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A Dynamical Model of Context Dependencies for the Vestibulo-Ocular Reflex Olivier J.M.D. Coenen* Terrence J. Sejnowskit Computational Neurobiology Laboratory Howard Hughes Medical Institute The Salk Institute for Biological Studies 10010 North Torrey Pines Road La Jolla, CA 92037, U.S.A. Departments oftBiology and *tPhysics University of California, San Diego La Jolla, CA 92093, U.S.A {olivier,terry}@salk.edu Abstract The vestibulo-ocular reflex (VOR) stabilizes images on the retina during rapid head motions. The gain of the VOR (the ratio of eye to head rotation velocity) is typically around -1 when the eyes are focused on a distant target. However, to stabilize images accurately, the VOR gain must vary with context (eye position, eye vergence and head translation). We first describe a kinematic model of the VOR which relies solely on sensory information available from the semicircular canals (head rotation), the otoliths (head translation), and neural correlates of eye position and vergence angle. We then propose a dynamical model and compare it to the eye velocity responses measured in monkeys. The dynamical model reproduces the observed amplitude and time course of the modulation of the VOR and suggests one way to combine the required neural signals within the cerebellum and the brain stem. It also makes predictions for the responses of neurons to multiple inputs (head rotation and translation, eye position, etc.) in the oculomotor system. 1 Introduction The VOR stabilizes images on the retina during rapid head motions: Rotations and translations of the head in three dimensions must be compensated by appropriate rotations of the eye. Because the head's rotation axis is not the same as the eye's rotation axis, the calculations for proper image stabilization of an object must take into account diverse variables such as object distance from each eye, 90 O. J. M. D. COENEN, T. J. SEJNOWSKI gaze direction, and head translation (Viire et al., 1986). The stabilization is achieved by integrating infonnation from different sources: head rotations from the semicircular canals of the inner ear, head translations from the otolith organs, eye positions, viewing distance, as well as other context infonnation, such as posture (head tilts) or activity (walking, running) (Snyder and King, 1992; Shelhamer et al.,1992; Grossman et al., 1989). In this paper we concentrate on the context modulation of the VOR which can be described by the kinematics of the reflex, i.e. eye position, eye vergence and head translation. 2 The Vestibulo-Ocular Reflex: Kinematic Model Definition of Vectors Coordinate System Eye position Vector Head Top View Target Object Gaze Vector Gaze Angle Interocular Distance Rotation Axis Semicircular Canals and Otoliths ¥ ~_--+_ Origin of coordinate syste,,-, (arbitrary) Figure 1: Diagram showing the definition of the vectors used in the equation of the kinematic model of the vestibulo-ocular reflex. The ideal VOR response is a compensatory eye movement which keeps the image fixed on the retina for any head rotations and translations. We therefore derived an equation for the eye rotation velocity by requiring that a target remains stationary on the retina. The velocity of the resulting compensatory eye rotation can be written as (see fig. 1): w = -Oe + 1:1 x [Dej x Oe - To;] (1) where Oe is the head rotation velocity sensed by the semicircular canals, TOj is the head translation velocity sensed by the otoliths, Dej == (e - OJ), e is a constant vector specifying the location of an eye in the head, OJ is the position of either the left or right otolith, fJ and Igl are the unit vector and amplitude of the gaze vector: fJ gives the eye position (orientation of the eye relative to the head), and Igl gives the distance from the eye to the object, and the symbol x indicates the cross-product between two vectors. wand Oe are rotation vectors which describe the instantaneous angUlar velocity of the eye and head, respectively. A rotation vector lies along the instantaneous axis of rotation; its magnitude indicates the speed of rotation around the axis, and its direction is given by the righthand screw rule. A motion of the head combining rotation (0) and translation (T) is sensed as the combination of a rotation velocity Oe measured by the semicircular canals and a translation velocity To sensed by the otoliths. The rotation vectors are equal (0 = Oe), and the translation velocity vector as measured by the otoliths is given by: TOj = OOj x 0 + T, where OOj == (a - OJ), and a is the position vector of the axis of rotation. A Dynarnical Model of Context Dependencies for the Vestibula-Ocular Reflex 91 The special case where the gaze is horizontal and the rotation vector is vertical (horizontal head rotation) has been studied extensively in the literature. We used this special case in the sirnulations. In that case w rnay be sirnplify by writing its equation with dot products. Since 9 and slc are then perpendicular (9 . fie = 0). the first term of the following expression in brackets is zero: (2) The sernicircular canals decornpose and report acceleration and velocity of head rotation fi by its cornponents along the three canals on each side of the head fie : horizontal. anterior and posterior. The two otolith organs on each side report the dynamical inertial forces generated during linear rnotion (translation) in two perpendicular plane. one vertical and the other horizontal relative to the head. Here we assurne that a translation velocity signal (To) derived frorn or reported by the otolith afferents is available. The otoliths encode as well the head orientation relative to the gravity vector force. but was not included in this study. To cornplete the correspondence between the equation and a neural correlate. we need to determine a physiological source for 9 and I!I. The eye position 9 is assurned to be given by the output of the velocity-to-position transformation or so-called "neural integrator" which provides eye position information and which is necessary for the activation of the rnotoneuron to sustain the eye in a fixed position. The integrator for horizontal eye position appears to be located in the nucleus prepositus hypoglossi in the pons. and the vertical integrator in the rnidbrain interstitial nucleus of Cajal. (Crawford. Cadera and Vilis. 1991; Cannon and Robinson. 1987). We assurne that the eye position is given as the coordinates of the unit vector 9 along the ~ and 1; of fig. 1. The eye position depends on the eye velocity according to '* = 9 x w. For the special case w(t) = w(t)z. i.e. for horizontal head rotation. the eye position coordinates are given by: 91 (t) = 91 (0) + f~ iJ2( r )w( r) dr 92(t) = 92(0) f~ 91(r)w(r)dr (3) This is a set of two negatively coupled integrators. The "neural integrator" therefore does not integrate the eye velocity directly but a product of eye position and eye velocity. The distance frorn eye to target I!I can be written using the gaze angles in the horizontal plane of the head: Right eye: 1 19RT (4) Left eye: 1 19LT (5) where «() R - () L) is the vergence angle. and I is the interocular distance; the angles are rneasured frorn a straight ahead gaze. and take on negative values when the eyes are turned towards the right. Within the oculornotor systern. the vergence angle and speed are encoded by the rnesencephalic reticular formation neurons (Judge and Curnrning. 1986; Mays. 1984). The nucleus reticularis tegrnenti pontis with reciprocal connections to the flocculus. oculornotor vermis. paravermis of the cerebellurn also contains neurons which activity varies linearly with vergence angle (Gamlin and Clarke. 1995). We conclude that it is possible to perform the cornputations needed to obtain an ideal VOR with signals known to be available physiologically. 92 Dynamical Model Overview Nod_ PftpoIItao IIyposIoooI O. J. M. D. COENEN, T. J. SEJNOWSKI Figure 2: Anatomical connections considered in the dynamical model. Only the left side is shown, the right side is identical and connected to the left side only for the calculation of vergence angle. The nucleus prepositus hypoglossi and the nucleus reticularis tegmenti pontis are meant to be representative of a class of nuclei in the brain stem carrying eye position or vergence signal. All connections are known to exist except the connection between the prepositus nucleus to the reticularis nucleus which has not been verified. Details of the cerebellum are in fig. 3 and of the vestibular nucleus in fig. 4. 3 Dynamical Model Snyder & King (1992) studied the effect of viewing distance and location of the axis of rotation on the VOR in monkeys; their main results are reproduced in fig. 5. In an attempt to reproduce their data and to understand how the signals that we have described in section 2 may be combined in time, we constructed a dynamical model based on the kinematic model. Its basic anatomical structure is shown in fig. 2. Details of the model are shown in fig. 3, and fig. 4 where all constants are written using a millisecond time scale. The results are presented in fig. 5. The dynamical variables represent the change of average firing rate from resting level of activity. The firing rate of the afferents has a tonic component proportional to the velocity and a phasic component proportional to the acceleration of movement. Physiologically, the afferents have a wide range of phasic and tonic amplitudes. This is reflected by a wide selection of parameters in the numerators in the boxes of fig. 3 and fig. 4. The Laplace transform of the integration operator in equation (3) of the eye position coordinates is ~. Following Robinson (1981), we modeled the neural integrator with a gain and a time constant of 20 seconds. We therefore replaced the pure integrator ~ with 20~~~~1 in the calculations of eye position. The term 1 in fig. 3 is calculated by using equations (4) and (5), and by using the integrator 9 20~o:!~1 on the eye velocity motor command to find the angles (h and (JR. The dynamical model is based on the assumption that the cerebellum is required for context modulation, and that because of its architecture, the cerebellum is more likely to implement complex functions of multiple signals than other relevant nuclei. The major contributions of vergence and eye position modulation on the VOR are therefore mediated by the cerebellum. Smaller and more transient contributions from eye position are assumed to be mediated through the vestibular nucleus as shown in fig. 4. The motivation for combining eye position as in fig. 4 are, first, the evidence for eye response oscillations; second, the theoretical consideration that linear movement information (To) is useless without eye position information for proper VOR. The parameters in the dynamical model were adjusted by hand after observing the behavior of the different components of the model and noting how these combine to produce the oscillations observed A Dynamical Model of Context Dependencies for the Vestibulo-Ocular Reflex Vestibular Semicirtular c..l O -----t 401+1 r-----®--f--..j 300+1 x OIolith 0Igan Cerebellum VHlibabr Nuc1tul 93 Figure 3: Contribution of the cerebellum to the dynamical model. Filtered velocity inputs from the canals and otoliths are combined with eye position according to equation (2). These calculations could be performed either outside the cerebellum in one or multiple brain stem nuclei (as shown) or possibly inside the cerebellum. The only output is to the vestibular nucleus. The Laplace notation is used in each boxes to represent a leaky integrator with a time constant. input derivative and input gain. The term oe are the coordinates of the vector oe shown in fig. 1. The x indicates a multiplication. The term! multiplies each inputs individually. The open arrows indicate inhibitory (negative) connections. VHlibalu Semicimtlu c.w Cere ... lIum O--'----t~l---+--®----t~~ X Figure 4: Contribution of the vestibular nucleus to the dynamical model. Three pathways in the vestibular nucleus process the canal and otolith inputs to drive the eye. The first pathway is modulated by the output of the cerebellum through a FIN (Flocculus Target Neuron). The second and third pathways report transient information from the inputs which are combined with eye position in a manner identical to fig. 3. The location of these calculations is hypothetical. in the data. Even though the number of parameters in the model is not small. it was not possible to fit any single response in fig. 5 without affecting most of the other eye responses. This puts severe limits on the set of parameters allowed in the model. The dynamical model suggests that the oscillations present in the data reflect: 1) important acceleration components in the neural signals. both rotational and linear, 2) different time delays between the canal and otolith signal processing. and 3) antagonistic or synergistic action of the canal and otolith signals with different axes of rotation, as described by the two terms in the bracket of equation (2). 4 Discussion By fitting the dynamical model to the data, we tested the hypothesis that the VOR has a response close to ideal taking into account the time constraints imposed by the sensory inputs and the neural networks performing the computations. The vector computations that we used in the model may not 94 ~ w O. J. M. D. COENEN, T. J. SEJNOWSKI Dynamical Model Responses vs Experimental Data 80 -20 LOMtIOftof .... 01 rotMIon -,a.-om -400~----~5~0------~ 10=0~ Time (m.) 80 60 40 20 -20 T .......... ~ .-40oL-----~ 5~ 0 ----~1~0~0- Time (m.) Figure 5: Comparison between the dynamical model and monkey data. The dotted lines show the effect of viewing distance and location of the axis of rotation on the VOR as recorded by Snyder & King (1992) from monkeys in the dark. The average eye velocity response (of left and right eye) to a sudden change in head velocity is shown for different target distances (left) and rotational axes (right). On the left, the location of the axis of rotation was in the midsagittal plane 12.5 cm behind the eyes (-12.5 cm), and the target distance was varied between 220 cm and 9 cm. On the right, the target di stance was kept constant at 9 cm in front of the eye, and the location of the axis of rotation was varied from 14 cm behind t04cm in front of the eyes (-14cm to 4cm) in the midsagittal plane. The solid lines show the model responses. The model replicates many characteristics of the data. On the left the model captures the eye velocity fluctuations between 20-50 ms, followed by a decrease and an increase which are both modulated with target distance (50-80 ms). The later phase of the response (80-100 ms) is almost exact for 220 cm, and one peak is seen at the appropriate location for the other distances. On the right the closest fits were obtained for the 4 cm and 0 cm locations. The mean values are in good agreement and the waveforms are close, but could be shifted in time for the other locations of the axis of rotations. Finally, the latest peak ( ..... lOOms) in the data appears in the model for -14 cm and 9 cm location. be the representation used in the oculomotor system. Mathematically, the vector representation is only one way to describe the computations involved. Other representations exist such as the quaternion representation which has been studied in the context of the saccadic system (Tweed and Vilis, 1987; see also Handzel and Flash, 1996 for a very general representation). Detailed comparisons between the model and recordings from neurons will be require to settle this issue. Direct comparison between Purkinje cell recordings (L.H. Snyder & W.M. King, unpublished data) and predictions of the model could be used to determine more precisely the different inputs to some Purkinje cells. The model can therefore be an important tool to gain insights difficult to obtain directly with experiments. The question of how the central nervous system learns the transformations that we described still remains. The cerebellum may be one site of learning for these transformations, and its output may modulate the VOR in real time depending on the context. This view is compatible with the results of Angelaki and Hess (1995) which indicate that the cerebellum is required to correctly perform an otolith transformation. It is also consistent with adaptation results in the VOR. To test this hypothesis, we have been working on a model of the cerebellum which learns to anticipate sensory inputs and feedbacks, and use these signals to modulate the VOR. The learning in the cerebellum and vestibular nuclei is mediated by the climbing fibers which report a reinforcement signal of the prediction error (Coenen and Sejnowski. in preparation). A Dynamical Model of Context Dependencies for the Vestibulo-Ocular Reflex 95 5 Conclusion Most research on the VOR has assumed forward gaze focussed at infinity. The kinematics of offcenter gaze and fixation at finite distance necessitates nonlinear corrections that require the integration of a variety of sensory inputs. The dynamical model studied here is a working hypothesis for how these corrections could be computed and is generally consistent with what is known about the cerebellum and brain stem nuclei. We are, however, far from knowing the mechanisms underlying these computations, or how they are learned through experience. 6 Acknowledgments The first author was supported by a McDonnell-Pew Graduate Fellowship during this research. We would like to thank Paul Viola for helpful discussions. References Angelaki, D. E. and Hess, B. J. (1995). Inertial representation of angular motion in the vestibular system of rhesus monkeyus. II. Otolith-controlled transformation that depends on an intact cerebellar nodulus. Journal of Neurophysiology, 73(5): 1729-1751. Cannon, S. C. and Robinson, D. A. (1987). Loss of the neural integrator of the oculomotor system from brain stem lesions in monkey. Journal of Neurophysiology, 57(5):1383-1409. Crawford, J. D., Cadera, W., and Vilis, T. (1991). Generation of torsional and vertical eye position signals by the interstitial nucleus of Cajal. Science, 252:1551-1553. Gamlin, P. D. R. and Clarke, R. J. (1995). Single-unit activity in the primate nucleus reticularis tegmenti pontis related to vergence and ocular accomodation. Journal of Neurophysiology, 73(5):2115-2119. Grossman, G. E., Leigh, R. J., Bruce, E. N., Huebner, W. P.,and Lanska, D.J. (1989). Performanceofthe human vestibu1oocu1ar reflex during locomotion. Journal of Neurophysiology, 62(1 ):264-272. Handzel, A. A. and Flash, T. (1996). The geometry of eye rotations and listing's law. In Touretzky, D., Mozer, M., and Hasselmo, M., editors, Advances in Neural Information Processing Systems 8, Cambridge, MA. MIT Press. Judge, S. J. and Cumming, B. G. (1986). Neurons in the monkey midbrain with activity related to vergence eye movement and accomodation. Journal of Neurophysiology, 55:915-930. Mays, L. E. (1984). Neural control of vergence eye movements: Convergence and divergence neurons in midbrain. Journal of Neurophysiology, 51:1091-1108. Robinson, D. A. (1981). The use of control systems analysis in the neurophysiology of eye movements. Ann. Rev. Neurosci., 4:463-503. Shelhamer, M., Robinson, D. A., and Tan, H. S. (1992). Context-specific adaptation of the gain of the vestibuloocular reflex in humans. Journal of Vestibular Research, 2:89-96. Snyder, L. H. and King, W. M. (1992). Effect of viewing distance and location ofthe axis of head rotation on the monkey's vestibuloocular reflex I. eye movement response. Journal of Neurophysiology, 67(4):861-874. Tweed, D. and Vilis, T. (1987). Implications of rotational kinematics for the oculomotor system in three dimensions. Journal of Neurophysiology, 58(4):832-849. Viire, E., Tweed, D., Milner, K., and Vilis, T. (1986). A reexamination of the gain ofthe vestibuloocular reflex. Journal of Neurophysiology, 56(2):439-450.
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Constructive Algorithms for Hierarchical Mixtures of Experts S.R.Waterhouse A.J.Robinson Cambridge University Engineering Department, Trumpington St., Cambridge, CB2 1PZ, England. Tel: [+44] 1223 332754, Fax: [+44] 1223 332662, Email: srw1001.ajr@eng.cam.ac.uk Abstract We present two additions to the hierarchical mixture of experts (HME) architecture. By applying a likelihood splitting criteria to each expert in the HME we "grow" the tree adaptively during training. Secondly, by considering only the most probable path through the tree we may "prune" branches away, either temporarily, or permanently if they become redundant. We demonstrate results for the growing and path pruning algorithms which show significant speed ups and more efficient use of parameters over the standard fixed structure in discriminating between two interlocking spirals and classifying 8-bit parity patterns. INTRODUCTION The HME (Jordan & Jacobs 1994) is a tree structured network whose terminal nodes are simple function approximators in the case of regression or classifiers in the case of classification. The outputs of the terminal nodes or experts are recursively combined upwards towards the root node, to form the overall output of the network, by "gates" which are situated at the non-terminal nodes. The HME has clear similarities with tree based statistical methods such as Classification and Regression Trees (CART) (Breiman, Friedman, Olshen & Stone 1984). We may consider the gate as replacing the set of "questions" which are asked at each branch of CART. From this analogy, we may consider the application of the splitting rules used to build CART. We start with a simple tree consisting of two experts and one gate. After partially training this simple tree we apply the splitting criterion to each terminal node. This evaluates the log-likelihood increase by splitting each expert into two experts and a gate. The split which yields the best increase in log-likelihood is then added permanently to the tree. This process of training followed by growing continues until the desired modelling power is reached. Constructive Algorithms for Hierarchical Mixtures of Experts 585 Figure 1: A simple mixture of experts. This approach is reminiscent of Cascade Correlation (Fahlman & Lebiere 1990) in which new hidden nodes are added to a multi-layer perceptron and trained while the rest of the network is kept fixed. The HME also has similarities with model merging techniques such as stacked regression (Wolpert 1993), in which explicit partitions of the training set are combined. However the HME differs from model merging in that each expert considers the whole input space in forming its output. Whilst this allows the network more flexibility since each gate may implicitly partition the whole input space in a "soft" manner, it leads to unnecessarily long computation in the case of near optimally trained models. At anyone time only a few paths through a large network may have high probability. In order to overcome this drawback, we introduce the idea of "path pruning" which considers only those paths from the root node which have probability greater than a certain threshold. CLASSIFICATION USING HIERARCHICAL MIXTURES OF EXPERTS The mixture of experts, shown in Figure 1, 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 architecture in a tree structured fashion. Each terminal node or "expert" may take on a variety of forms, depending on the application. In the case of multi-way classification, each expert outputs a vector Yj in which element m is the conditional probability of class m (m = 1 ... M) which is computed using the soft max function: P(CmI x(n>, Wj) = exp(w~jx(n») It exp(w~.kX(n») k=1 where Wj = [wlj W2j ... WMj] is the parameter matrix for expert j and Ci denotes class i. The outputs of the experts are combined using a "gate" which sits at the nonterminal nodes. The gate outputs are estimates of the conditional probability of selecting the daughters of the non-terminal node given the input and the path taken to that node from the root node. This is once again computed using the softmax function: P(Zj I ~ (.), ~) = exp( ~ J ~(.» It <xp( ~f ~(.» where'; = [';1';2 ... ';f] is the parameter matrix for the gate, and Zj denotes expert j . 586 S. R. WATERHOUSE, A. 1. ROBINSON The overall output is given by a probabilistic mixture in which the gate outputs are the mixture weights and the expert outputs are the mixture components. The probability of class m is then given by: ] P(cmlz(n),8) = I: P(zilz(n), ~)P(Cmlz(n), Wi). i=1 A straightforward extension of this model also gives us the conditional probability ht) of selecting expert j given input zen) and correct class Ck, In order to train the HME to perform classification we maximise the log likelihood L = l:~=1 l:~=1 t~) log P(cm Iz(n), 8), where the variable t~) is one if m is the correct class at exemplar (n) and zero otherwise. This is done via the expectation maximisation (EM) algorithm of Dempster, Laird & Rubin (1977), as described by Jordan & Jacobs (1994). TREE GROWING The standard HME differs from most tree based statistical models in that its architecture is fixed. By relaxing this constraint and allowing the tree to grow, we achieve a greater degree of flexibility in the network. Following the work on CART we start with a simple tree, for instance with two experts and one gate which we train for a small number of cycles. Given this semi-trained network, we then make a set of candidate splits {&} of terminal nodes {z;}. Each split involves replacin~ an expert Zi with a pair of new experts {Zu}j=1 and a gate, as shown in Figure 2. We wish to select eventually only the "best" split S out of these candidate splits. Let us define the best split as being that which maximises the increase in overall log-likelihood due to the split, IlL = L(P+1) L(P) where L(P) is the likelihood at the pth generation of the tree. If we make the constraint that all the parameters of the tree remain fixed apart from the paramL(P) \ \ L(P+I) Figure 2: Making a candidate split of a terminal node. \ \ eters of the new split whenever a candidate split is made, then the maximisation is simplified into a dependency on the increases in the local likelihoods {Li} of the nodes {Zi}. We thus constrain the tree growing process to be localised such that we find the node which gains the most by being split. max M(&) _ max M· = max(Ly*1) - L(P» i i I i I I Constructive Algorithms for Hierarchical Mixtures of Experts 587 Figure 3: Growing the HME. This figure shows the addition of a pair of experts to the partially grown tree. where n m L~+l) = L L t~) log L P(zijlz(n), c;;,zi)P(cmlz(n), zij, wij) n m j This splitting rule is similar in form to the CART splitting criterion which uses maximisation of the entropy of the node split, equivalent to our local increase in lop;-likelihood. TIle final growing algorithm starts with a tree of generation p and firstly fixes the parameters of all non-terminal nodes. All terminal nodes are then split into two experts and a gate. A split is only made if the sum of posterior probabilities En h~n), as described (1), at the node is greater than a small threshold. This prevents splits being made on nodes which have very little data assigned to them. In order to break symmetry, the new experts of a split are initialised by adding small random noise to the original expert parameters. The gate parameters are set to small random weights. For each node i, we then evaluate M; by training the tree using the standard EM method. Since all non-terminal node parameters are fixed the only changes to the log-likelihood are due the new splits. Since the parameters of each split are thus independent of one another, all splits can be trained at once, removing the need to train multiple trees separately. After each split has been evaluated, the best split is chosen. This split is kept and all other splits are discarded. The original tree structure is then recovered except for the additional winning split, as shown in Figure 3. The new tree, of generation p + I is then trained as usual using EM. At present the decision on when to add a new split to the tree is fairly straightforward: a candidate split is made after training the fixed tree for a set number of iterations. An alternative scheme we have investigated is to make a split when the overall log-likelihood of the fixed tree has not increased for a set number of cycles. In addition, splits are rejected if they add too little to the local log-likelihood. Although we have not discussed the issue of over-fitting in this paper, a number of techniques to prevent over-fitting can be used in the HME. The most simple technique, akin to those used in CART, involves growing a large tree and successively removing nodes from the tree until the performance on a cross validation set reaches an optimum. Alternatively the Bayesian techniques of Waterhouse, MacKay & Robinson (1995) could be applied. 588 S. R. WATERHOUSE, A. J. ROBINSON Tree growing simulations This algorithm was used to solve the 8-bit parity classification task. We compared the growing algorithm to a fixed HME with depth of 4 and binary branches. As can be seen in Figures 4(a) and (b), the factorisation enabled by the growing algorithm significantly speeds up computation over the standard fixed structure. The final tree shape obtained is shown in Figure 4(c). We showed in an earlier paper (Waterhouse & Robinson 1994) that the XOR problem may be solved using at least 2 experts and a gate. The 8 bit parity problem is therefore being solved by a series of XOR classifiers, each gated by its parent node, which is an intuitively appealing form with an efficient use of parameters. 8 ,E -200oL----1~0----2~0~--~~~--~4~0--~W Time (a) Evolution of log-likelihood vs. time in CPU seconds. -50 ~-100 "I .§' -150 -2000 1 2 3 4 5 6 Generation (b) Evolution of log-likelihood for (i) vs generations of tree. O'()OI 0.001 (c) Final tree structure obtained from (i), showing utilisation U; of each node where U; = L: P(z;, R;I:c(n») I N, and Ri is the path t~en from the root node to node i . Figure 4: HME GROWING ON THE 8 BIT PARITY PROBLEM;(i) growing HME with 6 generations; (ii) 4 deep binary branching HME (no growing). PATH PRUNING If we consider the HME to be a good model for the data generation process, the case for path pruning becomes clear. In a tree with sufficient depth to model the Constructive Algorithms for Hierarchical Mixtures of Experts 589 underlying sub-processes producing each data point, we would expect the activation of each expert to tend to binary values such that only one expert is selected at each time exemplar. The path pruning scheme is depicted in Figure 5. The pruning scheme utilises the "activation" of each node at each exemplar. The activation is defined as the product of node probabilities along a path from the root node to the current node, lin) = Li log P(zi/Ri, :.:(n»), where Ri is the path taken to node i from the root node. If .l}n) for node l at exemplar n falls below a threshold value, ft, then we ignore the subtree Sl and we backtrack up to the parent node of l. During training this involves not accumulating the statistics of the subtree Sl; during evaluation it involves setting the output of subtree Sl to zero. In addition to this path pruning scheme we can use the activation of the nodes to do more permanent pruning. If the overall utilisation Vi = Ln P(Zi, Rd:.:(n»)IN of a node falls below a small threshold, then a node is pruned completely from the tree. The sister subtrees of the removed node then subsume their parent nodes. This process is used solely to improve computational efficiency in this paper, although conceivably it could be used as a regularisation method, akin to the brain surgery techniques of Cun, Denker & Solla (1990). In such a scheme, however, a more useful measure of node utilisation would be the effective number of parameters (Moody 1992). Path pruning simulations ..... _---_ .. -.. Figure 5: Path pruning in the HME. Figure 6 shows the application of the pruning algorithm to the task of discriminating between two interlocking spirals. With no pruning the solution to the two-spirals takes over 4,000 CPU seconds, whereas with pruning the solution is achieved in 155 CPU seconds. One problem which we encountered when implementing this algorithm was in computing updates for the parameters of the tree in the case of high pruning thresholds. If a node is visited too few times during a training pass, it will sometimes have too little data to form reliable statistics and thus the new parameter values may be unreliable and lead to instability. This is particularly likely when the gates are saturated. To avoid this saturation we use a simplified version of the regularisation scheme described in Waterhouse et al. (1995). CONCLUSIONS We have presented two extensions to the standard HME architecture. By pruning branches either during training or evaluation we may significantly reduce the computational requirements of the HME. By applying tree growing we allow greater flexibility in the HME which results in faster training and more efficient use of parameters. 590 0 -20 "C -40 0 0 ;5 -60 ~ T -80 0> 0 ....J -100 -120 S. R. WATERHOUSE, A. J. ROBINSON (a) ,.,fi (iii " (iv) ,: .. .i .... ':1 ,I ,.,. , , , , I . / ( .'/ ."..:'" ."" ' ,,,.. ~ -,~ '.'' 10 100 1000 Time (5) (b) (c) Figure 6: The effect of pruning on the two spirals classification problem by a 8 deep binary branching hme:(a) Log-likelihood vs. Time (CPU seconds), with log pruning thresholds for experts and gates f: (i) f = -5. 6,(ii) f = -lO,(iii) f = -15,(iv) no pruning, (b) training set for two-spirals task; the two classes are indicated by crosses and circles, (c) Solution to two spirals problem. References Breiman, L., Friedman, J., Olshen, R. & Stone, C. J. (1984), Classification and Regression Trees, Wadswoth and Brooks/Cole. Cun, Y. L., Denker, J. S. & Solla, S. A. (1990), Optimal brain damage, in D. S. Touretzky, ed., 'Advances in Neural Information Processing Systems 2', Morgan Kaufmann, pp. 598-605. 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. Fahlman, S. E. & Lebiere, C. (1990), The Cascade-Correlation learning architecture, Technical Report CMU-CS-90-100, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213. Jordan, M. I. & Jacobs, R. A. (1994), 'Hierarchical Mixtures of Experts and the EM algorithm', Neural Computation 6, 181-214. Moody, J. E. (1992), The effective number of parameters: An analysis of generalization and regularization in nonlinear learning systems, in J. E. Moody, S. J. Hanson & R. P. Lippmann, eds, 'Advances in Neural Information Processing Systems 4', Morgan Kaufmann, San Mateo, California, pp. 847-854. Waterhouse, S. R. & Robinson, A. J . (1994), Classification using hierarchical mixtures of experts, in 'IEEE Workshop on Neural Networks for Signal Processing', pp. 177-186. Waterhouse, S. R., MacKay, D. J. C. & Robinson, A. J. (1995), Bayesian methods for mixtures of experts, in M. C. M. D. S. Touretzky & M. E. Hasselmo, eds, 'Advances in Neural Information Processing Systems 8', MIT Press. Wolpert, D. H. (1993), Stacked generalization, Technical Report LA-UR-90-3460, The Santa Fe Institute, 1660 Old Pecos Trail, Suite A, Santa Fe, NM, 87501.
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Discovering Structure in Continuous Variables Using Bayesian Networks Reimar Hofmann and Volker Tresp* Siemens AG, Central Research Otto-Hahn-Ring 6 81730 Munchen, Germany Abstract We study Bayesian networks for continuous variables using nonlinear conditional density estimators. We demonstrate that useful structures can be extracted from a data set in a self-organized way and we present sampling techniques for belief update based on Markov blanket conditional density models. 1 Introduction One of the strongest types of information that can be learned about an unknown process is the discovery of dependencies and -even more important- of independencies. A superior example is medical epidemiology where the goal is to find the causes of a disease and exclude factors which are irrelevant. Whereas complete independence between two variables in a domain might be rare in reality (which would mean that the joint probability density of variables A and B can be factored: p(A, B) = p(A)p(B)), conditional independence is more common and is often a result from true or apparent causality: consider the case that A is the cause of B and B is the cause of C, then p(CIA, B) = p(CIB) and A and C are independent under the condition that B is known. Precisely this notion of cause and effect and the resulting independence between variables is represented explicitly in Bayesian networks. Pearl (1988) has convincingly argued that causal thinking leads to clear knowledge representation in form of conditional probabilities and to efficient local belief propagating rules. Bayesian networks form a complete probabilistic model in the sense that they represent the joint probability distribution of all variables involved. Two of the powerful Reimar.Hofmann@zfe.siemens.de Volker.Tresp@zfe.siemens.de Discovering Structure in Continuous Variables Using Bayesian Networks 501 features of Bayesian networks are that any variable can be predicted from any subset of known other variables and that Bayesian networks make explicit statements about the certainty of the estimate of the state of a variable. Both aspects are particularly important for medical or fault diagnosis systems. More recently, learning of structure and of parameters in Bayesian networks has been addressed allowing for the discovery of structure between variables (Buntine, 1994, Heckerman, 1995). Most of the research on Bayesian networks has focused on systems with discrete variables, linear Gaussian models or combinations of both. Except for linear models, continuous variables pose a problem for Bayesian networks. In Pearl's words (Pearl, 1988): "representing each [continuous] quantity by an estimated magnitude and a range of uncertainty, we quickly produce a computational mess. [Continuous variables] actually impose a computational tyranny of their own." In this paper we present approaches to applying the concept of Bayesian networks towards arbitrary nonlinear relations between continuous variables. Because they are fast learners we use Parzen windows based conditional density estimators for modeling local dependencies. We demonstrate how a parsimonious Bayesian network can be extracted out of a data set using unsupervised self-organized learning. For belief update we use local Markov blanket conditional density models which - in combination with Gibbs sampling- allow relatively efficient sampling from the conditional density of an unknown variable. 2 Bayesian Networks This brief introduction of Bayesian networks follows closely Heckerman, 1995. Considering a joint probability density I p( X) over a set of variables {Xl, ••. , X N} we can decompose using the chain rule of probability N p(x) = IIp(xiIXI, ... ,Xi-I). (1) i=l For each variable Xi, let the parents of Xi denoted by Pi ~ {XI, . .. , Xi- d be a set of variables2 that renders Xi and {x!, ... , Xi-I} independent, that is (2) Note, that Pi does not need to include all elements of {XI, ... , Xi-Il which indicates conditional independence between those variables not included in Pi and Xi given that the variables in Pi are known. The dependencies between the variables are often depicted as directed acyclic3 graphs (DAGs) with directed arcs from the members of Pi (the parents) to Xi (the child). Bayesian networks are a natural description of dependencies between variables if they depict causal relationships between variables. Bayesian networks are commonly used as a representation of the knowledge of domain experts. Experts both define the structure of the Bayesian network and the local conditional probabilities. Recently there has been great 1 For simplicity of notation we will only treat the continuous case. Handling mixtures of continuous and discrete variables does not impose any additional difficulties. 2Usually the smallest set will be used. Note that in Pi is defined with respect to a given ordering of the variables. :li.e. not containing any directed loops. 502 R. HOFMANN. V. TRESP emphasis on learning structure and parameters in Bayesian networks (Heckerman, 1995). Most of previous work concentrated on models with only discrete variables or on linear models of continuous variables where the probability distribution of all continuous given all discrete variables is a multidimensional Gaussian. In this paper we use these ideas in context with continuous variables and nonlinear dependencies. 3 Learning Structure and Parameters in Nonlinear Continuous Bayesian Networks Many of the structures developed in the neural network community can be used to model the conditional density distribution of continuous variables p( Xi IPi). Under the usual signal-plus independent Gaussian noise model a feedforward neural network N N(.) is a conditional density model such that p(Xi IPi) = G(Xi; N N(Pi), 0-2 ), where G(x; c, 0-2 ) is our notation for a normal density centered at c and with variance 0-2• More complex conditional densities can, for example, be modeled by mixtures of experts or by Parzen windows based density estimators which we used in our experiments (Section 5). We will use pM (Xi IP;) for a generic conditional probability model. The joint probability model is then N pM (X) = II pM (xi/Pi). (3) i=l following Equations 1 and 2. Learning Bayesian networks is usually decomposed into the problems of learning structure (that is the arcs in the network) and of learning the conditional density models pM (Xi IPi) given the structure4 . First assume the structure of the network is given. If the data set only contains complete data, we can train conditional density models pM (Xi IPi ) independently of each other since the log-likelihood of the model decomposes conveniently into the individual likelihoods of the models for the conditional probabilities. Next, consider two competing network structures. We are basically faced with the well-known bias-variance dilemma: if we choose a network with too many arcs, we introduce large parameter variance and if we remove too many arcs we introduce bias. Here, the problem is even more complex since we also have the freedom to reverse arcs. In our experiments we evaluate different network structures based on the model likelihood using leave-one-out cross-validation which defines our scoring function for different network structures. More explicitly, the score for network structure S is Score = 10g(p(S)) + Lev, where p(S) is a prior over the network structures and Lev = ~f=llog(pM (xkIS, X - {xk})) is the leave-one-out cross-validation loglikelihood (later referred to as cv-Iog-likelihood). X = {xk}f=l is the set of training samples, and pM (xk IS, X - {xk}) is the probability density of sample Xk given the structure S and all other samples. Each of the terms pM (xk IS, X - {xk}) can be computed from local densities using Equation 3. Even for small networks it is computationally impossible to calculate the score for all possible network structures and the search for the global optimal network structure 4Differing from Heckerman we do not follow a fully Bayesian approach in which priors are defined on parameters and structure; a fully Bayesian approach is elegant if the occurring integrals can be solved in closed form which is not the case for general nonlinear models or if data are incomplete. Discovering Structure in Continuous Variables Using Bayesian Networks 503 is NP-hard. In the Section 5 we describe a heuristic search which is closely related to search strategies commonly used in discrete Bayesian networks (Heckerman, 1995). 4 Prior Models In a Bayesian framework it is useful to provide means for exploiting prior knowledge, typically introducing a bias for simple structures. Biasing models towards simple structures is also useful if the model selection criteria is based on cross-validation, as in our case, because of the variance in this score. In the experiments we added a penalty per arc to the log-likelihood i.e. 10gp(S) ex: -aNA where NA is the number of arcs and the parameter a determines the weight of the penalty. Given more specific knowledge in form of a structure defined by a domain expert we can alternatively penalize the deviation in the arc structure (Heckerman, 1995). Furthermore, prior knowledge can be introduced in form of a set of artificial training data. These can be treated identical to real data and loosely correspond to the concept of a conjugate prior. 5 Experiment In the experiment we used Parzen windows based conditional density estimators to model the conditional densities pM (Xj IPd from Equation 2, i.e. (4) where {xi }f=l is the training set. The Gaussians in the nominator are centered at (x7, Pf) which is the location of the k-th sample in the joint input/output (or parent/child) space and the Gaussians in the denominator are centered at (Pf) which is the location of the k-th sample in the input (or parent) space. For each conditional model, (J"j was optimized using leave-one-out cross validation5• The unsupervised structure optimization procedure starts with a complete Bayesian model corresponding to Equation 1, i.e. a model where there is an arc between any pair of variables6 • Next, we tentatively try all possible arc direction changes, arc removals and arc additions which do not produce directed loops and evaluate the change in score. After evaluating all legal single modifications, we accept the change which improves the score the most. The procedure stops if every arc change decreases the score. This greedy strategy can get stuck in local minima which could in principle be avoided if changes which result in worse performance are also accepted with a nonzero probability 7 (such as in annealing strategies, Heckerman, 1995). Calculating the new score at each step requires only local computation. The removal or addition of an arc corresponds to a simple removal or addition of the corresponding dimension in the Gaussians of the local density model. However, 5Note that if we maintained a global (7 for all density estimators, we would maintain likelihood equivalence which means that each network displaying the same independence model gets the same score on any test set. 6The order of nodes determining the direction of initial arcs is random. 7 In our experiments we treated very small changes in score as if they were exactly zero thus allowing small decreases in score. 504 R. HOFMANN. V. TRESP 15~-----------------------, 100~------~------~------~ 10 - - - --50 ~ ~ "T ~ I -5 -100 -10~----------~----------~ o 50 100 -150~--------------------~ o 5 10 15 Number of Iterations Number of inputs Figure 1: Left: evolution of the cv-log-Iikelihood (dashed) and of the log-likelihood on the test set (continuous) during structure optimization. The curves are averages over 20 runs with different partitions of training and test sets and the likelihoods are normalized with respect to the number of cv- or test-samples, respectively. The penalty per arc was a = 0.1. The dotted line shows the Parzen joint density model commonly used in statistics, i.e. assuming no independencies and using the same width for all Gaussians in all conditional density models. Right: log-likelihood of the local conditional Parzen model for variable 3 (pM (x3IP3)) on the test set (continuous) and the corresponding cv-log-likelihood (dashed) as a function of the number of parents (inputs). crime ra.te 2 percent land zoned for lots a percent nonretail business 4 located on Charles river? 5 nitrogen oxide concentration 6 Average number of rooms 7 percent built before 1940 8 weighted distance to employment center 9 access to radial highways 10 tax rate 11 pupil/teacher ratio 12 percent black 13 percent lower-status population 14 median value of homes Figure 2: Final structure of a run on the full data set. after each such operation the widths of the Gaussians O'i in the affected local models have to be optimized. An arc reversal is simply the execution of an arc removal followed by an arc addition. In our experiment, we used the Boston housing data set, which contains 506 samples. Each sample consists of the housing price and 14 variables which supposedly influence the housing price in a Boston neighborhood (Figure 2). Figure 1 (left) shows an experiment where one third of the samples was reserved as a test set to monitor the process. Since the algorithm never sees the test data the increase in likelihood of the model on the test data is an unbiased estimator for how much the model has improved by the extraction of structure from the data. The large increase in the log-likelihood can be understood by studying Figure 1 (right). Here we picked a single variable (node 3) and formed a density model to predict this variable from the remaining 13 variables. Then we removed input variables in the order of their significance. After the removal of a variable, 0'3 is optimized. Note that the cv-Iog-likelihood increases until only three input variables are left due to the fact Discovering Structure in Continuous Variables Using Bayesian Networks 505 that irrelevant variables or variables which are well represented by the remaining input variables are removed. The log-likelihood of the fully connected initial model is therefore low (Figure 1 left). We did a second set of 15 runs with no test set. The scores of the final structures had a standard deviation of only 0.4. However, comparing the final structures in terms of undirected arcs8 the difference was 18% on average. The structure from one of these runs is depicted in Figure 2 (right). In comparison to the initial complete structure with 91 arcs, only 18 arcs are left and 8 arcs have changed direction. One of the advantages of Bayesian networks is that they can be easily interpreted. The goal of the original Boston housing data experiment was to examine whether the nitrogen oxide concentration (5) influences the housing price (14). Under the structure extracted by the algorithm, 5 and 14 are dependent given all other variables because they have a common child, 13. However, if all variables except 13 are known then they are independent. Another interesting question is what the relevant quantities are for predicting the housing price, i.e. which variables have to be known to render the housing price independent from all other variables. These are the parents, children, and children's parents of variable 14, that is variables 8, 10, 11, 6, 13 and 5. It is well known that in Bayesian networks, different constellations of directions of arcs may induce the same independencies, i.e. that the direction of arcs is not uniquely determined. It can therefore not be expected that the arcs actually reflect the direction of causality. 6 Missing Data and Markov Blanket Conditional Density Model Bayesian networks are typically used in applications where variables might be missing. Given partial information (i. e. the states of a subset of the variables) the goal is to update the beliefs (i. e. the probabilities) of all unknown variables. Whereas there are powerful local update rules for networks of discrete variables without (undirected) loops, the belief update in networks with loops is in general NP-hard. A generally applicable update rule for the unknown variables in networks of discrete or continuous variables is Gibbs sampling. Gibbs sampling can be roughly described as follows: for all variables whose state is known, fix their states to the known values. For all unknown variables choose some initial states. Then pick a variable Xi which is not known and update its value following the probability distribution p(xil{Xl, ... , XN} \ {xd) ex: p(xilPd II p(xjIPj ). (5) x.E1'j Do this repeatedly for all unknown variables. Discard the first samples. Then, the samples which are generated are drawn from the probability distribution of the unknown variables given the known variables. Using these samples it is easy to calculate the expected value of any of the unknown variables, estimate variances, covariances and other statistical measures such as the mutual information between variables. 8 Since the direction of arcs is not unique we used the difference in undirected arcs to compare two structures. We used the number of arcs present in one and only one of the structures normalized with respect to the number of arcs in a fully connected network. 506 R. HOFMANN, V. TRESP Gibbs sampling requires sampling from the univariate probability distribution in Equation 5 which is not straightforward in our model since the conditional density does not have a convenient form. Therefore, sampling techniques such as importance sampling have to be used. In our case they typically produce many rejected samples and are therefore inefficient. An alternative is sampling based on Markov blanket conditional density models. The Markov blanket of Xi, Mi is the smallest set of variables such that P(Xi I{ Xb . .. , XN} \ Xi) = P(Xi IMi) (given a Bayesian network, the Markov blanket of a variable consists of its parents, its children and its children's parents.). The idea is to form a conditional density model pM (xilMd ~ p(xdMd for each variable in the network instead of computing it according to Equation 5. Sampling from this model is simple using conditional Parzen models: the conditional density is a mixture of Gaussians from which we can sample without rejection9 • Markov blanket conditional density models are also interesting if we are only interested in always predicting one particular variable, as in most neural network applications. Assuming that a signal-plus-noise model is a reasonably good model for the conditional density, we can train an ordinary neural network to predict the variable of interest. In addition, we train a model for each input variable predicting it from the remaining variables. In addition to having obtained a model for the complete data case, we can now also handle missing inputs and do backward inference using Gibbs sampling. 7 Conclusions We demonstrated that Bayesian models of local conditional density estimators form promising nonlinear dependency models for continuous variables. The conditional density models can be trained locally if training data are complete. In this paper we focused on the self-organized extraction of structure. Bayesian networks can also serve as a framework for a modular construction of large systems out of smaller conditional density models. The Bayesian framework provides consistent update rules for the probabilities i.e. communication between modules. Finally, consider input pruning or variable selection in neural networks. Note, that our pruning strategy in Figure 1 can be considered a form of variable selection by not only removing variables which are statistically independent of the output variable but also removing variables which are represented well by the remaining variables. This way we obtain more compact models. If input values are missing then the indirect influence of the pruned variables on the output will be recovered by the sampling mechanism. References Buntine, W. (1994). Operations for learning with graphical models. Journal of Artificial Intelligence Research 2: 159-225. Heckerman, D. (1995). A tutorial on learning Bayesian networks. Microsoft Research, TR. MSR-TR-95-06, 1995. Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems. San Mateo, CA: Morgan Kaufmann. 9There are, however, several open issues concerning consistency between the conditional models.
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Temporal coding in the sub-millisecond range: Model of barn owl auditory pathway Richard Kempter* Institut fur Theoretische Physik Physik-Department der TU Munchen D-85748 Garching bei Munchen Germany J. Leo van Hemmen Institut fur Theoretische Physik Physik-Department der TU Munchen 0-85748 Garching bei Munchen Germany Wulfram Gerstner Institut fur Theoretische Physik Physik-Department der TU Munchen D-85748 Garching bei Munchen Germany Hermann Wagner Institut fur Zoologie Fakultiit fur Chemie und Biologie D-85748 Garching bei Munchen Germany Abstract Binaural coincidence detection is essential for the localization of external sounds and requires auditory signal processing with high temporal precision. We present an integrate-and-fire model of spike processing in the auditory pathway of the barn owl. It is shown that a temporal precision in the microsecond range can be achieved with neuronal time constants which are at least one magnitude longer. An important feature of our model is an unsupervised Hebbian learning rule which leads to a temporal fine tuning of the neuronal connections. ·email: kempter.wgerst.lvh@physik.tu-muenchen.de Temporal Coding in the Submillisecond Range: Model of Bam Owl Auditory Pathway 125 1 Introduction Owls are able to locate acoustic signals based on extraction of interaural time difference by coincidence detection [1, 2]. The spatial resolution of sound localization found in experiments corresponds to a temporal resolution of auditory signal processing well below one millisecond. It follows that both the firing of spikes and their transmission along the so-called time pathway of the auditory system must occur with high temporal precision. Each neuron in the nucleus magnocellularis, the second processing stage in the ascending auditory pathway, responds to signals in a narrow frequency range. Its spikes are phase locked to the external signal (Fig. 1a) for frequencies up to 8 kHz [3]. Axons from the nucleus magnocellularis project to the nucleus laminaris where signals from the right and left ear converge. Owls use the interaural phase difference for azimuthal sound localization. Since barn owls can locate signals with a precision of one degree of azimuthal angle, the temporal precision of spike encoding and transmission must be at least in the range of some 10 J.lS. This poses at least two severe problems. First, the neural architecture has to be adapted to operating with high temporal precision. Considering the fact that the total delay from the ear to the nucleus magnocellularis is approximately 2-3 ms [4], a temporal precision of some 10 J.lS requires some fine tuning, possibly based on learning. Here we suggest that Hebbian learning is an appropriate mechanism. Second, neurons must operate with the necessary temporal precision. A firing precision of some 10 J.ls seems truly remarkable considering the fact that the membrane time constant is probably in the millisecond range. Nevertheless, it is shown below that neuronal spikes can be transmitted with the required temporal precision. 2 Neuron model We concentrate on a single frequency channel of the auditory pathway and model a neuron of the nucleus magnocellularis. Since synapses are directly located on the soma, the spatial structure of the neuron can be reduced to a single compartment. In order to simplify the dynamics, we take an integrate-and-fire unit. Its membrane potential changes according to d u -u = -- + 1(t) dt TO (1) where 1(t) is some input and TO is the membrane time constant. The neuron fires, if u(t) crosses a threshold {) = 1. This defines a firing time to. After firing u is reset to an initial value uo = O. Since auditory neurons are known to be fast, we assume a membrane time constant of 2 ms. Note that this is shorter than in other areas of the brain, but still a factor of 4 longer than the period of a 2 kHz sound signal. The magnocellular neuron receives input from several presynaptic neurons 1 ~ k ~ J{. Each input spike at time t{ generates a current pulse which decays exponentially with a fast time constant Tr = 0.02 ms. The magnitude of the current pulse depends on the coupling strength h. The total input is t tf 1(t) = L h: exp( --=-.!. ) O(t - t{) k,f Tr (2) where O(x) is the unit step function and the sum runs over all input spikes. 126 R. KEMPTER, W. GERSTNER, J. L. VAN HEMMEN, H. WAGNER a) foE- T~ /\ h b /\ vvt I I I I I I I I I I I o <p 21t b) t t Fig. 1. Principles of phase locking and learning. a) The stimulus consists of a sound wave (top). Spikes of auditory nerve fibers leading to the nucleus magnocellularis are phase-locked to the periodic wave, that is, they occur at a preferred phase in relation to the sound, but with some jitter 0". Three examples of phase-locked spike trains are indicated. b) Before learning (left), many auditory input fibers converge to a neuron of the nucleus magnocellularis. Because of axonal delays which vary between different fibers, spikes arrive incoherently even though they are generated in a phase locked fashion. Due to averaging over several incoherent inputs, the total postsynaptic potential (bottom left) of a magnocellular neuron follows a rather smooth trajectory with no significant temporal structure. After learning (right) most connections have disappeared and only a few strong contacts remain. Input spikes now arrive coherently and the postsynaptic potential exhibits a clear oscillatory structure. Note that firing must occur during the rising phase of the oscillation. Thus output spikes will be phase locked. Temporal Coding in the Submillisecond Range: Model of Bam Owl Auditory Pathway 127 All input signals belong to the same frequency channel with a carrier frequency of 2 kHz (period T = 0.5 ms), but the inputs arise from different presynaptic neurons (1 ~ k ~ K). Their axons have different diameter and length leading to a signal transmission delay ~k which varies between 2 and 3 ms [4]. Note that a delay as small as 0.25 ms shifts the signal by half a period. Each input signal consists of a periodic spike train subject to two types of noise. First, a presynaptic neuron may not fire regularly every period but, on average, every nth period only where n ~ 1/(vT) and v is the mean firing rate of the neuron. For the sake of simplicity, we set n = 1. Second, the spikes may occur slightly too early or too late compared to the mean delay~. Based on experimental results, we assume a typical shift (1 = ±0.05 ms [3]. Specifically we assume in our model that inputs from a presynaptic neuron k arrive with the probability density P( J) __ 1_ ~ [-(t{ -nT- ~k)2l tk . m= L...t exp 2 v2~(1 2(1 n=-OO (3) where ~k is the axonal transmission delay of input k (Fig. 1). 3 Temporal tuning through learning We assume a developmental period of unsupervised learning during which a fine tuning of the temporal characteristics of signal transmission takes place (Fig. Ib). Before learning the magnocellular neuron receives many inputs (K = 50) with weak coupling (Jk = 1). Due to the broad distribution of delays the tptal input (2) has, apart from fluctuations, no temporal structure. After learning, the magnocellular neuron receives input from two or three presynaptic neurons only. The connections to those neurons have become very effective; cf. Fig. 2. a) <f ,c) <f ,30 20 10 0 2.0 30 20 10 0 2.0 2.5 ~[ms) 2.5 ~[ms) 3.0 3.0 b) <f ,d) <f ,30 20 10 0 2.0 30 20 10 0 2.0 2.5 ~[ms] 2.5 ~[ms) 3.0 3.0 Fig. 2. Learning. We plot the number of synaptic contacts (y-axis) for each delay ~ (x-axis). (a) At the beginning, the neuron has contacts to 50 presynaptic neurons with delays 2ms ~ ~ ~ 3ms. (b) and (c) During learning, some presynaptic neurons increase their number of contacts, other contacts disappear. (d) After learning, contacts to three presynaptic neurons with delays 2.25, 2.28, and 2.8 ms remain. The remaining contacts are very strong. 128 R. KEMPfER, W. GERSTNER, J. L. VAN HEMMEN, H. WAGNER The constant h: measures the total coupling strength between a presynaptic neuron k and the postsynaptic neuron. Values of h: larger than one indicate that several synapses have been formed. It has been estimated from anatomical data that a fully developed magnocellular neuron receives inputs from as few as 1-4 presynaptic neurons, but each presynaptic axon shows multiple branching near the postsynaptic soma and makes up to one hundred synaptic contacts on the soma of the magnocellular neuron[5]. The result of our simulation study is consistent with this finding. In our model, learning leads to a final state with a few but highly effective inputs. The remaining inputs all have the same time delay modulo the period T of the stimulus. Thus, learning leads to reduction of the number of input neurons contacts with a nucleus magnocellularis neuron. This is the fine tuning of the neuronal connections necessary for precise temporal coding (see below, section 4). a) 0.2 X -3: 0.0 o b) 1.0 X -w 0.5 0.0 o t:: j 0.0 5 X [ms] 5 X [ms] 0.5 10 10 Fig. 3. (a) Time window of learning W(x). Along the x-axis we plot the time difference between presynaptic and postsynaptic fiing x = t{ tl:. The window function W(x) has a positive and a negative phase. Learning is most effective, if the postsynaptic spike is late by 0.08 ms (inset). (b) Postsynaptic potential {(x). Each input spike evoked a postsynaptic potential which decays with a time constant of 2 ms. Since synapses are located directly at the soma, the rise time is very fast (see inset). Our learning scenario requires that the rise time of {(x) should be approximately equal to the time x where W(x) has its maximum. In our model, temporal tuning is achieved by a variant of Hebbian learning. In standard Hebbian learning, synaptic weights are changed if pre- and postsynaptic activity occurs simultaneously. In the context of temporal coding by spikes, the concept of (simultaneous activity' has to be refined. We assume that a synapse k is Temporal Coding in the Submillisecond Range: Model of Barn Owl Auditory Pathway 129 changed, if a presynaptic spike t{ and a postsynaptic spike to occur within a time window W(t{ -to). More precisely, each pair of presynaptic and postsynaptic spikes changes a synapse Jk by an amount (4) with a prefactor , = 0.2. Depending on the sign of W( x), a contact to a presynaptic neuron is either increased or decreased. A decrease below Jk = 0 is not allowed. In our model, we assume a function W(x) with two phases; cf. Fig. 3. For x ~ 0, the function W(x) is positive. This leads to a strengthening (potentiation) of the contact with a presynaptic neuron k which is active shortly before or after a postsynaptic spike. Synaptic contacts which become active more than 3 ms later than the postsynaptic spike are decreased. Note that the time window spans several cycles of length T. The combination of decrease and increase balances the average effects of potentiation and depression and leads to a normalization of the number and weight of synapses. Learning is stopped after 50.000 cycles of length T. 4 Temporal coding after learning After learning contacts remain to a small number of presynaptic neurons. Their axonal transmission delays coincide or differ by multiples of the period T. Thus the spikes arriving from the few different presynaptic neurons have approximately the same phase and add up to an input signal (2) which retains, apart from fluctuations, the periodicity of the external sound signal (Fig.4a). a) -. 9-+'" CJ) ~ o b) 1t 21t o 1t 21t Fig. 4. (a) Distribution of input phases after learning. The solid line shows the number of instances that an input spike with phase <p has occured (arbitrary units). The input consists of spikes from the three presynaptic neurons which have survived after learning; cf. Fig. 1 d. Due to the different delays, the mean input phase v(lries slightly between the three input channels. The dashed curves show the phase distribution of the individual channels, the solid line is the sum of the three dashed curves. (b) Distribution of output phases after learning. The histogram of output phases is sharply peaked. Comparison of the position of the maxima of the solid curves in (a) and (b) shows that the output is phase locked to the input with a relative delay fl<p which is related to the rise time of the postsynaptic potential. 130 R. KEMPTER, W. GERSTNER, J. L. VAN HEMMEN, H. WAGNER Output spikes of the magnocellular neuron are generated by the integrate-and-fire process (1). In FigAb we show a histogram of the phases of the output spikes. We find that the phases have a narrow distribution around a peak value. Thus the output is phase locked to the external signal. The width of the phase distribution corresponds to a precision of 0.084 phase cycles which equals 42 jlS for a 2 kHz stimulus. Note that the temporal precision of the output has improved compared to the input where we had three channels with slightly different mean phases and a variation of (T = 50jls each. The increase in the precision is due to the average over three uncorrelated input signals. We assume that the same principles are used during the following stages along the auditory pathway. In the nucleus laminaris several hundred signals are combined. This improves the signal-to-noise ratio further and a temporal precision below 10 jlS could be achieved. 5 Discussion We have demonstrated that precise temporal coding in the microsecond range is possible despite neuronal time constants in the millisecond range. Temporal refinement has been achieved through a slow developmental learning rule. It is a correlation based rule with a time window W which spans several milliseconds. Nevertheless learning leads to a fine tuning of the connections supporting temporal coding with a resolution of 42 jlS. The membrane time constant was set to 2 ms. This is nearly two orders of magnitudes longer than the achieved resolution. In our model, there is only one fast time constant which describes the typical duration of a input current pulse evoked by a presynaptic spike. Our value of Tr = 20 jlS corresponds to a rise time of the postsynaptic potential of 100 jls. This seems to be realistic for auditory neurons since synaptic contacts are located directly on the soma of the postsynaptic neuron. The basic results of our model can also be applied to other areas of the brain and can shed new light on some aspects of temporal coding with slow neurons. Acknowledgments: R.K. holds scholarship of the state of Bavaria. W.G. has been supported by the Deutsche Forschungsgemeinschaft (DFG) under grant number He 1729/22. H.W. is a Heisenberg fellow of the DFG. References [1] L. A. Jeffress, J. Compo Physiol. Psychol. 41, 35 (1948). [2] M. Konishi, Trends Neurosci. 9, 163 (1986). [3] C. E. Carr and M. Konishi, J. Neurosci. 10,3227 (1990). [4] W. E. Sullivan and M. Konishi, J. Neurosci. 4,1787 (1984). [5] C. E. Carr and R. E. Boudreau, J. Compo Neurol. 314, 306 (1991).
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Stable Dynamic Parameter Adaptation Stefan M. Riiger Fachbereich Informatik, Technische Universitat Berlin Sekr. FR 5-9, Franklinstr. 28/29 10587 Berlin, Germany async~cs. tu-berlin.de Abstract A stability criterion for dynamic parameter adaptation is given. In the case of the learning rate of backpropagation, a class of stable algorithms is presented and studied, including a convergence proof. 1 INTRODUCTION All but a few learning algorithms employ one or more parameters that control the quality of learning. Backpropagation has its learning rate and momentum parameter; Boltzmann learning uses a simulated annealing schedule; Kohonen learning a learning rate and a decay parameter; genetic algorithms probabilities, etc. The investigator always has to set the parameters to specific values when trying to solve a certain problem. Traditionally, the metaproblem of adjusting the parameters is solved by relying on a set of well-tested values of other problems or an intensive search for good parameter regions by restarting the experiment with different values. In this situation, a great deal of expertise and/or time for experiment design is required (as well as a huge amount of computing time). 1.1 DYNAMIC PARAMETER ADAPTATION In order to achieve dynamic parameter adaptation, it is necessary to modify the learning algorithm under consideration: evaluate the performance of the parameters in use from time to time, compare them with the performance of nearby values, and (if necessary) change the parameter setting on the fly. This requires that there exist a measure of the quality of a parameter setting, called performance, with the following properties: the performance depends continuously on the parameter set under consideration, and it is possible to evaluate the performance locally, i. e., at a certain point within an inner loop of the algorithm (as opposed to once only at the end of the algorithm). This is what dynamic parameter adaptation is all about. 226 S.M.RUOER Dynamic parameter adaptation has several virtues. It is automatic; and there is no need for an extra schedule to find what parameters suit the problem best. When the notion of what the good values of a parameter set are changes during learning, dynamic parameter adaptation keeps track of these changes. 1.2 EXAMPLE: LEARNING RATE OF BACKPROPAGATION Backpropagation is an algorithm that implements gradient descent in an error function E: IRn ~ llt Given WO E IRn and a fixed '" > 0, the iteration rule is WH1 = wt - ",V E(wt). The learning rate", is a local parameter in the sense that at different stages of the algorithm different learning rates would be optimal. This property and the following theorem make", especially interesting. Trade-off theorem for backpropagation. Let E: JR1l ~ IR be the error function of a neural net with a regular minimum at w· E IRn , i. e., E is expansible into a Taylor series about w· with vanishing gradient V E( w·) and positive definite Hessian matrix H(w·) . Let A denote the largest eigenvalue of H(w·). Then, in general, backpropagation with a fixed learning rate", > 2/ A cannot converge to w· . Proof. Let U be an orthogonal matrix that diagonalizes H(w·), i. e., D := UT H ( w·) U is diagonal. U sing the coordinate transformation x = UT (w - w·) and Taylor expansion, E(w) - E(w·) can be approximated by F(x) := xT Dx/2. Since gradient descent does not refer to the coordinate system, the asymptotic behavior of backpropagation for E near w· is the same as for F near O. In the latter case, backpropagation calculates the weight components x~ = x~(I- Dii",)t at time step t. The diagonal elements Dii are the eigenvalues of H(w·); convergence for all geometric sequences t 1-7 x~ thus requires", < 2/ A. I The trade-off theorem states that, given "', a large class of minima cannot be found, namely, those whose largest eigenvalue of the corresponding Hessian matrix is larger than 2/",. Fewer minima might be overlooked by using a smaller "', but then the algorithm becomes intolerably slow. Dynamic learning-rate adaptation is urgently needed for backpropagation! 2 STABLE DYNAMIC PARAMETER ADAPTATION Transforming the equation for gradient descent, wt+l = wt - ",VE(wt), into a differential equation, one arrives at awt fat = -",V E(wt). Gradient descent with constant step size", can then be viewed as Euler's method for solving the differential equation. One serious drawback of Euler's method is that it is unstable: each finite step leaves the trajectory of a solution without trying to get back to it. Virtually any other differential-equation solver surpasses Euler's method, and there are even some featuring dynamic parameter adaptation [5]. However, in the context of function minimization, this notion of stability ("do not drift away too far from a trajectory") would appear to be too strong. Indeed, differential-equation solvers put much effort into a good estimation of points that are as close as possible to the trajectory under consideration. What is really needed for minimization is asymptotic stability: ensuring that the performance of the parameter set does not decrease at the end of learning. This weaker stability criterion allows for greedy steps in the initial phase of learning. There are several successful examples of dynamic learning-rate adaptation for backpropagation: Newton and quasi-Newton methods [2] as an adaptive ",-tensor; individual learning rates for the weights [3, 8]; conjugate gradient as a one-dimensional ",-estimation [4]; or straightforward ",-adaptation [1, 7]. Stable Dynamic Parameter Adaptation 227 A particularly good example of dynamic parameter adaptation was proposed by Salomon [6, 7]: let ( > 1; at every step t of the backpropagation algorithm test two values for 17, a somewhat smaller one, 17d(, and a somewhat larger one, 17t(; use as 17HI the value with the better performance, i. e., the smaller error: The setting of the new parameter (proves to be uncritical (all values work, especially sensible ones being those between 1.2 and 2.1). This method outperforms many other gradient-based algorithms, but it is nonetheless unstable. b) Figure 1: Unstable Parameter Adaptation The problem arises from a rapidly changing length and direction of the gradient, which can result in a huge leap away from a minimum, although the latter may have been almost reached. Figure 1a shows the niveau lines of a simple quadratic error function E: 1R2 -+ IR along with the weight vectors wo, WI , . .. (bold dots) resulting from the above algorithm. This effect was probably the reason why Salomon suggested using the normalized gradient instead of the gradient, thus getting rid of the changes in the length of the gradient. Although this works much better, Figure 1b shows the instability of this algorithm due to the change in the gradient's direction. There is enough evidence that these algorithms converge for a purely quadratic error function [6, 7]. Why bother with stability? One would like to prove that an algorithm asymptotically finds the minimum, rather than occasionally leaping far away from it and thus leaving the region where the quadratic Hessian term of a globally nonquadratic error function dominates. 3 A CLASS OF STABLE ALGORITHMS In this section, a class of algorithms is derived from the above ones by adding stability. This class provides not only a proof of asymptotic convergence, but also a significant improvement in speed. Let E: IRn -+ IR be an error function of a neural net with random weight vector W O E IRn. Let ( > 1, 170 > 0, 0 < c ~ 1, and 0 < a ~ 1 ~ b. At step t of the algorithm, choose a vector gt restricted only by the conditions gtV E(wt)/Igtllv Ewt I ~ c and that it either holds for all t that 1/1gtl E [a, b) or that it holds for all t that IV E(wt)I/lgtl E [a, b), i. e., the vectors g have a minimal positive projection onto the gradient and either have a uniformly bounded length or are uniformly bounded by the length of the gradient. Note that this is always possible by choosing gt as the gradient or the normalized gradient. Let e: 17 t-t E (wt - 17gt) denote a one-dimensional error function given by E, wt and gt. Repeat (until the gradient vanishes or an upper limit of t or a lower limit Emin 228 S.M.ROOER of E is reached) the iteration WH1 = wt - 'T/tHgt with 'T/* .'T/t(/2 if e(O) < e('T/t() .- 1 + e('T/t() - e(O) 'T/Hl = 'T/t(gt\1 E(wt) (1) 'T/d( if e('T/d() ::; e('T/t() ::; e(O) 'T/t( otherwise. The first case for 'T/Hl is a stabilizing term 'T/*, which definitely decreases the error when the error surface is quadratic, i. e., near a minimum. 'T/* is put into effect when the errOr e(T}t() , which would occur in the next step if'T/t+l = 'T/t( was chosen, exceeds the error e(O) produced by the present weight vector wt . By construction, 'T/* results in a value less than 'T/t(/2 if e('T/t() > e(O); hence, given ( < 2, the learning rate is decreased as expected, no matter what E looks like. Typically, (if the values for ( are not extremely high) the other two cases apply, where 'T/t( and 'T/d ( compete for a lower error. Note that, instead of gradient descent, this class of algorithms proposes a "gt descent," and the vectors gt may differ from the gradient. A particular algorithm is given by a specification of how to choose gt. 4 PROOF OF ASYMPTOTIC CONVERGENCE Asymptotic convergence. Let E: w f-t 2:~=1 AiW; /2 with Ai > O. For all ( > 1, o < c ::; 1, 0 < a ::; 1 ::; b, 'T/o > 0, and WO E IRn, every algorithm from Section :1 produces a sequence t f-t wt that converges to the minimum 0 of E with an at least exponential decay of t f-t E(wt). Proof. This statement follows if a constant q < 1 exists with E(WH1 ) ::; qE(wt) for all t. Then, limt~oo wt = 0, since w f-t ..jE(w) is a norm in IRn. Fix a wt, 'T/t, and a gt according to the premise. Since E is a positive definite quadratic form, e: 'T/ f-t E( wt - 'T/gt) is a one-dimensional quadratic function with a minimum at, say, 'T/*. Note that e(O) = E(wt) and e('T/tH) = E(wt+l). e is completely determined by e(O), e'(O) = -gt\1 E(wt), 'T/te and e('T/t(). Omitting the algebra, it follows that 'T/* can be identified with the stabilizing term of (1). e(O) .A'-~--I qe( 0) -...-...J'----+I (1 - q11)e(0) + q11e('T/*) e"----r-++--+j qee(O) __ ~<-+--+I 11t+~:11· e(O) + (1 11t±~:11· )e('T/*) e( 'T/*) 1--____ ---""' ...... ----A~-_+_--+t e( 'T/tH) o Figure 2: Steps in Estimating a Bound q for the Improvement of E. Stable Dynamic Parameter Adaptation 229 If e(17t() > e(O), by (1) 17t+l will be set to 17·; hence, Wt+l has the smallest possible error e(17·) along the line given by l. Otherwise, the three values 0, 17t!(, and 17t( cannot have the same error e, as e is quadratic; e(17t() or e(17t!() must be less than e(O), and the argument with the better performance is used as 17tH' The sequence t I-t E(wt) is strictly decreasing; hence, a q ~ 1 exists. The rest of the proof shows the existence of a q < 1. Assume there are two constants 0 < qe, qT/ < 1 with E [qT/,2 - qT/] ~ qee(O). Let 17tH ~ 17·; using first the convexity of e, then (2), and (3), one obtains < < < e(17tH -17· 2 • + (1- 17t+l -17·) .) 17. 17 17. 17 17t+l -17· e(O) + (1- 17tH -17· )e(17.) 17· 17· (1 - qT/)e(O) + qf/e(17·) (1- qT/(1 - qe))e(O). (2) (3) Figure 2 shows how the estimations work. The symmetric case 0 < 17tH ~ 17· has the same result E(wt+l) ~ qE(wt) with q := 1 - qT/(1 - qe) < 1. Let ,X < := minPi} and ,X> := max{'xi}. A straightforward estimation for qe yields ,X< qe := 1 - c2 ,X> < 1. Note that 17· depends on wt and gt. A careful analysis of the recursive dependence of 17t+l /17· (wt , gt) on 17t /17·( wt - 1 ,l-l) uncovers an estimation ._ min _2_ ~ ca ~ 17o (,X > 0 ( <) 3/2 < qT/ .{(2 + l' (2 + 1 b'x> , bmax{1, J2'x> E(WO)}} . 5 NON-GRADIENT DIRECTIONS CAN IMPROVE CONVERGENCE • It is well known that the sign-changed gradient of a function is not necessarily the best direction to look for a minimum. The momentum term of a modified backpropagation version uses old gradient directions; Newton or quasi-Newton methods explicitly or implicitly exploit second-order derivatives for a change of direction; another choice of direction is given by conjugate gradient methods [5]. The algorithms from Section 3 allow almost any direction, as long as it is not nearly perpendicular to the gradient. Since they estimate a good step size, these algorithms can be regarded as a sort of "trial-and-error" line search without bothering to find an exact minimum in the given direction, but utilizing any progress made so far. One could incorporate the Polak-Ribiere rule, cttH = \1 E( Wt+l) + a(3ctt, for conjugate directions with dO = \1 E (WO), a = 1, and (\1E(Wt+l) - \1E(wt))\1E(wt+l) (3 = (\1 E(Wt))2 230 S.M. RUOER to propose vectors gt := ett /Iettl for an explicit algorithm from Section 3. As in the conjugate gradient method, one should reset the direction ett after each n (the number of weights) updates to the gradient direction. Another reason for resetting the direction arises when gt does not have the minimal positive projection c onto the normalized gradient. a = 0 sets the descent direction gt to the normalized gradient "V E(wt)/I"V E(wt)lj this algorithm proves to exhibit a behavior very similar to Salomon's algorithm with normalized gradients. The difference lies in the occurrence of some stabilization steps from time to time, which, in general, improve the convergence. Since comparisons of Salomon's algorithm to many other methods have been published [7], this paper confines itself to show that significant improvements are brought about by non-gradient directions, e. g., by Polak-Ribiere directions (a = 1). Table 1: Average Learning Time for Some Problems PROBLEM Emin a = 0 a = 1 (a) 3-2-4 regression 10° 195± 95% 58 ± 70% (b) 3-2-4 approximation 10-4 1070 ± 140% 189± 115% (c) Pure square (n = 76) 10-16 464± 17% 118± 9% (d) Power 1.8 (n = 76) 10-4 486± 29% 84± 23% (e) Power 3.8 (n = 76) 10-16 28 ± 10% 37± 14% (f) 8-3-8 encoder 10-4 1380± 60% 300± 60% Table 1 shows the average number of epochs of two algorithms for some problems. The average was taken over many initial random weight vectors and over values of ( E [1.7,2.1]j the root mean square error of the averaging process is shown as a percentage. Note that, owing to the two test steps for ",t/( and "'t(, one epoch has an overhead of around 50% compared to a corresponding epoch of backpropagation. a f:. 0 helps: it could be chosen by dynamic parameter adaptation. Problems (a) and (b) represent the approximation of a function known only from some example data. A neural net with 3 input, 2 hidden, and 4 output nodes was used to generate the example dataj artificial noise was added for problem (a). The same net with random initial weights was then used to learn an approximation. These problems for feedforward nets are expected to have regular minima. Problem (c) uses a pure square error function E: w rt L:~1 ilwil P /2 with p = 2 and n = 76. Note that conjugate gradient needs exactly n epochs to arrive at the minimum [5]. However, the few additional epochs that are needed by the a = 1 algorithm to reach a fairly small error (here 118 as opposed to 76) must be compared to the overhead of conjugate gradient (one line search per epoch). Powers other than 2, as used in (d) or (e), work well as long as, say, p > 1.5. A power p < 1 will (if n ~ 2) produce a "trap" for the weight vector at a location near a coordinate axis, where, owing to an infinite gradient component, no gradient-based algorithm can escape1 . Problems are expected even for p near 1: the algorithms of Section 3 exploit the fact that the gradient vanishes at a minimum, which in turn is numerically questionable for a power like 1.1. Typical minima, however, employ powers 2,4, ... Even better convergence is expected and found for large powers. IDynamic parameter adaptation as in (1) can cope with the square-root singularity (p = 1/2) in one dimension, because the adaptation rule allows a fast enough decay of the learning rate; the ability to minimize this one-dimensional square-root singularity is somewhat overemphasized in [7]. Stable Dynamic Parameter Adaptation 231 The 8-3-8 encoder (f) was studied, because the error function has global minima at the boundary of the domain (one or more weights with infinite length). These minima, though not covered in Section 4, are quickly found. Indeed, the ability to increase the learning rate geometrically helps these algorithms to approach the boundary in a few steps. 6 CONCLUSIONS It has been shown that implementing asymptotic stability does help in the case of the backpropagation learning rate: the theoretical analysis has been simplified, and the speed of convergence has been improved. Moreover, the presented framework allows descent directions to be chosen flexibly, e. g., by the Polak-Ribiere rule. Future work includes studies of how to apply the stability criterion to other parametric learning problems. References [1] R. Battiti. Accelerated backpropagation learning: Two optimization methods. Complex Systems, 3:331-342, 1989. [2] S. Becker and Y. Ie Cun. Improving the convergence of back-propagation learning with second order methods. In D. Touretzky, G. Hinton, and T. Sejnowski, editors, Proceedings of the 1988 Connectionist Models Summer School, pages 29-37. Morgan Kaufmann, San Mateo, 1989. [3] R. Jacobs. Increased rates of convergence through learning rate adaptation. Neural Networks, 1:295-307, 1988. [4] A. Kramer and A. Sangiovanni-Vincentelli. Efficient parallel learning algorithms for neural networks. In D. Touretzky, editor, Advances in Neural Information Processing Systems 1, pages 40-48. Morgan Kaufmann, San Mateo, 1989. [5] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes in C. Cambridge University Press, 1988. [6] R. Salomon. Verbesserung konnektionistischer Lernverfahren, die nach der Gradientenmethode arbeiten. PhD thesis, TU Berlin, October 1991. [7] R. Salomon and J. L. van Hemmen. Accelerating backpropagation through dynamic self-adaptation. Neural Networks, 1996 (in press). [8] F. M. Silva and L. B. Almeida. Speeding up backpropagation. In Proceedings of NSMS - International Symposium on Neural Networks for Sensory and Motor Systems, Amsterdam, 1990. Elsevier.
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Optimizing Cortical Mappings Geoffrey J. Goodhill The Salk Institute 10010 North Torrey Pines Road La Jolla, CA 92037, USA Steven Finch Human Communication Research Centre University of Edinburgh, 2 Buccleuch Place Edinburgh EH8 9LW, GREAT BRITAIN Terrence J. Sejnowski The Howard Hughes Medical Institute The Salk Institute for Biological Studies 10010 North Torrey Pines Road, La Jolla, CA 92037, USA & Department of Biology, University of California San Diego La Jolla, CA 92037, USA Abstract "Topographic" mappings occur frequently in the brain. A popular approach to understanding the structure of such mappings is to map points representing input features in a space of a few dimensions to points in a 2 dimensional space using some selforganizing algorithm. We argue that a more general approach may be useful, where similarities between features are not constrained to be geometric distances, and the objective function for topographic matching is chosen explicitly rather than being specified implicitly by the self-organizing algorithm. We investigate analytically an example of this more general approach applied to the structure of interdigitated mappings, such as the pattern of ocular dominance columns in primary visual cortex. 1 INTRODUCTION A prevalent feature of mappings in the brain is that they are often "topographic". In the most straightforward case this simply means that neighbouring points on a two-dimensional sheet (e.g. the retina) are mapped to neighbouring points in a more central two-dimensional structure (e.g. the optic tectum). However a more complex case, still often referred to as topographic, is the mapping from an abstract space of features (e.g. position in the visual field, orientation, eye of origin etc) to Optimizing Cortical Mappings 331 the cortex (e.g. layer 4 of VI). In many cortical sensory areas, the preferred sensory stimuli of neighbouring neurons changes slowly, except at discontinuous jumps, suggestive of an optimization principle that attempts to match "similar" features to nearby points in the cortex. In this paper, we (1) discuss what might constitute an appropriate measure of similarity between features, (2) outline an optimization principle for matching the similarity structure of two abstract spaces (i.e. a measure of the degree of topography of a mapping), and (3) use these ideas to analyse the case where two equivalent input variables are mapped onto one target structure, such as the "ocular dominance" mapping from the right and left eyes to VI in the cat and monkey. 2 SIMILARITY MEASURES A much-investigated computational approach to the study of mappings in VI is to consider the input features as pOints in a multidimensional euclidean space [1,5,9]. The input dimensions then consist of e.g. spatial position, orientation, ocular dominance, and so on. Some distribution of points in this space is assumed which attempts, in some sense, to capture the statistics of these features in the visual world. For instance, in [5], distances between points in the space are interpreted as a decreasing function of the degree to which the corresponding features are correlated over an ensemble of images. Some self-organizing algorithm is then applied which produces a mapping from the high-dimensional feature space to a two-dimensional sheet representing the cortex, such that nearby points in the feature space map to nearby points in the two-dimensional sheet.l However, such approaches assume that the dissimilarity structure of the input features is well-captured by euclidean distances in a geometric space. There is no particular reason why this should be true. For instance, such a representation implies that the dissimilarity between features can become arbitrarily large, an unlikely scenario. In addition, it is difficult to capture higher-order relationships in such a representation, such as that two oriented line-segment detectors will be more correlated if the line segments are co-linear than if they are not. We propose instead that, for a set of features, one could construct directly from the statistics of natural stimuli a feature matrix representing similarities or dissimilarities, without regard to whether the resulting relationships can be conveniently captured by distances in a euclidean feature space. There are many ways this could be done; one example is given below. Such a similarity matrix for features can then be optimally matched (in some sense) to a similarity matrix for positions in the output space. A disadvantage from a computational point of view of this generalized approach is that the self-organizing algorithms of e.g. [6,2] can no longer be applied, and possibly less efficient optimization techniques are required. However, an advantage of this is that one may now explore the consequences of optimizing a whole range of objective functions for quantifying the quality of the mapping, rather than having to accept those given explicitly or implicitly by the particular self-organizing algorithm. lWe mean this in a rather loose sense, and wish to include here the principles of mapping nearby points in the sheet to nearby points in the feature space, mapping distant points in the feature space to distant points in the sheet, and so on. 332 G. J. GOODHILL. S. FINCH. T.J. SEJNOWSKI Vout M Figure 1: The mapping framework. 3 OPTIMIZATION PRINCIPLES We now outline a general framework for measuring to what degree a mapping matches the structure of one similarity matrix to that of another. It is assumed that input and output matrices are of the same (finite) dimension, and that the mapping is bijective. Consider an input space Yin and an output space Vout, each of which contains N points. Let M be the mapping from points in Yin to points in Vout (see figure 1). We use the word "space" in a general sense: either or both of Yin and Vout may not have a geometric interpretation. Assume that for each space there is a symmetric "similarity" function which, for any given pair of points in the space, specifies how similar (or dissimilar) they are. Call these functions F for Yin and G for Vout. Then we define a cost functional C as follows N C = L L F(i,j)G(M(i), MO)), (1) i=1 i<i where i and j label pOints in ViT\J and M(i) and M(j) are their respective images in Vout. The sum is over all possible pairs of points in Yin. Since M is a bijection it is invertible, and C can equivalently be written N C = LL F(M-1(i),M-1(j))G(i,j), (2) i=1 i<i where now i and j label points in Vout! and M - I is the inverse map. A good (i.e. highly topographic) mapping is one with a high value of C. However, if one of F or G were given as a dissimilarity function (i.e. increasing with decreasing similarity) then a good mapping would be one with a low value of C. How F and G are defined is problem-specific. C has a number of important properties that help to justify its adoption as a measure of the degree of topography of a mapping (for more details see [3]). For instance, it can be shown that if a mapping that preserves ordering relationships between two similarity matrices exists, then maximizing C will find it. Such maps are homeomorphisms. However not all homeomorphisms have this propert}j so we refer to such "perfect" maps as "topographic homeomorphisms". Several previously defined optimization principles, such as minimum path and minimum Optimizing Cortical Mappings 333 wiring [1], are special cases of C. It is also closely related (under the assumptions above) to Luttrell's minimum distortion measure [7], if F is euclidean distance in a geometric input space, and G gives the noise process in the output space. 4 INTERDIGITATED MAPPINGS As a particular application of the principles discussed so far, we consider the case where the similarity structure of Yin can be expressed in matrix form as where Qs and Qc are of dimension Nil. This means that Yin consists of two halves, each with the same internal similarity structure, and an in general different similarity structure between the two halves. The question is how best to match this dual similarity structure to a single similarity structure in Vout. This is of mathematical interest since it is one of the simplest cases of a mismatch between the similarity structures of V in and Vout! and of biological interest since it abstractly represents the case of input from two equivalent sets of receptors coming together in a single cortical sheet, e.g. ocular dominance columns in primary visual cortex (see e.g. [8, 5]). For simplicity we consider only the case of two one-dimensional retinae mapping to a one-dimensional cortex. The feature space approach to the problem presented in [5] says that the dissimilarities in Yin are given by squared euclidean distances between points arranged in two parallel rows in a two-dimensional space. That is, { I· '12 . . l.- J F(l., J) = Ii _ j _ NIll2 + k2 : i, j in same half of Yin : i, j in different halves of Yin (3) assuming that indices 1 ... Nil give points in one half and indices Nil + 1 ... N give pOints in the other half. G (i, j) is given by G (. .) _ {1 : i, j neighbouring l., J 0 : otherwise (4) It can be shown that the globally optimal mapping (i.e. minimum of C) when k > 1 is to keep the two halves of V in entirely separate in Vout [5]. However, there is also a local minimum for an interdigitated (or "striped") map, where the interdigitations have width n = lk. By varying the value of k it is thus possible to smoothly vary the periodicity of the locally optimal striped map. Such behavior predicted the outcome of a recent biological experiment [4]. For k < 1 the globally optimal map is stripes of width n = 1. However, in principle many alternative ways of measuring the similarity in Yin are possible. One obvious idea is to assume that similarity is given directly by the degree of correlation between points within and between the two eyes. A simple assumption about the form of these correlations is that they are a gaussian function of physical distance between the receptors (as in [8]). That is, { I· '12 . . e- ott-) F(l.,J)= ce-f3li-i-N/211 i, j in same half of Yin i, j in different halves of Yin (5) with c < 1. We assume for ease of analysis that G is still as given in equation 4. This directly implements an intuitive notion put forward to account for the interdigitation of the ocular dominance mapping [4]: that the cortex tries to represent 334 G. J. GOODHILL, S. FINCH, TJ. SEJNOWSKI similar inputs close together, that similarity is given by the degree of correlation between the activities of points (cells), and additionally that natural visual scenes impose a correlational structure of the same qualitative form as equation 5. We now calculate C analytically for various mappings (c.f. [5]), and compare the cost of a map that keeps the two halves of Yin entirely separate in Vout to those which interdigitate the two halves of Yin with some regular periodicity. The map of the first type we consider will be refered to as the "up and down" map: moving from one end of Vout to the other implies moving entirely through one half of ViT\l then back in the opposite direction through the other half. For this map, the cost Cud is given by Cud = 2(N - l)e- ct + c. (6) For an interdigitated (striped) map where the stripes are of width n ~ 2: Cs(n) = N [2 (1 - ~) e- ct + ~ (e-~f(n) + e-~g(n))] (7) where for n even f(n) = g(n) = (n"22)2 and for n odd f(n) = (n"2I)2, g(n) = (n"23 ) 2. To characterize this system we now analyze how the n for which C s ( n) has a local maximum varies with c, a., 13, and when this local maximum is also a global maximum. Setting dCci£n) = 0 does not yield analytically tractable expressions (unlike [5]). However, more direct methods can be used: there is a local maximum atnifCs(n-1) < Cs(n) > Cs(n+ 1). Using equation 7we derive conditions on C for this to be true. For n odd, we obtain the condition CI < C < C2 where CI = C2; that is, there are no local maxima at odd values of n. For n even, we also obtain CI < C < C2 where now 2e- ct CI = n-4 2 n-2 2 ne-~(-z) - (n 2)e-~(-z) and c2(n) = CI (n + 2). CI (n) and c2(n) are plotted in figure 2, from which one can see the ranges of C for which particular n are local maxima. As 13 increases, maxima for larger values of n become apparent, but the range of c for which they exist becomes rather small. It can be shown that Cud is always the global maximum, except when e- ct > c, when n = 2 is globally optimal. As C decreases the optimal stripe width gets wider, analogously to k increasing in the dissimilarities given by equation 3. When 13 is such that there is no local maximum the only optimum is stripes as wide as possible. This fits with the intuitive idea that if corresponding points in the two halves of Yin (Le. Ii - j I = N/2) are sufficiently similar then it is favorable to interdigitate the two halves in VoutJ otherwise the two halves are kept completely separate. The qualitative behavior here is similar to that for equation 3. n = 2 is a global optimum for large c (small k), then as C decreases (k increases) n = 2 first becomes a local optimum, then the position of the local optimum shifts to larger n. However, ~n important difference is that in equation 3 the dissimilarities increase without limit with distance, whereas in equation 5 the similarities tend to zero with distance. Thus for equation 5 the extra cost of stripes one unit wider rapidly becomes negligible, whereas for equation 3 this extra cost keeps on increasing by ever larger amounts. As n -+ 00, Cud'" Cs(n) for the similarities defined by equation 5 (i.e. there is the same cost for traversing the two blocks in the same direction as in the opposite direction), whereas for the dissimilarities defined by equation 3 there is a quite different cost in these two cases. That F and G should tend to a bounded value as i and j become ever more distant neighbors seems biologically more plausible than that they should be potentially unbounded. Optimizing Cortical Mappings (a) '" 1.0 "" ~ 0.9 <.> 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 2 · · • · • · · o D cl c:2 4 6 8 10 12 14 n (b) '" 1.0 .t< ~ 0.9 <.> 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 335 , • , , . , . • , , , • . . . ...... ,' cl c:2 0 n Figure 2: The ranges of c for which particular n are local maxima. (a) oc = f3 = 0.25. (b) oc = 0.25, i3 = 0.1. When the Cl (dashed) line is below the c, (solid) line no local maxima exist. For each (even) value of n to the left of the crossing point, the vertical range between the two lines gives the values of c for which that n is a local maximum. Below the solid line and to the right of the crossing point the only maximum is stripes as wide as possible. Issues such as those we have addressed regarding the transition from "striped" to "blocked" solutions for combining two sets of inputs distinguished by their intraand inter-population similarity structure may be relevant to understanding the spatial representation of functional attributes across cortex. The results suggest the hypothesis that two variables are interdigitated in the same area rather than being represented separately in two distinct areas if the inter-population similarity is sufficiently high. An interesting point is that the striped solutions are often only local optima. It is possible that in reality developmental constraints (e.g. a chemically defined bias towards overlaying the two projections) impose a bias towards finding a striped rather than blocked solution, even though the latter may be the global optimum. 5 DISCUSSION We have argued that, in order to understand the structure of mappings in the brain, it could be useful to examine more general measures of similarity and of topographic matching than those implied by standard feature space models. The consequences of one particular alternative set of choices has been examined for the case of an interdigitated map of two variables. Many alternative objective functions for topographic matching are of course possible; this topic is reviewed in [3]. Two issues we have not discussed are the most appropriate way to define the features of interest, and the most appropriate measures of similarity between features (see [10] for an interesting discussion). A next step is to apply these methods to more complex structures in VI than just the ocular dominance map. By examining more of the space of possibilities than that occupied by the current feature space models, we hope to understand more about the optimization strategies that might be being pursued by the cortex. Feature space models may still tum out to be more or less the right answer; however even if this is true, our approach will at least give a deeper level of understanding why. 336 G. 1. GOODHILL, S. FINCH, T.l. SEINOWSKI Acknowledgements We thank Gary Blasdel, Peter Dayan and Paul Viola for stimulating discussions. References [1] Durbin, R. & Mitchison, G. (1990). A dimension reduction framework for understanding cortical maps. Nature, 343, 644-647. [2] Durbin, R. & Willshaw, D.J. (1987). An analogue approach to the travelling salesman problem using an elastic net method. Nature, 326,689-691. [3] Goodhill, G. J., Finch, S. & Sejnowski, T. J. (1995). Quantifying neighbourhood preservation in topographic mappings. Institute for Neural Computation Technical Report Series, No. INC-9505, November 1995. Available from ftp:/ / salk.edu/pub / geoff/ goodhillJinch_sejnowski_tech95.ps.Z or http://cnl.salk.edu/ ""geoff. [4] Goodhill, G.J. & Lowel, S. (1995). Theory meets experiment: correlated neural activity helps determine ocular dominance column periodicity. Trends in Neurosciences, 18,437-439. [5] Goodhill, G.J. & Willshaw, D.J. (1990). Application of the elastic net algorithm to the formation of ocular dominance stripes. Network, 1, 41-59. [6] Kohonen, T. (1982). Self-organized formation of topologically correct feature maps. Bioi. Cybern., 43, 59-69. [7] Luttrell, S.P. (1990). Derivation of a class of training algorithms. IEEE Trans. Neural Networks, 1,229-232. [8] Miller, KD., Keller, J.B. & Stryker, M.P. (1989). Ocular dominance column development: Analysis and simulation. Science, 245, 605-615. [9] Obermayer, K, Blasdel, G.G. & Schulten, K (1992). Statistical-mechanical analysis of self-organization and pattern formation during the development of visual maps. Phys. Rev. A, 45, 7568-7589. [10] Weiss, Y. & Edelman, S. (1995). Representation of similarity as a goal of early sensory coding. Network, 6, 19-41.
