indeed a compromise between bagging, taking all networks into acount, and bumping, keeping just one network. We also compared these methods on the well-known Boston housing data set concerning the median housing price in several tracts based on 13 mainly socio-economic predictor variables (see e.g. [1] for more information). We left out 50 of the 506 available cases for assessment of the generalization performance. All other 456 cases were used for training and stopping neural networks with 4 hidden units. The average individual mean squared error over all 300 bootstrap runs is 16.2, which is comparable to the mean squared error reported in [1]. To study how the performance depends on the number of bootstrap replicates, we randomly drew sets of n = 5,10,20,40 and 80 bootstrap replicates out of our ensemble of 300 replicates and applied the combination methods on these sets. For each n we did this 48 times. Figure 1 shows the mean decrease in the generalization error relative to the average individual generalization error and its standard deviation. Again, balancing comes out best, especially for a larger number of bootstrap replicates. It seems that beyond say 20 replicates both bumping and bagging are hardly helped by more runs, whereas both maximization of the ambiguity and balancing still increase their performance. Bagging, fully taking into account all network pre472 T. Heskes dictions, yields the smallest variation, bumping, keeping just one of them, by far the largest. Balancing and maximization of the ambiguity combine several predictions and thus yield a variation that is somewhere in between. 5 CONCLUSION AND DISCUSSION Balancing, a compromise between bagging and bumping, is an attempt to arrive at better performances on regression problems. The crux in all this is to obtain reasonable estimates for the quality of the different networks and to incorporate these estimates in the calculation of the proper weighting factors (see [5, 9] for similar ideas and related work in the context of stacked generalization). Obtaining several estimators is computationally expensive. However, the notorious instability offeedforward neural networks hardly leaves us a choice. Furthermore, an ensemble of bootstrapped neural networks can also be used to deduce (approximate) confidence and prediction intervals (see e.g. [4]), to estimate the relevance of input fields and so on. It has also been argued that combination of several estimators destroys the structure that may be present in a single estimator [8]. Having hardly any interpretable structure, neural networks do not seem to have a lot they can lose. It is a challenge to show that an ensemble of neural networks does not only give more accurate predictions, but also reveals more information than a single network. References [1] L. Breiman. Bagging predictors. Machine Learning, 24:123-140, 1996. [2] B. Efron and R. Tibshirani. An Introduction to the Bootstrap. Chapman & Hall, London, 1993. [3] L. Hansen and P. Salomon. Neural network ensembles. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12:993-1001, 1990. [4] T. Heskes. Practical confidence and prediction intervals. These proceedings, 1997. [5] R. Jacobs. Methods for combining experts' probability assessments. Neural Computation, 7:867-888, 1995. [6] A. Krogh and J. Vedelsby. Neural network ensembles, cross validation, and active learning. In G. Tesauro, D. Touretzky, and T. Leen, editors, Advances in Neural Information Processing Systems 7, pages 231-238, Cambridge, 1995. MIT Press. [7] P. Sollich and A. Krogh. Learning with ensembles: How over-fitting can be useful. In D. Touretzky, M. Mozer, and M. Hasselmo, editors, Advances in Neural Information Processing Systems 8, pages 190-196, San Mateo, 1996. Morgan Kaufmann. [8] R. Tibshirani and K. Knight. Model search and inference by bootstrap "bumping". Technical report, University of Toronto, 1995. [9] D. Wolpert and W. Macready. Combining stacking with bagging to improve a learning algorithm. Technical report, Santa Fe Institute, Santa Fe, 1996.	1996	143