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Experiments with Neural Networks for Real Time Implementation of Control P. K. Campbell, M. Dale, H. L. Ferra and A. Kowalczyk Telstra Research Laboratories 770 Blackburn Road Clayton, Vic. 3168, Australia {p.campbell, m.dale, h.ferra, a.kowalczyk}@trl.oz.au Abstract This paper describes a neural network based controller for allocating capacity in a telecommunications network. This system was proposed in order to overcome a "real time" response constraint. Two basic architectures are evaluated: 1) a feedforward network-heuristic and; 2) a feedforward network-recurrent network. These architectures are compared against a linear programming (LP) optimiser as a benchmark. This LP optimiser was also used as a teacher to label the data samples for the feedforward neural network training algorithm. It is found that the systems are able to provide a traffic throughput of 99% and 95%, respectively, of the throughput obtained by the linear programming solution. Once trained, the neural network based solutions are found in a fraction of the time required by the LP optimiser. 1 Introduction Among the many virtues of neural networks are their efficiency, in terms of both execution time and required memory for storing a structure, and their practical ability to approximate complex functions. A typical drawback is the usually "data hungry" training algorithm. However, if training data can be computer generated off line, then this problem may be overcome. In many applications the algorithm used to generate the solution may be impractical to implement in real time. In such cases a neural network substitute can become crucial for the feasibility of the project. This paper presents preliminary results for a non-linear optimization problem using a neural network. The application in question is that of capacity allocation in an optical communications network. The work in this area is continuing and so far we have only explored a few possibilities. 2 Application: Bandwidth Allocation in SDH Networks Synchronous Digital Hierarchy (SDH) is a new standard for digital transmission over optical fibres [3] adopted for Australia and Europe equivalent to the SONET (Synchronous Optical NETwork) standard in North America. The architecture of the particular SDH network researched in this paper is shown in Figure 1 (a). 1) Nodes at the periphery of the SDH network are switches that handle individual calls. 974 P. CAMPBELL, M. DALE, H. L. FERRA, A. KOWALCZYK 2) Each switch concentrates traffic for another switch into a number of streams. 3) Each stream is then transferred to a Digital Cross-Connect (DXC) for switching and transmission to its destination by allocating to it one of several alternative virtual paths. The task at hand is the dynamic allocation of capacities to these virtual paths in order to maximize SDH network throughput. This is a non-linear optimization task since the virtual path capacities and the constraints, i.e. the physical limit on capacity of links between DXC's, are quantized, and the objective function (Erlang blocking) depends in a highly non-linear fashion on the allocated capacities and demands. Such tasks can be solved 'optimally' with the use of classical linear programming techniques [5], but such an approach is time-consuming - for large SDH networks the task could even require hours to complete. One of the major features of an SDH network is that it can be remotely reconfigured using software controls. Reconfiguration of the SDH network can become necessary when traffic demands vary, or when failures occur in the DXC's or the links connecting them. Reconfiguration in the case of failure must be extremely fast, with a need for restoration times under 60 ms [1]. o DXC (Digital ® Switch Cross-Connect) Figure 1 (b) link offered capacities traffic output: path capacities synaptic weights (22302) hidden units: 'AND' gates (l10) thresholds (738,67 used) input (a) Example of an Inter-City SDH/SONET Network Topology used in experiments. (b) Example of an architecture of the mask perceptron generated in experiments. In our particular case, there are three virtual paths allocated between any pair of switches, each using a different set of links between DXC's of the SDH network. Calls from one switch to another can be sent along any of the virtual paths, leading to 126 paths in total (7 switches to 6 other switches, each with 3 paths). The path capacities are normally set to give a predefined throughput. This is known as the "steady state". If links in the SDH network become partially damaged or completely cut, the operation of the SDH network moves away from the steady state and the path capacities must be reconfigured to satisfy the traffic demands subject to the following constraints: (i) Capacities have integer values (between 0 and 64 with each unit corresponding to a 2 Mb/s stream, or 30 Erlangs), (ii) The total capacity of all virtual paths through anyone link of the SDH network Experiments with Neural Networks for Real Time Implementation of Control 975 cannot exceed the physical capacity of that link. The neural network training data consisted of 13 link capacities and 42 traffic demand values, representing situations in which the operation of one or more links is degraded (completely or partially). The output data consisted of 126 integer values representing the difference between the steady state path capacities and the final allocated path capacities. 3 Previous Work The problem of optimal SDH network reconfiguration has been researched already. In particular Gopal et. al. proposed a heuristic greedy search algorithm [4] to solve this nonlinear integer programming problem. Herzberg in [5] reformulated this non-linear integer optimization problem as a linear programming (LP) task, Herzberg and Bye in [6] investigated application of a simplex algorithm to solve the LP problem, whilst Bye [2] considered an application of a Hopfield neural network for this task, and finally Leckie [8] used another set of AI inspired heuristics to solve the optimization task. All of these approaches have practical deficiencies; the linear programming is slow, while the heuristic approaches are relatively inaccurate and the Hopfield neural network method (simulated on a serial computer) suffers from both problems. In a previous paper Campbell et al. [10] investigated application of a mask perceptron to the problem of reconfiguration for a "toy" SDH network. The work presented here expands on the work in that paper, with the idea of using a second stage mask perceptron in a recurrent mode to reduce link violationslunderutilizations. 4 The Neural Controller Architecture Instead of using the neural network to solve the optimization task, e.g. as a substitute for the simplex algorithm, it is taught to replicate the optimal LP solution provided by it. We decided to use a two stage approach in our experiments. For the first stage we developed a feedforward network able to produce an approximate solution. More precisely, we used a collection of 2000 random examples for which the linear programming solution of capacity allocations had been pre-computed to develop a feedforward neural network able to approximate these solutions. Then, for a new example, such an "approximate" neural network solution was rounded to the nearest integer, to satisfy constraint (i), and used to seed the second stage providing refinement and enforcement of constraint (ii). For the second stage experiments we initially used a heuristic module based on the Gopal et al. approach [4]. The heuristic firstly reduces the capacities assigned to all paths which cause a physical capacity violation on any links, then subsequently increases the capacities assigned to paths across links which are being under-utilized. We also investigated an approach for the second stage which uses another feedforward neural network. The teaching signal for the second stage neural network is the difference between the outputs from the first stage neural network alone and the combined first stage neural networkiheuristic solution. This time the input data consisted of 13 link usage values (either a link violation or underutilization) and 42 values representing the amount of traffic lost per path for the current capacity allocations. The second stage neural network had 126 outputs representing the correction to the first stage neural network's outputs. The second stage neural network is run in a recurrent mode, adjusting by small steps the currently allocated link capacities, thereby attempting to iteratively move closer to the combined neural-heuristic solution by removing the link violations and under-utilizations left behind by the first stage network. The setup used during simulation is shown in Figure 2. For each particular instance tested the network was initialised with the solution from the first stage neural network. The offered traffic (demand) and the available maximum link capacities were used to determine the extent of any link violations or underutilizations as well as the amount of lost traffic (demand satisfaction). This data formed the initial input to the second stage network. The outputs of the neural network were then used to check the quality of the 976 P. CAMPBELL, M. DALE, H. L. FERRA, A. KOWALCZYK solution, and iteration continued until either no link violations occurred or a preset maximum number of iterations had been performed. offered traffic link capacities computation of constraint -demand satisfaction [ ........ ~ ........ -..... -----~(+) ! solution (t-l) ! I ! initialization: solution (0) from stage 1 correction (t) demand satisfaction (t-l 42 inputs link capacities violation!underutilization (t-l) 13 inputs Figure 2. Recurrent Network used for second stage experiments. solution (t) When computing the constraint satisfaction the outputs of the neural network where combined and rounded to give integer link violations/under-utilizations. This means that in many cases small corrections made by the network are discarded and no further improvement is possible. In order to overcome this we introduced a scheme whereby errors (link violations/under-utilizations) are occasionally amplified to allow the network a chance of removing them. This scheme works as follows: 1) an instance is iterated until it has either no link violations or until 10 iterations have been performed; 2) if any link violations are still present then the size of the errors are multiplied by an amplification factor (> 1); 3) a further maximum of 10 iterations are performed; 4) if subsequently link violations persist then the amplification factor is increased; the procedure repeats until either all link violations are removed or the amplification factor reaches some fixed value. S Description of Neural Networks Generated The first stage feedforward neural network is a mask perceptron [7], c.f. Figure 1 (b). Each input is passed through a number of arbitrarily chosen binary threshold units. There were a total of 738 thresholds for the 55 inputs. The task for the mask perceptron training algorithm [7] is to select a set of useful thresholds and hidden units out of thousands of possibilities and then to set weights to minimize the mean-square-error on the training set. The mask perceptron training algorithm automatically selected 67 of these units for direct connection to the output units and a further 110 hidden units ("AND" gates) whose Experiments with Neural Networks for Real Time Implementation of Control 977 outputs are again connected to the neural network outputs, giving 22,302 connections in all. Such neural networks are very rapid to simulate since the only operations required are comparison and additions. For the recurrent network used in the second stage we also used a mask perceptron. The training algori thIn used for the recurrent network was the same as for the first stage, in particular note that no gradual adaptation was employed. The inputs to the network are passed through 589 arbitrarily chosen binary threshold units. Of these 35 were selected by the training algorithm for direct connection to the output units via 4410 weighted links. 6 Results The results are presented in Table 1 and Figure 3. The values in the table represent the traffic throughput of the SDH network, for the respective methods, as a percentage of the throughput determined by the LP solution. Both the neural networks were trained using 2000 instances and tested against a different set of 2000 instances. However for the recurrent network approximately 20% of these cases still had link violations after simulation so the values in Table 1 are for the 80% of valid solutions obtained from either the training or test set. Solution type Training Test Feedforward Net/Heuristic 99.08% 98.90%, Feedforward Net/Recurrent Net 94.93% (*) 94.76%(*) Gopal-S 96.38% 96.20% Gopal-O 85.63% 85.43% (*) these numbers are for the 1635 training and 1608 test instances (out of 2000) for which the recurrent network achieved a solution with no link violations after simulation as described in Section 3. Table 1. Efficiency of solutions measured by average fraction of the 'optimal' throughput of the LP solution As a comparison we implemented two solely heuristic algorithms. We refer to these as Gopal-S and Gopal-O. Both employ the same scheme described earlier for the Gopal et al. heuristic. The difference between the two is that Gopal-S uses the steady state solution as an initial starting point to determine virtual path capacities for a degraded network, whereas Gopal-O starts from a point where all path capacities are initially set to zero. Referring to Figure 3, link capacity ratio denotes the total link capacity of the degraded SDH network relative to the total link capacity of the steady state SDH network. A low value of link capacity ratio indicates a heavily degraded network. The traffic throughput ratio denotes the ratio between the throughput obtained by the method in question, and the throughput of the steady state solution. Each dot in the graphs in Figure 3 represents one of the 2000 test set cases. It is clear from the figure that the neural network/heuristic approach is able to find better solutions for heavily degraded networks than each of the other approaches. Overall the clustering of dots for the neural network/heuristic combination is tighter (in the y-direction) and closer to 1.00 than for any of the other methods. The results for the recurrent network are very encouraging being qUalitatively quite close to those for the Gopal-S algorithm. All experiments were run on a SPARCStation 20. The neural network training took a few minutes. During simulation the neural network took an average of 9 ms per test case with a further 36.5 ms for the heuristic, for a total of 45.5 ms. On average the Gopal-S algorithm required 55.3 ms and the Gopal-O algorithm required 43.7 ms per test case. The recurrent network solution required an average of 55.9 ms per test case. The optimal solutions calculated using the linear programming algorithm took between 2 and 60 seconds per case on a SPARCStation 10. 978 P. CAMPBELL, M. DALE, H. L. FERRA, A. KOWALCZYK Neural Network/Heuristic Recurrent Neural Network 1.00 .2 ~ 0.95 8. 0.90 .r: 0> is 0.85 .c t.!.! 0.80 ~ ~ 0.75 0.70 0.50 0.60 0.10 0.80 0.90 1.00 link Capacity Ratio 1.00 .2 ra 0.95 cr ~ 0.90 .r: 0> 6 0.85 .c t-,g 0.80 ~ ~ 0.75 Gopal-S ······:····:· · :,,~i~ffI~ .-. -,. " " - -' - "~':' ........ ~ ...... -... --- -... . 0.70 0.50 0.60 0.70 0.80 0.90 1.00 link Capacity Ratio •• , _ ._0 •• _ • •• • • • •• : • ••• :.' ••• :.~' •• : • •••• :. '0"" _ •• • •• _ •••••••• 0.70 0.50 0.60 0.70 0.80 0.90 1.00 Link Capacity Ratio 1.00 .2 r.; 0.95 cr ~ 0.90 .r: 0> 5 0.85 .c t. ~ 0.80 ~ ~ 0.75 Gopal-O 0.70 0.50 0.60 0.70 0.80 0.90 100 Link Capacily Ratio Figure 3. Experimental results for the Inter-City SDH network (Fig. 1) on the independent test set of 2000 random cases. On the x axis we have the ratio between the total link capacity of the degraded SDH network and the steady state SDH network. On the y axis we have the ratio between the throughput obtained by the method in question, and the throughput of the steady state solution. Fig 3. (a) shows results for the neural network combined with the heuristic second stage. Fig 3. (b) shows results for the recurrent neural network second stage. Fig 3. (c) shows results for the heuristic only, initialised by the steady state (Gopal-S) and Fig 3. (d) has the results for the heuristic initialised by zero (Gopal-O). 7 Discussion and Conclusions The combined neural network/heuristic approach performs very well across the whole range of degrees of SDH network degradation tested. The results obtained in this paper are consistent with those found in [10]. The average accuracy of -99% and fast solution generation times « ffJ ms) highlight this approach as a possible candidate for implementation in a real system, especially when one considers the easily achievable speed increase available from parallelizing the neural network. The mask perceptron used in these experiments is well suited for simulation on a DSP (or other hardware): the operations required are only comparisons, calculation of logical "AND" and the summation of synaptic weights (no multiplications or any non-linear transfonnations are required). The interesting thing to note is the relatively good perfonnance of the recurrent network, namely that it is able to handle over 80% of cases achieving very good perfonnance when compared against the neural network/heuristic solution (95% of the quality of the teacher). One thing to bear in mind is that the heuristic approach is highly tuned to producing a solution which satisfies the constraints, changing the capacity of one link at a time until the desired goal is achieved. On the other hand the recurrent network is generic and does not target the constraints in such a specific manner, making quite crude global changes in Experiments with Neural Networks for Real Time Implementation of Control 979 one hit, and yet is still able to achieve a reasonable level of performance. While the speed for the recurrent network was lower on average than for the heuristic solution in our experiments, this is not a major problem since many improvements are still possible and the results reported here are only preliminary, but serve to show what is possible. It is planned to continue the SOH network experiment in the future; with more investigation on the recurrent network for the second stage and also more complex SDH architectures. Acknowledgments The research and development reported here has the active support of various sections and individuals within the Telstra Research Laboratories (TRL), especially Dr. C. Leckie, Mr. P. Sember, Dr. M. Herzberg, Mr. A. Herschtal and Dr. L. Campbell. The permission of the Managing Director, Research and Information Technology, Telstra, to publish this paper is acknowledged. The research and development reported here has the active support of various sections and individuals within the Telstra Research Laboratories (TRL), especially Dr. C. Leckie and Mr. P. Sember who were responsible for the creation and trialling of the programs designed to produce the testing and training data. The SOH application was possible due to co-operation of a number of our colleagues in TRL, in particular Dr. L. Campbell (who suggested this particular application), Dr. M. Herzberg and Mr. A. Herschtal. The permission of the Managing Director, Research and Information Technology, Telstra, to publish this paper is acknowledged. References [1] E. Booker, Cross-connect at a Crossroads, Telephony, Vol. 215, 1988, pp. 63-65. [2] S. Bye, A Connectionist Approach to SDH Bandwidth Management, Proceedings of the 19th International Conference on Artificial Neural Networks (ICANN-93), Brighton Conference Centre, UK, 1993, pp. 286-290. [3] R. Gillan, Advanced Network Architectures Exploiting the Synchronous Digital Hierarchy, Telecommunications Journal of Australia 39, 1989, pp. 39-42. [4] G. Gopal, C. Kim and A. Weinrib, Algorithms for Reconfigurable Networks, Proceedings of the 13th International Teletraffic Congress (ITC-13), Copenhagen, Denmark, 1991, pp. 341-347. [5] M. Herzberg, Network Bandwidth Management - A New Direction in Network Management, Proceedings of the 6th Australian Teletraffic Research Seminar, Wollongong, Australia, pp. 218-225. [6] M. Herzberg and S. Bye, Bandwidth Management in Reconfigurable Networks, Australian Telecommunications Research 27, 1993, pp 57-70. [7] A. Kowalczyk and H.L. Ferra, Developing Higher Order Networks with Empirically Selected Units, IEEE Transactions on Neural Networks, pp. 698-711, 1994. [8] C. Leckie, A Connectionist Approach to Telecommunication Network Optimisation, in Complex Systems: Mechanism of Adaptation, R.J. Stonier and X.H. Yu, eds., lOS Press, Amsterdam, 1994. [9] M. Schwartz, Telecommunications Networks, Addison-Wesley, Readings, Massachusetts, 1987. [10] p. Campbell, H.L. Ferra, A. Kowalczyk, C. Leckie and P. Sember, Neural Networks in Real Time Decision Making, Proceedings of the International Workshop on Applications of Neural Networks to Telecommunications 2 (IWANNT-95), Ed. J Alspector et. al. Lawrence Erlbaum Associates, New Jersey, 1995, pp. 273-280.
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Cholinergic suppression of transmission may allow combined associative memory function and self-organization in the neocortex. Michael E. Hasselmo and Milos Cekic Department of Psychology and Program in Neurosciences, Harvard University, 33 Kirkland St., Cambridge, MA 02138 hasselmo@katIa.harvard.edu Abstract Selective suppression of transmission at feedback synapses during learning is proposed as a mechanism for combining associative feedback with self-organization of feed forward synapses. Experimental data demonstrates cholinergic suppression of synaptic transmission in layer I (feedback synapses), and a lack of suppression in layer IV (feedforward synapses). A network with this feature uses local rules to learn mappings which are not linearly separable. During learning, sensory stimuli and desired response are simultaneously presented as input. Feedforward connections form self-organized representations of input, while suppressed feedback connections learn the transpose of feedforward connectivity. During recall, suppression is removed, sensory input activates the self-organized representation, and activity generates the learned response. 1 INTRODUCTION The synaptic connections in most models of the cortex can be defined as either associative or self-organizing on the basis of a single feature: the relative infl uence of modifiable synapses on post-synaptic activity during learning (figure 1). In associative memories, postsynaptic activity during learning is determined by nonmodifiable afferent input connections, with no change in the storage due to synaptic transmission at modifiable synapses (Anderson, 1983; McNaughton and Morris, 1987). In self-organization, post-synaptic activity is predominantly influenced by the modifiable synapses, such that modification of synapses influences subsequent learning (Von der Malsburg, 1973; Miller et al., 1990). Models of cortical function must combine the capacity to form new representations and store associations between these representations. Networks combining self-organization and associative memory function can learn complex mapping functions with more biologically plausible learning rules (Hecht-Nielsen, 1987; Carpenter et al., 1991; Dayan et at., 132 M. E. HASSELMO, M. CEKIC 1995), but must control the influence of feedback associative connections on self-organization. Some networks use special activation dynamics which prevent feedback from influencing activity unless it coincides with feedforward activity (Carpenter et al., 1991). A new network alternately shuts off feedforward and feedback synaptic transmission (Dayan et al., 1995). A. c. Self-organizing Afferent Self-organizing feedforward Associative feedback Figure 1 - Defining characteristics of self-organization and associative memory. A. At self-organizing synapses, post-synaptic activity during learning depends predominantly upon transmission at the modifiable synapses. B. At synapses mediating associative memory function, post-synaptic activity during learning does not depend primarily on the modifiable synapses, but is predominantly influenced by separate afferent input. C. Selforganization and associative memory function can be combined if associative feedback synapses are selectively suppressed during learning but not recall. Here we present a model using selective suppression of feedback synaptic transmission during learning to allow simultaneous self-organization and association between two regions. Previous experiments show that the neuromodulator acetylcholine selectively suppresses synaptic transmission within the olfactory cortex (Hasselmo and Bower, 1992; 1993) and hippocampus (Hasselmo and Schnell, 1994). If the model is valid for neocortical structures, cholinergic suppression should be stronger for feedback but not feedforward synapses. Here we review experimental data (Hasselmo and Cekic, 1996) comparing cholinergic suppression of synaptic transmission in layers with predominantly feedforward or feedback synapses. 2. BRAIN SLICE PHYSIOLOGY As shown in Figure 2, we utilized brain slice preparations of the rat somatosensory neocortex to investigate whether cholinergic suppression of synaptic transmission is selective for feedback but not feedforward synaptic connections. This was possible because feedforward and feedback connections show different patterns of termination in neocortex. As shown in Figure 2, Layer I contains primarily feedback synapses from other cortical regions (Cauller and Connors, 1994), whereas layer IV contains primarily afferent synapses from the thalamus and feedforward synapses from more primary neocortical structures (Van Essen and Maunsell, 1983). Using previously developed techniques (Cauller and Connors, 1994; Li and Cauller, 1995) for testing of the predominantly feedback connections in layer I, we stimulated layer I and recorded in layer I (a cut prevented spread of Cholinergic Suppression of Transmission in the Neocortex 133 activity from layers II and III). For testing the predominantly feedforward connections terminating in layer IV, we elicited synaptic potentials by stimulating the white matter deep to layer VI and recorded in layer IV. We tested suppression by measuring the change in height of synaptic potentials during perfusion of the cholinergic agonist carbachol at lOOJ,1M. Figure 3 shows that perfusion of carbachol caused much stronger suppression of synaptic transmission in layer I as compared to layer IV (Hasselmo and Cekic, 1996), suggesting that cholinergic suppression of transmission is selective for feedback synapses and not for feedforward synapses. I White matter / stimulation 1/ Layer IV recording I ~~ I ll-ill ll-ill IV .1 Foedback IV V-VI V-VI Region 1 Region 2 Figure 2. A. Brain slice preparation of somatosensory cortex showing location of stimulation and recording electrodes for testing suppression of synaptic transmission in layer I and in layer IV. Experiment based on procedures developed by Cauller (Cauller and Connors, 1994; Li and Cauller, 1995). B. Anatomical pattern of feedforward and feedback connectivity within cortical structures (based on Van Essen and Maunsell, 1983). Feedforward -layer IV Control Carbachol (1 OOJlM) Wash I~ -0 5ms Feedback - layer I '!oi' Control Carbachol (1 OOJlM) Wash Figure 3 - Suppression of transmission in somatosensory neocortex. Top: Synaptic potentials recorded in layer IV (where feedforward and afferent synapses predominate) show little effect of l00J.tM carbachol. Bottom: Synaptic potentials recorded in layer I (where feedback synapses predominate) show suppression in the presence of lOOJ,1M carbachol. 134 M. E. HASSELMO, M. CEKIC 3. COMPUTATIONAL MODELING These experimental results supported the use of selective suppression in a computational model (Hasselmo and Cekic, 1996) with self-organization in its feedforward synaptic connections and associative memory function in its feedback synaptic connections (Figs 1 and 4). The proposed network uses local, Hebb-type learning rules supported by evidence on the physiology of long-tenn potentiation in the hippocampus (Gustafsson and Wigstrom, 1986). The learning rule for each set of connections in the network takes the fonn: tlWS:'Y) = 11 (a?) - 9(Y» g (ar» Where W(x. Y) designates the connections from region x to region y, 9 is the threshold of synaptic modification in region y, 11 is the rate of modification, and the output function is g(a;.(x~ = [tanh(~(x) - J.1(x~]+ where []+ represents the constraint to positive values only. Feedforward connections (Wi/x<y» have self-organizing properties, while feedback connections (Wir>=Y~ have associative memory properties. This difference depends entirely upon the selective suppression of feedback synapses during learning, which is implemented in the activation rule in the form (I-c). For the entire network, the activation rule takes the fonn: M II(X) N II(X) II(Y) a?) = A?) + 2, 2, Wi~<Y) g (a~x» + 2, 2, (1- c) Wi~~Y) g (a~x» - 2, Hi~) (g (af») x=lk=l x=lk=l k=l where a;.(y) represents the activity of each of the n(y) neurons in region y, ~ (x) is the activity of each of the n(x) neurons in other regions x, M is the total number of regions providing feedforward input, N is the total number of regions providing feedback input, Aj(y) is the input pattern to region y, H(Y) represents the inhibition between neurons in region y, and (1 - c) represents the suppression of synaptic transmission. During learning, c takes a value between 0 and 1. During recall, suppression is removed, c = O. In this network, synapses (W) between regions only take positive values, reflecting the fact that long-range connections between cortical regions consist of excitatory synapses arising from pyramidal cells. Thus, inhibition mediated by the local inhibitory interneurons within a region is represented by a separate inhibitory connectivity matrix H. After each step of learning, the total weight of synaptic connections is nonnalized pre-synaptically for each neuron j in each region: ~-------------Wij (t+l) = [Wij(t) + l1W;j(t)]I( .i [Wij(t) +l1Wij (t)] 2) 1= 1 Synaptic weights are then normalized post-synaptically for each neuron i in each region (replacing i with j in the sum in the denominator in equation 3). This nonnalization of synaptic strength represents slower cellular mechanisms which redistribute pre and postsynaptic resources for maintaining synapses depending upon local influences. In these simulations, both the sensory input stimuli and the desired output response to be learned are presented as afferent input to the neurons in region 1. Most networks using error-based learning rules consist of feedforward architectures with separate layers of input and output units. One can imagine this network as an auto-encoder network folded back on itself, with both input and output units in region 1, and hidden units in region 2. Cholinergic Suppression of Transmission in the Neocortex 135 As an example of its functional properties, the network presented here was trained on the XOR problem. The XOR problem has previously been used as an example of the capability of error based training schemes for solving problems which are not linearly separable. The specific characteristics of the network and patterns used for this simulation are shown in figure 4. The two logical states of each component of the XOR problem are represented by two separate units (designated on or off in figures 4 and 5), ensuring that activation of the network is equal for each input condition. The problem has the appearance of two XOR problems with inverse logical states being solved simultaneously. As shown in figure 4, the input and desired output of the network are presented simultaneously during learning to region 1. The six neurons in region 1 project along feedforward connections to four neurons in region 2, the hidden units of the network. These four neurons project along feedback connections to the six neurons in region 1. All connections take random initial weights. During learning, the feedforward connections undergo self-organization which ultimately causes the hidden units to become feature detectors responding to each of the four patterns of input to region 1. Thus, the rows of the feedforward synaptic connectivity matrix gradually take the form of the individual input patterns. STIMULUS RESPONSE on off on off yes no 1. oeeo eo 2. oeoe oe Afferent eooe eo input 3. 4. 9 9 9 "'-Region 1 Figure 4 - Network for learning the XOR problem, with 6 units in region 1 and 4 units in region 2. Four different patterns of afferent input are presented successively to region 1. The input stimuli of the XOR problem are represented by the four units on the left, and the desired output designation of XOR or not-XOR is represented by the two units on the right. The XOR problem has four basic states: on-off and off-on on the input is categorized by yes on the output, while on-on and off-off on the input is categorized by no on the output. Modulation is applied during learning in the form of selective suppression of synaptic transmission along feedback connections (this suppression need not be complete), giving these connections associative memory function. Hebbian synaptic modification causes these connections to link each of the feature detecting hidden units in region 2 with the cells in region 1 activated by the pattern to which the hidden unit responds. Gradually, the feedback synaptic connectivity matrix becomes the transpose of the feedforward connectivity matrix. (parameters used in simulation: Aj(l) = 0 or I, h = 2.0, q(l) = 0.5, q(2) = 0.6, (1) = 0.2, (2) = 0.5, c = 1.0 and Hik(2) = 0.6). Function was similar and convergence was obtained more rapidly with c = 0.5. Feedback synaptic transmission prevented con136 M. E. HASSELMO. M. CEKIC vergence during learning when c = 0.367). During recall, modulation of synaptic transmission is removed, and the various input stimuli of the XOR problem are presented to region 1 without the corresponding output pattern. Activity spreads along the self-organized feedforward connections to activate the specific hidden layer unit responding to that pattern. Activity then spreads back along feedback connections from that particular unit to activate the desired output units. The activity in the two regions settles into a final pattern of recall. Figure 5 shows the settled recall of the network at different stages of learning. It can be seen that the network initially may show little recall activity, or erroneous recall activity, but after several cycles of learning, the network settles into the proper response to each of the XOR problem states. Convergence during learning and recall have been obtained with other problems, including recognition of whether on units were on the left or right, symmetry of on units, and number of on units. In addition, larger scale problems involving multiple feedforward and feedback layers have been shown to converge. 1. 2. 3. 4. ~ on on no off off no off on yes on off yes -- --- - --- §L 3 -- -R ----- - ------- - - -- -· · . -----. · · :-=.:: - · - ---- - - · - -· · . 11----. · · _ .. ---- · - .-- . - - · - -· · --- --· · - · - -- - - · - --· - ----- · · -- ---- · - --- - - · - -. · - ----- ::=:: - -- --- - - · -- -. · - 11---- 1 - - =-- - - -- -. · - ---- ----- - - .- - - .: ••• - 11:11:: :11-== - - .-- - -. ----- - - .- - - - -- · - ----- - -- -- - -- · - ----- ----- - - -.- -- - -- · - ----- · ----- - --.--- - -- - - --.- • • > ----- - - ---- • .: - .~.: - -:==~: - - ---0 - ---- - -- - -~. =-=: --:11 := -- - -- ----- • .: - • • < -. -• • - .... -< I. - - - - =: - - - - --- - --• - - ---- - -- -- -- -- --- - - =: --- -- - - --- - - -- -- -• • - - • .: - • • •• -- -- --- - - =: --- :=. -- ---- --- -• • -- • .: - • • •• -- - = -- - - -- --- == - -- - - -- --- -- --- - - -- -= -- -- --- - - -- -- - • :. - - • .: - • • •• ---= -- - - =: --= :== - .- - - --- -.- -- - -- --- 26 -- - - --- - - -- --- en :;0 I ~. 0 t 3 ~ Region 1 Region 2 Figure 5 - Output neuronal activity in the network shown at different learning steps. The four input patterns are shown at top. Below these are degraded patterns presented during recall, missing the response components of the input pattern. The output of the 6 region 1 units and the 4 region 2 units are shown at each stage of learning. As learning progresses, gradually one region 2 unit starts to respond selectively to each input pattern, and the correct output unit becomes active in response to the degraded input. Note that as learning progresses the response to pattern 4 changes gradually from incorrect (yes) to correct (no). Cholinergic Suppression of Transmission in the Neocortex 137 References Anderson, 1.A. (1983) Cognitive and psychological computation with neural models. IEEE Trans. Systems, Man, Cybem. SMC-13,799-815. Carpenter, G.A., Grossberg, S. and Reynolds, 1.H. (1991) ARTMAP: Supervised realtime learning and classification of nonstationary data by a self-organizing neural network. Neural Networks 4: 565-588. Cauller, LJ. and Connors, B.W. (1994) Synaptic physiology of horizontal afferents to layer I in slices of rat SI neocortex. 1. Neurosci. 14: 751-762. Dayan, P., Hinton, G.E., Neal, RM. and Zemel, RS. (1995) The Helmholtz machine. Neural Computation. Gustafsson, B. and Wigstrom, H. (1988) Physiological mechanisms underlying long-term potentiation. Trends Neurosci. 11: 156-162. Hasselmo, M.E. (1993) Acetylcholine and learning in a cortical associative memory. Neural Computation. 5(1}: 32-44. Hasselmo M.E. and Bower 1.M. (1992) Cholinergic suppression specific to intrinsic not afferent fiber synapses in rat piriform (olfactory) cortex. 1. Neurophysiol. 67: 1222-1229. Hasselmo, M.E. and Bower, 1.M. (1993) Acetylcholine and Memory. Trends Neurosci. 26: 218-222. Hasselmo, M.E. and Cekic, M. (1996) Suppression of synaptic transmission may allow combination of associative feedback and self-organizing feed forward connections in the neocortex. Behav. Brain Res. in press. Hasselmo M.E., Anderson B.P. and Bower 1.M. (1992) Cholinergic modulation of cortical associative memory function. 1. Neurophysiol. 67: 1230-1246. Hasselmo M.E. and Schnell, E. (1994) Laminar selectivity of the cholinergic suppression of synaptic transmission in rat hippocampal region CAl: Computational modeling and brain slice physiology. 1. Neurosci. 15: 3898-3914. Hecht-Nielsen, R (1987) Counterpropagation networks. Applied Optics 26: 4979-4984. Li, H. and Cauller, L.l. (1995) Acetylcholine modulation of excitatory synaptic inputs from layer I to the superficial layers of rat somatosensory neocortex in vitro. Soc. Neurosci. Abstr. 21: 68. Linsker, R (1988) Self-organization in a perceptual network. Computer 21: 105-117. McNaughton B.L. and Morris RG.M. (1987) Hippocampal synaptic enhancement and information storage within a distributed memory system. Trends in Neurosci. 10:408-415. Miller, K.D., Keller, 1.B. and Stryker, M.P. (1989) Ocular dominance column development Analysis and simulation. Science 245: 605-615. van Essen, D.C. and Maunsell, 1.H.R. (1983) Heirarchical organization and functional streams in the visual cortex. Trends Neurosci. 6: 370-375. von der Malsburg, C. (1973) Self-organization of orientation sensitive cells in the striate cortex. Kybemetik 14: 85-100.
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Dynamics of Attention as Near Saddle-Node Bifurcation Behavior Hiroyuki Nakahara" General Systems Studies U ni versi ty of Tokyo 3-8-1 Komaba, Meguro Tokyo 153, Japan nakahara@vermeer.c.u-tokyo.ac.jp Kenji Doya ATR Human Information Processing Research Laboratories 2-2 Hikaridai, Seika, Soraku Kyoto 619-02, Japan doya@hip.atr.co.jp Abstract In consideration of attention as a means for goal-directed behavior in non-stationary environments, we argue that the dynamics of attention should satisfy two opposing demands: long-term maintenance and quick transition. These two characteristics are contradictory within the linear domain. We propose the near saddlenode bifurcation behavior of a sigmoidal unit with self-connection as a candidate of dynamical mechanism that satisfies both of these demands. We further show in simulations of the 'bug-eat-food' tasks that the near saddle-node bifurcation behavior of recurrent networks can emerge as a functional property for survival in nonstationary environments. 1 INTRODUCTION Most studies of attention have focused on the selection process of incoming sensory cues (Posner et al., 1980; Koch et al., 1985; Desimone et al., 1995). Emphasis was placed on the phenomena of causing different percepts for the same sensory stimuli. However, the selection of sensory input itself is not the final goal of attention. We consider attention as a means for goal-directed behavior and survival of the animal. In this view, dynamical properties of attention are crucial. While attention has to be maintained long enough to enable robust response to sensory input, it also has to be shifted quickly to a novel cue that is potentially important. Long-term maintenance and quick transition are critical requirements for attention dynamics. ·currently at Dept. of Cognitive Science and Institute for Neural Computation, U. C. San Diego, La Jolla CA 92093-0515. hnakahar@cogsci.ucsd.edu Dynamics of Attention as Near Saddle-node Bifurcation Behavior 39 We investigate a possible neural mechanism that enables those dynamical characteristics of attention. First, we analyze the dynamics of a network of sigmoidal units with self-connections. We show that both long-term maintenance and quick transition can be achieved when the system parameters are near a "saddle-node bifurcation" point. Then, we test if such a dynamical mechanism can actually be helpful for an autonomously behaving agent in simulations of a 'bug-eat-food' task. The result indicates that near saddle-node bifurcation behavior can emerge in the course of evolution for survival in non-stationary environments. 2 NEAR SADDLE-NODE BIFURCATION BEHAVIOR When a pulse-like input is given to a linear system, the rising and falling phases of the response have the same time constants. This means that long-term maintenance and quick transition cannot be simultaneously achieved by linear dynamics. Therefore, it is essential to consider a nonlinear dynamical mechanism to achieve these two demands. 2.1 DYNAMICS OF A SELF-RECURRENT UNIT First, we consider the dynamics of a single sigmoidal unit with the self-connection weight a and the bias b. y(t + 1) F(x) F(ay(t) + b) , 1 1 + exp( -x)' (1) (2) The parameters (a, b) determine the qualitative behavior of the system such as the number of fixed points and their stabilities. As we change the parameters, the qualitative behavior of the system may suddenly change. This is referred to as "bifurcation" (Guckenheimer, et al., 1983). A typical example is a "saddle-node bifurcation" in which a pair of fixed points, one stable and one unstable, emerges. In our system, this occurs when the state transition curve y(t + 1) = F(ay(t) + b) is tangent to y(t + 1) = y(t). Let y* be this point of tangency. We have the following condi tion for saddle-node bifurcation. F(ay* + b) dF(ay + b) I dy y=y. y* 1 These equations can be solved, by noting F'(x) = F(x)(l- F(x)), as a b = 1 y* (1 - y*) 1 F-1(y*) - ay* = F-l(y*) - -I- y* (3) ( 4) (5) (6) By changing the fixed point value y* between a and 1, we can plot a curve in the parameter space (a, b) on which saddle-node bifurcation occurs, as shown in Figure 1 (left). A pair of a saddle point and a stable fixed point emerges or disappears when the parameters pass across the cusp like curve (cases 2 and 4) . The system has only one stable fixed point when the parameters are outside the cusp (case 1) and three fixed points inside the cusp (case 3). 40 H.NAKAHARA,K.DOYA y ( t+ l ) CASE 1 y(t +1 J CASE 2 " b Bifurcation Diagr am C. 8f::.x.d Pt •. , ~' 1 ~ 8:tixed pts,' 06, ", 0 61 ,,' 041 " 0 "i " O~/ ../ 0'1/ H 0.20. 40.60 8 ly(tJ ""0~ 1O:".,,,,0 ';;;-0;;-; 8 lYlt) ., -'0 Y'~tL · " CASE 3 Y',t." CASE • . ' . o. a: ,,/ 0 B " 't1x.d p t ., "' 3 f lxed p ts " . 2 (I 6 ' 0 6 ' , . OJ' '' 0 " ,,' , . o ,/ 0 2: ' '0 20 40 60 8 1 y( t ) 0 2(' 4~ 60 8: 1 y (tJ - 15 - lO Figure 1: Bifurcation Diagram of a Self-Recurrent Unit. Left: the curve in the parameter space (a , b) on which saddle-node bifurcation is seen. Right: state transition diagrams for four different cases. y ( t · l) o'~=Lil l. 1111 b = - 7. 9 0.6 I 0. 4 " 0 .2 I a ... " y et) 0.20. 40 . 60. 81 y et ) o .~ o. o. o. o 5 10 15 2(fi me (t) Y1d (t:l)ll.ll11 b = ~9 o. I O. I O. " I O. I , o yet) o . 20 . 4 0 60. 8 1 yet) dL o. & 0 . 41 0.21 o 5 10 l S 2a:'irne l t ) Figure 2: Temporal Responses of Self-Recurrent Units. Left: near saddle-node bifurcation. Right: far from bifurcation. An interesting behavior can be seen when the parameters are just outside the cusp, as shown in Figure 2 (left). The system has only one fixed point near Y = 0, but once the unit is activated (y ~ 1), it stays "on" for many time steps and then goes back to the fixed point quickly. Such a mechanism may be useful in satisfying the requirements of attention dynamics: long-term maintenance and quick transition. 2.2 NETWORK OF SELF-RECURRENT UNITS Next, we consider the dynamics of a network of the above self-recurrent units. Yi(t + 1) = F[aYi(t) + b + L CijYj(t) + diUi(t)], (7) j,jti where a is the self connection weight , b is the bias, Cij is the cross connection weight, and di is the input connection weight, and Ui(t) is the external input. The effect of lateral and external inputs is equivalent to the change in the bias, which slides the sigmoid curve horizontally without changing the slope. For example, one parameter set of the bifurcation at y* = 0.9 is a = 11.11 and b ~ -7.80. Let b = -7.90 so that the unit has a near saddle-node bifurcation behavior when there is no lateral or external inputs. For a fixed a = 11.11, as we increase b, the qualitative behavior of the system appears as case 3 in Figure 1, and Dynamics of Attention as Near Saddle-node Bifurcation Behavior 41 Sensory Inputs Actions Network Structure , , 'olr lood " - '--,~ ,- \, :/~ - - - . r-.~ "==:-:"~'on:'oocl ~~ ....... ~~ .... ... ..... Creature ... Creature Inpol. IJrI .111'2 Figure 3: A Creature's Sensory Inputs(Left), Motor System(Center) and Network Architecture(Right) then, it changes again at b:::::: -3.31, where the fixed point at Y = 0.1, or another bifurcation point , appears as case 4 in Figure L Therefore, ifthe input sum is large enough, i.e. Lj ,j;Ci CijYj + diuj > -3.31- (-7.90) :::::: 4.59, the lower fixed point at Y = 0.1 disappears and the state jumps up to the upper fixed point near Y = 1, quickly turning the unit "on". If the lateral connections are set properly, this can in turn suppress the activation of other units. Once the external input goes away, as we see in Figure 2 (left), the state stays "on" for a long time until it returns to the fixed point near Y = O. 3 EVOLUTION OF NEAR BIFURCATION DYNAMICS In the above section, we have theoretically shown the potential usefulness of near saddle-node bifurcation behavior for satisfying demands for attention dynamics. We further hypothesize that such behavior is indeed useful in animal behaviors and can be found in the course of learning and evolution of the neural system. To test our hypothesis, we simulated a 'bug-eat-food' task. Our purpose in t.his simulation was to see whether the attention dynamics discussed in the previous section would help obtain better performance in a non-stationary environment. Vve used evolutionary programming (Fogel et aI, 1990) to optimize the performance of recurrent networks and feedforward networks. 3.1 THE BUG AND THE WORLD In our simulation, a simple creature traveled around a non-stationary environment. In the world, there were a certain number of food items. Each item was fixed at a certain place in the world but appeared or disappeared in a stochastic fashion, as determined by a two-state Markov system. In order to survive, A creature looked for food by traveling the world . The amount of food a creature found in a certain time period was the measure of its performance. A creature had five sensory inputs, each of which detected food in the sector of 45 degrees (Figure 3, right). Its output level was given by L J' .l.., where Tj ,"vas the r J distance to the j-th food item within the sector. Note that the format of the input contained information about distance and also that the creature could only receive the amount of the input but could not distinguish each food from others. For the sake of simplicity, we assumed that the creature lived in a grid-like world. On each time step, it took one of three motor commands: L: turn left (45 degrees), 42 H. NAKAHARA, K. DOYA Density of Food 0.05 0.10 Markov Transition Matrix .5 .5 .8 .8 .5 .5 .8 .8 of each food .5 .5 .2 .2 .5 .5 .2 .2 Random Walk 7.0 6.9 13.8 13.9 Nearest Visible 42.7 18.6 65.3 32.4 FeedForward 58.6 37.3 84.8 60.0 Recurrent 65.7 43.6 94.0 66.1 Nearest Visible/Invisible 97.7 97.1 129.1 128.8 Table 1: Performances of the Recurrent Network and Other Strategies. C: step forward, and R: turn right (Figure 3, center). Simulations were run with different Markov transition matrices of food appearance and with different food densities. A creature got the food when it reached the food, whether it was visible or invisible. When a creature ate a food item, a new food item was placed randomly. The size of the world was 10x10 and both ends were connected as a torus. A creature was composed of two layers: visual layer and motor layer (Figure 3, left). There were five units1 in visual layer, one for each sensory input, and their dynamics were given by Equation (7). The self-connection a, the bias b and the input weight di were the same for all units. There were three units in motor layer, each coding one of three motor commands, and their state was given by ek + L: fkiYi(t), exp(xk(t)) L:/ exp(x/(t)) ' (8) (9) where ek was the bias and fki was the feedforward connection weight. 2 One of the three motor commands (L,C,R) was chosen stochastically with the probability Pk (k=L,C,R). The activation pattern in visual layer was shifted when the creature made a turn, which should give proper mapping between the sensory input and the working memory. 3.2 EVOLUTIONARY PROGRAMMING Each recurrent network was characterized by the parameters (a,b,Cij,di,ek,lkd, some of which were symmetrically shared, e.g. C12 = C21. For comparison, we also tested feedforward networks where recurrent connections were removed, i.e. a = Cij = O. A population of 60 creatures was tested on each generation. The initial population was generated with random parameters. Each of the top twenty scoring creatures produced three offspring; one identical copy of the parameters of the parent's and two copies of these parameters with a Gaussian fluctuation. In this paper, we report the result after 60 generations. 3.3 PERFORMANCE 1 We denote each unit in visual layer by Ul, U2, U3, U4, Us from the left to the right for the later convenience 2In this simulation reported here, we set ek = O. Dynamics of Attention as Near Saddle-node Bifurcation Behavior 43 -, -, -, , . . " .... : .... -, , -, , - 7 , _7 , - 10 - 12 . 5 -L25 a b "Transition matrix = ( : ~ .5 ) .5 bTransition matrix = ( :~ :~ ) Figure 4: The Convergence of the Parameter of (a , b) by Evolutionary Programming Plotted in the Bifurcation Diagram. The food density is 0.10 in both examples above. Table 1 shows the average of food found after 60 generations. As a reference of performance level, we also measured the performances of three other simple algorithms: 1) random walk: one of the three motor commands is taken randomly with equal probability. 2) nearest visible: move toward the nearest food visible at the time within the creature's field of view of (U2, U3, U4). 3) nearest visible/invisible: move toward the nearest food within the view of (U2, U3, U4) no matter if it is visible or not, which gives an upper bound of performance. The performance of recurrent network is better than that of feedforward network and 'nearest visible'. This suggests that the ability of recurrent network to remember the past is advantageous. The performance of feedforward network is better than that of 'nearest visible'. One reason is that feedforward network could cover a broader area to receive inputs than 'nearest visible'. In addition, two factors, the average time in which a creature reaches the food and the average time in which the food disappears, may influence the performance of feedforward network and 'nearest visible'. Feedforward network could optimize its output to adapt two factors with its broader view in evolution while 'nearest visible' did not have such adaptability. It should be noted that both of 'nearest visible/invisible' and 'nearest visible' explicitly assumed the higher-order sensory processing: distinguishing each food item from the others and measuring the distance between each food and its body. Since its performance is so different regardless of its higher-order sensory processing, it implies the importance of remembering the past. We can regard recurrent network as compromising two characteristics, remembering the past as 'nearest visible/invisible' did and optimizing the sensitivity as feedforward network did, although recurrent network did not have a perfect memory as 'nearest visible/invisible'. 3.4 CONVERGENCE TO NEAR-BIFURCATION REGIME We plotted the histogram of the performance in each generation and the history of the performance of a top-scoring creature over generations. Though they are not shown here, the performance was almost optimal after 60 generations. Figure 4 shows that two examples of a graph in which we plotted the parameter 44 H. NAKAHARA, K. DOYA set (a , b) of top twenty scoring creatures in the 60th generation in the bifurcation diagram. In the left graph, we can see the parameter set has converged to a regime that gives a near saddle-node bifurcation behavior. On the other hand, in the right graph, the parameter set has converged into the inside of cusp. It is interesting to note that the area inside of the cusp gives bistable dynamics. Hence, if the input is higher than a repelling point, it goes up and if the input is lower, it goes down. The reason of the convergence to that area is because of the difference of the world setting, that is, a Markov transition matrix. Since food would disappear more quickly and stay invisible longer in the setting of the right graph, it should be beneficial for a creature to remember the direction of higher inputs longer. In most of cases reported in Table 1, we obtained the convergence into our predicted regime and/or the inside of the cusp. 4 DISCUSSION Near saddle-node bifurcation behavior can have the long-term maintenance and quick transition, which characterize attention dynamics. A recurrent network has better performance than memoryless systems for tasks in our simulated nonstationary environment. Clearly, near saddle-node bifurcation behavior helped a creature's survival and in fact, creatures actually evolved to our expected parameter regime. However, we also obtained the convergence into another unexpected regime which gives bistable dynamics. How the bistable dynamics are used remains to be investigated. Acknowledgments H.N. is grateful to Ed Hutchins for his generous support, to John Batali and David Fogel for their advice on the implementation of evolutionary programming and to David Rogers for his comments on the manuscript of this paper. References R. Desimone, E. K. Miller, L. Chelazzi, & A. Lueschow. (1995) Multiple Memory Systems in the Visual Cortex. In M. Gazzaniga (ed.), The Cognitive Neurosciences, 475-486. MIT Press. D. B. Fogel, L. J. Fogel, & V. W. Porto. (1990) Evolving Neural Networks. Biological cybernetics 63:487-493. J. Guckenheimer & P. Homes. (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields C. Koch & S. Ullman. (1985) Shifts in selective visual attention:towards the underlying neural circuitry. Human Neurobiology 4:219-227. M. Posner, C .. R .R. Snyder, & B. J. Davidson. (1980) Attention and the detection of signals. Journal of Experimental Psychology: General 109:160-174
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Learning with ensembles: How over-fitting can be useful Peter Sollich Department of Physics University of Edinburgh, U.K. P.SollichGed.ac.uk Anders Krogh'" NORDITA, Blegdamsvej 17 2100 Copenhagen, Denmark kroghGsanger.ac.uk Abstract We study the characteristics of learning with ensembles. Solving exactly the simple model of an ensemble of linear students, we find surprisingly rich behaviour. For learning in large ensembles, it is advantageous to use under-regularized students, which actually over-fit the training data. Globally optimal performance can be obtained by choosing the training set sizes of the students appropriately. For smaller ensembles, optimization of the ensemble weights can yield significant improvements in ensemble generalization performance, in particular if the individual students are subject to noise in the training process. Choosing students with a wide range of regularization parameters makes this improvement robust against changes in the unknown level of noise in the training data. 1 INTRODUCTION An ensemble is a collection of a (finite) number of neural networks or other types of predictors that are trained for the same task. A combination of many different predictors can often improve predictions, and in statistics this idea has been investigated extensively, see e.g. [1, 2, 3]. In the neural networks community, ensembles of neural networks have been investigated by several groups, see for instance [4, 5, 6, 7]. Usually the networks in the ensemble are trained independently and then their predictions are combined. In this paper we study an ensemble of linear networks trained on different but overlapping training sets. The limit in which all the networks are trained on the full data set and the one where all the data sets are different has been treated in [8] . In this paper we treat the case of intermediate training set sizes and overlaps ·Present address: The Sanger Centre, Hinxton, Cambs CBIO IRQ, UK. Learning with Ensembles: How Overfitting Can Be Useful 191 exactly, yielding novel insights into ensemble learning. Our analysis also allows us to study the effect of regularization and of having different predictors in an ensemble. 2 GENERAL FEATURES OF ENSEMBLE LEARNING We consider the task of approximating a target function fo from RN to R. It will be assumed that we can only obtain noisy samples of the function, and the (now stochastic) target function will be denoted y(x). The inputs x are taken to be drawn from some distribution P(x). Assume now that an ensemble of K independent predictors fk(X) of y(x) is available. A weighted ensemble average is denoted by a bar, like lex) = L,wkfk(X), (1) k which is the final output of the ensemble. One can think of the weight Wk as the belief in predictor k and we therefore constrain the weights to be positive and sum to one. For an input x we define the error of the ensemble c(x), the error of the kth predictor ck(X), and its ambiguity ak(x) c(x) (y(x) -lex)? (2) ck(X) (y(x) - fk(X)? (3) (fk(X) -1(x»2. (4) The ensemble error can be written as c(x) = lex) - a(x) [7], where lex) = L,k Wkck(X) is the average error over the individual predictors and a(x) = L,k Wkak(X) is the average of their ambiguities, which is the variance of the output over the ensemble. By averaging over the input distribution P(x) (and implicitly over the target outputs y(x», one obtains the ensemble generalization error (5) where c(x) averaged over P(x) is simply denoted c, and similarly for land a. The first term on the right is the weighted average of the generalization errors of the individual predictors, and the second is the weighted average of the ambiguities, which we refer to as the ensemble ambiguity. An important feature of equation (5) is that it separates the generalization error into a term that depends on the generalization errors of the individual students and another term that contains all correlations between the students. The latter can be estimated entirely from unlabeled data, i. e., without any knowledge of the target function to be approximated. The relation (5) also shows that the more the predictors differ, the lower the error will be, provided the individual errors remain constant. In this paper we assume that the predictors are trained on a sample of p examples of the target function, (xt',yt'), where yt' = fo(xt') + TJt' and TJt' is some additive noise (Jl. = 1, ... ,p). The predictors, to which we refer as students in this context because they learn the target function from the training examples, need not be trained on all the available data. In fact, since training on different data sets will generally increase the ambiguity, it is possible that training on subsets of the data will improve generalization. An additional advantage is that, by holding out for each student a different part of the total data set for the purpose of testing, one can use the whole data set for training the ensemble while still getting an unbiased estimate of the ensemble generalization error. Denoting this estimate by f, one has (6) where Ctest = L,k WkCtest,k is the average of the students' test errors. As already pointed out, the estimate ft of the ensemble ambiguity can be found from unlabeled data. 192 P. SOLLICH, A. KROGH So far, we have not mentioned how to find the weights Wk. Often uniform weights are used, but optimization of the weights in some way is tempting. In [5, 6] the training set was used to perform the optimization, i.e., the weights were chosen to minimize the ensemble training error. This can easily lead to over-fitting, and in [7] it was suggested to minimize the estimated generalization error (6) instead. If this is done, the estimate (6) acquires a bias; intuitively, however, we expect this effect to be small for large ensembles. 3 ENSEMBLES OF LINEAR STUDENTS In preparation for our analysis of learning with ensembles of linear students we now briefly review the case of a single linear student, sometimes referred to as 'linear perceptron learning'. A linear student implements the input-output mapping 1 T J(x) = ..JNw x parameterized in terms of an N-dimensional parameter vector w with real components; the scaling factor 1/..JN is introduced here for convenience, and . .. T denotes the transpose of a vector. The student parameter vector w should not be confused with the ensemble weights Wk. The most common method for training such a linear student (or parametric inference models in general) is minimization of the sum-of-squares training error E = L:(y/J - J(x/J))2 + Aw2 /J where J.L = 1, ... ,p numbers the training examples. To prevent the student from fitting noise in the training data, a weight decay term Aw2 has been added. The size of the weight decay parameter A determines how strongly large parameter vectors are penalized; large A corresponds to a stronger regularization of the student. For a linear student, the global minimum of E can easily be found. However, in practical applications using non-linear networks, this is generally not true, and training can be thought of as a stochastic process yielding a different solution each time. We crudely model this by considering white noise added to gradient descent updates of the parameter vector w. This yields a limiting distribution of parameter vectors P(w) ex: exp(-E/2T), where the 'temperature' T measures the amount of noise in the training process. We focus our analysis on the 'thermodynamic limit' N -t 00 at constant normalized number of training examples, ex = p/ N. In this limit, quantities such as the training or generalization error become self-averaging, i.e., their averages over all training sets become identical to their typical values for a particular training set. Assume now that the training inputs x/J are chosen randomly and independently from a Gaussian distribution P(x) ex: exp( ~x2), and that training outputs are generated by a linear target function corrupted by additive noise, i.e., y/J = w'f x/J /..IN + 1]/J, where the 1]/J are zero mean noise variables with variance u2 • Fixing the length of the parameter vector of the target function to w~ = N for simplicity, the generalization error of a linear student with weight decay A and learning noise T becomes [9] 8G (; = (u2 + T)G + A(U2 - A) 8A . (7) On the r.h.s. of this equation we have dropped the term arising from the noise on the target function alone, which is simply u2 , and we shall follow this convention throughout. The 'response function' Gis [10, 11] G = G(ex, A) = (1 - ex - A + )(1 - ex - A)2 + 4A)/2A. (8) Learning with Ensembles: How Overfitting Can Be Useful 193 For zero training noise, T = 0, and for any a, the generalization error (7} is minimized when the weight decay is set to A = (T2j its value is then (T2G(a, (T2), which is the minimum achievable generalization error [9]. 3.1 ENSEMBLE GENERALIZATION ERROR We now consider an ensemble of K linear students with weight decays Ak and learning noises Tk (k = 1 . . . K). Each ,student has an ensemble weight Wk and is trained on N ak training examples, with students k and I sharing N akl training examples (of course, akk = ak). As above, we consider noisy training data generated by a linear target function. The resulting ensemble generalization error can be calculated by diagrammatic [10] or response function [11] methods. We refer the reader to a forthcoming publication for details and only state the result: (9) where (10) Here Pk is defined as Pk = AkG(ak, Ak). The Kronecker delta in the last term of (10) arises because the training noises of different students are uncorrelated. The generalization errors and ambiguities of the individual students are ak = ckk - 2 LWlckl + LWIWmclm; I 1m the result for the Ck can be shown to agree with the single student result (7). In the following sections, we shall explore the consequences of the general result (9). We will concentrate on the case where the training set of each student is sampled randomly from the total available data set of size NO', For the overlap of the training sets of students k and I (k 'II) one then has akl/a = (ak/a)(al/a) and hence ak/ = akal/a (11) up to fluctuations which vanish in the thermodynamic limit. For finite ensembles one can construct training sets for which akl < akal/a. This is an advantage, because it results in a smaller generalization error, but for simplicity we use (11). 4 LARGE ENSEMBLE LIMIT We now use our main result (9) to analyse the generalization performance of an ensemble with a large number K of students, in particular when the size of the training sets for the individual students are chosen optimally. If the ensemble weights Wk are approximately uniform (Wk ~ 1/ K) the off-diagonal elements of the matrix (ckl) dominate the generalization error for large K, and the contributions from the training noises n are suppressed. For the special case where all students are identical and are trained on training sets of identical size, ak = (1 - c)a, the ensemble generalization error is shown in Figure 1(left). The minimum at a nonzero value of c, which is the fraction of the total data set held out for testing each student, can clearly be seen. This confirms our intuition: when the students are trained on smaller, less overlapping training sets, the increase in error of the individual students can be more than offset by the corresponding increase in ambiguity. The optimal training set sizes ak can be calculated analytically: _ 1 - Ak/(T2 Ck = 1 - ak/a = 1 + G(a, (T2) ' (12) 194 w 1.0 r---,-----,r---.,----,.----:. 0.8 0.6 0.4 0.2 ,...------/ , / , / , 0.0 / , 0.0 0.2 0.4 0.6 0.8 1.0 C w P. SOLLICH, A. KROGH 1.0 r---,-----,---.----r----" 0.8 .' 0.6 0.2 ------0.0 ..... 0.0 0.2 0.4 0.6 0.8 1.0 C Figure 1: Generalization error and ambiguity for an infinite ensemble of identical students. Solid line: ensemble generalization error, fj dotted line: average generalization error of the individual students, l; dashed line: ensemble ambiguity, a. For both plots a = 1 and (72 = 0.2. The left plot corresponds to under-regularized students with A = 0.05 < (72. Here the generalization error of the ensemble has a minimum at a nonzero value of c. This minimum exists whenever>' < (72. The right plot shows the case of over-regularized students (A = 0.3 > (72), where the generalization error is minimal at c = O. The resulting generalization error is f = (72G(a, (72) + 0(1/ K), which is the globally minimal generalization error that can be obtained using all available training data, as explained in Section 3. Thus, a large ensemble with optimally chosen training set sizes can achieve globally optimal generalization performance. However, we see from (12) that a valid solution Ck > 0 exists only for Ak < (72, i.e., if the ensemble is under-regularized. This is exemplified, again for an ensemble of identical students, in Figure 1 (right) , which shows that for an over-regularized ensemble the generalization error is a: monotonic function of c and thus minimal at c = o. We conclude this section by discussing how the adaptation of the training set sizes could be performed in practice, for simplicity confining ourselves to an ensemble of identical students, where only one parameter c = Ck = 1- ak/a has to be adapted. If the ensemble is under-regularized one expects a minimum of the generalization error for some nonzero c as in Figure 1. One could, therefore, start by training all students on a large fraction of the total data set (corresponding to c ~ 0), and then gradually and randomly remove training examples from the students' training sets. Using (6), the generalization error of each student could be estimated by their performance on the examples on which they were not trained, and one would stop removing training examples when the estimate stops decreasing. The resulting estimate of the generalization error will be slightly biased; however, for a large enough ensemble the risk of a strongly biased estimate from systematically testing all students on too 'easy' training examples seems small, due to the random selection of examples. 5 REALISTIC ENSEMBLE SIZES We now discuss some effects that occur in learning with ensembles of 'realistic' sizes. In an over-regularized ensemble nothing can be gained by making the students more diverse by training them on smaller, less overlapping training sets. One would also Learning with Ensembles: How Overfitting Can Be Useful 195 Figure 2: The generalization error of an ensemble with 10 identical students as a function of the test set fraction c. From bottom to top the curves correspond to training noise T = 0,0.1,0.2, ... ,1.0. The star on each curve shows the error of the optimal single perceptron (i. e., with optimal weight decay for the given T) trained on all examples, which is independent of c. The parameters for this example are: a = 1, A = 0.05, 0'2 = 0.2. 0.2 0.0 L-_--'-_---' __ -'--_--'-_~ 0.0 0.2 0.4 0.6 0.8 1.0 C expect this kind of 'diversification' to be unnecessary or even counterproductive when the training noise is high enough to provide sufficient 'inherent' diversity of students. In the large ensemble limit, we saw that this effect is suppressed, but it does indeed occur in finite ensembles. Figure 2 shows the dependence of the generalization error on c for an ensemble of 10 identical, under-regularized students with identical training noises Tk = T. For small T, the minimum of f. at nonzero c persists. For larger T, f. is monotonically increasing with c, implying that further diversification of students beyond that caused by the learning noise is wasteful. The plot also shows the performance of the optimal single student (with A chosen to minimize the generalization error at the given T), demonstrating that the ensemble can perform significantly better by effectively averaging out learning noise. For realistic ensemble sizes the presence of learning noise generally reduces the potential for performance improvement by choosing optimal training set sizes. In such cases one can still adapt the ensemble weights to optimize performance, again on the basis of the estimate of the ensemble generalization error (6). An example is 1.0 1.0 I I 0.8 , 0.8 / , ,,I 0.6 I 0.6 tV tV 0.4 0.4 ..... -_ ................. 0.2 ---0.2 0.0 ....... 0.0 0.001 0.010 0 2 0.100 1.000 0.001 0.010 0 2 0.100 1.000 Figure 3: The generalization error of an ensemble of 10 students with different weight decays (marked by stars on the 0'2-axis) as a function of the noise level 0'2. Left: training noise T = 0; right: T = 0.1. The dashed lines are for the ensemble with uniform weights, and the solid line is for optimized ensemble weights. The dotted lines are for the optimal single perceptron trained on all data. The parameters for this example are: a = 1, c = 0.2. 196 P. SOu...ICH, A. KROGH shown in Figure 3 for an ensemble of size 1< = 10 with the weight decays >'k equally spaced on a logarithmic axis between 10-3 and 1. For both of the temperatures T shown, the ensemble with uniform weights performs worse than the optimal single student. With weight optimization, the generalization performance approaches that of the optimal single student for T = 0, and is actually better at T = 0.1 over the whole range of noise levels rr2 shown. Even the best single student from the ensemble can never perform better than the optimal single student, so combining the student outputs in a weighted ensemble average is superior to simply choosing the best member of the ensemble by cross-validation, i.e., on the basis of its estimated generalization error. The reason is that the ensemble average suppresses the learning noise on the individual students. 6 CONCLUSIONS We have studied ensemble learning in the simple, analytically solvable scenario of an ensemble of linear students. Our main findings are: In large ensembles, one should use under-regularized students in order to maximize the benefits of the variance-reducing effects of ensemble learning. In this way, the globally optimal generalization error on the basis of all the available data can be reached by optimizing the training set sizes of the individual students. At the same time an estimate of the generalization error can be obtained. For ensembles of more realistic size, we found that for students subjected to a large amount of noise in the training process it is unnecessary to increase the diversity of students by training them on smaller, less overlapping training sets. In this case, optimizing the ensemble weights can still yield substantially better generalization performance than an optimally chosen single student trained on all data with the same amount of training noise. This improvement is most insensitive to changes in the unknown noise levels rr2 if the weight decays of the individual students cover a wide range. We expect most of these conclusions to carryover, at least qualitatively, to ensemble learning with nonlinear models, and this correlates well with experimental results presented in [7]. References [1] C. Granger, Journal of Forecasting 8, 231 (1989). [2] D. Wolpert, Neural Networks 5, 241 (1992). [3] L. Breimann, Tutorial at NIPS 7 and personal communication. [4] L. Hansen and P. Salamon, IEEE Trans. Pattern Anal. and Mach. Intell. 12, 993 (1990). [5] M. P. Perrone and L. N. Cooper, in Neural Networks for Speech and Image processing, ed. R. J. Mammone (Chapman-Hall, 1993). [6] S. Hashem: Optimal Linear Combinations of Neural Networks. Tech. Rep. PNL-SA-25166, submitted to Neural Networks (1995). [7] A. Krogh and J. Vedelsby, in NIPS 7, ed. G. Tesauro et al., p. 231 (MIT Press, 1995). [8] R. Meir, in NIPS 7, ed. G. Tesauro et al., p. 295 (MIT Press, 1995). [9] A. Krogh and J. A. Hertz, J. Phys. A 25,1135 (1992). [10] J. A. Hertz, A. Krogh, and G. I. Thorbergsson, J. Phys. A 22, 2133 (1989). [11] P. Sollich, J. Phys. A 27, 7771 (1994).
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Softassign versus Softmax: Benchmarks in Combinatorial Optimization Steven Gold Department of Computer Science Yale University New Haven, CT 06520-8285 Anand Rangarajan Dept. of Diagnostic Radiology Yale University New Haven, CT 06520-8042 Abstract A new technique, termed soft assign, is applied for the first time to two classic combinatorial optimization problems, the traveling salesman problem and graph partitioning. Soft assign , which has emerged from the recurrent neural network/statistical physics framework, enforces two-way (assignment) constraints without the use of penalty terms in the energy functions. The soft assign can also be generalized from two-way winner-take-all constraints to multiple membership constraints which are required for graph partitioning. The soft assign technique is compared to the softmax (Potts glass). Within the statistical physics framework, softmax and a penalty term has been a widely used method for enforcing the two-way constraints common within many combinatorial optimization problems. The benchmarks present evidence that soft assign has clear advantages in accuracy, speed, parallelizabilityand algorithmic simplicity over softmax and a penalty term in optimization problems with two-way constraints. 1 Introduction In a series of papers in the early to mid 1980's, Hopfield and Tank introduced techniques which allowed one to solve combinatorial optimization problems with recurrent neural networks [Hopfield and Tank, 1985]. As researchers attempted to reproduce the original traveling salesman problem results of Hopfield and Tank, problems emerged, especially in terms of the quality of the solutions obtained. More recently however, a number of techniques from statistical physics have been adopted to mitigate these problems. These include deterministic annealing which convexifies the energy function in order help avoid some local minima and the Potts glass approximation which results in a hard enforcement of a one-way (one set of) winner-take-all (WTA) constraint via the softmax. In Softassign versus Softmax: Benchmarks in Combinatorial Optimization 627 the late 80's, armed with these techniques optimization problems like the traveling salesman problem (TSP) [Peterson and Soderberg, 1989] and graph partitioning [Peterson and Soderberg, 1989, Van den Bout and Miller III, 1990] were reexamined and much better results compared to the original Hopfield-Tank dynamics were obtained. However, when the problem calls for two-way interlocking WTA constraints, as do TSP and graph partitioning, the resulting energy function must still include a penalty term when the softmax is employed in order to enforce the second set of WTA constraints. Such penalty terms may introduce spurious local minima in the energy function and involve free parameters which are hard to set. A new technique, termed soft assign, eliminates the need for all such penalty terms. The first use of the soft assign was in an algorithm for the assignment problem [Kosowsky and Yuille, 1994]. It has since been applied to much more difficult optimization problems, including parametric assignment problems-point matching [Gold et aI., 1994, Gold et aI., 1995, Gold et aI., 1996] and quadratic assignment problems-graph matching [Gold et aI., 1996, Gold and Rangarajan, 1996, Gold, 1995]. Here, we for the first time apply the soft assign to two classic combinatorial optimization problems, TSP and graph partitioning. Moreover, we show that the soft assign can be generalized from two-way winner-take-all constraints to multiple membership constraints, which are required for graph partitioning (as described below). We then run benchmarks against the older softmax (Potts glass) methods and demonstrate advantages in terms of accuracy, speed, parallelizability, and simplicity of implementation. It must be emphasized there are other conventional techniques, for solving some combinatorial optimization problems such as TSP, which remain superior to this method in certain ways [Lawler et aI., 1985]. (We think for some problems-specifically the type of pattern matching problems essential for cognition [Gold, 1995]-this technique is superior to conventional methods.) Even within neural networks, elastic net methods may still be better in certain cases. However, the elastic net uses only a one-way constraint in TSP. The main goal of this paper is to provide evidence, that when minimizing energy functions within the neural network framework, which have two-way constraints, the soft assign should be the technique of choice. We therefore compare it to the current dominant technique, softmax with a penalty term. 2 Optimizing With Softassign 2.1 The Traveling Salesman Problem The traveling salesman problem may be defined in the following way. Given a set of intercity distances {hab} which may take values in R+ , find the permutation matrix M such that the following objective function is minimized. 1 N N N E 1(M) = 2 LLL habMai Mb(i6H) a==lb==li=l (1) subject to Va L~l Mai = 1 , Vi L~=l Mai = 1 , Vai Mai E {O, 1}. In the above objective hab represents the distance between cities a and b. M is a permutation matrix whose rows represent cities, and whose columns represent the day (or order) the city was visited and N is the number of cities. (The notation i EEl 1 628 S.GOLD,A.RANGARAJAN is used to indicate that subscripts are defined modulo N, i.e. Ma(N+I) = Mal.) So if Mai = 1 it indicates that city a was visited on day i. Then, following [Peterson and Soderberg, 1989, Yuille and Kosowsky, 1994] we employ Lagrange multipliers and an x log x barrier function to enforce the constraints, as well as a 'Y term for stability, resulting in the following objective: 1 N N N N N E2(M, 1',11) = 2 L L L babMaiMb(ieJ I ) ~ L L M;i a=l b=l i=l a=l i=l INN N N N N +p I: I: Mai(10g M ai - 1) + I: J.la(I: Mai - 1) + I: lIi(I: Mai - 1) (2) a=l i=l a=l i=l i=l a=l In the above we are looking for a saddle point by minimizing with respect to M and maximizing with respect to I' and 11, the Lagrange multipliers. 2.2 The Soft assign In the above formulation of TSP we have two-way interlocking WTA constraints. {Mai} must be a permutation matrix to ensure that a valid tour-one in which each city is visited once and only once-is described. A permutation matrix means all the rows and columns must add to one (and the elements must be zero or one) and therefore requires two-way WTA constraints-a set of WTA constraints on the rows and a set of WTA constraints on the columns. This set of two-way constraints may also be considered assignment constraints, since each city must be assigned to one and only one day (the row constraint) and each day must be assigned to one and only one city (the column constraint). These assignment constraints can be satisfied using a result from [Sinkhorn, 1964]. In [Sinkhorn, 1964] it is proven that any square matrix whose elements are all positive will converge to a doubly stochastic matrix just by the iterative process of alternatively normalizing the rows and columns. (A doubly stochastic matrix is a matrix whose elements are all positive and whose rows and columns all add up to one-it may roughly be thought of as the continuous analog of a permutation matrix). The soft assign simply employs Sinkhorn's technique within a deterministic annealing context. Figure 1 depicts the contrast between the soft assign and the softmax. In the softmax, a one-way WTA constraint is strictly enforced by normalizing over a vector. [Kosowsky and Yuille, 1994] used the soft assign to solve the assignment problem, i.e. minimize: 2:~=1 2:{=1 MaiQai. For the special case of the quadratic assignment problem, being solved here, by setting Q ai = -:J:i' and using the values of M from the previous iteration, we can at each iteration produce a new assignment problem for which the soft assign then returns a doubly stochastic matrix. As the temperature is lowered a series of assignment problems are generated, along with the corresponding doubly stochastic matrices returned by each soft assign , until a permutation matrix is reached. The update with the partial derivative in the preceding may be derived using a Taylor series expansion. See [Gold and Rangarajan, 1996, Gold, 1995] for details. The algorithm dynamics then become: Softassign versus Softmax: Benchmarks in Combinatorial Optimization 629 Softassign Softmax Positivity M.i = exP(I3Q.) 1 Positivity Mi = exP(I3Qi) Two-way constraints Row Normalization 1 ( ~) Mai--_ 1 l:M.i 0<;. """"'~_ M.i- l:~. a 1 One-way constraint M· M· ___ 1_ 1 l:M. i ) Figure 1: Softassign and softmax. This paper compares these two techniques. (3) Mai = Softassignai (Q) (4) E2 is E2 without the {3, J.l or II terms of (2), therefore no penalty terms are now included. The above dynamics are iterated as (3, the inverse temperature, is gradually increased. These dynamics may be obtained by evaluating the saddle points of the objective in (2). Sinkhorn's method finds the saddle points for the Lagrange parameters. 2.3 Graph Partitioning The graph partitioning problem maybe defined in the following way. Given an unweighted graph G, find the membership matrix M such that the following objective function is minimized. A I I E3(M) = - I:L:L:GijMaiMaj (5) a=1 i=1 j=1 subject to Va E;=1 Mai = IIA, Vi E:=1 Mai = 1, Vai Mai E to, I} where graph G has I nodes which should be equally partitioned into A bins. {Gij} is the adjacency matrix of the graph, whose elements must be 0 or 1. M is a membership matrix such that Mai = 1 indicates that node i is in bin a. The permutation matrix constraint present in TSP is modified to the membership constraint. Node i is a member of only bin a and the number of members in each bin is fixed at IIA. When the above objective is at a minimum, then graph G will be partitioned into A equal sized bins, such that the cutsize is minimum for all possible partitionings of G into A equal sized bins. We assume IIA is an integer. Then following the treatment for TSP, we derive the following objective: 630 S.GOLD,A. RANGARAJAN A I I A I E4(M,p,v) = - I: I:L: CijMaiMaj ~ L:L:M;i a=l i=l j=l a=l i=l 1AI A I I A +:8 I: I: Mai(lOgMai - 1) + I:Pa(2: Mai - [fA) + 2: Vi (2: Mai -1) (6) a=li=l a=l i=l i=l a=l which is minimized with a similar algorithm employing the softassign. Note however now in the soft assign the columns are normalized to [j A instead of 1. 8 Experimental Results Experiments on Euclidean TSP and graph partitioning were conducted. For each problem three different algorithms were run. One used the soft assign described above. The second used the Potts glass dynamics employing synchronous update as described in [Peterson and Soderberg, 1989]. The third used the Potts glass dynamics employing serial update as described in [Peterson and Soderberg, 1989]. Originally the intention was to employ just the synchronous updating version of the Potts glass dynamics, since that is the dynamics used in the algorithms employing soft assign and is the method that is massively parallelizable. We believe massive parallelism to be such a critical feature of the neural network architecture [Rumelhart and McClelland, 1986] that any algorithm that does not have this feature loses much of the power of the neural network paradigm. Unfortunately the synchronous updating algorithms just worked so poorly that we also ran the serial versions in order to get a more extensive comparison. Note that the results reported in [Peterson and Soderberg, 1989] were all with the serial versions. 3.1 Euclidean TSP Experiments Figure 2 shows the results of the Euclidean TSP experiments. 500 different 100city tours from points uniformly generated in the 2D unit square were used as input. The asymptotic expected length of an optimal tour for cities distributed in the unit square is given by L( n) = J( Vn where n is the number of cities and 0.765 ~ J( ~ 0.765 +.1 [Lawler et al., 1985]. This gives the interval [7.65,8.05] for the 100 city TSP. 95<70 of the tour lengths fall in the interval [8,11] when using the soft assign approach. Note the large difference in performance between the soft assign and the Potts glass algorithms. The serial Potts glass algorithm ran about 5 times slower than the soft assign version. Also as noted previously the serial version is not massively parallelizable. The synchronous Potts glass ran about 2 times slower. Also note the softassign algorithm is much simpler to implement-fewer parameters to tune. 3.2 Graph Partitioning Experiments Figure 3 shows the results of the graph partitioning experiments. 2000 different randomly generated 100 node graphs with 10% connectivity were used as input. These graphs were partitioned into four bins. The soft assign performs better than the Potts glass algorithms, however here the difference is more modest than in the TSP experiments. However the serial Potts glass algorithm again ran about 5 times slower then the soft assign version and as noted previously the serial version is not massively parallelizable. The synchronous Potts glass ran about 2 times slower. Softassign versus Softmax: Benchmarks in Combinatorial Optimization 631 r-"' • r• "' r-• ,. r,. • ,. •• • r• • • 11 It ,. " 11 • n Inn • ..--:I .r-I!'--,,:,:...u:~~-""'=---!;-~,........_---!,.. I II 11 "' n "I 11 ,.1 ,. '.1 1.~' " It II • . .. .. --"' ,. -......... ... ..... Figure 2: 100 City Euclidean TSP. 500 experiments. Left: Softassign .. Middle: Softmax (serial update). Right: Softmax (synchronous update). Also again note the softassign algorithm was much simpler to implement-fewer parameters to tune. .. ',.. e, r0.. til e.. "' rr,. , •• rInn. .n "' ,. .. til ,. • .n n_ ,. nn. • ~n ~ .. ... "' "' til _ ~ .. _ M .. "' _ .. Figure 3: 100 node Graph Partitioning, 4 bins. 2000 experiments. Left: Softassign •. Middle: Softmax (serial update). Right: Softmax (synchronous update). A relatively simple version of graph partitioning was run. It is likely that as the number of bins are increased the results on graph partitioning will come to resemble more closely the TSP results, since when the number of bins equal the number of nodes, the TSP can be considered a special case of graph partitioning (there are some additional restrictions). However even in this simple case the softassign has clear advantages over the softmax and penalty term. 4 Conclusion For the first time, two classic combinatorial optimization problems, TSP and graph partitioning, are solved using a new technique for constraint satisfaction, the soft assign. The softassign, which has recently emerged from the statistical physics/neural networks framework, enforces a two-way (assignment) constraint, without penalty terms in the energy function. We also show that the softassign can be generalized from two-way winner-take-all constraints to multiple membership constraints, which are required for graph partitioning. Benchmarks against the Potts glass methods, using softmax and a penalty term, clearly demonstrate its advantages in terms of accuracy, speed, parallelizability and simplicity of implementation. Within the neural network/statistical physics framework, soft assign should be considered the technique of choice for enforcing two-way constraints in energy functions. 632 S. GOLD,A. RANGARAJAN References [Gold, 1995] Gold, S ~ (1995). Matching and Learning Structural and Spatial Representations with Neural Networks. PhD thesis, Yale University. [Gold et al., 1995] Gold, S., Lu, C. P., Rangarajan, A., Pappu, S., and Mjolsness, E. (1995). New algorithms for 2-D and 3-D point matching: pose estimation and correspondence. In Tesauro, G., Touretzky, D. S., and Leen, T. K., editors, Advances in Neural Information Processing Systems 7, pages 957-964. MIT Press, Cambridge, MA. [Gold et al. , 1994] Gold, S., Mjolsness, E., and Rangarajan, A. (1994). Clustering with a domain specific distance measure. In Cowan, J., Tesauro, G., and AIspector, J., editors, Advances in Neural Information Processing Systems 6, pages 96-103. Morgan Kaufmann, San Francisco, CA. [Gold and Rangarajan, 1996] Gold, S. and Rangarajan, A. (1996). A graduated assignment algorithm for graph matching. IEEE Transactions on Pattern Analysis and Machine Intelligence, (in press). [Gold et al., 1996] Gold, S., Rangarajan, A., and Mjolsness, E. (1996). Learning with preknowledge: clustering with point and graph matching distance measures. Neural Computation, (in press). [Hopfield and Tank, 1985] Hopfield, J. J. and Tank, D. (1985). 'Neural' computation of decisions in optimization problems. Biological Cybernetics, 52:141-152. [Kosowsky and Yuille, 1994] Kosowsky, J . J . and Yuille, A. L. (1994). The invisible hand algorithm: Solving the assignment problem with statistical physics. Neural Networks, 7(3):477-490. [Lawler et al., 1985] Lawler, E. L., Lenstra, J. K., Kan, A. H. G. R., and Shmoys, D. B., editors (1985). The Traveling Salesman Problem. John Wiley and Sons, Chichester. [Peterson and Soderberg, 1989] Peterson, C. and Soderberg, B. (1989). A new method for mapping optimization problems onto neural networks. Inti. Journal of Neural Systems, 1(1):3-22. [Rumelhart and McClelland, 1986] Rumelhart, D. and McClelland, J. L. (1986). Parallel Distributed Processing, volume 1. MIT Press, Cambridge, MA. [Sinkhorn, 1964] Sinkhorn, R. (1964). A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Statist., 35:876-879. [Van den Bout and Miller III, 1990] Van den Bout, D. E. and Miller III, T . K. (1990). Graph partitioning using annealed networks. IEEE Trans. Neural Networks, 1(2):192-203. [Yuille and Kosowsky, 1994] Yuille, A. L. and Kosowsky, J. J. (1994). Statistical physics algorithms that converge. Neural Computation, 6(3):341-356.
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From Isolation to Cooperation: An Alternative View of a System of Experts Stefan Schaal:!:* Christopher C. Atkeson:!: sschaal@cc.gatech.edu cga@cc.gatech.edu http://www.cc.gatech.eduifac/Stefan.Schaal http://www.cc.gatech.eduifac/Chris.Atkeson +College of Computing, Georgia Tech, 801 Atlantic Drive, Atlanta, GA 30332-0280 * A TR Human Infonnation Processing, 2-2 Hikaridai, Seiko-cho, Soraku-gun, 619-02 Kyoto Abstract We introduce a constructive, incremental learning system for regression problems that models data by means of locally linear experts. In contrast to other approaches, the experts are trained independently and do not compete for data during learning. Only when a prediction for a query is required do the experts cooperate by blending their individual predictions. Each expert is trained by minimizing a penalized local cross validation error using second order methods. In this way, an expert is able to find a local distance metric by adjusting the size and shape of the receptive field in which its predictions are valid, and also to detect relevant input features by adjusting its bias on the importance of individual input dimensions. We derive asymptotic results for our method. In a variety of simulations the properties of the algorithm are demonstrated with respect to interference, learning speed, prediction accuracy, feature detection, and task oriented incremental learning. 1. INTRODUCTION Distributing a learning task among a set of experts has become a popular method in computationallearning. One approach is to employ several experts, each with a global domain of expertise (e.g., Wolpert, 1990). When an output for a given input is to be predicted, every expert gives a prediction together with a confidence measure. The individual predictions are combined into a single result, for instance, based on a confidence weighted average. Another approach-the approach pursued in this paper-of employing experts is to create experts with local domains of expertise. In contrast to the global experts, the local experts have little overlap or no overlap at all. To assign a local domain of expertise to each expert, it is necessary to learn an expert selection system in addition to the experts themselves. This classifier determines which expert models are used in which part of the input space. For incremental learning, competitive learning methods are usually applied. Here the experts compete for data such that they change their domains of expertise until a stable configuration is achieved (e.g., Jacobs, Jordan, Nowlan, & Hinton, 1991). The advantage of local experts is that they can have simple parameterizations, such as locally constant or locally linear models. This offers benefits in terms of analyzability, learning speed, and robustness (e.g., Jordan & Jacobs, 1994). For simple experts, however, a large number of experts is necessary to model a function. As a result, the expert selection system has to be more complicated and, thus, has a higher risk of getting stuck in local minima and/or of learning rather slowly. In incremental learning, another potential danger arises when the input distribution of the data changes. The expert selection system usually makes either implicit or explicit prior assumptions about the input data distribution. For example, in the classical mixture model (McLachlan & Basford, 1988) which was employed in several local expert approaches, the prior probabilities of each mixture model can be interpreted as 606 S. SCHAAL. C. C. ATKESON the fraction of data points each expert expects to experience. Therefore, a change in input distribution will cause all experts to change their domains of expertise in order to fulfill these prior assumptions. This can lead to catastrophic interference. In order to avoid these problems and to cope with the interference problems during incremental learning due to changes in input distribution, we suggest eliminating the competition among experts and instead isolating them during learning. Whenever some new data is experienced which is not accounted for by one of the current experts, a new expert is created. Since the experts do not compete for data with their peers, there is no reason for them to change the location of their domains of expertise. However, when it comes to making a prediction at a query point, all the experts cooperate by giving a prediction of the output together with a confidence measure. A blending of all the predictions of all experts results in the final prediction. It should be noted that these local experts combine properties of both the global and local experts mentioned previously. They act like global experts by learning independently of each other and by blending their predictions, but they act like local experts by confining themselves to a local domain of expertise, i.e., their confidence measures are large only in a local region. The topic of data fitting with structurally simple local models (or experts) has received a great deal of attention in nonparametric statistics (e.g., Nadaraya, 1964; Cleveland, 1979; Scott, 1992, Hastie & Tibshirani, 1990). In this paper, we will demonstrate how a nonparametric approach can be applied to obtain the isolated expert network (Section 2.1), how its asymptotic properties can be analyzed (Section 2.2), and what characteristics such a learning system possesses in terms of the avoidance of interference, feature detection, dimensionality reduction, and incremental learning of motor control tasks (Section 3). 2. RECEPTIVE FIELD WEIGHTED REGRESSION This paper focuses on regression problems, i.e., the learning of a map from 9tn ~ 9tm • Each expert in our learning method, Receptive Field Weighted Regression (RFWR), consists of two elements, a locally linear model to represent the local functional relationship, and a receptive field which determines the region in input space in which the expert's knowledge is valid. As a result, a given data set will be modeled by piecewise linear elements, blended together. For 1000 noisy data points drawn from the unit interval of the function z == max[exp(-10x 2),exp(-50l),1.25exp(-5(x2 + l)], Figure 1 illustrates an example of function fitting with RFWR. This function consists of a narrow and a wide ridge which are perpendicular to each other, and a Gaussian bump at the origin. Figure 1 b shows the receptive fields which the system created during the learning process. Each experts' location is at the center of its receptive field, marked by a $ in Figure 1 b. The recep0 . 5 0 -0.5 -1 1.5 ,1 10. 5% 0 I 1- 0 .5 1 0 - 0 .5 -1 x (a) 1.5 0.5 ,., 0 -0.5 -1 -1.5 -1.5 (b) -1 -0.5 o x 0.5 1.5 Figure 1: (a) result of function approximation with RFWR. (b) contour lines of 0.1 iso-activation of each expert in input space (the experts' centers are marked by small circles). From Isolation to Cooperation: An Alternative View of a System of Experts 607 tive fields are modeled by Gaussian functions, and their 0.1 iso-activation lines are shown in Figure 1 b as well. As can be seen, each expert focuses on a certain region of the input space, and the shape and orientation of this region reflects the function's complexity, or more precisely, the function's curvature, in this region. It should be noticed that there is a certain amount of overlap among the experts, and that the placement of experts occurred on a greedy basis during learning and is not globally optimal. The approximation result (Figure 1 a) is a faithful reconstruction of the real function (MSE = 0.0025 on a test set, 30 epochs training, about 1 minute of computation on a SPARC1O). As a baseline comparison, a similar result with a sigmoidal 3-layer neural network required about 100 hidden units and 10000 epochs of annealed standard backpropagation (about 4 hours on a SPARC1O). 2.1 THE ALGORITHM li'Iear ..•... '. ~"" " Galng Unrt WeighBd' / ~:~~ ConnectIOn Average centered at e Output y, Figure 2: The RFWR network RFWR can be sketched in network form as shown in Figure 2. All inputs connect to all expert networks, and new experts can be added as needed. Each expert is an independent entity. It consists of a two layer linear subnet and a receptive field subnet. The receptive field subnet has a single unit with a bell-shaped activation profile, centered at the fixed location c in input space. The maximal output of this unit is "I" at the center, and it decays to zero as a function of the distance from the center. For analytical convenience, we choose this unit to be Gaussian: (1) x is the input vector, and D the distance metric, a positive definite matrix that is generated from the upper triangular matrix M. The output of the linear subnet is: A Tb b -Tf3 y=x + o=x (2) The connection strengths b of the linear subnet and its bias bO will be denoted by the d-dimensional vector f3 from now on, and the tilde sign will indicate that a vector has been augmented by a constant "I", e.g., i = (x T , Il. In generating the total output, the receptive field units act as a gating component on the output, such that the total prediction is: (3) The parameters f3 and M are the primary quantities which have to be adjusted in the learning process: f3 forms the locally linear model, while M determines the shape and orientation of the receptive fields. Learning is achieved by incrementally minimizing the cost function: (4) The first term of this function is the weighted mean squared cross validation error over all experienced data points, a local cross validation measure (Schaal & Atkeson, 1994). The second term is a regularization or penalty term. Local cross validation by itself is consistent, i.e., with an increasing amount of data, the size of the receptive field of an expert would shrink to zero. This would require the creation of an ever increasing number of experts during the course of learning. The penalty term introduces some non-vanishing bias in each expert such that its receptive field size does not shrink to zero. By penalizing the squared coefficients of D, we are essentially penalizing the second derivatives of the function at the site of the expert. This is similar to the approaches taken in spline fitting 608 S. SCHAAL, C. C. A TI(ESON (deBoor, 1978) and acts as a low-pass filter: the higher the second derivatives, the more smoothing (and thus bias) will be introduced. This will be analyzed further in Section 2.2. The update equations for the linear subnet are the standard weighted recursive least squares equation with forgetting factor A (Ljung & SOderstrom, 1986): 1 ( pn- -Tpn ) f3n+1 =f3n+wpn+lxe wherepn+1 =_ pn_ xx ande =(y-xT f3n) cv' A Ajw + xTpnx cv (5) This is a Newton method, and it requires maintaining the matrix P, which is size 0.5d x (d + 1) . The update of the receptive field subnet is a gradient descent in J: Mn+l=Mn- a dJ!aM (6) Due to space limitations, the derivation of the derivative in (6) will not be explained here. The major ingredient is to take this derivative as in a batch update, and then to reformulate the result as an iterative scheme. The derivatives in batch mode can be calculated exactly due to the Sherman-Morrison-Woodbury theorem (Belsley, Kuh, & Welsch, 1980; Atkeson, 1992). The derivative for the incremental update is a very good approximation to the batch update and realizes incremental local cross validation. A new expert is initialized with a default M de! and all other variables set to zero, except the matrix P. P is initialized as a diagonal matrix with elements 11 r/, where the ri are usually small quantities, e.g., 0.01. The ri are ridge regression parameters. From a probabilistic view, they are Bayesian priors that the f3 vector is the zero vector. From an algorithmic view, they are fake data points of the form [x = (0, ... , '12 ,o, ... l,y = 0] (Atkeson, Moore, & Schaal, submitted). Using the update rule (5), the influence of the ridge regression parameters would fade away due to the forgetting factor A. However, it is useful to make the ridge regression parameters adjustable. As in (6), rj can be updated by gradient descent: 1'n+1 = 1'n - a aJ/ar I I I (7) There are d ridge regression parameters, one for each diagonal element of the P matrix. In order to add in the update of the ridge parameters as well as to compensate for the forgetting factor, an iterative procedure based on (5) can be devised which we omit here. The computational complexity of this update is much reduced in comparison to (5) since many computations involve multiplications by zero. Initialize the RFWR network. with no expert; For every new training sample (x,y): a) For k= I to #experts: b) c) d) e) end; - calculate the activation from (I) - update the expert's parameters according to (5), (6), and (7) end; Ir no expert was activated by more than W gen: - create a new expert with c=x end; Ir two experts are acti vated more than W pn .. ~ - erase the expert with the smaller receptive field end; calculate the mean, err ""an' and standard de viation errslIl of the incrementally accumulated error er,! of all experts; For k.= I to #experts: Ir (Itrr! - err_I> 9 er'Sld) reinitialize expert k with M = 2 • Mdef end; In sum, a RFWR expert consists of three sets of parameters, one for the locally linear model, one for the size and shape of the receptive fields, and one for the bias. The linear model parameters are updated by a Newton method, while the other parameters are updated by gradient descent. In our implementations, we actually use second order gradient descent based on Sutton (1992), since, with minor extra effort, we can obtain estimates of the second derivatives of the cost function with respect to all parameters. Finally, the logic of RFWR becomes as shown in the pseudo-code above. Point c) and e) of the algorithm introduce a pruning facility. Pruning takes place either when two experts overlap too much, or when an expert has an exceptionally large mean squared error. The latter method corresponds to a simple form of outlier detection. Local optimization of a distance metric always has a minimum for a very large receptive field size. In our case, this would mean that an expert favors global instead of locally linear regression. Such an expert will accumulate a very large error which can easily be detected From Isolation to Cooperation: An Alternative View of a System of Experts 609 in the given way. The mean squared error term, err, on which this outlier detection is based, is a bias-corrected mean squared error, as will be explained below. 2.2 ASYMPTOTIC BIAS AND PENALTY SELECTION The penalty term in the cost function (4) introduces bias. In order to assess the asymptotic value of this bias, the real function f(x) , which is to be learned, is assumed to be represented as a Taylor series expansion at the center of an expert's receptive field. Without loss of generality, the center is assumed to be at the origin in input space. We furthermore assume that the size and shape of the receptive field are such that terms higher than 0(2) are negligible. Thus, the cost (4) can be written as: J ~ (1w(f. +fTX+~XTFX-bo -bTx Y dx )/(1 wdx )+r~Dnm (8) where fo' f, and F denote the constant, linear, and quadratic terms of the Taylor series expansion, respectively. Inserting Equation (1), the integrals can be solved analytically after the input space is rotated by an orthonormal matrix transforming F to the diagonal matrix F'. Subsequently, bo' b, and D can be determined such that J is minimized: 0.25 ( ) ~ b~ = fa + bias = fa + ~075 ~ sgn(F:')~IF;,:I, b' = f, D:: = (2r)2 (9) This states that the linear model will asymptotically acquire the correct locally linear model, while the constant term will have bias proportional to the square root of the sum of the eigenvalues of F, i.e., the F:n • The distance metric D, whose diagonalized counterpart is D', will be a scaled image of the Hessian F with an additional square root distortion. Thus, the penalty term accomplishes the intended task: it introduces more smoothing the higher the curvature at an expert's location is, and it prevents the receptive field of an expert shrinking to zero size (which would obviously happen for r ~ 0). Additionally, Equation (9) shows how to determine rfor a given learning problem from an estimate of the eigenvalues and a permissible bias. Finally, it is possible to derive estimates of the bias and the mean squared error of each expert from the current distance metric D: biasesl = ~0 .5r IJeigenvalues(D)l.; en,,~, = r L D;m (10) n.m The latter term was incorporated in the mean squared error, err, in Section 2.1. Empirical evaluations (not shown here) verified the validity of these asymptotic results. 3. SIMULA TION RESULTS This section will demonstrate some of the properties of RFWR. In all simulations, the threshold parameters of the algorithm were set to e = 3.5, w prune = 0.9, and w min = 0.1. These quantities determine the overlap of the experts as well as the outlier removal threshold; the results below are not affected by moderate changes in these parameters. 3.1 AVOIDING INTERFERENCE In order to test RFWR's sensitivity with respect to changes in input data distribution, the data of the example of Figure 1 was partitioned into three separate training sets 1; = {(x, y, z) 1-1.0 < x < -O.2} , 1; = {(x, y, z) 1-0.4 < x < OA}, 1; = {(x, y, z) I 0.2 < x < 1.0}. These data sets correspond to three overlapping stripes of data, each having about 400 uniformly distributed samples. From scratch, a RFWR network was trained first on I; for 20 epochs, then on T2 for 20 epochs, and finally on 1; for 20 epochs. The penalty was chosen as in the example of Figure 1 to be r = I.e - 7 , which corresponds to an asymptotic bias of 610 S. SCHAAL, C. C. ATKESON 0.1 at the sharp ridge of the function. The default distance metric D was 50*1, where I is the identity matrix. Figure 3 shows the results of this experiment. Very little interference can be found. The MSE on the test set increased from 0.0025 (of the original experiment of Figure 1) to 0.003, which is still an excellent reconstruction of the real function. y 0 .5 -0 . 5 - 0 . 5 (a) (b) (c) -1 Figure 3: Reconstructed function after training on (a) 7;, (b) then ~,(c) and finally 1;. 3.2 LOCAL FEATURE DETECTION The examples of RFWR given so far did not require ridge regression parameters. Their importance, however, becomes obvious when dealing with locally rank deficient data or with irrelevant input dimensions. A learning system should be able to recognize irrelevant input dimensions. It is important to note that this cannot be accomplished by a distance metric. The distance metric is only able to decide to what spatial extent averaging over data in a certain dimension should be performed. However, the distance metric has no means to exclude an input dimension. In contrast, bias learning with ridge regression parameters is able to exclude input dimensions. To demonstrate this, we added 8 purely noisy inputs (N(0,0.3)) to the data drawn from the function of Figure 1. After 30 epochs of training on a 10000 data point training set, we analyzed histograms of the order of magnitude of the ridge regression parameters in all 100bias input dimensions over all the 79 experts that had been generated by the learning algorithm. All experts recognized that the input dimensions 3 to 8 did not contain relevant information, and correctly increased the corresponding ridge parameters to large values. The effect of a large ridge regression parameter is that the associated regression coefficient becomes zero. In contrast, the ridge parameters of the inputs 1, 2, and the bias input remained very small. The MSE on the test set was 0.0026, basically identical to the experiment with the original training set. 3.3 LEARNING AN INVERSE DYNAMICS MODEL OF A ROBOT ARM Robot learning is one of the domains where incremental learning plays an important role. A real movement system experiences data at a high rate, and it should incorporate this data immediately to improve its performance. As learning is task oriented, input distributions will also be task oriented and interference problems can easily arise. Additionally, a real movement system does not sample data from a training set but rather has to move in order to receive new data. Thus, training data is always temporally correlated, and learning must be able to cope with this. An example of such a learning task is given in Figure 4 where a simulated 2 DOF robot arm has to learn to draw the figure "8" in two different regions of the work space at a moderate speed (1.5 sec duration). In this example, we assume that the correct movement plan exists, but that the inverse dynamics model which is to be used to control this movement has not been acquired. The robot is first trained for 10 minutes (real movement time) in the region of the lower target trajectory where it performs a variety of rhythmic movements under simple PID control. The initial performance of this controller is shown in the bottom part of Figure 4a. This training enables the robot to learn the locally appropriate inverse dynamics model, a ~6 ~ ~2 continuous mapping. Subsequent perFrom Isolation to Cooperation: An Alternative View of a System of Experts 611 0.5 0.' t GralMy 0.' 0.2 ~ 8 0.1 ~t Z 8 8 ..,. ~. ·0.4 ~.5 (a) (b) (0) 0 0.1 0.2 0.3 0.4 0.!5 Figure 4: Learning to draw the figure "8" with a 2-joint arm: (a) Performance of a PID controller before learning (the dimmed lines denote the desired trajectories, the solid lines the actual performance); (b) Performance after learning using a PD controller with feedforward commands from the learned inverse model; (c) Performance of the learned controller after training on the upper "8" of (b) (see text for more explanations). formance using this inverse model for control is depicted in the bottom part of Figure 4b. Afterwards, the same training takes place in the region of the upper target trajectory in order to acquire the inverse model in this part of the world. The figure "8" can then equally well be drawn there (upper part of Figure 4a,b). Switching back to the bottom part of the work space (Figure 4c), the first task can still be performed as before. No interference is recognizable. Thus, the robot could learn fast and reliably to fulfill the two tasks. It is important to note that the data generated by the training movements did not always have locally full rank. All the parameters of RFWR were necessary to acquire the local inverse model appropriately. A total of 39 locally linear experts were generated. 4. DISCUSSION We have introduced an incremental learning algorithm, RFWR, which constructs a network of isolated experts for supervised learning of regression tasks. Each expert determines a locally linear model, a local distance metric, and local bias parameters by incrementally minimizing a penalized local cross validation error. Our algorithm differs from other local learning techniques by entirely avoiding competition among the experts, and by being based on nonparametric instead of parametric statistics. The resulting properties of RFWR are a) avoidance of interference in the case of changing input distributions, b) fast incremental learning by means of Newton and second order gradient descent methods, c) analyzable asymptotic properties which facilitate the selection of the fit parameters, and d) local feature detection and dimensionality reduction. The isolated experts are also ideally suited for parallel implementations. Future work will investigate computationally less costly delta-rule implementations of RFWR, and how well RFWR scales in higher dimensions. 5. REFERENCES Atkeson, C. G., Moore, A. W., & Schaal, S. (submitted). "Locally weighted learning." Artificial Intelligence Review. Atkeson, C. G. (1992). "Memory-based approaches to approximating continuous functions." In: Casdagli, M., & Eubank, S. (Eds.), Nonlinear Modeling and Forecasting, pp.503-521. Addison Wesley. Belsley, D. A., Kuh, E., & Welsch, R. E. (1980). Regression diagnostics: Identifying influential data and sources ofcollinearity. New York: Wiley. Cleveland, W. S. (1979). "Robust locally weighted regression and smoothing scatterplots." J. American Stat. Association, 74, pp.829-836. de Boor, C. (1978). A practical guide to splines. New York: Springer. Hastie, T. J., & Tibshirani, R. J. (1990). Generalized additive models. London: Chapman and Hall. 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. (1994). "Hierarchical mixtures of experts and the EM algorithm." Neural Computation, 6, pp.79-87. Ljung, L., & S_derstr_m, T. (1986). Theory and practice of recursive identification. Cambridge, MIT Press. McLachlan, G. J., & Basford, K. E. (1988). Mixture models. New York: Marcel Dekker. Nadaraya, E. A. (1964). "On estimating regression." Theor. Prob. Appl., 9, pp.141-142. Schaal, S., & Atkeson, C. G. (l994b). "Assessing the quality of learned local models." In: Cowan, J. ,Tesauro, G., & Alspector, J. (Eds.), Advances in Neural Information Processing Systems 6. Morgan Kaufmann. Scott, D. W. (1992). Multivariate Density Estimation. New York: Wiley. Sutton, R. S. (1992). "Gain adaptation beats least squares." In: Proc. of 7th Yale Workshop on Adaptive and Learning Systems, New Haven, CT. Wolpert, D. H. (1990). "Stacked genealization." Los Alamos Technical Report LA-UR-90-3460.
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Fast Learning by Bounding Likelihoods in Sigmoid Type Belief Networks Tommi Jaakkola tommi@psyche.mit.edu Lawrence K. Saul lksaul@psyche.mit.edu Michael I. Jordan jordan@psyche.mit.edu Department of Brain and Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 Abstract Sigmoid type belief networks, a class of probabilistic neural networks, provide a natural framework for compactly representing probabilistic information in a variety of unsupervised and supervised learning problems. Often the parameters used in these networks need to be learned from examples. Unfortunately, estimating the parameters via exact probabilistic calculations (i.e, the EM-algorithm) is intractable even for networks with fairly small numbers of hidden units. We propose to avoid the infeasibility of the E step by bounding likelihoods instead of computing them exactly. We introduce extended and complementary representations for these networks and show that the estimation of the network parameters can be made fast (reduced to quadratic optimization) by performing the estimation in either of the alternative domains. The complementary networks can be used for continuous density estimation as well. 1 Introduction The appeal of probabilistic networks for knowledge representation, inference, and learning (Pearl, 1988) derives both from the sound Bayesian framework and from the explicit representation of dependencies among the network variables which allows ready incorporation of prior information into the design of the network. The Bayesian formalism permits full propagation of probabilistic information across the network regardless of which variables in the network are instantiated. In this sense these networks can be "inverted" probabilistically. This inversion, however, relies heavily on the use of look-up table representations Fast Learning by Bounding Likelihoods in Sigmoid Type Belief Networks 529 of conditional probabilities or representations equivalent to them for modeling dependencies between the variables. For sparse dependency structures such as trees or chains this poses no difficulty. In more realistic cases of reasonably interdependent variables the exact algorithms developed for these belief networks (Lauritzen & Spiegelhalter, 1988) become infeasible due to the exponential growth in the size of the conditional probability tables needed to store the exact dependencies. Therefore the use of compact representations to model probabilistic interactions is unavoidable in large problems. As belief network models move away from tables, however, the representations can be harder to assess from expert knowledge and the important role of learning is further emphasized. Compact representations of interactions between simple units have long been emphasized in neural networks. Lacking a thorough probabilistic interpretation, however, classical feed-forward neural networks cannot be inverted in the above sense; e.g. given the output pattern of a feed-forward neural network it is not feasible to compute a probability distribution over the possible input patterns that would have resulted in the observed output. On the other hand, stochastic neural networks such as Boltzman machines admit probabilistic interpretations and therefore, at least in principle, can be inverted and used as a basis for inference and learning in the presence of uncertainty. Sigmoid belief networks (Neal, 1992) form a subclass of probabilistic neural networks where the activation function has a sigmoidal form - usually the logistic function. Neal (1992) proposed a learning algorithm for these networks which can be viewed as an improvement ofthe algorithm for Boltzmann machines. Recently Hinton et al. (1995) introduced the wake-sleep algorithm for layered bi-directional probabilistic networks. This algorithm relies on forward sampling and has an appealing coding theoretic motivation. The Helmholtz machine (Dayan et al., 1995), on the other hand, can be seen as an alternative technique for these architectures that avoids Gibbs sampling altogether. Dayan et al. also introduced the important idea of bounding likelihoods instead of computing them exactly. Saul et al. (1995) subsequently derived rigorous mean field bounds for the likelihoods. In this paper we introduce the idea of alternative - extended and complementary - representations of these networks by reinterpreting the nonlinearities in the activation function. We show that deriving likelihood bounds in the new representational domains leads to efficient (quadratic) estimation procedures for the network parameters. 2 The probability representations Belief networks represent the joint probability of a set of variables {S} as a product of conditional probabilities given by n P(St, ... , Sn) = IT P(Sk Ipa[k]), (1) k=l where the notation pa[k], "parents of Sk", refers to all the variables that directly influence the probability of Sk taking on a particular value (for equivalent representations, see Lauritzen et al. 1988). The fact that the joint probability can be written in the above form implies that there are no "cycles" in the network; i.e. there exists an ordering of the variables in the network such that no variable directly influences any preceding variables. In this paper we consider sigmoid belief networks where the variables S are binary 530 T. JAAKKOLA, L. K. SAUL, M. I. JORDAN (0/1), the conditional probabilities have the form P(Ss:lpa[i]) = g( (2Ss: - 1) L WS:jSj) j (2) and the weights Wij are zero unless Sj is a parent of Si, thus preserving the feedforward directionality of the network. For notational convenience we have assumed the existence of a bias variable whose value is clamped to one. The activation function g(.) is chosen to be the cumulative Gaussian distribution function given by 1 jX .l ~ 1 1 00 .l( )~ g(x) = -e- 2 Z dz = -e- 2 z-x dz (3) ..j2; - 00 ..j2; 0 Although very similar to the standard logistic function, this activation function derives a number of advantages from its integral representation. In particular, we may reinterpret the integration as a marginalization and thereby obtain alternative representations for the network. We consider two such representations. We derive an extended representation by making explicit the nonlinearities in the activation function. More precisely, P(Silpa[i]) g( (2Si - 1) L WijSj) j (4) This suggests defining the extended network in terms of the new conditional probabilities P(Si, Zs:lpa[i]). By construction then the original binary network is obtained by marginalizing over the extra variables Z. In this sense the extended network is (marginally) equivalent to the binary network. We distinguish a complementary representation from the extended one by writing the probabilities entirely in terms of continuous variables!. Such a representation can be obtained from the extended network by a simple transformation of variables. The new continuous variables are defined by Zs: = (2Si - l)Zi, or, equivalently, by Zi = IZs: I and Si = O( Zs:) where 0(·) is the step function. Performing this transformation yields P(Z-'I [.]) - _1_ -MZi-L: . Wij9(Zj)1~ I pa z rn=e J V 211" (5) which defines a network of conditionally Gaussian variables. The original network in this case can be recovered by conditional marginalization over Z where the conditioning variables are O(Z). Figure 1 below summarizes the relationships between the different representations. As will become clear later, working with the alternative representations instead of the original binary representation can lead to more flexible and efficient (leastsquares) parameter estimation. 3 The learning problem We consider the problem of learning the parameters of the network from instantiations of variables contained in a training set. Such instantiations, however, need not 1 While the binary variables are the outputs of each unit the continuous variables pertain to the inputs - hence the name complementary. Fast Learning by Bounding Likelihoods in Sigmoid Type Belief Networks Extended network ___ --~-::a-z_.::~ ~ _::s. Z) Original network over {S} "tr:sfonnation of ~ariables Complementary network over {Z} Figure 1: The relationship between the alternative representations. 531 be complete; there may be variables that have no value assignments in the training set as well as variables that are always instantiated. The tacit division between hidden (H) and visible (V) variables therefore depends on the particular training example considered and is not an intrinsic property of the network. To learn from these instantiations we adopt the principle of maximum likelihood to estimate the weights in the network. In essence, this is a density estimation problem where the weights are chosen so as to match the probabilistic behavior of the network with the observed activities in the training set. Central to this estimation is the ability to compute likelihoods (or log-likelihoods) for any (partial) configuration of variables appearing in the training set. In other words, if we let XV be the configuration of visible or instantiated variables2 and XH denote the hidden or uninstantiated variables, we need to compute marginal probabilities of the form (6) XH If the training samples are independent, then these log marginals can be added to give the overall log-likelihood of the training set 10gP(training set) = L:logP(XVt) (7) Unfortunately, computing each of these marginal probabilities involves summing (integrating) over an exponential number of different configurations assumed by the hidden variables in the network. This renders the sum (integration) intractable in all but few special cases (e.g. trees and chains). It is possible, however, to instead find a manageable lower bound on the log-likelihood and optimize the weights in the network so as to maximize this bound. To obtain such a lower bound we resort to Jensen's inequality: 10gP(Xv) 10gL p(XH,XV) = 10gLQ(XH)P(XH,;V) XH XH Q(X ) > ~Q(XH)1 p(XH,XV) (8) f; og Q(XH) Although this bound holds for all distributions Q(X) over the hidden variables, the accuracy of the bound is determined by how closely Q approximates the posterior distribution p(XH IXv) in terms of the Kullback-Leibler divergence; if the approximation is perfect the divergence is zero and the inequality is satisfied with equality. Suitable choices for Q can make the bound both accurate and easy to compute. The feasibility of finding such Q, however, is highly dependent on the choice of the representation for the network. 2To postpone the issue of representation we use X to denote 5, {5, Z}, or Z depending on the particular representation chosen. 532 T. JAAKKOLA, L. K. SAUL, M. I. JORDAN 4 Likelihood bounds in different representations To complete the derivation of the likelihood bound (equation 8) we need to fix the representation for the network. Which representation to select, however, affects the quality and accuracy of the bound. In addition, the accompanying bound of the chosen representation implies bounds in the other two representational domains as they all code the same distributions over the observables. In this section we illustrate these points by deriving bounds in the complementary and extended representations and discuss the corresponding bounds in the original binary domain. Now, to obtain a lower bound we need to specify the approximate posterior Q. In the complementary representation the conditional probabilities are Gaussians and therefore a reasonable approximation (mean field) is found by choosing the posterior approximation from the family of factorized Gaussians: Q(Z) = IT _1_e-(Zi-hi)~/2 (9) i..;?:; Substituting this into equation 8 we obtain the bound log P(S*) ~ -~ L (hi - Ej Jij g(hj»2 - ~ L Ji~g(hj )g(-hj ) (10) i ij The means hi for the hidden variables are adjustable parameters that can be tuned to make the bound as tight as possible. For the instantiated variables we need to enforce the constraints g( hi) = S: to respect the instantiation. These can be satisfied very accurately by setting hi = 4(2S: - 1). A very convenient property of this bound and the complementary representation in general is the quadratic weight dependence - a property very conducive to fast learning. Finally, we note that the complementary representation transforms the binary estimation problem into a continuous density estimation problem. We now turn to the interpretation of the above bound in the binary domain. The same bound can be obtained by first fixing the inputs to all the units to be the means hi and then computing the negative total mean squared error between the fixed inputs and the corresponding probabilistic inputs propagated from the parents. The fact that this procedure in fact gives a lower bound on the log-likelihood would be more difficult to justify by working with the binary representation alone. In the extended representation the probability distribution for Zi is a truncated Gaussian given Si and its parents. We therefore propose the partially factorized posterior approximation: (11) where Q(ZiISi) is a truncated Gaussian: Q(Zi lSi) = 1 _1_e- t(Zi-(2S,-1)hi)~ g« 2Si- 1)hi ) ..;?:; (12) As in the complementary domain the resulting bound depends quadratically on the weights. Instead of writing out the bound here, however, it is more informative to see its derivation in the binary domain. A factorized posterior approximation (mean field) Q(S) = n. q~i(1 - qi)l-S, for the binary network yields a bound I I 10gP(S*) > L {(Si 10gg(Lj J,jSj») + (1- Si) 10g(l- 9(L; Ji;S;»)} i Fast Learning by Bounding Likelihoods in Sigmoid Type Belief Networks 533 (13) where the averages (.) are with respect to the Q distribution. These averages, however, do not conform to analytical expressions. The tractable posterior approximation in the extended domain avoids the problem by implicitly making the following Legendre transformation: 1 2 1 2 1 2 logg(x) = ["2x + logg(x)] -"2x ~ AX - G(A) - "2x (14) which holds since x 2/2 + logg(x) is a convex function. Inserting this back into the relevant parts of equation 13 and performing the averages gives 10gP(S*) > L {[qjAj - (1- qj),Xd Lhjqj - qjG(Ai) - (1- qj)G('xi)} j I", 2 1",2 ( -"2(L.,.. Jijqj) -"2 L.,.. Jjjqj 1- gj) j ij (15) which is quadratic in the weights as expected. The mean activities q for the hidden variables and the parameters A can be optimized to make the bound tight. For the instantiated variables we set qi = S; . 5 Numerical experiments To test these techniques in practice we applied the complementary network to the problem of detecting motor failures from spectra obtained during motor operation (see Petsche et al. 1995). We cast the problem as a continuous density estimation problem. The training set consisted of 800 out of 1283 FFT spectra each with 319 components measured from an electric motor in a good operating condition but under varying loads. The test set included the remaining 483 FFTs from the same motor in a good condition in addition to three sets of 1340 FFTs each measured when a particular fault was present. The goal was to use the likelihood of a test FFT with respect to the estimated density to determine whether there was a fault present in the motor. We used a layered 6 -+ 20 -+ 319 generative model to estimate the training set density. The resulting classification error rates on the test set are shown in figure 2 as a function of the threshold likelihood. The achieved error rates are comparable to those of Petsche et al. (1995). 6 Conclusions Network models that admit probabilistic formulations derive a number of advantages from probability theory. Moving away from explicit representations of dependencies, however, can make these properties harder to exploit in practice. We showed that an efficient estimation procedure can be derived for sigmoid belief networks, where standard methods are intractable in all but a few special cases (e.g. trees and chains). The efficiency of our approach derived from the combination of two ideas. First, we avoided the intractability of computing likelihoods in these networks by computing lower bounds instead. Second, we introduced new representations for these networks and showed how the lower bounds in the new representational domains transform the parameter estimation problem into 534 0.0 ..... 0.8 0.7 0.8 ',_ fo.s "\ D:' ' , ... .. d.. ''', 0.3 " --0.2 . .. '0.1 , . , , , , , , , '. " , , . T. JAAKKOLA, L. K. SAUL, M. 1. JORDAN Figure 2: The probability of error curves for missing a fault (dashed lines) and misclassifying a good motor (solid line) as a function of the likelihood threshold. quadratic optimization. Acknowledgments The authors wish to thank Peter Dayan for helpful comments. This project was supported in part by NSF grant CDA-9404932, by a grant from the McDonnellPew Foundation, by a grant from ATR Human Information Processing Research Laboratories, by a grant from Siemens Corporation, and by grant N00014-94-10777 from the Office of Naval Research. Michael I. Jordan is a NSF Presidential Young Investigator. References P. Dayan, G. Hinton, R. Neal, and R. Zemel (1995). The helmholtz machine. Neural Computation 7: 889-904. A. Dempster, N. Laird, and D. Rubin. Maximum likelihood from incomplete data via the EM algorithm (1977). J. Roy. Statist. Soc. B 39:1-38. G. Hinton, P. Dayan, B. Frey, and R. Neal (1995). The wake-sleep algorithm for unsupervised neural networks. Science 268: 1158-1161. S. L. Lauritzen and D. J. Spiegelhalter (1988). Local computations with probabilities on graphical structures and their application to expert systems. J. Roy. Statist. Soc. B 50:154-227. R. Neal. Connectionist learning of belief networks (1992). Artificial Intelligence 56: 71-113. J. Pearl (1988). Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann: San Mateo. T. Petsche, A. Marcantonio, C. Darken, S. J. Hanson, G. M. Kuhn, I. Santoso (1995). A neural network autoassociator for induction motor failure prediction. In Advances in Neural Information Processing Systems 8. MIT Press. 1. K. Saul, T. Jaakkola, and M. I. Jordan (1995). Mean field theory for sigmoid belief networks. M.l. T. Computational Cognitive Science Technical Report 9501.
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Neural Control for Nonlinear Dynamic Systems Ssu-Hsin Yu Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, MA 02139 Email: hsin@mit.edu Anuradha M. Annaswamy Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, MA 02139 Email: aanna@mit.edu Abstract A neural network based approach is presented for controlling two distinct types of nonlinear systems. The first corresponds to nonlinear systems with parametric uncertainties where the parameters occur nonlinearly. The second corresponds to systems for which stabilizing control structures cannot be determined. The proposed neural controllers are shown to result in closed-loop system stability under certain conditions. 1 INTRODUCTION The problem that we address here is the control of general nonlinear dynamic systems in the presence of uncertainties. Suppose the nonlinear dynamic system is described as x= f(x , u , 0) , y = h(x, u, 0) where u denotes an external input, y is the output, x is the state, and 0 is the parameter which represents constant quantities in the system. The control objectives are to stabilize the system in the presence of disturbances and to ensure that reference trajectories can be tracked accurately, with minimum delay. While uncertainties can be classified in many different ways, we focus here on two scenarios. One occurs because the changes in the environment and operating conditions introduce uncertainties in the system parameter O. As a result, control objectives such as regulation and tracking, which may be realizable using a continuous function u = J'(x, 0) cannot be achieved since o is unknown. Another class of problems arises due to the complexity of the nonlinear function f. Even if 0, f and h can be precisely determined, the selection of an appropriate J' that leads to stabilization and tracking cannot be made in general. In this paper, we present two methods based on neural networks which are shown to be applicable to both the above classes of problems. In both cases, we clearly outline the assumptions made, the requirements for adequate training of the neural network, and the class of engineering problems where the proposed methods are applicable. The proposed approach significantly expands the scope of neural controllers in relation to those proposed in (Narendra and Parthasarathy, 1990; Levin and Narendra, 1993; Sanner and Slotine, 1992; Jordan and Rumelhart, 1992). Neural Control for Nonlinear Dynamic Systems 1011 The first class of problems we shall consider includes nonlinear systems with parametric uncertainties. The field of adaptive control has addressed such a problem, and over the past thirty years, many results have been derived pertaining to the control of both linear and nonlinear dynamic systems (Narendra and Annaswamy, 1989). A common assumption in almost all of the published work in this field is that the uncertain parameters occur linearly. In this paper, we consider the control of nonlinear dynamic systems with nonlinear parametrizations. We design a neural network based controller that adapts to the parameter o and show that closed-loop system stability can be achieved under certain conditions. Such a controller will be referred to as a O-adaptive neural controller. Pertinent results to this class are discussed in section 2. The second class of problems includes nonlinear systems, which despite being completely known, cannot be stabilized by conventional analytical techniques. The obvious method for stabilizing nonlinear systems is to resort to linearization and use linear control design methods. This limits the scope of operation of the stabilizing controller. Feedback linearization is another method by which nonlinear systems can be stably controlled (lsidori, 1989). This however requires fairly stringent set of conditions to be satisfied by the functions! and h. Even after these conditions are satisfied, one cannot always find a closed-form solution to stabilize the system since it is equivalent to solving a set of partial differential equations. We consider in this paper, nonlinear systems, where system models as well as parameters are known, but controlIer structures are unknown. A neural network based controller is shown to exist and trained so that a stable closed-loop system is achieved. We denote this class of controllers as a stable neural controller. Pertinent results to this class are discussed in section 3. 2 O-ADAPTIVE NEURAL CONTROLLER The focus of the nonlinear adaptive controller to be developed in this paper is on dynamic systems that can be written in the d-step ahead predictor form as follows: Yt+d = !r(Wt,Ut,O) (I) where wi = [Yt," . ,Yt-n+l, Ut-I, ' ", Ut-m-d+l], n ~ I, m ~ 0, d ~ I, m + d = n, YI, UI C ~ containing the origin and 8 1 C ~k are open, ir : Y1 x U;n+d x 8 1 -t ~, Yt and Ut are the output and the input of the system at time t respectively, and 0 is an unknown parameter and occurs nonlinearly in ir.1 The goal is to choose a control input 'It such that the system in (1) is stabilized and the plant output is regulated around zero. Letxi ~ [Yt+d-I , '" ,Yt+l ,wil T , Am = [e2,"', en+d-I, 0, en+d+I,"', en+m+2d-2, 0], Bm = [el' en+d], where e, is an unit vector with the i-th term equal to I. The following assumptions are made regarding the system in Eq. (I ). (AI) For every 0 E 8 1, ir(O,O, O) = 0. CA2) There exist open and convex neighborhoods of the origin Y2 C YI and U2 C UI, an open and convex set 82 C 8 1, and a function K : 0.2 x Y2 x 8 2 ---> U I such that for every Wt E 0.2, Yt+d E Y2 and 0 E 8 2, Eq. (1) can be written as Ut = K(wt, Yt+d, 0), where 0.2 ~ Y2 X u;,,+d-I. (A3) K is twice differentiable and has bounded first and second derivatives on EI ~ 0.2 X Y2 X 8 2, while ir is differentiable and has a bounded derivative on 0.2 x I{ (E I ) x 8 2. (A4) There exists bg > ° such that for every YI E ir(o.2, K(o.2' 0, 8 2), 8 2), W E 0.2 and o BE 8 11 - (8K(w,y ,O) _ 8K(w,y,9))1 _ . 8f,(w ,u ,O) I - I > b ,2, ay ay Y YI au U-UI g' 1 Here, as well as in the following sections, An denotes the n-th product space of the set A. 1012 S. YU, A. M. ANNASW AMY (A5) There exist positive definite matrices P and Q of dimensions (n + m + 2d - 2) such T T' -T T T T T that xt (AmPAm - P)Xt+ J( BrnPBmK + 2xt ArnPBmK ~ -Xt QXt, where [( = [0, K(wt, 0, O)]T. Since the objective is to control the system ~n (1) where 0 is unknown, in order to stabilize the output Y at the origin with an estimate Of, we choose the control input as (2) 2.1 PARAMETER ESTIMATION SCHEME Suppose the estimation algorithm for updating Ot is defined recursively as /10t ~ OtOt-I = R(Yt,Wt-d,Ut-d,Ot-l) the problem is to determine the function R such that Ot converges to 0 asymptotical1y. In general, R is chosen to depend on Yt, Wt-d, 1£t-d and Ot-l since they are measurable and contain information regarding O. For example, in the case of linear systems which can be cast in the input predictor form, 1£t = <b[ 0, a wel\known linear parameter estimation method is to adjust /10 as (Goodwin and Sin, 1984) /10t = 1+4>'t~~t-'/ [1£t-d - ¢LdOt-d· In other words, the mechanism for carrying out parameter estimation is realized by R. In the case of general nonlinear systems, the task of determining such a function R is quite difficult, especial\y when the parameters occur nonlinearly. Hence, we propose the use of a neural network parameter estimation algorithm denoted O-adaptive neural network (TANN) (Annaswamy and Yu, 1996). That is, we adjust Ot as if /1Vd, < -f otherwise (3) where the inputs of the neural network are Yt, Wt-d, 1£t-d and Ot-I, the output is /10t, and f defines a dead-zone where parameter adaptation stops. The neural network is to be trained so that the resulting network can improve the parameter estimation over time for any possible 0 in a compact set. In addition, the trained network must ensure that the overal1 system in Eqs. (1), (2) and (3) is stable. Toward this end, N in TANN algorithm is required to satisfy the fol1owing two properties: (PI) IN(Yt,wt- d,1£t- d,Ot-l)12 ~ a(llfJt;~~,~1~2)2uLd' and (P2) /1Vt -/1Vd, < fl, > 0 where A Tf Iii 12 _ Iii 12 ii II _ II AV, -a 2+IC( <f;,_,,)1 1 1£-2 fl , L.l.Vt Ut Ut- I, Ut Ut u, L.l. d, ( IC( )12)2 t- d' 1+ , 4>,-./ C(¢t) = (~~ (Wt,Yt+d,O)lo=oo)T, Ut = Ut - K(Wt,Yt+d,Ot+d- I), (fit = [WT,Yt+djT, a E (0, I) and 00 is the point where K is linearized and often chosen to be the mean value of parameter variation. 2.2 TRAINING OF TANN FOR CONTROL In the previous section, we proposed an algorithm using a neural network for adjusting the control parameters. We introduced two properties (PI) and (P2) of the identification algorithm that the neural network needs to possess in order to maintain stability of the closed-loop system. In this section, we discuss the training procedure by which the weights of the neural network are to be adjusted so that the network retains these properties, The training set is constructed off-line and should compose of data needed in the training phase. If we wan..!. the algorithm in Eq. (3) to be valid on the specified sets Y3 and U3 for various 0 and 0 in 83, the training set should cover those variables appearing in Eq. (3) in their respective ranges. Hence, we first sample W in the set Y;- x U;:+d-I, Neural Control for Nonlinear Dynamic Systems 1013 and B, 8 in the set 83. Their values are, say, WI, BI and 81 respectively. For the particular fh and BI we sample 8 again in the set {B E 8 31 IB BII :s: 181 BI I}, and its value is, say, 8t. Once WI, BI , 81 and 8t are sampled, other data can then be calculated, such as UI = K(WI' 0, 8d and YI = fr(WI, UI, Bd. We can also ob. th d· C(A-) BK ( B) All: 2+IC(¢dI2 ( ~)2 d tam ecorrespon mg '1'1 - ao WI , YI, 0, il d l -a(I+IC(¢I)i2)2 UI -'ttl an _ IC(¢IW ~ 2 _ T T ~ _ ~d LI a (1+IC(<I>I)I2)2 (UI UI) ,where ¢I [WI ,yJ} and UI K(WI' YI,( 1 )· A data element can then be formed as (Yl ,WI ,UI, 8t, BI , ~ Vd l , Ld. Proceeding in the same manner, by choosing various ws , Bs , 1f. and 8~ in their respective ranges, we form a typical training set Ttram = { (Ys , W s, us,1f~ , Bs, ~ Vd d Ls) 11 :s: s :s: M}, where M denotes the total number of patterns in the training set. If the quadratic penalty function method (Bertsekas, 1995) is used, properties (PI) and (P2) can be satisfied by training the network on the training set to minimize the following cost function: M mJpl ~ mJ,n~~{(max{0 , ~VeJ)2+ ;2 (max{0, INi(W)12 - L t})2} (4) To find a W which minimizes the above unconstrained cost function 1, we can apply algorithms such as the gradient method and the Gauss-Newton method. 2.3 STABILITY RESULT With the plant given by Eq. (1), the controller by Eq. (2), and the TANN parameter estimation algorithm by Eq. (3), it can be shown that the stability of the closed-loop system is guaranteed. Based on the assumptions of the system in (1) and properties (PI) and (P2) that TANN satisfies, the stability result of the closed-loop system can be concluded in the following theorem. We refer the reader to (Yu and Annaswamy, 1996) for further detail. Theorem 1 Given the compact sets Y;+ I X U:;+d x 8 3 where the neural network in Eq. (3) is trained. There exist EI, E > 0 such that for any interior point B of 8 3, there exist open sets Y4 C Y3, U4 C U3 and a neighborhood 8 4 of B such that if Yo , ... , Yn+d-2 E Y4, Uo, .. . , U n -2 E U4, and 8n - l , ... ,8n+d - 2 E 8 4, then all the signals in the closed-loop system remain bounded and Yt converges to a neighborhood of the origin. 2.4 SIMULATION RESULTS In this section, we present a simulation example of the TANN controller proposed in this . Th . f h f lIy, ( I-y,) h B· h b sectton. e system IS 0 t e orm Yt+1 = I+e U.USH", + Ut, were IS t e parameter to e determined on-line. Prior information regarding the system is that () E [4, 10]. Based on 8 (I) ~ Eq. (2), the controller was chosen to be Ut = ,y, 0 ,-;'Y' ' where Bt denotes the parameter I+e- · "" estimate at time t. According to Eq. (3), B was estimated using the TANN algorithm with inputs YHI, Yt. Ut and~, and E = 0.01. N is a Gaussian network with 700 centers. The training set and the testing set were composed of 6,040 and 720 data elements respectively. After the training was completed, we tested the TANN controller on the system with six different values of B, 4.5, 5.5, 6.5, 7.~, 8.5 and 9.5, while the initial parameter estimate and the initial output were chosen as BI = 7 and Yo = -0.9 respectively. The results are plotted in Figure 1. It can be seen that Yt can be stabilized at the origin for all these values of B. For comparison, we also simulated the system under the same conditions but with 8 1014 -1 -2 o 50 10 100 4 ~ 0 -1 -2 o 50 S. YU, A. M. ANNASWAMY 10 100 4 Figure 1: Yt (TANN Controller) Figure 2: Yt (Extended Kalman Filter) estimated using the extended Kalman filter (Goodwin and Sin, 1984). Figure 2 shows the output responses. It is not surprising that for some values of fJ, especially when the initial estimation error is large, the responses either diverge or exhibit steady state error. 3 STABLE NEURAL CONTROLLER 3.1 STATEMENT OF THE PROBLEM Consider the following nonlinear dynamical system X= j(x,u), Y = h(x) (5) where x E Rn and u E RTn. Our goal is to construct a stabilizing neural controller as u = N(y; W) where N is a neural network with weights W, and establish the conditions under which the closed-loop system is stable. The nonlinear system in (5) is expressed as a combination of a higher-order linear part and a nonlinear part as x= Ax + Bu + RI (x, u) and y = Cx + R 2(x), where j(O,O) = 0 and h(O) = O. We make the following assumptions: (AI) j, h are twice continuously differentiable and are completely known. (A2) There exists a K such that (A - BKC) is asymptotically stable. 3.2 TRAINING OF THE STABLE NEURAL CONTROLLER In order for the neural controller in Section 3.1 to result in an asymptotically stable c1osedloop system, it is sufficient to establish that a continuous positive definite function of the state variables decreases monotonically through output feedback. In other words, if we can find a scalar definite function with a negative definite derivative of all points in the state space, we can guarantee stability of the overall system. Here, we limit our choices of the Lyapunov function candidates to the quadratic form, i.e. V = xT Px, where P is positive definite, and the goal is to choose the controller so that V < 0 where V = 2xT P j(x, N(h(x), W)). Based on the above idea, we define a "desired" time-derivative V d as V d= -xTQx where Q = QT > O. We choose P and Q matrices as follows. First, according to (AI), we can find a matrix K to make (A - BKC) asymptotically stable. We can then find a (P, Q) pair by choosing an arbitrary positive definite matrix Q and solving the Lyapunov equation, (A - BKC)T P + P(A - BKC) = -Q to obtain a positive definite P. Neural Control for Nonlinear Dynamic Systems 1015 With the contro\1er of the form in Section 3.1, the goal is to find W in the neural network which yields V:::; V d along the trajectories in a neighborhood X C ~n of the origin in the state space. Let Xi denote the value of a sample point where i is an index to the sample variable X E X in the state space. To establish V:::; V d, it is necessary that for every Xl in a neighborhood X C ~n of the origin, Vi:::;Vd" where Vi= 2x;Pf(x l ,N(h(:rl ) , W)) and V d, = -x; QXi . That is, the goal is to find a W such that the inequality constraints tlVe , :::; 0, where i = 1,··· , M, is satisfied, where tlVe , =V l V d, and M denotes the total number of sample points in X. As in the training of TANN controller, this can also be posed as an optimization problem. If the same quadratic penalty function method is used, the problem is to find W to minimize the fo\1owing cost function over the training set, which is described as Ttrain = {(Xl' Yi, V d.)\l :::; i :::; M}: 1M rwn J 6. mJp 2 I: (max{O, tlVe,})2 (6) i= 1 3.3 STABILITY OF THE CLOSED-LOOP SYSTEM Assum~tions (A 1) and (A2) imply that a stabilizing controller u = - J( y exists so that V = X Px is a candidate Lyapunov function. More genera\1y, suppose a continuous but unknown function ,,((y) exists such that for V = xT Px, a control input 1t = "((y) leads to V:::; -xT Qx, then we can find a neural network N (y) which approximates "((y) arbitrarily closely in a compact set leading to closed-loop stability. This is summarized in Theorem 2 (Yu and Annaswamy, 1995). Theorem 2 Let there be a continuous function "((h(x)) such that 2xT P f(x , "((h(x))) + xT Qx :::; 0 for every X E X where X is a compact set containing the origin as an interior point. Then, given a neighborhood 0 C X of the origin, there exists a neural controlierH = N(h(x); W) and a compact set Y E X such that the solutions of x= f(x , N(h(x); W)) converge to 0, for every initial condition x(to) E y. 3.4 SIMULATION RESULTS In this section, we show simulation results for a discrete-time nonlinear systems using the proposed neural network contro\1er in Section 3, and compare it with a linear contro\1er to illustrate the difference. The system we considered is a second-order nonlinear system Xt = f(xt-I , Ut-I), where f = [II, 12]T, h = Xl t _ 1 X (1 +X2'_1 )+X2t-1 x (l-ut- I +uLI) and 12 = XT'_I + 2X2'_1 +Ut-I (1 + X2'_I)· It was assumed that X is measurable, and we wished to stabilize the system around the origin. The controller is of the form Ht = N (x It, X2 t ). The neural network N used is a Gaussian network with 120 centers. The training set and the testing set were composed of 441 and 121 data elements respectively. After the training was done, we plotted the actual change of the Lyapunov function, tl V, using the linear controller U = - K x and the neural network controller in Figures 3 and 4 respectively. It can be observed from the two figures that if the neural network contro\1er is used, tl V is negative definite except in a small neighborhood of the origin, which assures that the closed-loop system would converge to vicinity of the origin; whereas, if the linear controller is used, tl V becomes positive in some region away from the origin, which implies that the system can be unstable for some initial conditions. Simulation results confirmed our observation. 1016 S. YU, A. M. ANNASW AMY -0 01 - 0 J - 0 I -0 J -()2 -O J Figure 3: ~V(u = -Kx ) Figure 4: ~V(u = N(x)) Acknowledgments This work is supported in part by Electrical Power Research Institute under contract No. 8060-13 and in part by National Science Foundation under grant No. ECS-9296070. References [1] A. M. Annaswamy and S. Yu. O-adaptive neural networks: A new approach to parameter estimation. IEEE Transactions on Neural Networks, (to appear) 1996. [2] D. P. Bertsekas. Nonlinear Programming. Athena Scientific, Belmont, MA, 1995. [3] G. C. Goodwin and K. S. Sin. Adaptive Filtering Prediction and Control. PrenticeHall, Inc., 1984. [4] A. Isidori. Nonlinear Control Systems. Springer-Verlag, New York, NY, 1989. [5] M. L Jordan and D. E. Rumelhart. Forward models: Supervised learning with a distal teacher. Cognitive Science, 16:307-354, 1992. [6] A. U. Levin and K. S. Narendra. Control of nonlinear dynamical systems using neural networks: Controllability and stabilization. IEEE Transactions on Neural Networks, 4(2): 192-206, March 1993. [7] K. S. Narendra and A. M. Annaswamy. Stable Adaptive Systems. Prentice-Hall, Inc., 1989. [8] K. S. Narendra and K. Parthasarathy. Identification and control of dynamical systems using neural networks. IEEE Transactions on Neural Networks, 1 (I ):4-26, March 1990. [9] R. M. Sanner and J.-J. E. Slotine. Gaussian networks for direct adaptive control. IEEE Transactions on Neural Networks, 3(6):837-863, November 1992. [10] S. Yu and A. M. Annaswamy. Adaptive control of nonlinear dynamic systems using O-adaptive neural networks. Technical Report 9601 , Adaptive Control Laboratory, Department of Mechanical Engineering, M.LT., 1996. [11] S.-H. Yu and A. M. Annaswamy. Control of nonlinear dynamic systems using a stability based neural network approach. In Technical report 9501, Adaptive Control Laboratory, MIT, Submitted to Proceedings of the 34th IEEE Conference on Decision and Control, New Orleans, LA, 1995.
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Sample Complexity for Learning Recurrent Percept ron Mappings Bhaskar Dasgupta Department of Computer Science University of Waterloo Waterloo, Ontario N2L 3G 1 CANADA bdasgupt~daisy.uwaterloo.ca Eduardo D. Sontag Department of Mathematics Rutgers University New Brunswick, NJ 08903 USA sontag~control.rutgers.edu Abstract Recurrent perceptron classifiers generalize the classical perceptron model. They take into account those correlations and dependences among input coordinates which arise from linear digital filtering. This paper provides tight bounds on sample complexity associated to the fitting of such models to experimental data. 1 Introduction One of the most popular approaches to binary pattern classification, underlying many statistical techniques, is based on perceptrons or linear discriminants; see for instance the classical reference (Duda and Hart, 1973). In this context, one is interested in classifying k-dimensional input patterns V=(Vl, . . . ,Vk) into two disjoint classes A + and A -. A perceptron P which classifies vectors into A + and A - is characterized by a vector (of "weights") C E lR k, and operates as follows. One forms the inner product C.V = CIVI + ... CkVk . If this inner product is positive, v is classified into A +, otherwise into A - . In signal processing and control applications, the size k of the input vectors v is typically very large, so the number of samples needed in order to accurately "learn" an appropriate classifying perceptron is in principle very large. On the other hand, in such applications the classes A + and A-often can be separated by means of a dynamical system of fairly small dimensionality. The existence of such a dynamical system reflects the fact that the signals of interest exhibit context dependence and Sample Complexity for Learning Recurrent Perceptron Mappings 205 correlations, and this prior information can help in narrowing down the search for a classifier. Various dynamical system models for classification appear from instance when learning finite automata and languages (Giles et. al., 1990) and in signal processing as a channel equalization problem (at least in the simplest 2-level case) when modeling linear channels transmitting digital data from a quantized source, e.g. (Baksho et. al., 1991) and (Pulford et. al., 1991). When dealing with linear dynamical classifiers, the inner product c. v represents a convolution by a separating vector c that is the impulse-response of a recursive digital filter of some order n ~ k. Equivalently, one assumes that the data can be classified using a c that is n-rec'Ursive, meaning that there exist real numbers TI, ... , Tn SO that n Cj = 2: Cj-iTi, j = n + 1, ... , k . i=1 Seen in this context, the usual perceptrons are nothing more than the very special subclass of "finite impulse response" systems (all poles at zero); thus it is appropriate to call the more general class "recurrent" or "IIR (infinite impulse response)" perceptrons. Some authors, particularly Back and Tsoi (Back and Tsoi, 1991; Back and Tsoi, 1995) have introduced these ideas in the neural network literature. There is also related work in control theory dealing with such classifying, or more generally quantized-output, linear systems; see (Delchamps, 1989; Koplon and Sontag, 1993). The problem that we consider in this paper is: if one assumes that there is an n-recursive vector c that serves to classify the data, and one knows n but not the particular vector, how many labeled samples v(i) are needed so as to be able to reliably estimate C? More specifically, we want to be able to guarantee that any classifying vector consistent with the seen data will classify "correctly with high probability" the unseen data as well. This is done by computing the VC dimension of the related concept class and then applying well-known results from computational learning theory. Very roughly speaking, the main result is that the number of samples needed is proportional to the logarithm of the length k (as opposed to k itself, as would be the case if one did not take advantage of the recurrent structure). Another application of our results, again by appealing to the literature from computational learning theory, is to the case of "noisy" measurements or more generally data not exactly classifiable in this way; for example, our estimates show roughly that if one succeeds in classifying 95% of a data set of size logq, then with confidence ~ lone is assured that the prediction error rate will be < 90% on future (unlabeled) samples. Section 5 contains a result on polynomial-time learnability: for n constant, the class of concepts introduced here is PAC learnable. Generalizations to the learning of real-valued (as opposed to Boolean) functions are discussed in Section 6. For reasons of space we omit many proofs; the complete paper is available by electronic mail from the authors. 2 Definitions and Statements of Main Results Given a set X, and a subset X of X, a dichotomy on X is a function fJ: X - {-I, I}. Assume given a class F of functions X {-I, I}, to be called the class of classifier functions. The subset X ~ X is shattered by F if each dichotomy on X is the restriction to X of some <P E F. The Vapnik-Chervonenkis dimension vc (F) is the supremum (possibly infinite) of the set of integers K for which there is some subset 206 B. DASGUPTA,E.D.SONTAG x ~ X of cardinality K, which can be shattered by:F. Due to space limitations, we omit any discussion regarding the relevance of the VC dimension to learning problems; the reader is referred to the excellent surveys in (Maass, 1994; Thran, 1994) regarding this issue. Pick any two integers n>O and q~O. A sequence C= (Cl, ... , cn+q) E lR.n+q is said to be n-recursive if there exist real numbers r1, .. . , rn so that n cn+j = 2: cn+j-iri, j = 1, . .. , q. i=l (In particular, every sequence of length n is n-recursive, but the interesting cases are those in which q i= 0, and in fact q ~ n.) Given such an n-recursive sequence C, we may consider its associated perceptron classifier. This is the map ¢c: lR.n+q --+{-1,1}: (X1, ... ,Xn+q) H sign (I:CiXi) .=1 where the sign function is understood to be defined by sign (z) = -1 if z ~ 0 and sign (z) = 1 otherwise. (Changing the definition at zero to be + 1 would not change the results to be presented in any way.) We now introduce, for each two fixed n, q as above, a class of functions: :Fn,q := {¢cl cE lR.n+q is n-recursive}. This is understood as a function class with respect to the input space X = lR. n+q, and we are interested in estimating vc (:Fn,q). Our main result will be as follows (all logs in base 2): Theorem 1 Imax {n, nLlog(L1 + ~ J)J} ~ vc (:Fn ,q) ~ min {n + q, 18n + 4n log(q + 1)} I Note that, in particular, when q> max{2 + n2 , 32}, one has the tight estimates n "2 logq ~ vc (:Fn ,q) ~ 8n logq . The organization of the rest of the paper is as follows. In Section 3 we state an abstract result on VC-dimension, which is then used in Section 4 to prove Theorem 1. Finally, Section 6 deals with bounds on the sample complexity needed for identification of linear dynamical systems, that is to say, the real-valued functions obtained when not taking "signs" when defining the maps ¢c. 3 An Abstract Result on VC Dimension Assume that we are given two sets X and A, to be called in this context the set of inputs and the set of parameter values respectively. Suppose that we are also given a function F: AxX--+{-1,1}. Associated to this data is the class of functions :F := {F(A,·): X --+ {-1, 1} I A E A} Sample Complexity for Learning Recurrent Perceptron Mappings 207 obtained by considering F as a function of the inputs alone, one such function for each possible parameter value A. Note that, given the same data one could, dually, study the class F*: {F(-,~) : A-{-I,I}I~EX} which obtains by fixing the elements of X and thinking of the parameters as inputs. It is well-known (and in any case, a consequence of the more general result to be presented below) that vc (F) ~ Llog(vc (F*»J, which provides a lower bound on vc (F) in terms of the "dual VC dimension." A sharper estimate is possible when A can be written as a product of n sets A = Al X A2 X • • . x An and that is the topic which we develop next. (1) We assume from now on that a decomposition of the form in Equation (1) is given, and will define a variation of the dual VC dimension by asking that only certain dichotomies on A be obtained from F*. We define these dichotomies only on "rectangular" subsets of A, that is, sets of the form L = Ll X .•. x Ln ~ A with each Li ~ Ai a nonempty subset. Given any index 1 ::; K ::; n, by a K-axis dichotomy on such a subset L we mean any function 6 : L {-I, I} which depends only on the Kth coordinate, that is, there is some function ¢ : Lit - {-I, I} so that 6(Al, . . . ,An ) = ¢(AIt) for all (Al, . . . ,An ) E L; an axis dichotomy is a map that is a K-axis dichotomy for some K. A rectangular set L will be said to be axisshattered if every axis dichotomy is the restriction to L of some function of the form F(·,~): A - {-I, I}, for some ~ EX. Theorem 2 If L = Ll X ... x Ln ~ A can be axis-shattered and each set Li has cardinality ri, then vc (F) ~ Llog(rt)J + ... + Llog(rn)J . (In the special case n=1 one recovers the classical result vc (F) ~ Llog(vc (F*)J.) The proof of Theorem 2 is omitted due to space limitations. 4 Proof of Main Result We recall the following result; it was proved, using Milnor-Warren bounds on the number of connected components of semi-algebraic sets, by Goldberg and Jerrum: Fact 4.1 (Goldberg and Jerrum, 1995) Assume given a function F : A x X {-I, I} and the associated class of functions F:= {F(A,·): X - {-I, I} I A E A} . Suppose that A = ~ k and X = ~ n, and that the function F can be defined in terms of a Boolean formula involving at most s polynomial inequalities in k + n variables, each polynomial being of degree at most d. Then, vc (F) ::; 2k log(8eds). 0 Using the above Fact and bounds for the standard "perceptron" model, it is not difficult to prove the following Lemma. Lemma 4.2 vc (Fn,q) ::; min{n + q, 18n + 4nlog(q + I)} Next, we consider the lower bound of Theorem 1. Lemma 4.3 vc (Fn,q) ~ maxin, nLlog(Ll + q~1 J)J} 208 B. DASGUPTA, E. D. SONTAG Proof As Fn,q contains the class offunctions <Pc with c= (C1, ... , cn, 0, ... ,0), which in turn being the set of signs of an n-dimensional linear space of functions, has VC dimension n, we know that vc (Fn,q) ~ n. Thus we are left to prove that if q > n then vc(Fn,q) ~ nLlog(l1 + ~J)J. The set of n-recursive sequences of length n + q includes the set of sequences of the following special form: n ~. 1 Cj = L.Jlf, i=l j=I, ... ,n+q (2) where ai, h E lR for each i = 1, ... , n. Hence, to prove the lower bound, it is sufficient to study the class of functions induced by F : lll.n x lll.n+. ~ {-I, I}, (~I"'" ~n, XI,···, x n+.) >-> sign (t, ~ ~i-I Xj) . Let r = L q+~-l J and let L1, ... ,Ln be n disjoint sets of real numbers (if desired, integers), each of cardinality r. Let L = U:'::l Lj . In addition, if rn < q+n-1, then select an additional set B of (q+n-rn-1) real numbers disjoint from L. We will apply Theorem 2, showing that the rectangular subset L1 x ... x Ln can be axis-shattered. Pick any,.. E {1, ... , n} and any <P : L,. ~ {-1, 1}. Consider the ( unique) interpolating polynomial n+q peA) = L XjAj- 1 j=l in A of degree q + n - 1 such that peA) = { ~(A) if A E L,. if A E (L U B) - L,.. Now pick e = (Xl, ... , Xn +q-1). Observe that F(lt, I" ... , In, Xl, .. . , xn+.) = sign (t, P(I'») = ¢(I.) for all (11, ... , In) E L1 X .•• X Ln, since p(l) = 0 fori ¢ L,. and p(l) = <P(I) otherwise. It follows from Theorem 2 that vc (Fn,q) ~ nLlog(r)J, as desired. • [) The Consistency Problem We next briefly discuss polynomial time learnability of recurrent perceptron mappings. As discussed in e.g. (Turan, 1994), in order to formalize this problem we need to first choose a data structure to represent the hypotheses in Fn,q. In addition, since we are dealing with complexity of computation involving real numbers, we must also clarify the meaning of "finding" a hypothesis, in terms of a suitable notion of polynomial-time computation. Once this is done, the problem becomes that of solving the consistency problem: Given a set ofs ~ s(c,8) inputs6,6, ... ,e& E lR n +q, and an arbitrary dichotomy ~ : {e1, 6, ... , e&} ~ {-I, I} find a representation of a hypothesis <Pc E Fn,q such that the restriction of <Pc to the set {e1,6, ... ,e&} is identical to the dichotomy ~ (or report that no such hypothesis exists). Sample Complexity for Learning Recurrent Perceptron Mappings 209 The representation to be used should provide an "efficient encoding" of the values of the parameters rl, • .. , rn , Cl , . . . , cn: given a set of inputs (Xl" ' " Xn+q) E jRn+q, one should be able to efficiently check concept membership (that is, compute sign (L:7~l CjXj)). Regarding the precise meaning of polynomial-time computation, there are at least two models of complexity possible: the unit cost model which deals with algebraic complexity (arithmetic and comparison operations take unit time) and the logarithmic cost model (computation in the Turing machine sense; inputs (Xl , . . . , X n+q ) are rationals, and the time involved in finding a representation of rl , . .. , r n , Cl, . .. , Cn is required to be polynomial on the number of bits L. Theorem 3 For each fixed n > 0, the consistency problem for :Fn,q can be solved in time polynomial in q and s in the unit cost model, and time polynomial in q, s, and L in the logarithmic cost model. Since vc (:Fn ,q) = O(n + nlog(q + 1)), it follows from here that the class :Fn,q is learnable in time polynomial in q (and L in the log model). Due to space limitations, we must omit the proof; it is based on the application of recent results regarding computational complexity aspects of the first-order theory of real-closed fields. 6 Pseudo-Dimension Bounds In this section, we obtain results on the learnability of linear systems dynamics, that is, the class of functions obtained if one does not take the sign when defining recurrent perceptrons. The connection between VC dimension and sample complexity is only meaningful for classes of Boolean functions; in order to obtain learnability results applicable to real-valued functions one needs metric entropy estimates for certain spaces of functions. These can be in turn bounded through the estimation of Pollard's pseudo-dimension. We next briefly sketch the general framework for learning due to Haussler (based on previous work by Vapnik, Chervonenkis, and Pollard) and then compute a pseudo-dimension estimate for the class of interest. The basic ingredients are two complete separable metric spaces X and If (called respectively the sets of inputs and outputs), a class :F of functions f : X -+ If (called the decision rule or hypothesis space), and a function f : If x If -+ [0, r] C jR (called the loss or cost function). The function f is so that the class of functions (x, y) ~ f(f(x), y) is "permissible" in the sense of Haussler and Pollard. Now, one may introduce, for each f E :F, the function AJ,l : X x If x jR -+ {-I, I} : (x, y, t) ~ sign (f(f(x) , y) - t) as well as the class A.1",i consisting of all such A/,i ' The pseudo-dimension of :F with respect to the loss function f, denoted by PO [:F, f], is defined as: PO [:F,R] := vc (A.1",i). Due to space limitations, the relationship between the pseudo-dimension and the sample complexity of the class :F will not be discussed here; the reader is referred to the references (Haussler, 1992; Maass, 1994) for details. For our application we define, for any two nonnegative integers n, q, the class :F~ , q := {¢<! ICE jRn+q is n-recursive} where ¢c jRn+q -+ jR: (Xl , .. . , Xn+q) ~ L:7~l CjXj . The following Theorem can be proved using Fact 4.1. Theorem 4 Let p be a positive integer and assume that the loss function f is given byf(Yl,Y2) = IYl- Y2IP • Then, PO [:F~ , q,f] ~ 18n+4nlog(p(q+ 1)) . 210 B. DASGUPTA, E. D. SONTAG Acknowledgements This research was supported in part by US Air Force Grant AFOSR-94-0293. References A.D. BACK AND A.C. TSOI, FIR and IIR synapses, a new neural network architecture for time-series modeling, Neural Computation, 3 (1991), pp. 375-385. A .D. BACK AND A .C. TSOI, A comparison of discrete-time operator models for nonlinear system identification, Advances in Neural Information Processing Systems (NIPS'94), Morgan Kaufmann Publishers, 1995, to appear. A.M . BAKSHO, S. DASGUPTA, J .S. GARNETT, AND C.R. JOHNSON, On the similarity of conditions for an open-eye channel and for signed filtered error adaptive filter stability, Proc. IEEE Conf. Decision and Control, Brighton, UK, Dec. 1991, IEEE Publications, 1991, pp. 1786-1787. A. BLUMER, A. EHRENFEUCHT, D. HAUSSLER, AND M . WARMUTH, Learnability and the Vapnik-Chervonenkis dimension, J. of the ACM, 36 (1989), pp. 929-965. D.F. DELCHAMPS, Extracting State Information from a Quantized Output Record, Systems and Control Letters, 13 (1989), pp. 365-372. R .O. DUDA AND P.E. HART, Pattern Classification and Scene Analysis, Wiley, New York, 1973. C.E. GILES, G.Z. SUN, H.H. CHEN, Y.C. LEE, AND D. CHEN, Higher order recurrent networks and grammatical inference, Advances in Neural Information Processing Systems 2, D.S. Touretzky, ed., Morgan Kaufmann, San Mateo, CA, 1990. P . GOLDBERG AND M. JERRUM, Bounding the Vapnik-Chervonenkis dimension of concept classes parameterized by real numbers, Mach Learning, 18, (1995): 131-148. D. HAUSSLER, Decision theoretic generalizations of the PAC model for neural nets and other learning applications, Information and Computation, 100, (1992): 78-150. R. KOPLON AND E.D. SONTAG, Linear systems with sign-observations, SIAM J. Control and Optimization, 31(1993): 1245 - 1266. W. MAASS, Perspectives of current research about the complexity of learning in neural nets, in Theoretical Advances in Neural Computation and Learning, V.P. Roychowdhury, K.Y. Siu, and A. Orlitsky, eds., Kluwer, Boston, 1994, pp. 295-336. G.W. PULFORD, R.A. KENNEDY, AND B.D.O. ANDERSON, Neural network structure for emulating decision feedback equalizers, Proc. Int. Conf. Acoustics, Speech, and Signal Processing, Toronto, Canada, May 1991, pp. 1517-1520. E.D. SONTAG, Neural networks for control, in Essays on Control: Perspectives in the Theory and its Applications (H.L. Trentelman and J .C. Willems, eds.), Birkhauser, Boston, 1993, pp. 339-380. GYORGY TURAN, Computational Learning Theory and Neural Networks:A Survey of Selected Topics, in Theoretical Advances in Neural Computation and Learning, V.P. Roychowdhury, K.Y. Siu,and A. Orlitsky, eds., Kluwer, Boston, 1994, pp. 243-293. L.G. VALIANT A theory of the learnable, Comm. ACM, 27, 1984, pp. 1134-1142. V.N .VAPNIK, Estimation of Dependencies Based on Empirical Data, Springer, Berlin, 1982.
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Is Learning The n-th Thing Any Easier Than Learning The First? Sebastian Thrun I Computer Science Department Carnegie Mellon University Pittsburgh, PA 15213-3891 World Wide Web: http://www.cs.cmu.edul'''thrun Abstract This paper investigates learning in a lifelong context. Lifelong learning addresses situations in which a learner faces a whole stream of learning tasks. Such scenarios provide the opportunity to transfer knowledge across multiple learning tasks, in order to generalize more accurately from less training data. In this paper, several different approaches to lifelong learning are described, and applied in an object recognition domain. It is shown that across the board, lifelong learning approaches generalize consistently more accurately from less training data, by their ability to transfer knowledge across learning tasks. 1 Introduction Supervised learning is concerned with approximating an unknown function based on examples. Virtually all current approaches to supervised learning assume that one is given a set of input-output examples, denoted by X, which characterize an unknown function, denoted by f. The target function f is drawn from a class of functions, F, and the learner is given a space of hypotheses, denoted by H, and an order (preference/prior) with which it considers them during learning. For example, H might be the space of functions represented by an artificial neural network with different weight vectors. While this formulation establishes a rigid framework for research in machine learning, it dismisses important aspects that are essential for human learning. Psychological studies have shown that humans often employ more than just the training data for generalization. They are often able to generalize correctly even from a single training example [2, 10]. One of the key aspects of the learning problem faced by humans, which differs from the vast majority of problems studied in the field of neural network learning, is the fact that humans encounter a whole stream of learning problems over their entire lifetime. When faced with a new thing to learn, humans can usually exploit an enormous amount of training data and I also affiliated with: Institut fur Informatik III, Universitat Bonn, Romerstr. 164, Germany Is Learning the n-th Thing Any Easier Than Learning the First? 641 experiences that stem from other, related learning tasks. For example, when learning to drive a car, years of learning experience with basic motor skills, typical traffic patterns, logical reasoning, language and much more precede and influence this learning task. The transfer of knowledge across learning tasks seems to play an essential role for generalizing accurately, particularly when training data is scarce. A framework for the study of the transfer of knowledge is the lifelong learning framework. In this framework, it is assumed that a learner faces a whole collection of learning problems over its entire lifetime. Such a scenario opens the opportunity for synergy. When facing its n-th learning task, a learner can re-use knowledge gathered in its previous n - 1 learning tasks to boost the generalization accuracy. In this paper we will be interested in the most simple version of the lifelong learning problem, in which the learner faces a family of concept learning tasks. More specifically, the functions to be learned over the lifetime of the learner, denoted by 11 , 12, 13, .. . E F , are all of the type I : I --+ {O, I} and sampled from F. Each function I E {II , h ,13, ... } is an indicator function that defines a particular concept: a pattern x E I is member of this concept if and only if I(x) = 1. When learning the n-th indicator function, In , the training set X contains examples of the type (x , In(x)) (which may be distorted by noise). In addition to the training set, the learner is also given n - 1 sets of examples of other concept functions, denoted by Xk (k = 1, .. . , n - I). Each Xk contains training examples that characterize Ik. Since this additional data is desired to support learning In, Xk is called a support set for the training set X . An example of the above is the recognition of faces [5, 7]. When learning to recognize the n-th person, say IBob, the learner is given a set of positive and negative example of face images of this person. In lifelong learning, it may also exploit training information stemming from other persons, such as I E {/Rieh, IMike , IDave , ... }. The support sets usually cannot be used directly as training patterns when learning a new concept, since they describe different concepts (hence have different class labels). However, certain features (like the shape of the eyes) are more important than others (like the facial expression, or the location of the face within the image). Once the invariances of the domain are learned, they can be transferred to new learning tasks (new people) and hence improve generalization. To illustrate the potential importance of related learning tasks in lifelong learning, this paper does not present just one particular approach to the transfer of knowledge. Instead, it describes several, all of which extend conventional memory-based or neural network algorithms. These approaches are compared with more traditional learning algorithms, i.e., those that do not transfer knowledge. The goal of this research is to demonstrate that, independent of a particular learning approach, more complex functions can be learned from less training data iflearning is embedded into a lifelong context. 2 Memory-Based Learning Approaches Memory-based algorithms memorize all training examples explicitly and interpolate them at query-time. We will first sketch two simple, well-known approaches to memory-based learning, then propose extensions that take the support sets into account. 2.1 Nearest Neighbor and Shepard's Method Probably the most widely used memory-based learning algorithm is J{ -nearest neighbor (KNN) [15]. Suppose x is a query pattern, for which we would like to know the output y. KNN searches the set of training examples X for those J{ examples (Xi, Yi) E X whose input patterns Xi are nearest to X (according to some distance metric, e.g., the Euclidian distance). It then returns the mean output value k 2:= Yi of these nearest neighbors. Another commonly used method, which is due to Shepard [13], averages the output values 642 s. THRUN of all training examples but weights each example according to the inverse distance to the query :~~~t x. ( ) ( I) -I L Ilx - ~: II + E· L Ilx - Xi II + E (x"y.)EX (x. ,y.)EX (1) Here E > 0 is a small constant that prevents division by zero. Plain memory-based learning uses exclusively the training set X for learning. There is no obvious way to incorporate the support sets, since they carry the wrong class labels. 2.2 Learning A New Representation The first modification of memory-based learning proposed in this paper employs the support sets to learn a new representation of the data. More specifically, the support sets are employed to learn a function, denoted by 9 : I --+ I', which maps input patterns in I to a new space, I' . This new space I' forms the input space for a memory-based algorithm. Obviously, the key property of a good data representations is that multiple examples of a single concept should have a similar representation, whereas the representation of an example and a counterexample of a concept should be more different. This property can directly be transformed into an energy function for g: n-I ( ) E:= ~ (X ,y~EXk (X"y~EXk Ilg(x)-g(x')11 (X"y~EXk Ilg(x)-g(x')11 (2) Adjusting 9 to minimize E forces the distance between pairs of examples of the same concept to be small, and the distance between an example and a counterexample of a concept to be large. In our implementation, 9 is realized by a neural network and trained using the Back-Propagation algorithm [12]. Notice that the new representation, g, is obtained through the support sets. Assuming that the learned representation is appropriate for new learning tasks, standard memory-based learning can be applied using this new representation when learning the n-th concept. 2.3 Learning A Distance Function An alternative way for exploiting support sets to improve memory-based learning is to learn a distance function [3, 9]. This approach learns a function d : I x I --+ [0, I] which accepts two input patterns, say x and x', and outputs whether x and x' are members of the same concept, regardless what the concept is. Training examples for d are ((x, x'),I) ify=y'=l ((x, x'), 0) if(y=IAy'=O)or(y=OAy'=I). They are derived from pairs of examples (x , y) , (x', y') E Xk taken from a single support set X k (k = 1, . .. , n I). In our implementation, d is an artificial neural network trained with Back-Propagation. Notice that the training examples for d lack information concerning the concept for which they were originally derived. Hence, all support sets can be used to train d. After training, d can be interpreted as the probability that two patterns x, x' E I are examples of the same concept. Once trained, d can be used as a generalized distance function for a memory-based approach. Suppose one is given a training set X and a query point x E I. Then, for each positive example (x' , y' = I) EX, d( x, x') can be interpreted as the probability that x is a member of the target concept. Votes from multiple positive examples (XI, I) , (X2' I), ... E X are combined using Bayes' rule, yielding Prob(fn(x)=I) .1- (I + II I:(~(::~,))-I (3) (x' ,y'=I)EXk Is Learning the n-th Thing Any Easier Than Learning the First? 643 Notice that d is not a distance metric. It generalizes the notion of a distance metric, because the triangle inequality needs not hold, and because an example of the target concept x' can provide evidence that x is not a member of that concept (if d(x, x') < 0.5). 3 Neural Network Approaches To make our comparison more complete, we will now briefly describe approaches that rely exclusively on artificial neural networks for learning In. 3.1 Back-Propagation Standard Back-Propagation can be used to learn the indicator function In, using X as training set. This approach does not employ the support sets, hence is unable to transfer knowledge across learning tasks. 3.2 Learning With Hints Learning with hints [1, 4, 6, 16] constructs a neural network with n output units, one for each function Ik (k = 1,2, .. . , n). This network is then trained to simultaneously minimize the error on both the support sets {Xk} and the training set X. By doing so, the internal representation of this network is not only determined by X but also shaped through the support sets {X k }. If similar internal representations are required for al1 functions Ik (k = 1,2, .. . , n), the support sets provide additional training examples for the internal representation. 3.3 Explanation-Based Neural Network Learning The last method described here uses the explanation-based neural network learning algorithm (EBNN), which was original1y proposed in the context of reinforcement learning [8, 17]. EBNN trains an artificial neural network, denoted by h : I ----+ [0, 1], just like Back-Propagation. However, in addition to the target values given by the training set X, EBNN estimates the slopes (tangents) of the target function In for each example in X. More specifically, training examples in EBNN are of the sort (x, In (x), \7 xl n (x)), which are fit using the Tangent-Prop algorithm [14]. The input x and target value In(x) are taken from the trai ning set X. The third term, the slope \7 xl n ( X ), is estimated using the learned distance function d described above. Suppose (x', y' = 1) E X is a (positive) training example. Then, the function dx ' : I ----+ [0, 1] with dx ' (z) := d(z , x') maps a single input pattern to [0, 1], and is an approximation to In. Since d( z, x') is represented by a neural network and neural networks are differentiable, the gradient 8dx ' (z) /8z is an estimate of the slope of In at z. Setting z := x yields the desired estimate of \7 xln (x) . As stated above, both the target value In (x) and the slope vector \7 x In (x) are fit using the Tangent-Prop algorithm for each training example x EX. The slope \7 xln provides additional information about the target function In. Since d is learned using the support sets, EBNN approach transfers knowledge from the support sets to the new learning task. EBNN relies on the assumption that d is accurate enough to yield helpful sensitivity information. However, since EBNN fits both training patterns (values) and slopes, misleading slopes can be overridden by training examples. See [17] for a more detailed description of EBNN and further references. 4 Experimental Results All approaches were tested using a database of color camera images of different objects (see Fig. 3.3). Each of the object in the database has a distinct color or size. The n-th 644 l1 .... 'I I't' 'I • < , > '" -.... . ~ 1:1 ,I , , c_. ML~._ ... , I '''!!!i!~, =' ;~~~ , ...... ~.. ' . ~ ~ <.:t " ~~- -_,1~ ~_ ~,-l/> ;' ;'j III ... '1 ~' ''',ll t! ~[~ .d!t~)ltI!{iH-"" :. ~~~ -~""":::.~ ~ -,:~~,} I , £ ».~ <.~ ,,, ~ -... ~ ~_l_~ __ E~ '~~ II _e·m;, ;1 ~ t ~,~,AA( , ~ " :R;1-; , ""111':'i, It f4~ r S. THRUN Figure 1: The support sets were compiled out of a hundred images of a bottle, a hat, a hammer, a coke can, and a book. The n-th learning tasks involves distinguishing the shoe from the sunglasses. Images were subsampled to a 100x 100 pixel matrix (each pixel has a color, saturation, and a brightness value), shown on the right side. learning task was the recognition of one of these objects, namely the shoe. The previous n 1 learning tasks correspond to the recognition of five other objects, namely the bottle, the hat, the hammer, the coke can, and the book. To ensure that the latter images could not be used simply as additional training data for In, the only counterexamples of the shoe was the seventh object, the sunglasses. Hence, the training set for In contained images of the shoe and the sunglasses, and the support sets contained images of the other five objects. The object recognition domain is a good testbed for the transfer of knowledge in lifelong learning. This is because finding a good approximation to In involves recognizing the target object invariant of rotation, translation, scaling in size, change of lighting and so on. Since these invariances are common to all object recognition tasks, images showing other objects can provide additional information and boost the generalization accuracy. Transfer of knowledge is most important when training data is scarce. Hence, in an initial experiment we tested all methods using a single image of the shoe and the sunglasses only. Those methods that are able to transfer knowledge were also provided 100 images of each of the other five objects. The results are intriguing. The generalization accuracies KNN Shepard repro g+Shep. distanced Back-Prop hints EBNN 60.4% 60.4% 74.4% 75.2% 59.7% 62.1% 74.8% ±8.3% ±8.3% ±18.5% ±18.9% ±9.0% ±10.2% ±11.1% illustrate that all approaches that transfer knowledge (printed in bold font) generalize significantly better than those that do not. With the exception of the hint learning technique, the approaches can be grouped into two categories: Those which generalize approximately 60% of the testing set correctly, and those which achieve approximately 75% generalization accuracy. The former group contains the standard supervised learning algorithms, and the latter contains the "new" algorithms proposed here, which are capable of transferring knOWledge. The differences within each group are statistically not significant, while the differences between them are (at the 95% level). Notice that random guessing classifies 50% of the testing examples correctly. These results suggest that the generalization accuracy merely depends on the particular choice of the learning algorithm (memory-based vs. neural networks). Instead, the main factor determining the generalization accuracy is the fact whether or not knowledge is transferred from past learning tasks. Is Learning the n-th Thing Any Easier Than Learning the First? 645 95% 85% ~ 80% ~ 15% 70% 65% 60% distance function d hepard 's method with representation g Shepard's method 55% 50%~2--~~----~10~~1~2~1~4--1~6--1~.--~20 training example. 95% , ,,'~ . . </'~ 70% if . 65% /f Back-Propagauon 60% ;;./ 55% ~%~2--~~----~1~O~1~2--1~4--1~6--~1B--~20 training exampletl Figure 2: Generalization accuracy as a function of training examples, measured on an independent test set and averaged over 100 experiments. 95%-confidence bars are also displayed. What happens as more training data arrives? Fig. 2 shows generalization curves with increasing numbers of training examples for some of these methods. As the number of training examples increases, prior knowledge becomes less important. After presenting 20 training examples, the results KNN Shepard repro g+Shep. distance d Back-Prop hints EBNN 81.0% 70.5% 81.7% 87.3% 88.4% n_avail. 90.8% ±3.4% ±4.9% ±2.7% ±O_9% ±2.5% ±2.7% illustrate that some of the standard methods (especially Back-Propagation) generalize about as accurately as those methods that exploit support sets. Here the differences in the underlying learning mechanisms becomes more dominant. However, when comparing lifelong learning methods with their corresponding standard approaches, the latter ones are stiIl inferior: BackPropagation (88.4%) is outperformed by EBNN (90.8%), and Shepard's method (70.5%) generalizes less accurately when the representation is learned (81.7%) or when the distance function is learned (87.3%). All these differences are significant at the 95% confidence level. 5 Discussion The experimental results reported in this paper provide evidence that learning becomes easier when embedded in a lifelong learning context. By transferring knowledge across related learning tasks, a learner can become "more experienced" and generalize better. To test this conjecture in a more systematic way, a variety of learning approaches were evaluated and compared with methods that are unable to transfer knowledge. It is consistently found that lifelong learning algorithms generalize significantly more accurately, particularly when training data is scarce. Notice that these results are well in tune with other results obtained by the author. One of the approaches here, EBNN, has extensively been studied in the context of robot perception [11], reinforcement learning for robot control, and chess [17]. In all these domains, it has consistently been found to generalize better from less training data by transferring knowledge from previous learning tasks. The results are also consistent with observations made about human learning [2, 10], namely that previously learned knowledge plays an important role in generalization, particularly when training data is scarce. [18] extends these techniques to situations where most support sets are not related.w However, lifelong learning rests on the assumption that more than a single task is to be learned, and that learning tasks are appropriately related. Lifelong learning algorithms are particularly well-suited in domains where the costs of collecting training data is the dominating factor in learning, since these costs can be amortized over several learning tasks. Such domains include, for example, autonomous service robots which are to learn and improve over their entire lifetime. They include personal software assistants which have 646 S. THRUN to perform various tasks for various users. Pattern recognition, speech recognition, time series prediction, and database mining might be other, potential application domains for the techniques presented here. References [1] Y. S. Abu-Mostafa. Learning from hints in neural networks. Journal of Complexity, 6: 192-198, 1990. [2] W-K. Ahn and W F. Brewer. Psychological studies of explanation-based learning. In G. Dejong, editor, Investigating Explanation-Based Learning. Kluwer Academic Publishers, BostonlDordrechtILondon, 1993. [3] c. A. Atkeson. Using locally weighted regression for robot learning. In Proceedings of the 1991 1EEE International Conference on Robotics and Automation, pages 958-962, Sacramento, CA, April 1991. [4] J. Baxter. Learning internal representations. In Proceedings of the Conference on Computation Learning Theory, 1995. [5] D. Beymer and T. Poggio. Face recognition from one model view. In Proceedings of the International Conference on Computer Vision, 1995. [6] R. Caruana. MuItitask learning: A knowledge-based of source of inductive bias. In P. E. Utgoff, editor, Proceedings of the Tenth International Conference on Machine Learning, pages 41-48, San Mateo, CA, 1993. Morgan Kaufmann. [7] M. Lando and S. Edelman. Generalizing from a single view in face recognition. Technical Report CS-TR 95-02, Department of Applied Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 76100, Israel, January 1995. [8] T. M. Mitchell and S. Thrun. Explanation-based neural network learning for robot control. In S. J. Hanson, J. Cowan, and C. L. Giles, editors, Advances in Neural Information Processing Systems 5, pages 287-294, San Mateo, CA, 1993. Morgan Kaufmann. [9] A. W Moore, D. 1. Hill, and M. P. Johnson. An Empirical Investigation of Brute Force to choose Features, Smoothers and Function Approximators. In S. Hanson, S. Judd, and T. Petsche, editors, Computational Learning Theory and Natural Learning Systems, Volume 3. MIT Press, 1992. [10] Y. Moses, S. Ullman, and S. Edelman. Generalization across changes in illumination and viewing position in upright and inverted faces. Technical Report CS-TR 93-14, Department of Applied Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 76100, Israel, 1993. [11] J. O'Sullivan, T. M. Mitchell, and S. Thrun. Explanation-based neural network learning from mobile robot perception. In K. Ikeuchi and M. Veloso, editors, Symbolic Visual Learning. Oxford University Press, 1995. [12] D. E. Rumelhart, G. E. Hinton, and R. J. Williams. Learning internal representations by error propagation. In D. E. Rumelhart and 1. L. McClelland, editors, Parallel Distributed Processing. Vol. I + II. MIT Press, 1986. [13] D. Shepard. A two-dimensional interpolation function for irregularly spaced data. In 23rd National Conference ACM, pages 517-523, 1968. [14] P. Simard, B. Victorri, Y. LeCun, and J. Denker. Tangent prop - a formalism for specifying selected invariances in an adaptive network. In 1. E. Moody, S. J. Hanson, and R. P. Lippmann, editors, Advances in Neural Information Processing Systems 4, pages 895-903, San Mateo, CA, 1992. Morgan Kaufmann. [15] c. Stanfill and D. Waltz. Towards memory-based reasoning. Communications of the ACM, 29(12): 1213-1228, December 1986. [16] S. C. Suddarth and A. Holden. Symbolic neural systems and the use of hints for developing complex systems. International Journal of Machine Studies, 35, 1991. [17] S. Thrun. Explanation-Based Neural Network Learning: A Lifelong Learning Approach. Kluwer Academic Publishers, Boston, MA, 1996. to appear. [18] S. Thrun and J. O'Sullivan. Clustering learning tasks and the selective cross-task transfer of knowledge. Technical Report CMU-CS-95-209, Carnegie Mellon University, School of Computer Science, Pittsburgh, PA 15213, November 1995.
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Parallel analog VLSI architectures for computation of heading direction and time-to-contact Giacomo Indiveri giacomo@klab.caltech.edu Jorg Kramer kramer@klab.caltech.edu Division of Biology California Institute of Technology Pasadena, CA 91125 Abstract Christof Koch koch@klab.caltech.edu We describe two parallel analog VLSI architectures that integrate optical flow data obtained from arrays of elementary velocity sensors to estimate heading direction and time-to-contact. For heading direction computation, we performed simulations to evaluate the most important qualitative properties of the optical flow field and determine the best functional operators for the implementation of the architecture. For time-to-contact we exploited the divergence theorem to integrate data from all velocity sensors present in the architecture and average out possible errors. 1 Introduction We have designed analog VLSI velocity sensors invariant to absolute illuminance and stimulus contrast over large ranges that are able to achieve satisfactory performance in a wide variety of cases; yet such sensors, due to the intrinsic nature of analog processing, lack a high degree of precision in their output values. To exploit their properties at a system level, we developed parallel image processing architectures for applications that rely mostly on the qualitative properties of the optical flow, rather than on the precise values of the velocity vectors. Specifically, we designed two parallel architectures that employ arrays of elementary motion sensors for the computation of heading direction and time-to-contact. The application domain that we took into consideration for the implementation of such architectures, is the promising one of vehicle navigation. Having defined the types of images to be analyzed and the types of processing to perform, we were able to use a priori inforVLSI Architectures for Computation of Heading Direction and Time-to-contact 721 mation to integrate selectively the sparse data obtained from the velocity sensors and determine the qualitative properties of the optical flow field of interest. 2 The elementary velocity sensors A velocity sensing element, that can be integrated into relatively dense arrays to estimate in parallel optical flow fields, has been succesfully built [Kramer et al., 1995]. Unlike most previous implementations of analog VLSI motion sensors, it unambiguously encodes 1-D velocity over considerable velocity, contrast, and illuminance ranges, while being reasonably compact. It implements an algorithm that measure'3 the time of travel of features (here a rapid temporal change in intensity) stimulus between two fixed locations on the chip. In a first stage, rapid dark-to-bright irradiance changes or temporal ON edges are converted into short current pulses. Each current pulse then gives rise to a sharp voltage spike and a logarithmically-decaying voltage signal at each edge detector location. The sharp spike from one location is used to sample the analog voltage of the slowly-decaying signal from an adjacent location. The sampled output voltage encodes the relative time delay of the two signals, and therefore velocity, for the direction of motion where the onset of the slowly-decaying pulse precedes the sampling spike. In the other direction, a lower voltage is sampled. Each direction thus requires a separate output stage. '05 ~ OIlS j " > o. Oil!> O' 40 .. VIIIoaIy (mmsec) Figure 1: Output voltage of a motion sensing element for the preferred direction of motion of a sharp high-contrast ON edge versus image velocity under incandescent room illumination. Each data point represents the average of 5 successive measurements. As implemented with a 2 J.tm CMOS process, the size of an elementary bi-directional motion element (including 30 transistors and 8 capacitances) is 0.045 mm2. Fig. 1 shows that the experimental data confirms the predicted logarithmic encoding of velocity by the analog output voltage. The data was taken by imaging a moving high-contrast ON edge onto the chip under incandescent room illumination. The calibration of the image velocity in the focal plane is set by the 300 J.tm spacing of adjacent photoreceptors on the chip. 3 Heading direction computation To simplify the computational complexity of the problem of heading direction detection we restricted our analysis to pure translational motion, taking advantage of the 722 G. INDIVERI, 1. KRAMER, C. KOCH fact that for vehicle navigation it is possible to eliminate the rotational component of motion using lateral accelerometer measurements from the vehicle. Furthermore, to analyze the computational properties of the optical flow for typical vehicle navigation scenes, we performed software simulations on sequences of images obtained from a camera with a 64 x 64 pixel silicon retina placed on a moving truck (courtesy of B. Mathur at Rockwell Corporation). The optical flow fields have been computed Figure 2: The sum of the horizontal components of the optical flow field is plotted on the bottom of the figure. The presence of more than one zero-crossing is due to different types of noise in the optical flow computation (e.g. quantization errors in software simulations or device mismatch in analog VLSI circuits). The coordinate of the heading direction is computed as the abscissa of the zero-crossing with maximum steepness and closest to the abscissa of the previously selected unit. by implementing an algorithm based on the image brightness consia'f)cy equation [Verri et al., 1992] [Barron et al., 1994]. For the application domain considered and the types of optical flow fields obtained from the simulations, it is reasonable to assume that the direction of heading changes smoothly in time. Furthermore, being interested in determining, and possibly controlling, the heading direction mainly along the horizontal axis, we can greatly reduce the complexity of the problem by considering one-dimensional arrays of velocity sensors. In such a case, if we assign positive values to vectors pointing in one direction and negative values to vectors pointing in the opposite direction, the heading direction location will correspond to the point closest to the zero-crossing. Under these assumptions, the computation of the horizontal coordInate of the heading direction has been carried out using the following functional operators: thresholding on the horizontal components of the optical flow vectors; spatial smoothing on the resulting values; detection and evaluation of the steepness of the zero-crossings present in the array and finally selection of the zero-crossing with maximum steepness. The zero-crossing with maximum steepness is selected only if its position is in the neighborhood of the previously selected zero-crossing. This helps to eliminate errors due to noise and device mismatch and assures that the computed heading direction location will shift smoothly in time. Fig. 2 shows a result of the software simulations, on an image of a road VLSI Architectures for Computation of Heading Direction and Time-to-contact 723 with a shadow on the left side. All of the operators used in the algorithm have been implemented with analog circuits (see Fig. 3 for a block diagram of the architecture). Specifically, we have WIIb .. Figure 3: Block diagram of the architecture for detecting heading direction: the first layer of the architecture computes the velocity of the stimulus; the second layer converts the voltage output of the velocity sensors into a positive/negative current performing a threshold operation; the third layer performs a linear smoothing operation on the positive and negative halfs of the input current; the fourth layer detects zero-crossings by comparing the intensity of positive currents from one pixel with negative currents from the neighboring pixel; the top layer implements a winner-take-all network with distributed excitation, which selects the zero-crossing with maximum steepness. designed test chips in which the thresholding function has been implemented using a transconductance amplifier whose current represents the output signal [Mead, 1989], spatial smoothing has been obtained using a circuit that separates positive currents and negative currents into two distinct paths and feeds them into two layers of current-mode diffuser networks [Boahen and Andreou, 1992], the zerocrossing detection and evaluation of its steepness has been implemented using a newly designed circuit block based on a modification of the simple current-correlator [Delbriick, 1991], and the selection of the zero-crossing with maximum steepness closest to the previously selected unit has been implemented using a winner-takeall circuit with distributed excitation [Morris et al., 1995]. The schematics of the former three circuits, which implement the top three layers of the diagram of Fig. 3, are shown in Fig. 4. Fig. 5 shows the output of a test chip in which all blocks up to the diffuser network (without the zero-crossing detection stages) were implemented. The velocity sensor layout was modified to maximize the number of units in the 1-D array. Each velocity sensor measures 60pm x 802pm. On a (2.2mm)2 size chip we were able to fit 23 units. The shown results have been obtained by imaging on the chip expanding or contracting stimuli using black and white edges wrapped around a rotating drum and reflected by an adjacent tilted mirror. The point of contact between drum and mirror corresponding to the simulated heading direction has been imaged approximately onto the 15th unit of the array. As shown, the test chip considered does not achieve 100% correct performance due to errors that arise mainly from the presence of parasitic capacitors in the modified part of the velocity sensor circuits; nonetheless, at least from a qualitative point of view, the data confirms the results obtained from software simulations and demonstrates the validity of the approach considered. 724 O. INDNERI. J. KRAMER. C. KOCH - - - - - ~- - - - - -Ir - - - - 1 ~ II rln II ~ I ~ '" ,.. II ~ Cd •• II cp I II II I II II ""'>---t~;-------'~:-----<-' II I.. II *"J>--........,'t--t--t-<.,fJ "'.>----t ....... -----i'-----<,..t II ill !P II II II II II II II .. _"II II I , .. , .. II I I ------------------- ______ 1 ______ ---Figure 4: Circuit schematics of the smoothing, zero-detection and winner-take-all blocks respectively. " .................. -r-~...,..... ........ ......-........ -.-................. -r-.,.......,.--.-, " .B .0, .0. 'O·.~~'~~~~7~.~.~'.~'~,~I2~"~'~.~'.~"~'~"~.~,,~~~'~ ' 22 lk1I P9ton (a) .0' "!-, ~ , ~,~,~. ~'~'''''7-' ~,~,."""~' ~,,~,,~,,~,"". ~,.""",,:-',~, .,.." ~'O~"~22' """""'''''' (b) Figure 5: Zero crossings computed as difference between smoothed positive currents and smoothed negative currents: (a) for expanding stimuli; (b) for contracting stimuli. The "zero" axis is shifted due to is a systematic offset of 80 nA. 4 Time-to-contact The time-to-contact can be computed by exploiting qualitative properties of the optical flow field such as expansion or contraction [Poggio et al., 1991]. The divergence theorem, or Gauss theorem, as applied to a plane, shows that the integral over a surface patch of the divergence of a vector field is equal to the line integral along the patch boundary of the component of the field normal to the boundary. Since a camera approaching a rigid object sees a linear velocity field, where the velocity vectors are proportional to their distance from the focus-of-expansion, the divergence is constant over the image plane. By integrating the radial component of the optical flow field along the circumference of a circle, the time-to-contact can thus be estimated, independently of the position of the focus-of-expansion. We implemented this algorithm with an analog integrated circuit, where an array of twelve motion sensing elements is arranged on a circle, such that each element measures velocity radially. According to the Gauss theorem, the time-to-contact is VLSI Architectures for Computation of Heading Direction and Time-to-contact 725 then approximated by N·R T= N ' 2:k=l Vk (1) where N denotes the number of elements, R the radius of the circle, and Vk the radial velocity components at the locations of the elements. For each clement, temporal aliasing is prevented by comparing the output voltages of the two directions of motion and setting the lower one, corresponding to the null direction, to zero. The output voltages are then used to control subthreshold transistor currents. Since these voltages are logarithmically dependent on velocity, the transistor currents are proportional to the measured velocities. The sum of the velocity components is thus calculated by aggregating the currents from all elements on two lines, one for outward motion and one for inward motion, and taking the difference of the total currents. The resulting bi-directional output current is an inverse function of the signed time-to-contact. ;: 1 o~--------~----~~ I -, -0 25 OOS 01 015 02 025 Time-1c>ContKt (sec) Figure 6: Output current of the time-to-contact sensor as a function of simulated time-to-contact under incandescent room illumination. The theoretical fit predicts an inverse relationship. The circuit has been implemented on a chip with a size of (2.2mm)2 using 2 pm technology. The photo diodes of the motion sensing elements are arranged on two concentric circles with radii of 400 pm and 600 pm respectively. In order to simulate an approaching or withdrawing object, a high-contrast spiral stimulus was printed onto a rotating disk. Its image was projected onto the chip with a microscope lens under incandescent room illumination. The focus-of-expansion was approximately centered with respect to the photo diode circles. The averaged output current is shown as a function of simulated time-to-contact with a theoretical fit in Fig. 6. The expected inverse relationship is qualitatively observed and the sign (expansion or contraction) is robustly encoded. However, the deviation of the output current from its average can be substantial: Since the output voltage of each motion sensing element decays slowly due to leak currents and since the spiral stimulus causes a serial update of the velocity values along the array, a step change in the output current is observed upon each update, followed by a slow decay. The effect is aggravated, if the individual motion sensing elements measure significantly differing velocities. This is generally the case, because the focus-of-expansion is usually not centered with respect to the sensor and because of inaccuracies in the velocity measurements due to circuit offsets, noise, and the aperture problem [Verri et al., 1992]. The integrative property of the algorithm is thus highly desirable, and more robust data would be obtained from an array with more elements and stimuli with higher edge densities. 726 G. INDIVERI. J. KRAMER. C. KOCH 5 Conclusions We have developed parallel architectures for motion analysis that bypass the problem of low precision in analog VLSI technology by exploiting qualitative properties of the optical flow. The correct functionality of the devices built, at least from a qualitative point of view, have confirmed the validity of the approach followed and induced us to continue this line of research. We are now in the process of designing more accurate circuits that implement the operators used in the architectures proposed. Acknowledgments This work was supported by grants from ONR, ERe and Daimler-Benz AG. The velocity sensor was developed in collaboration with R. Sarpeshkar. The chips were fabricated through the MOSIS VLSI Fabrication Service. References [Barron et al., 1994] J.1. Barron, D.J. Fleet, and S.S. Beauchemin. Performance of optical flow techniques. International Journal on Computer Vision, 12(1):43-77, 1994. [Boahen and Andreou, 1992] K.A. Boahen and A.G. Andreou. A contrast sensitive silicon retina with reciprocal synapses. In NIPS91 Proceedings. IEEE, 1992. [Delbriick, 1991] T. Delbriick. "Bump" circuits for computing similarity and dissimilarity of analog voltages. In Proc. IJCNN, pages 1-475-479, June 1991. [Kramer et al., 1995] J. Kramer, R. Sarpeshkar, and C. Koch. An analog VLSI velocity sensor. In Proc. Int. Symp. Circuit and Systems ISCAS '95, pages 413-416, Seattle, WA, May 1995. [Mead, 1989] C.A. Mead. Analog VLSI and Neural Systems. Addison-Wesley, Reading, 1989. [Morris et al., 1995] T .G. Morris, D.M. Wilson, and S.P. DeWeerth. Analog VLSI circuits for manufacturing inspection. In Conference for Advanced Research in VLSI-Chapel Hill, North Carolina, March 1995. [Poggio et al., 1991] T. Poggio, A. Verri, and V. Torre. Green theorems and qualitative properties of the optical flow. Technical report, MIT, 1991. Internal Lab. Memo 1289. [Verri et al., 1992] A. Verri, M. Straforini, and V. Torre. Computational aspects of motion perception in natural and artificial vision systems. Phil. Trans. R. Soc. Lond. B, 337:429-443, 1992. PART VI SPEECH AND SIGNAL PROCESSING
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A Dynamical Systems Approach for a Learnable Autonomous Robot J un Tani and N aohiro Fukumura Sony Computer Science Laboratory Inc. Takanawa Muse Building, 3-14-13 Higashi-gotanda, Shinagawa-ku,Tokyo, 141 JAPAN Abstract This paper discusses how a robot can learn goal-directed navigation tasks using local sensory inputs. The emphasis is that such learning tasks could be formulated as an embedding problem of dynamical systems: desired trajectories in a task space should be embedded into an adequate sensory-based internal state space so that an unique mapping from the internal state space to the motor command could be established. The paper shows that a recurrent neural network suffices in self-organizing such an adequate internal state space from the temporal sensory input. In our experiments, using a real robot with a laser range sensor, the robot navigated robustly by achieving dynamical coherence with the environment. It was also shown that such coherence becomes structurally stable as the global attractor is self-organized in the coupling of the internal and the environmental dynamics. 1 Introd uction Conventionally, robot navigation problems have been formulated assuming a global view of the world. Given a detailed map of the workspace, described in a global coordinate system, the robot navigates to the specified goal by following this map. However, in situations where robots have to acquire navigational knowledge based on their own behaviors, it is important to describe the problems from the internal views of the robots. [Kuipers 87], [Mataric 92] and others have developed an approach based on landmark detection. The robot acquires a graph representation of landmark types as a topological modeling of the environment through its exploratory travels using the local sensory inputs. In navigation, the robot can identify its topological position by anticipating the landmark types in the graph representation obtained. It is, however, considered that this navigation strategy might be susceptible to erroneous landmark-matching. If the robot is once lost by such a catastrophe, its recoverance of the positioning might be difficult. We need certain mechanisms by which the 990 J. TANI, N. FUKUMURA robot can recover autonomously from such failures. We study the above problems by using the dynamical systems approach, expecting that this approach would provide an effective representational and computational framework. The approach focuses on the fundamental dynamical structure that arises from coupling the internal and the environmental dynamics [Beer 95]. Here, the objective of learning is to adapt the internal dynamical function such that the resultant dynamical structure might generate the desired system behavior. The system's performance becomes structurally stable if the dynamical structure maintains a sufficiently large basin of attraction against possible perturbations. We verify our claims through the implementation of our scheme on YAMABICO mobile robot equipped with a laser range sensor. The robot conducts navigational tasks under the following assumptions and conditions. (1) The robot cannot access its global position, but it navigates depending on its local sensory (range image) input. (2) There is no explicit landmarks accessible to the robot in the adopted workspace. (3) The robot learns tasks of cyclic routing by following guidance of a trainer. (4) The navigation should be robust enough against possible noise in the environment. 2 NAVIGATION ARCHITECTURE The YAMABICO mobile robot [Yuta and Iijima 90] was used as an experimental platform. The robot can obtain range images by a range finder consisting of laser projectors and three CCD cameras. The ranges for 24 directions, covering a 160 degree arc in front of the robot, are measured every 150 milliseconds. In our formulation, maneuvering commands are generated as the output of a composite system consisting of two levels [Tani and Fukumura 94]. The control level generates a collision-free, smooth trajectory using the range image, while the navigation level directs the control level in a macroscopic sense, responding to the sequential branching that appears in the sensory flows. The control level is fixed; the navigation level, on the other hand, can be adapted through learning. Firstly, let us describe the control level. The robot can sense the forward range readings of the surrounding environment, given in robot-centered polar coordinates by ri (1 ~ i ~ N). The angular range profile Ri is obtained by smoothing the original range readings through applying an appropriate Gaussian filter. The maneuvering focus of the robot is the maximum (the angular direction of the largest range) in this range profile. The robot proceeds towards the maximum of the profile (an open space in the environment). The navigation level focuses on the topological changes in the range profile as the robot moves. As the robot moves through a given workspace, the profile gradually changes until another local peak appears when the robot reaches a branching point. At this moment of branching the navigation level decides whether to transfer the focus to the new local peak or to remain with the current one. It is noted that this branching could be quite undeterministic one if applied to rugged obstacle environment. The robot is likely to fail to detect branching points frequently in such environment. The navigation level determines the branching by utilizing the range image obtained at branch points. Since the pertinent information in the range profile at a given moment is assumed to be only a small fraction of the total, we employ a vector quantization technique, known as the Kohonen network [Kohonen 82]' so that the information in the profile may be compressed into specific lower-dimensional data. The Kohonen network employed here consists of an I-dimensional lattice with m nodes along each dimension (l=3 and m=6 for the experiments with YAMABICO). The range image consisting of 24 values is input to the lattice, then the most A Dynamical Systems Approach for a Learnable Autonomous Robot Pn : sensory inputs TPM's output space • of (6.6.6) • , , , , , , , , F\,: range profile cn: context units Figure 1: Neural architecture for skill-based learning. 991 highly activated unit in the lattice, the "winner" unit, is found. The address of the winner unit in the lattice denotes the output vector of the network. Therefore, the navigation level receives the sensory input compressed into three dimensional data. The next section will describe how the robot can generate right branching sequences upon receiving the compressed range image. 3 Formulation 3.1 Learning state-action map The neural adaptation schemes are applied to the navigation level so that it can generate an adequate state-action map for a given task. Although some might consider that such map can be represented by using a layered feed-forward network with the inputs of the sensory image and the outputs of the motor command, this is not always true. The local sensory input does not always correspond uniquely to the true state of the robot (the sensory inputs could be the same for different robot positions). Therefore, there exists an ambiguity in determining the motor command solely from sensory inputs. This is a typical example of so-called non-Markovian problems which have been discussed by Lin and Mitchell [Lin and Mitchell 92]. In order to solve this ambiguity, a representation of contexts which are memories of past sensory sequences is required. For this purpose, a recurrent neural network (RNN) [Elman 90] was employed since its recurrent context states could represent the memory of past sequences. The employed neural architecture is shown in Figure. 1. The sensory input Pn and the context units en determine the appropriate motor command Xn+l' The motor command Xn takes a binary value of 0 (staying at the current branch) or 1 (a transit to a new branch). The RNN learning of sensorymotor (Pn,xn+d sequences, sampled through the supervised training, can build the desired state-action map by self-organizing adequate internal representation in time. 992 J. TANI, N. FUKUMURA (a) task space internal state space (b) task space internal state space Figure 2: The desired trajectories in the task space and its mapping to the internal state space. 3.2 Embedding problem The objective of the neural learning is to embed a task into certain global attractor dynamics which are generated from the coupling of the internal neural function and the environment. Figure 2 illustrates this idea. We define the internal state of the robot by the state of the RNN. The internal dynamics, which are coupled with the environmental dynamics through the sensory-motor loop, evolve as the robot travels in the task space. We assume that the desired vector field in the task space forms a global attractor, such as a fixed point for a homing task or limit cycling for a cyclic routing task. All that the robot has to do is to follow this vector flow by means of its internal state-action map. This requires a condition: the vector field in the internal state space should be self-organized as being topologically equivalent to that in the task space in order that the internal state determine the action (motor command) uniquely. This is the embedding problem from the task space to the internal state space, and RNN learning can attain this, using various training trajectories. This analysis conjectured that the trajectories in the task space can always converge into the desired one as long as the task is embedded into the global attractor in the internal state space. 4 Experiment 4.1 Task and training procedure Figure 3 shows an example of the navigation task, (which is adopted for the physical experiment in a later section). The task is for the robot to repeat looping of a figure of '8' and '0' in sequence. The task is not trivial because at the branching position A the robot has to decide whether to go '8' or '0' depending on its memory of the last sequence. The robot learns this navigation task through supervision by a trainer. The trainer repeatedly guides the robot to the desired loop from a set of arbitrarily selected A Dynamical Systems Approach for a Learnable Autonomous Robot 993 CJ Figure 3: Cyclic routing task, in which YAMABICO has to trace a figure of eight followed by a single loop. Figure 4: Trace of test travels for cyclic routing. initial locations. (The training was conducted with starting the robot from 10 arbitrarily selected initial locations in the workspace.) In actual training, the robot moves by the navigation of the control level and stops at each branching point, where the branching direction is taught by the trainer. The sequence of range images and teaching branching commands at those bifurcation points are fed into the neural architecture as training data. The objective of training RNN is to find the optimal weight matrix that minimizes the mean square error of the training output (branching decision) sequences associating with sensory inputs (outputs of Kohonen network). The weight matrix can be obtained through an iterative calculation of back-propagation through time (BPTT) [Rumelhart et al. 86]. 4.2 Results After the training, we examined how the robot achieves the trained task. The robot was started from arbitrary initial positions for this test. Fig. 4 shows example test travels. The result showed that the robot always converged to the desired loop regardless of its starting position. The time required to converge, however, took a 994 J. TANI. N. FUKUMURA AA Jt.JLGlLrr1L .dLA[L[L O:t....an..lIIlITIl. nlLllil..lliLrriL lliLalLlliLlliL lllllLllll11 Q. oIL .fiL .fiL .ilL .. (A') 'iii '" lLlLlLlL c :2 0 rn:L 0....... rn:L 0....... c ~ .Q dlLdILdlLdIL dLdLdLdL rrllrrllrrllrrll J1..J1..JlJ1.. .ill. .ill. JlI. JlI. (A) [hJ[hJ[hJ[hJ ~uJrr.J~ • cycle Figure 5: The sequence of activations in input and context units during the cycling travel. certain period that depended on the case. The RNN initially could not function correctly because of the arbitrary initial setting of the context units. However, while the robot wandered around the workspace, the RNN became situated (recovered the context) as it encountered pre-learned sensory sequences. Thereafter, its navigation converged to the cycling loop. Even after convergence, the robot could, by chance, leave the loop, under the influence of noise. However, the robot always came back to the loop after a while. These observations indicate that the robot learned the objective navigational task as embedded in a global attractor of limit cycling. It is interesting to examine how the task is encoded in the internal dynamics of the RNN. We investigated the activation patterns of RNN after its convergence into the loop. The results are shown in Fig. 5. The input and context units at each branching point are shown as three white and two black bars, respectively. One cycle (the completion of two routes of '0' and '8') are aligned vertically as one column. The figure shows those of four continuous cycles. It can be seen that robot navigation is exposed to much noise; the sensing input vector becomes unstable at particular locations, and the number of branchings in one cycle is not constant (i.e. some branching points are undeterministic). The rows labeled as (A) and (A') are branches to the routes of '0' and '8', respectively. In this point, the sensory input receives noisy chattering of different patterns independent of (A) or (A'). The context units, on the other hand, is completely identifiable between (A) and (A'), which shows that the task sequence between two routes (a single loop and an eight) is rigidly encoded internally, even in a noisy environment. In further experiments in more rugged obstacle environments, we found that this sort of structural stability could not be always assured. When the undeterministicity in the branching exceeds a certain limit, the desired dynamical structure cannot be preserved. A Dynamical Systems Approach for a Learnable Autonomous Robot 995 5 Summary and Discussion The navigation learning problem was formulated from the dynamical systems perspective. Our experimental results showed that the robot can learn the goal-directed navigation by embedding the desired task trajectories in the internal state space through the RNN training. It was also shown that the robot achieves the navigational tasks in terms of convergence of attract or dynamics which emerge in the coupling of the internal and the environmental dynamics. Since the dynamical coherence arisen in this coupling leads to the robust navigation of the robot, the intrinsic mechanism presented here is characterized by the term "autonomy". Finally, it is interesting to study how robots can obtain analogical models of the environment rather than state-action maps for adapting to flexibly changed goals. We discuss such formulation based on the dynamical systems approach elsewhere [Tani 96]. References [Beer 95] R.D. Beer. A dynamical systems perspective on agent-environment interaction. Artificial Intelligence, Vol. 72, No.1, pp.173- 215, 1995. [Elman 90] J.L. Elman. Finding structure in time. Cognitive Science, Vol. 14, pp.179- 211, 1990. [Kohonen 82] T. Kohonen. Self-Organized Formation of Topographically Correct Feature Maps. Biological Cybernetics, Vol. 43, pp.59- 69, 1982. [Kuipers 87] B. Kuipers. A Qualitative Approach to Robot Exploration and Map Learning. In AAAI Workshop Spatial Reasoning and Multi-Sensor Fusion {Chicago),1987. [Lin and Mitchell 92] L.-J. Lin and T.M. Mitchell. Reinforcement learning with hidden states. In Proc. of the Second Int. Conf. on Simulation of Adaptive Behavior, pp. 271- 280, 1992. [Mataric 92] M. Mataric. Integration of Representation into Goal-driven Behaviorbased Robot. IEEE Trans. Robotics and Automation, Vol. 8, pp.304- 312, 1992. [Rumelhart et al. 86] D.E. Rumelhart, G.E. Hinton, and R.J. Williams. Learning Internal Representations by Error Propagation. In Parallel Distributed Processing. MIT Press, 1986. [Tani 96] J. Tani. Model-Based Learning for Mobile Robot Navigation from the Dynamical Systems Perspective. IEEE Trans. System, Man and Cybernetics Part B, Special issue on robot learning, Vol. 26, No.3, 1996. [Tani and Fukumura 94] J. Tani and N. Fukumura. Learning goal-directed sensorybased navigation of a mobile robot. Neural Networks, Vol. 7, No.3, pp.553- 563, 1994. [Yuta and Iijima 90] S. Yuta and J. Iijima. State Information Panel for InterProcessor Communication in an Autonomous Mobile Robot Controller. In proc. of IROS'90, 1990.