1,199	Self-Organizing and Adaptive Algorithms for Generalized Eigen-Decomposition Chanchal Chatterjee Newport Corporation 1791 Deere Avenue, Irvine, CA 92606 Vwani P. Roychowdhury Electrical Engineering Department UCLA, Los Angeles, CA 90095 ABSTRACT The paper is developed in two parts where we discuss a new approach to self-organization in a single-layer linear feed-forward network. First, two novel algorithms for self-organization are derived from a two-layer linear hetero-associative network performing a one-of-m classification, and trained with the constrained least-mean-squared classification error criterion. Second, two adaptive algorithms are derived from these selforganizing procedures to compute the principal generalized eigenvectors of two correlation matrices from two sequences of random vectors. These novel adaptive algorithms can be implemented in a single-layer linear feed-forward network. We give a rigorous convergence analysis of the adaptive algorithms by using stochastic approximation theory. As an example, we consider a problem of online signal detection in digital mobile communications. 1. INTRODUCTION We study the problems of hetero-associative trammg, linear discriminant analysis, generalized eigen-decomposition and their theoretical connections. The paper is divided into two parts. In the first part, we study the relations between hetero-associative training with a linear feed-forward network, and feature extraction by the linear discriminant analysis (LOA) criterion. Here we derive two novel algorithms that unify the two problems. In the second part, we generalize the self-organizing algorithm for LOA to obtain adaptive algorithms for generalized eigen-decomposition, for which we provide a rigorous proof of convergence by using stochastic approximation theory. 1.1 HETERO-ASSOCIATION AND LINEAR DISCRIMINANT ANALYSIS In this discussion, we consider a special case of hetero-association that deals with the classification problems. Here the inputs belong to a finite m-set of pattern classes, and the Self-Organizing and Adaptive Generalized Eigen-Decomposition 397 outputs indicate the classes to which the inputs belong. Usually, the ith standard basis vector ei is chosen to indicate that a particular input vector x belongs to class i. The LDA problem, on the other hand, aims at projecting a multi-class data in a lower dimensional subspace such that it is grouped into well-separated clusters for the m classes. The method is based upon a set of scatter matrices commonly known as the mixture scatter Sm and between class scatter Sb (Fukunaga, 1990). These matrices are used to formulate criteria such as tr(Sm-ISb) and det(Sb)1 det(Sm) which yield a linear transform <1> that satisfy the generalized eigenvector problem Sb<1>=Sm<1>A, where A is the generalized eigenvalue matrix. If Sm is positive definite, we obtain a <1> such that <1>TSm<1> =1 and <1>TSb<1>=A. Furthermore, the significance of each eigenvector (for class separability) is determined by the corresponding generalized eigenvalue. A relation between hetero-association and LDA was demonstrated by Gallinari et al. (1991). Their work made explicit that for a linear multi-layer perceptron performing a one-from-m classification that minimized the total mean square error (MSE) at the network output, also maximized a criterion det(Sb)/det(Sm) for LDA at the final hidden layer. This study was generalized by Webb and Lowe (1990) by using a nonlinear transform from the input data to the final hidden units, and a linear transform in the final layer. This has been further generalized by Chatterjee and Roychowdhury (1996) by including the Bayes cost for misclassification into the criteria tr(Sm -ISb). Although the above studies offer useful insights into the relations between heteroassociation and LDA, they do not suggest an algorithm to extract the optimal LDA transform <1>. Since the criteria for class separability are insensitive to multiplication by nonsingular matrices, the above studies suggest that any training procedure that minimizes the MSE at the network output will yield a nonsingular transformation of <1>; i.e., we obtain Q<1> where Q is a nonsingular matrix. Since Q<1> does not satisfy the generalized