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Optimization Principles for the Neural Code Michael DeWeese Sloan Center, Salk Institute La Jolla, CA 92037 deweese@salk.edu Abstract Recent experiments show that the neural codes at work in a wide range of creatures share some common features. At first sight, these observations seem unrelated. However, we show that these features arise naturally in a linear filtered threshold crossing (LFTC) model when we set the threshold to maximize the transmitted information. This maximization process requires neural adaptation to not only the DC signal level, as in conventional light and dark adaptation, but also to the statistical structure of the signal and noise distributions. We also present a new approach for calculating the mutual information between a neuron's output spike train and any aspect of its input signal which does not require reconstruction of the input signal. This formulation is valid provided the correlations in the spike train are small, and we provide a procedure for checking this assumption. This paper is based on joint work (DeWeese [1], 1995). Preliminary results from the LFTC model appeared in a previous proceedings (DeWeese [2], 1995), and the conclusions we reached at that time have been reaffirmed by further analysis of the model. 1 Introduction Most sensory receptor cells produce analog voltages and currents which are smoothly related to analog signals in the outside world. Before being transmitted to the brain, however, these signals are encoded in sequences of identical pulses called action potentials or spikes. We would like to know if there is a universal principle at work in the choice of these coding strategies. The existence of such a potentially powerful theoretical tool in biology is an appealing notion, but it may not turn out to be useful. Perhaps the function of biological systems is best seen as a complicated compromise among constraints imposed by the properties of biological materials, the need to build the system according to a simple set of development rules, and 282 M. DEWEESE the fact that current systems must arise from their ancestors by evolution through random change and selection. In this view, biology is history, and the search for principles (except for evolution itself) is likely to be futile. Obviously, we hope that this view is wrong, and that at least some of biology is understandable in terms of the same sort of universal principles that have emerged in the physics of the inanimate world. Adrian noticed in the 1920's that every peripheral neuron he checked produced discrete, identical pulses no matter what input he administered (Adrian, 1928). From the work of Hodgkin and Huxley we know that these pulses are stable non-linear waves which emerge from the non-linear dynamics describing the electrical properties of the nerve cell membrane These dynamics in turn derive from the molecular dynamics of specific ion channels in the cell membrane. By analogy with other nonlinear wave problems, we thus understand that these signals have propagated over a long distance e.g. ~ one meter from touch receptors in a finger to their targets in the spinal cord so that every spike has the same shape. This is an important observation since it implies that all information carried by a spike train is encoded in the arrival times of the spikes. Since a creature's brain is connected to all of its sensory systems by such axons, all the creature knows about the outside world must be encoded in spike arrival times. Until recently, neural codes have been studied primarily by measuring changes in the rate of spike production by different input signals. Recently it has become possible to characterize the codes in information-theoretic terms, and this has led to the discovery of some potentially universal features of the code (Bialek, 1996) (or see (Bialek, 1993) for a brief summary). They are: 1. Very high information rates. The record so far is 300 bits per second in a cricket mechanical sensor. 2. High coding efficiency. In cricket and frog vibration sensors, the information rate is within a factor of 2 of the entropy per unit time of the spike train. 3. Linear decoding. Despite evident non-linearities ofthe nervous system, spike trains can be decoded by simple linear filters. Thus we can write an estimate of the analog input signal s(t) as Sest (t) = Ei Kl (t - td, with Kl chosen to minimize the mean-squared errors (X2 ) in the estimate. Adding non-linear K2(t - ti, t - tj) terms does not significantly reduce X2 . 4. Moderate signal-to-noise ratios (SNR). The SNR in these experiments was defined as the ratio of power spectra of the input signal to the noise referred back to the input; the power spectrum of the noise was approximated by X2 defined above. All these examples of high information transmission rates have SNR of order unity over a broad bandwidth, rather than high SNR in a narrow band. We will try to tie all of these observations together by elevating the first to a principle: The neural code is chosen to maximize information transmission where information is quantified following Shannon. We apply this principle in the context of a simple model neuron which converts analog signals into spike trains. Before we consider a specific model, we will present a procedure for expanding the information rate of any point process encoding of an analog signal about the limit where the spikes are uncorrelated. We will briefly discuss how this can be used to measure information rates in real neurons. Optimization Principles for the Neural Code 283 This work will also appear in Network. 2 Information Theory In the 1940's, Shannon proposed a quantitative definition for "information" (Shannon, 1949). He argued first that the average amount of information gained by observing some event Z is the entropy of the distribution from which z is chosen, and then showed that this is the only definition consistent with several plausible requirements. This definition implies that the amount of information one signal can provide about some other signal is the difference between the entropy of the first signal's a priori distribution and the entropy of its conditional distribution. The average of this quantity is called the mutual (or transmitted) information. Thus, we can write the amount of information that the spike train, {td, tells us about the time dependent signal, s(t), as (1) where I1Jt; is shorthand for integration over all arrival times {til from 0 to T and summation over the total number of spikes, N (we have divided the integration measure by N! to prevent over counting due to equivalent permutations of the spikes, rather than absorb this factor into the probability distribution as we did in (DeWeese [1], 1995)). < ... >8= I1JsP[sO]'" denotes integration over the space offunctions s(t) weighted by the signal's a priori distribution, P[{t;}ls()] is the probability distribution for the spike train when the signal is fixed and P[{t;}] is the spike train's average distribution. 3 Arbitrary Point Process Encoding of an Analog Signal In order to derive a useful expression for the information given by Eq. (1), we need an explicit representation for the conditional distribution of the spike train. If we choose to represent each spike as a Dirac delta function, then the spike train can be defined as N p(t) = L c5(t - t;). (2) ;=1 This is the output spike train for our cell, so it must be a functional of both the input signal, s(t), and all the noise sources in the cell which we will lump together and call '7(t). Choosing to represent the spikes as delta functions allows us to think of p(t) as the probability of finding a spike at time t when both the signal and noise are specified. In other words, if the noise were not present, p would be the cell's firing rate, singular though it is. This implies that in the presence of noise the cell's observed firing rate, r(t), is the noise average of p(t): r(t) = J 1J'7P ['70Is0]p(t) = (p(t))'1' (3) Notice that by averaging over the conditional distribution for the noise rather than its a priori distribution as we did in (DeWeese [1], 1995), we ensure that this expression is still valid if the noise is signal dependent, as is the case in many real neurons. For any particular realization of the noise, the spike train is completely specified which means that the distribution for the spike train when both the signal and 284 M. DEWEESE noise are fixed is a modulated Poisson process with a singular firing rate, p(t). We emphasize that this is true even though we have assumed nothing about the encoding of the signal in the spike train when the noise is not fixed. One might then assume that the conditional distribution for the spike tra.in for fixed signal would be the noise average of the familiar formula for a modulated Poisson process: (4) However, this is only approximately true due to subtleties arising from the singular nature of p(t). One can derive the correct expression (DeWeese [1], 1995) by carefully taking the continuum limit of an approximation to this distribution defined for discrete time. The result is the same sum of noise averages over products of p's produced by expanding the exponential in Eq. (4) in powers of f dtp(t) except that all terms containing more than one factor of p(t) at equal times are not present. The exact answer is: (5) where the superscripted minus sign reminds us to remove all terms containing products of coincident p's after expanding everything in the noise average in powers of p. 4 Expanding About the Poisson Limit An exact solution for the mutual information between the input signal and spike train would be hopeless for all but a few coding schemes. However, the success of linear decoding coupled with the high information rates seen in the experiments suggests to us that the spikes might be transmitting roughly independent information (see (DeWeese [1], 1995) or (Bialek, 1993) for a more fleshed out argument on this point). If this is the case, then the spike train should approximate a Poisson process. We can explicitly show this relationship by performing a cluster expansion on the right hand side of Eq. (5): (6) where we have defined ~p(t) == p(t)- < p(t) >'1= p(t) - r(t) and introduced C'1(m) which collects all terms containing m factors of ~p. For example, C (2) == ~ ,,(~Pi~Pj}q - J dt' ~ (~p' ~Pi}q + ~ J dt'dt"(~ '~ ")-. '1 2 L..J r·r · L..J r · 2 p P '1 i¢j , J i=l' (7) Clearly, if the correlations between spikes are small in the noise distribution, then the C'1 's will be small, and the spike train will nearly approximate a modulated Poisson process when the signal is fixed. Optimization Principles for the Neural Code 285 Performing the cluster expansion on the signal average of Eq. (5) yields a similar expression for the average distribution for the spike train: (8) where T is the total duration of the spike train, r is the average firing rate, and C'1. 8 (m) is identical to C'1(m) with these substitutions: r(t) --+ r, ~p(t) --+ ap(t) == p(t) - f, and ( ... ); --+ {{ .. ·);)8. In this case, the distribution for a homogeneous Poisson process appears in front of the square brackets, and inside we have 1 + corrections due to correlations in the average spike train. 5 The Transmitted Information Inserting these expressions for P[ {til IsO] and P[ {til] (taken to all orders in ~p and ap, respectively) into Eq. (1), and expanding to second non-vanishing order in fTc results in a useful expression for the information (DeWeese [1], 1995): (9) where we have suppressed the explicit time notation in the correction term inside the double integral. If the signal and noise are stationary then we can replace the I; dt in front of each of these terms by T illustrating that the information does indeed grow linearly with the duration of the spike train. The leading term, which is exact if there are no correlations between the spikes, depends only on the firing rate, and is never negative. The first correction is positive when the correlations between pairs of spikes are being used to encode the signal, and negative when individual spikes carry redundant information. This correction term is cumbersome but we present it here because it is experimentally accessible, as we now describe. This formula can be used to measure information rates in real neurons without having to assume any method of reconstructing the signal from the spike train. In the experimental context, averages over the (conditional) noise distribution become repeated trials with the same input signal, and averages over the signal are accomplished by summing over all trials. r(t), for example, is the histogram of the spike trains resulting from the same input signal, while f(t) is the histogram of all spike trains resulting from all input signals. If the signal and noise are stationary, then f will not be time dependent. {p(t)p(t'))'1 is in general a 2-dimensional histogram which is signal dependent: It is equal to the number of spike trains resulting from some specific input signal which simultaneously contain a spike in the time bins containing t and t'. If the noise is stationary, then this is a function of only t - t', and it reduces to a 1-dimensional histogram. In order to measure the full amount of information contained in the spike train, it is crucial to bin the data in small enough time bins to resolve all of the structure in 286 M. DEWEESE r(t), (p(t)p(t'))'l' and so on. We have assumed nothing about the noise or signal; in fact, they can even be correlated so that the noise averages are signal dependent without changing the experimental procedure. The experimenter can also choose to fix only some aspects of the sensory data during the noise averaging step, thus measuring the mutual information between the spike train and only these aspects of the input. The only assumption we have made up to this point is that the spikes are roughly uncorrelated which can be checked by comparing the leading term to the first correction, just as we do for the model we discuss in the next section. 6 The Linear Filtered Threshold Crossing Model As we reported in a previous proceedings (DeWeese [2], 1995) (and see (DeWeese [1], 1995) for details), the leading term in Eq. (9) can be calculated exactly in the case of a linear filtered threshold crossing (LFTC) model when the signal and noise are drawn from independent Gaussian distributions. Unlike the Integrate and Fire (IF) model, the LFTC model does not have a "renewal process" which resets the value of the filtered signal to zero each time the threshold is reached. Stevens and Zador have developed an alternative formulation for the information transmission which is better suited for studying the IF model under some circumstances (Stevens, 1995), and they give a nice discussion on the way in which these two formulations compliment each other. For the LFTC model, the leading term is a function of only three variables: 1) The threshold height; 2) the ratio of the variances of the filtered signal and the filtered noise, (s2(t)),/(7J2(t))'l' which we refer to as the SNR; 3) and the ratio of correlation times ofthe filtered signal and the filtered noise, T,/T'l' where T; == (S2(t)),/(S2(t)), and similarly for the noise. In the equations in this last sentence, and in what follows, we absorb the linear filter into our definitions for the power spectra of the signal and noise. Near the Poisson limit, the linear filter can only affect the information rate through its generally weak influence on the ratios of variances and correlation times of the signal and noise, so we focus on the threshold to understand adaptation in our model cell. When the ratio of correlation times of the signal and noise is moderate, we find a maximum for the information rate near the Poisson limit the leading term ~ lOx the first correction. For the interesting and physically relevant case where the noise is slightly more broadband than the signal as seen through the cell's prefiltering, we find that the maximum information rate is achieved with a threshold setting which does not correspond to the maximum average firing rate illustrating that this optimum is non-trivial. Provided the SNR is about one or less, linear decoding does well a lower bound on the information rate based on optimal linear reconstruction of the signal is within a factor of two of the total available information in the spike train. As SNR grows unbounded, this lower bound asymptotes to a constant. In addition, the required timing resolution for extracting the information from the spike train is quite modest discretizing the spike train into bins which are half as wide as the correlation time of the signal degrades the information rate by less than 10%. However, at maximum information transmission, the information per spike is low Rmaz/r ~ .7 bits/spike, much lower than 3 bits/spike seen in the cricket. This low information rate drives the efficiency down to 1/3 of the experimental values despite the model's robustness to timing jitter. Aside from the low information rate, the optimized model captures all the experimental features we set out to explain. Optimization Principles for the Neural Code 287 7 Concluding Remarks We have derived a useful expression for the transmitted information which can be used to measure information rates in real neurons provided the correlations between spikes are shorter range than the average inter-spike interval. We have described a method for checking this hypothesis experimentally. The four seemingly unrelated features that were common to several experiments on a variety of neurons are actually the natural consequences of maximizing the transmitted information. Specifically, they are all due to the relation between if and Tc that is imposed by the optimization. We reiterate our previous prediction (DeWeese [2], 1995; Bialek, 1993): Optimizing the code requires that the threshold adapt not only to cancel DC offsets, but it must adapt to the statistical structure of the signal and noise. Experimental hints at adaptation to statistical structure have recently been seen in the fly visual system (de Ruyter van Steveninck, 1994) and in the salamander retina (Warland, 1995). 8 References M. DeWeese 1995 Optimization Principles for the Neural Code (Dissertation, Princeton University) M. DeWeese and W. Bialek 1995 Information flow in sensory neurons II Nuovo Cimento l7D 733-738 E. D. Adrian 1928 The Basis of Sensation (New York: W. W. Norton) F. Rieke, D. Warland, R. de Ruyter van Steveninck, and W. Bialek 1996 Neural Coding (Boston: MIT Press) W. Bialek, M. DeWeese, F. Rieke, and D. Warland 1993 Bits and Brains: Information Flow in the Nervous System Physica A 200 581-593 C. E. Shannon 1949 Communication in the presence of noise, Proc. I. R. E. 37 10-21 C. Stevens and A. Zador 1996 Information Flow Through a Spiking Neuron in M. Hasselmo ed Advances in Neural Information Processing Systems, Vol 8 (Boston: MIT Press) (this volume) R.R. de Ruyter van Steveninck, W. Bialek, M. Potters, R.H. Carlson 1994 Statistical adaptation and optimal estimation in movement computation by the blowfly visual system, in IEEE International Conference On Systems, Man, and Cybernetics pp 302-307 D. Warland, M. Berry, S. Smirnakis, and M. Meister 1995 personal communication
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A Framework for Non-rigid Matching and Correspondence Suguna Pappu, Steven Gold, and Anand Rangarajan1 Departments of Diagnostic Radiology and Computer Science and the Yale Neuroengineering and Neuroscience Center Yale University New Haven, CT 06520-8285 Abstract Matching feature point sets lies at the core of many approaches to object recognition. We present a framework for non-rigid matching that begins with a skeleton module, affine point matching, and then integrates multiple features to improve correspondence and develops an object representation based on spatial regions to model local transformations. The algorithm for feature matching iteratively updates the transformation parameters and the correspondence solution, each in turn. The affine mapping is solved in closed form, which permits its use for data of any dimension. The correspondence is set via a method for two-way constraint satisfaction, called softassign, which has recently emerged from the neural network/statistical physics realm. The complexity of the non-rigid matching algorithm with multiple features is the same as that of the affine point matching algorithm. Results for synthetic and real world data are provided for point sets in 2D and 3D, and for 2D data with multiple types of features and parts. 1 Introduction A basic problem of object recognition is that of matching- how to associate sensory data with the representation of a known object. This entails finding a transformation that maps the features of the object model onto the image, while establishing a correspondence between the spatial features. However, a tractable class of transformation, e.g., affine, may not be sufficient if the object is non-rigid or has relatively independent parts. If there is noise or occlusion, spatial information alone may not be adequate to determine the correct correspondence. In our previous work in spatial point matching [1], the 2D affine transformation was decomposed into its Ie-mail address of authors: lastname-firstname@cs.yale.edu 796 S. PAPPU, S. GOLD, A. RANGARAJAN physical component elements, which does not generalize easily to 3D, and so, only a rigid 3D transformation was considered. We present a framework for non-rigid matching that begins with solving the basic affine point matching problem. The algorithm iteratively updates the affine parameters and correspondence in turn, each as a function of the other. The affine transformation is solved in closed form, which lends tremendous flexibility- the formulation can be used in 2D or 3D. The correspondence is solved by using a softassign [1] procedure, in which the two-way assignment constraints are solved without penalty functions. The accuracy of the correspondence is improved by the integration of multiple features. A method for non-rigid parameter estimation is developed, based on the assumption of a well-articulated model with distinct regions, each of which may move in an affine fashion, or can be approximated as such. Umeyama [3] has done work on parameterized parts using an exponential time tree search technique, and Wakahara [4] on local affine transforms, but neither integrates multiple features nor explicitly considers the non-rigid matching case, while expressing a one-to-one correspondence between points. 2 Affine Point Matching The affine point matching problem is formulated as an optimization problem for determining the correspondence and affine transformation between feature points. Given two sets of data points Xj E Rn-l, n = 3,4 .. . , i = 1, ... , J, the image, and Yk E Rn-l, n = 3,4, ... , k = 1, ... , K, the model, find the correspondence and associated affine transformation that best maps a subset of the image points onto a subset of the model point set. These point sets are expressed in homogeneous coordinates, Xj = (l,Xj), Yk = (1, Yk). {aij} = A E Rnxn is the affine transformation matrix. Note that{alj = 0 Vi} because of the homogeneous coordinates. Define the match variable Mjk where Mjk E [0,1]. For a given match matrix {Mjd, transformation A and I, an identity matrix of dimension n, Lj,k MjkllXj - (A + I)Yk112 expresses the similarity between the point sets. The term -a Lj,k Mjk, with parameter a > 0 is appended to this to encourage matches (else Mjk = 0 V i, k minimizes the function). To limit the range of transformations, the terms of the affine matrix are regularized via a term Atr(AT A) in the objective function, with parameter A, where tr(.) denotes the trace of the matrix. Physically, Xj may fully match to one Yk, partially match to several, or may not match to any point. A similar constraint holds for Yk. These are expressed as the constraints in the following optimization problem: (1) s.t. LMjk::S 1, Vk, LMjk::S 1, Vi and Mjk ~ 0 j k To begin, slack variables Mj,K+l and MJ+l,k are introduced so that the inequality constraints can be transformed into equality constraints: Lf~t Mjk = 1, Vk and Lf:/ Mjk = 1, Vi. Mj,K+l = 1 indicates that Xj does not match to any point in Yk. An equivalent unconstrained optimization problem to (2) is derived by relaxing the constraints via Lagrange parameters Ilj, l/k, and introducing an x log x barrier function, indexed by a parameter {3. A similar technique was used A Framework for Nonrigid Matching and Correspondence 797 [2] to solve the assignment problem. The energy function used is: J K+1 min max LMjkllXj - (A+ J)Yk112 + Atr(AT A) - a LMjk + LJLj(L Mjk -1) A,M ~,v . . . ),k ),k ) k=l K J+1 1 J+1 K+1 + LlIk(LMjk -1) + (j L L Mjk(1ogMjk -1) k j=l j=l k=l This is to be minimized with respect to the match variables and affine parameters while satisfying the constraints via Lagrange parameters. Using the recently developed soft assign technique, we satisfy the constraints explicitly. When A is fixed, we have an assignment problem. Following the development in [1], the assignment constraints are satisfied using soft assign , a technique for satisfying two-way (assignment) constraints without a penalty term that is analogous to softmax which enforces a one-way constraint. First, the match variables are initialized: (2) This is followed by repeated row-column normalization of the match variables until a stopping criterion is reached: M)"k = Mjk Mjk '""' then M j k = '""' M L--j' Mj'k L--k' jk' (3) When the correspondence between the two point sets is fixed, A can be solved in closed form, by holding M fixed in the objective function, and differentiating and solving for A: A = A*(M) = (L Mjk(Xj Y[ - YkY{»(L MjkYkY[ + AI)-l (4) j,k j,k The algorithm is summarized as: 1. INITIALIZE: Variables: A = 0, M = 0 Parameters: .Binitial, .Bupdate, .Bfinal T = Inner loop iterations, A 2. ITERATE: Do T times for a fixed value of .B Softassign: Re-initialize M*(A) and then (Eq. 2) until ilM small A*(M) updated (Eq. 4) 3. UPDATE: While.B < .Bfinal, .B ~.B * .Bupdate, Return to 2. The complexity of the algorithm is O(J K). Starting with small .Binitial permits many partial correspondences in the initial solution for M. As.B increases the correspondence becomes more refined. For large .Bfinal, M approaches a permutation matrix (adjusting appropriately for the slack variables). 3 Nonrigid Feature Matching: Affine Quilts Recognition of an object requires many different types of information working in concert. Spatial information alone may not be sufficient for representation, especially in the presence of noise. Additionally the affine transformation is limited in its inability to handle local variation in an object, due to the object's non-rigidity or to the relatively independent movement of its parts, e.g., in human movement. The optimization problem (2) easily generalizes to integrate multiple invariant features. A representation with multiple features has a spatial component indicating 798 S. PAPPU, S. GOLD, A. RANGARAJAN the location of a feature element. At that location, there may be invariant geometric characteristics, e.g., this point belongs on a curve, or non-geometric invariant features such as color, and texture. Let Xjr be the value of feature r associated with point Xj. The location of point Xj is the null feature. There are R features associated with each point Xj and Yk. Note that the match variable remains the same. The new objective function is identical to the original objective function, (2), appended by the term "£j,k,r MjkWr(Xjr - Ykr)2. The (Xjr - Ykr)2 quantity captures the similarity between invariant types of features, with Wr a weighting factor for feature r. Non-invariant features are not considered. In this way, the point matching algorithm is modified only in the re-initialization of M(A): Mjk = exp(-,8(IIXj - (I + A)Yk112 + "£rWr(Xjr - ykr )2 - a)) The rest of the algorithm remains unchanged. Decomposition of spatial transformations motivates classification of the B individual regions of an object and use of a "quilt" of local affine transformations. In the multiple affine scenario, membership to a region is known on the well-articulated model, but not on the image set. It is assumed that all points that are members of one region undergo the same affine transformation. The model changes by the addition of one subscript to the affine matrix, Ab(k) where b(k) is an operator that indicates which transformation operates on point k. In the algorithm, during the A(M) update, instead of a single update, B updates are done. Denote K(b) = {klb(k) = b}, i.e., all the points that are within region b. Then in the affine update, Ab = Ab(M) = (L:j, kEK(b) Mjk(Xj Y{ - YkY{))("£j, kEK(b) MjkYkY{ + AbI)-l However, the theoretical complexity does not change, since the B updates still only require summing over the points. 4 Experimental Results: Hand Drawn and Synthetic The speed for matching point sets of 50 points each is around 20 seconds on an SGI workstation with a R4400 processor. This is true for points in 2D, 3D and with extra features . This can be improved with a tradeoff in accuracy by adopting a looser schedule for the parameter ,8 or by changing the stopping criterion. In the hand drawn examples, the contours of the images are drawn, discretized and then expressed as a set of points in the plane. In Figure (1), the contours of the boy's face were drawn in two different positions, and a subset of the points were extracted to make up the point sets. In each set this was approximately 250 points. Note that even with the change in mood in the two pictures, the corresponding parts of the face are found. However, in Figure (2) spatial information alone is Figure 1: Correspondence with simple point features insufficient. Although the rotation ofthe head is not a true affine transformation, it A Framework for Nonrigid Matching and Correspondence 799 is a weak perspective projection for which the approximation is valid. Each photo is outlined, generating approximately 225 points in each face. A point on a contour Figure 2: Correspondence with multiple features has associated with it a feature marker indicating the incident textures. For a human face, we use a binary 4-vector, with a 1 in position r if feature r is present. Specifically, we have used a vector with elements [skin, hair, lip, eye]. For example, a point on the line marking the mouth segment the lip from the skin has a feature vector [1,0,1,0]. Perceptual organization of the face motivates this type of feature marking scheme. The correspondence is depicted in Figure (2) for a small subset of matches. Next, we demonstrate how the multiple affine works in recovering the correct correspondence and transformation. The points associated with the standing figure have a marker indicating its part membership. There are six parts in this figure: head, torso, each arm and each leg. The correspondence is shown in Figure (3). For synthetic data, all 2D and 3D single part experiments used this protocol: The model set was generated uniformly on a unit square. A random affine matrix is generated, whose parameters, aij are chosen uniformly on a certain interval, which is used to generate the image set. Then, Pd image points are deleted, and Gaussian noise, N(O, u) is added. Finally, spurious points, Ps are added. For the multiple feature scenario, the elements of the feature vector are randomly mislabelled with probability, Pr , to represent distortion. For these experiments, 50 model points were generated, and aij are uniform on an interval of length 1.5. u E {0.01, 0.02, ... , 0.08}. Point deletions and spurious additions range from 0% to 50% of the image points. The random feature noise associated with nonspatial features has a probability of Pr = 0.05. The error measure we use is ea = C Li,j laij -a.ij I where c = # par:meters interva~ length· aij and a.ij are the correct parameter and the computed value, respectively. The constant term c normalizes the measure so that the error equals 1 in the case that the aij and aij are chosen at random on this interval. The factor 3 in the numerator of this formula follows since 800 S. PAPPU, S. GOLD, A. RANGARAJAN ~·~--··············~-··-····· ·········R Il · ·······························~ '-' . . _ ................... , ---................ ~ • . . .;.;;:...':":-:-:~~ ... -... ~~:~~.~.:~~g ~··~·~-·-··-···············:······<:"!»>,.~Il ;']~f Figure 3: Articulated Matching: Figure with six parts Elx - yl = ~, when x and yare chosen randomly on the unit interval, and we want to normalize the error. The parameters used in all experiments were: ,Binitial = .091, ,Bfmal = 100, ,Bupdate = 1.075, and T = 4. The model has four regions, 24 parameters. Points corresponding to part 1 were centered at (.5, .5), and generated randomly with a diameter of 1.0. For the image set, an affine transformation was applied with a translation diameter of .5, i.e., for a21, an, and the remaining four parameters have a diameter of 1. Points corresponding to regions 2, 3, and 4 were centered at (-.5, .5), (-.5, -.5), (.5, -.5) with model points and transformations generated in a similar fashion. 120 points were generated for the model point set, divided equally among the four parts. Image points were deleted with equal probability from each region. Spurious point were not explicitly added, since the overlapping of parts provides implicit spurious points. Results for the 2D and 3D (simple point) experiments are in Figure (4). Each data point represents 500 runs for a different randomly generated affine transformation. In all experiments, note that the error for small amounts of noise is approximately equal to that when there is no noise. We performed similar experiments for point sets that are 3-dimensional (12 parameters), but without any feature information. For the experiments with features, shown in Figure (5) we used R = 4 features, and Wr = 0.2, Vr. Each data point represents 500 runs.As expected, the inclusion of feature information reduces the error, especially for large u. Additionally, Figure (5) details synthetic results for experiments with multiple affines (2D). Each data point represents 70 runs. 5 Conclusion We have developed an affine point matching module, robust in the presence of noise and able to accommodate data of any dimension. The module forms the basis for a non-rigid feature matching scheme in which multiple types of features interact to establish correspondence. Modeling an object in terms of its spatial regions and then using multiple affines to capture local transformations results in a tractable method for non-rigid matching. This non-rigid matching framework arising out of A Framework for Nonrigid Matching and Correspondence 20 Results 0 .25r--~--~-~--..., ! 0.2 ~ 0.15 ..e-g 0.1 Q) ~0 .05 o o o o x o x o x O~----~-----~ 0.02 0.04 0.06 0.08 Standard deviation: Jitter -. : Pd = 0%,P8 = 0%, + : Pd = 10%,P8 = 10%, 3D Resutts 0.25~~--~-~-~-, ! 0.2 ~ 0.15 0 0 .e0 g 0.1 0 0 0 0 0 x x : x x x x x + + ~0 .05 + + _ .~ . J_!.-+- . - ' -' o~----~-----~ 0.02 0.04 0.06 0.08 Standard deviation: Jitter 0: Pd = 50%,P8 = 10% X: Pd = 30%,P8 = 10% Figure 4: Synthetic Experiments: 2D and 3D iii "ai ~ 0.1 ., Q. .." o 4 Features --ai 0.05 " • • '''' ... . -., ...... _.-' • • .-iii 0.25 "ai ~ 0.2 c;; %0.15 e iii 0.1 ~0 .05 4 Parts x x 0 x 0 ." . "" ~ .9·-" . " -.x x 0 0 ..0 ..o~----~-----~ 0.02 0.04 0.06 0.08 o~----~-----~ 0.02 0.04 0.06 0.08 Standard deviation: Jitter Standard deviation: Jitter .- :Pd = 0%,P8 = 0% * : Pd = 10%,P8 = 10% 0: Pd = 10%,P8 = 0% . : Pd = 30%,P8 = 10%, X: Pd = 25%,P8 = 0% -- : Pd = 50%,P8 = 10%, - : Pd = 40%,P8 = 0% Figure 5: Synthetic Experiments: Multiple features and parts 801 neural computation is widely applicable in object recognition. Acknowledgements: Our thanks to Eric Mjolsness for many interesting discussions related to the present work. References [1] S. Gold, C. P. Lu, A. Rangarajan, S. Pappu, and E. Mjolsness. New algorithms for 2D and 3D point matching: Pose estimation and correspondence. In G. Tesauro, D. Touretzky, and J. Alspector, editors, Advances in Neural Information Processing Systems, volume 7, San Francisco, CA, 1995. Morgan Kaufmann Publishers. [2] J. Kosowsky and A. Yuille. The invisible hand algorithm: Solving the assignment problem with statistical physics. Neural Networks, 7:477-490, 1994. [3] S. Umeyama. Parameterized point pattern matching and its application to recognition of object families. IEEE Trans. on Pattern Analysis and Machine Intelligence, 15:136-144,1993. [4] T . Wakahara. Shape matching using LAT and its application to handwritten numeral recognition. IEEE Trans. in Pattern Analysis and Machine Intelligence, 16:618- 629, 1994.
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Analog VLSI Processor Implementing the Continuous Wavelet Transform R. Timothy Edwards and Gert Cauwenberghs Department of Electrical and Computer Engineering Johns Hopkins University 3400 North Charles Street Baltimore, MD 21218-2686 {tim,gert}@bach.ece.jhu.edu Abstract We present an integrated analog processor for real-time wavelet decomposition and reconstruction of continuous temporal signals covering the audio frequency range. The processor performs complex harmonic modulation and Gaussian lowpass filtering in 16 parallel channels, each clocked at a different rate, producing a multiresolution mapping on a logarithmic frequency scale. Our implementation uses mixed-mode analog and digital circuits, oversampling techniques, and switched-capacitor filters to achieve a wide linear dynamic range while maintaining compact circuit size and low power consumption. We include experimental results on the processor and characterize its components separately from measurements on a single-channel test chip. 1 Introduction An effective mathematical tool for multiresolution analysis [Kais94], the wavelet transform has found widespread use in various signal processing applications involving characteristic patterns that cover multiple scales of resolution, such as representations of speech and vision. Wavelets offer suitable representations for temporal data that contain pertinent features both in the time and frequency domains; consequently, wavelet decompositions appear to be effective in representing wide-bandwidth signals interfacing with neural systems [Szu92]. The present system performs a continuous wavelet transform on temporal one-dimensional analog signals such as speech, and is in that regard somewhat related to silicon models of the cochlea implementing cochlear transforms [Lyon88], [Liu92] , [Watt92], [Lin94]. The multiresolution processor we implemented expands on the architecture developed in [Edwa93], which differs from the other analog auditory processors in the way signal components in each frequency band are encoded. The signal is modulated with the center Analog VLSI Processor Implementing the Continuous Wavelet Transform 693 _: '\/'V 1m - I s'(t) set) LPF ~ Lff ~ LPF x(t) ~ yet) x(t) x ~ ~ yet) get) h(t) Multiplier h(t) Prefilter Multiplexer (a) (b) Figure 1: Demodulation systems, (a) using multiplication, and (b) multiplexing. frequency of each channel and subsequently lowpass filtered, translating signal components taken around the center frequency towards zero frequency. In particular. we consider wavelet decomposition and reconstruction of analog continuous-time temporal data with a complex Gaussian kernel according to the following formulae: Yk(t) {teo x(e) exp (jWke- Q(Wk(t - e))2) de (decomposition) (1) x'(t) C 2::y\(t) exp(-jwkt) k (reconstruction) where the center frequencies Wk are spaced on a logarithmic scale. The constant Q sets the relative width of the frequency bins in the decomposition, and can be adjusted (together with C) alter the shape of the wavelet kernel. Successive decomposition and reconstruction transforms yield an approximate identity operation; it cannot be exact as no continuous orthonormal basis function exists for the CWT [Kais94]. 2 Architecture The above operations are implemented in [Edwa93] using two demodulator systems per channel, one for the real component of (1), and another for the imaginary component, 90° out of phase with the first. Each takes the form of a sinusoidal modulator oscillating at the channel center frequency, followed by a Gaussian-shaped lowpass filter, as shown in Figure 1 (a). This arrangement requires a precise analog sine wave generator and an accurate linear analog multiplier. In the present implementation, we circumvent both requirements by using an oversampled binary representation of the modulation reference signal. 2.1 Multiplexing vs. MUltiplying Multiplication of an analog signal x(t) with a binary (± 1) sequence is naturally implemented with high precision using a mUltiplexer, which alternates between presenting either the input or its inverse -x(t) to the output. This principle is applied to simplify harmonic modulation. and is illustrated in Figure 1 (b). The multiplier has been replaced by an analog inverter followed by a multiplexer, where the multiplexer is controlled by an oversampled binary periodic sequence representing the sine wave reference. The oversampled binary sequence is chosen to approximate the analog sine wave as closely as possible. disregarding components at high frequency which are removed by the subsequent lowpass filter. The assumption made is that no high frequency components are present in the input signal 694 SiglUll ,---- • ________ ----I In : eLK/: , , , , In Seltrt CLK2 , ' I. ________________ : Reconstruction Input Mult;pl;er R. T. EDWARDS, G. CAUWENBERGHS eLK] f ---<.-iK4 ----1:CLKS---: ;-----------------1 II • I II " E I ' 'I .---f-..., :: : : Ret'onJlructed : : cmLy, 0.,. i " , .--...L..--,:: : I I : ~~--+,~, , '-==:.J : ~ ____ __________ __ J Gaussian Filter Output Muxing Wavelet Reconstruction Figure 2: Block diagram of a single channel in the wavelet processor, showing test points A through E. under modulation, which otherwise would convolve with corresponding high frequency components in the binary sequence to produce low frequency distortion components at the output. To that purpose, an additionallowpass filter is added in front of the multiplexer. Residual low-frequency distortion at the output is minimized by maximizing roll-off of the filters, placing proper constraints on their cutofffrequencies, and optimally choosing the bit sequence in the oversampled reference [Edwa95]. Clearly, the signal accuracy that can be achieved improves as the length N of the sequence is extended. Constraints on the length N are given by the implied overhead in required signal bandwidth, power dissipation, and complexity of implementation. 2.2 Wavelet Gaussian Function The reason for choosing a Gaussian kernel in (l) is to ensure optimal support in both time and frequency [Gros89]. A key requirement in implementing the Gaussian filter is linear phase, to avoid spectral distortion due to non-uniform group delays. A worryfree architecture would be an analog FIR filter; however the number of taps required to accommodate the narrow bandwidth required would be prohibitively large for our purpose. Instead, we approximate a Gaussian filter by cascading several first-order lowpass filters. From probabilistic arguments, the obtained lowpass filter approximates a Gaussian filter increasingly well as the number of stages increases [Edwa93]. 3 Implementation Two sections of a wavelet processor, each containing 8 parallel channels, were integrated onto a single 4 mm x 6 mm die in 2 /lm CMOS technology. Both sections can be configured to perform wavelet decomposition as well as reconstruction. The block diagram for one of the channels is shown in Figure 2. In addition, a separate test chip was designed which performs one channel of the wavelet function. Test points were made available at various points for either input or output, as indicated in boldface capitals, A through E, in Figure 2. Each channel performs complex harmonic modulation and Gaussian lowpass filtering, as defined above. At the front end of the chip is a sample-and-hold section to sample timemultiplexed wavelet signals for reconstruction. In cases of both signal decomposition and reconstruction, each channel removes the input DC component removed, filters the result through the premultiplication lowpass (PML) filter, inverts the result, and passes both non-inverted and inverted signals onto the multiplexer. The multiplexer output is passed through a postmultiplication lowpass filter (PML, same architecture) to remove high frequency components of the oversampled sequence, and then passed through the Gaussianshaped lowpass filter. The cutoff frequencies of all filters are controlled by the clock rates Analog VLSI Processor Implementing the Continuous Wavelet Transform 695 (CLKI to CLK4 in Figure 2). The remainder of the system is for reconstruction and for time-multiplexing the output. 3.1 MUltiplier The multiplier is implemented by use of the above multiplexing scheme, driven by an oversampled binary sequence representing a sine wave. The sequence we used was 256 samples in length, created from a 64-sample base sequence by reversal and inversion. The sequence length of256 generates a modulator wave of 4 kHz (useful for speech applications) from a clock of about 1 MHz. We derived a sequence which, after postfiltering through a 3rd-order lowpass filter of the fonn of the PML prefilter (see below), produces a sine wave in which all hannonics are more than 60 dB down from the primary [Edwa95]. The optimized 64-bit base sequence consists of 11 zeros and 53 ones, allowing a very simple implementation in which an address decoder decodes the "zero" bits. The binary sequence is shown in Figure 4. The magnitude of the prime hannonic of the sequence is approximately 1.02, within 2% of unity. The process of reversing and inverting the sequence is simplified by using a gray code counter to produce the addresses for the sequence, with only a small amount of combinatorial logic needed to achieve the desired result [Edwa95]. It is also straightforward to generate the addresses for the cosine channel, which is 90° out of phase with the original. 3.2 Linear Filtering All filters used are implemented as linear cascades of first-order, single-pole filter sections. The number of first-order sections for the PML filters is 3. The number of sections for the "Gaussian" filter is 8, producing a suitable approximation to a Gaussian filter response for all frequencies of interest (Figure 5). Figure 3 shows one first-order lowpass section of the filters as implemented. This standard >-.+--o va"' v,,, + Figure 3: Single discrete-time lowpass filter section. switched-capacitor circuit implements a transfer function containing a single pole, approximately located in the Laplace domain at s = Is / a for large values of the parameter a, with Is being the sampling frequency. The value for this parameter a is fixed at the design stage as the ratio of two capacitors in Figure 3, and was set to be 15 for the The PML filters and 12 for the Gaussian filters. 4 Measured Results 4.1 Sine wave modulator We tested the accuracy of the sine wave modulation signal by applying two constant voltages at test points A and B, such that the sine wave modulation signal is effectively multiplied 696 R. T. EDWARDS, G. CAUWENBERGHS Sine sequence and filtered sine wave output Binary sine sequence Simulated filtered output x Measured output -1.5 L-___ --' ____ -'-____ ........ ____ -'-____ .J...J o 50 100 150 200 250 Time (us) Figure 4: Filtered sine wave output. by a constant. The output of the mUltiplier is filtered and the output taken at test point D, before the Gaussian filter. Figure 4 shows the (idealized) multiplexer output at test point C, which accurately creates the desired binary sequence. Figure 4 also shows the measured sine wave after filtering with the PML filter and the expected output from the simulation model, using a deviating value of 8.0 for the capacitor ratio a, as justified below. FFT analysis of Figure 4 has shown that the resulting sine wave has all harmonics below about -49 dB. This is in good agreement with the simulation model, provided a correction is made for the value of the capacitor ratio a to account for fringe and (large) parasitic capacitances. The best fit for the measured data from the postmultiplication filter is a = 8.0, compared to the desired value of a = 15.0. The transform of the simulated output shown in the figure takes into account the smaller value of a. Because the postmultiplication filter is followed by the Gaussian filter, the bandwidth of the output can be directly controlled by proper clocking ofthe Gaussian filter, so the distortion in the sine wave is ultimately much smaller than that measured at the output of the postmultiplication filter. 4.2 Gaussian filter The Gaussian filter was tested by applying a signal at test point D and measuring the response at test point E. Figure 5 shows the response of the Gaussian filter as compared to expected responses. There are two sets of curves, one for a filter clocked at 64 kHz, and the other clocked at 128 kHz; these curves are normalized by plotting time relative to the clock frequency is. The solid line indicates the best match for an 8th-order lowpass filter, using the capacitor ratio, a, as a fitting parameter. The best-fit value of a is approximately 6.8. This is again much lower than the capacitor area ratio of 12 on the chip. The dotted line is the response of the ideal Gaussian characteristic exp ( _w 2 / (2aw~)) approximated by the cascade of first-order sections with capacitor ratio a. Figure 5 (b) shows the measured phase response of the Gaussian filter for the 128 kHz clock. The phase response is approximately linear throughout the passband region. Analog VLSI Processor Implementing the Continuous Wavelet Transform 697 Gaussian filter response o~~~~--~-=~~~~~~~--~ x x Chip data at 64kHz clock o 0 Chip data at 128kHz clock iii'-10 ~ <lJ ]1 -20 ~ E « -30 "0 <lJ N ~ -40 § o Z -50 0.01 8th-order filter ideal response Gaussian filter ideal response 0.07 0.08 Frequency (units fs) 500 Theoretical 8-stage phase o Measured response 0.0 I 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Frequency (units fs) Figure 5: Gaussianfilter transfer functions: theoretical and actual. (a) Relative amplitude; (b) Phase. 4.3 Wavelet decomposition Figure 6 shows the test chip performing a wavelet transform on a simple sinusoidal input, illustrating the effects of (oversampled) sinusoidal modulation followed by lowpass filtering through the Gaussian filter. The chip multiplier system is clocked at 500 kHz. The input wave is approximately 3.1 kHz, close to the center frequency of the modulator signal, which is the clock rate divided by 128, or about 3.9 kHz (a typical value for the highestfrequency channel in an auditory application). The top trace in the figure shows the filtered and inverted input, taken from test point B. The middle trace shows the output of the multiplexer (test point C), wherein the output is multiplexed between the signal and its inverse. The bottom trace is taken from the system output (labeled Cosine Out in Figure 2) and shows the demodulated signal of frequency 800 Hz (= 3.9 kHz - 3.1 kHz). Not shown is the cosine output, which is 90° out of phase with the one shown. This demonstrates the proper operation of complex demodulation in a single channel configured for wavelet decomposition. In addition, we have tested the full16-channel chip decomposition, and all individual parts function properly. The total power consumption of the 16-channel wavelet chip was measured to be less than 50mW, of which a large fraction can be attributed to external interfacing and buffering circuitry at the periphery of the chip. 5 Conclusions We have demonstrated the full functionality of an analog chip performing the continuous wavelet transform (decomposition). The chip is based on mixed analog/digital signal processing principles, and uses a demodulation scheme which is accurately implemented using oversampling methods. Advantages of the architecture used in the chip are an increased dynamic range and a precise control over lateral synchronization of wavelet components. An additional advantage inherent to the modulation scheme used is the potential to tune the channel bandwidths over a wide range, down to unusually narrow bands, since the cutoff frequency of the Gaussian filter and the center frequency of the modulator are independently adjustable and precisely controllable parameters. References G. Kaiser, A Friendly Guide to Wavelets, Boston, MA: Birkhauser, 1994. T. Edwards and M. Godfrey, "An Analog Wavelet Transform Chip," IEEE Int'l Can! on 698 R. T. EDWARDS, G. CAUWENBERGHS Figure 6: Scope trace of the wavelet transform: filtered input (top), multiplexed signal (middle), and wavelet output (bottom). Neural Networks, vol. III, 1993, pp. 1247-1251. T. Edwards and G. Cauwenberghs, "Oversampling Architecture for Analog Harmonic Modulation," to appear in Electronics Letters, 1996. A Grossmann, R Kronland-Martinet, and J. MorIet, "Reading and understanding continuous wavelet transforms," Wavelets: Time-Frequency Methods and Phase Space. SpringerVerlag, 1989, pp. 2-20. W. Liu, AG. Andreou, and M.G. Goldstein, "Voiced-Speech Representation by an Analog Silicon Model ofthe Auditory Periphery," IEEE T. Neural Networks, vol. 3 (3), pp 477-487, 1992. J. Lin, W.-H. Ki, T. Edwards, and S. Shamma, "Analog VLSI Implementations of Auditory Wavelet Transforms Using Switched-Capacitor Circuits," IEEE Trans. Circuits and Systems-I, vol.41 (9), pp. 572-583, September 1994. A Lu and W. Roberts, ''A High-Quality Analog Oscillator Using Oversampling D/A Conversion Techniques," IEEE Trans. Circuits and Systems-II, vol.41 (7), pp. 437-444, July 1994. RF. Lyon and C.A Mead, "An Analog Electronic Cochlea," IEEE Trans. Acoustics, Speech and Signal Proc., vol. 36, pp 1119-1134, 1988. H.H. Szu, B. Tefter, and S. Kadembe, "Neural Network Adaptive Wavelets for Signal Representation and Classification," Optical Engineering, vol. 31 (9), pp. 1907-1916, September 1992. L. Watts, D.A Kerns, and RF. Lyon, "Improved Implementation of the Silicon Cochlea," IEEE Journal of Solid-State Circuits, vol. 27 (5), pp 692-700,1992.
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Hierarchical Recurrent Neural Networks for Long-Term Dependencies Salah El Hihi Dept. Informatique et Recherche Operationnelle Universite de Montreal Montreal, Qc H3C-3J7 elhihiGiro.umontreal.ca Yoshua Bengio· Dept. Informatique et Recherche Operationnelle Universite de Montreal Montreal, Qc H3C-3J7 bengioyGiro.umontreal.ca Abstract We have already shown that extracting long-term dependencies from sequential data is difficult, both for determimstic dynamical systems such as recurrent networks, and probabilistic models such as hidden Markov models (HMMs) or input/output hidden Markov models (IOHMMs). In practice, to avoid this problem, researchers have used domain specific a-priori knowledge to give meaning to the hidden or state variables representing past context. In this paper, we propose to use a more general type of a-priori knowledge, namely that the temporal dependencIes are structured hierarchically. This implies that long-term dependencies are represented by variables with a long time scale. This principle is applied to a recurrent network which includes delays and multiple time scales. Experiments confirm the advantages of such structures. A similar approach is proposed for HMMs and IOHMMs. 1 Introduction Learning from examples basically amounts to identifying the relations between random variables of interest. Several learning problems involve sequential data, in which the variables are ordered (e.g., time series). Many learning algorithms take advantage of this sequential structure by assuming some kind of homogeneity or continuity of the model over time, e.g., bX sharing parameters for different times, as in Time-Delay Neural Networks (TDNNs) tLang, WaIbel and Hinton, 1990), recurrent neural networks (Rumelhart, Hinton and Williams, 1986), or hidden Markov models (Rabiner and Juang, 1986). This general a-priori assumption considerably simplifies the learning problem. In previous papers (Bengio, Simard and Frasconi, 1994· Bengio and Frasconi, 1995a), we have shown for recurrent networks and Markovian models that, even with this assumption, dependencies that span longer intervals are significantly harder to learn. In all of the systems we have considered for learning from sequential data, some form of representation of context ( or state) is required (to summarize all "useful" past information). The "hard learning" problem IS to learn to represent context, which involves performing the proper ° also, AT&T Bell Labs, Holmdel, NJ 07733 494 S. E. HIHI. Y. BENGIO credit assignment through time. Indeed, in practice, recurrent networks (e.g., injecting prior knowledge for grammar inference (Giles and Omlin, 1992; Frasconi et al., 1993)) and HMMs (e.g., for speech recognition (Levinson, Rabiner and Sondhi, 1983; Rabiner and Juang, 1986)) work quite well when the representation of context (the meaning of the state variable) is decided a-priori. The hidden variable is not any more completely hidden. Learning becomes much easier. Unfortunately, this requires a very precise knowledge of the appropriate state variables, which is not available in many applications. We have seen that the successes ofTDNNs, recurrent networks and HMMs are based on a general assumption on the sequential nature of the data. In this paper, we propose another, simple, a-priori assumption on the sequences to be analyzed: the temporal dependencies have a hierarchical structure. This implies that dependencies spanning long intervals are "robust" to small local changes in the timing of events, whereas dependencies spanning short intervals are allowed to be more sensitive to the precise timing of events. This yields a multi-resolution representation of state information. This general idea is not new and can be found in various approaches to learning and artificial intelligence. For example, in convolutional neural networks, both for sequential data with TDNNs (Lang, Waibel and Hinton, 1990), and for 2-dimensional data with MLCNNs (LeCun et al., 1989; Bengio, LeCun and Henderson, 1994), the network is organized in layers representing features of increasing temporal or spatial coarseness. Similarly, mostly as a tool for analyzing and preprocessing sequential or spatial data, wavelet transforms (Daubechies, 1990) also represent such information at mUltiple resolutions. Multi-scale representations have also been proposed to improve reinforcement learning systems (Singh, 1992; Dayan and Hinton, 1993; Sutton, 1995) and path planning systems. However, with these algorithms, one generally assumes that the state of the system is observed, whereas, in this paper we concentrate on the difficulty of learning what the state variable should represent. A related idea using a hierarchical structure was presented in (Schmidhuber, 1992). On the HMM side, several researchers (Brugnara et al., 1992; Suaudeau, 1994) have attempted to improve HMMs for speech recognition to better model the different types of var1ables, intrmsically varying at different time scales in speech. In those papers, the focus was on setting an a-priori representation, not on learning how to represent context. In section 2, we attempt to draw a common conclusion from the analyses performed on recurrent networks and HMMs to learn to represent long-term dependencies. This will justify the proposed approach, presented in section 3. In section 4 a specific hierarchical model is proposed for recurrent networks, using different time scales for different layers of the network. EXp'eriments performed with this model are described in section 4. Finally, we discuss a sim1lar scheme for HMMs and IOHMMs in section 5. 2 Too Many Products In this section, we take another look at the analyses of (Bengio, Simard and Frasconi, 1994) and (Bengio and Frasconi, 1995a), for recurrent networks and HMMs respectively. The objective 1S to draw a parallel between the problems encountered with the two approaches, in order to guide us towards some form of solution, and justify the proposals made here. First, let us consider the deterministic dynamical systems (Bengio, Simard and Frasconi, 1994) (such as recurrent networks), which map an input sequence U1 l . .. , UT to an output sequence Y1, ... , ftr· The state or context information is represented at each time t by a variable Xt, for example the activities of all the hidden units of a recurrent network: (1) where Ut is the system input at time t and 1 is a differentiable function (such as tanh(Wxt_1 + ut)). When the sequence of inputs U1, U2, • .. , UT is given, we can write Xt = It(Xt-d = It(/t-1( .. . l1(xo)) . .. ). A learning criterion Ct yields gradients on outputs, and therefore on the state variables Xt. Since parameters are shared across time, learning using a gradient-based algorithm depends on the influence of parameters W on Ct through an time steps before t : aCt _ " aCt OXt oXT oW L...J OXt OX T oW T (2) Hierarchical Recurrent Neural Networks for Long-term Dependencies 495 The Jacobian matrix of derivatives .!!:U{J{Jx can further be factored as follows: Xr (3) Our earlier analysis (Bengio, Simard and Frasconi, 1994) shows that the difficulty revolves around the matrix product in equation 3. In order to reliably "store" informatIOn in the dynamics of the network, the state variable Zt must remain in regions where If:! < 1 (i.e., near enough to a stable attractor representing the stored information). However, the above products then rapidly converge to 0 when t T increases. Consequently, the sum in 2 is dominated by terms corresponding to short-term dependencies (t T is small). Let us now consider the case of Markovian models (including HMMs and IOHMMs (Bengio and Frasconi, 1995b)). These are probabilistic models, either of an "output" sequence P(YI . . . YT) (HMMs) or of an output sequence given an input sequence P(YI ... YT lUI ... UT) (IOHMMs). Introducing a discrete state variable Zt and using Markovian assumptIOns of independence this probability can be factored in terms of transition probabilities P(ZtIZt-d (or P(ZtIZt-b ut}) and output probabilities P(ytlZt) (or P(ytiZt, Ut)) . According to the model, the distribution of the state Zt at time t given the state ZT at an earlier time T is given by the matrix P(ZtlZT) = P(ZtiZt-I)P(Zt-Ilzt-2) . .. P(zT+dzT) (4) where each of the factors is a matrix of transition probabilities (conditioned on inputs in the case of IOHMMs) . Our earlier analysis (Bengio and Frasconi, 1995a) shows that the difficulty in representing and learning to represent context (i.e., learning what Zt should represent) revolves around equation 4. The matrices in the above equations have one eigenvalue equal to 1 (because of the normalization constraint) and the others ~ 1. In the case in which all eIgenvalues are 1 the matrices have only i's and O's, i.e, we obtain deterministic dynamics for IOHMMs or pure cycles for HMMs (which cannot be used to model most interesting sequences). Otherwise the above product converges to a lower rank matrix (some or most of the eigenvalues converge toward 0). Consequently, P(ZtlZT) becomes more and more independent of ZT as t - T increases. Therefore, both representing and learning context becomes more difficult as the span of dependencies increases or when the Markov model is more non-deterministic (transition probabilities not close to 0 or 1). Clearly, a common trait of both analyses lies in taking too many products, too many time steps, or too many transformations to relate the state variable at time T with the state variable at time t > T, as in equations 3 and 4. Therefore the idea presented in the next section is centered on allowing several paths between ZT and Zt, some with few "transformations" and some with many transformations. At least through those with few transformations, we expect context information (forward), and credit assignment (backward) to propagate more easily over longer time spans than through "paths" lDvolving many tralIBformations. 3 Hierarchical Sequential Models Inspired by the above analysis we introduce an assumption about the sequential data to be modeled, although it will be a very simple and general a-priori on the structure of the data. Basically, we will assume that the sequential structure of data can be described hierarchically: long-term dependencies (e.g., between two events remote from each other in time) do not depend on a precise time scale (Le., on the precise timing of these events). Consequently, in order to represent a context variable taking these long-term dependencies into account, we will be able to use a coarse time scale (or a Slowly changing state variable). Therefore, instead of a single homogeneous state variable, we will introduce several levels of state variables, each "working" at a different time scale. To implement in a discretetime system such a multi-resolution representation of context, two basic approaches can be considered. Either the higher level state variables change value less often or they are constrained to change more slowly at each time step. In our ex~eriments, we have considered input and output variables both at the shortest time scale highest frequency), but one of the potential advantages of the approach presented here is t at it becomes very 496 S. E. IDHI, Y. BENOIO Figure 1: Four multi-resolution recurrent architectures used in the experiments. Small sguares represent a discrete delay, and numbers near each neuron represent its time scale. The architectures B to E have respectively 2, 3, 4, and 6 time scales. simple to incorporate input and output variables that operate at different time scales. For example, in speech recognition and synthesis, the variable of interest is not only the speech signal itself (fast) but also slower-varying variables such as prosodic (average energy, pitch, etc ... ) and phonemic (place of articulation, phoneme duration) variables. Another example is in the application of learning algorithms to financial and economic forecasting and decision taking. Some of the variables of interest are given daily, others weekly, monthly, etc ... 4 Hierarchical Recurrent Neural Network: Experiments As in TDNNs (Lang, Waibel and Hinton, 1990) and reverse-TDNNs (Simard and LeCun, 1992), we will use discrete time delays and subsampling (or oversampling) in order to implement the multiple time scales. In the time-unfolded network, paths going through the recurrences in the slow varying units (long time scale) will carry context farther, while paths going through faster varying units (short time scale) will respond faster to changes in input or desired changes in output. Examples of such multi-resolution recurrent neural networks are shown in Figure 1. Two sets of simple experiments were performed to validate some of the ideas presented in this paper. In both cases, we compare a hierarchical recurrent network with a single-scale fully-connected recurrent network. In the first set of experiments, we want to evaluate the performance of a hierarchical recurrent network on a problem already used for studying the difficulty in learning longterm dependencies (Bengio, Simard and Frasconi, 1994; Bengio and Frasconi, 1994). In this 2-class J?roblem, the network has to detect a pattern at the beginning of the sequence, keeping a blt of information in "memory" (while the inputs are noisy) until the end of the sequence (supervision is only a the end of the sequence). As in (Bengio, Simard and Frasconi, 1994; Bengio and Frasconi, 1994) only the first 3 time steps contain information about the class (a 3-number pattern was randomly chosen for each class within [-1,1]3). The length of the sequences is varied to evaluate the effect of the span of input/output dependencies. Uniformly distributed noisy inputs between -.1 and .1 are added to the initial patterns as well as to the remainder of the sequence. For each sequence length, 10 trials were run with different initial weights and noise patterns, with 30 training sequences. Experiments were performed with sequence of lengths 10, 20,40 and 100. Several recurrent network architectures were compared. All were trained with the same algorithm (back-propagation through time) to minimize the sum of squared differences between the final output and a desired value. The simplest architecture (A) is similar to architecture B in Figure 1 but it is not hierarchical: it has a single time scale. Like the Hierarchical Recurrent Neural Networks for Long-term Dependencies eo 50 40 l ~ 30 20 ABCDE ABCDE ABCDE ABCDE seq.1engIh 10 20 40 100 1.4 1.3 1.2 1.1 1.0 0.9 Is 0.8 Ii 0.7 I 0.6 0.5 0.4 ~~ '-----'-I....L.~~~....L.-.W....L.mL.......J....I.ll ABCDE ABCDE ABCDE ABCDE 1IIq. 1engIh 10 20 40 100 497 Figure 2: Average classification error after training for 2-sequence problem (left, classification error) and network-generated data (right, mean squared error), for varying sequence lengths and architectures. Each set of 5 consecutive bars represents the performance of 5 architectures A to E, with respectively 1, 2, 3, 4 and 6 time scales (the architectures B to E are shown in Figure 1). Error bars show the standard deviation over 10 trials. other networks, it has however a theoretically "sufficient" architecture, i.e., there exists a set of weights for which it classifies perfectly the trainin~ sequences. Four of the five architectures that we compared are shown in Figure 1, wIth an increasing number of levels in the hierarchy. The performance of these four architectures (B to E) as well as the architecture with a single time-scale (A) are compared in Figure 2 (left, for the 2sequence problem). Clearly, adding more levels to the hierarchy has significantly helped to reduce the difficulty in learning long-term dependencies. In a second set of experiments, a hierarchical recurrent network with 4 time scales was initialized with random (but large) weights and used to generate a data set. To generate the inputs as well as the outputs, the network has feedback links from hidden to input units. At the initial time step as well as at 5% of the time steps (chosen randomly), the input was clamped with random values to introduce some further variability. It is a regression task, and the mean squared error is shown on Figure 2. Because of the network structure, we expect the data to contain long-term dependencies that can be modeled with a hierarchical structure. 100 training sequences of length 10, 20,40 and 100 were generated by this network. The same 5 network architectures as in the previous experiments were compared (see Figure 1 for architectures B to E), with 10 training trials per network and per sequence length. The results are summarized in Figure 2 (right). More high-level hierarchical structure appears to have improved performance for long-term dependencies. The fact that the simpler I-level network does not achieve a good performance suggests that there were some difficult long-term dependencies in the the artificially generated data set. It is interesting to compare those results with those reported in (Lin et al., 1995) which show that using longer delays in certain recurrent connections helps learning longer-term dependencies. In both cases we find that introducing longer time scales allows to learn dependencies whose span is proportionally longer. 5 Hierarchical HMMs How do we represent multiple time scales with a HMM? Some solutions have already been proposed in the speech recognition literature, motivated by the obvious presence of different time scales in the speech phenomena. In (Brugnara et al., 1992) two Markov chains are coupled in a "master/slave" configuration. For the "master" HMM, the observations are slowly varying features (such as the signal energy), whereas for the "slave" HMM the observations are t.he speech spectra themselves. The two chains are synchronous and operate at the same time scale, therefore the problem of diffusion of credit in HMMs would probably also make difficult the learning of long-term dependencies. Note on the other 498 S. E. HIHI, Y. BENOIO hand that in most applications of HMMs to speech recognition the meaning of states is fixed a-priori rather than learned from the data (see (Bengio and Frasconi, 1995a) for a discussion). In a more recent contribution, Nelly Suaudeau (Suaudeau, 1994) proposes a "two-level HMM" in which the higher level HMM represents "segmental" variables (such as phoneme duration). The two levels operate at different scales: the higher level state varIable represents the phonetic identity and models the distributions of the average energy and the duration within each phoneme. Again, this work is not geared towards learning a representation of context, but rather, given the traditional (phoneme-based) representation of context in speech recognition, towards building a better model of the distribution of "slow" segmental variables such as phoneme duration and energy. Another promising approach was recently proposed in (Saul and Jordan, 1995). Using decimation techniques from statistical mechanics, a polynomial-time algorithm is derived for parallel Boltzmann chains (which are similar to parallel HMMs), which can operate at different time scales. The ideas presented here point toward a HMM or IOHMM in which the (hidden) state variable Xt is represented by the Cartesian product of several state variables Xt, each "working" at a different time scale: Xt = (x;, x~, ... I xf).. To take advantage of the decomposition, we propose to consider that tbe state dIstrIbutions at the different levels are conditionally independent (given the state at the previous time step and at the current and previous levels). Transition probabilities are therefore factored as followed: (5) To force the state variable at a each level to effectively work at a given time scale, selftransition probabilities are constrained as follows (using above independence assumptions): P(x:=i3Ixt_l=iI,.· ., x:_l=i3" .. , xt-l=is) = P(x:=i3Ix:_1 =i3, X::t=i3-d = W3 6 Conclusion Motivated by the analysis of the problem of learning long-term dependencies in sequential data, i.e., of learning to represent context, we have proposed to use a very general assumption on the structure of sequential data to reduce the difficulty of these learning tasks. Following numerous previous work in artificial intelligence we are assuming that context can be represented with a hierarchical structure. More precisely, here, it means that long-term dependencies are insensitive to small timing variations, i.e., they can be represented with a coarse temporal scale. This scheme allows context information and credit information to be respectively propagated forward and backward more easily. Following this intuitive idea, we have proposed to use hierarchical recurrent networks for sequence processing. These networks use multiple-time scales to achieve a multi-resolution representation of context. Series of experiments on artificial data have confirmed the advantages of imposing such structures on the network architecture. Finally we have proposed a similar application of this concept to hidden Markov models (for density estimation) and input/output hidden Markov models (for classification and regression). References Bengio, Y. and Frasconi, P. (1994). Credit assignment through time: Alternatives to backpropagation. In Cowan, J., Tesauro, G., and Alspector, J., editors, Advances in Neural Information Processing Systems 6. Morgan Kaufmann. Bengio, Y. and Frasconi, P. (1995a). Diffusion of context and credit information in markovian models. Journal of Artificial Intelligence Research, 3:223-244. Bengio, Y. and Frasconi, P. (1995b). An input/output HMM architecture. In Tesauro, G., Touretzky, D., and Leen, T., editors, Advances in Neural Information Processmg Systems 7, pages 427-434. MIT Press, Cambridge, MA. Bengio, Y., LeCun, Y., and Henderson, D. (1994). Globally trained handwritten word recognizer using spatial representation, space displacement neural networks and hidden Markov models. In Cowan, J ., Tesauro, G., and Alspector, J., editors, Advances in Neural Information Processing Systems 6, pages 937- 944. Hierarchical Recurrent Neural Networks for Long-term Dependencies 499 Bengio, Y., Simard, P., and Frasconi, P. (1994). Learning long-term dependencies with gradient descent is difficult. IEEE Transactions on Neural Networks, 5(2):157-166. Brugnara, F., DeMori, R, Giuliani, D., and Omologo, M. (1992). A family of parallel hidden markov models. In International Conference on Acoustics, Speech and Signal Processing, pages 377-370, New York, NY, USA. IEEE. Daubechies, I. (1990). The wavelet transform, time-frequency localization and signal analysis. IEEE Transaction on Information Theory, 36(5):961-1005 . Dayan, P. and Hinton, G. (1993). Feudal reinforcement learning. In Hanson, S. J., Cowan, J. D., and Giles, C. L., edItors, Advances in Neural Information Processing Systems 5, San Mateo, CA. Morgan Kaufmann. Frasconi, P., Gori, M., Maggini, M., and Soda, G. (1993). Unified integration of explicit rules and learning by example in recurrent networks. IEEE Transactions on Knowledge and Data Engineering. (in press). Giles, C. 1. and amlin, C. W. (1992). Inserting rules into recurrent neural networks. In Kung, Fallside, Sorenson, and Kamm, editors, Neural Networks for Signal Processing II, Proceedings of the 1992 IEEE workshop, pages 13-22. IEEE Press. Lang, K. J., Waibel, A. H., and Hinton, G. E. (1990). A time-delay neural network architecture for isolated word recognition. Neural Networks, 3:23-43. LeCun, Y., Boser, B., Denker, J., Henderson, D., Howard, R, Hubbard, W., and Jackel, L. (1989) . Backpropagation applied to handwritten zip code recognition. Neural Computation, 1:541-551. Levinson, S., Rabiner, 1., and Sondhi, M. (1983). An introduction to the application ofthe theory of probabilistic functions of a Markov process to automatic speech recognition. Bell System Technical Journal, 64(4):1035-1074. Lin, T ., Horne, B., Tino, P., and Giles, C. (1995). Learning long-term dependencies is not as difficult with NARX recurrent neural networks. Techmcal Report UMICAS-TR95-78, Institute for Advanced Computer Studies, University of Mariland. Rabiner, L. and Juang, B. (1986). An introduction to hidden Markov models. IEEE A SSP Magazine, pages 257-285. Rumelhart, D., Hinton, G., and Williams, R (1986). Learning internal representations by error propagation. In Rumelhart, D. and McClelland, J., editors, Parallel Distributed Processing, volume 1, chapter 8, pages 318-362. MIT Press, Cambridge. Saul, L. and Jordan, M. (1995). Boltzmann chains and hidden markov models. In Tesauro, G., Touretzky, D., and Leen, T., editor~ Advances in Neural Information Processing Systems 7, pages 435--442. MIT Press, vambridge, MA. Schmidhuber, J. (1992). Learning complex, extended sequences using the principle of history compression. Neural Computation, 4(2):234-242. Simard, P. and LeCun, Y. (1992). Reverse TDNN: An architecture for trajectory generation. In Moody, J., Hanson, S., and Lipmann, R, editors, Advances in Neural Information Processing Systems 4, pages 579-588, Denver, CO. Morgan Kaufmann, San Mateo. Singh, S. (1992). Reinforcement learning with a hierarchy of abstract models. In Proceedings of the 10th National Conference on Artificial Intelligence, pages 202-207. MIT / AAAI Press. Suaudeau, N. (1994). Un modele probabiliste pour integrer la dimension temporelle dans un systeme de reconnaissance automatique de la parole. PhD thesis, Universite de Rennes I, France. Sutton, RjI995). TD models: modeling the world at a mixture of time scales. In Proceedings 0 the 12th International Conference on Machine Learning. Morgan Kaufmann.
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Finite State Automata that Recurrent Cascade-Correlation Cannot Represent Stefan C. Kremer Department of Computing Science University of Alberta Edmonton, Alberta, CANADA T6H 5B5 Abstract This paper relates the computational power of Fahlman' s Recurrent Cascade Correlation (RCC) architecture to that of fInite state automata (FSA). While some recurrent networks are FSA equivalent, RCC is not. The paper presents a theoretical analysis of the RCC architecture in the form of a proof describing a large class of FSA which cannot be realized by RCC. 1 INTRODUCTION Recurrent networks can be considered to be defmed by two components: a network architecture, and a learning rule. The former describes how a network with a given set of weights and topology computes its output values, while the latter describes how the weights (and possibly topology) of the network are updated to fIt a specifIc problem. It is possible to evaluate the computational power of a network architecture by analyzing the types of computations a network could perform assuming appropriate connection weights (and topology). This type of analysis provides an upper bound on what a network can be expected to learn, since no system can learn what it cannot represent. Many recurrent network architectures have been proven to be fInite state automaton or even Turing machine equivalent (see for example [Alon, 1991], [Goudreau, 1994], [Kremer, 1995], and [Siegelmann, 1992]). The existence of such equivalence proofs naturally gives confIdence in the use of the given architectures. This paper relates the computational power of Fahlman's Recurrent Cascade Correlation architecture [Fahlman, 1991] to that of fInite state automata. It is organized as follows: Section 2 reviews the RCC architecture as proposed by Fahlman. Section 3 describes fInite state automata in general and presents some specifIc automata which will play an important role in the discussions which follow. Section 4 describes previous work by other Finite State Automata that Recurrent Cascade-Correlation Cannot Represent 613 authors evaluating RCC' s computational power. Section 5 expands upon the previous work, and presents a new class of automata which cannot be represented by RCC. Section 6 further expands the result of the previous section to identify an infinite number of other unrealizable classes of automata. Section 7 contains some concluding remarks. 2 THE RCC ARCHITECTURE The RCC architecture consists of three types of units: input units, hidden units and output units. After training, a RCC network performs the following computation: First, the activation values of the hidden units are initialized to zero. Second, the input unit activation values are initialized based upon the input signal to the network. Third, each hidden unit computes its new activation value. Fourth, the output units compute their new activations. Then, steps two through four are repeated for each new input signal. The third step of the computation, computing the activation value of a hidden unit, is accomplished according to the formula: a(t+l) = a( t W.a(t+l) + w.a(t)]. J ;=\ 'J' JJ J Here, ai(t) represents the activation value of unit i at time t, a(e) represents a sigmoid squashing function with fInite range (usually from 0 to 1), and Wij represents the weight of the connection from unit ito unitj. That is, each unit computes its activation value by mUltiplying the new activations of all lowered numbered units and its own previous activation by a set of weights, summing these products, and passing the sum through a logistic activation function. The recurrent weight Wjj from a unit to itself functions as a sort of memory by transmitting a modulated version of the unit's old activation value. The output units of the RCC architecture can be viewed as special cases of hidden units which have weights of value zero for all connections originating from other output units. This interpretation implies that any restrictions on the computational powers of general hidden units will also apply to the output units. For this reason, we shall concern ourselves exclusively with hidden units in the discussions which follow. Finally, it should be noted that since this paper is about the representational power of the RCC architecture, its associated learning rule will not be discussed here. The reader wishing to know more about the learning rule, or requiring a more detailed description of the operation of the RCC architecture, is referred to [Fahlman, 1991]. 3 FINITE STATE AUTOMATA A Finite State Automaton (FSA) [Hopcroft, 1979] is a formal computing machine defmed by a 5-tuple M=(Q,r.,8,qo,F), where Q represents a fmite set of states, r. a fmite input alphabet, 8 a state transition function mapping Qxr. to Q, qoEQ the initial state, and FcQ a set of fmal or accepting states. FSA accept or reject strings of input symbols according to the following computation: First, the FSA' s current state is initialized to qo. Second, the next inut symbol of the str ing, selected from r., is presented to the automaton by the outside world. Third, the transition function, 8, is used to compute the FSA' s new state based upon the input symbol, and the FSA's previous state. Fourth, the acceptability of the string is computed by comparing the current FSA state to the set of valid fmal states, F. If the current state is a member of F then the automaton is said to accept the string of input symbols presented so far. Steps two through four are repeated for each input symbol presented by the outside world. Note that the steps of this computation mirror the steps of an RCC network's computation as described above. It is often useful to describe specifIc automata by means of a transition diagram [Hopcroft, 1979]. Figure 1 depicts the transition diagrams of fIve FSA. In each case, the states, Q, 614 S.C. KREMER are depicted by circles, while the transitions defmed by 0 are represented as arrows from the old state to the new state labelled with the appropriate input symbol. The arrow labelled "Start" indicates the initial state, qo; and fmal accepting states are indicated by double circles. We now defme some terms describing particular FSA which we will require for the following proof. The first concerns input signals which oscillate. Intuitively, the input signal to a FSA oscillates if every pm symbol is repeated for p> 1. More formally, a sequence of input symbols, s(t), s(t+ 1), s(t+ 2), ... , oscillates with a period of p if and only if p is the minimum value such that: Vt s(t)=s(t+p). Our second definition concerns oscillations of a FSA's internal state, when the machine is presented a certain sequence of input signals. Intuitively, a FSA' s internal state can oscillate in response to a given input sequence if there is some starting state for which every subsequent <.>th state is repeated. Formally, a FSA' s state can oscillate with a period of <.> in response to a sequence of input symbols, s(t), s(t+ 1), s(t+2), ... , if and only if <.> is the minimum value for which: 3qo S.t. Vt o(qo, s(t» = o( .. , o( o( o(qo, s(t», s(t+ 1», s(t+2», ... , s(t+<.>)) The recursive nature of this formulation is based on the fact a FSA' s state depends on its previous state, which in tum depends on the state before, etc .. We can now apply these two defmitions to the FSA displayed in Figure 1. The automaton labelled "a)" has a state which oscillates with a period of <.>=2 in response to any sequence consisting of Os and Is (e.g. "00000 ... ", "11111.. .. ", "010101. .. ", etc.). Thus, we can say that it has a state cycle of period <.>=2 (Le. qoqtqoqt ... ), when its input cycles with a period of p= 1 (Le. "0000 ... If). Similarly, when automaton b)'s input cycles with period p= 1 (Le. ''000000 ... "), its state will cycle with period <.>=3 (Le. qOqtq2qOqtq2' .. ). For automaton c), things are somewhat more complicated. When the input is the sequence "0000 .. . ", the state sequence will either be qoqo%qo .. ' or fA fA fA fA .. . depending on the initial state. On the other hand, when the input is the sequence "1111 ... ", the state sequence will alternate between qo and qt. Thus, we say that automaton c) has a state cycle of <.> = 2 when its input cycles with period p = 1. But, this automaton can also have larger state cycles. For example, when the input oscillates with a period p=2 (Le. "01010101. .. If), then the state of the automaton will oscillate with a period <.>=4 (Le. qoqoqtqtqoqoqtqt ... ). Thus, we can also say that automaton c) has a state cycle of <.>=4 when its input cycles with period p =2. The remaining automata also have state cycles for various input cycles, but will not be discussed in detail. The importance of the relationship between input period (P) and the state period (<.» will become clear shortly. 4 PREVIOUS RESULTS CONCERNING THE COMPUTATIONAL POWEROFRCC The first investigation into the computational powers of RCC was performed by Giles et. al. [Giles, 1995]. These authors proved that the RCC architecture, regardless of connection weights and number of hidden units, is incapable of representing any FSA which "for the same input has an output period greater than 2" (p. 7). Using our oscillation defmitions above, we can re-express this result as: if a FSA' s input oscillates with a period of p= 1 (Le. input is constant), then its state can oscillate with a period of at most <.>=2. As already noted, Figure Ib) represents a FSA whose state oscillates with a period of <.>=3 in response to an input which oscillates with a period of p=1. Thus, Giles et. al.'s theorem proves that the automaton in Figure Ib) cannot be implemented (and hence learned) by a RCC network. Finite State Automata that Recurrent Cascade-Correlation Cannot Represent 615 a) Start b) Start 0, I c) Start d) Start o o o o o e) Start Figure I: Finite State Automata. Giles et. al. also examined the automata depicted in Figures la) and lc). However, unlike the formal result concerning FSA b), the authors' conclusions about these two automata were of an empirical nature. In particular, the authors noted that while automata which oscillated with a period of 2 under constant input (Le. Figure la» were realizable, the automaton of Ic) appeared not be be realizable by RCC. Giles et. al. could not account for this last observation by a formal proof. 616 S.C.KREMER 5 AUTOMATA WITH CYCLES UNDER ALTERNATING INPUT We now turn our attention to the question: why is a RCC network unable to learn the automaton of lc)? We answer this question by considering what would happen if lc) were realizable. In particular, suppose that the input units of a RCC network which implements automaton lc) are replaced by the hidden units of a RCC network implementing la). In this situation, the hidden units of la) will oscillate with a period of 2 under constant input. But if the inputs to lc) oscillate with a period of 2, then the state of Ic) will oscillate with a period of 4. Thus, the combined network's state would oscillate with a period of four under constant input. Furthermore, the cascaded connectivity scheme of the RCC architecture implies that a network constructed by treating one network's hidden units as the input units of another, would not violate any of the connectivity constraints of RCC. In other words, if RCC could implement the automaton of lc), then it would also be able to implement a network which oscillates with a period of 4 under constant input. Since Giles et. al. proved that the latter cannot be the case, it must also be the case that RCC cannot implement the automaton of lc). The line of reasoning used here to prove that the FSA of Figure lc) is unrealizable can also be applied to many other automata. In fact, any automaton whose state oscillates with a period of more than 2 under input which oscillates with a period 2, could be used to construct one of the automata proven to be illegal by Giles. This implies that RCC cannot implement any automaton whose state oscillates with a period of greater than <.>=2 when its input oscillates with a period of p=2. 6 AUTOMATA WITH CYCLES UNDER OSCILLATING INPUT Giles et. aI.' s theorem can be viewed as defining a class of automata which cannot be implemented by the RCC architecture. The proof in Section 5 adds another class of automata which also cannot be realized. More precisely, the two proofs concern inputs which oscillate with periods of one and two respectively. It is natural to ask whether further proofs for state cycles can be developed when the input oscillates with a period of greater than two. We now present the central theorem of this paper, a unified defmition of unrealizable automata: Theorem: If the input signal to a RCC network oscillates with a period, p, then the network can represent only those FSA whose outputs form cycles of length <.>, where pmod<.>=O if p is even and 2pmod<.> =0 if p is odd. To prove this theorem we will first need to prove a simpler one relating the rate of oscillation of the input signal to one node in an RCC network to the rate of oscillation of that node's output signal. By "the input signal to one node" we mean the weighted sum of all activations of all connected nodes (Le. all input nodes, and all lower numbered hidden nodes), but not the recurrent signal. I. e . : j - I A(t+ 1) == " W .. a .(t+ 1) . L.J IJ I 1=1 Using this defmition, it is possible to rewrite the equation to compute the activation of node j (given in Section 2) as: ap+l) == a( A(t+l)+Wha/t) ) . But if we assume that the input signal oscillates with a period of p, then every value of A(t+ 1) can be replaced by one of a fmite number of input signals (.to, AI, A 2, .,. Ap. I ) . In other words, A(t+ 1) = Atmodp ' Using this substitution, it is possible to repeatedly expand the addend of the previous equation to derive the formula: ap+ 1) = a( Atmodp + '")j . a( A(t-I)modp + Wp . a( A(t-2)modp + '")j .... a( A(t-p+I)modp + ,")/ait-p+ 1) ) ... ) ) ) Finite State Automata that Recurrent Cascade-Correlation Cannot Represent 617 The unravelling of the recursive equation now allows us to examine the relationship between ap+ 1) and t;(t-p+ 1). Specifically, we note that if ~ >0 or if p is even then aj{t+ 1) = ft.ap-p+ 1» implies that/is a monotonically increasing function. Furthermore, since 0' is a function with finite range,f must also have finite range. It is well known that for any monotonically increasing function with [mite range, /, the sequence, ft.x), fif(x» , fift.j{x») , ... , is guaranteed to monotonically approach a fixed point (whereft.x)=x). This implies that the sequence, ap+l), t;(t+p+l), q(t+2p+l), ... , must also monotonically approach a fixed point (where ap+ 1) = q.(t-p+ 1». In other words, the sequence does not oscillate. Since every prh value of ~{t) approaches a fixed point, the sequence ap), ap+ 1), ap+2), '" can have a period of at most p, and must have a period which divides p evenly. We state this as our first lemma: Lemma 1: If A.(t) oscillates with even period, p, or if Wu > 0, then state unit j's activation value must oscillate with a period c..>, where pmodc..> =0. We must now consider the case where '"11 < 0 and p is odd. In this case, ap+ 1) = ft.ap-p+ 1» implies that/is a monotonically decreasing function. But, in this situation the function/ 2(x)=ft.f{x» must be monotonically increasing with finite range. This implies that the sequence: ap+ 1), a;<t+2p+ 1), a;<t+4p+ 1), ... , must monotonically approach a fixed point (where a;<t+ 1)=ap-2p+ 1». This in turn implies that the sequence ap), ap+ 1), ap+2), ... , can have a period of at most 2p, and must have a period which divides 2p evenly. Once again, we state this result in a lemma: Lemma 2: If A.(t) oscillates with odd period p, and if Wii<O, then state unit j must oscillate with a period c..>, where 2pmodc..>=0. Lemmas 1 and 2 relate the rate of oscillation of the weighted sum of input signals and lower numbered unit activations, A.(t) to that of unitj. However, the theorem which we wish to prove relates the rate of oscillation of only the RCC network's input signal to the entire hidden unit activations. To prove the theorem, we use a proof by induction on the unit number, i: Basis: Node i= 1 is connected only to the network inputs. Therefore, if the input signal oscillates with period p, then node i can only oscillate with period c..>, where pmodc..> =0 if P is even and 2pmodc..> =0 if P is odd. (This follows from Lemmas 1 and 2). Assumption: If the input signal to the network oscillates with period p, then node i can only oscillate with period c..>, where pmodc..> =0 if p is even and 2pmodc..>=0 if p is odd. Proof: If the Assumption holds for all nodes i, then Lemmas 1 and 2 imply that it must also hold for node i+ 1.0 This proves the theorem: Theorem: If the input signal to a RCC network oscillates with a period, p, then the network can represent only those FSA whose outputs form cycles of length c..>, where pmodc..>=O ifp is even and 2pmodc..> =0 ifp is odd. 7 CONCLUSIONS It is interesting to note that both Giles et. al. 's original proof and the constructive proof by contradiction described in Section 5 are special cases of the theorem. Specifically, Giles et. al. I S original proof concerns input cycles of length p = 1. Applying the theorem of Section 6 proves that an RCC network can only represent those FSA whose state transitions form cycles of length c..>, where 2(I)modc..>=0, implying that state cannot oscillate with a period of greater than 2. This is exactly what Giles et. al concluded, and proves that (among others) the automaton of Figure Ib) cannot be implemented by RCC. 618 S.C.KREMER Similarly, the proof of Section 5 concerns input cycles of length p=2. Applying our theorem proves that an RCC network can only represent those machines whose state transitions form cycles of length <.>, where (2)modw=O. This again implies that state cannot oscillate with a period greater than 2, which is exactly what was proven in Section 5. This proves that the automaton of Figure lc) (among others) cannot be implemented by RCC. In addition to unifying both the results of Giles et. al. and Section 5, the theorem of Section 6 also accounts for many other FSA which are not representable by RCC. In fact, the theorem identifies an inflnite number of other classes of non-representable FSA (for p = 3, P =4, P = 5, ... ). Each class itself of course contains an infinite number of machines. Careful examination of the automaton illustrated in Figure ld) reveals that it contains a state cycle of length 9 (QcIJ.IQ2QIQ2Q3Q2Q3Q4QcIJ.IQ2Qlq2q3q2q3q4"') in response to an input cycle of length 3 ("001001... "). Since this is not one of the allowable input/state cycle relationships defined by the theorem, it can be concluded that the automaton of Figure Id) (among others) cannot be represented by RCC. Finally, it should be noted that it remains unknown if the classes identified by this paper IS theorem represent the complete extent of RCC's computational limitations. Consider for example the automaton of Figure Ie). This device has no input/state cycles which violate the theorem, thus we cannot conclude that it is unrepresentable by RCC. Of course, the issue of whether or not this particular automaton is representable is of little interest. However, the class of automata to which the theorem does not apply, which includes automaton Ie), requires further investigation. Perhaps all automata in this class are representable; perhaps there are other subclasses (not identified by the theorem) which RCC cannot represent. This issue will be addressed in future work. References N. Alon, A. Dewdney, and T. Ott, Efficient simulation of flnite automata by neural nets, Journal of the Association for Computing Machinery, 38 (2) (1991) 495-514. S. Fahlman, The recurrent cascade-correlation architecture, in: R. Lippmann, J. Moody and D. Touretzky, Eds., Advances in Neural Information Processing Systems 3 (Morgan Kaufmann, San Mateo, CA, 1991) 190-196. C.L. Giles, D. Chen, G.Z. Sun, H.H. Chen, Y.C. Lee, and M.W. Goudreau, Constructive Learning of Recurrent Neural Networks: Limitations of Recurrent Cascade Correlation and a Simple Solution, IEEE Transactions on Neural Networks, 6 (4) (1995) 829-836. M. Goudreau, C. Giles, S. Chakradhar, and D. Chen, First-order v.S. second-order single layer recurrent neural networks, IEEE Transactions on Neural Networks, 5 (3) (1994) 511513. J.E. Hopcroft and J.D. Ullman, Introduction to Automata Theory, Languages and Computation (Addison-Wesley, Reading, MA, 1979). S.C. Kremer, On the Computational Power of Elman-style Recurrent Networks, IEEE Transactions on Neural Networks, 6 (4) (1995) 1000-1004. H.T. Siegelmann and E.D. Sontag, On the Computational Power of Neural Nets, in: Proceedings of the Fifth ACM Workshop on Computational Learning Theory, (ACM, New York, NY, 1992) 440-449.
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Memory-based Stochastic Optimization Andrew W. Moore and Jeff Schneider School of Computer Science Carnegie-Mellon University Pittsburgh, PA 15213 Abstract In this paper we introduce new algorithms for optimizing noisy plants in which each experiment is very expensive. The algorithms build a global non-linear model of the expected output at the same time as using Bayesian linear regression analysis of locally weighted polynomial models. The local model answers queries about confidence, noise, gradient and Hessians, and use them to make automated decisions similar to those made by a practitioner of Response Surface Methodology. The global and local models are combined naturally as a locally weighted regression. We examine the question of whether the global model can really help optimization, and we extend it to the case of time-varying functions. We compare the new algorithms with a highly tuned higher-order stochastic optimization algorithm on randomly-generated functions and a simulated manufacturing task. We note significant improvements in total regret, time to converge, and final solution quality. 1 INTRODUCTION In a stochastic optimization problem, noisy samples are taken from a plant. A sample consists of a chosen control u (a vector ofreal numbers) and a noisy observed response y. y is drawn from a distribution with mean and variance that depend on u. y is assumed to be independent of previous experiments. Informally the goal is to quickly find control u to maximize the expected output E[y I u). This is different from conventional numerical optimization because the samples can be very noisy, there is no gradient information, and we usually wish to avoid ever performing badly (relative to our start state) even during optimization. Finally and importantly: each experiment is very expensive and there is ample computational time (often many minutes) for deciding on the next experiment. The following questions are both interesting and important: how should this computational time best be used, and how can the data best be used? Stochastic optimization is of real industrial importance, and indeed one of our reasons for investigating it is an association with a U.S. manufacturing company Memory-based Stochastic Optimization 1067 that has many new examples of stochastic optimization problems every year. The discrete version of this problem, in which u is chosen from a discrete set, is the well known k-armed bandit problem. Reinforcement learning researchers have recently applied bandit-like algorithms to efficiently optimize several discrete problems [Kaelbling, 1990, Greiner and Jurisica, 1992, Gratch et al., 1993, Maron and Moore, 1993]. This paper considers extensions to the continuous case in which u is a vector of reals. We anticipate useful applications here too. Continuity implies a formidable number of arms (uncountably infinite) but permits us to assume smoothness of E[y I u] as a function of u. The most popular current techniques are: • Response Surface Methods (RSM). Current RSM practice is described in the classic reference [Box and Draper, 1987]. Optimization proceeds by cautious steepest ascent hill-climbing. A region of interest (ROI) is established at a starting point and experiments are made at positions within the region that can best be used to identify the function properties with low-order polynomial regression. A large portion of the RSM literature concerns experimental design-the decision of where to take data points in order to acquire the lowest variance estimate of the local polynomial coefficients in a fixed number of experiments. When the gradient is estimated with sufficient confidence, the ROI is moved accordingly. Regression of a quadratic locates optima within the ROI and also diagnoses ridge systems and saddle points. The strength of RSM is that it is careful not to change operating conditions based on inadequate evidence, but moves once the data justifies. A weakness of RSM is that human judgment is needed: it is not an algorithm, but a manufacturing methodology . • Stochastic Approximation methods. The algorithm of [Robbins and Monro, 1951] does root finding without the use of derivative estimates. Through the use of successively smaller steps convergence is proven under broad assumptions about noise. Keifer-Wolfowitz (KW) [Kushner and Clark, 1978] is a related algorithm for optimization problems. From an initial point it estimates the gradient by performing an experiment in each direction along each dimension of the input space. Based on the estimate, it moves its experiment center and repeats. Again, use of decreasing step sizes leads to a proof of convergence to a local optimum. The strength of KW is its aggressive exploration, its simplicity, and that it comes with convergence guarantees. However, it has more of a danger of attempting wild experiments in the presence of noise, and effectively discards the data it collects after each gradient estimate is made. In practice, higher order versions of KW are available in which convergence is accelerated by replacing the fixed step size schedule with an adaptive one [Kushner and Clark, 1978]. Later we compare the performance of our algorithms to such a higher-order KW. 2 MEMORY-BASED OPTIMIZATION Neither KW nor RSM uses old data. After a gradient has been identified the control u is moved up the gradient and the data that produced the gradient estimate is discarded. Does this lead to inefficiencies in operation? This paper investigates one way of using old data: build a global non-linear plant model with it. We use locally weighted regression to model the system [Cleveland and Delvin, 1988, Atkeson, 1989, Moore, 1992]. We have adapted the methods to return posterior distributions for their coefficients and noise (and thus, indirectly, their predictions) 1068 A. W. MOORE, J. SCHNEIDER based on very broad priors, following the Bayesian methods for global linear regression described in [DeGroot, 1970]. We estimate the coefficients f3 = {,8I ... ,8m} of a local polynomial model in which the data was generated by the polynomial and corrupted with gaussian noise of variance u2, which we also estimate. Our prior assumption will be that f3 is distributed according to a multivariate gaussian of mean 0 and covariance matrix E. Our prior on u is that 1/u2 has a gamma distribution with parameters a and ,8. Assume we have observed n pieces of data. The jth polynomial term for the ith data point is Xij and the output response of the ith data point is Ii. Assume further that we wish to estimate the model local to the query point X q , in which a data point at distance di from the the query point has weight Wi = exp( -dl! K). K, the kernel width is a fixed parameter that determines the degree of localness in the local regression. Let W = Diag(wl,w2 . .. Wn). The marginal posterior distribution of f3 is' a t distribution with mean 13 = (E- 1 + X T W 2X)-1(XT W 2y) covariance (2,8 + (yT - f3T XT)W2yT)(E-l + X TW 2 X)-l / (2a + I:~=l wi) (1) and a + I:~=l w'f degrees of freedom. We assume a wide, weak, prior E = Diag(202,202, ... 202), a = 0.8,,8 = 0.001, meaning the prior assumes each regression coefficient independently lies with high probability in the range -20 to 20, and the noise lies in the range 0.01 to 0.5. Briefly, we note the following reasons that Bayesian locally weighted polynomial regression is particularly suited to this application: • We can directly obtain meaningful confidence estimates of the joint pdf of the regressed coefficients and predictions. Indirectly, we can compute the probability distribution of the steepest gradient, the location of local optima and the principal components of the local Hessian. • The Bayesian approach allows meaningful regressions even with fewer data points than regression coefficients-the posterior distribution reveals enormous lack of confidence in some aspects of such a model but other useful aspects can still be predicted with confidence. This is crucial in high dimensions, where it may be more effective to head in a known positive gradient without waiting for all the experiments that would be needed for a precise estimate of steepest gradient. • Other pros and cons of locally weighted regression in the context of control can be found in [Moore et ai., 1995]. Given the ability to derive a plant model from data, how should it best be used? The true optimal answer, which requires solving an infinite-dimensional Markov decision process, is intractable. We have developed four approximate algorithms that use the learned model, described briefly below. • AutoRSM. Fully automates the (normally manual) RSM procedure and incorporates weighted data from the model; not only from the current design. It uses online experimental design to pick ROI design points to maximize information about local gradients and optima. Space does not permit description of the linear algebraic formulations of these questions. • PMAX. This is a greedy, simpler approach that uses the global non-linear model from the data to jump immediately to the model optimum. This is similar to the technique described in [Botros, 1994], with two extensions. First, the Bayesian Memory-based Stochastic Optimization 1069 Figure 1: Three examples of 2-d functions used in optimization experiments priors enable useful decisions before the regression becomes full-rank. Second, local quadratic models permit second-order convergence near an optimum. • IEMAX. Applies Kaelbling's IE algorithm [Kaelbling, 1990] in the continuous case using Bayesian confidence intervals. argmax; () llchosen = u J opt U (2) where iopt(u) is the top of the 95th %-ile confidence interval. The intuition here is that we are encouraged to explore more aggressively than PMAX, but will not explore areas that are confidently below the best known optimum . • COMAX. In a real plant we would never want to apply PMAX or IEMAX. Experiments must be cautious for reasons of safety, quality control, and managerial peace of mind. COMAX extends IEMAX thus: argmax A A . llchosen = fopt(u);U E SAFE{=} f,pess(U) > dIsaster threshold (3) u E SAFE Analysis of these algorithms is problematic unless we are prepared to make strong assumptions about the form of E[Y I u]. To examine the general case we rely on Monte Carlo simulations, which we now describe. The experiments used randomly generated nonlinear unimodal (but not necessarily convex) d-dimensional functions from [0, l]d -+ [0,1]. Figure 1 shows three example 2-d functions. Gaussian noise (0- = 0.1) is added to the functions. This is large noise, and means several function evaluations would be needed to achieve a reliable gradient estimate for a system using even a large step size such as 0.2. The following optimization algorithms were tested on a sample of such functions. Vary-KW The best performing KW algorithm we could find varied step size and adapted gradient estimation steps to avoid undue regret at optima. Fixed-KW A version of KW that keeps its gradient-detecting step size fixed. This risks causing extra regret at a true optima, but has less chance of becoming delayed by a non-optimum. Auto-RSM The best performing version thereof. Passlve-RSM Auto-RSM continues to identify the precise location of the optimum when it's arrived at that optimum. When Passive-RSM is confident (greater than 99%) that it knows the location of the optimum to two significant places, it stops experimenting. Linear RSM A linear instead of quadratic model, thus restricted to steepest ascent. CRSM Auto-RSM with conservative parameters, more typical of those recommended in the RSM literature. Pmax, IEmax As described above. and Comax Figures 2a and 2b show the first sixty experiments taken by AutoRSM and KW respectively on their journeys to the goal. 1070 (a) (b) (0) RetroI_d ............ _ A. W. MOORE, J. SCHNEIDER Figure 2a: The path taken (start at (0.8,0.2)) by AutoRSM optimizing the given function with added noise of standard deviation 0.1 at each experiment. Figure 2b: The path taken (start at (0.8,0.2)) by KW. KW's path looks deceptively bad, but remember it is continually buffeted by considerable noise. te) No. of", YntII wllhln 0,05 rtf optimum let) ............ of FINAL ... tepe Figure 3: Comparing nine stochastic optimization algorithms by four criteria: (a) Regret, (b) Disasters, (c) Speed to converge (d) Quality at convergence. The partial order depicted shows which results are significant at the 99% level (using blocked pairwise comparisons). The outputs of the random functions range between 0-1 over the input domain. The numbers in the boxes are means over fifty 5-d functions. (a) Regret is defined as the mean Yopt - Yi-the cost incurred during the optimization compared with performance if we had known the optimum location and used it from the beginning. With the exception of IEMAX, model-based methods perform significantly better than KW, with reduced advantage for cautious and linear methods. (b) The %-age of steps which tried experiments with more than 0.1 units worse performance than at the search start. This matters to a risk averse manager. AutoRSM has fewer than 1% disasters, but COMAX and the modelfree methods do better still. PMAX's aggressive exploration costs it. (c) The number of steps until we reach within 0.05 units of optimal. PMAX's aggressiveness wins. (d) The quality of the "final" solution between steps 50 and 60 of the optimization. Results for 50 trials of each optimization algorithms for five-dimensional randomly generated functions are depicted in Figure 3. Many other experiments were performed in other dimensionalities and for modified versions of the algorithm, but space does not permit detailed discussion here. Finally we performed experiments with the simulated power-plant process in Figure 4. The catalyst controller adjusts the flow rate of the catalyst to achieve the goal chemical A content. Its actions also affect chemical B content. The temperature controller adjusts the reaction chamber temperature to achieve the goal chemical B content. The chemical contents are also affected by the flow rate which is determined externally by demand for the product. The task is to find the optimal values for the six controller parameters that minimize the total squared deviation from desired values of chemical A and chemical B contents. The feedback loops from sensors to controllers have significant delay. The controller gains on product demand are feedforward terms since there is significant delay in the effects of demand on the process. Finally, the performance of the system may also depend on variations over time in the composition of the input chemicals which can not be directly sensed. Memory-based Stochastic Optimization Catalyst Supply Sensor A Raw Input Chemicals Optimize 6 Controller Parameters To Minimize Squared Deviation from Goal Chemical A and B Content Te Catalyst Controller base lenns: Base temperature rccdback term,,; Sen.for B gain Product Demand f'L-_---'----'----~orwvd tem\s: Product demand gain Base input rate Sensor A gain Product demand gain REACTION CHAMBER Chemical A content sensor Pumps governed by demand for product Chemical B content sensor Product output 1071 Figure 4: A Simulated Chemical Process The total summed regrets of the optimization methods on 200 simulated steps were: Stay AtStart 10.86 FixedKW 2.82 AutoRSM 1.32 PMAX 3.30 COMAX 4.50 In this case AutoRSM is best, considerably beating the best KW algorithm we could find. In contrast PM AX and COMAX did poorly: in this plant wild experiments are very costly to PMAX and COMAX is too cautious. Stay AtStart is the regret that would be incurred if all 200 steps were taken at the initial parameter setting. 3 UNOBSERVED DISTURBANCES An apparent danger of learning a model is that if the environment changes, the out of date model will mean poor performance and very slow adaptation. The modelfree methods, which use only recent data, will react more nimbly. A simple but unsatisfactory answer to this is to use a model that implicitly (e.g. a neural net) or explicitly (e.g. local weighted regression of the fifty most recent points) forgets. An interesting possibility is to learn a model in a way that automatically determines whether a disturbance has occurred, and if so, how far back to forget. The following "adaptive forgetting" (AF) algorithm was added to the AutoRSM algorithm: At each step, use all the previous data to generate 99% confidence intervals on the output value at the current step. If the observed output is outside the intervals assume that a large change in the system has occured and forget all previous data. This algorithm is good for recognizing jumps in the plant's operating characteristics and allows AutoRSM to respond to them quickly, but is not suitable for detecting and handling process drift. We tested our algorithm's performance on the simulated plant for 450 steps. Operation began as before, but at step 150 there was an unobserved change in the composition of the raw input chemicals. The total regrets of the optimization methods were: StayAtStart FixedKW AutoRSM PMAX AutoRSM/AF 11.90 5.31 8.37 9.23 2.75 AutoRSM and PMAX do poorly because all their decisions after step 150 are based partially on the invalid data collected before then. The AF addition to AutoRSM solves the problem while beating the best KW by a factor of 2. Furthermore, AutoRSMj AF gets 1.76 on the invariant task, thus demonstrating that it can be used safely in cases where it is not known if the process is time varying. 1072 A. W. MOORE, J. SCHNEIDER 4 DISCUSSION Botros' thesis [Botros, 1994] discusses an algorithm similar to PMAX based on local linear regression. [Salganicoff and Ungar, 1995] uses a decision tree to learn a model. They use Gittins indices to suggest experiments: we believe that the memory-based methods can benefit from them too. They, however, do not use gradient information, and so require many experiments to search a 2D space. IEmax performed badly in these experiments, but optimism-gl1ided exploration may prove important in algorithms which check for potentially superior local optima. A possible extension is self tuning optimization. Part way through an optimization, to estimate the best optimization parameters for an algorithm we can run montecarlo simulations which run on sample functions from the posterior global model given the current data. This paper has examined the question of how much can learning a Bayesian memorybased model accelerate the convergence of stochastic optimization. We have proposed four algorithms for doing this, one based on an autonomous version of RSM; the other three upon greedily jumping to optima of three criteria dependent on predicted output and uncertainty. Empirically the model-based methods provide significant gains over a highly tuned higher order model-free method. References [Atkeson, 1989] C . G . Atkeson. Using Local Models to Control Movement. In Proceedings of Neural Information Processing Systems Conference, November 1989. [Botros, 1994] S. M. Botros. Model-Based Techniques in Motor Learning and Task Optimization. PhD. Thesis, MIT Dept. of Brain and Cognitive Sciences, February 1994. [Box and Draper, 1987] G. E . P. Box and N. R. Draper. Empirical Model-Building and Response Surfaces. Wiley, 1987. [Cleveland and Delvin, 1988] W. S. Cleveland and S. J . Delvin. Locally Weighted Regression: An Approach to Regression Analysis by Local Fitting. Journal of the American Statistical Association, 83(403):596-610, September 1988. [DeGroot, 1970] M. H . DeGroot. Optimal Statistical Decisions. McGraw-Hill, 1970. [Gratch et al., 1993] J. Gratch , S. Chien, and G. DeJong. Learning Search Control Knowledge for Deep Space Network Scheduling. In Proceedings of the 10th International Conference on Machine Learning. Morgan Kaufmann, June 1993. [Greiner and Jurisica, 1992] R. Greiner and I. Jurisica. A statistical approach to solving the EBL utility problem. In Proceedings of the Tenth International Conference on Artificial Intelligence (AAAI92). MIT Press, 1992. [Kaelbling, 1990] L. P. Kaelbling. Learning in Embedded Systems. PhD. Thesis; Technical Report No. TR-90-04, Stanford University, Department of Computer Science, June 1990. [Kushner and Clark, 1978] H. Kushner and D. Clark. Stochastic Approximation Methods for Constrained and Unconstrained Systems. Springer-Verlag, 1978. [Maron and Moore, 1993] O. Maron and A. Moore. Hoeffding Races: Accelerating Model Selection Search for Classification and Function Approximation. In Advances in Neural Information Processing Systems 6. Morgan Kaufmann, December 1993. [Moore et al., 1995] A . W . Moore, C. G . Atkeson, and S. Schaal. Memory-based Learning for Control. Technical report, CMU Robotics Institute, Technical Report CMU-RI-TR-95-18 (Submitted for Publication), 1995. [Moore, 1992] A. W . Moore. Fast, Robust Adaptive Control by Learning only Forward Models. In J . E . Moody, S. J . Hanson , and R . P. Lippman, editors, Advances in Neural Information Processing Systems 4. Morgan Kaufmann, April 1992. [Robbins and Monro, 1951] H . Robbins and S. Monro. A stochastic approximation method. Annals of Mathematical Statist2cs, 22:400-407, 1951. [Salganicoff and Ungar, 1995] M. Salganicoffand L. H. Ungar. Active Exploration and Learning in RealValued Spaces using Multi-Armed Bandit Allocation Indices. In Proceedings of the 12th International Conference on Machine Learning. Morgan Kaufmann, 1995.
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Handwritten Word Recognition using Contextual Hybrid Radial Basis Function NetworklHidden Markov Models Bernard Lemarie La Poste/SRTP 10, Rue de l'lle-Mabon F-44063 Nantes Cedex France lemarie@srtp.srt-poste.fr Michel Gilloux La Poste/SRTP 10, Rue de l'1le-Mabon F-44063 Nantes Cede x France gilloux@srtp.srt-poste.fr Manuel Leroux La Poste/SRTP 10, Rue de l'lle-Mabon F-44063 Nantes Cedex France leroux@srtp.srt-poste.fr Abstract A hybrid and contextual radial basis function networklhidden Markov model off-line handwritten word recognition system is presented. The task assigned to the radial basis function networks is the estimation of emission probabilities associated to Markov states. The model is contextual because the estimation of emission probabilities takes into account the left context of the current image segment as represented by its predecessor in the sequence. The new system does not outperform the previous system without context but acts differently. 1 INTRODUCTION Hidden Markov models (HMMs) are now commonly used in off-line recognition of handwritten words (Chen et aI., 1994) (Gilloux et aI., 1993) (Gilloux et al. 1995a). In some of these approaches (Gilloux et al. 1993), word images are transformed into sequences of image segments through some explicit segmentation procedure. These segments are passed on to a module which is in charge of estimating the probability for each segment to appear when the corresponding hidden state is some state s (state emission probabilities). Model probabilities are generally optimized for the Maximum Likelihood Estimation (MLE) criterion. MLE training is known to be sub-optimal with respect to discrimination ability when the underlying model is not the true model for the data. Moreover, estimating the emission probabilities in regions where examples are sparse is difficult and estimations may not be accurate. To reduce the risk of over training, images segments consisting of bitmaps are often replaced by feature vector of reasonable length (Chen et aI., 1994) or even discrete symbols (Gilloux et aI., 1993). Handwritten Word Recognition Using HMMlRBF Networks 765 In a previous paper (Gilloux et aI., 1995b) we described a hybrid HMMlradial basis function system in which emission probabilities are computed from full-fledged bitmaps though the use of a radial basis function (RBF) neural network. This system demonstrated better recognition rates than a previous one based on symbolic features (Gilloux et aI., 1995b). Yet, many misclassification examples showed that some of the simplifying assumptions made in HMMs were responsible for a significant part of these errors. In particular, we observed that considering each segment independently from its neighbours would hurt the accuracy of the model. For example, figure 1 shows examples of letter a when it is segmented in two parts. The two parts are obviously correlated. al Figure 1: Examples of segmented a. We propose a new variant of the hybrid HMMIRBF model in which emission probabilities are estimated by taking into account the context of the current segment. The context will be represented by the preceding image segment in the sequence. The RBF model was chosen because it was proven to be an efficient model for recognizing isolated digits or letters (Poggio & Girosi, 1990) (Lemarie, 1993). Interestingly enough, RBFs bear close relationships with gaussian mixtures often used to model emission probabilities in markovian contexts. Their advantage lies in the fact that they do not directly estimate emission probabilities and thus are less prone to errors in this estimation in sparse regions. They are also trained through the Mean Square Error (MSE) criterion which makes them more discriminant. The idea of using a neural net and in particular a RBF in conjunction with a HMM is not new. In (Singer & Lippman, 1992) it was applied to a speech recognition task. The use of context to improve emission probabilities was proposed in (Bourlard & Morgan, 1993) with the use of a discrete set of context events. Several neural networks are there used to estimate various relations between states, context events and current segment. Our point is to propose a different method without discrete context based on a adapted decomposition of the HMM likelihood estimation.This model is next applied to off-line handwritten word recognition. The organization of this paper is as follows. Section 1 is an overview of the architecture of our HMM. Section 2 describes the justification for using RBF outputs in a contextual hidden Markov model. Section 3 describes the radial basis function network recognizer. Section 4 reports on an experiment in which the contextual model is applied to the recognition of handwritten words found on french bank or postal cheques. 2 OVERVIEW OF THE HIDDEN MARKOV MODEL In an HMM model (Bahl et aI., 1983), the recognition scores associated to words ware likelihoods L(wli) ... in) = P(i1 ···inlw)xP(W) in which the first term in the product encodes the probability with which the model of each word w generates some image (some sequence of image segments) ij ... in- In the HMM paradigm, this term is decomposed into a sum on all paths (i.e. sequence of hidden states) of products of the probability of the hidden path by the probability that the path generated the Image sequence: p(i) ... inlw) = 766 B. LEMARIE. M. GILLOUX. M. LEROUX It is often assumed that only one path contributes significantly to this term so that In HMMs, each sequence element is assumed to depend only on its corresponding state: n p(il···i ISI···s ) = ITp(i·ls .) n n } } j=1 Moreover, first-order Markov models assume that paths are generated by a first-order Markov chain so that n P(sl ···s ) = ITp(s . ls. I) n } }j = I We have reported in previous papers (Gilloux et aI., 1993) (Gilloux et aI., 1995a) on several handwriting recognition systems based on this assumption.The hidden Markov model architecture used in all systems has been extensively presented in (Gilloux et aI., 1995a). In that model, word letters are associated to three-states models which are designed to account for the situations where a letter is realized as 0, 1 or 2 segments. Word models are the result of assembling the corresponding letter models. This architecture is depicted on figure 2. We used here transition emission rather than state emission. However, this does not E, 0.05 E,0.05 I a val Figure 2: Outline of the model for "laval". change the previous formulas if we replace states by transitions, i.e. pairs of states. One of these systems was an hybrid RBFIHMM model in which a radial basis function network was used to estimate emission probabilities p (i. Is.) . The RBF outputs are introduced by applying Bayes rule in the expression of p (i I .~. i; I s I ... S n) : n p(s.1 i.) xp(i.) p(il ···i IsI· ··s) = IT }} } n n. p (s.) } = 1 } Since the product of a priori image segments probabilities p (i.) does not depend on the word hypothesis w, we may write: } n p (s. Ii.) p(il···inlsl···sn)oc.IT p~s./ } = 1 } In the above formula, terms of form p (s . Is. _ I) are transition probabilities which may be estimated through the Baum-Welch re-istirhatlOn algorithm. Terms of form p (s.) are a priori probabilities of states. Note that for Bayes rule to apply, these probabilitid have and only have to be estimated consistently with terms of form p (s. Ii.) since p (i. Is.) is independent of the statistical distribution of states. } } } } It has been proven elsewhere (Richard & Lippman, 1992) that systems trained through the MSE criterion tend to approximate Bayes probabilities in the sense that Bayes probaHandwritten Word Recognition Using HMMlRBF Networks 767 bilities are optimal for the MSE criterion. In practice, the way in which a given system comes close to Bayes optimum is not easily predictable due to various biases of the trained system (initial parameters, local optimum, architecture of the net, etc.). Thus real output scores are generally not equal to Bayes probabilities. However, there exist different procedures which act as a post-processor for outputs of a system trained with the MSE and make them closer to Bayes probabilities (Singer & Lippman, 1992). Provided that such a postprocessor is used, we will assume that terms p (s. Ii.) are well estimated by the post-processed outputs of the recognition system. Then, u~~ p (s .) are just the a priori probabilities of states on the set used to train the system or post-prbcess the system outputs. This hybrid handwritten word recognition system demonstrated better performances than previous systems in which word images were represented through sequences of symbolic features instead of full-fledged bitmaps (Gilloux et aI., 1995b). However, some recognition errors remained, many of which could be explained by the simplifying assumptions made in the model. In particular, the fact that emission probabilities depend only on the state corresponding to the current bitmap appeared to be a poor choice. For example, on figure 3 the third and fourth segment are classified as two halves of the letter i. For letters Figure 3: An image of trente classified as mille. segmented in two parts, the second half is naturally correlated to the first (see figure 1). Yet, our Markov model architecture is designed so that both halves are assumed uncorrelated. This has two effects. Two consecutive bitmaps which cannot be the two parts of a unique letter are sometimes recognized as such like on figure 3. Also, the emission probability of the second part of a segmented letter is lower than if the first part has been considered for estimating this probability. The contextual model described in the next section is designed so has to make a different assumption on emission probabilities. 3 THE HYBRID CONTEXTUAL RBFIHMM MODEL The exact decomposition of the emission part of word likelihoods is the following: n p(i1 ···inls1···sn) = P(il ls 1··· sn) x ITp(ijlsl ... sn,il ... ij_l) j=2 We assume now that bitmaps are conditioned by their state and the previous image in the sequence: n P(il··· in I sl· ·· sn) ==p(i11 sl) x IT p (ij I sj'ij _ l ) j=2 The RBF is again introduced by applying Bayes rule in the following way: P(s1 1 il ) xp(i l ) n p(s . 1 i ., i . 1) xp (i . I i . 1) p (i 1··· in lSI·· · s n) == () x IT }}} (I · /) P sl . P s. !. 1 J=2 J JSince terms of form p (i . Ii . _ 1) do not contribute to the discrimination of word hypotheses, we may write: J J p (s 1 IiI) n p (s . Ii., i . 1 ) ( . . I) IT } J JP 11· · ·ln sl··· sn oc () x I.) pSI . P (s . I. 1 J=2 J J768 B. LEMARIE, M. GILLOUX, M. LEROUX The RBF has now to estimate not only terms of form p (s. Ii ., i. _ 1) but also terms like p (s . Ii. 1) which are no longer computed by mere countind. 'two radial basis function netJoris-are then used to estimate these probabilities. Their common architecture is described in the next section. 4 THE RADIAL BASIS FUNCTION MODEL The radial basis function model has been described in (Lemarie, 1993). RBF networks are inspired from the theory of regularization (Poggio & Girosi, 1990). This theory studies how multivariate real functions known on a finite set of points may be approximated at these points in a family of parametric functions under some bias of regularity. It has been shown that when this bias tends to select smooth functions in the sense that some linear combination of their derivatives is minimum, there exist an analytical solution which is a linear combination of gaussians centred on the points where the function is known (Poggio & Girosi, 1990). It is straightforward to transpose this paradigm to the problem of learning probability distributions given a set of examples. In practice, the theory is not tractable since it requires one gaussian per example in the training set. Empirical methods (Lemarie, 1993) have been developed which reduce the number of gaussian centres. Since the theory is no longer applicable when the number of centres is reduced, the parameters of the model (centres and covariance matrices for gaussians, weights for the linear combination) have to be trained by another method, in that case the gradient descent method and the MSE criterion. Finally, the resulting RBF model may be looked at like a particular neural network with three layers. The first is the input layer. The second layer is completely connected to the input layer through connections with unit weights. The transfer functions of cells in the second layer are gaussians applied to the weighed distance between the corresponding centres and the weighed input to the cell. The weight of the distance are analogous to the parameters of a diagonal covariance matrix. Finally, the last layer is completely connected to the second one through weighted connections. Cells in this layer just output the sum of their input. In our experiments, inputs to the RBF are feature vectors of length 138 computed from the bitmaps of a word segment (Lemarie, 1993). The RBF that estimates terms of form p (s. Ii., i. 1) uses to such vectors as input whereas the second RBF (terms p (/ I/_il) ) is only fed with the vector associated to ij _l . These vectors are inspired from "cha'rac{eristic loci" methods (Gluksman, 1967) and encode the proportion of white pixels from which a bitmap border can be reached without meeting any black pixel in various of directions. 5 EXPERIMENTS The model has been assessed by applying it to the recognition of words appearing in legal amounts of french postal or bank cheques. The size of the vocabulary is 30 and its perplexity is only 14.3 (Bahl et aI., 1983). The training and test bases are made of images of amount words written by unknown writers on real cheques. We used 7 191 images during training and 2 879 different images for test. The image resolution was 300 dpi. The amounts were manually segmented into words and an automatic procedure was used to separate the words from the preprinted lines of the cheque form. The training was conducted by using the results of the former hybrid system. The segmentation module was kept unchanged. There are 48 140 segments in the training set and 19577 in the test set. We assumed that the base system is almost always correct when aligning segments onto letter models. We thus used this alignment to label all the segments in the training set and took these labels as the desired outputs for the RBF. We used a set of 63 different labels since 21 letters appear in the amount vocabulary and 3 types of segments are possible for each letter. The outputs of the RBF are directly interpreted as Bayes probHandwritten Word Recognition Using HMMJRBF Networks 769 abilities without further post-processing. First of all, we assessed the quality of the system by evaluating its ability to recognize the class of a segment through the value of p (s . Ii., i. 1) and compared it with that of the previous hybrid system. The results of this e'xpdrirhent are reported on table 1 for the test set. They demonstrate the importance of the context and thus its potential interest for a Table 1: Recognition and confusion rates for segment classifiers Recognition rate Confusion rate Mean square error RBF system without context 32.6% 67.4% 0.828 RBF system with context 41.7% 58.3% 0.739 word recognition system. We next compare the performance on word recognition on the data base of 2878 images of words. Results are shown in table 2. The first remark is that the system without context Table 2: Recognition and confusion rates for the word recognition systems Recognition rate Confusion rate # Confusions RBF system without context 81,3% 16,7% 536 RBF system with context 76,3% 23,7% 683 present better results than the contextual system. Some of the difference between the systems with and without context are shown below in figures 4 and 5 and may explain why the contextual system remains at a lower level of performance. The word "huit" and "deux" of figure 4 are well recognized by the system without context but badly identified by the contextual system respectively as "trente" and "franc". The image of the word "huit", for example, is segmented into eight segments and each of the four letters of the word is thus necessarily considered as separated in two parts. The fifth and sixth segments are thus recognized as two halves of the letter "i" by the standard system while the contextual system avoids this decomposition of the letter "i". On the next image, the contextual system proposes "ra" for the second and third segments mainly because of the absence of information on the relative position ofthese segments. On the other hand, figure 5 shows examples where the contextual system outperforms the system without context. In the first case the latter proposed the class "trois" with two halves on the letter "i" on the fifth and sixth segments. In the second case the context is clearly useful for the recognition on the first letter of the word. Forthcoming experiments will try to combine the two systems so as to benefit of their respective characteritics. Figure 4 : some new confusions produced by the contextual system. Experiments have also revealed that the contextual system remains very sensible to the numerical output values for the network which estimates p (s. Ii. _ 1) . Several approaches for solving this problem are currently under investigation. Ffrst'results have yet been obtained by trying to approximate the network which estimates p (Sj I ij _ 1) from the network which estimates p (Sj I ij' ij _ 1) . 770 B. LEMARIE, M. GILLOUX, M. LEROUX 6 CONCLUSION We have described a new application of a hybrid radial basis function/hidden Markov model architecture to the recognition of off-line handwritten words. In this architecture, the estimation of emission probabilities is assigned to a discriminant classifier. The estimation of emission probabilities is enhanced by taking into account the context as represented by the previous bitmap in the sequence to be classified. A formula have been derived introducing this context in the estimation of the likelihood of word scores. The ratio of the output values of two networks are now used so as to estimate the likelihood. The reported experiments reveal that the use of context, if profitable at the segment recognition level, is not yet useful at the word recognition level. Nevertheless, the new system acts differently from the previous system without context and future applications will try to exploit this difference. The dynamic of the ratio networks output values is also very unstable and some solutions to stabilize it which will be deeply tested in the forthcoming experiences. References Bahl L, Jelinek F, Mercer R, (1983). A maximum likelihood approach to speech recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence 5(2): 179-190. Bahl LR, Brown PF, de Souza PV, Mercer RL, (1986). Maximum mutual information estimation of hidden Markov model parameters for speech recognition. In: Proc of the Int Conf on Acoustics, Speech, and Signal Processing (ICASSP'86):49-52. Bourlard, H., Morgan, N., (1993). Continuous speech recognition by connectionist statistical methods, IEEE Trans. on Neural Networks, vol. 4, no. 6, pp. 893-909, 1993. Chen, M.-Y., Kundu, A., Zhou, J., (1994). Off-line handwritten word recognition using a hidden Markov model type stochastic network, IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 16, no. 5:481-496. Gilloux, M., Leroux, M., Bertille, J.-M., (1993). Strategies for handwritten words recognition using hidden Markov models, Proc. of the 2nd Int. Conf. on Document Analysis and Recognition:299-304. Gilloux, M., Leroux, M., Bertille, J.-M., (1995a). "Strategies for Cursive Script Recognition Using Hidden Markov Models", Machine Vision & Applications, Special issue on Handwriting recognition, R Plamondon ed., accepted for publication. Gilloux, M., Lemarie, B., Leroux, M., (l995b). "A Hybrid Radial Basis Function Network! Hidden Markov Model Handwritten Word Recognition System", Proc. of the 3rd Int. Conf. on Document Analysis and Recognition:394-397. Gluksman, H.A., (1967). Classification of mixed font alphabetics by characteristic loci, 1 st Annual IEEE Computer Conf.: 138-141. Lemarie, B., (1993). Practical implementation of a radial basis function network for handwritten digit recognition, Proc. of the 2nd Int. Conf. on Document Analysis and Recognition:412-415. Poggio, T., Girosi, F., (1990). Networks for approximation and learning, Proc. of the IEEE, vol 78, no 9. Richard, M.D., Lippmann, RP., (1991). "Neural network classifiers estimate bayesian a posteriori probabilities", Neural Computation, 3:461-483. Singer, E, Lippmann, RP., (1992). A speech recognizer using radial basis function networks in an HMM framework, Proc. of the Int. Conf. on acoustics, Speech, and Signal Processing.
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Universal Approximation and Learning of Trajectories Using Oscillators Pierre Baldi* Division of Biology California Institute of Technology Pasadena, CA 91125 pfbaldi@juliet.caltech.edu Kurt Hornik Technische Universitat Wien Wiedner Hauptstra8e 8-10/1071 A-1040 Wien, Austria Kurt.Hornik@tuwien.ac.at Abstract Natural and artificial neural circuits must be capable of traversing specific state space trajectories. A natural approach to this problem is to learn the relevant trajectories from examples. Unfortunately, gradient descent learning of complex trajectories in amorphous networks is unsuccessful. We suggest a possible approach where trajectories are realized by combining simple oscillators, in various modular ways. We contrast two regimes of fast and slow oscillations. In all cases, we show that banks of oscillators with bounded frequencies have universal approximation properties. Open questions are also discussed briefly. 1 INTRODUCTION: TRAJECTORY LEARNING The design of artificial neural systems, in robotics applications and others, often leads to the problem of constructing a recurrent neural network capable of producing a particular trajectory, in the state space of its visible units. Throughout evolution, biological neural systems, such as central pattern generators, have also been faced with similar challenges. A natural approach to tackle this problem is to try to "learn" the desired trajectory, for instance through a process of trial and error and subsequent optimization. Unfortunately, gradient descent learning of complex trajectories in amorphous networks is unsuccessful. Here, we suggest a possible approach where trajectories are realized, in a modular and hierarchical fashion, by combining simple oscillators. In particular, we show that banks of oscillators have universal approximation properties. * Also with the Jet Propulsion Laboratory, California Institute of Technology. 452 P. BALDI, K. HORNIK To begin with, we can restrict ourselves to the simple case of a network with one! visible linear unit and consider the problem of adjusting the network parameters in a way that the output unit activity u(t) is equal to a target function I(t), over an interval of time [0, T]. The hidden units of the network may be non-linear and satisfy, for instance, one of the usual neural network charging equations such as dUi Ui ~ dt = - Ti + L..JjWij/jUj(t Tij), (1) where Ti is the time constant of the unit, the Tij represent interaction delays, and the functions Ij are non-linear input/output functions, sigmoidal or other. In the next section, we briefly review three possible approaches for solving this problem, and some of their limitations. In particular, we suggest that complex trajectories can be synthesized by proper combination of simple oscillatory components. 2 THREE DIFFERENT APPROACHES TO TRAJECTORY LEARNING 2.1 GRADIENT DESCENT APPROACHES One obvious approach is to use a form of gradient descent for recurrent networks (see [2] for a review), such as back-propagation through time, in order to modify any adjustable parameters of the networks (time constants, delays, synaptic weights and/or gains) to reduce a certain error measure, constructed by comparing the output u(t) with its target I(t). While conceptually simple, gradient descent applied to amorphous networks is not a successful approach, except on the most simple trajectories. Although intuitively clear, the exact reasons for this are not entirely understood, and overlap in part with the problems that can be encountered with gradient descent in simple feed-forward networks on regression or classification tasks. There is an additional set of difficulties with gradient descent learning offixed points or trajectories, that is specific to recurrent networks, and that has to do with the bifurcations of the system being considered. In the case of a recurrent2 network, as the parameters are varied, the system mayor may not undergo a series of bifurcations, i.e., of abrupt changes in the structure of its trajectories and, in particular, of its at tractors (fixed points, limit cycles, ... ). This in turn may translate into abrupt discontinuities, oscillations or non-convergence in the corresponding learning curve. At each bifurcation, the error function is usually discontinuous, and therefore the gradient is not defined. Learning can be disrupted in two ways: when unwanted abrupt changes occur in the flow of the dynamical system, or when desirable bifurcations are prevented from occurring. A classical example of the second type is the case of a neural network with very small initial weights being trained to oscillate, in a symmetric and stable fashion, around the origin. With small initial weights, the network in general converges to its unique fixed point at the origin, with a large error. If we slightly perturb the weights, remaining away from any bifurcation, the network continues to converge to its unique fixed point which now may be slightly displaced from the origin, and yield an even greater error, so that learning by gradient descent becomes impossible (the starting configuration of zero weights is a local minimum of the error function). 1 All the results to be derived can be extended immediately to the case of higherdimensional trajectories. 2In a feed-forward network, where the transfer functions of the units are continuous, the output is a continuous function of the parameters and therefore there are no bifurcations. Universal Approximation and Learning of Trajectories Using Oscillators 453 8 o Figure 1: A schematic representation of a 3 layer oscillator network for double figure eight. Oscillators with period T in a given layer gate the corresponding oscillators, with period T /2, in the previous layer. 2.2 DYNAMICAL SYSTEM APPROACH In the dynamical system approach, the function /(t) is approximated in time, over [0, T] by a sequence of points Yo, Yl, .... These points are associated with the iterates of a dynamical system, i.e., Yn+l = F(Yn) = Fn(yo), for some function F. Thus the network implementation requires mainly a feed-forward circuit that computes the function F. It has a simple overall recursive structure where, at time n, the output F(Yn) is calculated, and fed back into the input for the next iteration. While this approach is entirely general, it leaves open the problem of constructing the function F. Of course, F can be learned from examples in a usual feed-forward connectionist network. But, as usual, the complexity and architecture of such a network are difficult to determine in general. Another interesting issue in trajectory learning is how time is represented in the network, and whether some sort of clock is needed. Although occasionally in the literature certain authors have advocated the introduction of an input unit whose output is the time t, this explicit representation is clearly not a suitable representation, since the problem of trajectory learning reduces then entirely to a regression problem. The dynamical system approach relies on one basic clock to calculate F and recycle it to the input layer. In the next approach, an implicit representation of time is provided by the periods of the oscillators. 2.3 OSCILLATOR APPROACH A different approach was suggested in [1] where, loosely speaking, complex trajectories are realized using weakly pre-structured networks, consisting of shallow hierarchical combinations of simple oscillatory modules. The oscillatory modules can consist, for instance, of simple oscillator rings of units satisfying Eq. 1, with two or three high-gain neurons, and an odd number of inhibitory connections ([3]). To fix the ideas, consider the typical test problem of constructing a network capable of producing a trajectory associated with a double figure eight curve (i.e., a set of four loops joined at one point), see Fig. 1. In this example, the first level of the hierarchy could contain four oscillator rings, one for each loop of the target trajectory. The parameters in each one of these four modules can be adjusted, for instance by gradient descent, to match each of the loops in the target trajectory. 454 P. BALDI, K. HORNIK The second level of the pyramid should contain two control modules. Each of these modules controls a distinct pair of oscillator networks from the first level, so that each control network in the second level ends up producing a simple figure eight. Again, the control networks in level two can be oscillator rings and their parameters can be adjusted. In particular, after the learning process is completed, they should be operating in their high-gain regimes and have a period equal to the sum of the periods of the circuits each one controls. Finally, the third layer consists of another oscillatory and adjustable module which controls the two modules in the second level, so as to produce a double figure eight. The third layer module must also end up operating in its high-gain regime with a period equal to four times the period of the oscillators in the first layer. In general, the final output trajectory is also a limit cycle because it is obtained by superposition of limit cycles in the various modules. If the various oscillators relax to their limit cycles independently of one another, it is essential to provide for adjustable delays between the various modules in order to get the proper phase adjustments. In this way, a sparse network with 20 units or so can be constructed that can successfully execute a double figure eight. There are actually different possible neural network realizations depending on how the action of the control modules is implemented. For instance, if the control units are gating the connections between corresponding layers, this amounts to using higher order units in the network. If one high-gain oscillatory unit, with activity c(t) always close to 0 or 1, gates the oscillatory activities of two units Ul(t) and U2(t) in the previous layer, then the overall output can be written as out(t) = C(t)Ul (t) + (1 - C(t))U2(t) . (2) The number of layers in the network then becomes a function of the order of the units one is willing to use. This approach could also be described in terms of a dynamic mixture of experts architecture, in its high gain regime. Alternatively, one could assume the existence of a fast weight dynamics on certain connections governed by a corresponding set of differential equations. Although we believe that oscillators with limit cycles present several attractive properties (stability, short transients, biological relevance, . . . ), one can conceivably use completely different circuits as building blocks in each module. 3 GENERALIZATION AND UNIVERSAL APPROXIMATION We have just described an approach that combines a modular hierarchical architecture, together with some simple form of learning, enabling the synthesis of a neural circuit suitable for the production of a double figure eight trajectory. It is clear that the same approach can be extended to triple figure eight or, for that matter, to any trajectory curve consisting of an arbitrary number of simple loops with a common period and one common point. In fact it can be extended to any arbitrary trajectory. To see this, we can subdivide the time interval [0, T] into n equal intervals of duration f = Tin . Given a certain level of required precision, we can always find n oscillator networks with period T (or a fraction of T) and visible trajectory Ui(t), such that for each i, the i-th portion of the trajectory u(t) with if ~ t ~ (i + l)f can be well approximated by a portion of Ui(t) , the trajectory of the i-th oscillator. The target trajectory can then be approximated as (3) Universal Approximation and Learning of Trajectories Using Oscillators 455 As usual, the control coefficient Cj(t) must have also period T and be equal to 1 for i{ :5 t :5 (i + 1){, and 0 otherwise. The control can be realized with one large high-gain oscillator, or as in the case described above, by a hierarchy of control oscillators arranged, for instance, as a binary tree of depth m if n = 2m , with the corresponding multiple frequencies. We can now turn to a slightly different oscillator approach, where trajectories are to be approximated with linear combinations of oscillators, with constant coefficients. What we would like to show again is that oscillators are universal approximators for trajectories. In a sense, this is already a well-known result of Fourier theory since, for instance, any reasonable function f with period T can be expanded in the form3 A.k = kiT. (4) For sufficiently smooth target functions, without high frequencies in their spectrum, it is well known that the series in Eq. 4 can be truncated. Notice, however, that both Eqs. 3 and 4 require having component oscillators with relatively high frequencies, compared to the final trajectory. This is not implausible in biological motor control, where trajectories have typical time scales of a fraction of a second, and single control neurons operate in the millisecond range. A rather different situation arises if the component oscillators are "slow" with respect to the final product. The Fourier representation requires in principle oscillations with arbitrarily large frequencies (0, liT, 2IT, .. . , niT, .. . ). Most likely, relatively small variations in the parameters (for instance gains, delays andlor synaptic weights) of an oscillator circuit can only lead to relatively small but continuous variations of the overall frequency. For instance, in [3] it is shown that the period T of an oscillator ring with n units obeying Eq. 1 must satisfy Thus, we need to show that a decomposition similar in flavor to Eq. 4 is possible, but using oscillators with frequencies in a bounded interval. Notice that by varying the parameters of a basic oscillator, any frequency in the allowable frequency range can be realized, see [3]. Such a linear combination is slightly different in spirit from Eq. 2, since the coefficients are independent of time, and can be seen as a soft mixture of experts. We have the following result. Theorem 1 Let a < b be two arbitrary real numbers and let f be a continuous function on [0, T]. Then for any error level { > 0, there exist n and a function 9n of the form such that the uniform distance Ilf - 9n 1100 is less than {. In fact, it is not even necessary to vary the frequencies A. over a continuous band [a, b]. We have the following. Theorem 2 Let {A.k} be an infinite sequence with a finite accumulation point, and let f be a continuous function on [0,7]. Then for any error level { > 0, there exist n and a function 9n(t) = 2:~=10:'ke27rjAkt such that Ilf - 9nll00 < {. 3In what follows, we use the complex form for notational convenience. 456 P. BALDI, K. HORNIK Thus, we may even fix the oscillator frequencies as e.g. Ak = l/k without losing universal approximation capabilities. Similar statements can be made about meansquare approximation or, more generally, approximation in p-norm LP(Il), where 1 ~ p < 00 and Il is a finite measure on [0, T]: Theorem 3 For all p and f in LP(Il) and for all { > 0, we can always find nand gn as above such that Ilf - gn IILP{Jl) < {. The proof of these results is surprisingly simple. Following the proofs in [4], if one of the above statements was not true, there would exist a nonzero, signed finite measure (T with support in [0, T] such that hO,T] e21fi>.t d(T(t) = ° for all "allowed" frequencies A. Now the function z t-+ !rO,T] e21fizt d(T(t) is clearly analytic on the whole complex plane. Hence, by a well-known result from complex variables, if it vanishes along an infinite sequence with a finite accumulation point, it is identically zero. But then in particular the Fourier transform of (T vanishes, which in turn implies that (T is identically zero by the uniqueness theorem on Fourier transforms, contradicting the initial assumption. Notice that the above results do not imply that f can exactly be represented as e.g. f(t) = f: e21fi>.t dV(A) for some signed finite measure v-such functions are not only band-limited, but also extremely smooth (they have an analytic extension to the whole complex plane). Hence, one might even conjecture that the above approximations are rather poor in the sense that unrealistically many terms are needed for the approximation. However, this is not true-one can easily show that the rates of approximation cannot be worse that those for approximation with polynomials. Let us briefly sketch the argument, because it also shows how bounded-frequency oscillators could be constructed. Following an idea essentially due to Stinchcombe & White [5], let, more generally, 9 be an analytic function in a neighborhood of the real line for which no derivative vanishes at the origin (above, we had g(t) = e21fit ). Pick a nonnegative integer n and a polynomial p of degree not greater than n - 1 arbitrarily. Let us show that for any { > 0, we can always find a gn of the form gn(t) = E~=l Cl'kg(Akt) with Ak arbitrarily small such that lip - gn 1100 < {. To do so, note that we can write L n - l p(t) = is,t', 1=0 where rn(At) is of the order of An, as A -t 0, uniformly for t in [0, T] . Hence, L:=l Cl'kg(Akt) L:=l Cl'k (L~=-ol fil (At)l + rn (At)) = L~=~l (L:: 1 Cl'kAi) filtl + L:=l Cl'krn (Akt). Now fix n distinct numbers el, ... ,en, let Ak = Ak(p) = pek, and choose the Cl'k = Cl'k(p) such that E;=lCl'k(p)Ak(p)' = iSl/fil for I = 0, ... , n 1. (This is possible because, by assumption, all fil are non-zero.) It is readily seen that Cl'k (p) is of the order of pl-n as p -t ° (in fact, the j-th row of the inverse of the coefficient matrix of the linear system is given by the coefficients of the polynomial nktj (A Ak)/(Aj -Ak)). Hence, as p -t 0, the remainder term EZ=lCl'k(p)rn(Ak(p)t) is ofthe order of p, and thus E~=lCl'k(p)g(Adp)t) -t E~=-oliS,t' = p(t) uniformly on [0, T]. Note that using the above method, the coefficients in the approximation grow quite rapidly when the approximation error tends to 0. In some sense, this was to be Universal Approximation and Learning of Trajectories Using Oscillators 457 expected from the observation that the classes of small-band-limited functions are rather "small". There is a fundamental tradeoff between the size of the frequencies, and the size of the mixing coefficients. How exactly the coefficients scale with the width of the allowed frequency band is currently being investigated. 4 CONCLUSION The modular oscillator approach leads to trajectory architectures which are more structured than fully interconnected networks, with a general feed-forward flow of information and sparse recurrent connections to achieve dynamical effects. The sparsity of units and connections are attractive features for hardware design; and so is also the modular organization and the fact that learning is much more circumscribed than in fully interconnected systems. We have shown in different ways that such architectures have universal approximation properties. In these architectures, however, some form of learning remains essential, for instance to fine tune each one of the modules. This, in itself, is a much easier task than the one a fully interconnected and random network would have been faced with. It can be solved by gradient or random descent or other methods. Yet, fundamental open problems remain in the overall organization of learning across modules, and in the origin of the decomposition. In particular, can the modular architecture be the outcome of a simple internal organizational process rather than an external imposition and how should learning be coordinated in time and across modules (other than the obvious: modules in the first level learn first, modules in the second level second, .. . )? How successful is a global gradient descent strategy applied across modules? How can the same modular architecture be used for different trajectories, with short switching times between trajectories and proper phases along each trajectory? Acknowledgments The work of PB is in part supported by grants from the ONR and the AFOSR. References [1] Pierre Baldi. A modular hierarchical approach to learning. In Proceedings of the 2nd International Conference on Fuzzy Logic and Neural Networks, volume II, pages 985-988, IIzuka, Japan, 1992. [2] Pierre F. Baldi. Gradient descent learning algorithm overview: a general dynamic systems perspective. IEEE Transactions on Neural Networks, 6(1}:182195, January 1995. [3] Pierre F. Baldi and Amir F. Atiya. How delays affect neural dynamics and learning. IEEE Transactions on Neural Networks, 5(4):612-621, July 1994. [4] Kurt Hornik. Some new results on neural network approximation. Neural Networks, 6:1069-1072,1993. [5] Maxwell B. Stinchcombe and Halbert White. Approximating and learning unknown mappings using multilayer feedforward networks with bounded weights. In International Joint Conference on Neural Networks, volume III, pages 7-16, Washington, 1990. Lawrence Earlbaum, Hillsdale.
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Correlated Neuronal Response: Time Scales and Mechanisms Wyeth Bair Howard Hughes Medical Inst. NYU Center for Neural Science 4 Washington PI., Room 809 New York, NY 10003 Ehud Zohary Dept. of Neurobiology Institute of Life Sciences The Hebrew University, Givat Ram Jerusalem, 91904 ISRAEL Christof Koch Computation and Neural Systems Caltech, 139-74 Pasadena, CA 91125 Abstract We have analyzed the relationship between correlated spike count and the peak in the cross-correlation of spike trains for pairs of simultaneously recorded neurons from a previous study of area MT in the macaque monkey (Zohary et al., 1994). We conclude that common input, responsible for creating peaks on the order of ten milliseconds wide in the spike train cross-correlograms (CCGs), is also responsible for creating the correlation in spike count observed at the two second time scale of the trial. We argue that both common excitation and inhibition may play significant roles in establishing this correlation. 1 INTRODUCTION In a previous study of pairs of MT neurons recorded using a single extracellular electrode, it was found that the spike count during two seconds of visual motion stimulation had an average correlation coefficient of r = 0.12 and that this correlation could significantly limit the usefulness of pooling across increasingly large populations of neurons (Zohary et aI., 1994). However, correlated spike count between two neurons could in principle occur at several time-scales. Correlated drifts Correlated Neuronal Response: Time Scales and Mechanisms 69 in the excitability of the cells, for example due to normal biological changes or electrode induced changes, could cause correlation at a time scale of many minutes. Alternatively, attentional or priming effects from higher areas could change the responsivity of the cells at the time scale of an experimental trial. Or, as suggested here, common input that changes on the order of milliseconds could cause correlation in spike count. The first section determines the time scale at which the neurons are correlated by analyzing the relationship between the peak in the spike train cross-correlograms (CCGs) and the correlation between the spike counts using a construct we call the trial CCG. The second section examines temporal structure that is indicative of correlated suppression of firing, perhaps due to inhibition, which may also contribute to the spike count correlation. 2 THE TIME SCALE OF CORRELATION At the time scale of the single trial, the correlation, r se, of spike counts x and y from two neurons recorded during nominally identical two second stimuli was computed using Pearson's correlation coefficient, E[xy] - ExEy rse = , uxuy (1) where E is expected value and u2 is variance. If spike counts are converted to z-scores, i.e., zero mean and unity variance, then rse = E[xy], and rse may be interpreted as the zero-lag value of the cross-correlation of the z-scored spike counts. The trial CCGs resulting from this procedure are shown for two pairs of neurons in Fig. l. To distinguish between cases like the two shown in Fig. 1, the correlation was broken into a long-term component, rlt, the average value (computed using a Gaussian window of standard deviation 4 trials) surrounding the zero-lag value, and a shortterm component, rst, the difference between the zero-lag value and rlt. Across 92 pairs of neurons from three monkeys, the average rst was 0.10 (s.d. 0.17) while rlt was not significantly different from zero (mean 0.01, s.d. 0.11). The mean of rst was similar to the overall correlation of 0.12 reported by Zohary et al. (1994). Under certain assumptions, including that the time scale of correlation is less than the trial duration, rst can be estimated from the area under the spike train CCG and the areas under the autocorrelations (derivation omitted). Under the additional assumption that the spike trains are individually Poisson and have no peak in the autocorrelation except that which occurs by definition at lag zero, the correlation coefficient for spike count can be estimated by rpeak ~ j.AA.ABArea, (2) where .AA and .AB are the mean firing rates of neurons A and B, and Area is the area under the spike train CCG peak, like that shown in Fig. 2 for one pair of neurons. Taking Area to be the area under the CCG between ±32 msec gives a good estimate of short-term rst, as shown in Fig. 3. In addition to the strong correlation (r = 0.71) between rpeak and rst, rpeak is a less noisy measure, having standard deviation (not shown) on average one fourth as large as those of rst. We conclude that the common input that causes the peaks in the spike train CCGs is also responsible for the correlation in spike count that has been previously reported. 70 0.3 0.2 d U U 0.1 ";3 o 80 160 240 320 0 Trial Number .~ 0 ~ ~~------------------~~ -0.1 +'-r-~..:...-,:......-~-.-,-~~---,-,~,.--,--, W. BAIR. E. ZOHARY. C. KOCH 400 800 1200 Trial Number ernu090 -100 -50 0 50 100 -50 -25 0 25 50 Lag (Trials) Lag (Trials) Figure 1: Normalized responses for two pairs of neurons and their trial crosscorrelograms (CCGs). The upper traces show the z-scored spike counts for all trials in the order they occurred. Spikes were counted during the 2 sec stimulus, but trials occurred on average 5 sec apart, so 100 trials represents about 2.5 minutes. The lower traces show the trial CCGs. For the pair of cells in the left panel, responsivity drifts during the experiment. The CCG (lower left) shows that the drift is correlated between the two neurons over nearly 100 trials. For the pair of cells in the right panel, the trial CCG shows a strong correlation only for simultaneous trials. Thus, the measured correlation coefficient (trial CCG at zero lag) seems to occur at a long time scale on the left but a short time scale (less than or equal to one trial) on the right. The zero-lag value can be broken into two components, T st and Tlt (short term and long term, respectively, see text). The short-term component, T st, is the value at zero lag minus the weighted average value at surrounding lag times. On the left, Tst ~ 0, while on the right, Tlt ~ O. Correlated Neuronal Response: Time Scales and Mechanisms 71 5 o 8 16 24 32 Width at HH (msec) 1 o emu064P -100 -50 o 50 100 Time Lag (msec) Figure 2: A spike train CCG with central peak. The frequency histogram of widths at half-height is shown (inset) for 92 cell pairs from three monkeys. The area of the central peak measured between ±32 msec is used to predict the correlation coefficients, rpeak. plotted in Fig. 3. The y-axis indicates the probability of a coincidence relative to that expected for Poisson processes at the measured firing rates. 0.8 0.6 ~ 0.4 ~ (1) ~ 0.2 • '-" ~ • 0 • • -0.2 -0.2 • • • • • • • • • ..... • • • • • • • • • o 0.2 0.4 0.6 0.8 r (Short Term) Figure 3: The area of the peak of the spike train CCG yields a prediction, rpeak (see Eqn. 2), that is strongly correlated (r = 0.71, p < 0.00001), with the short-term spike count correlation coefficient, rst . The absence of points in the lower right corner of the plot indicates that there are no cases of a pair of cells being strongly correlated without having a peak in the spike train CCG. 72 w. BAIR, E. ZOHARY, C. KOCH In Fig. 3, there are no pairs of neurons that have a short-term correlation and yet do not have a peak in the ±32 msec range of the spike train CCG. 3 CORRELATED SUPPRESSION There is little doubt that common excitatory input causes peaks like the one shown in Fig. 2 and therefore results in the correlated spike count at the time scale of the trial. However, we have also observed correlated periods of suppressed firing that may point to inhibition as another contribution to the CCG peaks and consequently to the correlated spike count. Fig. 4 A and B show the response of one neuron to coherent preferred and null direction motion, respectively. Excessively long inter-spike intervals (ISIs), or gaps, appear in the response to preferred motion, while bursts appear in the response to null motion. Across a database of 84 single neurons from a previous study (Britten et aI., 1992), the occurrence of the gaps and bursts has a symmetrical time course-both are most prominent on average from 600-900 msec post-stimulus onset, although there are substantial variations from cell to cell (Bair, 1995). The gaps, roughly 100 msec long, are not consistent with the slow, steady adaptation (presumably due to potassium currents) which is observed under current injection in neocortical pyramidal neurons, e.g., the RS 1 and RS2 neurons of Agmon and Connors (1992). Fig. 4 C shows spike trains from two simultaneously recorded neurons stimulated with preferred direction motion. The longest gaps appear to occur at about the same time. To assess the correlation with a cross-correlogram, we first transform the spike trains to interval trains, shown in Fig. 4 D for the spike trains in C. This emphasizes the presence of long ISIs and removes some of the information regarding the precise occurrence times of action potentials. The interval crosscorrelation (ICC) between each pair of interval trains is computed and averaged over all trials, and the average shift predictor is subtracted. Fig. 4 E and F show ICCs (thick lines) for two different pairs of neurons. In 17 of 31 pairs (55%), there were peaks in the raw ICC that were at least 4 standard errors above the level of the shift predictor. The peaks were on average centered (mean 4.3 msec, SD 54 msec) and had mean width at half-height of 139 msec (SD 59 msec). To isolate the cause of the peaks, the long intervals in the trains were set to the mean of the short intervals. Long intervals were defined as those that accounted for 30% of the duration of the data and were longer than all short intervals. Note that this is only a small fraction of the number of ISIs in the spike train (typically less than about 10%), since a few long intervals consume the same amount of time as many short intervals. Data from 300-1950 msec was processed, avoiding the on-transient and the lack of final interval. With the longest intervals neutralized, the peaks were pushed down to the level of the noise in the ICC (thin lines, Fig. 4 E, F). Thus, 90% of the action potentials may serve to set a mean rate, while a few periods of long ISIs dominate the ICC peaks. The correlated gaps are consistent with common inhibition to neurons in a local region of cortex, and this inhibition adds area to the spike train CCG peaks in the form of a broader base (not shown). The data analyzed here is from behaving animals, so the gaps may be related to small saccades (within the 0.5 degree Correlated Neuronal Response: Time Scales and Mechanisms 73 1 2 I I 0 II II I I 1111 """111111111111"'"" 11111'""111111111"11 ""11"1'"''''''"' III ""'11111"'"'' 1111111 '"' ""II.!IIIIIIIIIIII "11"""1111111110111111111111111111111""111'1 II IlIg" "' 111111111111111"11111 111"'."111111111111 11111111'"11111111111111111111 111111111'"'"111 IIItIlI! 111111 1111111111'"1111111 1111111111 '"IIIIIII!!I1I11'"1I1I 1I1"'"tll ""UIU ""'"'" '"""" 111'"""""1111 1111"111'"'"1111111 '"""""'"11'"111111' 11111111'"1111111111'"'"""'"' 11""'"1111111 II' ""1111111111.11. 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"" 11"" 11111111111"1 "'"111'"'" III'"!II " I II! 1111111.111111111111111111111' 11111111""'"1111111111111'1111111111 '1IIIII'"IIIIIIII""""""YlI.""111I ,""'',''1''','' ,,'.', '11I'1~I'M'i"IlI'III"'I,IIII"II'"III,III' t""'IIIIII "II'"'~IIIIII,II!'"'''',,\ I III I "' II "' II I I II 1111 ,. 'I " I I "" ta' II! .111 I'" I I "' • 1111 III " "11 I "" III II .1 I' N 1 I I II • '"~ '111' I! II "' 11 II 11111 , I II. I" I III III III I' 'I! • II II i I A B 500 1000 1500 2000 11111111111111 11.1111 1111 1 E msec 1111111111111111 11111111 II 01 n 11111111 11111111111111111111111111111111111111111111111111118 11111 c 1111 1 11111111 111111 I 1I11 III I 11I1111I111I111I111I111111I11Im1l11111 II D 1000 Time (msec) 2000 F -1000 -500 o 500 1000 -1000 -500 Time Lag (msec) o 500 1000 Figure 4: (A) The brisk response to coherent preferred direction motion is interrupted by occasional excessively long inter-spike intervals, i.e., gaps. (B) The suppressed response to null direction motion is interrupted by bursts of spikes. (C) Simultaneous spike trains from two neurons show correlated gaps in the preferred direction response. (D) The interval representation for the spike trains in C. (E,F) Interval cross-correlograms have peaks indicating that the gaps are correlated (see text). 74 w. BAIR. E. ZOHARY. C. KOCH fixation window) or eyelid blink. It has been hypothesized that blink suppression and saccadic visual suppression may operate through the same pathways and are of neuronal origin (Ridder and Tomlinson, 1993). An alternative hypothesis is that the gaps and bursts arise in cortex from intrinsic circuitry arranged in an opponent fashion. 4 CONCLUSION Common input that causes central peaks on the order of tens of milliseconds wide in spike train CCGs is also responsible for causing the correlation in spike count at the time scale of two second long trials. Long-term correlation due to drifts in responsivity exists but is zero on average across all cell pairs and may represent a source of noise which complicates the accurate measurement of cell-to-cell correlation. The area of the peak of the spike train CCG within a window of ±32 msec is the basis of a good prediction of the spike count correlation coefficient and provides a less noisy measure of correlation between neurons. Correlated gaps observed in the response to coherent preferred direction motion is consistent with common inhibition and contributes to the area of the spike train CCG peak, and thus to the correlation between spike count. Correlation in spike count is an important factor that can limit the useful pool-size of neuronal ensembles (Zohary et al., 1994; Gawne and Richmond, 1993). Acknowledgements We thank William T. Newsome, Kenneth H. Britten, Michael N. Shadlen, and J. Anthony Movshon for kindly providing data that was recorded in previous studies and for helpful discussion. This work was funded by the Office of Naval Research and the Air Force Office of Scientific Research. W. B. was supported by the L. A. Hanson Foundation and the Howard Hughes Medical Institute. References Agmon A, Connors BW (1992) Correlation between intrinsic firing patterns and thalamocortical synaptic responses of neurons in mouse barrel cortex. J N eurosci 12:319-329. Bair W (1995) Analysis of Temporal Structure in Spike Trains of Visual Cortical Area MT. Ph.D. thesis, California Institute of Technology. Britten KH, Shadlen MN, Newsome WT, Movshon JA (1992) The analysis of visual motion: a comparison of neuronal and psychophysical performance. J Neurosci 12:4745-4765. Gawne T J, Richmond BJ (1993) How independent are the messages carried by adjacent inferior temporal cortical neurons? J Neurosci 13:2758-2771. Ridder WH, Tomlinson A (1993) Suppression of contrasts sensitivity during eyelid blinks. Vision Res 33: 1795- 1802. Zohary E, Shadlen MN, Newsome WT (1994) Correlated neuronal discharge rate and its implications for psychophysical performance. Nature 370:140-143.
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Worst-case Loss Bounds for Single Neurons David P. Helmbold Department of Computer Science University of California, Santa Cruz Santa Cruz, CA 95064 USA Jyrki Kivinen Department of Computer Science P.O. Box 26 (Teollisuuskatu 23) FIN-00014 University of Helsinki Finland Manfred K. Warmuth Department of Computer Science University of California, Santa Cruz Santa Cruz, CA 95064 USA Abstract We analyze and compare the well-known Gradient Descent algorithm and a new algorithm, called the Exponentiated Gradient algorithm, for training a single neuron with an arbitrary transfer function . Both algorithms are easily generalized to larger neural networks, and the generalization of Gradient Descent is the standard back-propagation algorithm. In this paper we prove worstcase loss bounds for both algorithms in the single neuron case. Since local minima make it difficult to prove worst-case bounds for gradient-based algorithms, we must use a loss function that prevents the formation of spurious local minima. We define such a matching loss function for any strictly increasing differentiable transfer function and prove worst-case loss bound for any such transfer function and its corresponding matching loss. For example, the matching loss for the identity function is the square loss and the matching loss for the logistic sigmoid is the entropic loss. The different structure of the bounds for the two algorithms indicates that the new algorithm out-performs Gradient Descent when the inputs contain a large number of irrelevant components. 310 D. P. HELMBOLD, J. KIVINEN, M. K. WARMUTH 1 INTRODUCTION The basic element of a neural network, a neuron, takes in a number of real-valued input variables and produces a real-valued output. The input-output mapping of a neuron is defined by a weight vector W E RN, where N is the number of input variables, and a transfer function ¢. When presented with input given by a vector x E RN, the neuron produces the output y = ¢(w . x). Thus, the weight vector regulates the influence of each input variable on the output, and the transfer function can produce nonlinearities into the input-output mapping. In particular, when the transfer function is the commonly used logistic function, ¢(p) = 1/(1 + e-P), the outputs are bounded between 0 and 1. On the other hand, if the outputs should be unbounded, it is often convenient to use the identity function as the transfer function, in which case the neuron simply computes a linear mapping. In this paper we consider a large class of transfer functions that includes both the logistic function and the identity function, but not discontinuous (e.g. step) functions. The goal of learning is to come up with a weight vector w that produces a desirable input-output mapping. This is achieved by considering a sequence S = ((X1,yt}, ... ,(Xl,Yl» of examples, where for t = 1, ... ,i the value Yt E R is the desired output for the input vector Xt, possibly distorted by noise or other errors. We call Xt the tth instance and Yt the tth outcome. In what is often called batch learning, alli examples are given at once and are available during the whole training session. As noise and other problems often make it impossible to find a weight vector w that would satisfy ¢(w· Xt) = Yt for all t, one instead introduces a loss function L, such as the square loss given by L(y, y) = (y - y)2/2, and finds a weight vector w that minimizes the empirical loss (or training error) l Loss(w,S) = LL(Yt,¢(w . xt}) . (1) t=l With the square loss and identity transfer function ¢(p) = p, this is the well-known linear regression problem. When ¢ is the logistic function and L is the entropic loss given by L(y, y) = Y In(yJY) + (1 - y) In((l - y)/(l - y)), this can be seen as a special case oflogistic regression. (With the entropic loss, we assume 0 ~ Yt, Yt ~ 1 for all t, and use the convention OlnO = Oln(O/O) = 0.) In this paper we use an on-line prediction (or life-long learning) approach to the learning problem. It is well known that on-line performance is closely related to batch learning performance (Littlestone, 1989; Kivinen and Warmuth, 1994). Instead of receiving all the examples at once, the training algorithm begins with some fixed start vector W1, and produces a sequence W1, ... , w l+1 of weight vectors. The new weight vector Wt+1 is obtained by applying a simple update rule to the previous weight vector Wt and the single example (Xt, Yt). In the on-line prediction model, the algorithm uses its tth weight vector, or hypothesis, to make the prediction Yt = ¢(Wt . xt). The training algorithm is then charged a loss L(Yt, Yt) for this tth trial. The performance of a training algorithm A that produces the weight vectors Wt on an example sequence S is measured by its total (cumulative) loss l Loss(A, S) = L L(Yt, ¢(Wt . Xt» . (2) t=l Our main results are bounds on the cumulative losses for two on-line prediction algorithms. One of these is the standard Gradient Descent (GO) algorithm. The other one, which we call EG±, is also based on the gradient but uses it in a different Worst-case Loss Bounds for Single Neurons 311 manner than GD. The bounds are derived in a worst-case setting: we make no assumptions about how the instances are distributed or the relationship between each instance Xt and its corresponding outcome Yt. Obviously, some assumptions are needed in order to obtain meaningful bounds. The approach we take is to compare the total losses, Loss(GD,5) and Loss(EG±, 5), to the least achievable empirical loss, infw Loss( w, 5). If the least achievable empirical loss is high, the dependence between the instances and outcomes in 5 cannot be tracked by any neuron using the transfer function, so it is reasonable that the losses of the algorithms are also high. More interestingly, if some weight vector achieves a low empirical loss, we also require that the losses of the algorithms are low. Hence, although the algorithms always predict based on an initial segment of the example sequence, they must perform almost as well as the best fixed weight vector for the whole sequence. The choice of loss function is crucial for the results that we prove. In particular, since we are using gradient-based algorithms, the empirical loss should not have spurious local minima. This can be achieved for any differentiable increasing transfer function ¢ by using the loss function L¢ defined by r1(y) L¢(y, fj) = f (¢(z) - y) dz . J ¢-l(y) (3) For y < fj the value L¢(y, fj) is the area in the z X ¢(z) plane below the function ¢(z), above the line ¢(z) = y, and to the left of the line z = ¢-l(fj). We call L¢ the matching loss function for transfer function ¢, and will show that for any example sequence 5, if L = L¢ then the mapping from w to Loss(w, 5) is conveX. For example, if the transfer function is the logistic function, the matching loss function is the entropic loss, and ifthe transfer function is the identity function, the matching loss function is the square loss. Note that using the logistic activation function with the square loss can lead to a very large number of local minima (Auer et al., 1996). Even in the batch setting there are reasons to use the entropic loss with the logistic transfer function (see, for example, Solla et al. , 1988). How much our bounds on the losses of the two algorithms exceed the least empirical loss depends on the maximum slope of the transfer function we use. More importantly, they depend on various norms of the instances and the vector w for which the least empirical loss is achieved. As one might expect, neither of the algorithms is uniformly better than the other. Interestingly, the new EG± algorithm is better when most of the input variables are irrelevant, i.e., when some weight vector w with Wi = 0 for most indices i has a low empirical loss. On the other hand, the GD algorithm is better when the weight vectors with low empirical loss have many nonzero components, but the instances contain many zero components. The bounds we derive concern only single neurons, and one often combines a number of neurons into a multilayer feedforward neural network. In particular, applying the Gradient Descent algorithm in the multilayer setting gives the famous back propagation algorithm. Also the EG± algorithm, being gradient-based, can easily be generalized for multilayer feedforward networks. Although it seems unlikely that our loss bounds will generalize to multilayer networks, we believe that the intuition gained from the single neuron case will provide useful insight into the relative performance of the two algorithms in the multilayer case. Furthermore, the EG± algorithm is less sensitive to large numbers of irrelevant attributes. Thus it might be possible to avoid multilayer networks by introducing many new inputs, each of which is a non-linear function of the original inputs. Multilayer networks remain an interesting area for future study. Our work follows the path opened by Littlestone (1988) with his work on learning 312 D. P. HELMBOLD, J. KIVINEN, M. K. WARMUTH thresholded neurons with sparse weight vectors. More immediately, this paper is preceded by results on linear neurons using the identity transfer function (CesaBianchi et aI., 1996; Kivinen and Warmuth, 1994). 2 THE ALGORITHMS This section describes how the Gradient Descent training algorithm and the new Exponentiated Gradient training algorithm update the neuron's weight vector. For the remainder of this paper, we assume that the transfer function </J is increasing and differentiable, and Z is a constant such that </J'(p) ~ Z holds for all pER. For the loss function LcjJ defined by (3) we have aLcjJ(Y, </J(w . x» = (</J(w . x) - Y)Xi . aWi (4) Treating LcjJ(Y, </J(w·x» for fixed x and Y as a function ofw, we see that the Hessian H of the function is given by Hij = </J'(W·X)XiXj. Then v T Hv = </J'(w·x)(v.x)2, so H is positive definite. Hence, for an arbitrary fixed 5, the empirical loss Loss( w, 5) defined in (1) as a function of W is convex and thus has no spurious local minima. We first describe the Gradient Descent (GO) algorithm, which for multilayer networks leads to the back-propagation algorithm. Recall that the algorithm's prediction at trial t is Yt = </J(Wt . Xt), where Wt is the current weight vector and Xt is the input vector. By (4), performing gradient descent in weight space on the loss incurred in a single trial leads to the update rule Wt+l = Wt - TJ(Yt Yt)Xt . The parameter TJ is a positive learning rate that multiplies the gradient of the loss function with respect to the weight vector Wt. In order to obtain worst-case loss bounds, we must carefully choose the learning rate TJ. Note that the weight vector Wt of GO always satisfies Wt = Wi + E!:; aixi for some scalar coefficients ai. Typically, one uses the zero initial vector Wi = O. A more recent training algorithm, called the Exponentiated Gradient (EG) algorithm (Kivinen and Warmuth, 1994), uses the same gradient in a different way. This algorithm makes multiplicative (rather than additive) changes to the weight vector, and the gradient appears in the exponent. The basic version of the EG algorithm also normalizes the weight vector, so the update is given by N Wt+i,i = Wt,ie-IJ(Yt-Yt)Xt" / L Wt,je-IJ(Yt-Y,)Xt,i j=i The start vector is usually chosen to be uniform, Wi = (1/ N, ... ,1/ N). Notice that it is the logarithms of the weights produced by the EG training algorithm (rather than the weights themselves) that are essentially linear combinations of the past examples. As can be seen from the update, the EG algorithm maintains the constraints Wt,i > 0 and Ei Wt,i = 1. In general, of course, we do not expect that such constraints are useful. Hence, we introduce a modified algorithm EG± by employinj a linear transformation of the inputs. In addition to the learning rate TJ, the EG algorithm has a scaling factor U > 0 as a parameter. We define the behavior of EG± on a sequence of examples 5 = ((Xi,Yi), .. . ,(Xl,Yl» in terms of the EG algorithm's behavior on a transformed example sequence 5' = ((xi, yd, .. . , (x~, Yl» Worst-case Loss Bounds for Single Neurons 313 where x' = (U Xl , ... , U XN , -U Xl, ... , -U XN) ' The EG algorithm uses the uniform start vector (1/(2N), . .. , 1/(2N» and learning rate supplied by the EG± algorithm. At each time time t the N-dimensional weight vector w of EG± is defined in terms of the 2N -dimensional weight vector Wi of EG as Wt,i = U(W~ , i W~ ,N+i ) ' Thus EG± with scaling factor U can learn any weight vector w E RN with Ilwlll < U by having the embedded EG algorithm learn the appropriate 2N-dimensional (nonnegative and normalized) weight vector Wi. 3 MAIN RESULTS The loss bounds for the GO and EG± algorithms can be written in similar forms that emphasize how different algorithms work well for different problems. When L = L¢n we write Loss¢(w, S) and Loss¢(A, S) for the empirical loss of a weight vector wand the total loss of an algorithm A, as defined in (1) and (2). We give the upper bounds in terms of various norms. For x E RN, the 2-norm Ilxl b is the Euclidean length of the vector x, the I-norm Ilxlll the sum of the absolute values of the components of x , and the (X)-norm Ilxlloo the maximum absolute value of any component of x . For the purposes of setting the learning rates, we assume that before training begins the algorithm gets an upper bound for the norms of instances. The GO algorithm gets a parameter X2 and EG a parameter Xoo such that IIxtl12 ~ X 2 and Ilxtl loo ~ X oo hold for all t. Finally, recall that Z is an upper bound on ¢/(p). We can take Z = 1 when ¢ is the identity function and Z = 1/4 when ¢ is the logistic function. Our first upper bound is for GO. For any sequence of examples S and any weight vector u ERN, when the learning rate is TJ = 1/(2X?Z) we have Loss¢(GO,S) ~ 2Loss¢(u,S) + 2(llulbX2)2Z . Our upper bounds on the EG± algorithm require that we restrict the one-norm of the comparison class: the set of weight vectors competed against. The comparison class contains all weight vectors u such that Ilulh is at most the scaling factor, U. For any scaling factor U , any sequence of examples S, and any weight vector u ERN with Ilulll ~ U, we have 4 16 Loss¢(EG± , S) ~ 3Loss¢(u,S)+ 3(UXoo )2Z1n(2N) when the learning rate is TJ = 1/(4(UXoo )2Z). Note that these bounds depend on both the unknown weight vector u and some norms of the input vectors. If the algorithms have some further prior information on the sequence S they can make a more informed choice of TJ. This leads to bounds with a constant of 1 before the the Loss¢(u, S) term at the cost of an additional square-root term (for details see the full paper, Helmbold et al. , 1996). It is important to realize that we bound the total loss of the algorithms over any adversarially chosen sequence of examples where the input vectors satisfy the norm bound. Although we state the bounds in terms of loss on the data, they imply that the algorithms must also perform well on new unseen examples, since the bounds still hold when an adversary adds these additional examples to the end of the sequence. A formal treatment of this appears in several places (Littlestone, 1989; 314 D. P. HELMBOLD, J. KIVINEN, M. K. WARMUTH Kivinen and Warmuth, 1994). Furthermore, in contrast to standard convergence proofs (e.g. Luenberger, 1984), we bound the loss on the entire sequence of examples instead of studying the convergence behavior of the algorithm when it is arbitrarily close to the best weight vector. Comparing these loss bounds we see that the bound for the EG± algorithm grows with the maximum component of the input vectors and the one-norm of the best weight vector from the comparison class. On the other hand, the loss bound for the GD algorithm grows with the tWo-norm (Euclidean length) of both vectors. Thus when the best weight vector is sparse, having few significant components, and the input vectors are dense, with several similarly-sized components, the bound for the EG± algorithm is better than the bound for the GD algorithm. More formally, consider the noise-free situation where Lossr/>(u, S) = 0 for some u. Assume Xt E { -1, I}N and U E {-I, 0, I}N with only k nonzero components in u. We can then take X 2 = ..,(N, Xoo = 1, IIuI12 = Vk, and U = k. The loss bounds become (16/3)k 2Z1n(2N) for EG± and 2kZN for GD, so for N ~ k the EG± algorithm clearly wins this comparison. On the other hand, the GD algorithm has the advantage over the EG algorithm when each input vector is sparse and the best weight vector is dense, having its weight distributed evenly over its components. For example, if the inputs Xt are the rows of an N x N unit matrix and U E { -1, 1 } N , then X2 = Xoo = 1, IIuI12 = ..,(N, and U = N. Thus the upper bounds become (16/3)N 2Z1n(2N) for EG± and 2NZ for GD, so here GD wins the comparison. Of course, a comparison of the upper bounds is meaningless unless the bounds are known to be reasonably tight. Our experiments with artificial random data suggest that the upper bounds are not tight. However, the experimental evidence also indicates that EG± is much better than G D when the best weight vector is sparse. Thus the upper bounds do predict the relative behaviors of the algorithms. The bounds we give in this paper are very similar to the bounds Kivinen and Warmuth (1994) obtained for the comparison class of linear functions and the square loss. They observed how the relative performances of the GD and EG± algorithms relate to the norms of the input vectors and the best weight vector in the linear case. Our methods are direct generalizations of those applied for the linear case (Kivinen and Warmuth, 1994). The key notion here is a distance function d for measuring the distance d( u, w) between two weight vectors U and w. Our main distance measures are the Squared Euclidean distance ~ II u - w II ~ and the Relative Entropy distance (or Kullback-Leibler divergence) L~l Ui In(ui/wi). The analysis exploits an invariant over t and u of the form aLr/>(Yt, Wt . Xt) - bLr/>(Yt, U· Xt) ~ d(u, Wt) - d(u, Wt+l) , where a and b are suitably chosen constants. This invariant implies that at each trial, if the loss of the algorithm is much larger than that of an arbitrary vector u, then the algorithm updates its weight vector so that it gets closer to u. By summing the invariant over all trials we can bound the total loss of the algorithms in terms of Lossr/>(u, S) and d(u, wI). Full details will be contained in a technical report (Helmbold et al., 1996). 4 OPEN PROBLEMS Although the presence of local minima in multilayer networks makes it difficult to obtain worst case bounds for gradient-based algorithms, it may be possible to Worst-case Loss Bounds for Single Neurons 315 analyze slightly more complicated settings than just a single neuron. One likely candidate is to generalize the analysis to logistic regression with more than two classes. In this case each class would be represented by one neuron. As noted above, the matching loss for the logistic transfer function is the entropic loss, so this pair does not create local minima. No bounded transfer function matches the square loss in this sense (Auer et aI., 1996), and thus it seems impossible to get the same kind of strong loss bounds for a bounded transfer function and the square loss as we have for any (increasing and differentiable) transfer function and its matching loss function . As the bounds for EG± depend only logarithmically on the input dimension, the following approach may be feasible. Instead of using a multilayer net, use a single (linear or sigmoided) neuron on top of a large set of basis functions. The logarithmic growth of the loss bounds in the number of such basis functions mean that large numbers of basis functions can be tried. Note that the bounds of this paper are only worst-case bounds and our experiments on artificial data indicate that the bounds may not be tight when the input values and best weights are large. However, we feel that the bounds do indicate the relative merits of the algorithms in different situations. Further research needs to be done to tighten the bounds. Nevertheless, this paper gives the first worst-case upper bounds for neurons with nonlinear transfer functions. References P. Auer, M. Herbster, and M. K. Warmuth (1996). Exponentially many local minima for single neurons. In Advances in Neural Information Processing Systems 8. N. Cesa-Bianchi, P. Long, and M. K. Warmuth (1996). Worst-case quadratic loss bounds for on-line prediction of linear functions by gradient descent. IEEE Transactions on Neural Networks. To appear. An extended abstract appeared in COLT '93, pp. 429-438. D. P. Helmbold, J . Kivinen, and M. K. Warmuth (1996). Worst-case loss bounds for single neurons. Technical Report UCSC-CRL-96-2, Univ. of Calif. Computer Research Lab, Santa Cruz, CA, 1996. In preparation. J . Kivinen and M. K. Warmuth (1994). Exponentiated gradient versus gradient descent for linear predictors. Technical Report UCSC-CRL-94-16, Univ. of Calif. Computer Research Lab, Santa Cruz, CA, 1994. An extended abstract appeared in STOC '95, pp. 209-218. N. Littlestone (1988). Learning when irrelevant attributes abound: A new linearthreshold algorithm. Machine Learning, 2:285-318. N. Littlestone (1989). From on-line to batch learning. In Proc. 2nd Annual Workshop on Computational Learning Theory, pages 269-284. Morgan Kaufmann, San Mateo, CA. D. G. Luenberger (1984). Linear and Nonlinear Programming. Addison-Wesley, Reading, MA. S. A. Solla, E. Levin, and M. Fleisher (1988). Accelerated learning in layered neural networks. Complex Systems, 2:625- 639 .