eigenvector problem Sb<1>=Sm<1>A for any arbitrary nonsingular matrix Q, we need to determine an algorithm that will yield Q=I. In order to obtain the optimum linear transform <1>, we constrain the training of a twolayer linear feed-forward network, such that at convergence, the weights for the first layer simultaneously diagonalizes Sm and Sb. Thus, the hetero-associative network is trained by minimizing a constrained MSE at the network output. This training procedure yields two novel algorithms for LDA. 1.2 LDA AND GENERALIZED EIGEN-DECOMPOSITION Since the LDA problem is a generalized eigen-decomposition problem for the symmetric-definite case, the self-organizing algorithms derived from the heteroassociative networks lead us to construct adaptive algorithms for generalized eigendecomposition. Such adaptive algorithms are required in several applications of image and signal processing. As an example, we consider the problem of online interference cancellation in digital mobile communications. Similar to the LDA problem Sb<1>=Sm<1>A, the generalized eigen-decomposition problem A<1>=B<1>A involves the matrix pencil (A ,B), where A and B are assumed to be real, symmetric and positive definite. Although a solution to the problem can be obtained by a conventional method, there are several applications in image and signal processing where an online solution of generalized eigen-decomposition is desired. In these real-time situations, the matrices A and B are themselves unknown. Instead, there are available two 398 C. Chatterjee and V. P Roychowdhury sequences of random vectors {xk} and {Yk} with limk~ooE[x~/J =A and limk~oo E[Yky/'I=B, where xk and Yk represent the online observations of the application. For every sample (x/C>Yk), we need to obtain the current estimates <1>k and Ak of <1> and A respectively, such that <1>k and Ak converge strongly to their true values. The conventional approach for evaluating <1> and A requires the computation of (A,B) after collecting all of the samples, and then the application of a numerical procedure; i.e., the approach works in a batch fashion. There are two problems with this approach. Firstly, the dimension of the samples may be large so that even if all of the samples are available, performing the generalized eigen-decomposition may take prohibitively large amount of computational time. Secondly, the conventional schemes can not adapt to slow or small changes in the data. So the approach is not suitable for real-time applications where the samples come in an online fashion. Although the adaptive generalized eigen-decomposition algorithms are natural generalizations of the self-organizing algorithms for LDA, their derivations do not constitute a proof of convergence. We, therefore, give a rigorous proof of convergence by stochastic approximation theory, that shows that the estimates obtained from our adaptive algorithms converge with probability one to the generalized eigenvectors. In summary, the study offers the following contributions: (1) we present two novel algorithms that unify the problems of hetero-associative training and LDA feature extraction; and (2) we discuss two single-stage adaptive algorithms for generalized eigendecomposition from two sequences of random vectors. In our experiments, we consider an example of online interference cancellation in digital mobile communications. In this problem, the signal from a desired user at a far distance from the receiver is corrupted by another user very near to the base. The optimum linear transform w for weighting the signal is the first principal generalized eigenvector of the signal correlation matrix with respect to the interference correlation matrix. Experiments with our algorithm suggest a rapid convergence within four bits of transmitted signal, and provides a significant advantage over many current methods. 2. HETERO-ASSOCIATIVE TRAINING AND LDA We consider a two-layer linear network performing a one-from-m classification. Let XE 9tn be an input to the network to be classified into one out of m classes ro l'''''rom. If x E ro j then the desired output d=ej (ith std. basis vector). Without loss of generality, we assume the inputs to be a zero-mean stationary process with a nonsingular covariance matrix. 