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Stock Selection via Nonlinear Multi-Factor Models Asriel U. Levin BZW Barclays Global Investors Advanced Strategies and Research Group 45 Fremont Street San Francisco CA 94105 email: asriel.levin@bglobal.com Abstract This paper discusses the use of multilayer feed forward neural networks for predicting a stock's excess return based on its exposure to various technical and fundamental factors. To demonstrate the effectiveness of the approach a hedged portfolio which consists of equally capitalized long and short positions is constructed and its historical returns are benchmarked against T-bill returns and the S&P500 index. 1 Introduction Traditional investment approaches (Elton and Gruber, 1991) assume that the return of a security can be described by a multifactor linear model: (1) where Hi denotes the return on security i, Fl are a set of factor values and Uil are security i exposure to factor I, ai is an intercept term (which under the CAPM framework is assumed to be equal to the risk free rate of return (Sharpe, 1984)) and ei is a random term with mean zero which is assumed to be uncorrelated across securities. The factors may consist of any set of variables deemed to have explanatory power for security returns. These could be aspects of macroeconomics, fundamental security analysis, technical attributes or a combination of the above. The value of a factor is the expected excess return above risk free rate of a security with unit exposure to the factor and zero exposure to all other factors. The choice offactors can be viewed as a proxy for the" state of the world" and their selection defines a metric imposed on the universe of securities: Once the factors are set, the model assumption is that, Stock Selection via Nonlinear Multi-factor Models 967 on average, two securities with similar factor loadings (Uil) will behave in a similar manner. The factor model (1) was not originally developed as a predictive model, but rather as an explanatory model, with the returns It; and the factor values Pi assumed to be contemporaneous. To utilize (1) in a predictive manner, each factor value must be replaced by an estimate, resulting in the model A A A It; = ai + UilFl + Ui2 F 2 + ... + UiLFL + ei (2) where Ri is a security's future return and F/ is an estimate of the future value of factor 1, based on currently available information. The estimation of Fl can be approached with varying degree of sophistication ranging from a simple use of the historical mean to estimate the factor value (setting Fl(t) = Fi), to more elaborate approaches attempting to construct a time series model for predicting the factor values. Factor models of the form (2) can be employed both to control risk and to enhance return. In the first case, by capturing the major sources of correlation among security returns, one can construct a well balanced portfolio which diversifies specific risk away. For the latter, if one is able to predict the likely future value of a factor, higher return can be achieved by constructing a portfolio that tilts toward "good" factors and away from "bad" ones. While linear factor models have proven to be very useful tools for portfolio analysis and investment management, the assumption of linear relationship between factor values and expected return is quite restrictive. Specifically, the use of linear models assumes that each factor affects the return independently and hence, they ignore the possible interaction between different factors. Furthermore, with a linear model, the expected return of a security can grow without bound as its exposure to a factor increases. To overcome these shortcomings of linear models, one would have to consider more general models that allow for nonlinear relationship among factor values, security exposures and expected returns. Generalizing (2), while maintaining the basic premise that the state of the world can be described by a vector of factor values and that the expected return of a security is determined through its coordinates in this factor world, leads to the nonlinear model: It; = j(Uil' Ui2,···, UiL, Fl , F2, ... , FL ) + ei (3) where JO is a nonlinear function and ei is the noise unexplained by the model, or "security specific risk" . The prediction task for the nonlinear model (3) is substantially more complex than in the linear case since it requires both the estimation of future factor values as well as a determination of the unknown function j. The task can be somewhat simplified if factor estimates are replaced with their historical means: It; J(Uil, Ui2, ... , UiL, lA, F2, ... , FL) + ei (4) where now Uil are the security's factor exposure at the beginning of the period over which we wish to predict. To estimate the unknown function t(-), a family of models needs to be selected, from which a model is to be identified. In the following we propose modeling the relationship between factor exposures and future returns using the class of multilayer feedforward neural networks (Hertz et al., 1991). Their universal approximation 968 A. U. LEVIN capabilities (Cybenko, 1989; Hornik et al., 1989), as well as the existence of an effective parameter tuning method (the backpropagation algorithm (Rumelhart et al., 1986)) makes this family of models a powerful tool for the identification of nonlinear mappings and hence a natural choice for modeling (4). 2 The stock selection problem Our objective in this paper is to test the ability of neural network based models of the form (4) to differentiate between attractive and unattractive stocks. Rather than trying to predict the total return of a security, the objective is to predict its performance relative to the market, hence eliminating the need to predict market directions and movements. The data set consists of monthly historical records (1989 through 1995) for the largest 1200-1300 US companies as defined by the BARRA HiCap universe. Each data record (::::::1300 per month) consists of an input vector composed of a security's factor exposures recorded at the beginning of the month and the corresponding output is the security's return over the month. The factors used to build the model include Earning/Price, Book/Price, past price performance, consensus of analyst sentiments etc, which have been suggested in the financial literature as having explanatory power for security returns (e.g. (Fama and French, 1992)). To minimize risk, exposure to other unwarranted factors is controlled using a quadratic optimizer. 3 Model construction and testing Potentially, changes in a price of a security are a function of a very large number of forces and events, of which only a small subset can be included in the factor model (4). All other sources of return play the role of noise whose magnitude is probably much larger than any signal that can be explained by the factor exposures. When this information is used to train a neural network, the network attempts to replicate the examples it sees and hence much of what it tries to learn will be the particular realizations of noise that appeared in the training set. To minimize this effect, both a validation set and regularization are used in the training. The validation set is used to monitor the performance of the model with data on which it has not been trained on. By stopping the learning process when validation set error starts to increase, the learning of noise is minimized. Regularization further limits the complexity of the function realized by the network and, through the reduction of model variance, improves generalization (Levin et al., 1994). The stock selection model is built using a rolling train/test window. First, M "two layer" feedforward networks are built for each month of data (result is rather insensitive to the particular choice of M). Each network is trained using stochastic gradient descent with one quarter of the monthly data (randomly selected) used as a validation set. Regularization is done using principal component pruning (Levin et al., 1994). Once training is completed, the models constructed over N consecutive month of data (again, result is insensitive to particular choice of N) are combined (thus increasing the robustness of the model (Breiman, 1994)) to predict the returns in the following month. Thus the predicted (out of sample) return of stock i in month k is given by (5) Stock Selection via Nonlinear Multi-factor Models 0.4 0.35 0.3 0.25 c 0 :;:::; 0.2 ctI Q) .... .... 0 0.15 () 0.1 ,-, , , 0.05 , ! i 0 i 0 5 Nonlinear Linear ~ -., I : i ! r---: ! ! 1! i r---l b -I I ! r- -I i ~mf ----_]'-T_-I ': : -rl -l-.+-r-+-,-...L-J.-l- -L J. , I L _J 10 15 20 Cell 969 Figure 1 : Average correlation between predicted alphas and realized returns for linear and nonlinear models where k(k) is stock's i predicted return, N Nk-j(·) denoted the neural network model built in month k - j and u71 are stock's i factor exposures as measured at the beginning of month k. 4 Benchmarking to linear As a first step in evaluating the added value of the nonlinear model, its performance was benchmarked against a generalized least squares linear model. Each model was run over three universes: all securities in the HiCap universe, the extreme 200 stocks (top 100, bottom 100 as defined by each model), and the extreme 100 stocks. As a comparative performance measure we use the Sharpe ratio (Elton and Gruber, 1991). As shown in Table 4, while the performance of the two models is quite comparable over the whole universe of stocks, the neural network based model performs better at the extremes, resulting in a substantially larger Sharpe ratio (and of course, when constructing a portfolio, it is the extreme alphas that have the most impact on performance). I Portfolio\Model II Linear Nonlinear II All HiCap 6.43 6.92 100 long/100 short 4.07 5.49 50 long/50 short 3.07 4.23 Table 1: Ex ante Sharpe ratios: Neural network vs. linear While the numbers in the above table look quite impressive, it should be emphasised that they do not represent returns of a practical strategy: turnover is huge and the figures do not take transaction costs into account. The main purpose of the table 970 A. U. LEVIN is to compare the information that can be captured by the different models and specifically to show the added value of the neural network at the extremes. A practical implementation scheme and the associated performance will be discussed in the next section. Finally, some insight as to the reason for the improved performance can be gained by looking at the correlation between model predictions and realized returns for different values of model predictions (commonly referred to as alphas). For that, the alpha range was divided to 20 cells, 5% of observations in each and correlations were calculated separately for each cell. As is shown in figure 1, while both neural network and linear model seem to have more predictive power at the extremes, the network's correlations are substantially larger for both positive and negative alphas. 5 Portfolio construction Given the superior predictive ability of the nonlinear model at the extremes, a natural way of translating its predictions into an investment strategy is through the use of a long/short construct which fully captures the model information on both the positive as well as the negative side. The long/short portfolio (Jacobs and Levy, 1993) is constructed by allocating equal capital to long and short positions. By monitoring and controlling the risk characteristics on both sides, one is able to construct a portfolio that has zero correlation with the market ((3 = 0) - a "market neutral" portfolio. By construction, the return of a market neutral portfolio is insensitive to the market up or down swings and its only source of return is the performance spread between the long and short positions, which in turn is a direct function of the model (5) discernment ability. Specifically, the translation of the model predictions into a realistically implementable strategy is done using a quadratic optimizer. Using the model predicted returns and incorporating volatility information about the various stocks, the optimizer is utilized to construct a portfolio with the following characteristics: • Market neutral (equal long and short capitalization). • Total number of assets in the portfolio <= 200. • Average (one sided) monthly turnover ~ 15%. • Annual active risk ~ 5%. In the following, all results are test set results (out of sample), net of estimated transaction costs (assumed to be 1.5% round trip). The standard benchmark for a market neutral portfolio is the return on 3 month T-bill and as can be seen in Table 2, over the test period the market neutral portfolio has consistently and decisively outperformed its benchmark. Furthermore, the results reported for 1995 were recorded in real-time (simulated paper portfolio). An interesting feature of the long/short construct is its ease of transportability (Jacobs and Levy, 1993). Thus, while the base construction is insensitive to market movement, if one wishes, full exposure to a desired market can be achieved through the use of futures or swaps (Hull, 1993). As an example, by adding a permanent S&P500 futures overlay in an amount equal to the invested capital, one is fully exposed to the equity market at all time, and returns are the sum of the long/short performance spread and the profits or losses resulting from the market price movements. This form of a long/short strategy is referred to as an "equitized" strategy and the appropriate benchmark will be overlayed index. The relative performance Stock Selection via Nonlinear Multi-factor Models 971 I Statistics II T-Bill I Neutral II S&P500 I Equitized II Total Return~%) 27.8 131.5 102.0 264.5 Annual total(Yr%) 4.6 16.8 10.4 27.0 Active Return(%) 103.7 162.5 Annual active(Yr%) 12.2 16.6 Active risk(Yr%) 4.8 4.8 Max draw down(%) 3.2 13.9 10.0 Turnover(Y r%) 198.4 198.4 Table 2: Comparative summary of ex ante portfolio performance (net of transaction costs) 8/90 - 12/95 4 3.5 Equitized --SP500 -+--3 Neutral .-0 . .. . T-bill ...... _. (I) ;:) 2.5 «i > .2 ~ 2 0 0.. C]) > ~ 1.5 "S E ::I () 91 92 93 94 95 96 Year Figure 2: Cumulative portfolio value 8/90 - 12/95 (net of estimated transaction costs) of the equitized strategy with an S&P500 futures overlay is presented in Table 2. Summary of the accumulated returns over the test period for the market neutral and equitized portfolios compared to T-bill and S&P500 are given in Figure 2. Finally, even though the performance of the model is quite good, it is very difficult to convince an investor to put his money on a "black box". A rather simple way to overcome this problem of neural networks is to utilize a CART tree (Breiman et aI., 1984) to explain the model's structure. While the performance of the tree on the raw data in substantially inferior to the network's, it can serve as a very effective tool for analyzing and interpreting the information that is driving the model. 6 Conclusion We presented a methodology by which neural network based models can be used for security selection and portfolio construction. In spite of the very low signal to noise ratio of the raw data, the model was able to extract meaningful relationship 972 A. U. LEVIN between factor exposures and expected returns. When utilized to construct hedged portfolios, these predictions achieved persistent returns with very favorable risk characteristics. The model is currently being tested in real time and given its continued consistent performance, is expected to go live soon. References Anderson, J. and Rosenfeld, E., editors (1988) . Neurocomputing: Foundations of Research. MIT Press, Cambridge. Breiman, L. (1994). Bagging predictors. Technical Report 416, Department of Statistics, VCB, Berkeley, CA. Breiman, L., Friedman, J ., Olshen, R., and Stone, C. (1984). Classification and Regression Trees. Chapman & Hall. Cybenko, G. (1989) . Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals, and Systems, 2:303-314. Elton, E. and Gruber, M. (1991). Modern Portfolio Theory and Investment Analysis. John Wiley. Fama, E. and French, K. (1992). The cross section of expected stock returns. Journal of Finance, 47:427- 465 . Hertz, J., Krogh, A., and Palmer, R. (1991). Introduction to the theory of neural computation, volume 1 of Santa Fe Institute studies in the sciences ofcomplexity. Addison Wesley Pub. Co. Hornik, K. , Stinchcombe, M., and White, H. (1989). Multilayer feedforward networks are universal approximators. Neural Networks, 2:359-366. Hull, J . (1993). Options, Futures and Other Derivative Securities. Prentice-Hall. Jacobs, B. and Levy, K. (1993). Long/short equity investing. Journal of Portfolio Management, pages 52-63. Levin, A. V., Leen, T. K., and Moody, J . E. (1994). Fast pruning using principal components. In Cowan, J . D., Tesauro, G., and Alspector, J., editors, Advances in Neural Information Processing Systems, volume 6. Morgan Kaufmann. to apear. Rumelhart, D., Hinton, G., and Williams, R. (1986) . Learning representations by back-propagating errors. Nature, 323:533- 536. Reprinted in (Anderson and Rosenfeld, 1988). Sharpe, W. (1984) . Factor models, CAPMs and the APT. Journal of Portfolio Management, pages 21-25.
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The Geometry of Eye Rotations and Listing's Law Amir A. Handzel* Tamar Flasht Department of Applied Mathematics and Computer Science Weizmann Institute of Science Rehovot, 76100 Israel Abstract We analyse the geometry of eye rotations, and in particular saccades, using basic Lie group theory and differential geometry. Various parameterizations of rotations are related through a unifying mathematical treatment, and transformations between co-ordinate systems are computed using the Campbell-BakerHausdorff formula. Next, we describe Listing's law by means of the Lie algebra so(3). This enables us to demonstrate a direct connection to Donders' law, by showing that eye orientations are restricted to the quotient space 80(3)/80(2). The latter is equivalent to the sphere S2, which is exactly the space of gaze directions. Our analysis provides a mathematical framework for studying the oculomotor system and could also be extended to investigate the geometry of mUlti-joint arm movements. 1 INTRODUCTION 1.1 SACCADES AND LISTING'S LAW Saccades are fast eye movements, bringing objects of interest into the center of the visual field. It is known that eye positions are restricted to a subset of those which are anatomically possible, both during saccades and fixation (Tweed & Vilis, 1990). According to Donders' law, the eye's gaze direction determines its orientation uniquely, and moreover, the orientation does not depend on the history of eye motion which has led to the given gaze direction. A precise specification of the "allowed" subspace of position is given by Listing's law: the observed orientations of the eye are those which can be reached from the distinguished orientation called primary *hand@wisdom.weizmann.ac.il t tamar@wisdom.weizmann.ac.il 118 A. A. HANDZEL, T. FLASH position through a single rotation about an axis which lies in the plane perpendicular to the gaze direction at the primary position (Listing's plane). We say then that the orientation of the eye has zero torsion. Recently, the domain of validity of Listing's law has been extended to include eye vergence by employing a suitable mathematical treatment (Van Rijn & Van Den Berg, 1993). Tweed and Vilis used quaternion calculus to demonstrate, in addition, that in order to move from one allowed position to another in a single rotation, the rotation axis itself lies outside Listing's plane (Tweed & Vilis, 1987). Indeed, normal saccades are performed approximately about a single axis. However, the validity of Listing's law does not depend on the rotation having a single axis, as was shown in double-step target displacement experiments (Minken, Van Opstal & Van Gisbergen, 1993): even when the axis of rotation itself changes during the saccade, Listing's law is obeyed at each and every point along the trajectory which is traced by the eye. Previous analyses of eye rotations (and in particular of Listing's law) have been based on various representations of rotations: quaternions (Westheimer, 1957), rotation vectors (Hepp, 1990), spinors (Hestenes, 1994) and 3 x 3 rotation matrices; however, they are all related through the same underlying mathematical object the three dimensional (3D) rotation group. In this work we analyse the geometry of saccades using the Lie algebra of the rotation group and the group structure. Next, we briefly describe the basic mathematical notions which will be needed later. This is followed by Section 2 in which we analyse various parameterizations of rotations from the point of view of group theory; Section 3 contains a detailed mathematical analysis of Listing's law and its connection to Donders' law based on the group structure; in Section 4 we briefly discuss the issue of angular velocity vectors or axes of rotation ending with a short conclusion. 1.2 THE ROTATION GROUP AND ITS LIE ALGEBRA The group of rotations in three dimensions, G = 80(3), (where '80' stands for special orthogonal transformations) is used both to describe actual rotations and to denote eye positions by means of a unique virtual rotation from the primary position. The identity operation leaves the eye at the primary position, therefore, we identify this position with the unit element of the group e E 80(3). A rotation can be parameterized by a 3D axis and the angle of rotation about it. Each axis "generates" a continuous set of rotations through increasing angles. Formally, if n is a unit axis of rotation, then EXP(O· n) (1) is a continuous one-parameter subgroup (in G) of rotations through angles () in the plane that is perpendicular to n. Such a subgroup is denoted as 80(2) C 80(3). We can take an explicit representation of n as a matrix and the exponent can be calculated as a Taylor series expansion. Let us look, for example, at the one parameter subgroup of rotations in the y- z plane, i.e. rotations about the x axis which is represented in this case by the matrix o o -1 A direct computation of this rotation by an angle () gives o cos () - sin () o sin () cos () (2) ) (3) The Geometry of Eye Rotations and Listing's Law 119 where I is the identity matrix. Thus, the rotation matrix R( 0) can be constructed from the axis and angle of rotation. The same rotation, however, could also be achieved using A Lx instead of Lx, where A is any scalar, while rescaling the angle to 0/ A. The collection of matrices ALx is a one dimensional linear space whose elements are the generators of rotations in the y-z plane. The set of all the generators constitutes the Lie algebra of a group. For the full space of 3D rotations, the Lie algebra is the three dimensional vector space that is spanned by the standard orthonormal basis comprising the three direction vectors of the principal axes: (4) Every axis n can be expressed as a linear combination of this basis. Elements of the Lie algebra can also be represented in matrix form and the corresponding basis for the matrix space is L.= 0 0 D L, = ( 0 0 n L, = ( ~1 1 D; 0 0 0 0 -1 -1 0 0 (5) hence we have the isomorphism ( -~, Oz 8, ) U: ) 0 Ox +-------t -Oy -Ox 0 (6) Thanks to its linear structure, the Lie algebra is often more convenient for analysis than the group itself. In addition to the linear structure, the Lie algebra has a bilinear antisymmetric operation defined between its elements which is called the bracket or commutator. The bracket operation between vectors in g is the usual vector cross product. When the elements of the Lie algebra are written as matrices, the bracket operation becomes a commutation relation, i.e. [A,B] == AB - BA. (7) As expected, the commutation relations of the basis matrices of the Lie algebra (of the 3D rotation group) are equivalent to the vector product: (8) Finally, in accordance with (1), every rotation matrix is obtained by exponentiation: R(8) = EXP(OxLx +OyLy +OzLz). where 8 stands for the three component angles. 2 CO-ORDINATE SYSTEMS FOR ROTATIONS (9) In linear spaces the "position" of a point is simply parameterized by the co-ordinates w.r.t. the principal axes (a chosen orthonormal basis). For a non-linear space (such as the rotation group) we define local co-ordinate charts that look like pieces of a vector space ~ n. Several co-ordinate systems for rotations are based on the fact that group elements can be written as exponents of elements of the Lie algebra (1). The angles 8 appearing in the exponent serve as the co-ordinates. The underlying property which is essential for comparing these systems is the noncommutativity of rotations. For usual real numbers, e.g. Cl and C2, commutativity implies expCI expC2 = expCI +C2. A corresponding equation for non-commuting elements is the Campbell-Baker-Hausdorff formula (CBH) which is a Taylor series 120 A. A. HANDZEL. T. FLASH expansion using repeated commutators between the elements of the Lie algebra. The expansion to third order is (Choquet-Bruhat et al., 1982): EXP(Xl)EXP(X2) = EXP (Xl + X2 + ~[Xl' X2] + 112 [Xl - X2, [Xl, X2]]) (10) where Xl, X2 are variables that stand for elements of the Lie algebra. One natural parameterization uses the representation of a rotation by the axis and the angle of rotation. The angles which appear in (9) are then called canonical co-ordinates of the first kind (Varadarajan, 1974). Gimbal systems constitute a second type of parameterization where the overall rotation is obtained by a series of consecutive rotations about the principal axes. The component angles are then called canonical co-ordinates of the second kind. In the present context, the first type of co-ordinates are advantageous because they correspond to single axis rotations which in turn represent natural eye movements. For convenience, we will use the name canonical co-ordinates for those of the first kind, whereas those of the second type will simply be called gimbals. The gimbals of Fick and Helmholtz are commonly used in the study of oculomotor control (Van Opstal, 1993). A rotation matrix in Fick gimbals is RF(Bx,Oy,Oz) = EXP(OzLz) . EXP(ByLy) . EXP(OxLx), (11) and in Helmholtz gimbals the order of rotations is different: RH(Ox, By,Oz) = EXP(ByLy) . EXP(OzLz) . EXP(OxLx). (12) The CBH formula (10) can be used as a general tool for obtaining transformations between various co-ordinate systems (Gilmore, 1974) such as (9,11,12). In particular, we apply (10) to the product of the two right-most terms in (11) and then again to the product of the result with the third term. We thus arrive at an expression whose form is the same as the right hand side of (10). By equating it with the expression for canonical angles (9) and then taking the log of the exponents on both sides of the equation, we obtain the transformation formula from Fick angles to canonical angles. Repeating this calculation for (12) gives the equivalent formula for Helmholtz anglesl . Both transformations are given by the following three equations where OF,H stands for an angle either in Fick or in Helmholtz co-ordinates; for Helmholtz angles there is a plus sign in front of the last term of the first equation and a minus sign in the case of Fick angles: Be - OF,H (1 _ ...L ((BF,H)2 + (OF,H)2)) ± lOF,H OF,H x x 12 Y z 2 Y z Of = O:,H ( 1 - /2 (( O;,H)2 + (O:,H)2) ) + ~O;,H O:,H Of = O;,H ( 1 - /2 (( B;,H? + (B:,H)2)) - !O;,H O:,H (13) The error caused by the above approximation is smaller than 0.1 degree within most of the oculomotor range. We mention in closing two additional parameterizations, namely quaternions and rotation vectors. Unit quaternions lie on the 3D sphere S3 (embedded in lR 4) which constitutes the same manifold as the group of unitary rotations SU(2). The latter is the double covering group of SO(3) having the same local structure. This enables to use quaternions to parameterize rotations. The popular rotation vectors (written as tan(Oj2)n, n being the axis of rotation and B its angle) are closely related to 1 In contrast to this third order expansion, second order approximations usually appear in the literature; see for example equation B2 in (Van Rijn & Van Den Berg, 1993). The Geometry of Eye Rotations and Listing's Law 121 quaternions because they are central (gnomonic) projections of a hemisphere of S3 onto the 3D affine space tangent to the quaternion qe = (1,0,0,0) E ]R4. 2 3 LISTING'S LAW AND DONDERS' LAW A customary choice of a head fixed coordinate system is the following: ex IS III the straight ahead direction in the horizontal plane, ey is in the lateral direction and ez points upwards in the vertical direction. ex and ez thus define the midsagittal plane; ey and ez define the coronal plane. The principal axes of rotations (Lx, Ly, Lz) are set parallel to the head fixed co-ordinate system. A reference eye orientation called the primary position is chosen with the gaze direction being (1,0,0) in the above co-ordinates. How is Listing's law expressed in terms of the Lie algebra of SO(3)? The allowed positions are generated by linear combinations of Lz and Ly only. This 2D subspace of the Lie algebra, 1 = Span{Ly, Lz }, (14) is Listing's plane. Denoting Span{ Lx} by h, we have a decomposition of the Lie algebra so(3) into a direct sum of two linear subspaces: 9 = 1 EB h. (15) Every vector v E 9 can be projected onto its component which is in I: proj. V = VI + Vh ----t VI. (16) Until now, only the linear structure has been considered. In addition, h is closed under the bracket operation: (17) and because h is closed both under vector addition and the Lie bracket, it is a sub algebra of g. In contrast, I is not a sub algebra because it is not closed under commutation (8) . The fact that h stands as an algebra on its own implies that it has a corresponding group H, just as 9 = so(3) corresponds to G = SO(3). The subalgebra h generates rotations about the x axis, and therefore H is SO(2), the group of rotations in a plane. The group G = SO(3) does not have a linear structure. We may still ask whether some kind of decomposition and projection can be achieved in G in analogy to (15,16). The answer is positive and the projection is performed as follows: take any element of the group, a E G, and multiply it by all the elements of the subgroup H. This gives a subset in G which is considered as a single object a called a coset: a = {ab I bEH} . (18) The set of all cosets constitutes the quotient space. It is written as S == G / H = SO(3)/ SO(2) (19) because mapping the group to the quotient space can be understood as dividing G by H. The quotient space is not a group , and this corresponds to the fact that the subspace I above (14) is not a sub algebra. The quotient space has been constructed algebraically but is difficult to visualize; however, it is mathematically equivalent 2 Geometrically, each point q E S3 can be connected to the center of the sphere by a line. Another line runs from qe in the direction parallel to the vector part of q within the tangent space. The intersection of the two lines is the projected point. Numerically, one simply takes the vector part of q divided by its scalar part. 122 A. A.HANDZEL,T. FLASH Table 1: Summary table of biological notions and the corresponding mathematical representation, both in terms of the rotation group and its Lie algebra. Biological notion Lie Algebra Rotation Group general eye position 9 = so(3) = h El71 G = SO(3) primary position O.q E 9 eE G eye torsion h = Span{Lx} H = SO(2) "allowed" eye 1= Span{ Ly, LzJ S = ~/H = SO(3)/SO(2) positions (Listing's plane) ~ S2 (Donders' sphere of gaze directions) to another space the unit sphere S2 (embedded in ~3). This equivalence can be seen in the following way: a unit vector in ~3, e.g. e = (1,0,0), can be rotated so that its head reaches every point on the unit sphere S2; however, for any such point there are infinitely many rotations by which the point can be reached. Moreover, all the rotations around the x axis leave the vector e above invariant. We therefore have to "factor out" these rotations (of H =SO(2» in order to eliminate the above degeneracy and to obtain a one-to-one correspondence between the required subset of rotations and the sphere. This is achieved by going to the quotient space. The matrix of a torsion less rotation (generated by elements in Listing's plane) is obtained by setting Ox = 0 in (9): ( cosO R = - sin 0 sin ljJ - sin 0 cos ljJ sin 0 sin ljJ cos 0 + (1 - cos 0) cos2 ljJ cos ljJ sin ljJ(l - cos 0) cos ljJ sin ljJ(l - cos 0) ,(20) sin 0 cos ljJ ) cos 0 + (1 - cos 0) sin 2 ljJ where 0 = .)0;+0; is the total angle of rotation and ljJ is the angle between 0 and the y axis in the Oy -Oz plane, i.e. (0, ljJ) are polar co-ordinates in Listing's plane. Notice that the first column on the left constitutes the Cartesian co-ordinates of a point on a sphere of unit radius (Gilmore, 1974). As we have just seen, there is an exact correspondence between the group level and the Lie algebra level. In fact, the two describe the same reality, the former in a global manner and the latter in an infinitesimal one. Table 1 summarizes the important biological notions concerning Listing's law together with their corresponding mathematical representations. The connection between Donders' law and Listing's law can now be seen in a clear and intuitive way. The sphere, which was obtained by eliminating torsion, is the space of gaze directions. Recall that Donders' law states that the orientation of the eye is determined uniquely by its gaze direction. Listing's law implies that we need only take into consideration the gaze direction and disregard torsion. In order to emphasize this point, we use the fact that locally, SO(3) looks like a product of topological spaces: 3 p = u x SO(2) where (21) U parameterizes gaze direction and SO(2) torsion. Donders' law restricts eye orientation to an unknown 2D submanifold of the product space P. Listing's law shows that the submanifold is U, a piece of the sphere. This representation is advantageous for biological modelling, because it mathematically sets apart the degrees of freedom of gaze orientation from torsion, which also differ functionally. 350(3) is a principal bundle over S2 with fiber 50(2). The Geometry of Eye Rotations and Listing's Law 123 4 AXES OF ROTATION FOR LISTING'S LAW As mentioned in the introduction, moving between two (non-primary) positions requires a rotation whose axis (i.e. angular velocity vector) lies outside Listing's plane. This is a result of the group structure of SO(3). Had the axis of rotation been contained within Listing's plane, the matrices of the quotient space (20) should have been closed under multiplication so as to form a subgroup of SO(3). In other words, if ri and rJ are matrices representing the current and target orientations of the eye corresponding to axes in Listing's plane, then rJ . r;l should have been a matrix of the same form (20); however, as explained in Section 3, this condition is not fulfilled. Finally, since normal saccades involve rotations about a single axis, they are oneparameter subgroups generated by a single element of the Lie algebra (1). In addition, they have the property of being geodesic curves in the group manifold under the natural metric which is given by the bilinear Cartan-Killing form of the group (Choquet-Bruhat et al., 1982). 5 CONCLUSION We have analysed the geometry of eye rotations using basic Lie group theory and differential geometry. The unifying view presented here can serve to improve the understanding of the oculomotor system. It may also be extended to study the three dimensional rotations of the joints of the upper limb. Acknowledgements We would like to thank Stephen Gelbart, Dragana Todoric and Yosef Yomdin for instructive conversations on the mathematical background and Dario Liebermann for fruitful discussions. Special thanks go to Stan Gielen for conversations which initiated this work. References Choquet-Bruhat Y., De Witt-Morette C. & Dillard-Bleick M., Analysis, Manifolds and Physics, North-Holland (1982). Gilmore R.,LieGroups, Lie Algebras, and Some of Their Applications, Wiley (1974). Hepp K., Commun. Math. Phys. 132 (1990) 285-292. Hestenes D., Neural Networks 7, No.1 (1994) 65-77. Minken A.W.H. Van Opstal A.J. & Van Gisbergen J.A.M., Exp. Brain Research 93 (1993) 521-533. Tweed, D. & Vilis T., J. Neurophysiology 58 (1987) 832-849. Tweed D. & Vilis T., Vision Research 30 (1990) 111-127. Van Opstal J., "Representations of Eye Positions in Three Dimensions", in Multisensory Control of Movement, ed. Berthoz A., (1993) 27-4l. Van Rijn L.J. & Van Den Berg A.V., Vision Research 33, No. 5/6 (1993) 691-708. Varadarajan V.S., Lie Groups, Lie Algebras, and Their Reps., Prentice-Hall (1974). Westheimer G., Journal of the Optical Society of America 47 (1957) 967-974.
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When is an Integrate-and-fire Neuron like a Poisson Neuron? Charles F. Stevens Salk Institute MNL/S La Jolla, CA 92037 cfs@salk.edu Anthony Zador Salk Institute MNL/S La Jolla, CA 92037 zador@salk.edu Abstract In the Poisson neuron model, the output is a rate-modulated Poisson process (Snyder and Miller, 1991); the time varying rate parameter ret) is an instantaneous function G[.] of the stimulus, ret) = G[s(t)]. In a Poisson neuron, then, ret) gives the instantaneous firing rate-the instantaneous probability of firing at any instant t-and the output is a stochastic function of the input. In part because of its great simplicity, this model is widely used (usually with the addition of a refractory period), especially in in vivo single unit electrophysiological studies, where set) is usually taken to be the value of some sensory stimulus. In the integrate-and-fire neuron model, by contrast, the output is a filtered and thresholded function of the input: the input is passed through a low-pass filter (determined by the membrane time constant T) and integrated until the membrane potential vet) reaches threshold 8, at which point vet) is reset to its initial value. By contrast with the Poisson model, in the integrate-and-fire model the ouput is a deterministic function of the input. Although the integrate-and-fire model is a caricature of real neural dynamics, it captures many of the qualitative features, and is often used as a starting point for conceptualizing the biophysical behavior of single neurons. Here we show how a slightly modified Poisson model can be derived from the integrate-and-fire model with noisy inputs yet) = set) + net). In the modified model, the transfer function G[.] is a sigmoid (erf) whose shape is determined by the noise variance /T~. Understanding the equivalence between the dominant in vivo and in vitro simple neuron models may help forge links between the two levels. 104 c. F. STEVENS. A. ZADOR 1 Introduction In the Poisson neuron model, the output is a rate-modulated Poisson process; the time varying rate parameter ret) is an instantaneous function G[.] of the stimulus, ret) = G[s(t)]. In a Poisson neuron, then, ret) gives the instantaneous firing rate-the instantaneous probability of firing at any instant t-and the output is a stochastic function of the input. In part because of its great simplicity, this model is widely used (usually with the addition of a refractory period), especially in in vivo single unit electrophysiological studies, where set) is usually taken to be the value of some sensory stimulus. In the integrate-and-fire neuron model, by contrast, the output is a filtered and thresholded function of the input: the input is passed through a low-pass filter (determined by the membrane time constant T) and integrated until the membrane potential vet) reaches threshold 0, at which point vet) is reset to its initial value. By contrast with the Poisson model, in the integrate-and-fire model the ouput is a deterministic function of the input. Although the integrate-and-fire model is a caricature of real neural dynamics, it captures many of the qualitative features, and is often used as a starting point for conceptualizing the biophysical behavior of single neurons (Softky and Koch, 1993; Amit and Tsodyks, 1991; Shadlen and Newsome, 1995; Shadlen and Newsome, 1994; Softky, 1995; DeWeese, 1995; DeWeese, 1996; Zador and Pearlmutter, 1996). Here we show how a slightly modified Poisson model can be derived from the integrate-and-fire model with noisy inputs yet) = set) + net). In the modified model, the transfer function G[.] is a sigmoid (erf) whose shape is determined by the noise variance (j~ . Understanding the equivalence between the dominant in vivo and in vitro simple neuron models may help forge links between the two levels. 2 The integrate-and-fire model Here we describe the the forgetful leaky integrate-and-fire model. Suppose we add a signal set) to some noise net), yet) = net) + set), and threshold the sum to produce a spike train z(t) = F[s(t) + net)], where F is the thresholding functional and z(t) is a list of firing times generated by the input. Specifically, suppose the voltage vet) of the neuron obeys vet) = - vet) + yet) (1) T where T is the membrane time constant. We assume that the noise net) has O-mean and is white with variance (j~. Thus yet) can be thought of as a Gaussian white process with variance (j~ and a time-varying mean set). If the voltage reaches the threshold 00 at some time t, the neuron emits a spike at that time and resets to the initial condition Vo. This is therefore a 5 parameter model: the membrane time constant T, the mean input signal Il, the variance of the input signal 172 , the threshold 0, and the reset value Vo. Of course, if net) = 0, we recover a purely deterministic integrate-and-fire model. When Is an Integrate-and-fire Neuron like a Poisson Neuron? 105 In order to forge the link between the integrate-and-fire neuron dynamics and the Poisson model, we will treat the firing times T probabilistically. That is, we will express the output of the neuron to some particular input set) as a conditional distribution p(Tls(t», i.e. the probability of obtaining any firing time T given some particular input set). Under these assumptions, peT) is given by the first passage time distribution (FPTD) of the Ornstein-Uhlenbeck process (Uhlenbeck and Ornstein, 1930; Tuckwell, 1988). This means that the time evolution of the voltage prior to reaching threshold is given by the Fokker-Planck equation (FPE), 8 u; 82 8 vet) 8t g(t, v) = 2 8v2 get, v) - av [(set) - --;:- )g(t, v)], (2) where uy = Un and get, v) is the distribution at time t of voltage -00 < v ::; (}o. Then the first passage time distribution is related to g( v, t) by 81 90 peT) = - at -00 get, v)dv. (3) The integrand is the fraction of all paths that p.ave not yet crossed threshold. peT) is therefore just the interspike interval (lSI) distribution for a given signal set). A general eigenfunction expansion solution for the lSI distribution is known, but it converges slowly and its terms offer little insight into the behavior (at least to us). We now derive an expression for the probability of crossing threshold in some very short interval ~t, starting at some v. We begin with the "free" distribution of g (Tuckwell, 1988): the probability of the voltage jumping to v' at time t' = t + ~t, given that it was at v at time t, assuming von Neumann boundary conditions at plus and minus infinity, get', v'lt, v) = exp y, 1 [ (v' - m( ~t; u »)2] J27r q(~t;Uy) 2 q(~t;Uy) (4) with and m(~t) = ve-at/ T + set) * T(l _ e-at/ T ), where * denotes convolution. The free distribution is a Gaussian with a timedependent mean m(~t) and variance q(~t; uy). This expression is valid for all ~t. The probability of making a jump ~v = v' - v in a short interval ~t ~ T depends only on ~v and ~t, ga(~t, ~v; uy) = 1 exp [_ ~~2 )]. ..j27r qa(uy) 2 qa uy (5) For small ~t, we expand to get qa(uy) :::::: 2u;~t, which is independent of T, showing that the leak can be neglected for short times. 106 c. F. STEVENS, A. ZADOR Now the probability Pt>, that the voltage exceeds threshold in some short Ilt, given that it started at v, depends on how far v is from threshold; it is Thus Pr[v + Ilv ~ 0] = Pr[llv ~ 0 - v]. (Xl dvgt>,(llt, v; O"y) J9-v -erfc 1 (o-v) 2 J2qt>,(O"y) -erfc 1 (o-v) 2 O"yJ21lt (6) where erfc(x) = 1 - -j; I; e-t~ dt goes from [2 : 0]. This then is the key result: it gives the instantaneous probability of firing as a function of the instantaneous voltage v. erfc is sigmoidal with a slope determined by O"y, so a smaller noise yields a steeper (more deterministic) transfer function; in the limit of 0 noise, the transfer function is a step and we recover a completely deterministic neuron. Note that Pt>, is actually an instantaneous function of v(t), not the stimulus itself s(t). If the noise is large compared with s(t) we must consider the distribution g$ (v, t; O"y) of voltages reached in response to the input s(t): Py(t) (7) 3 Ensemble of Signals What if the inputs s(t) are themselves drawn from an ensemble? If their distribution is also Gaussian and white with mean Jl and variance 0";, and if the firing rate is low (E[T] ~ T), then the output spike train is Poisson. Why is firing Poisson only in the slow firing limit? The reason is that, by assumption, immediately following a spike the membrane potential resets to 0; it must then rise (assuming Jl > 0) to some asymptotic level that is independent of the initial conditions. During this rise the firing rate is lower than the asymptotic rate, because on average the membrane is farther from threshold, and its variance is lower. The rate at which the asymptote is achieved depends on T. In the limit as t ~ T, some asymptotic distribution of voltage qoo(v), is attained. Note that if we make the reset Vo stochastic, with a distribution given by qoo (v), then the firing probability would be the same even immediately after spiking, and firing would be Poisson for all firing rates. A Poisson process is characterized by its mean alone. We therefore solve the FPE (eq. 2) for the steady-state by setting °tg(t, v) = 0 (we consider only threshold crossings from initial values t ~ T; negYecting the early events results in only a small error, since we have assumed E{T} ~ T). Thus with the absorbing boundary When Is an Integrate-and-fire Neuron like a Poisson Neuron? 107 at 0 the distribution at time t ~ T (given here for JJ = 0) is g~(Vj uy) = kl (1 - k2erfi [uyfi]) exp [~i:] , (8) where u; = u; + u~, erfi(z) = -ierf(iz), kl determines the normalization (the sign of kl determines whether the solution extends to positive or negative infinity) and k2 = l/erfi(O/(uy Vr)) is determined by the boundary. The instantaneous Poisson rate parameter is then obtained through eq. (7), (9) Fig. 1 tests the validity of the exponential approximation. The top graph shows the lSI distribution near the "balance point" , when the excitation is in balance with the inhibition and the membrane potential hovers just subthreshold. The bottom curves show the lSI distribution far below the balance point. In both cases, the exponential distribution provides a good approximation for t ~ T. 4 Discussion The main point of this paper is to make explicit the relation between the Poisson and integrate-and-fire models of neuronal acitivity. The key difference between them is that the former is stochastic while the latter is deterministic. That is, given exactly the same stimulus, the Poisson neuron produces different spike trains on different trials, while the integrate-and-fire neuron produces exactly the same spike train each time. It is therefore clear that if some degree of stochasticity is to be obtained in the integrate-and-fire model, it must arise from noise in the stimulus itself. The relation we have derived here is purely formalj we have intentionally remained agnostic about the deep issues of what is signal and what is noise in the inputs to a neuron. We observe nevertheless that although we derive a limit (eq. 9) where the spike train of an integrate-and-fire neuron is a Poisson process-i.e. the probability of obtaining a spike in any interval is independent of obtaining a spike in any other interval (except for very short intervals )-from the point of view of information processing it is a very different process from the purely stochastic rate-modulated Poisson neuron. In fact, in this limit the spike train is deterministically Poisson if u y = u., i. e. when n( t) = OJ in this case the output is a purely deterministic function of the input, but the lSI distribution is exponential. 108 C. F. STEVENS, A. ZADOR References Amit, D. and Tsodyks, M. (1991). Quantitative study of attractor neural network retrieving at low spike rates. i. substrate-spikes, rates and neuronal gain. Network: Computation in Neural Systems, 2:259-273. DeWeese, M. (1995). Optimization principles for the neural code. PhD thesis, Dept of Physics, Princeton University. DeWeese, M. (1996). Optimization principles for the neural code. In Hasselmo, M., editor, Advances in Neural Information Processing Systems, vol. 8. MIT Press, Cambridge, MA. Shadlen, M. and Newsome, W. (1994). Noise, neural codes and cortical organization. Current Opinion in Neurobiology, 4:569-579. Shadlen, M. and Newsome, W. (1995). Is there a signal in the noise? [comment]. Current Opinion in Neurobiology, 5:248-250. Snyder, D. and Miller, M. (1991). Random Point Processes in Time and Space, 2nd edition. Springer-Verlag. Softky, W. (1995). Simple codes versus efficient codes. Current Opinion in Neurobiology, 5:239-247. Softky, W. and Koch, C. (1993). The highly irregular firing of cortical cells is inconsistent with temporal integration of random epsps. J. Neuroscience., 13:334-350. Tuckwell, H. (1988). Introduction to theoretical neurobiology (2 vols.). Cambridge. Uhlenbeck, G. and Ornstein, L. (1930). On the theory of brownian motion. Phys. Rev., 36:823-84l. Zador, A. M. and Pearlmutter, B. A. (1996). VC dimension of an integrate and fire neuron model. Neural Computation, 8(3). In press. When Is an Integrate-and-fire Neuron like a Poisson Neuron? 109 lSI distributions at balance point and the exponential limit 0.02 0.015 .~ 15 0.01 .8 e a. 0.005 50 100 150 200 250 300 350 400 450 500 Time (msec) 2 x 10-3 1.5 ~ ~ 1 .0 0 ... a. 0.5 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 lSI (msec) Figure 1: lSI distributions. (A; top) lSI distribution for leaky integrate-and-fire model at the balance point, where the asymptotic membrane potential is just subthreshold, for two values of the signal variance (1'2 . Increasing (1'2 shifts the distribution to the left. For the left curve, the parameters were chosen so that E{T} ~ T, giving a nearly exponential distribution; for the right curve, the distribution would be hard to distinguish experimentally from an exponential distribution with a refractory period. (T = 50 msec; left: E{T} = 166 msec; right: E{T} = 57 msec). (B; bottom) In the subthreshold regime, the lSI distribution (solid} is nearly exponential (dashed) for intervals greater than the membrane time constant. (T = 50 msec; E{T} = 500 msec)
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A Novel Channel Selection System in Cochlear Implants Using Artificial Neural Network Marwan A. Jabri & Raymond J. Wang Systems Engineering and Design Automation Laboratory Department of Electrical Engineering The University of Sydney NSW 2006, Australia {marwan,jwwang}Osedal.usyd.edu.au Abstract State-of-the-art speech processors in cochlear implants perform channel selection using a spectral maxima strategy. This strategy can lead to confusions when high frequency features are needed to discriminate between sounds. We present in this paper a novel channel selection strategy based upon pattern recognition which allows "smart" channel selections to be made. The proposed strategy is implemented using multi-layer perceptrons trained on a multispeaker labelled speech database. The input to the network are the energy coefficients of N energy channels. The output of the system are the indices of the M selected channels. We compare the performance of our proposed system to that of spectral maxima strategy, and show that our strategy can produce significantly better results. 1 INTRODUCTION A cochlear implant is a device used to provide the sensation of sound to those who are profoundly deaf by means of electrical stimulation of residual auditory neurons. It generally consists of a directional microphone, a wearable speech processor, a head-set transmitter and an implanted receiver-stimulator module with an electrode A Novel Channel Selection System in Cochlear Implants 911 array which all together provide an electrical representation of the speech signal to the residual nerve fibres of the peripheral auditory system (Clark et ai, 1990). Electrode Array Figure 1: A simplified schematic diagram ofthe cochlear implants Brain A simplified schematic diagram of the cochlear implants is shown in Figure 1. Speech sounds are picked up by the directional microphone and sent to the speech processor. The speech processor amplifies, filters and digitizes these signals, and then selects and codes the appropriate sound information. The coded signal contains information as to which electrode to stimulate and the intensity level required to generate the appropriate sound sensations. The signal is then sent to the receiver/stimulator via the transmitter coil. The receiver/stimulator delivers electrical impulses to the appropriate electrodes in the cochlea. These stimulated electrodes then directly activate the hearing nerve in the inner ear, creating the sensation of sound, which is then forwarded to the brain for interpretation. The entire process happens in milliseconds. For multi-channel cochlear implants, the task of the speech processor is to compute the spectral energy of the electrical signals it receives, and to quantise them into different levels. The energy spectrum is commonly divided into separate bands using a filter bank of N (typically 20) bandpass filters with centre frequencies ranging from 250 Hz to 10 KHz. The bands of energy are allocated t.o electrodes in the patient's implant on a one-to-one basis. Usually the most-apical bipolar electrode pairs are allocated to the channels in tonotopic order. The limitations of implant systems usually require only a selected number of the quantised energy levels to be fed to the implanted electrode array (Abbas, 1993; Schouten, 1992). The state-of-the-art speech processor for multi-channel implants performs channel selection using spectral maxima strategy (McDermott et ai, 1992; Seligman & McDermott, 1994). The maxima strategy selects the M (about 6) largest spectral energy of the frequency spectrum as stimulation channels from a filter bank of N (typically 20) bandpass. It is believed that compared to other channel selection techniques (FOF2, FOF1F2, MPEAK ... ), the maxima strategy increases the amount of spectral information and improves the speech perception and recognition performance. However, maxima strategy relies heavily on the highest energies. This often leads to the same levels being selected for different sounds, as the energy levels that distinguish them are not high enough to be selected. For some speech signals, 912 M. A. JABRI, R. J. WANG it does not cater for confusions and cannot discriminate between high frequency features. We present in this paper Artificial Neural Networks (ANN) techniques for implementing "smart" channel selection for cochlear implant systems. The input to the proposed channel selection system consists of the energy coefficients (18 in our experiments) and the output the indices of the selected channels (6 in our experiments). The neural network based selection system is trained on a multi-speaker labelled speech and has been evaluated on a separated multi-speaker database not used in the training phase. The most important feature of our ANN based channel selection system is its ability to select the channels for stimulation on the basis of the overall morphology of the energy spectrum and not only on the basis of the maximal energy values. 2 THE PATTERN RECOGNITION BASED CHANNEL SELECTION STRATEGY Speech is the most natural form of human communication. The speech information signal can be divided into phonemes, which share some common acoustic properties with one another for a short interval of time. The phonemes are typically divided into two broad classes: (a) vowels, which allow unrestricted airflow in the vocal tract, and (b) consonants, which restrict airflow at some point and are weaker than vowels. Different phonemes have different morphology in the energy spectrum. Moreover, for different speakers and different speech sentences, the same phonemes have different energy spectrum morphologies (Kent & Read, 1992). Therefore, simple methods to select some of the most important channels for all the phoneme patterns will not perform as good as the method that considers the spectrum in its entirety. The existing maxima strategy only refers to the spectrum amplitudes found in the entire estimated spectrum without considering the morphology. Typically several of the maxima results can be obtained from a single spectral peak. Therefore, for some phoneme patterns, the selection result is good enough to represent the original phoneme. But for some others, some important features of the phoneme are lost. This usually happens to those phonemes with important features in the high frequency region. Due to the low amplitude of the high frequency in the spectrum morphology, maxima methods are not capable to extract those high frequency features. The relationship between the desired M output channels and the energy spectrum patterns is complex, and depending on the conditions, may be influenced by many factors. As mentioned in the Introduction, channel selection methods that make use of local information only in the energy spectrum are bound to produce channel sub-sets where sounds may be confused. The confusions can be reduced if "global" information of the energy spectrum is used in the selection process. The channel selection approach we are proposing makes use of the overall energy spectrum. This is achieved by turning the selection problem into that of a spectrum morphology pattern recognition one and hence, we call our approach Pattern Recognition based Channel Selection (PRCS). A Novel Channel Selection System in Cochlear Implants 913 2.1 PRCS STRATEGY The PRCS strategy is implemented using two cascaded neural networks shown in Figure 2: • Spectral morphological classifier: Its inputs are the spectrum energy amplitudes of all the channels and its outputs all the transformations of the inputs. The transformation between input and out.put can be seen as a recognition, emphasis, and/or decaying of the inputs. The consequence is that some inputs are amplified and some decayed, depending on the morphology of the spectrum. The classifier performs a non-linear mapping . • M strongest of N classifier: It receives the output of morphological classifier and applies a M strongest selection rule. C21 .. 'IR. ---Spectral Morphological CI ..... .., • • • ----- Labeia MStrongaat ofN CIanIf.., Figure 2: The pattern recognition based channel selection architecture 2.2 TRAINING AND TESTING DATA The most difficult task in developing the proposed PRCS is to set up the labelled training and testing data for the spectral morphological classifier. The training and testing data sets have been constructed using the process shown in Figure 3. r " Hlmmlng 18Ch8nnela Training Window + r-- Quantlsatlor Chlinnel r-&Teatlng 128 FFT & scaling labelling Sets 'Figure 3: The process of generating training and testing sets The sounds in the data sets are speech extracted from the DARPA TIMIT multispeaker speech corpus (Fisher et ai, 1987) which contains a total of 6300 sentences, 10 sentences spoken by each of 630 speakers. The speech signal is sampled at 16KHz rate with 16 bit precision. As the speech is nonstationary, to produce the energy spectrum versus channel numbers, a short-time speech analysis method is used. The Fast Fourier Transform with 8ms smooth Hamming window technique is applied to yield the energy spectrum. The hamming window has the shape of a raised 914 cosine pulse: h( n) = { ~.54 - 0.46 cos (J~n. ) M. A. JABRI, R. J. WANG for 0 ~ n ~ N-l otherwise The time frame on which the speech analysis is performed is 4ms long and the successive time frame windows overlap by 50%. Using frequency allocations similar to that used in commercial cochlear implant speech processors, the frequency range in the spectrum is divided into 18 channels with each channel having the center frequencies of 250, 450, 650, 850 1050, 1250, 1450, 1650, 1895, 2177, 2500, 2873, 3300, 3866, 4580, 5307, 6218 and 7285Hz respectively. Each energy spectrum from a time frame is quantised into these 18 frequency bands. The energy amplitude for each level is the sum of the amplitude value of the energy for all the frequency components in the level. The quantised energy spectrum is then labelled using a graphics based tool, called LABEL, developed specially for this application. LABEL displays the spectrum pattern including the unquantised spectrum, the signal source, speaker's name, speech sentence, phoneme, signal pre-processing method and FFT results. All these information assists labelling experts to allocate a score (1 to 18) to each channel. The score reflects the importance of the information provided by each of the bands. Hence, if six channels are only to be selected, the channels with the score 1 to 6 can be used and are highlighted. The labelling is necessary as a supervised neural network training method is being used. A total of 5000 energy spectrum patterns have been labelled. They are from 20 different speakers and different. spoken sentences. Of the 5000 example patterns, 4000 patterns are allocated for training and 1000 patterns for testing. 3 EXPERIMENTAL RESULTS We have implemented and tested the PH.CS system as described above and our experiments show that it has better performance than channel selection systems used in present cochlear implant processors. The PRCS system is effectively constructed as a multi-module neural network using MUME (Jabri et ai, 1994). The back-propagation algorithm in an on-line mode is used to train the MLP. The training patterns input components are the energy amplitudes of the 18 channels and the teacher component consists of a "I" for a channel to be selected and "0" for all others. The MLP is trained for up to 2000 epochs or when a minimum total mean squared error is reached. A learning rate 7J of 0.01 is used (no weight decay). We show the average performance of our PRCS in Table 1 where we also show the performance of a leading commercial spectral maxima strategy called SPEAK on the same test set. In the first column of this table we show the number of channels that matched out of the 6 desired channels. For example, the first row corresponds to the case where all 6 channels match the desired 6 channels in the test data base, and so on. As Table 1 shows, the PRCS produces a significantly better performance than the commercial strategy on the speech test set. The selection performance to different phonemes is listed in Table 2. It clearly A Novel Channel Selection System in Cochlear Implants 915 Table 1: The comparison of average performance between commercial and PRCS system II II The Channel Selections from the two different methods II PRCS results Commercial technique results Fully matched 22 % 4% 5 matched 80 % 25 % 4 matched 98 % 57 % 3 matched 100 % 93 % 2 matched 100 % 99 % 1 matched 100 % 100 % Table 2: PRCS channel selecting performance on different phoneme patterns The P RCS results for different phoneme patterns Phoneme Fully matched 5 matched 4 matched 3 matched Stops 19 % 69 % 96 % 100 % Fricatives 18 % 66 % 92 % 100 % Nasals 14 % 66 % 96 % 100 % Semivowels & Glides 14 % 79 % 95 % 100% Vowels 25 % 84 % 98 % 100 % \I shows that the PRCS strategy can cater for the features of all the speech spectrum patterns. To compare the practical performance of the PRes with the maxima strategies we have developed a direct performance test system which allows us to play the synthesized speech of the selected channels through post-speech synthesizer. Our test shows that the PRCS produces more intelligible speech to the normal ears. Sixteen different sentences spoken by sixteen people are tested using both maxima and PRCS methods. It is found that the synthesized speech from PRCS has much more high frequency features than that of the speech produced by the maxima strategy. All listeners who were asked to take the test agreed that the quality of the speech sound from PRCS is much better than those from the commercial maxima channel selection system. The tape recording of the synthesized speech will be available at the conference. 4 CONCLUSION A pattern recognition based channel selection strategy for Cochlear Implants has been presented. The strategy is based on a 18-72-18 MLP strongest selector. The proposed channel selection strategy has been compared to a leading commercial technique. Our simulation and play back results show that our machine learning based technique produces significantly better channel selections. 916 M. A. JABRI, R. J. WANG Reference Abbas, P. J. (1993) Electrophysiology. "Cochlear Implants: Audiological Foundations" edited by R. S. Tyler, Singular Publishing Group, pp.317-355. Clark, G. M., Tong, Y. C.& Patrick, J. F. (1990) Cochlear Prosthesis. Edi n borough: Churchill Living stone. Fisher, W. M., Zue, V., Bernstein, J. & Pallett, D. (1987) An Acoustic-Phonetic Data Base. In 113th Meeting of Acoust Soc Am, May 1987 Jabri, M. A., Tinker, E. A. & Leerink, L. (1994) MUME A Multi-Net MultiArchitecture Neural Simulation Environment. "Neural Network Simulation Environments", J. Skrzypek ed., Kluwer Academic Publishers. Kent, R. D. & Read, C. (1992) The Acoustic Analysis of Speech. Whurr Publishers. McDermott, H. J., McKay, C. M. & Vandali, A. E. (1992) A new portable sound processor for the University of Melbourne / Nucleus Limited multielectrode cochlear implant. J. Acoust. Soc. Am. 91(6), June 1992, pp.3367-3371 Schouten, M. E. H edited (1992) The Auditory Processing of Speech From Sounds to Words. Speech Research 10, Mouton de Groyter. Seligman, P. & McDermott, H. (1994) Architecture of the SPECTRA 22 Speech Processor. International Cochlear Implant, Speech and Hearing Symposium, Melbourne, October, 1994, p.254.
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Examples of learning curves from a modified VC-formalism. A. Kowalczyk & J. Szymanski Telstra Research Laboratories 770 Blackbtun Road, Clayton, Vic. 3168, Australia {akowalczyk,j.szymanski }@trl.oz.au) P.L. Bartlett & R.C. Williamson Department of Systems Engineering Australian National University Canberra, ACT 0200, Australia {bartlett, williams }@syseng.anu.edu.au Abstract We examine the issue of evaluation of model specific parameters in a modified VC-formalism. Two examples are analyzed: the 2-dimensional homogeneous perceptron and the I-dimensional higher order neuron. Both models are solved theoretically, and their learning curves are compared against true learning curves. It is shown that the formalism has the potential to generate a variety of learning curves, including ones displaying ''phase transitions." 1 Introduction One of the main criticisms of the Vapnik-Chervonenkis theory of learning [15] is that the results of the theory appear very loose when compared with empirical data. In contrast, theory based on statistical physics ideas [1] provides tighter numerical results as well as qualitatively distinct predictions (such as "phase transitions" to perfect generalization). (See [5, 14] for a fuller discussion.) A question arises as to whether the VC-theory can be modified to give these improvements. The general direction of such a modification is obvious: one needs to sacrifice the universality of the VC-bounds and introduce model (e.g. distribution) dependent parameters. This obviously can be done in a variety of ways. Some specific examples are VC-entropy [15], empirical VC-dimensions [16], efficient complexity [17] or (p., C)-uniformity [8, 9] in a VC-formalism with error shells. An extension of the last formalism is of central interest to this paper. It is based on a refinement of the "fundamental theorem of computational learning" [2] and its main innovation is to split the set of partitions of a training sample into separate "error shells", each composed of error vectors corresponding to the different error values. Such a split introduces a whole range of new parameters (the average number of patterns in each of a series of error shells) in addition to the VC dimension. The difficulty of determining these parameters then arises. There are some crude, "obvious" upper bounds Examples of Learning Curves from a Modified VC-fonnalism 345 on them which lead to both the VC-based estimates [2, 3, 15] and the statistical physics based formalism (with phase transitions) [5] as specific cases of this novel theory. Thus there is an obvious potential for improvement of the theory with tighter bounds. In particular we find that the introduction of a single parameter (order of uniformity), which in a sense determines shifts in relative sizes of error shells, leads to a full family of shapes of learning curves continuously ranging in behavior from decay proportional to the inverse of the training sample size to "phase transitions" (sudden drops) to perfect generalization in small training sample sizes. We present initial comparison of the learning curves from this new formalism with "true" learning curves for two simple neral networks. 2 Overview of the formalism The presentation is set in the typical PAC-style; the notation follows [2]. We consider a space X of samples with a probability measure J1., a subspace H of binary functions X -+ {O, 1} (dichotomies) (called the hypothesis space) and a target hypothesis t E H. Foreachh E H andeachm-samplez = (:el, ... , :em) E xm (m E {1, 2, ... }),wedenoteby €h,z d;j ~ E::llt-hl(:ei)theempiricalerrorofhonz,andbY€h d;j fx It- h l(:e)J1.(d:e) the expected error of h E H. For each m E {1, 2, ... } let us consider the random variable maa: (-) de! { O} € H :l: = max €h j €h z = hEH ' (1) defined as the maximal expected error of an hypothesis h E H consistent with t on z. The learning curve of H, defined as the expected value of tJiaa: , €j{(m) d;j Exm.[€Jiaa:] = f €Jiaa: (z)Jr (dz) (z E xm) (2) Jx = is of central interest to us. Upper bounds on it can be derived from basic PAC-estimates as de! follows. For € ~ ° we denote by HE = {h E H j €h ~ €} the subset of €-bad hypotheses and by Q;! d;j {z E Xm j 3hE H. €h,ri = O} = {z E Xm j 3hEH €h,ri = ° & €h ~ €} (3) the subset of m-samples for which there exists an €-bad hypothesis consistent with the target t. Lemmal IfJ1.m(Q;!) ~ 1J!(€,m), then€j{(m) ~ folmin(l,1J!(€,m))J1.(d€), and equality in the assumption implies equality in the conclusion. 0 Proof outline. If the assumption holds, then 'lr(€, m) d~ 1 - min(l, 1J!( €, m)) is a lower bound on the cumulative distribution of the random variable (1). Thus E x= [€Jiaa:] ~ f01 € tE 'lr( €, m)d€ and integration by parts yields the conclusion. o Givenz = (:el, ... ,:em) E Xm,letusintroducethetransformation(projection)1rt,ri: H-+ {O, l}m allocating to each h E H the vector 1rt,:i!(h) d;j (Ih(:el) - t(:el)l, ... , Ih(:em) - t(:em)l) called the error pattern of h on z. For a subset G C H, let 1rt,:i!(G) = {1rt,:i!(h) : hE G}. The space {o,l}m is the disjoint union of error shells £i d~ {(el, ... ,em) E {O,l}m j el + ... + em = i} for i = 0,1, ... , m, and l1rt,ri(HE) n £il is the number 346 A. KOWALCZYK, J. SZYMANSKI, P. L. BARTLETT, R. C. WILLIAMSON of different error patterns with i errors which can be obtained for h E HE' We shall emplOy the following notation for its average: IHEli d~ Ex ... [l1I't,z(HE) n t:in = r l'II't,z(HE) n t:ilJ.£m(dz). (4) Jx ... The central result of this paper, which gives a bOlUld on the probability of the set Qr;' as in Lemma 1 in terms of I HE Ii, will be given now. It is obtained by modification of the proof of [8, Theorem 1] which is a refinement of the proof of the ''ftmdamental theorem of computational learning" in [2]. It is a simplified version (to the consistent learning case) of the basic estimate discussed in [9, 7]. Theorem 2 For any integer Ie ~ 0 and 0 ::; E, 'Y ::; 1 I-'m(Q';")::; A f ,k,7 t (~) (m:- 1e)-lIHElj+A:, j~7k J J (5) whereA E,k,7 d~ (1- E}~~ O)Ej(l-E)k-j) -l,forle > OandA E,o,7 d~ 1.0 Since error shells are disjoint we have the following relation: PH(m) d~ 2-m i_I".(H)I!r(dZ) = 2-m t.IHli ~ IIH(m)/2m (6) where 1I'z(h) d~ 1I'0,z(h), IHli d~ IHoli and IIH(m) d~ maxz E x'" I 'll'z (H) I is the growth function [2] of H. (Note that assuming that the target t == 0 does not affect the cardinality of 1I't,z(H).) If the VC-dimension of H, d = dvc(H), is finite, we have the well-known estimate [2] IIH(m)::; ~(d,m) d~ t (rr:) ::; (em/d)d. j=O J (7) Corollary 3 (i) If the VC-dimension d of H is finite and m > 8/E, then J.£m(Qr;') ::; 22- mE/ 2(2em/ d)d. (ii) If H has finite cardinality, then J.£m (Qr;') ::; EhEH. (1 - Eh)m. Proof. (i) Use the estimate AE,k,E/2 ::; 2 for Ie ~ 8/E resulting from the Chernoff bound and set'Y = E /2 and Ie = m in (5). (ii) Substitute the following crude estimate: m m IHEli ::; L IHEli ::; L IHli ::; PH ::; (em/d)d, i=O i=O into the previous estimate. (iii) Set Ie = 0 into (i) and use the estimate IHli::; L Prx ... (Eh,z = i/m) = L (1- Eh)m-iEhi. 0 The inequality in Corollary 3.i (ignoring the factor of 2) is the basic estimate of the VCformalism (c.f. [2]); the inequality in Corollary 3.ii is the union bound which is the starting point for the statistical physics based formalism developed in [5]. In this sense both of these theories are unified in estimate (5) and all their conclusions (including the prediction Examples of Learning Curves from a Modified VC-formalism 347 100 (a) (b) , \ \ \ I I I I I 10- 1 I I I I I I I I I I I \ CJ = 3 : chain line 10-2 ...... _-' .... CJ = 3 and COl. = 1 : broken line 10-2 '-'-"'--'-.L...l.....L...l.....L...l....~~.L...l.....L...l.....L...l.....L...l.....L...l.....L...J 4 5 6 7 8 9 o 10 20 30 40 50 mid mi d Figure 1: (a) Examples of upper bounds on the learning curves for the case of finite VCdimension d = dvc(H) implied by Corollary 4.ii for Cw,m == const. They split into five distinct "bands" of four curves each, according to the values of the order of uniformity w = 2, 3,4,5, 10 (in the top-down order). Each band contains a solid line (Cw,m == 1, d = 100), a dotted line (Cw,m == 100, d = 100), a chain line (Cw,m == 1, d = 1000) and a broken line (Cw,m == 100, d = 1000). (b) Various learning curves for the 2-dimensional homogeneous perceptron. Solid lines (top to bottom): (i) - for the VC-theory bound (Corollary 3.ii) with VC-dimension d = 2; (ii) - for the bound (for Eqn, 5 and Lemma 1) with'Y = f, k = m and the upper bounds IHElr ~ IHlr = 2 for i = 1, " " m - 1 and IHElr ~ IHlr = 1 for i = 0, m ; (iii) - as in (ii) but with the exact values for IH Elr as in (11); (iv) - true learning curve (Eqn. 13). The w-uniformity bound for w = 2 (with the minimal C w,m satisfying (9), which turn out to be = const = 1) is shown by dotted line; for w = 3 the chain line gives the result for minimal Cw m and the broken line for Cw m set to 1. , , of phase transitions to perfect generalization for the Ising perceptron for a = mj d < 1.448 in the thermodynamic limit [5]) can be derived from this estimate, and possibly improved with the use of tighter estimates on IH E Ir. We now formally introduce a family of estimates on IHElr in order to discuss a potential of our formalism. For any m, f and w ~ 1.0 there exists Cw,m > 0 such that IH.lr s: IHlr s: Cw,m (7) PH(m)l-ll-2i/ml'" (for 0 s: i ~ m), (8) We shall call such an estimate an w-uniformity bound. Corollary 4 (i) If an w -lllliformity bolllld (8) holds, then ILm(Qm) < A C ~ (m)PH (2m)l-ll-j/ml"', rE _ Elm • .., W,m ~ . , j~",m J (9) (ii) if additionallyd = dvc(H) < 00, then m ( ) m m m 2m d l-Il-j/ml'" J1- (Q.) s: A.,m,,,,Cw,m L . (T (2emjd)) . 0 j~",m J (10) 3 Examples of learning curves In this section we evaluate the above formalism on two examples of simple neural networks. 348 A. KOWALCZYK, J. SZYMANSKI, P. L. BARTLETT, R. C. WILLIAMSON 20 I I I I (b) -'~ -' 15 f_.-r;-"= 3 u 2 10 ." <III " ------------. ..2 / C'oj = 2 / 10- 1 / ", 5r/ '/ J~ ~ C'oj = 2 : dolled line I,' Col = 3: chain line ,/ ' 0 I' I I I I 0 10 20 30 40 50 0 100 200 300 400 500 m/(d+l) m Figure 2: (a) Different learning curves for the higher order neuron (analogous to Fig. l.b). Solid lines (top to bottom)( i) - forthe VC-theory bound (Corollary 3.ii) with VC-dimension d + 1 = 21; (ii) - for the bound (5) with 'Y = € and the upper bounds I H E Ii ~ I H Ii with IHli given by (15); (iii) - true learning curve (the upper bound given by (18)). The wuniformity bound/approximation are plotted as chain and dotted lines for the minimal C w,m satisfying (8), and as broken (long broken) line for C w,m = const = 1 with w = 2 (w = 3). (b) Plots of the minimal value of Cw,m satisfying condition of w-uniformity bound (8) for higher order neuron and selected values of w. 3.1 2-dimensional homogeneous perceptron We consider X d.~ R2 and H defined as the family of all functions (el, 6) ~ 8(el Wl + 6W2)' where (Wl, W2) E R2 and 8(r) is defined as 1 if r ~ 0 and 0, otherwise, and the probability measure jJ. on R2 has rotational symmetry with respect to the origin. Fix an arbitrary target t E H . In such a case IH I~= 1 { 2(1 - €)m - (1 - 2€)m E, . () 22:;=0 j €i (1- €)m-; (for i = 0 and 0 ~ € ~ 1/2), (fori = m), ( otherwise). In particular we find that IHli = 1 for i = 0, m and IHli = 2, otherwise, and m (11) PH(m) = L IHli /2m = (1 + 2 + ... + 2 + 1)/2m = m/2m - l . (12) and the true learning curve is €j{ (m) = 1.5(m + 1)-1. (13) The latter expression results from Lemma 1 and the equality m(Qm) _ { 2(1 - €)m - (1 - 2€)m (for 0 ~ € ~ 1/2), jJ. f 2(1 - €)m (for 1/2 < € ~ 1), (14) Different learning curves (bounds and approximations) for homogeneous perceptron are plotted in Figure 1.b. 3.2 I-dimensional higher order neuron We consider X d.~ [0,1] c R with a continuous probability distribution jJ., Define the hypothesis space H C {O, l}X as the set of all functions of the form 8op(z) where p is a Examples of Learning Curves from a Modified VC-formalism 349 polynomial of degree :::; d on R. Let the target be constant, t == 1. It is easy to see that H restricted to a finite subset of [0,1] is exactly the restriction of the family of all fimctions iI c {O, 1 }[O,lj with up to d "jumps" from a to lor 1 toO and thus dvc(H) = d+ 1. With probability 1 an m-sample Z = (Zl' "" zm) from xm is such that Zi #- Zj for i #- j. For such a generic Z, l7rt,z(H) n t:il = const = IHli. This observation was used to derive the following relations for the computation of I H Ii: min(d,m-l) IHli = L liI(6)li + liI(6)1:_i, (15) 6=0 for ° :::; i :::; m, where liI(6)li, for 0 = 0,1, ... ,d, is defined as follows. We initialize liI(O)lo = liI(l)li d~ 1 fori = 1, .. " m-1, liI(1) 10 = liI(l)l~ d~ ° and liI(6)li d~ ° for i = 0, 1, ... , m, 0 = 2,3, .. " d, and then, recurrently, for 0 ~ 2 we set liI(6) Ii d~ ~m-l . liI(6-l)l~ if 0 is odd and liI(6)1~ d~ ~m-l liI(6-l)l~ ifo is even L.Jk=max(6,m-~) ~-m+k ~ L.Jk=6 ~ . (Here liI(6)li is defined by the relation (4) with the target t == 1 for the hypothesis space H(6) C iI composed of functions having the value 1 near a and exactly 0 jumps in (0,1), exactly at entries of z; similarly as for H, IH(6)li = l7rl,zH(6) n t:il for a generic m-sample z E (0, l)m.) Analyzing an embedding of R into Rd, and using an argument based on the Vandermonde determinant as in [6,13], itcan be proved that the partition function IIH is given by Cover's counting function [4], and that (16) For the uniform distribution on [0, 1] and a generic z E [0, l]m letAk(z) denote the sum of Ie largest segments of the partition of [0, 1] into m + 1 segments by the entries of Z. Then Ald/:lJ(Z):::; e'J;arz:(z):::; Ald/:lJ+l(Z), (17) An explicit expression for the expected value of Ak is known [11], thus a very tight bound on the true learning curve eH (m) defined by (2) can be obtained: ~/2J1 (1 + E ~):::; eH(m):::; Ld~2J : 1 (1 + E ~), (18) + i=ld/:lJ+1 J + i=ld/:lJ+:l J Numerical results are shown in Figure 2. 4 Discussion and conclusions The basic estimate (5) of Theorem 1 has been used to produce upper bounds on the learning curve (via Lemma 1) in three different ways: (i) using the exact values of coefficients IHEli (Fig. 1a), (ii) using the estimate IHEli :::; IHli and the values of IHli and (iii) using the w-uniformity bound (8) with minimal value of Cw,m and as an "apprOximation" with Cw,m = const = 1. Both examples of simple learning tasks considered in the paper allowed us to compare these results with the true learning curves (or their tight bounds) which can serve as benchmarks. Figure 1.a implies that values of parameter w in the w-uniformity bound (approximation) governing a distribution of error patterns between different error shells (c.f, [10)) has a 350 A. KOWALCZYK, J. SZYMANSKI, P. L. BARTLETT, R. C. WILLIAMSON significant impact on learning curve shapes, changing from slow decrease to rapid jumps (''phase transitions',) in generalization. Figure l.b proves that one loses tightness of the bound by using I HI i rather than I HE Ii , and even more is lost if w-unifonnity bounds (with variable C W,17l) are employed. Inspecting Figures l.b and 2.a we find that approximate approaches consisting of replacing IHElr by a simple estimate (w-uniforrnity) can produce learning curves very close to IHlilearning curves suggesting that an application of this formalism to learning systems where neither IHElr nor IHlr can by calculated might be possible. This could lead to a sensible approximate theory capturing at least certain qualitative properties of learning curves for more complex learning tasks. Generally, the results of this paper show that by incorporating the limited knowledge of the statistical distribution of error patterns in the sample space one can dramatically improve bounds on the learning curve with respect to the classical universal estimates of the VCtheory. This is particularly important for "practical" training sample sizes (m ~ 12 x VC-dimension) where the VC-bounds are void. Acknowledgement. The permission of Director, Telstra Research Laboratories, to publish this paper is gratefully acknowledged. A.K. acknowledges the support of the Australian Research Council. References (1) S. Amari, N. Fujita, and S. Shinomoto. Four types of learning curves. Neural Computation, 4(4):605-618, 1992. (2) M. Anthony and N. Biggs. Computational Learning Theory. Cambridge University Press, 1992. (3) A. Blumer, A. Ehrenfeucht, D. Haussler, and M.K. Warmuth. Learnability and the VapnikChervonenkis dimensions. Journal of the ACM, 36:929-965, (Oct. 1989). (4) T.M. Cover. Geometrical and statistical properties of linear inequalities with applications to pattern recognition. IEEE Trans. Elec. Comp., EC-14:326-334, 1965. (5) D. Haussler, M. Keams, H.S. Seung, and N. Tishby. Rigorous learning curve bounds from statistical mechanics. In Proc. 7th Ann. ACM Con[. on Compo Learn. Theory, pages 76-87, 1994. (6) A. Kowalczyk. Estimates of storage capacity of multi-layer perceptron with threshold logic hidden units. Neural Networks, to appear. (7) A. Kowalczyk. VC-formalism with explicit bounds on error shells size distribution. A manuscript, 1994. (8) A. Kowalczykand H. Ferra. Generalisation in feedforward networks. Adv. in NIPS 7, The MIT Press, Cambridge, 1995. (9) A. Kowalczyk, J. Szymanski, and H. Ferra. Combining statistical physics with VC-bounds on generalisation in learning systems. In Proc. ACNN'95, Sydney, 1995. University of Sydney. (10) A. Kowalczyk, J. Szymanski, and R.C. Williamson. Learning curves from a modified vcformalism: a case stUdy. In Proceedings of ICNN'95, Perth (CD'ROM), volume VI, pages 2939-2943, Rundle Mall, South Australia, 1995. IEEE'J'Causal Production. (11) J.G. Mauldon. Random division of an interval. Proc. Cambridge Phil. Soc., 47:331-336,1951. (12) K.R. Muller, M. Finke, N. Murata, and S. Amari. On large scale simulations for learning curves. In Proc. ACNN'95, pages 45-48, Sydney, 1995. University of Sydney. (13) A. Sakurai. n-h-l networks store no less n h + 1 examples but sometimes no more. In Proceedings of the 1992 International Conference on Neural Networks,pagesill-936-ill-941. IEEE, June 1992. (14) H. Sompolinsky, H.S. Seung, and N. Tishby. Statistical mechanics of learning curves. Physical Reviews, A45:6056-6091, 1992. (15) V. Vapnik. Estimation of Dependences Based on Empirical Data. Springer-Verlag, 1982. (16) V. Vapnik, E. Levin, and Y. Le Cun. Measuring the VC-dimension ofa learning machine. Neural Computation, 6 (5):851-876, 1994. (17) C. Wang and S.S. Venkantesh. Temporal dynamics of generalisation in neural networks. Adv. in NIPS 7, The MIT Press, Cambridge, 1995.
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Using Unlabeled Data for Supervised Learning Geoffrey Towell Siemens Corporate Research 755 College Road East Princeton, N J 08540 Abstract Many classification problems have the property that the only costly part of obtaining examples is the class label. This paper suggests a simple method for using distribution information contained in unlabeled examples to augment labeled examples in a supervised training framework. Empirical tests show that the technique described in this paper can significantly improve the accuracy of a supervised learner when the learner is well below its asymptotic accuracy level. 1 INTRODUCTION Supervised learning problems often have the following property: unlabeled examples have little or no cost while class labels have a high cost. For example, it is trivial to record hours of heartbeats from hundreds of patients. However, it is expensive to hire cardiologists to label each of the recorded beats. One response to the expense of class labels is to squeeze the most information possible out of each labeled example. Regularization and cross-validation both have this goal. A second response is to start with a small set of labeled examples and request labels of only those currently unlabeled examples that are expected to provide a significant improvement in the behavior of the classifier (Lewis & Catlett, 1994; Freund et al., 1993). A third response is to tap into a largely ignored potential source of information; namely, unlabeled examples. This response is supported by the theoretical work of Castelli and Cover (1995) which suggests that unlabeled examples have value in learning classification problems. The algorithm described in this paper, referred to as SULU (Supervised learning Using Labeled and Unlabeled examples), takes this third 648 G. TOWELL path by using distribution information from unlabeled examples during supervised learning. Roughly, SULU uses the centroid of labeled and unlabeled examples in the neighborhood of a labeled example as a new training example. In this way, SULU extracts information about the local variability of the input from unlabeled data. SULU is described in Section 2. In its use of unlabeled examples to alter labeled examples, SULU is reminiscent of techniques for adding noise to networks during training (Hanson, 1990; Matsuoka, 1992). SULU is also reminiscent of instantiations of the EM algorithm that attempt to fill in missing parts of examples (Ghahramani & Jordan, 1994). The similarity of SULU to these, and other, works is explored in Section 3. SULU is intended to work on classification problems for which there is insufficient labeled training data to allow a learner to approach its asymptotic accuracy level. To explore this problem, the experiments described in Section 4 focus on the early parts of the learning curves of six datasets (described in Section 4.1). The results show that SULU consistently, and statistically significantly, improves classification accuracy over systems trained with only the labeled data. Moreover, SULU is consistently more accurate than an implementation of the EM-algorithm that was specialized for the task of filling in missing class labels. From these results, it is reasonable to conclude that SULU is able to use the distribution information in unlabeled examples to improve classification accuracy. 2 THE ALGORITHM SULU uses standard neural-network supervised training techniques except that it occasionally replaces a labeled example with a synthetic example. in addition, the criterion to stop training is slightly modified to require that the network correctly classify almost every labeled example and a majority of the synthetic examples. For instance, the experiments reported in Section 4 generate synthetic examples 50% of the time; the stopping criterion requires that 80% of the examples seen in a single epoch are classified correctly. The main function in Table 1 provides psuedocode for this process. The synthesize function in Table 1 describes the process through which an example is synthesized. Given a labeled example to use as a seed, synthesize collects neighboring examples and returns an example that is the centroid of the collected examples with the label of the starting point. synthesize collects neighboring examples until reaching one of the following three stopping points. First, the maximum number of points is reached; the goal of SULU is to get information about the local variance around known points, this criterion guarantees locality. Second, the next closest example to the seed is a labeled example with a different label; this criterion prevents the inclusion of obviously incorrect information in synthetic examples. Third, the next closest example to the seed is an unlabeled example and the closest labeled example to that unlabeled example has a different label from the seed; this criterion is intended to detect borders between classification areas in example space. The call to synthesize from main effectively samples with replacement from a space defined by a labeled example and its neighbors. As such, there are many ways in which main and synthesize could be written. The principle consideration in this implementation is memory; the space around the labeled examples can be huge. Using Unlabeled Data for Supervised Learning 649 Table 1: Pseudocode for SULU RANDOH(min,max): return a uniformly distributed random integer between min and max, inclusive HAIN(B,H): /* B - in [0 .. 100], controls the rate of example synthesis */ /* H - controls neighborhood size during synthesis */ Let: E /* a set of labeled examples */ U /* a set of unlabeled examples */ N /* an appropriate neural network */ Repeat Permute E Foreach e in E if random(0,100) > B then e (- SYNTHESIZE(e,E,U,random(2,M» TRAIN N using e Until a stopping criterion is reached SYNTHESIZE(e,E,U,m): Let: C /* will hold a collection of examples */ For i from 1 to m c (- ith nearest neighbor of e in E union U if «c is labeled) and (label of c not equal to label of e» then STOP if c is not labeled cc (- nearest neighbor of c in E if label of cc not equal to label of e then STOP add c to C return an example whose input is the centroid of the inputs of the examples in C and has the class label of e. 3 RELATED WORK SULU is similar to two methods of exploring the input space beyond the boundaries of the labeled examples; example generation and noise addition. Example generation commonly uses a model of how a space deforms and an example of the space to generate new examples. For instance, in training a vehicle to turn, Pomerleau (1993) used information about how the scene shifts when a car is turned to gener·ate examples of turns. The major problem with example generation is that deformation models are uncommon. By contrast to example generation, noise addition is a model-free procedure. In general, the idea is to add a small amount of noise to either inputs (Matsuoka, 1992), link weights (Hanson, 1990), or hidden units (Judd & Munro, 1993). For example, Hanson (1990) replaces link weights with a Gaussian. During a forward pass, the Gaussian is sampled to determine the link weight. Training affects both the mean and the variance of the Gaussian. In so doing, Hanson's method uses distribution information in the labeled examples to estimate the global variance of each input dimension. By contrast, SULU uses both labeled and unlabeled examples to make local variance estimates. (Experiments, results not shown, with Hanson's method indicate that it cannot improve classification results as much as SULU.) Finally, there has been some other work on using unclassified examples during training. de Sa (1994) uses the co-occurrence of inputs in multiple sensor modali650 G. TOWELL ties to substitute for missing class information. However, sensor data from multiple modalities is often not available. Another approach is to use the EM algorithm (Ghahramani & Jordan, 1994) which iteratively guesses the value of missing information (both input and output) and builds structures to predict the missing information. Unlike SULU, EM uses global information in this process so it may not perform well on highly disjunctive problems. Also SULU may have an advantage over EM in domains in which only the class label is missing as that is SULU'S specific focus. 4 EXPERIMENTS The experiments reported in this section explore the behavior of SULU on six datasets. Each of the datasets has been used previously so they are only briefly described in the first subsection. The results of the experiments reported in the last part of this section show that SULU significantly and consistently improves classification results. 4.1 DATASETS The first two datasets are from molecular biology. Each take a DNA sequence and encode it using four bits per nucleotide. The first problem, promoter recognition (Opitz & Shavlik, 1994), is: given a sequence of 57 DNA nucleotides, determine if a promoter begins at a particular position in the sequence. Following Opitz and Shavlik, the experiments in this paper use 234 promoters and 702 non promoters. The second molecular biology problem, splice-junction determination (Towell & Shavlik, 1994), is: given a sequence of 60 DNA nucleotides, determine if there is a splice-junction (and the type of the junction) at the middle of the sequence. The data consist of 243 examples of one junction type (acceptors), 228 examples of the other junction type (donors) and 536 examples of non-junctions. For both of these problems, the best randomly initialized neural networks have a small number of hidden units in a single layer (Towell & Shavlik, 1994). The remaining four datasets are word sense disambiguation problems (Le. determine the intended meaning of the word "pen" in the sentence "the box is in the pen"). The problems are to learn to distinguish between six noun senses of "line" or four verb senses of "serve" using either topical or local encodings (Leacock et al., 1993) of a context around the target word. The line dataset contains 349 examples of each sense. Topical encoding, retaining all words that occur more than twice, requires 5700 position vectors. Local encoding, using three words on either side of line, requires 4500 position vectors. The serve dataset contains 350 examples of each sense. Under the same conditions as line, topical encoding requires 4400 position vectors while local encoding requires 4500 position vectors. The best neural networks for these problems have no hidden units (Leacock et al., 1993). 4.2 METHODOLOGY The following methodology was used to test SULU on each dataset. First, the data was split into three sets, 25 percent was set aside to be used for assessing generalization, 50 percent had the class labels stripped off, and the remaining 25 percent was to be used for training. To create learning curves, the training set was Using Unlabeled Data for Supervised Learning 651 Table 2: Endpoints of the learnings curves for standard neural networks and the best result for each of the six datasets. Training Splice Serve Line Set size Promoter Junction Local Topical Local Topical smallest 74.7 66.4 53.9 41.8 38.7 40.6 largest 90.3 85.4 71.7 63.0 58.8 63.3 asymptotic 95.8 94.4 83.1 75.5 70.1 79.2 further subdivided into sets containing 5, 10, 15, 20 and 25 percent of the data such that smaller sets were always subsets of larger sets. Then, a single neural network was created and copied 25 times. At each training set size, a new copy of the network was trained under each of the following conditions: 1) using SULU, 2) using SULU but supplying only the labeled training examples to synthesize, 3) standard network training, 4) using a variant of the EM algorithm that has been specialized to the task of filling in missing class labels, and 5) using standard network training but with the 50% unlabeled prior to stripping the labels. This procedure was repeated eleven times to average out the effects of example selection and network initialization. When SULU was used, synthetic examples replaced labeled examples 50 percent of the time. Networks using the full SULU (case 1) were trained until 80 percent of the examples in a single epoch were correctly classified. All other networks were trained until at least 99.5% of the examples were correctly classified. Stopping criteria intended to prevent overfitting were investigated, but not used because they never improved generalization. 4.3 RESULTS & DISCUSSION Figure 1 and Table 2 summarize the results of these experiments. The graphs in Figure 1 show the efficacy of each algorithm. Except for the largest training set on the splice junction problem, SULU always results in a statistically significant improvement over the standard neural network with at least 97.5 percent confidence (according to a one-tailed paired-sample t-test). Interestingly, SULU'S improvement is consistently between :t and ~ of that achieved by labeling the unlabeled examples. This result contrasts Castelli and Cover's (1995) analysis which suggests that labeled examples are exponentially more valuable than unlabeled examples. In addition, SUL U is consistently and significantly superior to the instantiation of the EM-algorithm when there are very few labeled samples. As the number of labeled samples increases the advantage of SULU decreases. At the largest training set sizes tested, the two systems are roughly equally effective. A possible criticism of SULU is that it does not actually need the unlabeled examples; the procedure may be as effective using only the labeled training data. This hypothesis is incorrect, As shown in Figure 1, SULU when given no unlabeled examples is consistently and significantly inferior ti SULU when given a large number of unlabeled examples. In addition, SULU with no unlabeled examples is consistently, although not always significantly, inferior to a standard neural network. The failure of SULU with only labeled examples points to a significant weakness 652 ~r-----------~~_~_~.~ ~7_~~~~ ~--' --SI..l.U Wl1h _un~ --EM""'_~ SUlU .... O~ + SIIl&IIIcII't'-..penorto5U.U o SIIIblIlcllly _ new .. SULU ~r---~c---~,~---,~--~~--~ Size oIlraining sel ~~~ ____ ~~~~~_Ud_ ~~U~~_Dh~~~~~~-' '-'-,== :-~-:oo~= ' ........ - - - --SlJ..UWllhOurNbe'-d ......... ......... + Stabs\ll::aly' M.lpenor to SlLU ......... ~o $tabstlcaly' ...... m SlJ..U ---..... _---. -SLl..U wrIh 1046 un1 .... ' .......... EM"lrI l04&~ ........... ~----su. Uwrttl ()lA'l~ ......... '+---- __ + SlatlSt.c.Ity alpena, m SUlU __ ....... -.!! _stat\s\lealyn1erD'tDSlJ..U - --+-----. G. TOWELL ~r---~------~~ _~_ ~~---+~~-~----~ " --8ll.u .... 50'2un~ ~ g "\. ~~~ :.~== ..fi~ ................ + Strolllk:lllJ~tDSLl.U _ ....... ~~..: lIIIaaly .. nor » SULU '8~ """-+-_ j:r---4~~ "-" -"~"'~" ~"~"-" -"~"-" --~-" -"~~~':~--~ 0-------G----_---~ ~r---~--~ l ~ O --~ l ~~~--~~~{· Size oIlralnlng sel ~ __ -..e--~o 0.. .... C'" __ __ -Ouu - - _u __ -e-. __ ~ ~ ~o~----~~----~-----&------~ <>. Figure 1: The effect of five training procedures on each of six learning problems. In each of the above graphs, the effect of standard neural learning has been subtracted from all results to suppress the increase in accuracy that results simply from an increase in the number of labeled training examples. Observations marked by a '0' or a '+' respectively indicate that the point is statistically significantly inferior or superior to a network trained using SULU. in its current implementation. Specifically, SULU finds the nearest neighbors of an example using a simple mismatch counting procedure. Tests of this procedure as an independent classification technique (results not shown) indicate that it is consistently much worse than any of the methods plotted in in Figure 1. Hence, its use imparts a downward bias to the generalizatio~ results. A second indication of room for improvement in SULU is the difference in generalization between SULU and a network trained using data in which the unlabeled examples provided to SULU have labels (case 5 above). On every dataset, the gain from labeling the examples is statistically significant. The accuracy of a network trained with all labeled examples is an upper bound for SULU, and one that is likely not reachable. However, the distance between the upper bound and SULU'S current performance indicate that there is room for improvement. Using Unlabeled Data for Supervised Learning 653 5 CONCLUSIONS This paper has presented the SULU algorithm that combines aspects of nearest neighbor classification with neural networks to learn using both labeled and unlabeled examples. The algorithm uses the labeled and unlabeled examples to construct synthetic examples that capture information about the local characteristics of the example space. In so doing, the range of examples seen by the neural network during its supervised learning is greatly expanded which results in improved generalization. Results of experiments on six real-work datasets indicate that SULU can significantly improve generalization when when there is little labeled data. Moreover, the results indicate that SULU is consistently more effective at using unlabeled examples than the EM-algorithm when there is very little labeled data. The results suggest that SULU will be effective given the following conditions: 1) there is little labeled training data, 2) unlabeled training data is essentially free, 3) the accuracy of the classifier when trained with all of the available data is below the level which is expected to be achievable. On problems with all of these properties SULU may significantly improve the generalization accuracy of inductive classifiers. References Castelli, V. & Cover, T. (1995). The relative value of labeled and unlabeled samples in pattern recognition with an unknown mixing parameter. (Technical Report 86), Department of Statistics: Stanford University. de Sa, V. (1994). Learning classification with unlabeled data. Advances in Neural Information Processing Systems, 6. Freund, Y., Seung, H. S., Shamit, E., & Tishby, N. (1993). Information, prediction and query by committee. Advances in Neural Information Processing Systems, 5. Ghahramani, Z. & Jordan, M. I. (1994). Supervised learning from incomplete data via an EM approach. Advances in Neural Information Processing Systems, 6. Hanson, S. J. (1990). A stochastic version of the delta rule. Physica D, 42, 265-272. Judd, J. S. & Munro, P. W. (1993). Nets with unreliable hidden units learn errOrcorrecting codes. Advances in Neural Information Processing Systems, 5. Leacock, C., Towell, G., & Voorhees, E. M. (1993). Towards building contextual representations of word senses using statistical models. Proceedings of SIGLEX Workshop: Acquisition of Lexical Knowledge from Text. Association for Computational Linguistics. Lewis, D. D. & Catlett, J. (1994). Heterogeneous uncertainty sampling for supervised learning. Eleventh International Machine Learning Conference. Matsuoka, K. (1992). Noise injection into inputs in back-propagation learning. IEEE Transactions on Systems, Man and Cybernetics, 22, 436-440. Opitz, D. W. & Shavlik, J. W. (1994). Using genetic search to refine knowledge-based neural networks. Eleventh International Machine Learning Conference. Pomerleau, D. A. (1993). Neural Network Perception for Mobile Robot Guidance. Boston: Kluwer. Towell, G. G. & Shavlik, J. W. (1994). Knowledge-based artificial neural networks. Artificial Intelligence, 70, 119-165.
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Implementation Issues in the Fourier Transform Algorithm Yishay Mansour" Sigal Sahar t Computer Science Dept. Tel-Aviv University Tel-Aviv, ISRAEL Abstract The Fourier transform of boolean functions has come to play an important role in proving many important learnability results. We aim to demonstrate that the Fourier transform techniques are also a useful and practical algorithm in addition to being a powerful theoretical tool. We describe the more prominent changes we have introduced to the algorithm, ones that were crucial and without which the performance of the algorithm would severely deteriorate. One of the benefits we present is the confidence level for each prediction which measures the likelihood the prediction is correct. 1 INTRODUCTION Over the last few years the Fourier Transform (FT) representation of boolean functions has been an instrumental tool in the computational learning theory community. It has been used mainly to demonstrate the learnability of various classes of functions with respect to the uniform distribution. The first connection between the Fourier representation and learnability of boolean functions was established in [6] where the class ACo was learned (using its FT representation) in O(nPoly-log(n)) time. The work of [5] developed a very powerful algorithmic procedure: given a function and a threshold parameter it finds in polynomial time all the Fourier coefficients of the function larger than the threshold. Originally the procedure was used to learn decision trees [5], and in [8, 2, 4] it was used to learn polynomial size DNF. The FT technique applies naturally to the uniform distribution, though some of the learnability results were extended to product distribution [1, 3] . .. e-mail: manSQur@cs.tau.ac.il t e-mail: gales@cs.tau.ac.il Implementation Issues in the Fourier Transform Algorithm 261 A great advantage of the FT algorithm is that it does not make any assumptions on the function it is learning. We can apply it to any function and hope to obtain "large" Fourier coefficients. The prediction function simply computes the sum of the coefficients with the corresponding basis functions and compares the sum to some threshold. The procedure is also immune to some noise and will be able to operate even if a fraction of the examples are maliciously misclassified. Its drawback is that it requires to query the target function on randomly selected inputs. We aim to demonstrate that the FT technique is not only a powerful theoretical tool, but also a practical one. In the process of implementing the Fourier algorithm we enhanced it in order to improve the accuracy of the hypothesis we generate while maintaining a desirable run time. We have added such feartures as the detection of inaccurate approximations "on the fly" and immediate correction of the errors incurred at a minimal cost. The methods we devised to choose the "right" parameters proved to be essential in order to achieve our goals. Furthermore, when making predictions, it is extremely beneficial to have the prediction algorithm supply an indicator that provides the confidence level we have in the prediction we made. Our algorithm provides us naturally with such an indicator as detailed in Section 4.1. The paper is organized as follows: section 2 briefly defines the FT and describes the algorithm. In Section 3 we describe the experiments and their outcome and in Section 4 the enhancements made. We end with our conclusions in Section 5. 2 FOURIER TRANSFORM (FT) THEORY In this section we briefly introduce the FT theory and algorithm. its connection to learning and the algorithm that finds the large coefficients. A comprehensive survey of the theoretical results and proofs can be found in [7]. We consider boolean functions of n variables: f : {O, l}n -t {-I, I}. We define the inner product: < g, f >= 2-n L::XE{O,l}R f(x)g(x) = E[g . f], where E is the expected value with respect to the uniform distribution. The basis is defined as follows: for each z E {O,l}n, we define the basis function :\:z(Xl,···,Xn) = (_1)L::~=lx;z •. Any function of n boolean inputs can be uniquely expressed as a linear combination of the basis functions. For a function f, the zth Fourier coefficient of f is denoted by j(z), i.e., f(x) = L::zE{O,l}R j(z)XAx). The Fourier coefficients are computed by j(z) =< f, Xz > and we call z the coefficient-name of j(z). We define at-sparse function to be a function that has at most t non-zero Fourier coefficients. 2.1 PREDICTION Our aim is to approximate the target function f by a t-sparse function h. In many cases h will simply include the "large" coefficients of f. That is, if A = {Zl' ... , zm} is the set of z's for which j(Zi) is "large", we set hex) = L::z;EA aiXz;(x), where at is our approximation of j(Zi). The hypothesis we generate using this process, hex), does not have a boolean output. In order to obtain a boolean prediction we use Sign(h(x)), i.e., output +1 if hex) 2 0 and -1 if hex) < o. We want to bound the error we get from approximating f by h using the expected error squared, E[(J - h )2]. It can be shown that bounding it bounds the boolean prediction error probability, i.e., Pr[f(x) f. sign(h(x))] ~ E[(J - h)2]. For a given t, the t-sparse 262 Y. MANSOUR, S. SAHAR hypothesis h that minimizes E[(J - h)2] simply includes the t largest coefficients of f. Note that the more coefficients we include in our approximation and the better we approximate their values, the smaller E[(J - h )2] is going to be. This provides us with the motivation to find the "large" coefficients. 2.2 FINDING THE LARGE COEFFICIENTS The algorithm that finds the "large" coefficients receives as inputs a function 1 (a black-box it can query) and an interest threshold parameter (J > 0. It outputs a list of coefficient-names that (1) includes all the coefficients-names whose corresponding coefficients are "large", i.e., at least (J , and (2) does not include "too many" coefficient-names. The algorithm runs in polynomial time in both 1/() and n. SUBROUTINE search( a) IF TEST[J, a, II] THEN IF lal = n THEN OUTPUT a ELSE search(aO); search(al); Figure 1: Subroutine search The basic idea of the algorithm is to perform a search in the space of the coefficientnames of I. Throughout the search algorithm (see Figure (1)) we maintain a prefix of a coefficient-name and try to estimate whether any of its extensions can be a coefficient-name whose value is "large". The algorithm commences by calling search(A) where A is the empty string. On each invocation it computes the predicate TEST[/, a, (J]. If the predicate is true, it recursively calls search(aO) and search(al). Note that if TEST is very permissive we may reach all the coefficients, in which case our running time will not be polynomial; its implementation is therefore of utmost interest. Formally, T EST[J, a, (J] computes whether Exe{O,l}n-"E;e{O,lP.[J(YX)Xa(Y)] 2: (J2, where k = Iiali . (1) Define la(x) = L:,ae{O,l}n-" j(aj3)x.,a(x). It can be shown that the expected value in (1) is exactly the sum of the squares of the coefficients whose prefix is a , i.e., Exe{o,l}n-"E;e{o,l}d/(yx)x.a(Y)] = Ex[/~(x)] = L:,ae{o,l}n-" p(aj3), implying that if there exists a coefficient Ii( a,8)1 2: (), then E[/;] 2: (J2 . This condition guarantees the correctness of our algorithm, namely that we reach all the "large" coefficients. We would like also to bound the number of recursive calls that search performs. We can show that for at most 1/(J2 of the prefixes of size k, TEST[!, a , (J] is true. This bounds the number of recursive calls in our procedure by O(n/(J2). In TEST we would like to compute the expected value, but in order to do so efficiently we settle for an approximation of its value. This can be done as follows: (1) choose ml random Xi E {a, l}n-k, (2) choose m2 random Yi,j E {a, l}k , (3) query 1 on Yi,jXi (which is why we need the query model-to query f on many points with the same prefix Xi) and receive I(Yi,j xd, and (4) compute the estimate as, Ba = ';1 L:~\ (~~ L:~l I(Yi,iXdXa(Yi,j)f . Again, for more details see [7]. 3 EXPERIMENTS We implemented the FT algorithm (Section 2.2) and went forth to run a series of experiments. The parameters of each experiment include the target function, (J , ml Implementation Issues in the Fourier Transform Algorithm 263 and m2. We briefly introduce the parameters here and defer the detailed discussion. The parameter () determines the threshold between "small" and "large" coefficients, thus controlling the number of coefficients we will output. The parameters wI and w2 determine how accurately we approximate the TEST predicate. Failure to approximate it accurately may yield faulty, even random, results (e.g., for a ludicrous choice of m1 = 1 and m2 = 1) that may cause the algorithm to fail (as detailed in Section 4.3). An intelligent choice of m1 and m2 is therefore indispensable. This issue is discussed in greater detail in Sections 4.3 and 4.4. Figure 2: Typical frequency plots and typical errors. Errors occur in two cases: (1) the algorithm predicts a +1 response when the actual response is -1 (the lightly shaded area), and (2) the algorithm predicts a -1 response, while the true response is +1 (the darker shaded area) . Figures (3)-(5) present representative results of our experiments in the form of graphs that evaluate the output hypothesis of the algorithm on randomly chosen test points. The target function, I, returns a boolean response, ±1, while the FT hypothesis returns a real response. We therefore present, for each experiment, a graph constituting of two curves: the frequency of the values of the hypothesis, h( x), when I( x) = +1, and the second curve for I( x) = -1. If the two curves intersect, their intersection represents the inherent error the algorithm makes. Figure 3: Decision trees of depth 5 and 3 with 41 variables. The 5-deep (3-deep) decision tree returns -1 about 50% (62.5%) of the time. The results shown above are for values (J = 0.03, ml = 100 and m2 = 5600 «(J = 0.06, ml = 100 and m2 = 1300). Both graphs are disjoint, signifying 0% error. 4 RESULTS AND ALGORITHM ENHANCEMENTS 4.1 CONFIDENCE LEVELS One of our most consistent and interesting empirical findings was the distribution of the error versus the value of the algorithm's hypothesis: its shape is always that of a bell shaped curve. Knowing the error distribution permits us to determine with a high (often 100%) confidence level the result for most of the instances, yielding the much sought after confidence level indicator. Though this simple logic thus far has not been supported by any theoretical result, our experimental results provide overwhelming evidence that this is indeed the case. Let us demonstrate the strength of this technique: consider the results of the 16-term DNF portrayed in Figure (4). If the algorithm's hypothesis outputs 0.3 (translated 264 Y. MANSOUR, S. SAHAR Figure 4: 16 terlD DNF. This (randomly generated) DNF of 40 variables returns -1 about 61 % of the time. The results shown above are for the values of 9 = 0.02 , m2 = 12500 and ml = 100. The hypothesis uses 186 non-zero coefficients. A total of 9.628% error was detected. into 1 in boolean terms by the Sign function), we know with an 83% confidence level that the prediction is correct. If the algorithm outputs -0.9 as its prediction, we can virtually guarantee that the response is correct. Thus, although the total error level is over 9% we can supply a confidence level for each prediction. This is an indispensable tool for practical usage of the hypothesis. 4.2 DETERMINING THE THRESHOLD Once the list of large coefficients is built and we compute the hypothesis h( x), we still need to determine the threshold, a, to which we compare hex) (i.e., predict +1 iff hex) > a). In the theoretical work it is assumed that a = 0, since a priori one cannot make a better guess. We observed that fixing a's value according to our hypothesis, improves the hypothesis. a is chosen to minimize the error with respect to a number of random examples. Figure 5: 8 terlD DNF. This (randomly generated) DNF of 40 variables returns -1 about 43% of the time. The results shown above are for the values of 9 = 0.03, m2 = 5600 and ml = 100. The hypothesis consists of 112 non-zero coefficients. For example, when trying to learn an 8-term DNF with the zero threshold we will receive a total of 1.22% overall error as depicted in Figure (5). However, if we choose the threshold to be 0.32, we will get a diminished error of 0.068%. 4.3 ERROR DETECTION ON THE FLY - RETRY During our experimentations we have noticed that at times the estimate Ba for E[J~] may be inaccurate. A faulty approximation may result in the abortion of the traversal of "interesting" subtreees, thus decreasing the hypothesis' accuracy, or in traversal of "uninteresting" subtrees, thereby needlessly increasing the algorithm's runtime. Since the properties of the FT guarantee that E[J~] = E[f~o] + E[J~d, we expect Ba :::::: Bao + Bal . Whenever this is not true, we conclude that at least one of our approximations is somewhat lacking. We can remedy the situation by Implementation Issues in the Fourier Transform Algorithm 265 running the search procedure again on the children, i.e., retry node a. This solution increases the probability of finding all the "large" coefficients. A brute force implementation may cost us an inordinate amount of time since we may retraverse subtrees that we have previously visited. However, since any discrepancies between the parent and its children are discovered-and corrected-as soon as they appear, we can circumvent any retraversal. Thus, we correct the errors without any superfluous additions to the run time. --J: ,-" i\ o " ....... Figure 6: Majority function of 41 variables. The result portrayed are for values m1 = 100, m2 = 800 and (J = 0.08. Note the majority-function characteristic distribution of the results1 . We demonstrate the usefulness of this approach with an example of learning the majority function of 41 boolean variables. Without the retry mechanism, 8 (of a total of 42) large coefficients were missed, giving rise to 13.724% error represented by the shaded area in Figure (6). With the retries all the correct coefficients were found, yielding perfect (flawless) results represented in the dotted curve in Figure (6). 4.4 DETERMINING THE PARAMETERS One of our aims was to determine the values of the different parameters, m1, m2 and (}. Recall that in our algorithm we calculate Ba , the approximation of Ex[f~(x)] where m1 is the number of times we sample x in order to make this approximation. We sample Y randomly m2 times to approximate fa(Xi) = Ey[f(YXih:a(Y)), for each Xi · This approximation of fa(Xi) has a standard deviation of approximately A . Assume that the true value is 13i, i.e. f3i = fa(Xi), then we expect the contribution of the ith element to Ba to be (13i ± )n;? = 131 ± J&; + rr!~. The algorithm tests Ba = rr!1 L 131 ? (}2, therefore, to ensure a low error, based on the above argument, we choose m2 = (J52 • Choosing the right value for m2 is of great importance. We have noticed on more than one occasion that increasing the value of m2 actually decreases the overall run time. This is not obvious at first: seemingly, any increase in the number of times we loop in the algorithm only increases the run time. However, a more accurate value for m2 means a more accurate approximation of the TEST predicate, and therefore less chance of redundant recursive calls (the run time is linear in the number of recursive calls). We can see this exemplified in Figure (7) where the number of recursive calls increase drastically as m2 decreases. In order to present Figure (7), 1The "peaked" distribution of the results is not coincidental. The FT of the majority function has 42 large equal coefficients, labeled cmaj' one for each singleton (a vector of the form 0 .. 010 .. 0) and one for parity (the all-ones vector). The zeros of an input vector with z zeros we will contribute ±1(2z - 41). cmajl to the result and the parity will contribute ±cma) (depending on whether z is odd or even), so that the total contribution is an even factor of cma)' Since cma) = (~g);tcr - 0 .12, we have peaks around factors of 0.24. The distribution around the peaks is due to the f~ct we only approximate each coefficient and get a value close to cma)' 266 Y. MANSOUR, S. SAHAR we learned the same 3 term DNF always using e = 0.05 and mr * m2 The trials differ in the specific values chosen in each trial for m2. 100000. Figure 7: Deter01ining 012' Note that the number of recursive calls grows dramatically as m2 's value decreases. For example, for m2 = 400, the number of recursive calls is 14,433 compared with only 1,329 recursive calls for m2 = 500. SPECIAL CASES: When k = 110'11 is either very small or very large, the values we choose for ml and m2 can be self-defeating: when k ,..... n we still loop ml (~ 2n - k ) times, though often without gaining additional information. The same holds for very small values of k, and the corresponding m2 (~ 2k) values. We therefore add the following feature: for small and large values of k we calculate exactly the expected value thereby decreasing the run time and increasing accuracy. 5 CONCLUSIONS In this work we implemented the FT algorithm and showed it to be a useful practical tool as well as a powerful theoretical technique. We reviewed major enhancements the algorithm underwent during the process. The algorithm successfully recovers functions in a reasonable amount of time. Furthermore, we have shown that the algorithm naturally derives a confidence parameter. This parameter enables the user in many cases to conclude that the prediction received is accurate with extremely high probability, even if the overall error probability is not negligible. Acknowledgements This research was supported in part by The Israel Science Foundation administered by The Israel Academy of Science and Humanities and by a grant of the Israeli Ministry of Science and Technology. References [1) Mihir Bellare. A technique for upper bounding the spectral norm with applications to learning. In 5th Annual Work&hop on Computational Learning Theory, pages 62-70, July 1992. (2) Avrim Blum, Merrick Furst, Jeffrey Jackson, Michael Kearns, Yishay Mansour, and Steven Rudich. Weakly learning DNF and characterizing statistical query learning using fourier analysis. In The 26th Annual AC M Sympo&ium on Theory of Computing, pages 253 - 262, 1994. (3) Merrick L. Furst , Jeffrey C. Jackson, and Sean W. Smith. Improved learning of ACO functions. In 4th Annual Work&hop on Computational Learning Theory, pages 317-325, August 1991. (4) J. Jackson. An efficient membership-query algorithm for learning DNF with respect to the uniform distribution. In Annual Sympo&ium on Switching and Automata Theory, pages 42 - 53, 1994. (5) E. Kushilevitz and Y. Mansour. Learning decision trees using the fourier spectrum. SIAM Journal on Computing 22(6): 1331-1348, 1993. (6) N. Linial, Y. Mansour, and N . Nisan. Constant depth circuits, fourier transform and learnability. JACM 40(3):607-620, 1993. (7) Y. Mansour. Learning Boolean Functions via the Fourier Transform. Advance& in Neural Computation, edited by V.P. Roychodhury and K-Y. Siu and A. Orlitsky, Kluwer Academic Pub. 1994. Can be accessed via Up:/ /ftp.math.tau.ac.iJ/pub/mansour/PAPERS/LEARNING/fourier-survey.ps.Z. (8) Yishay Mansour. An o(nlog log n) learning algorihm for DNF under the uniform distribution. J. of Computer and Sy&tem Science, 50(3):543-550, 1995.
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A Bound on the Error of Cross Validation Using the Approximation and Estimation Rates, with Consequences for the Training-Test Split 1 INTRODUCTION Michael Kearns AT&T Research We analyze the performance of cross validation 1 in the context of model selection and complexity regularization. We work in a setting in which we must choose the right number of parameters for a hypothesis function in response to a finite training sample, with the goal of minimizing the resulting generalization error. There is a large and interesting literature on cross validation methods, which often emphasizes asymptotic statistical properties, or the exact calculation of the generalization error for simple models. Our approach here is somewhat different, and is pri mari I y inspired by two sources. The first is the work of Barron and Cover [2], who introduced the idea of bounding the error of a model selection method (in their case, the Minimum Description Length Principle) in terms of a quantity known as the index of resolvability. The second is the work of Vapnik [5], who provided extremely powerful and general tools for uniformly bounding the deviations between training and generalization errors. We combine these methods to give a new and general analysis of cross validation performance. In the first and more formal part of the paper, we give a rigorous bound on the error of cross validation in terms of two parameters of the underlying model selection problem: the approximation rate and the estimation rate. In the second and more experimental part of the paper, we investigate the implications of our bound for choosing 'Y, the fraction of data withheld for testing in cross validation. The most interesting aspect of this analysis is the identification of several qualitative properties of the optimal 'Y that appear to be invariant over a wide class of model selection problems: • When the target function complexity is small compared to the sample size, the performance of cross validation is relatively insensitive to the choice of 'Y. • The importance of choosing 'Y optimally increases, and the optimal value for 'Y decreases, as the target function becomes more complex relative to the sample size. • There is nevertheless a single fixed value for'Y that works nearly optimally for a wide range of target function complexity. 2 THE FORMALISM We consider model selection as a two-part problem: choosing the appropriate number of parameters for the hypothesis function, and tuning these parameters. The training sample is used in both steps of this process. In many settings, the tuning of the parameters is determined by a fixed learning algorithm such as backpropagation, and then model selection reduces to the problem of choosing the architecture. Here we adopt an idealized version of this division of labor. We assume a nested sequence of function classes Hl C ... C H d ••• , called the structure [5], where Hd is a class of boolean functions of d parameters, each IPerhaps in conflict with accepted usage in statistics, here we use the term "cross validation" to mean the simple method of saving out an independent test set to perform model selection. Precise definitions will be stated shortly. 184 M.KEARNS function being a mapping from some input space X into {O, I}. For simplicity, in this paper we assume that the Vapnik-Chervonenkis (VC) dimension [6, 5] of the class Hd is O( d). To remove this assumption, one simply replaces all occurrences of d in our bounds by the VC dimension of H d • We assume that we have in our possession a learning algorithm L that on input any training sample 8 and any value d will output a hypothesis function hd E H d that minimizes the training error over H d that is, £ t ( hd) = minhE H" { £ t (h)}, where EtCh) is the fraction of the examples in 8 on which h disagrees with the given label. In many situations, training error minimization is known to be computationally intractable, leading researchers to investigate heuristics such as backpropagation. The extent to which the theory presented here applies to such heuristics will depend in part on the extent to which they approximate training error minimization for the problem under consideration. Model selection is thus the problem of choosing the best value of d. More precisely, we assume an arbitrary target function I (which mayor may not reside in one of the function classes in the structure H1 C ... C H d ••• ), and an input distribution P; I and P together define the generalization error function £g(h) = PrzEP[h(x) =f I(x)]. We are given a training sample 8 of I, consisting of m random examples drawn according to P and labeled by I (with the labels possibly corrupted by a noise process that randomly complements each label independently with probability TJ < 1/2). The goal is to minimize the generalization error of the hypothesis selected. In this paper, we will make the rather mild but very useful assumption that the structure has the property that for any sample size m, there is a value dm.u:(m) such that £t(hdm.u:(m)) = o for any labeled sample 8 of m examples. We call the function dmaz(m) the fitting number of the structure. The fitting number formalizes the simple notion that with enough parameters, we can always fit the training data perfectly, a property held by most sufficiently powerful function classes (including multilayer neural networks). We typically expect the fitting number to be a linear function of m, or at worst a polynomial in m. The significance of the fitting number for us is that no reasonable model selection method should choose hd for d ~ dmaz(m), since doing so simply adds complexity without reducing the training error. In this paper we concentrate on the simplest version of cross validation. We choose a parameter "( E [0, 1], which determines the split between training and test data. Given the input sample 8 of m examples, let 8' be the subsample consisting of the first (1 - "()m examples in 8, and 8" the subsample consisting of the last "(mexamples. In cross validation, rather than giving the entire sample 8 to L, we give only the smaller sample 8', resulting in the sequence h1' ... , hdmaz ((1-"I)m) of increasingly complex hypotheses. Each hypothesis is now obtained by training on only (I - "()m examples, which implies that we will only consider values of d smaller than the corresponding fitting number dmaz((1 - "()m); let us introduce the shorthand d"!naz for dmaz((1 - "()m). Cross validation chooses the hd satisfying hd = mini E {1, ... ,d~az} { £~' (~)} where £~' (~) is the error of hi on the subsample 8". Notice that we are not considering multifold cross validation, or other variants that make more efficient use of the sample, because our analyses will require the independence of the test set. However, we believe that many of the themes that emerge here may apply to these more sophisticated variants as well. We use £ ClI ( m) to denote the generalization error £ g( hd) ofthe hypothesis hd chosen by cross validation when given as input a sample 8 of m random examples of the target function. Obviously, £clI(m) depends on 8, the structure, I, P, and the noise rate. When bounding £cv (m), we will use the expression "with high probability" to mean with probability 1 ~ over the sample 8, for some small .fixed constant ~ > O. All of our results can also be stated with ~ as a parameter at the cost of a loge 1 /~) factor in the bounds, or in terms of the expected value of £clI(m). 3 THE APPROXIMATION RATE It is apparent that any nontrivial bound on £ cv (m) must take account of some measure of the "complexity" of the unknown target function I. The correct measure of this complexity is less obvious. Following the example of Barron and Cover's analysis of MDL performance A Bound on the Error of Cross Validation 185 in the context of density estimation [2], we propose the approximation rate as a natural measure of the complexity of I and P in relation to the chosen structure HI C ... C H d •••• Thus we define the approximation rate function Eg(d) to be Eg(d) = minhEH .. {Eg(h)}. The function E 9 (d) tells us the best generalization error that can be achieved in the class H d, and it is a nonincreasing function of d. If Eg(S) = 0 for some sufficiently large s, this means that the target function I, at least with respect to the input distribution, is realizable in the class H., and thus S is a coarse measure of how complex I is. More generally, even if Eg(d) > 0 for all d, the rate of decay of Eg(d) still gives a nice indication of how much representational power we gain with respect to I and P by increasing the complexity of our models. StilI missing, of course, is some means of determining the extent to which this representational power can be realized by training on a finite sample of a given size, but this will be added shortly. First we give examples of the approximation rate that we will examine following the general bound on E ClI ( m). The Intervals Problem. In this problem, the input space X is the real interval [0,1], and the class Hd of the structure consists of all boolean step functions over [0,1] of at most d steps; thus, each function partitions the interval [0, 1] into at most d disjoint segments (not necessarily of equal width), and assigns alternating positive and negative labels to these segments. The input space is one-dimensional, but the structure contains arbitrarily complex functions over [0, 1]. It is easily verified that our assumption that the VC dimension of Hd is Oed) holds here, and that the fitting number obeys dmllZ(m) S m. Now suppose that the input density P is uniform, and suppose that the target function I is the function of S alternating segments of equal width 1/ s, for some s (thUS, I lies in the class H.). We will refer to these settings as the intervals problem. Then the approximation rate is Eg(d) = (1/2)(1 - dis) for 1 S d < sand Eg(d) = 0 for d ~ s (see Figure 1). The Perceptron Problem. In this problem, the input space X is RN for some large natural number N. The class Hd consists of all perceptrons over the N inputs in which at most d weights are nonzero. If the input density is spherically symmetric (for instance, the uniform density on the unit ball in RN ), and the target function is the function in H. with all s nonzero weights equal to 1, then it can be shown that the approximation rate is Eg(d) = (1/11") cos-I(..jd/s) for d < s [4], and of course Eg(d) = 0 for d ~ s (see Figure 1). Power Law Decay. In addition to the specific examples just given, we would also like to study reasonably natural parametric forms of Eg( d), to determine the sensitivity of our theory to a plausible range of behaviors for the approximation rate. This is important, because in practice we do not expect to have precise knowledge of Eg(d), since it depends on the target function and input distribution. Following the work of Barron [1], who shows a c/dbound on Eg(d) for the case of neural networks with one hidden layer under a squared error generalization measure (where c is a measure of target function complexity in terms of a Fourier transform integrability condition) 2, we can consider approximation rates of the form Eg(d) = (c/d)a + Emin, where Emin ~ 0 is a parameter representing the "degree of unreal izability" of I with respect to the structure, and c, a > 0 are parameters capturing the rate of decay to Emin (see Figure 1). 4 THE ESTIMATION RATE For a fixed I, P and HI C .. . C Hd· .. , we say that a function p( d, m) is an estimation rate boundifforall dand m, with high probability over the sampleSwehave IEt(hd)-Eg(hct)1 S p(d, m), where as usual hd is the result of training error minimization on S within Hd. Thus p( d, m) simply bounds the deviation between the training error and the generalization error of hd • Note that the best such bound may depend in a complicated way on all of the elements of the problem: I, P and the structure. Indeed, much of the recent work on the statistical physics theory of learning curves has documented the wide variety of behaviors that such deviations may assume [4, 3]. However, for many natural problems 2Since the bounds we will give have straightforward generalizations to real-valued function learning under squared error, examining behavior for Eg( d) in this setting seems reasonable. 186 M. KEARNS it is both convenient and accurate to rely on a universal estimation rate bound provided by the powerful theory of unifonn convergence: Namely, for any I, P and any structure, the function p(d, m) = ..j(d/m) log(m/d) is an estimation rate bound [5]. Depending upon the details of the problem, it is sometimes appropriate to omit the loge m/ d) factor, and often appropriate to refine the J dim behavior to a function that interpolates smoothly between dim behavior for small Et to Jd/m for large Et. Although such refinements are both interesting and important, many of the qualitative claims and predictions we will make are invariant to them as long as the deviation kt(hd) - Eg(hd)1 is well-approximated by a power law (d/m)a (0 > 0); it will be more important to recognize and model the cases in which power law behavior is grossly violated. Note that this universal estimation rate bound holds only under the assumption that the training sample is noise-free, but straightforward generalizations exist. For instance, if the training data is corrupted by random label noise at rate 0 ~ TJ < 1/2, then p( d, m) ..j(d/(1 - 2TJ)2m)log(m/d) is again a universal estimation rate bound. 5 THE BOUND Theorem 1 Let HI C ... C Hd · .. be any structure, where the VC dimension 0/ Hd is Oed). Let I and P be any target function and input distribution, let Eg(d) be the approximation rate/unction/or the structure with respect to I and P, and let p(d, m) be an estimation rate bound/or the structure with respect to I and P. Then/or any m, with high probability Ecv(m) ~ min {Eg(d) + p(d, (1 - ,)m)} + 0 ( I~d~di... (1) where, is the/raction o/the training sample used/or testing, and lfYmax is thefitting number dmax( (1 -,)m). Using the universal estimation bound rate and the rather weak assumption that dmax(m) is polynomial in m, we obtain that with high probability 10g«I-,)m)) . ,m (2) Straightforward generalizations 0/ these bounds/or the case where the data is corrupted by classification noise can be obtained, using the modified estimation rate bound given in Section 4 3. We delay the proof of this theorem to the full paper due to space considerations. However, the central idea is to appeal twice to uniform convergence arguments: once within each class Hd to bound the generalization error of the resulting training error minimizer hd E Hd, and a second time to bound the generalization error of the hd minimizing the error on the test set of ,m examples. In the bounds given by (1) and (2), themin{· } expression is analogous to Barron and Cover's index of resolvability [2]; the final tenn in the bounds represents the error introduced by the testing phase of cross validation. These bounds exhibit tradeoff behavior with respect to the parameter,: as we let, approach 0, we are devoting more of the sample to training the hd, and the estimation rate bound tenn p(d, (1 - ,)m) is decreasing. However, the test error tenn O( Jlog(~,u:)/(Tm)) is increasing, since we have less data to accurately estimate the Eg(hd). The reverse phenomenon occurs as we let, approach 1. While we believe Theorem 1 to be enlightening and potentially useful in its own right, we would now like to take its interpretation a step further. More precisely, suppose we ~e main effect of classification noise at rate '1 is the replacement of occurrences in the bound of the sample size m by the smaller "effective" sample size (1 - '1)2m. A Bound on the Error of Cross Validation 187 assume that the bound is an approximation to the actual behavior of EClI(m). Then in principle we can optimize the bound to obtain the best value for "Y. Of course, in addition to the assumptions involved (the main one being that p(d, m) is a good approximation to the training-generalization error deviations of the hd), this analysis can only be carried out given information that we should not expect to have in practice (at least in exact form)in particular, the approximation rate function Eg(d), which depends on f and P. However. we argue in the coming sections that several interesting qualitative phenomena regarding the choice of"Y are largely invariant to a wide range of natural behaviors for Eg (d). 6 A CASE STUDY: THE INTERVALS PROBLEM We begin by performing the suggested optimization of"Y for the intervals problem. Recall that the approximation rate here is Eg(d) = (1/2)(1 - d/8) for d < 8 and Ey(d) = 0 for d ~ 8, where 8 is the complexity of the target function. Here we analyze the behavior obtained by assuming that the estimation rate p(d, m) actually behaves as p(d, m) = Jd/(l - "Y)m (so we are omitting the log factor from the universal bound), and to simplify the formal analysis a bit (but without changing the qualitative behavior) we replace the term Jlog«1 - "Y)m)/bm) by the weaker Jlog(m)/m. Thus, if we define the function F(d, m, "Y) = Ey(d) + Jd/(1 - "Y)m + Jlog(m)/bm) then following Equation (1), we are approximating EclI(m) by EclI (m) ~ min1<d<d" {F(d, m, "Yn 4. __ maa: The first step of the analysis is to fix a value for"Y and differentiate F( d, m, "Y) with respect to d to discover the minimizing value of d; the second step is to differentiate with respect to "Y. It can be shown (details omitted) that the optimal choice of"Y under the assumptions is "Yopt = (log (m)/ 8)1/3/(1 + (Iog(m)/ 8 )1/3). It is importantto remember at this point that despite the fact that we have derived a precise expression for "Yopt. due to the assumptions and approximations we have made in the various constants, any quantitative interpretation of this expression is meaningless. However, we can reasonably expect that this expression captures the qualitative way in which the optimal "Y changes as the amount of data m changes in relation to the target function complexity 8. On this score the situation initially appears rather bleak, as the function (log( m)/ 8)1/3 /(1 + (log(m)/ 8 )1/3) is quite sensitive to the ratio log(m)/8, which is something we do not expect to have the luxury of knowing in practice. However, it is both fortunate and interesting that "Yopt does not tell the entire story. In Figure 2, we plot the function F ( 8, m, "Y) as a function of"Y for m = 10000 and for several different values of 8 (note that for consistency with the later experimental plots, the z axis of the plot is actually the training fraction 1 - "Y). Here we can observe four important qualitative phenomena, which we list in order of increasing subtlety: (A) When 8 is small compared to m, the predicted error is relatively insensitive to the choice of "Y: as a function of "Y, F( 8, m, "Y) has a wide, flat bowl, indicating a wide range of "Y yielding essentially the same near-optimal error. (B) As s becomes larger in comparison to the fixed sample size m, the relative superiority of "Yopt over other values for"Y becomes more pronounced. In particular, large values for"Y become progressively worse as s increases. For example, the plots indicate that for s = 10 (again, m = 10000), even though "Yopt = 0.524 ... the choice "Y = 0.75 will result in error quite near that achieved using "Yopt. However, for s = 500, "Y = 0.75 is predicted to yield greatly suboptimal error. Note that for very large s, the bound predicts vacuously large error for all values of "Y, so that the choice of "Y again becomes irrelevant. (C) Because of the insensitivity to "Y for s small compared to m, there is a fixed value of "Y which seems to yield reasonably good performance for a wide range of values for s. This value is essentially the value of "Yopt for the case where 8 is large but nontrivial generalization is still possible, since choosing the best value for "Y is more important there than for the small 8 case. (D) The value of "Yopt is decreasing as 8 increases. This is slightly difficult to confirm from the plot, but can be seen clearly from the precise expression for "Yopt. 4 Although there are hidden constants in the 0(.) notation of the bounds. it is the relative weights of the estimation and test error terms that is important. and choosing both constants equal to 1 is a reasonable choice (since both terms have the same Chernoff bound origins). 188 M.KEARNS In Figure 3, we plot the results of experiments in which labeled random samples of size m = 5000 were generated for a target function of s equal width intervals, for s = 10,100 and 500. The samples were corrupted by random label noise at rate TJ = 0.3. For each value of 'Y and each value of d, (1 - 'Y)m of the sample was given to a program performing training error minimization within Hd.; the remaining 'Ym examples were used to select the best hd. according to cross validation. The plots show the true generalization error of the hd. selected by cross validation as a function of'Y (the generalization error can be computed exactly for this problem). Each point in the plots represents an average over 10 trials. While there are obvious and significant quantitative differences between these experimental plots and the theoretical predictions of Figure 2, the properties (A), (B) and (C) are rather clearly borne out by the data: (A) In Figure 3, when s is small compared to m, there is a wide range of acceptable 'Y; it appears that any choice of'Y between 0.10 and 0.50 yields nearly optimal generalization error. (B) By the time s = 100, the sensitivity to'Y is considerably more pronounced. For example, the choice 'Y = 0.50 now results in clearly suboptimal performance, and it is more important to have 'Y close to 0.10. (C) Despite these complexities, there does indeed appear to be single value of'Y approximately 0.10that performs nearly optimally for the entire range of s examined. The property (D) namely, that the optimal 'Y decreases as the target function complexity is increased relative to a fixed m is certainly not refuted by the experimental results, but any such effect is simply too small to be verified. It would be interesting to verify this prediction experimentally, perhaps on a different problem where the predicted effect is more pronounced. 7 CONCLUSIONS For the cases where the approximation rate Eg(d) obeys either power law decay or is that derived for the perceptron problem discussed in Section 3, the behavior of EClI(m) as a function of 'Y predicted by our theory is largely the same (for example, see Figure 4). In the full paper, we describe some more realistic experiments in which cross validation is used to determine the number of backpropogation training epochs. Figures similar to Figures 2 through 4 are obtained, again in rough accordance with the theory. In summary, our theory predicts that although significant quantitative differences in the behavior of cross validation may arise for different model selection problems, the properties (A), (B), (C) and (D) should be present in a wide range of problems. At the very least, the behavior of our bounds exhibits these properties for a wide range of problems. It would be interesting to try to identify natural problems for which one or more of these properties is strongly violated; a potential source for such problems may be those for which the underlying learning curve deviates from classical power law behavior [4, 3]. Acknowledgements: I give warm thanks to Yishay Mansour, Andrew Ng and Dana Ron for many enlightening conversations on cross validation and model selection. Additional thanks to Andrew Ng for his help in conducting the experiments. References [1] A. Barron. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transaclions on Information Theory. 19:930-944. 1991. [2] A. R. Barron and T. M. Cover. Minimum complexity density estimation. IEEE Transaclions on Information Theory, 37:1034-1054, 1991. [3] D. Haussler, M. Kearns. H.S. Seung, and N. Tishby. Rigourous learning curve bounds from statistical mechanics. In Proceedings of the Seventh Annual ACM Confernce on Compulalional Learning Theory. pages 76-87. 1991l. [4] H. S. Seung, H. Sompolinsky. and N. Tishby. Statistical mechanics of learning from examples. Physical Review, A45:6056-6091, 1992. [5] V. N. Vapnik:. Estimalion of Dependences Based on Empirical Dala. Springer-Verlag, New York, 1982. [6] V. N. Vapnik: and A. Y. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and ils Applicalions. 16(2):264-280, 1971. A Bound on the Error of Cross Validation rgg pprox mat on a as v grror .. .., rr vs ra n sat s za, sno SQ, m.. . ~ .~ .. ..,J .... ¢i. CV Dun. (c ) or c rom . to ., m' .1 ... 189 Figure 1: Plots of three approximation rates: for the intervals problem with target complexity II = 250 intervals (linear plot intersecting d-axis at 250), for the perceptron problem with target complexity II = 150 nonzero weights (nonlinear plot intersecting d-axis at 150), and for power law decay asymptoting at E",," = 0.05. Figure 2: Plot of the predicted generalization error of cross validation for the intervals model selection problem, as a function of the fraction 1 "'( of data used for training. (In the plot, the fraction of training data is 0 on the left (-y = 1) and 1 on the right ("'( = 0». The fixed sample size m = 10,000 was used, and the 6 plots show the error predicted by the theory for target function complexity values II = 10 (bottom plot), 50, 100, 250, 500, and 1000 (top plot) . Figure 3: Experimental plots of cross validation generalization error in the intervals problem as a function of training set size (1-"'() m. Experiments with the three target complexity values II = 10,100 and 500 (bottom plot to top plot) are shown. Each point represents performance averaged over 10 trials . Figure 4: Plot of the predicted generalization error of cross validation for the power law case E,( d) = (c/d), as a function of the fraction 1-",(ofdata used for training. The fixed sample size m = 25,000 was used, and the 6 plots show the error predicted by the theory for target function complexity values c = 1 (bottom plot), 25,50,75, 100. and 150 (top plot).
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Classifying Facial Action Marian Stewart Bartlett, Paul A. Viola, Terrence J. Sejnowski, Beatrice A. Golomb Howard Hughes Medical Institute The Salk Institute, La Jolla, CA 92037 marni, viola, terry, beatrice @salk.edu Jan Larsen The Niels Bohr Institute 2100 Copenhagen Denmark jlarsen@fys.ku.dk Paul Ekman Joseph C. Hager Network Information Research Corp Salt Lake City, Utah jchager@ibm.net University of California San Francisco San Francisco, CA 94143 ekmansf@itsa.ucsf.edu Abstract The Facial Action Coding System, (FACS), devised by Ekman and Friesen (1978), provides an objective meanS for measuring the facial muscle contractions involved in a facial expression. In this paper, we approach automated facial expression analysis by detecting and classifying facial actions. We generated a database of over 1100 image sequences of 24 subjects performing over 150 distinct facial actions or action combinations. We compare three different approaches to classifying the facial actions in these images: Holistic spatial analysis based on principal components of graylevel images; explicit measurement of local image features such as wrinkles; and template matching with motion flow fields. On a dataset containing six individual actions and 20 subjects, these methods had 89%, 57%, and 85% performances respectively for generalization to novel subjects. When combined, performance improved to 92%. 1 INTRODUCTION Measurement of facial expressions is important for research and assessment psychiatry, neurology, and experimental psychology (Ekman, Huang, Sejnowski, & Hager, 1992), and has technological applications in consumer-friendly user interfaces, interactive video and entertainment rating. The Facial Action Coding System (FACS) is a method for measuring facial expressions in terms of activity in the underlying facial muscles (Ekman & Friesen, 1978). We are exploring ways to automate FACS. 824 BARTLETI, VIOLA, SEJNOWSKI, GOLOMB, LARSEN, HAGER, EKMAN Rather than classifying images into emotion categories such as happy, sad, or surprised, the goal of this work is instead to detect the muscular actions that comprise a facial expression. FACS was developed in order to allow researchers to measure the activity of facial muscles from video images of faces. Ekman and Friesen defined 46 distinct action units, each of which correspond to activity in a distinct muscle or muscle group, and produce characteristic facial distortions which can be identified in the images. Although there are static cues to the facial actions, dynamic information is a critical aspect of facial action coding. FACS is currently used as a research tool in several branches of behavioral science, but a major limitation to this system is the time required to both train human experts and to manually score the video tape. Automating the Facial Action Coding System would make it more widely accessible as a research tool, and it would provide a good foundation for human-computer interactions tools. Why Detect Facial Actions? Most approaches to facial expression recognition by computer have focused on classifying images into a small set of emotion categories such as happy, sad, or surprised (Mase, 1991; Yacoob & Davis, 1994; Essa & Pentland, 1995). Real facial signals, however, consist ofthousands of distinct expressions, that differ often in only subtle ways. These differences can signify not only which emotion is occurring, but whether two or more emotions have blended together, the intensity of the emotion(s), and if an attempt is being made to control the expression of emotion (Hager & Ekman, 1995). An alternative to training a system explicitly on a large number of expression categories is to detect the facial actions that comprise the expressions. Thousands of facial expressions can be defined in terms of this smaller set of structural components. We can verify the signal value of these expressions by reference to a large body of behavioral data relating facial actions to emotional states which have already been scored with FACS. FACS also provides a meanS for obtaining reliable training data. Other approaches to automating facial measurement have mistakenly relied upon voluntary expressions, which tend to contain exaggerated and redundant cues, while omitting some muscular actions altogether (Hager & Ekman, 1995). 2 IMAGE DATABASE We have collected a database of image sequences of subjects performing specified facial actions. The full database contains over 1100 sequences containing over 150 distinct actions, or action combinations, and 24 different subjects. The sequences contain 6 images, beginning with a neutral expression and ending with a high intensity muscle contraction (Figure 1). For our initial investigation we used data from 20 subjects and attempted to classify the six individual upper face actions illustrated in Figure 2. The information that is available in the images for detecting and discriminating these actions include distortions in the shapes and relative positions of the eyes and eyebrows, the appearance of wrinkles, bulges, and furrows, in specific regions of the face, and motion of the brows and eyelids. Prior to classifying the images, we manually located the eyes, and we used this information to crop a region around the upper face and scale the images to 360 x 240. The images were rotated so that the eyes were horizontal, and the luminance was normalized. Accurate image registration is critical for principal components based approaches. For the holistic analysis and flow fields, the images were further scaled Classifying Facial Action 825 to 22 x 32 and 66 x 96, respectively. Since the muscle contractions are frequently asymmetric about the face, we doubled the size of our data set by reflecting each image about the vertical axis, giving a total of 800 images. Figure 1: Example action sequences from the database. AU 1 AU2 AU4 AU5 AU6 AU7 Figure 2: Examples of the six actions used in this study. AU 1: Inner brow raiser. 2: Outer brow raiser. 4: Brow lower. 5: Upper lid raiser (widening the eyes). 6: Cheek raiser. 7: Lid tightener (partial squint). 3 HOLISTIC SPATIAL ANALYSIS The Eigenface (Thrk & Pentland, 1991) and Holon (Cottrell & Metcalfe, 1991) representations are holistic representations based on principal components, which can be extracted by feed forward networks trained by back propagation. Previous work in our lab and others has demonstrated that feed forward networks taking such holistic representations as input can successfully classify gender from facial images (Cottrell & Metcalfe, 1991; Golomb, Lawrence, & Sejnowski, 1991). We evaluated the ability of a back propagation network to classify facial actions given principal components of graylevel images as input. The primary difference between the present approach and the work referenced above is that we take the principal components of a set of difference images, which we obtained by subtracting the first image in the sequence from the subsequent images (see Figure 3). The variability in our data set is therefore due to the facial distortions and individual differences in facial distortion, and we have removed variability due to surface-level differences in appearance. We projected the difference images onto the first N principal components of the dataset, and these projections comprised the input to a 3 layer neural network with 10 hidden units, and six output units, one per action (Figure 3.) The network is feed forward and fully connected with a hyperbolic tangent transfer function, and was trained with conjugate gradient descent. The output of the network was determined using winner take all, and generalization to novel subjects was determined by using the leave-one-out, or jackknife, procedure in which we trained the network on 19 subjects and reserved all of the images from one subject for testing. This process was repeated for each of the subjects to obtain a mean generalization performance across 20 test cases. 826 BARTLETI, VIOLA, SEJNOWSKI, GOLOMB, LARSEN, HAGER, EKMAN We obtained the best performance with 50 component projections, which gave 88.6% correct across subjects. The benefit obtained by using principal components over the 704-dimensional difference images themselves is not large. Feeding the difference images directly into the network gave a performance of 84% correct. 6 OUtputs I WT A Figure 3: Left: Example difference image. Input values of -1 are mapped to black and 1 to white. Right: Architecture of the feed forward network. 4 FEATURE MEASUREMENT We turned next to explicit measurement of local image features associated with these actions. The presence of wrinkles in specific regions of the face is a salient cue to the contraction of specific facial muscles. We measured wrinkling at the four facial positions marked in Figure 4a, which are located in the image automatically from the eye position information. Figure 4b shows pixel intensities along the line segment labeled A, and two major wrinkles are evident. We defined a wrinkle measure P as the sum of the squared derivative of the intensity values along the segment (Figure 4c.) Figure 4d shows P values along line segment A, for a subject performing each of the six actions. Only AU 1 produces wrinkles in the center of the forehead. The P values remain at zero except for AU 1, for which it increases with increases in action intensity. We also defined an eye opening measure as the area of the visible sclera lateral to the iris. Since we were interested in changes in these measures from baseline, we subtract the measures obtained from the neutral image. a c p b 0'---____ --' Pixel d 3....----------, ....-----, 2 c( c.. 1 ~ _~ .... ----to ~.~~ -.... --']t;. ... ~.t(=-: .... -.... ="!1t:-:-:-..:-::!:: 2 o 1 234 5 Image in Seqence AU1 __ AU2 -+-. AU4 -E!-. AU5 -K-· AU6-4-· AU7-ll--Figure 4: a) Wrinkling was measured at four image locations, A-D. b) Smoothed pixel intensities along the line labeled A. c) Wrinkle measure. d) P measured at image location A for one subject performing each of the six actions. We classified the actions from these five feature measures using a 3-layer neural net with 15 hidden units. This method performs well for some subjects but not for Classifying Facial Action 827 Figure 5: Example flow field for a subject performing AU 7, partial closure of the eyelids. Each flow vector is plotted as an arrow that points in the direction of motion. Axes give image location. others, depending on age and physiognomy. It achieves an overall generalization performance of 57% correct. 5 OPTIC FLOW The motion that results from facial action provides another important source of information. The third classifier attempts to classify facial actions based only on the pattern of facial motion. Motion is extracted from image pairs consisting of a neutral image and an image that displays the action to be classified. An approximation to flow is extracted by implementing the brightness constraint equation (2) where the velocity (vx,Vy) at each image point is estimated from the spatial and temporal gradients of the image I. The velocities can only be reliably extracted at points of large gradient, and we therefore retain only the velocities from those locations. One of the advantages of this simple local estimate of flow is speed. It takes 0.13 seconds on a 120 MHz Pentium to compute one flow field. A resulting flow image is illustrated in Figure 5. 8I(x, y, t) 8I(x, y, t) 8I(x, y, t) _ 0 Vx 8x + Vy 8y + 8t (2) We obtained weighted templates for each of the actions by taking mean flow fields from 10 subjects. We compared novel flow patterns, r to the template ft by the similarity measure S (3). S is the normalized dot product of the novel flow field with the template flow field. This template matching procedure gave 84.8% accuracy for novel subjects. Performance was the same for the ten subjects used in the training set as for the ten in the test set. (3) 6 COMBINED SYSTEM Figure 6 compares performance for the three individual methods described in the previous sections. Error bars give the standard deviation for the estimate of generalization to novel subjects. We obtained the best performance when we combined all three sources of information into a single neural network. The classifier is a 828 BAR1LETI, VIOLA, SEJNOWSKI, GOLOMB, LARSEN, HAGER, EKMAN I 6 Output I WTA 11Ol:i Classifier Figure 6: Left: Combined system architecture. Right: Performance comparisons. Holistic v. Flow r :0.52 • i g ~ ~ 50 Feature v. Row r :0.26 • • • • 60 70 80 90 • 100 Feature v. Holistic i r:O.OO Ii! ~~--...;......,..".~ ~:-----50 60 70 80 90 100 Figure 7: Performance correlations among the three individual classifiers. Each data point is performance for one of the 20 subjects. feed forward network taking 50 component projections, 5 feature measures, and 6 template matches as input (see Figure 6.) The combined system gives a generalization performance of 92%, which is an improvement over the best individual method at 88.6%. The increase in performance level is statistically significant by a paired t-test. While the improvement is small, it constitutes about 30% of the difference between the best individual classifier and perfect performance. Figure 6 also shows performance of human subjects on this same dataset. Human non-experts can correctly classify these images with about 74% accuracy. This is a difficult classification problem that requires considerable training for people to be able to perform well. We can examine how the combined system benefits from multiple input sources by looking at the cprrelations in performance of the three individual classifiers. Combining estimators is most beneficial when the individual estimators make very different patterns of errors.1 The performance of the individual classifiers are compared in Figure 7. The holistic and the flow field classifiers are correlated with a coefficient of 0.52. The feature based system, however, has a more independent pattern of errors from the two template-based methods. Although the stand-alone performance of the featurebased system is low, it contributes to the combined system because it provides estimates that are independent from the two template-based systems. Without the feature measures, we lose 40% of the improvement. Since we have only a small number of features, this data does not address questions about whether templates are better than features, but it does suggest that local features plus templates may be superior to either one alone, since they may have independent patterns of errors. iTom Dietterich, Connectionists mailing list, July 24, 1993. Classifying Facial Action 829 7 DISCUSSION We have evaluated the performance of three approaches to image analysis on a difficult classification problem. We obtained the best performance when information from holistic spatial analysis, feature measurements, and optic flow fields were combined in a single system. The combined system classifies a face in less than a second on a 120 MHz Pentium. Our initial results are promising since the upper facial actions included in this study represent subtle distinctions in facial appearance that require lengthy training for humans to make reliably. Our results compare favorably with facial expression recognition systems developed by Mase (1991), Yacoob and Davis (1994), and Padgett and Cottrell (1995), who obtained 80%, 88%, and 88% accuracy respectively for classifying up to six full face expressions. The work presented here differs from these systems in that we attempt to detect individual muscular actions rather than emotion categories, we use a dataset of labeled facial actions, and our dataset includes low and medium intensity muscular actions as well as high intensity ones. Essa and Pentland (1995) attempt to relate facial expressions to the underlying musculature through a complex physical model of the face. Since our methods are image-based, they are more adaptable to variations in facial structure and skin elasticity in the subject population. We intend to apply these techniques to the lower facial actions and to action combinations as well. A completely automated method for scoring facial actions from images would have both commercial and research applications and would reduce the time and expense currently required for manual scoring by trained observers. Acknow ledgments This research was supported by Lawrence Livermore National Laboratories, IntraUniversity Agreement B291436, NSF Grant No. BS-9120868, and Howard Hughes Medical Institute. We thank Claudia Hilburn for image collection. References Cottrell, G.,& Metcalfe, J. (1991): Face, gender and emotion recognition using holons. In Advances in Neural Information Processing Systems 9, D. Touretzky, (Ed.) San Mateo: Morgan & Kaufman. 564 - 571. Ekman, P., & Friesen, W. (1978): Facial Action Coding System: A Technique for the Measurement of Facial Movement. Palo Alto, CA: Consulting Psychologists Press. Ekman, P., Huang, T., Sejnowski, T., & Hager, J. (1992): Final Report to NSF of the Planning Workshop on Facial Expression Understanding. Available from HIL-0984, UCSF, San Francisco, CA 94143. Essa, I., & Pentland, A. (1995). Facial expression recognition using visually extracted facial action parameters. Proceedings of the International Workshop on Automatic Face- and Gesture-Recognition. University of Zurich, Multimedia Laboratory. Golomb, B., Lawrence, D., & Sejnowski, T. (1991). SEXnet: A neural network identifies sex from human faces. In Advances in Neural Information Processing Systems 9, D. Touretzky, (Ed.) San Mateo: Morgan & Kaufman: 572 - 577. Hager, J., & Ekman, P., (1995). The essential behavioral science of the face and gesture that computer scientists need to know. Proceedings of the International Workshop on Automatic Face- and Gesture-Recognition. University of Zurich, Multimedia Laboratory. Mase, K. (1991): Recognition of facial expression from optical flow. IEICE Transactions E 74(10): 3474-3483. Padgett, C., Cottrell, G., (1995). Emotion in static face images. Proceedings of the Institute for Neural Computation Annual Research Symposium, Vol 5. La Jolla, CA. Turk, M., & Pentland, A. (1991): Eigenfaces for Recognition. Journal of Cognitive Neuroscience 3(1): 71 - 86. Yacoob, Y., & Davis, L. (1994): Recognizin~ human facial expression. University of Maryland Center for Automation Research Technical Report No. 706.
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1995
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135
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1,038
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Gradient and Hamiltonian Dynamics Applied to Learning in Neural Networks James W. Howse Chaouki T. Abdallah Gregory L. Heileman Department of Electrical and Computer Engineering University of New Mexico Albuquerque, NM 87131 Abstract The process of machine learning can be considered in two stages: model selection and parameter estimation. In this paper a technique is presented for constructing dynamical systems with desired qualitative properties. The approach is based on the fact that an n-dimensional nonlinear dynamical system can be decomposed into one gradient and (n - 1) Hamiltonian systems. Thus, the model selection stage consists of choosing the gradient and Hamiltonian portions appropriately so that a certain behavior is obtainable. To estimate the parameters, a stably convergent learning rule is presented. This algorithm has been proven to converge to the desired system trajectory for all initial conditions and system inputs. This technique can be used to design neural network models which are guaranteed to solve the trajectory learning problem. 1 Introduction A fundamental problem in mathematical systems theory is the identification of dynamical systems. System identification is a dynamic analogue of the functional approximation problem. A set of input-output pairs {u(t), y(t)} is given over some time interval t E [7i, 1j]. The problem is to find a model which for the given input sequence returns an approximation of the given output sequence. Broadly speaking, solving an identification problem involves two steps. The first is choosing a class of identification models which are capable of emulating the behavior of the actual system. The second is selecting a method to determine which member of this class of models best emulates the actual system. In this paper we present a class of nonlinear models and a learning algorithm for these models which are guaranteed to learn the trajectories of an example system. Algorithms to learn given trajectories of a continuous time system have been proposed in [6], [8], and [7] to name only a few. To our knowledge, no one has ever proven that the error between the learned and desired trajectories vanishes for any of these algorithms. In our trajectory learning system this error is guaranteed to vanish. Our models extend the work in [1] by showing that Cohen's systems are one instance of the class of models generated by decomposing the dynamics into a component normal to some surface and a set of components tangent to the same surface. Conceptually this formalism can be used to design dynamical systems with a variety of desired qualitative properties. Furthermore, we propose a provably convergent learning algorithm which allows the parameters of Cohen's models to be learned from examples rather than being programmed in advance. The algorithm is Gradient and Hamiltonian Dynamics Applied to Learning in Neural Networks 275 convergent in the sense that the error between the model trajectories and the desired trajectories is guaranteed to vanish. This learning procedure is related to one discussed in [5] for use in linear system identification. 2 Constructing the Model First some terminology will be defined. For a system of n first order ordinary differential equations, the phase space of the system is the n-dimensional space of all state components. A solution trajectory is a curve in phase space described by the differential equations for one specific starting point. At every point on a trajectory there exists a tangent vector. The space of all such tangent vectors for all possible solution trajectories constitutes the vector field for this system of differential equations. The trajectory learning models in this paper are systems of first order ordinary differential equations. The form of these equations will be obtained by considering the system dynamics as motion relative to some surface. At each point in the state space an arbitrary system trajectory will be decomposed into a component normal to this surface and a set of components tangent to this surface. This approach was suggested to us by the results in [4], where it is shown that an arbitrary n-dimensional vector field can be decomposed locally into the sum of one gradient vector field and (n - 1) Hamiltonian vector fields. The concept of a potential function will be used to define these surfaces. A potential function V(:z:) is any scalar valued function of the system states :z: = [Xl, X2, ••• , Xn.] t which is at least twice continuously differentiable (Le. V(:z:) E or : r ~ 2). The operation [.]t denotes the transpose of the vector. If there are n components in the system state, the function V{:z:), when plotted with respect all of the state components, defines a surface in an (n + 1 )-dimensional space. There are two curves passing through every point on this potential surface which are of interest in this discussion, they are illustrated in Figure 1(a). The dashed curve is (z - zo)t \7 ... v (z)l ... o = 0 (a) (b) V(z) = KFigure 1: (a) The potential function V(z) = X~ (Xl _1)2 +x~ plotted versus its two dependent variables Xl and X2. The dashed curve is called a level surface and is given by V(z) = 0.5. The solid curve follows the path of steepest descent through Zo. (b) The partitioning of a 3-dimensional vector field at the point Zo into a 1dimensional portion which is normal to the surface V(z) = K- and a 2-dimensional portion which is tangent to V(z) = K-. The vector -\7 ... V(z) 1"'0 is the normal vector to the surface V(z) = K- at the point Zo. The plane (z - zo)t \7 ... V (z) 1"'0 = 0 contains all of the vectors which are tangent to V(z) = K- at Zo. Two linearly independent vectors are needed to form a basis for this tangent space, the pair Q2(z) \7 ... V (z)l ... o and Q3(Z) \7 ... V (z)l ... o that are shown are just one possibility. referred to as a level surface, it is a surface along which V(:z:) = K for some constant K. Note that in general this level surface is an n-dimensional object. The solid curve 276 J. W. HOWSE, C. T. ABDALLAH, G. L. HEILEMAN moves downhill along V (X) following the path of steepest descent through the point Xo. The vector which is tangent to this curve at Xo is normal to the level surface at Xo. The system dynamics will be designed as motion relative to the level surfaces of V(x). The results in [4] require n different local potential functions to achieve arbitrary dynamics. However, the results in [1] suggest that a considerable number of dynamical systems can be achieved using only a single global potential function. A system which is capable of traversing any downhill path along a given potential surface V(x), can be constructed by decomposing each element of the vector field into a vector normal to the level surface of V(x) which passes through each point and a set of vectors tangent to the level surface of V(x) which passes through the same point. So the potential function V(x) is used to partition the n-dimensional phase space into two subspaces. The first contains a vector field normal to some level surface V(x) = }( for }( E IR, while the second subspace holds a vector field tangent to V(x) = IC. The subspace containing all possible normal vectors to the n-dimensional level surface at a given point, has dimension one. This is equivalent to the statement that every point on a smooth surface has a unique normal vector. Similarly, the subspace containing all possible tangent vectors to the level surface at a given point has dimension (n - 1). An example of this partition in the case of a 3-dimensional system is shown in Figure 1 (b). Since the space of all tangent vectors at each point on a level surface is (n - I)-dimensional, (n - 1) linearly independent vectors are required to form a basis for this space. Mathematically, there is a straightforward way to construct dynamical systems which either move downhill along V(x) or remain at a constant height on V(x). In this paper, dynamical systems which always move downhill along some potential surface are called gradient-like systems. These systems are defined by differential equations of the form x = -P(x) VII: V(x), (1) where P(x) is a matrix function which is symmetric (Le. pt = P) and positive definite at every point x, and where V III V(x) = [g;: , g;: , ... , :z~]f. These systems are similar to the gradient flows discussed in [2]. The trajectories of the system formed by Equation (1) always move downhill along the potential surface defined by V(x). This can be shown by taking the time derivative of V(x) which is V(x) = -[VII: V (x)]t P(x) [VII: V(x)] :5 O. Because P(x) is positive definite, V(x) can only be zero where V II: V (x) = 0, elsewhere V(x) is negative. This means that the trajectories of Equation (1) always move toward a level surface of V(x) formed by "slicing" V(x) at a lower height, as pointed out in [2]. It is also easy to design systems which remain at a constant height on V(x). Such systems will be denoted Hamiltonian-like systems. They are specified by the equation x = Q(x) VII: V(x), (2) where Q(x) is a matrix function which is skew-symmetric (Le. Qt = -Q) at every point x. These systems are similar to the Hamiltonian systems defined in [2]. The elements of the vector field defined by Equation (2) are always tangent to some level surface of V (x). Hence the trajectories ofthis system remain at a constant height on the potential surface given by V(x). Again this is indicated by the time derivative of V(x), which in this case is V(x) = [VII: V(x)]f Q(x)[VII: V(x)] = o. This indicates that the trajectories of Equation (2) always remain on the level surface on which the system starts. So a model which can follow an arbitrary downhill path along the potential surface V(x) can be designed by combining the dynamics of Equations (1) and (2). The dynamics in the subspace normal to the level surfaces of V(x) can be Gradient and Hamiltonian Dynamics Applied to Learning in Neural Networks 277 defined using one equation of the form in Equation (1). Similarly the dynamics in the subspace tangent to the level surfaces of Vex) can be defined using (n - 1) equations of the form in Equation (2). Hence the total dynamics for the model are n z= -P(x)VIDV(x) + LQi(X)VIDV(x). (3) i=2 For this model the number and location of equilibria is determined by the function Vex), while the manner in which the equilibria are approached is determined by the matrices P(x) and Qi(x). If the potential function Vex) is bounded below (i.e. Vex) > Bl V x E IRn , where Bl is a constant), eventually increasing (i.e. limlllDlI-+oo Vex) ~ 00) , and has only a finite number of isolated local maxima and minima (i.e. in some neighborhood of every point where V III V (x) = 0 there are no other points where the gradient vanishes), then the system in Equation (3) satisfies the conditions of Theorem 10 in [1]. Therefore the system will converge to one of the points where V ID Vex) = 0, called the critical points of Vex), for all initial conditions. Note that this system is capable of all downhill trajectories along the potential surface only if the (n - 1) vectors Qi(X) VID Vex) V i = 2, ... , n are linearly independent at every point x. It is shown in [1] that the potential function V(z) = C ( 1:., (-y) d-y + t, [ ~ (XI - I:.,(xd)' + ~ J:' 1:., h )II:.: (-y)]' d-y 1 (4) satisfies these three criteria. In this equation £.i(Xt} Vi = 1, ... , n are interpolation polynomials, C is a real positive constant, Xi Vi = 1, ... , n are real constants chosen so that the integrals are positive valued, and £.Hxt} == f:-. 3 The Learning Rule In Equation (3) the number and location of equilibria can be controlled using the potential function Vex), while the manner in which the equilibria are approached can be controlled with the matrices P(x) and Qi(X). If it is assumed that the locations of the equilibria are known, then a potential function which has local minima and maxima at these points can be constructed using Equation (4). The problem of trajectory learning is thereby reduced to the problem of parameterizing the matrices P(x) and Qi(x) and finding the parameter values which cause this model to best emulate the actual system. If the elements P(x) and Qi(x) are correctly chosen, then a learning rule can be designed which makes the model dynamics converge to that of the actual system. Assume that the dynamics given by Equation (3) are a parameterized model of the actual dynamics. Using this model and samples of the actual system states, an estimator for states of the actual system can be designed. The behavior of the model is altered by changing its parameters, so a parameter estimator must also be constructed. The following theorem provides a form for both the state and parameter estimators which guarantees convergence to a set of parameters for which the error between the estimated and target trajectories vanishes. Theorem 3.1. Given the model system k Z = LAili(x) +Bg(u) (5) i=l where Ai E IRnxn and BE IRnxm are unknown, and li(') and g(.) are known smooth functions such that the system has bounded solutions for bounded inputs u(t). Choose 278 J. W. HOWSE, C. T. ABDALLAH, G. L. HEILEMAN a state estimator of the form k ~ = 'R.B (x - x) + L Ai fi(x) + iJ g(u) i=1 (6) where'R.B is an (n x n) matrix of real constants whose eigenvalues must all be in the left half plane, and Ai and iJ are the estimates of the actual parameters. Choose parameter estimators of the form ~ t Ai = -'R.p (x - x) [fi(x)] V i = 1, ... , k B = -'R.p (x - x) [g(u)]t (7) where 'R.p is an (n x n) matrix of real constants which is symmetric and positive definite, and (x - x) [.]t denotes an outer product. For these choices of state and parameter estimators limt~oo(x(t) -x(t» = 0 for all initial conditions. Furthermore, this remains true if any of the elements of Ai or iJ are set to 0, or if any of these matrices are restricted to being symmetric or skew-symmetric. The proof of this theorem appears in [3]. Note that convergence of the parameter estimates to the actual parameter values is not guaranteed by this theorem. The model dynamics in Equation (3) can be cast in the form of Equation (5) by choosing each element of P(x) and Qi(X) to have the form n I-I n I-I PrB = LL~rBjkt?k(Xj) and QrB = LLArBjk ek(Xj), (8) j=1 k=O j=1 k=O where {t?o(Xj), t?1 (Xj), ... ,t?I-1 (Xj)} and {eo(Xj), el (Xj), ... ,el-l (Xj)} are a set of 1 orthogonal polynomials which depend on the state Xj' There is a set of such polynomials for every state Xj, j = 1,2, ... , n. The constants ~rBjk and ArBjk determine the contribution of the kth polynomial which depends on the jth state to the value of Prs and Qrs respectively. In this case the dynamics in Equation (3) become :i: = t. ~ { S;. [11.(x;) V. V (z)j + t, A;;. [e;.(x;) v. V(z)j } + T g(u(t)) (9) where 8 jk is the (n x n) matrix of all values ~rsjk which have the same value of j and k. Likewise Aijk is the (n x n) matrix of all values Arsjk, having the same value of j and k, which are associated with the ith matrix Qi(X). This system has m inputs, which may explicitly depend on time, that are represented by the m-element vector function u(t). The m-element vector function g(.) is a smooth, possibly nonlinear, transformation of the input function. The matrix Y is an (n x m) parameter matrix which determines how much of input S E {I, ... , m} effects state r E {I, ... , n}. Appropriate state and parameter estimators can be designed based on Equations (6) and (7) respectively. 4 Simulation Results Now an example is presented in which the parameters of the model in Equation (9) are trained, using the learning rule in Equations (6) and (7), on one input signal and then are tested on a different input signal. The actual system has three equilibrium points, two stable points located at (1,3) and (3,5), and a saddle point located at (2 ~,4 + ~). In this example the dynamics of both the actual system and the model are given by (~1) = (1'1 + 1'2 Z~ +:3 Z~ O 2) (:~) + (0 - {1'7 + 1'8 Z1 + 1'9 Z2}) (:~ ) + (1'10) u(t) (10) Z2 0 1'4 + 1'5 Z1 + 1'6 Z2 8Y 'P7 + 'P8 ZI + 1'9 Z2 0 8Y 0 8Z2 8Z2 Gradient and Hamiltonian Dynamics Applied to Learning in Neural Networks 279 where V(x) is defined in Equation (4) and u(t) is a time varying input. For the actual system the parameter values were 'PI = 'P4 = -4, 'P2 = 'Ps = -2, 'P3 = 'P6 = -1, 'P7 = 1, 'Ps = 3, 'P9 = 5, and 'PIO = 1. In the model the 10 elements 'Pi are treated as the unknown parameters which must be learned. Note that the first matrix function is positive definite if the parameters 'PI-'P6 are all negative valued. The second matrix function is skew-symmetric for all values of 'P7-'P9. The two input signals used for training and testing were Ul = 10000 (sin! 1000t + sin ~ 1000t) and U2 = 5000 sin 1000 t. The phase space responses of the actual system to the inputs UI and U2 are shown by the solid curves in Figures 3(b) and 3(a) respectively. Notice that both of these inputs produce a periodic attractor in the phase space of Equation (10). In order to evaluate the effectiveness of the learning algorithm the Euclidean distance between the actual and learned state and parameter values was computed and plotted versus time. The results are shown in Figure 2. Figure 2(a) shows these statistics when {1I~zll, II~'PII} {1I~zll, II~'PII} 17.5 15 15 12.5 12.5 10 7.5 i ,., ~--.----... -... --....... ---2.5 50 100 150 200 250 300 t 50 100 150 200 250 300 t (a) (b) Figure 2: (a) The state and parameter errors for training using input signal Ut. The solid curve is the Euclidean distance between the state estimates and the actual states as a function of time. The dashed curve shows the distance between the estimated and actual parameter values versus time. (b) The state and parameter errors for training using input signal U2. training with input UI, while Figure 2(b) shows the same statistics for input U2. The solid curves are the Euclidean distance between the learned and actual system states, and the dashed curves are the distance between the learned and actual parameter values. These statistics have two noteworthy features. First, the error between the learned and desired states quickly converges to very small values, regardless of how well the actual parameters are learned. This result was guaranteed by Theorem 3.1. Second, the final error between the learned and desired parameters is much lower when the system is trained with input UI. Intuitively this is because input Ul excites more frequency modes of the system than input U2. Recall that in a nonlinear system the frequency modes excited by a given input do not depend solely on the input because the system can generate frequencies not present in the input. The quality of the learned parameters can be qualitatively judged by comparing the phase plots using the learned and actual parameters for each input, as shown in Figure 3. In Figure 3(a) the system was trained using input Ul and tested with input U2, while in Figure 3(b) the situation was reversed. The solid curves are the system response using the actual parameter values, and the dashed curves are the response for the learned parameters. The Euclidean distance between the target and test trajectories in Figure 3(a) is in the range (0,0.64) with a mean distance of 0.21 and a standard deviation of 0.14. The distance between the the target and test trajectories in Figure 3(b) is in the range (0,4.53) with a mean distance of 0.98 and a standard deviation of 1.35. Qualitatively, both sets of learned parameters give an accurate response for non-training inputs. 280 -l -1 1 Xl (a) 1. W. HOWSE, C. T. ABDALLAH, G. L. HEILEMAN 5 I I o -------r--------------{i - 5 -10 -15 - 2 -1 4 (b) Figure 3: (a) A phase plot of the system response when trained with input UI and tested with input U2. The solid line is the response to the test input using the actual parameters. The dotted line is the system response using the learned parameters. (b) A phase plot of the system response when trained with input U2 and tested with input UI. Note that even when the error between the learned and actual parameters is large, the periodic attractor resulting from the learned parameters appears to have the same "shape" as that for the actual parameters. 5 Conclusion We have presented a conceptual framework for designing dynamical systems with specific qualitative properties by decomposing the dynamics into a component normal to some surface and a set of components tangent to the same surface. We have presented a specific instance of this class of systems which converges to one of a finite number of equilibrium points. By parameterizing these systems, the manner in which these equilibrium points are approached can be fitted to an arbitrary data set. We present a learning algorithm to estimate these parameters which is guaranteed to converge to a set of parameter values for which the error between the learned and desired trajectories vanishes. Acknowledgments This research was supported by a grant from Boeing Computer Services under Contract W-300445. The authors would like to thank Vangelis Coutsias, Tom Caudell, and Bill Home for stimulating discussions and insightful suggestions. References [1] M.A. Cohen. The construction of arbitrary stable dynamics in nonlinear neural networks. Neural Networks, 5(1):83-103, 1992. [2] M.W. Hirsch and S. Smale. Differential equations, dynamical systems, and linear algebra, volume 60 of Pure and Applied Mathematics. Academic Press, Inc., San Diego, CA, 1974. [3] J.W. Howse, C.T. Abdallah, and G.L. Heileman. A gradient-hamiltonian decomposition for designing and learning dynamical systems. Submitted to Neural Computation, 1995. [4] R.V. Mendes and J.T. Duarte. Decomposition of vector fields and mixed dynamics. Journal of Mathematical Physics, 22(7):1420-1422, 1981. [5] K.S. Narendra and A.M. Annaswamy. Stable adaptitJe systems. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1989. [6] B.A. Pearlmutter. Learning state space trajectories in recurrent neural networks. Neural Computation, 1(2):263-269, 1989. [7] D. Saad. Training recurrent neural networks via trajectory modification. Complex Systems, 6(2):213-236, 1992. [8] M.-A. Sato. A real time learning algorithm for recurrent analog neural networks. Biological Cybernetics, 62(2):237-241, 1990.
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A New Learning Algorithm for Blind Signal Separation s. Amari* University of Tokyo Bunkyo-ku, Tokyo 113, JAPAN amari@sat.t. u-tokyo.ac.jp A. Cichocki Lab. for Artificial Brain Systems FRP, RIKEN Wako-Shi, Saitama, 351-01, JAPAN cia@kamo.riken.go.jp H. H. Yang Lab. for Information Representation FRP, RIKEN Wako-Shi, Saitama, 351-01, JAPAN hhy@koala.riken.go.jp Abstract A new on-line learning algorithm which minimizes a statistical dependency among outputs is derived for blind separation of mixed signals. The dependency is measured by the average mutual information (MI) of the outputs. The source signals and the mixing matrix are unknown except for the number of the sources. The Gram-Charlier expansion instead of the Edgeworth expansion is used in evaluating the MI. The natural gradient approach is used to minimize the MI. A novel activation function is proposed for the on-line learning algorithm which has an equivariant property and is easily implemented on a neural network like model. The validity of the new learning algorithm are verified by computer simulations. 1 INTRODUCTION The problem of blind signal separation arises in many areas such as speech recognition, data communication, sensor signal processing, and medical science. Several neural network algorithms [3, 5, 7] have been proposed for solving this problem. The performance of these algorithms is usually affected by the selection of the activation functions for the formal neurons in the networks. However, all activation ·Lab. for Information Representation, FRP, RIKEN, Wako-shi, Saitama, JAPAN 758 S. AMARI, A. CICHOCKI, H. H. YANG functions attempted are monotonic and the selections of the activation functions are ad hoc. How should the activation function be determined to minimize the MI? Is it necessary to use monotonic activation functions for blind signal separation? In this paper, we shall answer these questions and give an on-line learning algorithm which uses a non-monotonic activation function selected by the independent component analysis (ICA) [7]. Moreover, we shall show a rigorous way to derive the learning algorithm which has the equivariant property, i.e., the performance of the algorithm is independent of the scaling parameters in the noiseless case. 2 PROBLEM Let us consider unknown source signals Si(t), i = 1"", n which are mutually independent. It is assumed that the sources Si(t) are stationary processes and each source has moments of any order with a zero mean. The model for the sensor output is x(t) = As(t) where A E R nxn is an unknown non-singular mixing matrix, set) [Sl(t),· .. , sn(t)]T and x(t) = [Xl(t), .. ·, xn(t)JT. Without knowing the source signals and the mixing matrix, we want to recover the original signals from the observations x(t) by the following linear transform: yet) = Wx(t) where yet) = [yl(t), ... , yn(t)]T and WE Rnxn is a de-mixing matrix. It is impossible to obtain the original sources Si(t) because they are not identifiable in the statistical sense. However, except for a permutation of indices, it is possible to obtain CiSi(t) where the constants Ci are indefinite nonzero scalar factors. The source signals are identifiable in this sense. So our goal is to find the matrix W such that [yl, ... , yn] coincides with a permutation of [Sl, ... ,sn] except for the scalar factors. The solution W is the matrix which finds all independent components in the outputs. An on-line learning algorithm for W is needed which performs the ICA. It is possible to find such a learning algorithm which minimizes the dependency among the outputs. The algorithm in [6] is based on the Edgeworth expansion[8] for evaluating the marginal negentropy. Both the Gram-Charlier expansion[8] and the Edgeworth expansion[8] can be used to approximate probability density functions. We shall use the Gram-Charlier expansion instead of the Edgeworth expansion for evaluating the marginal entropy. We shall explain the reason in section 3. 3 INDEPENDENCE OF SIGNALS The mathematical framework for the ICA is formulated in [6]. The basic idea of the ICA is to minimize the dependency among the output components. The dependency is measured by the Kullback-Leibler divergence between the joint and the product of the marginal distributions of the outputs: J p(y) D(W) = p(y) log rr ( a) dy a=lPa y (1) where Pa(ya) is the marginal probability density function (pdf). Note the KullbackLeibler divergence has some invariant properties from the differential-geometrical point of view[l]. A New Learning Algorithm for Blind Signal Separation 759 It is easy to relate the Kullback-Leibler divergence D(W) to the average MI of y: n D(W) = -H(y) + LH(ya) (2) a=l where H(y) = - J p(y) logp(y)dy, H(ya) = - J Pa(ya)logPa(ya)dya is the marginal entropy. The minimization of the Kullback-Leibler divergence leads to an ICA algorithm for estimating W in [6] where the Edgeworth expansion is used to evaluate the negentropy. We use the truncated Gram-Charlier expansion to evaluate the KullbackLeibler divergence. The Edgeworth expansion has some advantages over the GramCharlier expansion only for some special distributions. In the case of the Gamma distribution or the distribution of a random variable which is the sum of iid random variables, the coefficients of the Edgeworth expansion decrease uniformly. However, there is no such advantage for the mixed output ya in general cases. To calculate each H(ya) in (2), we shall apply the Gram-Charlier expansion to approximate the pdf Pa(ya). Since E[y] = E[W As] = 0, we have E[ya] = 0. To simplify the calculations for the entropy H(ya) to be carried out later, we assume m2 = 1. We use the following truncated Gram-Charlier expansion to approximate the pdf Pa(ya): (3) where lI;a = ma, 11;4 = m4 - 3, mk = E[(ya)k] is the k-th order moment of ya, 2 a(y) = ~e-lIi-, and Hk(Y) are Chebyshev-Hermite polynomials defined by the identity We prefer the Gram-Charlier expansion to the Edgeworth expansion because the former clearly shows how lI;a and 11;4 affect the approximation of the pdf. The last term in (3) characterizes non-Gaussian distributions. To apply (3) to calculate H(ya), we need the following integrals: - / a(y)H2(y)loga(y)dy = ~ (4) J a(y)(H2(y))2 H4(y)dy = 24. (5) These integrals can be obtained easily from the following results for the moments of a Gaussian random variable N(O,l): / y2k+1a(y)dy = 0, / y2ka(y)dy = 1·3··· (2k - 1). (6) By using the expansion y2 log(l + y) ~ y - 2 + O(y3) and taking account of the orthogonality relations of the Chebyshev-Hermite polynomials and (4)-(5), the entropy H(ya) is expanded as 1 (lI;a)2 (lI;a)2 5 1 H(ya) ~ -log(27re) __ 3 ___ 4_ + _(lI;a)2I1;a + _(lI;a)3. (7) 2 2 . 3! 2 . 4! 8 3 4 16 4 760 S. AMARI, A. CICHOCKI, H. H. YANG It is easy to calculate -J a(y)loga(y)dy = ~ log(27re). From y = Wx, we have H(y) = H(x) + log Idet(W)I. Applying (7) and the above expressions to (2), we have n n (Ka)2 (Ka)2 D(W) ~ -H(x) -log Idet(W)1 + -log(27re) - "[_3_, + ~4' 2 ~ 2 ·3. 2·. a=l (8) 4 A NEW LEARNING ALGORITHM To obtain the gradient descent algorithm to update W recursively, we need to calculate 88.0.. where wk' is the (a,k) element of W in the a-th row and k-th column. WI. Let cof(wk) be the cofactor of wk' in W. It is not difficult to derive the followings: 8log [det(W)[ _ 8wI: 81t3 _ 8w;: 81t; _ 8wl: cof(wk') = (W-Tt det(W) k 3E[(ya)2xk] 4E[(ya)3xk] where (W-T)k' denotes the (a,k) element of (WT)-l. From (8), we obtain ;!!a ~ -(W-T)k' + f(K'3, K~)E[(ya)2xk] + g(K'3, K~)E[(ya)3xk] (9) k where f(y, z) = -~y + l1yz, g(y, z) = -~z + ~y2 + ~z2. From (9), we obtain the gradient descent algorithm to update W recursively: " oD d~1s = -TJ( t)-oWk' TJ(t){(W- T)k - f(K'3, K~)E[(ya)2xk]_ g(K'3, K~)E[(ya)3xk]} (10) where TJ(t) is a learning rate function. Replacing the expectation values in (10) by their instantaneous values, we have the stochastic gradient descent algorithm: d~k = TJ(t){(W-T)k' - f(K'3, K~)(ya)2xk - g(K'3, K~)(ya)3xk}. (11) We need to use the following adaptive algorithm to compute K'3 and K~ in (11): dKa dt = -J.'(t)(K'3 - (ya)3) dKa d/ = -J.'(t)(K~ - (ya)4 + 3) (12) where 1'( t) is another learning rate function. The performance of the algorithm (11) relies on the estimation of the third and fourth order cumulants performed by the algorithm (12). Replacing the moments A New Learning Algorithm for Blind Signal Separation 761 ofthe random variables in (11) by their instantaneous values, we obtain the following algorithm which is a direct but coarse implementation of (11): dwa dt = 1](t){(W-T)~ - f(ya)x k} (13) where the activation function f(y) is defined by f() 3 11 25 9 14 7 47 5 29 3 Y = 4Y + 4 Y -"3Y - 4 Y + 4Y . (14) Note the activation function f(y) is an odd function, not a monotonic function. The equation (13) can be written in a matrix form: (15) This equation can be further simplified as following by substituting xTWT = yT: (16) where f(y) = (f(yl), ... , f(yn))T. The above equation is based on the gradient descent algorithm (10) with the following matrix form: dW aD dt = -1](t) aw' (17) From information geometry perspective[l], since the mixing matrix A is nonsingular we had better replace the above algorithm by the following natural gradient descent algorithm: dW aD T dt = -1](t)aw W w. (18) Applying the previous approximation of the gradient :& to (18), we obtain the following algorithm: (19) which has the same "equivariant" property as the algorithms developed in [4, 5]. Although the on-line learning algorithms (16) and (19) look similar to those in [3, 7] and [5] respectively, the selection of the activation function in this paper is rational, not ad hoc. The activation function (14) is determined by the leA. It is a non-monotonic activation function different from those used in [3, 5, 7]. There is a simple way to justify the stability of the algorithm (19). Let Vec(·) denote an operator on a matrix which cascades the columns of the matrix from the left to the right and forms a column vector. Note this operator has the following property: Vec(ABC) = (CT 0 A)Vec(B). (20) Both the gradient descent algorithm and the natural gradient descent algorithm are special cases of the following general gradient descent algorithm: dVec(W) = _ (t)P aD dt 1] aVec(W) (21) where P is a symmetric and positive definite matrix. It is trivial that (21) becomes (17) when P = I. When P = WTW 0 I, applying (20) to (21), we obtain dVec(W) T aD aD T dt = -1]( t)(W W 0 I) aVec(W) = -1]( t)Vec( aw W W) 762 s. AMARI. A. CICHOCKI. H. H. YANG and this equation implies (18). So the natural gradient descent algorithm updates Wet) in the direction of decreasing the dependency D(W). The information geometry theory[l] explains why the natural gradient descent algorithm should be used to minimize the MI. Another on-line learning algorithm for blind separation using recurrent network was proposed in [2]. For this algorithm, the activation function (14) also works well. In practice, other activation functions such as those proposed in [2]-[6] may also be used in (19). However, the performance of the algorithm for such functions usually depends on the distributions of the sources. The activation function (14) works for relatively general cases in which the pdf of each source can be approximated by the truncated Gram-Charlier expansion. 5 SIMULATION In order to check the validity and performance of the new on-line learning algorithm (19), we simulate it on the computer using synthetic source signals and a random mixing matrix. The extensive computer simulations have fully confirmed the theory and the validity of the algorithm (19). Due to the limit of space we present here only one illustrative example. Example: Assume that the following three unknown sources are mixed by a random mixing matrix A: [SI (t), S2(t), S3(t)] = [n(t), O.lsin( 400t)cos(30t), 0.01sign[sin(500t + 9cos( 40t))] where net) is a noise source uniformly distributed in the range [-1, +1], and S2(t) and S3(t) are two deterministic source signals. The elements of the mixing matrix A are randomly chosen in [-1, +1]. The learning rate is exponentially decreaSing to zero as rJ(t) = 250exp( -5t). A simulation result is shown in Figure 1. The first three signals denoted by Xl, X2 and X3 represent mixing (sensor) signals: x l (t), x2(t) and x3(t). The last three signals denoted by 01, 02 and 03 represent the output signals: yl(t), y2(t), and y3(t). By using the proposed learning algorithm, the neural network is able to extract the deterministic signals from the observations after approximately 500 milliseconds. The performance index El is defined by El = tct IPijl - 1) + tct IPijl - 1) i=1 j=1 maxk IPikl j=l i=l maxk IPkjl where P = (Pij) = WA. 6 CONCLUSION The major contribution of this paper the rigorous derivation of the effective blind separation algorithm with equivariant property based on the minimization of the MI of the outputs. The ICA is a general principle to design algorithms for blind signal separation. The most difficulties in applying this principle are to evaluate the MI of the outputs and to find a working algorithm which decreases the MI. Different from the work in [6], we use the Gram-Charlier expansion instead of the Edgeworth expansion to calculate the marginal entropy in evaluating the MI. Using A New Learning Algorithm for Blind Signal Separation 763 the natural gradient method to minimize the MI, we have found an on-line learning algorithm to find a de-mixing matrix. The algorithm has equivariant property and can be easily implemented on a neural network like model. Our approach provides a rational selection of the activation function for the formal neurons in the network. The algorithm has been simulated for separating unknown source signals mixed by a random mixing matrix. Our theory and the validity of the new learning algorithm are verified by the simulations. o. 04 0' o I Figure 1: The mixed and separated signals, and the performance index Acknowledgment We would like to thank Dr. Xiao Yan SU for the proof-reading of the manuscript. References [1] S.-I. Amari. Differential-Geometrical Methods in Statistics, Lecture Notes in Statistics vol.28. Springer, 1985. [2] S. Amari, A. Cichocki, and H. H. Yang. Recurrent neural networks for blind separation of sources. In Proceedings 1995 International Symposium on Nonlinear Theory and Applications, volume I, pages 37-42, December 1995. [3] A. J. Bell and T. J . Sejnowski. An information-maximisation approach to blind separation and blind deconvolution. Neural Computation, 7:1129-1159, 1995. [4] J.-F. Cardoso and Beate Laheld. Equivariant adaptive source separation. To appear in IEEE Trans. on Signal Processing, 1996. [5] A. Cichocki, R. Unbehauen, L. MoszczyIiski, and E. Rummert. A new on-line adaptive learning algorithm for blind separation of source signals. In ISANN94, pages 406-411, Taiwan, December 1994. [6] P. Comon. Independent component analysis, a new concept? Signal Processing, 36:287-314, 1994. [7] C. Jutten and J. Herault. Blind separation of sources, part i: An adaptive algorithm based on neuromimetic architecture. Signal Processing, 24:1- 10, 1991. [8] A. Stuart and J. K. Ord. Kendall's Advanced Theory of Statistics. Edward Arnold, 1994.
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Adaptive Back-Propagation in On-Line Learning of Multilayer Networks Ansgar H. L. West 1,2 and David Saad2 1 Department of Physics, University of Edinburgh Edinburgh EH9 3JZ, U.K. 2Neural Computing Research Group, University of Aston Birmingham B4 7ET, U.K. Abstract An adaptive back-propagation algorithm is studied and compared with gradient descent (standard back-propagation) for on-line learning in two-layer neural networks with an arbitrary number of hidden units. Within a statistical mechanics framework , both numerical studies and a rigorous analysis show that the adaptive back-propagation method results in faster training by breaking the symmetry between hidden units more efficiently and by providing faster convergence to optimal generalization than gradient descent. 1 INTRODUCTION Multilayer feedforward perceptrons (MLPs) are widely used in classification and regression applications due to their ability to learn a range of complicated maps [1] from examples. When learning a map fo from N-dimensional inputs e to scalars ( the parameters {W} of the student network are adjusted according to some training algorithm so that the map defined by these parameters fw approximates the teacher fo as close as possible. The resulting performance is measured by the generalization error Eg, the average of a suitable error measure E over all possible inputs Eg = (E)e' This error measure is normally defined as the squared distance between the output of the network and the desired output, i.e., (1) One distinguishes between two learning schemes: batch learning , where training algorithms are generally based on minimizing the above error on the whole set of given examples, and on-line learning , where single examples are presented serially and the training algorithm adjusts the parameters after the presentation of each 324 A.H.L. VfEST,D.SAJU) example. We measure the efficiency of these training algorithms by how fast (or whether at all) they converge to an "acceptable" generalization error. This research has been motivated by recent work [2] investigating an on-line learning scenario of a general two-layer student network trained by gradient descent on a task defined by a teacher network of similar architecture. It has been found that in the early stages oftraining the student is drawn into a suboptimal symmetric phase, characterized by each student node imitating all teacher nodes with the same degree of success. Although the symmetry between the student nodes is eventually broken and the student converges to the minimal achievable generalization error, the majority of the training time may be spent with the system trapped in the symmetric regime, as one can see in Fig. 1. To investigate possible improvements we introduce an adaptive back-propagation algorithm, which improves the ability of the student to distinguish between hidden nodes of the teacher. We compare its efficiency with that of gradient descent in training two-layer networks following the framework of [2]. In this paper we present numerical studies and a rigorous analysis of both the breaking of the symmetric phase and the convergence to optimal performance. We find that adaptive back-propagation can significantly reduce training time in both regimes by breaking the symmetry between hidden units more efficiently and by providing faster exponential convergence to zero generalization error. 2 DERIVATION OF THE DYNAMICAL EQUATIONS The student network we consider is a soft committee machine [3] , consisting of J< hidden units which are connected to N-dimensional inputs e by their weight vectors W = {"Wi} (i = 1, .. . , J<). All hidden units are connected to the linear output unit by couplings of unit strength and the implemented mapping is therefore fw(e) = L~l g(Xi), where Xi = "Wi ·e is the activation of hidden unit i and gO is a sigmoidal transfer function. The map fo to be learned is defined by a teacher network of the same architecture except for a possible difference in the number of hidden units M and is defined by the weight vectors B = {Bn} (n = 1, ... , M). Training examples are of the form (e,(~), where the components of the input vectors e are drawn independently from a zero mean unit variance Gaussian distribution; the outputs are (~ = L~=l g(y~), where y~ = En·e is the activation of teacher hidden unit n. An on-line training algorithm A is defined by the update of each weight in response to the presentation of an example (e~, (~) , which can take the general form "Wi~+1 ="Wi~ +Ai({,}, W~ , e,(~) , where {,} defines parameters adjustable by the user. In the case of standard back-propagation, i.e., gradient descent on the error function defined in Eq. (1): Afd(7J, W~,e,(~) = (7JIN)8,/e with 8,/ = 8~g'(xn = [(~ - fw(e)] g'(xn, (2) where the only user adjustable parameter is the learning rate 7J scaled by 11 N. One can readily see that the only term that breaks the symmetry between different hidden units is g'(xn, i.e., the derivative ofthe transfer function gO. The fact that a prolonged symmetric phase can exist indicates that this term is not significantly different over the hidden units for a typical input in the symmetric phase. The rationale of the adaptive back-propagation algorithm defined below is therefore to alter the g'-term, in order to magnify small differences in the activation between hidden units. This can be easily achieved by altering g'(Xi) to g'({3xi), where {3 plays the role of an inverse "temperature". Varying {3 changes the range of hidden unit activations relevant for training, e.g., for {3 > 1 learning is more confined to Adaptive Back-Propagation in On-line Learning of Multilayer Networks 325 small activations, when compared to gradient descent (/3 = 1). The whole adaptive back-propagation training algorithm is therefore: A:bP(7], j3,W tl ,e,(tl) = ~8tlg'(/3xnetl= ~8te (3) with 8tl as in Eq. (2). To compare the adaptive back-propagation algorithm with normal gradient descent, we follow the statistical mechanics calculation in [2]. Here we will only outline the main ideas and present 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 (3) in "'i as equations in the order parameters describing the overlaps between student nodes Qij = "'i. Wj , student and teacher nodes R;,n = "'i' Bn and teacher nodes Tnm = Bn' Bm. The generalization error cg , measuring the typical performance, can be expressed in these variables only [2]. The order parameters Qij and Rin are the new dynamical variables, which are self-averaging with respect to the randomness in the training data in the thermodynamic limit (N --+ 00). If we interpret the normalized example number 0:' = p./ N as a continuous time variable, the update equations for the order parameters become first order coupled differential equations dR;,n dO:' dQ· · _'_J dO:' (4) All the integrals in Eqs. (4) and the generalization error can be calculated explicitly if we choose g( x) = erf( x / V2) as the sigmoidal activation function [2] . The exact form of the resulting dynamical equations for adaptive back-propagation is similar to the equations in [2] and will be presented elsewhere [4]. They can easily be integrated numerically for any number of K student and M teacher hidden units. For the remainder of the paper, we will however focus on the realizable case (K = M) and uncorrelated isotropic teachers of unit length Tnm = 8nm . The dynamical evolution of the overlaps Qij and R;,n follows from integrating the equations of motion (4) from initial conditions determined by the random initialization of the student weights W. Whereas the resulting norms Qii of the student vector will be order 0(1) , the overlaps Qij between student vectors, and studentteacher vectors Rin will be only order 0(1/.JFi) . The random initialization of the weights is therefore simulated by initializing the norms Qii and the overlaps Qij and Rin from uniform distributions in the [0, 0.5] and [0,10- 12] interval respectively. In Fig. 1 we show the difference of a typical evolution of the overlaps and the generalization error for j3 = 12 and /3 = 1 (gradient descent) for K = 3 and 7] = 0.01. In both cases, the student is drawn quickly into a suboptimal symmetric phase, characterized by a finite generalization error (Fig. Ie) and no differentiation between the hidden units of the student: the student norms Qii and overlaps Qij are similar (Figs. 1b,ld) and the overlaps of each student node with all teacher nodes Rin are nearly identical (Figs. 1a,lc). The student trained by gradient descent (Figs. 1c,ld) is trapped in this unstable suboptimal solution for most ofthe training time, whereas adaptive back-propagation (Figs. 1a,lb) breaks the symmetry significantly earlier. The convergence phase is characterized by a specialization of the different student nodes and the evolution of the overlap matrices Q and R to their optimal value T , except for the permutational symmetry due to the arbitrary labeling of the student nodes. Clearly, the choice /3 = 12 is suboptimal in this regime. The student trained with /3 = 1 converges faster to zero generalization error (Fig. Ie). In order to optimize /3 seperately for both the symmetric and the convergence phase, we will examine the equations of motions analytically in the following section. 326 1.0 (a) /' R 11 _ 0.8 R 12 ......... i R 13 ---_. 0.6 I R21 - _. Rin , R22 _.I R 23 ---0.4 R31 _ .. R 32 ··· · . 0.2 R33 _ ....• o 20000 40000 a 60000 80000 1.0,...-;::(b:"7)-------::::::===---, 0.8 0.6 Qij 0.4 0.2 '--..-._. Q11Q12 ........ . Q22 ----. Q23 - _. Q33 _.Q13 -.--O.O-+-T"'I---...-... ..,.......,:=r=;=-r-.,-,--r-..-r-I o 20000 40000 a 60000 80000 Figure 1: Dynamical evolution of the student-teacher overlaps Rin (a,c), the student-student overlaps Qij (b,d), and the generalization error (e) as a function of the normalized example number a for a student with three hidden nodes learning an isotropic three-node teacher (Tnm=c5nm ). The learning rate 7]=0.01 is fixed but the value of the inverse temperature varies (a,b): .8=12 and (c,d): .8=1 (gradient descent). A. H. L. WEST, D. SAAD 1.0 Rll (c) ....... R12 0.8 ---- R 13 -R21 _.- R22 0.6 ---_. R 23 Rin - .. R31 . . . . Rn 0.4 _ .... R33 0.2 o 20000 40000 a 60000 80000 1.0 _ Q11 (d) ......... Q12 0.8 ____ . Q22 - _. Q23 0.6 - .Q33 Qij --- Q13 0.4 o 20000 40000 a 60000 80000 .~-r-(~)----------------------, e f3 = 12 f3 = 1 ........ . .02-l"------..01 o 20000 40000 a 60000 80000 3 ANALYSIS OF THE DYNAMICAL EQUATIONS In the case of a realizable learning scenario (K=M) and isotropic teachers (Tnm=c5nm ) the order parameter space can be very well characterized by similar diagonal and off-diagonal elements of the overlap matrices Q and R, i.e., Qij = Qc5ij + C(1 c5ij ) for the student-student overlaps and, apart from a relabeling of the student nodes, by Rin = Rc5in + 5(1 c5in ) for the student-teacher overlaps. As one can see from Fig. 1, this approximation is particularly good in the symmetric phase and during the final convergence to perfect generalization. 3.1 SYMMETRIC PHASE AND ONSET OF SPECIALIZATION Numerical integration of the equations of motion for a range of learning scenarios show that the length of the symmetric phase is especially prolonged by isotropic teachers and small learning rates 7]. We will therefore optimize the dynamics (4) in Adaptive Back-Propagation in On-line Learning of Multilayer Networks 327 the symmetric phase with respect to {3 for isotropic teachers in the small 7] regime, where terms proportional to 7]2 can be neglected. The fixed point of the truncated equations of motion Q* * 1 = C = 2I< -1 and R* = S* = W = 1 V K JI«2I< - 1) (5) is independent of f3 and thus identical to the one obtained in [2]. However, the symmetric solution is an unstable fixed point of the dynamics and the small perturbations introduced by the generically nonsymmetric initial conditions will eventually drive the student towards specialization. To study the onset of specialization, we expand the truncated differential equations to first order in the deviations q = Q - Q*, c = C - C* , T' = R - R*, and s = S - S" from the fixed point values (5). The linearized equations of motion take the form dv/do: = M·v, where v = (q, c, T', s) and M is a 4 x 4 matrix whose elements are the first derivatives of the truncated update equations (4) at the fixed point with respect to v. Perturbations or modes which are proportional to the eigenvectors Vi of M will therefore decrease or increase exponentially depending on whether the corresponding eigenvalue Ai is negative or positive. For the onset of specialization only the modes are relevant which are amplified by the dynamics, i.e., the ones with positive eigenvalue. For them we can identify the inverse eigenvalue as a typical escape time Ti from the symmetric phase. We find only one relevant perturbation for q = c = 0 and s = -r/(I< - 1). This can be confirmed by a closer look at Fig. 1. The onset of specialization is signaled by the breaking of the symmetry between the student-teacher overlaps, whereas significant differences from the symmetric fixed point values of the student norms and overlaps occur later. The escape time T associated with the above perturbation is T({3) = ~ V2I< - 1(2I< + {3)3/2 27] I< f3 (6) Minimization of T with respect to f3 yields rr'pt = 4I<, i.e., the optimal f3 scales with the number of hidden units, and Topt = 97r V2I{ - 1 27] V6I< (7) Trapping in the symmetric phase is therefore always inversely proportional to the learning rate 7]. In the large I< limit it is proportional to the number of hidden nodes I< (T""'" 27rI</7]) for gradient descent, whereas it is independent of I< [T""'" 3V37r/(27])] for the optimized adaptive back-propagation algorithm. 3.2 CONVERGENCE TO OPTIMAL GENERALIZATION In order to predict the optimal learning rate 7]0pt and inverse temperature {30pt for the convergence, we linearize the full equations of motion (4) around the zero generalization error fixed point R* = Q* = 1 and S* = C* = O. The matrix M of the resulting system of four coupled linear differential equations in q = 1 - Q, c = C , r = 1 - R, and s = S is very complicated for arbitrary {3, I< and 7], and its eigenvalues and eigenvectors can therefore only be analysed numerically. We illustrate the solution space with I< = 3 and two {3 values in Fig. 2a. We find that the dynamics decompose into four modes: two slow modes associated with eigenvalues Al and A2 and two fast modes associated with eigenvalues A3 and A4, which are negative for all learning rates and whose magnitude is significantly larger. 328 A. H. L. WEST, D. SAAD 0.1-,------------..-.., 2.05 ......... ------------...., (a) _.- .\1(1) .\1 (fioPt) 0.05 ......... .\2(1) ----. .\2 (fiopt) O.O~..,.....-----------+--+--I ,\ -0.05 -0.1 0.0 0.5 1.0 1.5 2.0 1.95 !3 1.9 1.85 1.8 5 10 I{ 50 100 5001000 Figure 2: (a) The eigenvalues AI, A2 (see text) as a function of the learning rate 7] at f{ = 3 for two values of (3: (3 = 1, and (3 = (30pt = 1.8314. For comparison we plot 2A2 and find that the optimal learning rate 7]0pt is given by the condition Al = 2A2 for (30Pt, but by the minimum of Al for (3 = 1. (b) The optimal inverse temperature (30pt as a function of the number of hidden units f{ saturates for large f{. The fast modes decay quickly and their influence on the long-time dynamics is negligible. They are therefore excluded from Fig. 2a and the following discussion. The eigenvalue A2 is negative and linear in 7]. The eigenvalue Al is a non-linear function of both (3 and 7] and negative for small 7]. For large 7], Al becomes positive and training does not converge to the optimal solution defining the maximum learning rate 7]max as Al (7]max) = O. For all 7] < 7]rnax the generalization error decays exponentionally to Eg * = O. In order to identify the corresponding convergence time r, which is inversely proportional to the modulus of the eigenvalue associated with the slowest decay mode, we expand the generalization error to second order in q, c, rand s. We find that the mode associated with the linear eigenvalue A2 does not contribute to first order terms, and controls only second order term in a decay rate of 2A2 . The learning rate 7]0pt which provides the fastest asymptotic decay rate A opt of the generalization error is therefore either given by the condition Al(7]°Pt) = 2A2(7]°Pt) or alternatively by min7](Al) if Al > 2A2 at the minimum of Al (see Fig. 2a). We can further optimize convergence to optimal generalization by minimizing the decay rate Aopt((3) with respect to (3 (see Fig. 2b). Numerically, we find that the optimal inverse temperature (30pt saturates for large f{ at (30pt ~ 2.03. For large f{, we find an associated optimal convergence time r opt ((30pt) ,....., 2.90f{ for adaptive back-propagation optimized with respect to 7] and (3, which is an improvement by 17% when compared to ropt(l)""'" 3.48f{ for gradient descent optimized with respect to 7]. The optimal and maximal learning rates show an asymptotic 1/ f{ behaviour and 7]0pt((30Pt) ,....., 4.78/ f{, which is an increase by 20% compared to gradient descent. Both algorithms are quite stable as the maximal learning rates, for which the learning process diverges, are about 30% higher than the optimal rates. 4 SUMMARY AND DISCUSSION This research has been motivated by the dominance of the suboptimal symmetric phase in on-line learning of two-layer feedforward networks trained by gradient descent [2]. This trapping is emphasized for inappropriate small learning rates but exists in all training scenarios, effecting the learning process considerably. We Adaptive Back-Propagation in On-line Learning of Multilayer Networks 329 proposed an adaptive back-propagation training algorithm [Eq. (3)] parameterized by an inverse temperature /3, which is designed to improve specialization of the student nodes by enhancing differences in the activation between hidden units. Its performance has been compared to gradient descent for a soft-committee student network with J{ hidden units trying to learn a rule defined by an isotropic teacher (Tnm = Dnm) of the same architecture. A linear analysis of the equations of motion around the symmetric fixed point for small learning rates has shown that optimized adaptive back-propagation characterized by /3opt = 4J{ breaks the symmetry significantly faster. The effect is especially pronounced for large networks, where the trapping time of gradient descent grows T ex f{ /7] compared to T ex 1/7] for ~Pt . With increasing network size it seems to become harder for a student node trained by gradient descent to distinguish between the many teacher nodes and to specialize on one of them. In the adaptive back-propagation algorithm this effect can be eliminated by choosing /3opt ex J{. An open question is how the choice of the optimal inverse temperature is effected for large learning rates, where 7]2-terms cannot be neglected, as unbounded increase of the learning rate causes uncontrolled growth of the student norms. However, the full equations of motion are very difficult to analyse in the symmetric phase. Numerical studies indicate that ~Pt is smaller but still scales with J{ and yields an overall decrease in training time which is still significant. We also find that the optimal learning rate 7]0pt, which exhibits the shortest symmetric phase, is significantly lower in this regime than during convergence [4]. During convergence, independent of which algorithm is used, the time constant for decay to zero generalization error scales with J{, due to the necessary rescaling of the learning rate by 1/1< as the typical quadratic deviation between teacher and student output increases proportional to 1<. The reduction in training time with adaptive back-propagation is 17% and independent of the number of hidden units in contrast to the symmetric phase, where a factor 1< is gained. This can be explained by the fact that each student node is already specialized on one teacher node and the effect of other nodes in inhibiting further specialization is negligible. In fact, at first it seems rather surprising that anything can be gained by not changing the weights of the network according to their error gradient. The optimal setting of /3 > 1, together with training at a larger learning rate, speeds up learning for small activations and slows down learning for highly activated nodes. This is equivalent to favouring rotational changes of the weight vectors over pure length changes to a degree determined by /3. We believe that the adaptive back-propagation algorithm investigated here will be beneficial for any multilayer feedforward network and hope that this work will motivate further theoretical research into the efficiency of training algorithms and their systematic improvement. References [1] C. Cybenko, Math. Control Signals and Systems 2, 303 (1989). [2] D. Saad and S. A. Solla, Phys. Rev. E 52, 4225 (1995). [3] M. Biehl and H. Schwarze, 1. Phys. A 28, 643 (1995). [4] A. West and D. Saad, in preparation (1995).