2.1 EXTRACTING THE PRINCIPAL LDA COMPONENTS In the two-layer linear hetero-associative network, let there be p neurons in the hidden layer, and m output units. The aim is to develop an algorithm so that indi",idual weight vectors for the first layer converge to the first p~m generalized eigenvectors corresponding to the p significant generalized eigenvalues arranged in decreasing order. Let WjE9tn (i=I, ... ,n) be the weight vectors for the input layer, and VjE9tm (i=I, ... ,m) be the weight vectors for the output layer. The neurons are trained sequentially; i.e., the training of the jlh neuron is started only after the weight vector of the (j_I)fh neuron has converged. Assume that all the j-I previous neurons have already been trained and their weights have converged to the Self-Organizing and Adaptive Generalized Eigen-Decomposition 399 optimal weight vectors wi for i E (1 J-l]. To extract the J'h generalized eigenvector in the output of the /h neuron, the updating model for this neuron should be constructed by subtracting the results from all previously computed j-I generalized eigenvectors from the desired output dj as below j-I T d j = d j - L v i W i x. (1) i=1 This process is equivalent to the deflation of the desired output. The scatter matrices Sm and Sb can be obtained from x and d as Sm=E[xxT] and Sb= MMT, where M=E[xd1). We need to extract the j1h LOA transform Wj that satisfies the generalized eigenvector equation SbWj=AlmWj such that Aj is the J'h largest generalized eigenvalue. The constrained MSE criterion at the network output is Jh,Vj )=,lldj <~:v;wT x-vjWJxr]+ p{wJSmw j -I). (2) Using (2), we obtain the update equation for Wj as w(J) = w(J) + {Mv(J) - S w(J)(w(J)T Mv(J»)- S j~1 w(J)v(i)T v(J») hI k k m k k k m L.. k k k . (3) ;=1 Differentiating (2) with respect to vi' and equating it to zero, we obtain the optimum value ofvj as MTWj. Substituting this Vj in (3) we obtain w(J) = w(J) + {s w(J) - S w(J)(w(J)T S w(J») - S j~1 wU)w(i)TS w(J») k+1 k b k m k k b k m L.. k k b k . (4) i=1 Let Wk be the matrix whose ith column is w~). Then (4) can be written in matrix form as Wk+1 = Wk + r{SbWk -SmWkU~W[SbWk p, (5) where UT[·] sets all elements below the diagonal of its matrix argument to zero, thereby making it upper triangular. 2.2 ANOTHER SELF-ORGANIZING ALGORITHM FOR LDA In the previous analysis for a two-layer linear hetero-associative network, we observed that the optimum value for V=WTM, where the jlh column of Wand row of V are formed by Wi and Vi respectively. It is, therefore, worthwhile to explore the gradient descent procedure on the error function below instead of (2) J(W) = E[lld- MTWWTxI12} (6) By differentiating this error function with respect to W, and including the deflation process, we obtain the following update procedure for W instead of (5) Wk+1 = Wk + ~2SbWk - Sm WkUT[ W[ SbWk ] - SbWkUT[ W[ SmWk]). (7) 3. LDA AND GENERALIZED EIGEN-DECOMPOSITION Since LOA consists of solving the generalized eigenvector problem Sb<P=Sm<PA, we can naturally generalize algorithms (5) and (7) to obtain adaptive algorithms for the generalized eigen-decomposition problem A<P=B<PA, where A and B are assumed to be symmetric and positive definite. Here, we do not have the matrices A and B. Instead, 400 C. Chatterjee and V. P. Roychowdhury there are available two sequences of random vectors {xk} and {Yk} with limk~ooE[xp/] =A and limk~~[Yky/]=B, where xk and Yk represent the online observations. From (5), we obtain the following adaptive algorithm for generalized eigendecomposition (8) Here {17k} is a sequence of scalar gains, whose properties are described in Section 4. The sequences {Ak} and {Bk} are instantaneous values of the matrices A and B respectively. Although the Ak and Bk values can be obtained from xk and Yk as xp/ and YkY/ respectively, our algorithm requires that at least one of the {Ak} or {Bk} sequences have a dominated convergence property. Thus, the {Ak} and {Bk} sequences may be obtained from xp/ and YkY/ from the following algorithms Ak = Ak_1 +Yk(XkXk -Ak- I ) and Bk = Bk- I +Yk(YkYk -Bk-d, (9) where Ao and Bo are symmetric, and {Yk} is a scalar gain sequence. As done before, we can generalize (7) to obtain the following adaptive algorithm for generalized eigen-decomposition from a sequence of samples {Ak} and {Bk} Wk+1 = Wk + l7k(2AkWk - BkWkUT[ W[ AkWk ] - AkWkUT[ W[ BkWk ]). (10) Although algorithms (8) and (10) were derived from the network MSE by the gradient descent approach, this derivation does not guarantee their convergence. In order to prove their convergence, we use stochastic approximation theory. We give the convergence results only for algorithm (l0). 4. STOCHASTIC APPROX. CONVG. PROOF FOR ALG. (10) In order to prove the con vergence of (10), we use stochastic approximation theory due to Ljung (1977). In stochastic approximation theory, we study the asymptotic properties of (10) in terms of the ordinary differential equation (ODE) ~ W(t)= 1!!! E[2AkW - BkWUT[ W T AkW]- AkWUT[ W T BkW]], where W(t) is the continuous time counterpart of Wk with t denoting continuous time. The method of proof requires the following steps: (1) establishing a set of conditions to be imposed on A, B, A", B", and 17", (2) finding the stable stationary points of the ODE; and (3) demonstrating that Wk visits a compact subset of the domain of attraction of a stable stationary point infinitely often. We use Theorem 1 of Ljung (1977) for the convergence proof. The following is a general set of assumptions for the convergence proof of (10): Assumption (AI). Each xk and Yk is bounded with probability one, and limk~ooE[xp/] = A and limk~ooE[y kY k 1) = B, where A and B are positive definite. Assumption (A2). {l7kE9t+} satisfies l7kJ..O, Lk=Ol7k =OO,Lk=Ol7k <00 for some r>1 and limk~oo sup(l7il -l7i~l) <00. Assumption (A3). The p largest generalized eigenvalues of A with respect to B are each of unit mUltiplicity. Lemma 1. Let Al and A2 hold. Let w* be a locally asymptotically stable (in the sense of Liapunov) solution to the ordinary differential equation (ODE): Self-Organizing and Adaptive Generalized Eigen-Decomposition 401 ~ W(t) = 2AW(t) - BW(t)U4W(t/ AW(t)] - AW(t)U4W(t/ BW(t)], (11) with domain of attraction D(W). Then if there is a compact subset S of D(W) such that Wk E S infinitely often, then we have Wk ~ W with probability one as k ~ 00. • We denote A\ > ~ > ... > Ap ~ ... ~ An > 0 as the generalized eigenvalues of A with respect to B, and 4>; as the generalized eigenvector corresponding to A; such that 4>\, ... ,4>n are orthonormal with respect to B. Let <l>=[4>\ ... 4>nl and A=diag(A\, ... ,An) denote the matrix of generalized eigenvectors and eigenvalues of A with respect to B. Note that if 4>; is a generalized eigenvector, then d;4>; (ld;l= 1) is also a generalized eigenvector. In the next two lemmas, we first prove that all the possible equilibrium points ofthe ODE (11) are up to an arbitrary permutation of the p generalized eigenvectors of A with respect to B corresponding to the p largest generalized eigenvalues. We next prove that all these equilibrium points of the ODE (11) are unstable equilibrium points, except for [d\4>\ ... dn4>nl, where Id;I=1 for i=I, ... ,p. Lemma 2. For the ordinary differential equation (11), let Al and A3 hold Then W=<l>DP are equilibrium points of (11), where D=[D\IOV is a nXp matrix with DI being a pXp diagonal matrix with diagonal elements d; such that Id;l= 1 or d;=O, and P is a nXn arbitrary permutation matrix. • Lemma 3. Let Al and A3 hold Then W=<l>D (where D=[D\101~ D\ =diag(d\, ... ,dp )' Id;I=I) are stable equilibrium points of the ODE (11). In addition, W=<l>DP (d;=O for i~p or P~J) are unstable equilibrium points of the ODE (11) . • Lemma 4. For the ordinary differential equation (11), let Al and A3 hold Then the points W=<l>D (where D=[D\101~ D\ =diag(d\, ... ,dp )' Id;I=1 for i=I, ... ,p) are asymptotically stable. • Lemma 5. Let AI-A3 hold Then there exists a uniform upper boundfor 17k such that Wk is uniformly bounded w.p.I. • The convergence of alg. (10) can now be established by referring to Theorem 1 of Ljung. Theorem 1. Let A I-A3 hold Assume that with probability one the process {Wk} visits infinitely often a compact subset of the domain of attraction of one of the asymptotically stable points <l>D. Then with probability one lim Wk = <l>D. k~OCl Proof. By Lemma 2, <l>D (ld;I=I) are asymptotically stable points of the ODE (11). Since we assume that {Wk} visits a compact subset of the domain of attraction of <l>D infmitely often, Lemma 1 then implies the theorem. • 5. EXPERIMENT AL RESULTS We describe the performance of algorithms (8) and (10) with an example of online interference cancellation in a high-dimensional signal, in a digital mobile communication problem. The problem occurs when the desired user transmits a signal from a far distance to the receiver, while another user simultaneously transmits very near to the base. For common receivers, the quality of the received signal from the desired user is dominated by interference from the user close to the base. Due to the high rate and large dimension of the data, the system demands an accurate detection method for just a few data samples. 402 C. Chatterjee and V. P. Roychowdhury If we use conventional (numerical analysis) methods, signal detection will require a significant part of the time slot allotted to a receiver, accordingly reducing the effective communication rate. Adaptive generalized eigen-decomposition algorithms, on the other hand, allow the tracking of slow changes, and directly performs signal detection. The details of the data model can be found in Zoltowski et al. (1996). In this application, the duration for each transmitted code is 127 IlS, within which we have lOllS of signal and 1171ls of interference. We take 10 frequency samples equi-spaced between -O.4MHz to +O.4MHz. Using 6 antennas, the signal and interference correlation matrices are of dimension 60X60 in the complex domain. We use both algorithms (8) and (10) for the cancellation of the interference. Figure 1 shows the convergence of the principal generalized eigenvector and eigenvalue. The closed form solution is obtained after collecting all of the signal and interference samples. In order to measure the accuracy of the algorithms, we compute the direction cosine of the estimated principal generalized eigenvector and the generalized eigenvector computed by the conventional method. The optimum value is one. We also show the estimated principal generalized eigenvalue in Figure 1 b. The results show that both algorithms converge after the 4th bit of signal. Algonthm (10) Algonlhm (8) 1.1r--.----T---~--__r~r__-,--......., CLOSl!D FORM SOUlTlON 1.0 •• • • •••.... .• •••.........••••.•..••.•.. • •...... ... _09.rf lll .-- ---~ 08 ~ /-.... I :; 07 I ~~: /' ;::: 04 ~ 03 Iii Q OJ ~ 0.1 O .OD·'------=5DD:-----lI~m----:-:I5DD'::-' NUMBER OF SAMPLES (a) Algonthm (10) Algonlhm (8) 35 ...----.---r--r----r-....-,-..,--.......,--...----.---, 25 ~20 ~ 15 .... I 13 I:: Iii 10 III !ii ~ I Iii ~ I °D·'------5DD~---~I~----IJ5DD---~DOO NUMBER OF SAMPLES (b) Figure 1. (a) Direction Cosine of Estimated First Principal Generalized Eigenvector, and (b) Estimated First Principal Generalized Eigenvalue. References C.Chatterjee and V.Roychowdhury (1996), "Statistical Risk Analysis for Classification and Feature Extraction by Multilayer Perceptrons", Proceedings IEEE Int 'l Conference on Neural Networks, Washington D.C. K.Fukunaga (1990), Introduction to Statistical Pattern Recognition, 2nd Edition, New York: Academic Press. P.Gallinari, S.Thiria, F.Badran, F.Fogelman-Soulie (1991), "On the Relations Between Discriminant Analysis and Multilayer Perceptrons", Neural Networks, Vol. 4, pp. 349-360. L.Ljung (1977), "Analysis of Recursive Stochastic Algorithms", IEEE Transactions on Automatic Control, Vol. AC-22, No. 4, pp. 551-575. A.R.Webb and D.Lowe (1990), "The Optimised Internal Representation of Multilayer Classifier Networks Perfonns Nonlinear Discriminant Analysis", Neural Networks, Vol. 3, pp. 367-375. M.D.Zoltowski, C.Chatterjee, V.Roychowdhury and J.Ramos (1996), "Blind Adaptive 2D RAKE Receiver for CDMA Based on Space-Time MVDR Processing", submitted to IEEE Transactions on Signal Processing.	1996	144

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