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Neuron-MOS Temporal Winner Search Hardware for Fully-Parallel Data Processing Tadashi SHIBATA, Tsutomu NAKAI, Tatsuo MORIMOTO Ryu KAIHARA, Takeo YAMASHITA, and Tadahiro OHMI Department of Electronic Engineering Tohoku University Aza-Aoba, Aramaki, Aobaku, Sendai 980-77 JAPAN Abstract A unique architecture of winner search hardware has been developed using a novel neuron-like high functionality device called Neuron MOS transistor (or vMOS in short) [1,2] as a key circuit element. The circuits developed in this work can find the location of the maximum (or minimum) signal among a number of input data on the continuous-time basis, thus enabling real-time winner tracking as well as fully-parallel sorting of multiple input data. We have developed two circuit schemes. One is an ensemble of selfloop-selecting v M OS ring oscillators finding the winner as an oscillating node. The other is an ensemble of vMOS variable threshold inverters receiving a common ramp-voltage for competitive excitation where data sorting is conducted through consecutive winner search actions. Test circuits were fabricated by a double-polysilicon CMOS process and their operation has been experimentally verified. 1 INTRODUCTION Search for the largest (or the smallest) among a number of input data, Le., the winner-take-all (WTA) action, is an essential part of intelligent data processing such as data retrieval in associative memories [3], vector quantization circuits [4], Kohonen's self-organizing maps [5] etc. In addition to the maximum or minimum search, data sorting also plays an essential role in a number of signal processing such as median filtering in image processing, evolutionary algorithms in optimizing problems [6] and so forth. Usually such data processing is carried out by software running on general purpose computers, but the computation time increases explo686 T. SHIBATA, T. NAKAI, T. MORIMOTO, R. KAIHARA, T. YAMASHITA, T. OHMI sively with the increase in the volume of data. In order to build electronic systems having a real-time-response capability, the direct implementation of fully parallel algorithms on the integrated circuits hardware is critically demanded. A variety of WTA [4, 7, 8) circuits have been implemented so far based on analog current-mode circuit technologies. A number of cells, each composed of a current source, competitively share the total current specified by a global current sink and the winner is identified through the current concentration toward the cell via tacit positive feedback mechanisms. The circuit implementations using MOSFET's operating in the subthreshold regime [4, 7) are ideal for large scale integration due to its ultra low power nature. Although they are inherently slow at circuit levels, the performance at a system level is far superior to digital counterparts owing to the flexible computing algorithms of analog. In order to achieve a high speed operation, MOSFET's biased at strong inversion is also utilized in Ref. [8). However, cost must be traded off for increased power. What we are presenting in this paper is a unique WTA architecture implemented by vMOS technology [1,2]. In vMOS circuits the summation of multiples of voltage signals is conducted on the vMOS floating gate (or better be called "temporary floating gate" when used in a clocked scheme [9]) via charge sharing among capacitors, and the result of the summation controls the transistor action. The voltage-mode summation capability of vMOS has been uniquely utilized to produce the WTA action. No DC current flows for the sum operation itself in contrast to the Kirchhoff sum. In vMOS transistors, however, DC current flows in a CMOS inverter configuration when the floating gate is biased in the transition region. Therefore the power consumption is larger than in the subthreshold circuitries. However, the vMOS WTA's presented in this article will give an opportunity of high speed operation at much less power consumption than current-mode circuitries operating in the strong inversion mode. In the following we present two kinds of winner search hardware featuring very fast operation. The winner can be tracked in a continuous-time regime with a detection delay time of about lOOpsec, while the sorting of multiple data is conducted in a fixed frame of time of about 100nsec. 2 NEURON-MOS CONTINUOUS-TIME WTA Fig. 1(a) shows a schematic circuit diagram of a vMOS continuous-time WTA for four input signals. Each signal is fed to an input-stage vMOS inverter-A: a VA'-V ... V,.,-VA4 Vs ,ole 'Lc 0:71' ~ .. o~ ~ .. V •• 1 l (b) : Va V. V •• 1 : v. (c) VAI-VA4 ::~fw 'ole o~ ~ •• : v. Voc1 l (d) (a) Figure 1: (a) Circuit diagram of vMOS continuous-time WTA circuit. (b)lV(d) Response of VAl IV V A4 as & function of the floating-gate potential of vMOS inverterA. Neuron-MOS Temporal Winner Search Hardware for Fully-parallel Data Processing 687 CMOS inverter in which the common gate is made floating and its potential ,pFA is determined via capacitance coupling with three input terminals. VI ('" 'V4) and VR are equally coupled to the floating gate and a small capacitance pulls down the floating gate to ground. The vMOS inverter-B is designed to turn on when the number of l's in its inputs (VAl'" VA4) is more than 1. When a feedback loop is formed as shown in the figure, it becomes a ring oscillator composed of odd-numbers of inverter stages. When Vi '" V4 = 0, the circuit is stable with VR = 1 because inverter-A's do not turn on. This is because the small grounded capacitor pulls down the floating gate potential ,pFA a little smaller than its inverting threshold (VDD/2) (see Fig. l(b)). If non-zero signals are given to input terminals, more-than-one inverter-A's turn on (see Fig. l(c)) and the inverter-B also turns on, thus initiating the transition of VR from VDD to O. According to the decrease in VR, some of the inverter-A's turn off but the inverter-B (number 1 detector) still stays at on-state until the last inverterA turns off. When the last inverter-A, the one receiving the largest voltage input, turns off, the inverter-B also turns off and VR begins to increase. As a result, ring oscillation occurs only in the loop including the largest-input inverter-A(Fig. l(d)). In this manner, the winner is identified as an oscillating node. The inverter-B can be altered to a number "2" detector or a number "3" detector etc. by just reducing the input voltage to the largest coupling capacitor. Then it is possible for top two or top three to be winners. o 4() .0 lOD 120 140 ~~ TIME [JIaec] (a) 10 20 31) 40 50 60 70 :~ ', ..... 4fl; ... VAi ···f] ···· .. :. fl· ...... , ...... -. ~ 2~ .... . ... ... .. . .... ... . ! o~, .. . ; I " • I. . .1. ' w CJ < .... -' o > .. ~ .. ~ " .... - ~ .. "' o 4 ~ . J. 2" ...... ,. VA.: o t.. f ~..J.!~ ......... i ~-'-'-' ! ,~ ..... i~...J I l I ... l I l I .. j o 10 20 30 40 50 60 70 TIME [nsec) (b) Figure 2: (a) Measured wave forms of four-input WTA as depicted in Fig. 1(80) (bread board experoment) . (b) Simulation results for non-oscillating WTA explained in Fig. 3. Fig. 2(80) demonstrates the measured wave forms of a bread-board test circuit composed of discrete components for verifying the circuit idea. It is clearly seen that ring oscillation occurs only at the temporal winner. However, the ring oscillation increases the power dissipation, and therefore, non-oscillating circuitry would be preferred. An example of simulation results for such a non-oscillating circuit is demonstrated in Fig. 2(b). Fig. 3(80) gives the circuit diagram of a non-oscillating version of the vMOS 688 T. SHIBATA. T. NAKAI. T. MORIMOTO. R. KAIHARA. T. YAMASHITA. T. OHMI vMOS Inv., .. r-A vMOS Inv_r-B aD ~ '1 I Vt ~: I~I • No,,-olCillalinl mod, o Olcillatl,. mod, 0 1 aD 0 ;0,2 • Va 0 0 () ! • =-.; 00 f • Va ~T • 0 ,,0.1i.R.O 1111 V. 1[>.1>: V .. 0 a • • • 10 2000 4000 •• 0 COXT~ R VA RI'0) CUT/c... (a) (b) (c) Figure 3: (a) Circuit diagram of non-oscillating-mode WTA. HSPICE simulation results: (b) combinations of R and CEXT for non-oscillating mode; (c) winner detection delay as a function of capacitance load. continuous-time WTA. In order to suppress the oscillation, the loop gain is reduced by removing the two-stage CMOS inverters in front of the inverter-B and RC delay element is inserted in the feedback loop. The small grounded capacitors were removed in inverter-A's. The waveforms demonstrated in Fig. 2(b) are the HSPICE simulation results with R = 0 and CEXT = 20Cgote(Cgote: input capacitance of elemental CMOS inverter=5.16f.F) . The circuit was simulated assuming a typical double-poly 0.5-pm CMOS process. Fig. 3(b) indicates the combinations of Rand C EXT yielding the non-oscillating mode of operation obtained by HSPICE simulation. It is important to note that if CEXT ~ 15Cgote , non-oscillating mode appears with R = O. This me8JlS the output resistance of the inverter-B plays the role of R. When the number of inverter-A's is increased, the increased capacitance load serves as CEXT. Therefore, WTA having more than 19 input signals C8Jl operate in the non-oscillating mode. Fig. 3(c) represents the detection delay as a function of CEXT. It is known that the increase in CEXT, therefore the increase in the number of input signals to the WTA, does not significantly increase the detection delay and that the delay is only in the r8Jlge of 100 to 200psec. A photomicrograph of a test circuit of the non-oscillating mode WTA fabricated by Tohoku-University st8Jldard double-polysilicon CMOS process on 3-pm design rules, and the measurement results are shown in Fig. 4(80) and (b), respectively. I~ v "" V 1 Y¥. ~ V V "[\( V :---/ I""---' V r-I'--/ INPU T OAl ~ v. ~ ~ I-- ~ ~ ~ """ .... o OUTP TO ~TA VA' (a) (b) TIM E [2511uc/dlv) Figure 4: (a) Photomicrograph of a test circuit for 4-input continuous-time WTA. Chip size is 800pmx500pm including all peripherals (3-pm rules). The core circuit of Fig. 3(80) occupies approximately 0.12 mm2 • (b) Measured wave forms. Neuron-MOS Temporal Winner Search Hardware for Fully-parallel Data Processing 689 3 NEURON-MOS DATA SORTING CIRCUITRY The elemental idea of this circuit was first proposed at ISSCC '93 [3] as an application of the vMOS WTA circuit. In the present work, a clocked-vMOS technique [9] was introduced to enhance the accuracy and reliability of vMOS circuit operation and test circuits were fabricated and their operation have been verified. Fig. 5(80) shows the circuit diagram of a test circuit for sorting three analog data VA, VB, and Vc , and a photomicrograph of a fabricated test circuit designed on 3-pm rules is shown in Fig. 5(b). Each input stage is a vMOS inverter: a CMOS inverter in which the common gate is made floating and its potential fjJ F is determined by two input voltages via equa.lly-weighted capacitance coupling, namely fjJF = (VA + VRAMP)/2. The reset signal forces the floating node be grounded, thus cancelling the charge on the vMOS floating gate each time before sorting. This is quite essential in achieving long-term reliability of vMOS operation. In the second stage are flip-flop memory cells to store sorting results. The third stage is a circuit which counts the number of 1's at its three input terminals and outputs the result in binary code. The concept of the vMOS A/D converter design [10] has been utilized in the circuit. (a) (b) ............... ~ . (j) vMOS @ Data latch @ Counter Inverter Figure 5: (a) Circuit diagram of vMOS data-soring circuit. (b) Photomicrograph of a test circuit fabricated by Tohoku Univ. Standard double-polysillicon CMOS process (3-pm rules). Chip size is 1250pmxBOOpm including a.ll peripherals. The sorting circuit is activated by ramping up VRAMP from OV to VDD. Then the vMOS inverter receiving the largest input turns on first and the output data of the counter at this moment (0,0) is latched in the respective memory cells. The counter output changes to (0,1) after gate delays in the counter and this code is latched when the vMOS inverter receiving the second largest turns on. Then the counter counts up to (1,0). In this manner, the all input data are numbered according to the order of their magnitudes after a ramp voltage scan is completed. The measurement results are demonstrated in Fig. 6(80) in comparison with the HSPICE simulation results. Simulation was carried out on the same architecture circuit designed on O.5-pm design rules and operated under 3V power supply. For three analog input voltages: VA = 5V, VB = 4V, and Vc = 2V, (0,0), (0,1), 690 T. SHIBATA, T. NAKAI, T. MORIMOTO, R. KAIHARA, T. YAMASHITA, T. OHMI MEASUREMENT ~ r Ii L r r ' 10~/div (a) 20nsec/civ 40 ~30 -S 20 j 1: _100 ~80 -eo S40 c: ~ 20 0 3-INPUT SORnNG CIRCUIT 2 4 6 8 SortIng Accuracy (bit ] (b) 15-INPUT SORTING CIRCUIT 2 4 6 8 SortIng Accuracy (bit ] (c) Figure 6: (a) Wave forms of the test circuit shown in Fig. 5(a) measured without buffer circuitry (left) and simulation results of a circuit designed with 0.5-pm rules (right). (b) Minimum scan time vs. sorting accuracy for a three-input sorter. (c) Minimum scan time vs. sorting accuracy for a 15-input sorter. and (1,0) are latched, respectively, after the ramp voltage scan, thus accomplishing correct sorting. Slow operation of the test circuit is due to the loading effect caused by the direct probing of the node voltage without output buffer circuitries. The simulation with a 0.5-pm-design-rule circuit indicates the sorting is accomplished within the scan time of 4Onsec. In Fig. 6(b), the minimum scan time obtained by simulation is plotted as a function of the bit accuracy in sorting analog data. N -bit accuracy means the minimum voltage difference required for winner discrimination is VDD/22 • If the ramp rate is too fast, the vMOS inverter receiving the next largest data turns on before the correct counting results become available, leading to an erroneous operation. The scan time/accuracy relation in Fig. 6(b) is primarily determined by the response delay in the counter. It should be noted that the number of inverter stages in the counter (vMOS A/D converter) is always three indifferent to the number of output bits, namely, the delay would not increase significantly by the increase in the number of input data. In order to investigate this, a 15-input counter was designed and the delay time was evaluated by HSPICE simulation. It was 312 psec in comparison with 110 psec of the 3-input counter of Fig. 5(a). The scan time/accuracy relation for the 15-input sorting circuit is shown in Fig. 6( c), indicating the sorting of 15 input data can be accomplished in 100 nsec with 8-bit accuracy. Neuron-MOS Temporal Winner Search Hardware for Fully-parallel Data Processing 691 4 CONCLUSIONS A novel neuron-like functional device liMOS has been successfully utilized in constructing intelligent electronic circuits which can carry out search for the temporal winner. As a result, it has become possible to perform data sorting as well as winner search in an instance, both requiring very time-consuming sequential data processing on a digital computer. The hardware algorithms presented here are typical examples of the liMOS binary-multivalue-analog merged computation scheme, which would play an important role in the future flexible data processing. Acknowledgements This work was partially supported by Grant-in-Aid for Scientific Research (06402038) from the Ministry of Education, Science, Sports, and Culture, Japan. A part of this work was carried out in the Super Clean Room of Laboratory for Electronic Intelligent Systems, Research Institute of Electrical communication, Tohoku University. References [1] T. Shibata and T . Ohmi, "A functional MOS transistor featuring gate-level weighted sum and threshold operations," IEEE Trans. Electron Devices, Vol. 39, No.6, pp.1444-1455 (1992). [2] T. Shibata, K. Kotani, T. Yamashita, H. Ishii, H. Kosaka, and T. Ohmi, "Implementing interlligence on silicon using neuron-like functional MOS transistors," in Advances in Neural Information Processing Systems 6 (San Francisco, CA: Morgan Kaufmann 1994) pp. 919-926. [3] T. Yamashita, T. Shibata, and T. Ohmi, "Neuron MOS winner-take-all circuit and its application to associative memory," in ISSCC Dig. Tech. Papers, Feb. 1993, FA 15.2, pp. 236-237. [4] G. Gauwenberghs and V. Pedroni, " A charge-based CMOS parallel analog vector quantizer," in Advances in Neural Information Processing Systems 7 (Cambridge, MA: The MIT Press 1995) pp. 779-786. [5] T. Kohonen, Self-Organization and Associative Memory, 2nd ed. (New York: Springer-Verlag 1988). [6] M. Kawamata, M. Abe, and T. Higuchi, "Evolutionary digital filters," in Proc. Int. Workshop on Intelligent Signal Processing and Communication Systems, seoul, Oct., 1994, pp. 263-268. [7] J. Lazzaro, S. Ryckebusch, M. A. Mahowald, and C. A. Mead, "Winner-TakeAll networks of O(N) complexity," in Advances in Neural Information Processing Systems 1 (San Mateo, CA: Morgan Kaufmann 1989) pp. 703-711. [8] J . Choi and B. J. Sheu, "A high-precision VLSI winner-take-all circuit for selforganizing neural networks," IEEE J. Solid State Circuits, Vol. 28, No.5, pp.576584 (1993). [9] K. Kotani, T. Shibata, M. Imai, and T. Ohmi, "Clocked-Neuron-MOS logic circuits employing auto-threshold-adjustment," in ISSCC Dig. Technical Papers, Feb. 1995, FA 19.5, pp. 320-321. [10] T. Shibata and T. Ohmi, "Neuron MOS binary-logic integrated circuits: Part II, Simplifying techniques of circuit configuration and their practical applications," IEEE Trans. Electron Devices, Vol. 40, No.5, 974-979 (1993).
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Rapid Quality Estimation of Neural Network Input Representations Kevin J. Cherkauer Jude W. Shav lik Computer Sciences Department, University of Wisconsin-Madison 1210 W. Dayton St., Madison, WI 53706 {cherkauer,shavlik }@cs.wisc.edu Abstract The choice of an input representation for a neural network can have a profound impact on its accuracy in classifying novel instances. However, neural networks are typically computationally expensive to train, making it difficult to test large numbers of alternative representations. This paper introduces fast quality measures for neural network representations, allowing one to quickly and accurately estimate which of a collection of possible representations for a problem is the best. We show that our measures for ranking representations are more accurate than a previously published measure, based on experiments with three difficult, real-world pattern recognition problems. 1 Introduction A key component of successful artificial neural network (ANN) applications is an input representation that suits the problem. However, ANNs are usually costly to train, preventing one from trying many different representations. In this paper, we address this problem by introducing and evaluating three new measures for quickly estimating ANN input representation quality. Two of these, called [DBleaves and Min (leaves), consistently outperform Rendell and Ragavan's (1993) blurring measure in accurately ranking different input representations for ANN learning on three difficult, real-world datasets. 2 Representation Quality Choosing good input representations for supervised learning systems has been the subject of diverse research in both connectionist (Cherkauer & Shavlik, 1994; Kambhatla & Leen, 1994) and symbolic paradigms (Almuallim & Dietterich, 1994; 46 K. J. CHERKAUER, J. W. SHA VLIK Caruana & Freitag, 1994; John et al., 1994; Kira & Rendell, 1992). Two factors of representation quality are well-recognized in this work: the ability to separate examples of different classes (sufficiency of the representation) and the number of features present (representational economy). We believe there is also a third important component that is often overlooked, namely the ease of learning an accurate concept under a given representation, which we call transparency. We define transparency as the density of concepts that are both accurate (generalize well) and simple (of low complexity) in the space of possible concepts under a given input representation and learning algorithm. Learning an accurate concept will be more likely if the concept space is rich in accurate concepts that are also simple, because simple concepts require less search to find and less data to validate. In this paper, we introduce fast transparency measures for ANN input representations. These are orders of magnitude faster than the wrapper method (John et al., 1994), which would evaluate ANN representations by training and testing the ANN s themselves. Our measures are based on the strong assumption that, for a fixed input representation, information about the density of accurate, simple concepts under a (fast) decision-tree learning algorithm will transfer to the concept space of an ANN learning algorithm. Our experiments on three real-world datasets demonstrate that our transparency measures are highly predictive of representation quality for ANNs, implying that the transfer assumption holds surprisingly well for some pattern recognition tasks even though ANNs and decision trees are believed to work best on quite different types of problems (Quinlan, 1994).1 In addition, our Exper. 1 shows that transparency does not depend on representational sufficiency. Exper. 2 verifies this conclusion and also demonstrates that transparency does not depend on representational economy. Finally, Exper. 3 examines the effects of redundant features on the transparency measures, demonstrating that the ID31eaves measure is robust in the face of such features. 2.1 Model-Based Transparency Measures We introduce three new "model-based" measures that estimate representational transparency by sampling instances of roughly accurate concept models from a decision-tree space and measuring their complexities. If simple, accurate models are abundant, the average complexity of the sampled models will be low. If they are sparse, we can expect a higher complexity value. Our first measure, avg(leaves), estimates the expected complexity of accurate concepts as the average number of leaves in n randomly constructed decision trees that correctly classify the training set: avg(leaves) == ~ 2:;=11eaves(t) where leaves(t) is the number of leaves in tree t. Random trees are built top-down; features are chosen with uniform probability from those which further partition the training examples (ignoring example class). Tree building terminates when each leaf achieves class purity (Le., the tree correctly classifies all the training examples). High values of avg(leaves) indicate high concept complexity (i.e., low transparency). The second measure, min(leaves), finds the minimum number of leaves over the n randomly constructed trees instead of the average to reflect the fact that learning systems try to make intelligent, not random, model choices: min (leaves) == min {leaves(t)} t=l,n lWe did not preselect datasets based on whether our experiments upheld the transfer assumption. We report the results for all datasets that we have tested our transparency measures on. Rapid Quality Estimation of Neural Network Input Representations 47 Table 1: Summary of datasets used. Dataset II Examples Classes Cross Validation Folds DNA 20,000 6 4 NIST 3,471 10 10 Magellan 625 2 4 The third measure, ID31eaves, simply counts the number of leaves in the tree grown by Quinlan's (1986) ID3 algorithm: ID31eaves == leaves(ID3 tree) We always use the full ID3 tree (100% correct on the training set). This measure assumes the complexity of the concept ID3 finds depends on the density of simple, accurate models in its space and thus reflects the true transparency. All these measures fix tree training-set accuracy at 100%, so simpler trees imply more accurate generalization (Fayyad, 1994) as well as easier learning. This lets us estimate transparency without the multiplicative additional computational expense of cross validating each tree. It also lets us use all the training data for tree building. 2.2 "Blurring" as a Transparency Measure Rendell and Ragavan (1993) address ease of learning explicitly and present a metric for quantifying it called blurring. In their framework, the less a representation requires the use of feature interactions to produce accurate concepts, the more transparent it is. Blurring heuristically estimates this by measuring the average information content of a representation's individual features. Blurring is equivalent to the (negation of the) average information gain (Quinlan, 1986) of a representation's features with respect to a training set, as we show in Cherkauer and Shavlik (1995). 3 Evaluating the Transparency Measures We evaluate the transparency measures on three problems: DNA (predicting gene reading frames; Craven & Shavlik, 1993), NIST (recognizing handwritten digits; "FI3" distribution), and Magellan (detecting volcanos in radar images of the planet Venus; Burl et al., 1994).2 The datasets are summarized in Table l. To assess the different transparency measures, we follow these steps for each dataset in Exper. 1 and 2: 1. Construct several different input representations for the problem. 2. Train ANNs using each representation and test the resulting generalization accuracy via cross validation (CV). This gives us a (costly) ground-truth ranking of the relative qualities of the different representations. 3. For each transparency measure, compute the transparency score of each representation. This gives us a (cheap) predicted ranking of the representations from each measure. 4. For each transparency measure, compute Spearman's rank correlation coefficient between the ground-truth and predicted rankings. The higher this correlation, the better the transparency measure predicts the true ranking. 20n these problems, we have found that ANNs generalize 1- 6 percentage points better than decision trees using identical input representations, motivating our desire to develop fast measures of ANN input representation quality. 48 K. 1. CHERKAUER, J. W. SHAVLIK Table 2: User CPU seconds on a Sun SPARCstation 10/30 for the largest representation of each dataset. Parenthesized numbers are standard deviations over 10 runs. I Dataset \I Blurring I ID3leaves I Min! A vg(leaves) I Backprop I DNA 1.68 2.38 1,245 3.96) 13,444 56.25 212,900 NIST 2.69 2.31 221 2.75 1,558 5.00 501,400 Magellan 0.21 0.15 1 0.07 12 0.13) 6,300 In Exper. 3 we rank only two representations at a time, so instead of computing a rank correlation in step 4, we just count the number of pairs ranked correctly. We created input representations (step 1) with an algorithm we call RS ("Representation Selector"). RS first constructs a large pool of plausible, domain-specific Boolean features (5,460 features for DNA, 251,679 for NIST, 33,876 for Magellan). For each CV fold, RS sorts the features by information gain on the entire training set. Then it scans the list, selecting each feature that is not strongly pairwise dependent on any feature already selected according to a standard X2 independence test using the X 2 statistic. This produces a single reasonable input representation, Rl.3 To obtain the additional representations needed for the ranking experiments, we ran RS several times with successively smaller subsets of the initial feature pool, created by deleting features whose training-set information gains were above different thresholds. For each dataset, we made nine additional representations of varying qualities, labeled R2-RlO , numbered from least to most "damaged" initial feature pool. To get the ground-truth ranking (step 2), we trained feed-forward ANNs with backpropagation using each representation and one output unit per class. We tried several different numbers of hidden units in one layer and used the best CV accuracy among these (Fig. 1, left) to rank each input representation for ground truth. Each transparency measure also predicted a ranking of the representations (step 3). A CPU time comparison is in Table 2. This table and the experiments below report min (leaves) and avg(leaves) results from sampling 100 random trees, but sampling only 10 trees (giving a factor 10 speedup) yields similar ranking accuracy. Finally, in Exper. 1 and 2 we evaluate each transparency measure (step 4) using 6.Em d2 Spearman's rank correlation coefficient, rs = 1 m(";i.!:l)·' between the groundtruth and predicted rankings (m is the number of representations (10); di is the ground-truth rank (an integer between 1 and 10) minus the transparency rank). We evaluate the transparency measures in Exper. 3 by counting the number (out of ten) of representation pairs each measure orders the same as ground truth. 4 Experiment I-Transparency vs. Sufficiency This experiment demonstrates that our transparency measures are good predictors of representation quality and shows that transparency does not depend on representational sufficiency (ability to separate examples). In this experiment we used transparency to rank ten representations for each dataset and compared the rankings to the ANN ground truth using the rank correlation coefficient. RS created the representations by adding features until each representation could completely separate the training data into its classes. Thus, representational sufficiency was 3Though feature selection is not the focus of this paper, note that similar feature selection algorithms have been used by others for machine learning applications (Baim, 1988; Battiti, 1994). Rapid Quality Estimation of Neural Network Input Representations DNA Backprop Ground-Truth Cross-Validation 1 00 .---..---.----.--r--.---,.----.-....--,.---.---, ~ 90 !!! ~ ~ 80 ~ ~ 70 rfl Cii 60 ~ ~ 50 Experiment 1 Experiment 2 ......... . 40~~~~~~~~~~~~ R1 R2 R3 R4 R5 R6 R7 R8 R9R10 Representation Number NIST Backprop Ground-Truth Cross-Validation 1 00 r-"--"'--~--.--r--r--.---.----.---..--, ~ 90 ~ o !i 80 *OJ 70 (f) 1ii 60 ~ ~ 50 Experiment 1 Experiment 2 ......... . 40~~~~~~~~~~~~ R1 R2 R3 R4 R5 R6 R7 R8 R9R10 Representation Number Magellan Backprop Ground-Truth Cross-Validation 100.---..--...--~--.--r--r--.---.----.---..--, DNA Dataset Measure Exp1 rs ID3leaves 0.99 Min (leaves) 0.94 A vgJleaves) 0.78 Blurring 0.78 NIST Dataset Measure Exp1 rs ID3leaves 1.00 Min(leaves) 1.00 Avg(leaves) 1.00 Blurring 1.00 49 Exp2 rs 0.95 0.99 0.96 0.81 Exp2 rs 1.00 1.00 1.00 1.00 ~ 90 ~ o Magellan Dataset !i 80 *OJ 70 rfl Cii 60 ~ Experiment 1 Experiment 2 ......... . !li 50 40~~~~~~~~~~~~ R1 R2 R3 R4 R5 R6 R7 R8 R9R10 Representation Number Measure Exp1 rs Exp2 rs ID3leaves 0.81 0.78 Min(leaves) 0.83 0.76 Avg(leaves) 0.71 0.71 Blurring 0.48 0.73 Figure 1: Left: Exper. 1 and 2 ANN CV test-set accuracies (y axis; error bars are 1 SD) used to rank the representations (x axis). Right: Exper. 1 and 2, transparency rankings compared to ground truth. rs: rank correlation coefficient (see text). held constant. (The number of features could vary across representations.) The rank correlation results are shown in Fig. 1 (right). ID31eaves and min (leaves) outperform the less sophisticated avg(leaves) and blurring measures on datasets where there is a difference. On the NIST data, all measures produce perfect rankings. The confidence that a true correlation exists is greater than 0.95 for all measures and datasets except blurring on the Magellan data, where it is 0.85. The high rank correlations we observe imply that our transparency measures captUre a predictive factor of representation quality. This factor does not depend on representational sufficiency, because sufficiency was equal for all representations. 50 K. J. CHERKAUER. J. W. SHAVLIK Table 3: Exper. 3 results: correct rankings (out of 10) by the transparency measures of the corresponding representation pairs, Ri vs. R~, from Exper. 1 and Exper. 2. I Dataset II ID3leaves Min{leaves) Avg(leaves) Blurring I ~:naJi ~~ ~ ~ ~ 5 Experiment 2-Transparency vs. Economy This experiment shows that transparency does not depend on representational economy (number of features), and it verifies Exper. 1's conclusion that it does not depend on sufficiency. It also reaffirms the predictive power of the measures. In Exper. 1, sufficiency was held constant, but economy could vary. Exper. 2 demonstrates that transparency does not depend on economy by equalizing the number of features and redoing the comparison. In Exper. 2, RS added extra features to each representation used in in Exper. 1 until they all contained a fixed number of features (200 for DNA, 250 for NIST, 100 for Magellan). Each Exper. 2 representation, R~ (i = 1, ... , 10), is thus a proper superset of the corresponding Exper. 1 representation, Ri. All representations for a given dataset in Exper. 2 have an identical number of features and allow perfect classification of the training data, so neither economy nor sufficiency can affect the transparency scores now. The results (Fig. 1, right) are similar to Exper. 1's. The notable changes are that blurring is not as far behind ID3leaves and min (leaves) on the Magellan data as before, and avg(leaves) has joined the accuracy of the other two model-based measures on the DNA. The confidence that correlations exist is above 0.95 in all cases. Again, the high rank correlations indicate that transparency is a good predictor of representation quality. Exper. 2 shows that transparency does not depend on representational economy or sufficiency, as both were held constant here. 6 Experiment 3-Redundant Features Exper. 3 tests the transparency measures' predictions when the number of redundant features varies, as ANNs can often use redundant features to advantage (Sutton & Whitehead, 1993), an ability generally not attributed to decision trees. Exper. 3 reuses the representations Ri and R~ (i = 1, ... , 10) from Exper. 1 and 2. Recall that R~ => Ri . The extra features in each R~ are redundant as they are not needed to separate the training data. We show the number of Ri vs. R~ representation pairs each transparency measure ranks correctly for each dataset (Table 3). For DNA and NIST, the redundant representations always improved ANN generalization (Fig. 1, left; 0.05 significance). Only ID3leaves predicted this correctly, finding smaller trees with the increased flexibility afforded by the extra features. The other measures were always incorrect because the lower quality redundant features degraded the random trees (avg (leaves) , min (leaves)) and the average information gain (blurring). For Magellan, ANN generalization was only significantly different for one representation pair, and all measures performed near chance. 7 Conclusions We introduced the notion of transparency (the prevalence of simple and accurate concepts) as an important factor of input representation quality and developed inRapid Quality Estimation of Neural Network Input Representations 51 expensive, effective ways to measure it. Empirical tests on three real-world datasets demonstrated these measures' accuracy at ranking representations for ANN learning at much lower computational cost than training the ANNs themselves. Our next step will be to use transparency measures as scoring functions in algorithms that apply extensive search to find better input representations. Acknowledgments This work was supported by ONR grant N00014-93-1-099S, NSF grant CDA9024618 (for CM-5 use), and a NASA GSRP fellowship held by KJC. References Almuallim, H. & Dietterich, T. (1994). Learning Boolean concepts in the presence of many irrelevant features. Artificial Intelligence, 69(1- 2):279-305. Bairn, P. (1988). A method for attribute selection in inductive learning systems. IEEE Transactions on Pattern Analysis fj Machine Intelligence, 10(6):888-896. Battiti, R. (1994). Vsing mutual information for selecting features in supervised neural net learning. IEEE Transactions on Neural Networks, 5(4):537-550. Burl, M., Fayyad, V., Perona, P., Smyth, P., & Burl, M. (1994). Automating the hunt for volcanoes on Venus. In IEEE Computer Society Con! on Computer Vision fj Pattern Recognition: Proc, Seattle, WA. IEEE Computer Society Press. Caruana, R. & Freitag, D. (1994). Greedy attribute selection. In Machine Learning: Proc 11th Intl Con!, (pp. 28-36), New Brunswick, NJ. Morgan Kaufmann. Cherkauer, K. & Shavlik, J. (1994). Selecting salient features for machine learning from large candidate pools through parallel decision-tree construction. In Kitano, H. & Hendler, J., ecis., Massively Parallel Artificial Intel. MIT Press, Cambridge, MA. Cherkauer, K. & Shavlik, J. (1995). Rapidly estimating the quality of input representations for neural networks. In Working Notes, IJCAI Workshop on Data Engineering for Inductive Learning, (pp. 99-108), Montreal, Canada. Craven, M. & Shavlik, J. (1993). Learning to predict reading frames in E. coli DNA sequences. In Proc 26th Hawaii Intl Con! on System Science, (pp. 773-782), Wailea, HI. IEEE Computer Society Press. Fayyad, V. (1994). Branching on attribute values in decision tree generation. In Proc 12th Natl Con! on Artificial Intel, (pp. 601-606), Seattle, WA. AAAIjMIT Press. John, G., Kohavi, R., & Pfleger, K. (1994). Irrelevant features and the subset selection problem. In Machine Learning: Proc 11th Intl Con!, (pp. 121-129), New Brunswick, NJ. Morgan Kaufmann. Kambhatla, N. & Leen, T. (1994). Fast non-linear dimension reduction. In Advances in Neural In!o Processing Sys (vol 6), (pp. 152-159), San Francisco, CA. Morgan Kaufmann. Kira, K. & Rendell, L. (1992). The feature selection problem: Traditional methods and a new al~orithm. In Proc 10th Natl Con! on Artificial Intel, (pp. 129-134), San Jose, CA. AAAI/MIT Press. Quinlan, J. (1986). Induction of decision trees. Machine Learning, 1:81-106. Quinlan, J. (1994). Comparing connectionist and symbolic learning methods. In Hanson, S., Drastal, G., & Rivest, R., eds., Computational Learning Theory fj Natural Learning Systems (vol I: Constraints fj Prospects). MIT Press, Cambridge, MA. Rendell, L. & Ragavan, H. (1993). Improving the design of induction methods by analyzing algorithm functionality and data-based concept complexity. In Proc 13th Intl Joint Con! on Artificial Intel, (pp. 952-958), Chamhery, France. Morgan Kaufmann. Sutton, R. & Whitehead, S. (1993). Online learning with random representations. In Machine Learning: Proc 10th IntI Con/, (pp. 314-321), Amherst, MA. Morgan Kaufmann.
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Human Reading and the Curse of Dimensionality Gale L. Martin MCC Austin, TX 78613 galem@mcc.com Abstract Whereas optical character recognition (OCR) systems learn to classify single characters; people learn to classify long character strings in parallel, within a single fixation. This difference is surprising because high dimensionality is associated with poor classification learning. This paper suggests that the human reading system avoids these problems because the number of to-be-classified images is reduced by consistent and optimal eye fixation positions, and by character sequence regularities. An interesting difference exists between human reading and optical character recognition (OCR) systems. The input/output dimensionality of character classification in human reading is much greater than that for OCR systems (see Figure 1). OCR systems classify one character at time; while the human reading system classifies as many as 8-13 characters per eye fixation (Rayner, 1979) and within a fixation, character category and sequence information is extracted in parallel (Blanchard, McConkie, Zola, and Wolverton, 1984; Reicher, 1969). OCR (Low Dbnensionality) I Dorothy lived In the .... I [Q] ... _ .................................... "D" ~ ................................... "0" o ~ "R" HUnlan Reading (High Dbnensionality) I Dorothy lived In the midst of the ..... I Dorothy lil Ilived In the I ....... I midst of the I .............. "DOROTHY LI" .. . .. )00 "LIVED IN THE" ... "MIDST OF THE" Figure 1: Character classification versus character sequence classification. This is an interesting difference because high dimensionality is associated with poor classification learning-the so-called curse of dimensionality (Denker, et ali 1987; Geman, Bienenstock, & Doursat, 1992). OCR systems are designed to classify single characters to minimize such problems. The fact that most people learn to read quite well even with the high dimensional inputs and outputs, implies that variance 18 G. L. MARTIN is somehow lowered in this domain, thereby making accurate classification learning possible. The present paper reports on simulations of parallel character classification which suggest that variance is lowered through regularities in eye fixation positions and in character sequences making up valid words. 1 Training and Testing Materials Training and testing materials were drawn from the story The Wonderful Wizard of Oz by L. Frank Baum. Images of text lines were created from 120 pages of text (about 160,000 characters, 33,000 total words, or 2,600 different words), which were divided into 6 different font and case conditions of 20 pages each. Three different fonts (variable and constant-width fonts), and two different cases (all upper-case or mixed-case characters) were used. Text line images were normalized with respect to height, but not width. All training and test sets contained an equal mix of the six font/case conditions. Two generalization sets were used, for test and crossvalidation, and each consisted of about 14,000 characters. Dorothy lived in the JDidst of the great Kansas Prairies. DOROTHY LIVED IN THE MIDST OF THE GREAT KANSAS PRAIRIES. Dorothy lived in the midst of the great Kansas Prairies. DOROTHY LIVED IN THE MIDST OF THE GREAT KANSAS PRAIRIES. Dorothy 1~ved ~n the m~dst of the great Kansas Pra~r~es. DOROTHY LIVED IN THE MIDST OF THE GREAT KANSAS PRAIRIES. Figure 2: Samples of the type font and case conditions used in the simulations 2 Network Architectures The simulations used backpropagation networks (Rumelhart, Hinton & Williams, 1986) that extended the local receptive field, shared-weight architecture used in many character-based OCR neural networks (LeCun, et aI, 1989; Martin & Pittman, 1991). In the previous single character-based approach, the input to the net is an image of a single character. The output is a vector representing the category ofthe character. Hidden nodes have local receptive fields that receive input from a spatially local region, (e.g., a 6x6 area) in the preceding layer. Groups of hidden nodes share their weights. Corresponding weights in each receptive field are initialized to the same value and updated by the same value. Different hidden nodes within a group learn to detect the same feature at different locations. A group is depicted as hidden nodes within a single plane of a cube that corresponds to a hidden layer. Different groups occupy different planes in the cube, and learn to detect different features. This architecture biases learning by reducing the number of free parameters available for representing a function. The fact that these nets usually train and generalize well in this domain, and that the local feature detectors that emerge are similar to the oriented-edge and -line detectors found in mammalian visual cortex (Hubel & Wiesel, 1979), suggests that the bias is at least roughly appropriate. The extension of this character network to a character-sequence network is illustrated in Figure 3, where n (number of to-be-classified characters) is equal to 4. Each output node represents a character category (e.g., "D") in one of the nth ordinal positions (e.g., "First character on the left"). The size of the input window is expanded horizontally to cover at least the n widest characters ("WWWW"). When the character string is made up of relatively narrow characters, more than n characters will appear in the input window and the network must learn to ignore Human Reading and the Curse of Dimensionality 19 them. Increasing input/output dimensionality is accomplished by expanding the number of hidden nodes horizontally. Network capacity is described by the depth of each hidden layer (the number of different features detected), as well as by the width of each hidden layer (the spatial coverage of the network). The network is potentially sensitive to both local and global visual information. Local receptive fields build in a sensitivity to letter features. Shared weights make learning transfer possible across representations of the same character at different positions. Output nodes are globally connected to all the nodes in the second hidden layer, but not with one another or with any word-level representations. Networks were trained until the training set accuracy failed to improve by at least .1% over 5 epochs, or overfitting became evident from periodic testing with the generalization test set. ABC EFOHIJKLMNOP RSTUVWXYZ ABCDEFOHIJKLMN PQRSTUVWXYZ Local, sharr!d-weight rl'!ceptive /ielcls Figure 3: Net architecture for parallel character sequence classification, n=4 chars. 3 Effects of Dimensionality on Training Difficulty and Generalization Experiment 1 provides a baseline measure of the impact of dimensionality. Increases in dimensionality result in exponential increases in the number of input and output patterns and the number of mapping functions. As a result, training problems arise due to limitations in network capacity or search scope. Generalization problems arise because it becomes impractical to use training sets large enough to obtain a good estimate of the underlying function. Four different levels of dimensionality were used (see Figure 4), from an input window of 20x20 pixels, with 1 to-beclassified character to an 80x20 window, with 4 to-be-classified characters ). Input patterns were generated by starting the window at the left edge of the text line such that the first character was centered 10 pixels from the left of the window, and then successively scanning across the text line at each character position. Five training set sizes were used (about 700 samples to 50,000). Two relative network capacities were used (15 and 18 different feature detectors per hidden layer). Forty different 2Ox2O - 1 Character ~ ---lOo-"D" @] .-.,. "a" Low 40x20 - 2 Characters ~ _"DO" @!9 .-~ "OR" 60x20 - 3 Characters lDocol ... »"DaR" lorotij ~ "ORO" ................... Dimensionality . 80x20 - 4 Characters I Dorothl ~ "DORa" I orothy \ ._ ..... "OROT" . ............. ........ ..... ~ High Figure 4: Four levels of input/output dimensionality used in the experiment. 20 G. L. MARTIN networks were trained, one for each combination of dimensionality, training set size and relative network capacity (4x5x2). Training difficulty is described by asymptotic accuracy achieved on the training set and by amount of training required to reach the asymptote. Generalization is reported for both the test set (used to check for overfitting) and the cross-validation set. The results (see Figure 5) are consistent Lower Capa:ity Nets Higher Capdy Nets B 3~ a)1~~8~ (.) ~ &i t: 0 (.) ~ e) g) 4 Ch 94 ' , o 10lXl 4IlD :nm «XDJ !illll o 10lXl 4IlD :nm «XDJ !illll Training Set Size Training Set Size Amount of Training Required to Reach Asymptote d) 4 Ch 3 Ch 2 Ch 1 Ch o Geoeralization AtturIII:)' on Test Set ~~====~::::-. 1 ~ f) m~~~~.:o1 ~ 4 ~ .. _ ... __ .t.... 4 Ch 0 10lXl m:D :nm «XDJ !illll 0 10lXl 4IlD :nm «XDJ &nO Oeoeralization Aa:uraq on Validation Set 28R h) l ~~ 3 Ch 4 h 4 Ch 0 1croJ m:D :nm «XDJ !DJl) 0 10lXl 4IlD :nm «XDJ &nO Figure 5: Impact of dimensionality on training and generalization. with expectations. Increasing dimensionality results in increased training difficulty and lower generalization. Since the problems associated with high dimensionality occur in both training and test sets, and seem to be alleviated somewhat in the high capacity nets, they are presumably due to both capacity/search limitations and insufficient sample size, 4 Regularities in Window Positioning One way human reading might reduce the problems associated with high dimensionality is to constrain eye fixation positions during reading; thereby reducing the number of different input images the system must learn to classify. Eye movement Human Reading and the Curse of Dimensionality 21 studies suggest that, although fixation positions within words do vary, there are consistencies Rayner, 1979). Moreover, the particular locations fixated, slightly to the left of the middle of words, appear to be optimal. People are most efficient at recognizing words at these locations (O'Regan & Jacobs, 1992). These fixation positions reduce variance by reducing the average variability in the positions of ordered characters within a word. Position variability increases as a function of distance from the fixated character. The average distance of characters within a word is minimized when the fixation position is toward the center of a word, as compared to when it is at the beginning or end of a word. Experiment 2 simulated consistent and optimal positioning with an 80x20 input window fixated on the 3rd character. Only words of 3 or more characters were fixated (see Figure 6). The network learned to classify the first 4 characters in the word. This condition was compared to a consistent positioning only condition, in which the input window was fixated on the first character of a word. Two control conditions were also examined. They were replications ofthe 20x20-1Character and the 80x20-4 Character conditions of Experiment 1, except that in the first case, the network was trained and tested only on the first 4 characters in each word and in the second case, the network was trained as before but was tested with the window fixated on the first character of the word. Four levels of training set size were used and three replications of each training set size x window conditions were run (4 x 4 x 3 = 48 networks trained and tested). All networks employed 18 different feature detectors for each hidden layer. The results (see Figure 7) support the idea that Consistent & Optimal 80x20 - 4 Chars I DOfthi --.. "DORO" ~--""LIVE" Consistent Only High Dim. Control 80x20 - 4 Chars 80x20 - 4 Chars 'i0roth~ --.. "DORO" ~oroth~ --.. "DORO" Low Dim. Control 2Ox20 - 1 Char Figure 6: Window positioning and dimensionality manipulations in Experiment 2 consistent and optimal positioning reduces variance, as indicated by reductions in training difficulties and improved generalization. The consistent and optimal positioning networks achieved training and generalization results superior to the high dimensionality control condition, and equivalent to, or better than those for the low dimensionality control. They were also slightly better than the consistent positioning only nets. 5 Character Sequence Regularities Since only certain character sequences are allowed in words, character sequence regularities in words may also reduce the number of distinct images the system must learn to classify. The system may also reduce variance by optimizing accuracy on highest frequency words. These hypotheses were tested by determining whether or not the three consistent and optimal positioning networks trained on the largest training set in Experiment 2, were more accurate in classifying high frequency words, as compared to low frequency words; and more accurate in classifying words as compared to pronounceable non-words or random character strings. The control condition used the networks trained in the low dimensional control (20x20 -1 Character) condition from Experiment 2. Human reading exhibits increased efficiency I accuracy in classifying high frequency as compared to low frequency words 22 G. L.MARTIN Tralnln3 Dlmculty Asymptotic Accuracy on Tralning Set Amount or Training to Reach Asymptote 'OOf~ ---am --- -m .. """"" • "" _ ConsISt. tl 98 '. Cnlrl-20x20 0) ............ !3 ---- ----------.. Cnlrl-80x20 U96 ~ 94 J ! o 600J 10000 1600J 20000 'Itaining Set Size Generalization Acclll'acy Test Set 1 oo r----:::~~~==::::::;:==::!!I CCJ"ISist. & Opt. I Cn,Irl-20x20 Consist. _ _ __ , ___ • Cntrl-BOx20 ~ ~ ~~.,,'~'-----~ 26 .. ' °O~--~6OOJ~--1-0000~---16OOJ~----20000 Training Set Size Validation Set 100.-------__ -,--:xconslsl. & Opt , ' - - - CCJ"Islst. Cnlrl-20x20 ____ , __ , ____ • - -" Cntrl-BOx20 ~~~6OOJ~-~10000~-1~6OOJ~~20000~ 'Itaining Set Size Figure 7: Impact of consistent & optimal window positions. (Howes & Solomon, 1951; Solomon & Postman, 1952) , and in classifying characters in words as compared to pronounceable non-words or random character strings (Baron & Thurston, 1973; Reicher, 1969). Experiment 3 involved creating a list of 30 4-letter words drawn from the Oz text, of which 15 occurred very frequently in the text (e.g., SAID), and 15 occurred infrequently (e.g., PAID), and creating a list of 30 4-letter pronounceable non-words (e.g., TOlD) and a list of 30 4-letter random strings (e.g., SDIA). Each string was reproduced in each of the 6 font Icase conditions and labeled to create a test set. One further condition involved creating a version of the word list in which the cases of the characters aLtErN aTeD. Psychologists used this manipulation to demonstrate that the advantages in processing words can not simply be due to the use of word-shape feature detectors, since the word advantage carries over to the alternating case condition, which destroys word-level features (McClelland, 1976). Consistent with human reading (see Figure 8), the character-sequence-based networks were most accurate on high frequency words and least accurate for low frequency words. The character-sequence-based networks also showed a progressive decline in accuracy as the character string became less word-like. The advantage for word-like strings can not be due to the use of word shape feature detectors because accuracy on aLtErNaTiNg case words, where word shape is unfamiliar, remains quite high. Word Frequency Effect D Hgh Fraq Low Fraq Character-Sequence-Based Consistent & Optimal Positioning Word Superiority Effect WordS Pmn NonWorcts Random aLtErNaTINg • Control condition, 20x20 single character Figure 8: Sensitivity to word frequency and character sequence regularities Human Reading and the Curse of Dimensionality 23 The present results raise questions about the role played by high dimensionality in determining reading disabilities and difficulties. Reading difficulties have been associated with reduced perceptual spans (Rayner, 1986; Rayner, et al., 1989), and with irregular eye fixation patterns (Rayner & Pollatsek, 1989). This suggests that some reading difficulties and disorders may be related to problems in generating the precise eye movements necessary to maintain consistent and optimal eye fixations. More generally, these results highlight the importance of considering the role of character classification in learning to read, particularly since content factors, such as word frequency, appear to influence even low-level classification operations. References Blanchard, H., McConkie, G., Zola, D., & Wolverton, G. (1984) Time course of visual information utilization during fixations in reading. Jour. of Exp. Psych.: Human Perc. fj Perf, 10, 75-89. Denker, J., Schwartz, D., Wittner, B., Solla, S., Howard, R., Jackel, L., & Hopfield, J. (1987) Large automatic learning, rule extraction and generalization, Complex Systems, 1, 877-933. Geman, S., Bienenstock, E., and Doursat, R. (1992) Neural networks and the bias/variance dilemma. Neural Computation, 4, 1-58. Howes, D. and Solomon, R. L. (1951) Visual duration threshold as a function of word probability. Journal of Exp. Psych., 41, 401-410. Hubel, D. & Wiesel, T. (1979) Brain mechanisms of vision. Sci. Amer., 241, 150-162. LeCun, Y., Boser, B., Denker, J., Henderson, D., Howard, R., Hubbard, W., & Jackel, L. (1990) Handwritten digit recognition with a backpropagation network. In Adv. in Neural Inf Proc. Sys. 2, D. Touretzky (Ed) Morgan Kaufmann. Martin, G. L. & Pittman, J. A. (1991) Recognizing hand-printed letters and digits using backpropagation learning. Neural Computation, 3, 258-267. McClelland, J. L. (1976) Preliminary letter identification in the perception of words and nonwords. Jour. of Exp. Psych.: Human Perc. fj Perf, 2, 80-91. O'Regan, J. & Jacobs, A.(1992) Optimal viewing position effect in word recognition. Jour. of Exp. Psych.: Human Perc.fj Perf, 18, 185-197. Rayner, K. (1986) Eye movements and the perceptual span in beginning and skilled readers. Jour. of Exp. Child Psych., 41, 211-236. Rayner, K. (1979) Eye guidance in reading. Perception, 8, 21-30. Rayner, K., Murphy, 1., Henderson, J. & Pollatsek, A. (1989) Selective attentional dyslexia. Cognitive Neuropsych., 6, 357-378. Rayner, K. & Pollatsek, A. (1989) The Psychology of reading. Prentice Hall Reicher, G. (1969) Perceptual recognition as a function of meaningfulness of stimulus material. Jour. of Exp. Psych., 81, 274-280. Rumelhart, D., Hinton, G., and Williams, R. (1986) Learning internal representations by error propagation. In D. Rumelhart and J. McClelland, (Eds) Parallel Distributed Processing, 1. MIT Press. Solomon, R. & Postman, L. (1952) Frequency of usage as a determinant of recognition thresholds for words. Jour. of Exp. Psych., 43, 195-210.
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Unsupervised Pixel-prediction William R. Softky Math Resp.arch Branch NIDDK, NIH 9190 Wisconsin Ave #350 Bethesda, MD 20814 bill@homer.niddk.nih.gov Abstract When a sensory system constructs a model of the environment from its input, it might need to verify the model's accuracy. One method of verification is multivariate time-series prediction: a good model could predict the near-future activity of its inputs, much as a good scientific theory predicts future data. Such a predicting model would require copious top-down connections to compare the predictions with the input. That feedback could improve the model's performance in two ways: by biasing internal activity toward expected patterns, and by generating specific error signals if the predictions fail. A proof-of-concept model-an event-driven, computationally efficient layered network, incorporating "cortical" features like all-excitatory synapses and local inhibition- was constructed to make near-future predictions of a simple, moving stimulus. After unsupervised learning, the network contained units not only tuned to obvious features of the stimulus like contour orientation and motion, but also to contour discontinuity ("end-stopping") and illusory contours. 1 Introduction Somehow, brains make very accurate models of the outside world from their raw sensory input. How might brains check and improve those models? What signal is there to verify a model of the world? The scientific method faces a similar problem: how to verify theories. In science, theories are verified by predicting future data, using the implicit assumption that 810 W.R.SOFfKY good predictions can only result from good models. By analogy, it is possible that brains predict their afferent input (e.g. at the thalamus), and that making such predictions and using them as feedback is a unifying design principle of cortex. The proof-of-concept model presented here uses unsupervised Hebbian learning to predict, pixel-wise, the location of a moving pattern slightly in the future. Why try prediction? • Predicting future data usually requires a good generative model. For instance: to predict the brightness of individual TV pixels even a fraction of a second in advance, one would need models of contours, objects, motion, occlusion, shadow, etc. • A successful prediction can help filter out input noise, like a Kalman filter. • A failed prediction provides a specific, high-dimensional error signal. • Prediction is not only possible in cortex-which has massive feedback connections-but necessary as well, because those feedback fibers, their target dendrites, and synaptic integration impose inevitable delays. So for a feedback signal to arrive at the cell body "on time," it would need to have been generated tens of milliseconds earlier, as a prediction of imminent activity. • In this model, "prediction" means producing spikes in advance which will correlate with subsequent input spikes. Specifically, the network's goal is to produce at each grid point a train of spikes at times Pj which predicts the input train Ik, in the sense of maximizing their normalized cross-correlation. The objective function L ("likeness") can be expressed in terms of a smoothing "bump" function B(t:J;, ty) (of spikes at times t:J; and ty) and a correlation function C(trainl, train2, ~t): C(P,I, ~T) L(P,I,~T) exp ( -It:J; T- tyl ) L: L: B(Pj + ~t, Ik) j k C(P, I, ~T) JC(P, P, O)C(I, 1,0) • In order to avoid a trivial but useless prediction ("the weather tomorrow will be just like today;'), one must ensure that a unit cannot usually predict its own firing (for example, pick ~t ~ T greater than the autocorrelation time of a spike train). 2 Model The input to the network is a 16 x 16 array of spike trains, with toroidal array boundary conditions. The spikes are driven by a "stimulus" bar of excitation one unit wide and seven units long, which moves smoothly perpendicular to its orientation behind the array (in a broad circle, so that all orientations and directions are represented; Fig. 1A). The stimulus point transiently generates spikes at each grid point there according to a Poisson process: the whole array of spikes can be visualized as a twinkling, moving contour. Unsupervised Pixel-prediction 811 A B forward t - - - trigger & _ _ _ _ helper syn~ /.. _ _ _ delay _ _ _ _ tuned, precise, predictive inputs feedback Figure 1: A network predicts dynamic patterns. A A moving pattern on a grid of spiking pixels describes a slow circle, and drives activity in a network above. B The three-layer network learns to predict that activity just before it occurs. Forward connections, evolving by Hebbian rules, produce top-level units with coarse receptive fields and fine stimulus-tuning (e.g. contour orientation and motion). Each spike from a top unit is "bound" (by coincidence detection) with the particular spike which triggered it, to produce feedback which is both stimulustuned and spatially specific. A Hebb rule determines how the delayed, predictive feedback will drive middle-layer units and be compared to input-layer units. Because all connections are excitatory, winner-take-all inhibition within local groups of units prevents runaway excitation. 2.1 Network Structure The network has three layers. The bottom layer contains the spiking pixels, and the "surprise" units described below. The middle layer, having the same spatial resolution as the input, has four coarsely-tuned units per input pixel. And the top layer contains the most finely-tuned units, spaced at half the spatial resolution (at every fourth gridpoint, i.e. with coarser spatial resolution and larger receptive fields). The signal flow is bi-directional [10, 7], with both forward and feedback synaptic connections. All connections between units are excitatory, and excitation is kept in check by local winner-take-all inhibition (WTA). For example, a given input spike can only trigger one spike out of the 16 units directly above it in the top layer (Fig. IB). Unsupervised learning occurs through two local Hebb-like rules. Forward connections evolve to make nearby (competing) units strongly anticorrelated-for instance, units typically become tuned to different contour orientations and directions of motion-while feedback connections evolve to maximally correlate delayed feedback signals with their targets. 2.2 Binary multiplication in single units While some neural models implement multiplication as a nonlinear function of the sum of the inputs, the spiking model used here implements multiplication as a binary operation on two distinct classes of synapses. 812 A ~'I~ ~ ~ comc. helper inh trigger detector delay "helper" I I I I .m~ V --- *-------out I B W.R.SOFrKY prediction of X Figure 2: Multiplicative synapses and surprise detection. A A spiking unit multiplies two types of synaptic inputs: the "helper" type increments an internal bias without triggering a spike, and the "trigger" type can trigger a spike (*), without incrementing, but only if the bias is above a threshold. Spike propagation may be discretely delayed, and coincidences of two units fired by the same input spike can be detected. B Once the network has generated a (delayed) prediction of a given pixel's activity, the match of prediction and reality can be tested by specialpurpose units: one type which detects unpredicted input, the other which detects unfulfilled predictions. The firing of either type can drive the network's learning rules, so units above can become tuned to consistent patters of failed predictions, as occur at discontinuities and illusory contours. A helper synapse, when activated by a presynaptic spike, will increment or decrement the postsynaptic voltage without ever initiating a spike. A trigger synapse, on the other hand, can initiate a spike (if the voltage is above the threshold determined by its WTA neighbors), but cannot adjust the voltage (Fig. 2A; the helper type is loosely based on the weak, slow NMDA synapses on cortical apical dendrites, while triggers are based on strong, brief AMPA synapses on basal dendrites.) Thus, a unit can only fire when both synaptic types are active, so the output firing rate approximates the product of the rates of helpers and triggers. Each unit has two characteristic timescales: a slower voltage decay time, and the essentially instantaneous time necessary to trigger and propagate a spike. This scheme has two advantages. One is that a single cell can implement a relatively "pure" multiplication of distinct inputs, as required for computations like motiondetection. The other advantage is that feedback signals, restricted to only helper synapses, cannot by themselves drive a cell, so closed positive-feedback loops cannot "latch" the network into a fixed state, independent of the input. Therefore, all trigger synapses in this network are forward, while all delayed, lateral, and feedback connections are of the helper type. 2.3 Feedback There are two issues in feedback: How to construct tuned, specific feedback, and what to do with the feedback where it arrives. Unsupervised Pixel-prediction 813 An accurate prediction requires information about the input: both about its exact present state, and about its history over nearby space and recent time. In this model, those signals are distinct: spatial and temporal specificity is given by each input spike, and the spatia-temporal history is given by the stimulus-tuned responses of the slow, coarse-grained units in the top layer. Spatially-precise feedback requires recombining those signals. (Feedback from V1 cortical Layer VI to thalamus has recently been shown to fit these criteria, being both spatially refined and directionselective; [3] Grieve & Sillito, 1995). In this network, each feedback signal results from the AND of spikes from a inputlayer spike (spatially specific) and the resulting top-layer spike it produces (stimulustuned). This "binding" across levels of specificity requires single-spike temporal precision, and may even be one of the perceptual uses for spike timing in cortex [1, 9]. 2.4 Surprise detection Once predictive feedback is learned, it can be used in two ways: biasing units toward expected activity, and comparing predictions against actual input. Feedback to the middle layer is used as a bias signal through helper synapses, by adding the feedback to the bias signal. But feedback to the bottom, input-layer is compared with actual input by means of special "surprise" units which subtract prediction from input (and vice versa). Because both prediction and input are noisy signals, their difference is even noisier, and must be both temporally smoothed and thresholded to generate a mismatchspike. In this model, these prediction/input differences are accomplished pixel-bypixel using ad-hoc units designed for the purpose (Fig. 2B). There is no indication that cortex operates so simplistically, but there are indications that cortical cells are in general sensitive to mismatches between expectation and reality, such as discontinuities in space (edges), in time (on- and off-responses), and in context (saliency) . The resulting error vector can drive upper-layer units just as the input does, so that the network can learn patterns of failed predictions, which typically correspond to discontinuities in the stimulus. Learning consistent patterns of bad predictions is a completely generic recipe for discovering such discontinuitites, which often correspond closely to visually important features like contour ends, corners, illusory contours, and occlusion. 3 Results and Discussion After prolonged exposure to the stimulus, the network produces a blurred cloud of spikes which anticipates the actual input spikes, but which also consistently predicts input beyond the bar's ends (leading to small clouds of surprise-unit activity tracking the ends). The top-level units, driven both by input signals and by feedback, become tuned either to different motions of the bar itself (due to Hebbian learning of the input), or to different motions of its ends (due to Hebbian learning of the surprise-units); see Fig. 3. Cells tuned to contour ends ("end-stopped") have been found in visual cortex [11], although the principles of their genesis are not known. Using the same parameters but a different stimlus, the network can also evolve units 814 38 36 34 32 30 28 ........ - ... XX>IMIOQO( CD 26 a. 24 ~ 22 C\I 20 Q; 18 ~ 1 6 8+t!tHIIIIII __ IIIH!Io' ...J 14 12 10 8 6 4 2 14 CD 12 ~ 10 8 Q; 6 ! 4 2 o + <H-$iIIIx x x _ * - \=. \= + • _ ••• '\= X * W.R.SOFfKY x --.III.BijMl!mj. + >MC1M_cc_IIf-1 t ~ Figure 3: Single units are highly stimulus-specific. Spikes from all units at one location are shown (with time) as a stimlus bar (insets) passes them with six different relative positions and motions. Out of the many units available, only one or two are active in each layer for a given stimulus configuration. The inactive units are tuned to stimulus orientations not shown here. Some units are driven by "surprise" units (Figure 2 and text), and respond only to the bar's ends (. and x), but not to its center (+). Such responses lag behind those of ordinary units, because they must temporally integrate to determine whether a significant mismatch exists between the noisy prediction and the noisy input. Spikes from five passes have been summed to show the units' reliability. which detect the illusory contours present in certain moving gratings. Several researchers propose that cortex (or similar networks) might use feedback pathways to recreate or regenerate their (static) input [7,4, 10]. The approach here requires instead that the network forecast future (dynamic) input [8]. In a general sense, predicting the future is a better test of a model than predicting the present, in the same sense that scientific theories which predict future experimental data are more persuasive than theories which predict existing data. Prediction of the raw input has advantages over prediction of some higher-level signal [5, 6, 2]: the raw input is the only unprocessed "reality" available to the network, and comparing the prediction with that raw input yields the highest-dimensional error vector possible. Spiking networks are likewise useful. As in cortex, spikes both truncate small inputs and contaminate them with quantization-noise, crucial practical problems which real-valued networks avoid. Spike-driven units can implement purely correlative computations like motion-detection, and can avoid parasitic positive-feedback loops. Spike timing can identify which of many possible inputs fired a given unit, thereby making possible a more specific feedback signal. The most practical benefit is that interactions among rare events (like spikes) are much faster to compute than realUnsupervised Pixel-prediction 815 valued ones; this particular network of 8000 units and 200,000 synapses runs faster than the workstation can display it. This model is an ad-hoc network to illustrate some of the issues a brain might face in trying to predict its retinal inputs; it is not a model of cortex. Unfortunately, the hypothesis that cortex predicts its own inputs does not suggest any specific circuit or model to test. But two experimental tests may be sufficiently model-independent. One is that cortical "non-classical" receptive fields should have a temporal structure which reflects the temporal sequences of natural stimuli, so a given cell's activity will be either enhanced or suppressed when its input matches contextual expectations. Another is that feedback to a single cell in thalamus, or to an individual cortical apical dendrite, should arrive on average earlier than afferent input to the same cell. References [1] A. Engel, P. Koenig, A. Kreiter, T. Schillen, and W. Singer. Temporal coding in the visual cortex: New vistas on integration in the nervous system. TINS, 15:218-226, 1992. [2] K. Fielding and D. Ruck. Recognition of moving light displays using hidden markov models. Pattern Recognition, 28:1415-1421,1995. [3] K. 1. Grieve and A. M. Sillito. Differential properties of cells in the feline primary visual cortex providing the cortifugal feedback to the lateral geniculate nucleus and visual claustrum. J. Neurosci., 15:4868-4874,1995. [4] G. Hinton, P. Dayan, B. Frey, and R. Neal. The wake-sleep algorithm for unsupervised neural networks. Science, 268:1158-1161,1995. [5] P. R. Montague and T. Sejnowski. The predictive brain: Temporal coincidence and· temporal order in synaptic learning mechanisms. Learning and Memory, 1:1-33, 1994. [6] P. Read Montague, Peter Dayan, Christophe Person, and T. Sejnowski. Bee foraging in uncertain environments using predictive hebbian learning. Nature, 377:725-728, 1995. [7] D. Mumford. Neuronal architectures for pattern-theoretic problems. In C. Koch and J. Davis, editors, Large-scale theories of the cortex, pages 125-152. MIT Press, 1994. [8] W. Softky. Could time-series prediction assist visual processing? Soc. Neurosci. Abstracts, 21:1499, 1995. [9] W. Softky. Simple codes vs. efficient codes. Current Opinion in Neurbiology, 5:239-247, 1995. [10] S. Ullman. Sequence-seeking and counterstreams: a model for bidirectional information flow in cortex. In C. Koch and J . Davis, editors, Large-scale theories of the cortex, pages 257-270. MIT Press, 1994. [11] S. Zucker, A. Dobbins, and L. Iverson. Two stages of curve detection suggest two styles of visual computation. Neural Computation, 1:68-81, 1989.
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Control of Selective Visual Attention: Modeling the "Where" Pathway Ernst Niebur· Computation and Neural Systems 139-74 California Institute of Technology Christof Koch Computation and Neural Systems 139-74 California Institute of Technology Abstract Intermediate and higher vision processes require selection of a subset of the available sensory information before further processing. Usually, this selection is implemented in the form of a spatially circumscribed region of the visual field, the so-called "focus of attention" which scans the visual scene dependent on the input and on the attentional state of the subject. We here present a model for the control of the focus of attention in primates, based on a saliency map. This mechanism is not only expected to model the functionality of biological vision but also to be essential for the understanding of complex scenes in machine vision. 1 Introduction: "What" and "Where" In Vision It is a generally accepted fact that the computations of early vision are massively parallel operations, i.e., applied in parallel to all parts of the visual field. This high degree of parallelism cannot be sustained in in~ermediate and higher vision because of the astronomical number of different possible combination of features. Therefore, it becomes necessary to select only a part of the instantaneous sensory input for more detailed processing and to discard the rest. This is the mechanism of visual selective attention. • Present address: Zanvyl Krieger Mind/Brain Institute and Department of Neuroscience, 3400 N. Charles Street, The Johns Hopkins University, Baltimore, MD 21218_ Control of Selective Visual Attention: Modeling the "Where" Pathway 803 It is clear that similar selection mechanisms are also required in machine vision for the analysis of all but the simplest visual scenes. Attentional mechanisms are slowly introduced in this field; e.g. , Yamada and Cottrell (1995) used sequential scanning by a "focus of attention" in the context of face recognition. Another model for eye scan path generation, which is characterized by a strong top-down influence, is presented by Rao and Ballard (this volume). Sequential scanning can be applied to more abstract spaces, like the dynamics of complex systems in optimization problems with large numbers of minima (Tsioutsias and Mjolsness, this volume). Primate vision is organized along two major anatomical pathways. One of them is concerned mainly with object recognition. For this reason, it has been called the What- pathway; for anatomical reasons, it is also known as the ventral pathway. The principal task of the other major pathway is the determination of the location of objects and therefore it is called the Where pathway or, again for anatomical reasons, the dorsal pathway. In previous work (Niebur & Koch , 1994), we presented a model for the implementation of the What pathway. The underlying mechanism is "temporal tagging:" it is assumed that the attended region of the visual field is distinguished from the unattended parts by the temporal fine-structure of the neuronal spike trains. We have shown that temporal tagging can be achieved by introducing moderate levels of correlation1 between those neurons which respond to attended stimuli. How can such synchronicity be obtained? We have suggested a simple, neurally plausible mechanism, namely common input to all cells which respond to attended stimuli. Such (excitatory) input will increase the propensity of postsynaptic cells to fire for a short time after receiving this input, and thereby increase the correlation between spike trains without necessarily increasing the average firing rate. The subject of the present study is to provide a model of the control system which generates such modulating input. We will show that it is possible to construct an integrated system of attentional control which is based on neurally plausible elements and which is compatible with the anatomy and physiology of the primate visual system. The system scans a visual Scene and identifies its most salient parts. A possible task would be "Find all faces in this image." We are confident that this model will not only further our understanding of the function of biological vision but that it will also be relevant for technical applications. 2 A Simple Model of The Dorsal Pathway 2.1 Overall Structure Figure 1 shows an overview of the model Where pathway. Input is provided in the form of digitized images from an NTSC camera which is then analyzed in various feature maps. These maps are organized around the known operations in early visual cortices. They are implemented at different spatial scales and in a center-surround structure akin to visual receptive fields . Different spatial scales are implemented as Gaussian pyramids (Adelson, Anderson, Bergen, Burt, & Ogden, 1984). The center 1 In (Niebur, Koch, & Rosin, 1993), a similar model was developed using periodic "40Hz" modulation. The present model can be adapted mutatis mutandis to this type of modulation. 804 E. NIEBUR, C. KOCH of the receptive field corresponds to the value of a pixel at level n in the pyramid and the surround to the corresponding pixels at level n + 2, level 0 being the image in normal size. The features implemented so far are the thr~e principal components of primate color vision (intensity, red-green, blue-yellow), four orientations, and temporal change. Short descriptions of the different feature maps are presented in the next section (2.2). We then (section 2.3) address the question of the integration of the input in the "saliency map," a topographically organized map which codes for the instantaneous conspicuity of the different parts of the visual field. Feature Maps (multiscale) • • • Figure 1: Overview of the model Where pathway. Features are computed as centersurround differences at 4 different spatial scales (only 3 feature maps shown) . They are combined and integrated in the saliency map ("SM") which provides input to an array of integrate-and-fire neurons with global inhibition. This array ("WTA") has the functionality of a winner-take-all network and provides the output to the ventral pathway ("V2") as well as feedback to the saliency map (curved arrow) . 2.2 Input Features 2.2.1 Intensity Intensity information is obtained from the chromatic information of the NTSC signal. With R, G, and B being the red, green and blue channels, respectively, the intensity I is obtained as 1= (R + G + B)/3. The entry in the feature map is the modulus Control of Selective Visual Attention: Modeling the "Where" Pathway 805 of the contrast, i.e., IIcenter I~tirrotindl. This corresponds roughly to the sum of two single-opponent cells of opposite phase, i.e. bright-center dark-surround and vice-versa. Note, however, that the present version of the model does not reproduce the temporal behavior of ON and OFF subfields because we update the activities in the feature maps instantaneously with changing visual input. Therefore, we neglect the temporal filtering properties of the input neurons. 2.2.2 Chromatic Input Red, green and blue are the pixel values of the RGB signal. Yellow is computed as (R + G)/2. At each pixel, we compute a quantity corresponding to the doubleopponency cells in primary visual cortex. For instance, for the red-green filter, we firs't compute at each pixel the value of (red-green). From this, we then subtract (green-red) of the surround. Finally, we take the absolute value of the result. 2.2.3 Orientation The intensity image is convolved with four Gabor patches of angles 0,45,90, and 135 degrees, respectively. The result of these four convolutions are four arrays of scalars at every level of the pyramid. The average orientation is then computed as a weighted vector sum. The components in this sum are the four unit vectors iii, i = 1, .. .4 corresponding to the 4 orientations, each with the weight Wi. This weight is given by the result of its convolution of the respective Gabor patch with the image. Let c be this vector for the center pixel, then c = L;=l Wi iii . The average orientation vector for the surround, s, is computed analogously. What enters in the SM is the center-surround difference, i.e. the scalar product c( s - C). This is a scalar quantity which corresponds to the center-surround difference in orientation at every location, and which also takes into account the relative "strength" of the oriented edges. 2.2.4 Change The appearance of an object and the segregation of an object from its background have been shown to capture attention, even for stimuli which are equiluminant with the background (Hillstrom & Yantis, 1994). We incorporate the attentional capture by visual onsets and motion by adding the temporal derivative of the input image sequence, taking into account chromatic information. More precisely, at each pixel we compute at time t and for a time difference tit = 200ms: 1 '3{IR(t) - R(t - tit) I + IG(t) - G(t - tit) I + IB(t) - B(t - tit)!} (1) 2.2.5 Top-Down Information Our model implements essentially bottom-up strategies for the rapid selection of conspicuous parts of the visual field and does not pretend to be a model for higher cognitive functions. Nevertheless, it is straightforward to incorporate some topdown influence. For instance, in a "Posner task" (Posner, 1980), the subject is instructed to attend selectively to one part of the visual field. This instruction can be implemented by additional input to the corresponding part of the saliency map. 806 E. NIEBUR, C. KOCH 2.3 The Saliency Map The existence of a saliency map has been suggested by Koch and Ullman (1985); see also the "master map" of Treisman (1988). The idea is that of a topographically organized map which encodes information on where salient (conspicuous) objects are located in the visual field, but not what these objects are. The task of the saliency map is the computation of the salience at every location in the visual field and the subsequent selection of the most salient areas or objects. At any time, only one such area is selected. The feature maps provide current input to the saliency map. The output of the saliency map consists of a spike train from neurons corresponding to this selected area in the topographic map which project to the ventral ("What") pathway. By this mechanism, they are "tagged" by modulating the temporal structure of the neuronal signals corresponding to attended stimuli (Niebur & Koch, 1994). 2.3.1 Fusion Of Information Once all relevant features have been computed in the various feature maps, they have to be combined to yield the salience, i.e. a scalar quantity. In our model, we solve this task by simply adding the activities in the different feature maps, as computed in section 2.2, with constant weights. We choose all weights identical except for the input obtained from the temporal change. Because of the obvious great importance changing stimuli have for the capture of attention, we select this weight five times larger than the others. 2.3.2 Internal Dynamics And Trajectory Generation By definition, the activity in a given location of the saliency map represents the relative conspicuity of the corresponding location in the visual field. At any given time, the maximum of this map is therefore the most salient stimulus. As a consequence, this is the stimulus to which the focus of attention should be directed next to allow more detailed inspection by the more powerful "higher" process which are not available to the massively parallel feature maps. This means that we have to determine the instantaneous maximum of this map. This maximum is selected by application of a winner-take-all mechanism. Different mechanisms have been suggested for the implementation of neural winner-take-all networks (e.g., Koch & Ullman, 1985; Yuille & Grzywacz, 1989). In our model, we used a 2-dimensionallayer of integrate-and-fire neurons with strong global inhibition in which the inhibitory population is reliably activated by any neuron in the layer. Therefore, when the first of these cells fires, it will inhibit all cells (including itself), and the neuron with the strongest input will generate a sequence of action potentials. All other neurons are quiescent. For a static image, the system would so far attend continuously the most conspicuous stimulus. This is neither observed in biological vision nor desirable from a functional point of view; instead, after inspection of any point, there is usually no reason to dwell on it any longer and the next-most salient point should be attended. We achieve this behavior by introducing feedback from the winner-take-all array. When a spike occurs in the WTA network, the integrators in the saliency map Control of Selective Visual Attention: Modeling the "Where" Pathway 807 receive additional input with the spatial structure of an inverted Mexican hat, ie. a difference of Gaussians. The (inhibitory) center is at the location of the winner which becomes thus inhibited in the saliency map and, consequently, attention switches to the next-most conspicuous location. The function ofthe positive lobes ofthe inverted Mexican hat is to avoid excessive jumping of the focus of attention. If two locations are of nearly equal conspicuity and one of them is close to the present focus of attention, the next jump will go to the close location rather than to the distant one. 3 Results We have studied the system with inputs constructed analogously to typical visual psychophysical stimuli and obtained results in agreement with experimental data. Space limitations prevent a detailed presentation of these results in this report. Therefore, in Fig. 2, we only show one example of a "real-world image." We choose, as an example, an image showing the Caltech bookstore and the trajectory of the focus of attention follows in our model. The most salient feature in this image is the red banner on the the wall of the building (in the center of the image). The focus of attention is directed first to this salient feature. The system then starts to scan the image in the order of decreasing saliency. Shown are the 3 jumps following the initial focussing on the red banner. The jumps are driven by a strong inhibition-of-return mechanism. Experimental evidence for such a mechanims has been obtained recently in area 7a of rhesus monkeys (Steinmetz, Connor, Constantinidis, & McLaughlin, 1994). Figure 2: Example image. The black line shows the trajectory of the simulated focus of attention over a time of 140 ms which jumps from the center (red banner on wall of building) to three different locations of decreasing saliency. 4 Conclusion And Outlook We present in this report a prototype for an integrated system mimicking the control of visual selective attention. Our model is compatible with the known anatomy and physiology of the primate visual system, and its different parts communicate by signals which are neurally plausible. The model identifies the most salient points in a visual scenes one-by-one and scans the scene autonomously in the order of 808 E. NIEBUR, C. KOCH decreasing saliency. This allows the control of a subsequently activated processor which is specialized for detailed object recognition. At present, saliency is determined by combining the input from a set offeature maps with fixed weights. In future work, we will generalize our approach by introducing plasticity in these weights and thus adapting the system to the task at hand. Acknowledgements Work supported by the Office of Naval Research, the Air Force Office of Scientific Research, the National Science Foundation, the Center for Neuromorphic Systems Engineering as a part of the National Science Foundation Engineering Research Center Program, and by the Office of Strategic Technology of the California Trade and Commerce Agency. References Adelson, E., Anderson, C., Bergen, J., Burt, P., & Ogden, J. (1984). Pyramid methods in image processing. RCA Engineer, Nov-Dec. Hillstrom, A. & Yantis, S. (1994). Visual motion and attentional capture. Perception 8 Psychophysics, 55(4),399-411 . Koch, C. & Ullman, S. (1985). Shifts in selective visual attention: towards the underlying neural circuitry. Human Neurobiol., 4,219-227. Niebur, E. & Koch, C. (1994). A model for the neuronal implementation of selective visual attention based on temporal correlation among neurons. Journal of Computational Neuroscience, 1 (1), 141- 158. Niebur, E., Koch, C., & Rosin, C. (1993). An oscillation-based model for the neural basis of attention. Vision Research, 33, 2789-2802. Posner, M. (1980). Orienting of attention. Quart. 1. Exp. Psychol., 32, 3-25. Steinmetz, M., Connor, C., Constantinidis, C., & McLaughlin, J. (1994). Covert attention suppresses neuronal responses in area 7 A of the posterior parietal cortex. 1. Neurophysiology, 72, 1020-1023. Treisman, A. (1988). Features and objects: the fourteenth Bartlett memorial lecture. Quart. 1. Exp. Psychol., 40A, 201-237. Tsioutsias, D. 1. & Mjolsness, E. (1996). A Multiscale Attentional Framework for Relaxation Neural Networks. In Touretzky, D., Mozer, M. C., & Hasselmo, M. E. (Eds.), Advances in Neural Information Processing Systems, Vol. 8. MIT Press, Cambridge, MA. Yamada, K. & Cottrell, G. W. (1995). A model of scan paths applied to face recognition. In Proc. 17th Ann. Cog. Sci. Con/. Pittsburgh. Yuille, A. & Grzywacz, N. (1989). A winner-take-all mechanism based on presynaptic inhibition feedback. Neural Computation, 2, 334-344.
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Quadratic-Type Lyapunov Functions for Competitive Neural Networks with Different Time-Scales Anke Meyer-Base Institute of Technical Informatics Technical University of Darmstadt Darmstadt, Germany 64283 Abstract The dynamics of complex neural networks modelling the selforganization process in cortical maps must include the aspects of long and short-term memory. The behaviour of the network is such characterized by an equation of neural activity as a fast phenomenon and an equation of synaptic modification as a slow part of the neural system. We present a quadratic-type Lyapunov function for the flow of a competitive neural system with fast and slow dynamic variables. We also show the consequences of the stability analysis on the neural net parameters. 1 INTRODUCTION This paper investigates a special class of laterally inhibited neural networks. In particular, we have examined the dynamics of a restricted class of laterally inhibited neural networks from a rigorous analytic standpoint. The network models for retinotopic and somatotopic cortical maps are usually composed of several layers of neurons from sensory receptors to cortical units, with feedforward excitations between the layers and lateral (or recurrent) connection within the layer. Standard techniques include (1) Hebbian rule and its variations for modifying synaptic efficacies, (2) lateral inhibition for establishing topographical organization of the cortex, and (3) adiabatic approximation in decoupling the dynamics of relaxation (which is on the fast time scale) and the dynamics of learning (which is on the slow time scale) of the network. However, in most cases, only computer simulation results were obtained and therefore provided limited mathematical understanding of the self-organizating neural response fields. The networks under study model the dynamics of both the neural activity levels, 338 A. MEYER-BASE the short-term memory (STM), and the dynamics of synaptic modifications, the long-term memory (LTM). The actual network models under consideration may be considered extensions of Grossberg's shunting network [Gr076] or Amari's model for primitive neuronal competition [Ama82]. These earlier networks are considered pools of mutually inhibitory neurons with fixed synaptic connections. Our results extended these earlier studies to systems where the synapses can be modified by external stimuli. The dynamics of competitive systems may be extremely complex, exhibiting convergence to point attractors and periodic attractors. For networks which model only the dynamic of the neural activity levels Cohen and Grossberg [CG83] found a Lyapunov function as a necessary condition for the convergence behavior to point attractors. In this paper we apply the results of the theory of Lyapunov functions for singularly perturbed systems on large-scale neural networks, which have two types of state variables (LTM and STM) describing the slow and the fast dynamics of the system. So we can find a Lyapunov function for the neural system with different time-scales and give a design concept of storing desired pattern as stable equilibrium points. 2 THE CLASS OF NEURAL NETWORKS WITH DIFFERENT TIME-SCALES This section defines the network of differential equations characterizing laterally inhibited neural networks. We consider a laterally inhibited network with a deterministic signal Hebbian learning law [Heb49] and is similar to the spatiotemporal system of Amari [Ama83]. The general neural network equations describe the temporal evolution of the STM (activity modification) and LTM states (synaptic modification). For the jth neuron of aN-neuron network these equations are: N Xj = -ajxj + L D ij!(Xi ) + BjSj (1) i=l (2) where Xj is the current activity level, aj is the time constant of the neuron, Bj is the contribution of the external stimulus term, !(Xi) is the neuron's output, D ij is the . lateral inhibition term and Yi is the external stimulus. The dynamic variable Sj represents the synaptic modification state and lyl21 is defined as lyl2 = yTy. We will assume that the input stimuli are normalized vectors of unit magnitude lyl2 = 1. These systems will be subject to our analysis considerations regarding the stability of their equilibrium points. 3 ASYMPTOTIC STABILITY OF NEURAL NETWORKS WITH DIFFERENT TIME-SCALES We show in this section that it is possible to determine the asymptotic stability of this class of neural networks interpreting them as nonlinear singularly perturbed systems. While singular perturbation theory, a traditional tool of fluid dynamics and nonlinear mechanics, embraces a wide variety of dynamic phenomena possesing slow and fast modes, we show that singular perturbations are present in many Quadratic-type Lyapunov Functions for Competitive Neural Networks 339 neurodynamical problems. In this sense we apply in this paper the results of this valuable analysis tool on the dynamics of laterally inhibited networks. In [SK84] is shown that a quadratic-type Lyapunov function for a singularly perturbed system is obtained as a weighted sum of quadratic-type Lyapunov functions of two lower order systems: the so-called reduced and the boundary-layer systems. Assuming that each of the two systems is asymptotically stable and has a Lyapunov function, conditions are derived to guarantee that, for a sufficiently small perturbation parameter, asymptotic stability of the singularly perturbed system can be established by means of a Lyapunov function which is composed as a weighted sum of the Lyapunov functions of the reduced and boundary-layer systems. Adopting the notations from [SK84] we will consider the singularly perturbed system 2 x = f(x, y) x E Bx C Rn (3) (4) We assume that, in Bx and By, the origin (x = y = 0) is the unique equilibrium point and (3) and (4) has a unique solution. A reduced system is defined by setting c = ° in (3) and (4) to obtain x = f(x,y) (5) O=g(x,y,O) (6) Assuming that in Bx and By, (6) has a unique root y = h(x), the reduced system is rewritten as x = f(x, h(x)) = fr(x) (7) A boundary-layer system is defined as ay aT = g(X,y(T),O) (8) where T = tic is a stretching time scale. In (8) the vector x E R n is treated as a fixed unknown parameter that takes values in Bx. The aim is to establish the stability properties of the singularly perturbed system (3) and (4), for small c, from those of the reduced system (7) and the boundary-layer system (8). The Lyapunov functions for system 7 and 8 are of quadratic-type. In [SK84] it is shown that under mild assumptions, for sufficiently small c, any weighted sum of the Lyapunov functions of the reduced and boundary-layer system is a quadratic-type Lyapunov function for the singularly perturbed system (3) and (4). The necessary assumptions are stated now [SK84]: 1. The reduced system (7) has a Lyapunov function V : R n -+ R+ such that for all xE Bx (9) where t/I(x) is a scalar-valued function of x that vanishes at x = 0 and is different from zero for all other x E Bx. This condition guarantees that x = 0 is an asymptotically stable equilibrium point of the reduced system (7). 2The symbol Bx indicates a closed sphere centered at x = OJ By is defined in the same way. 340 A. MEYER-BASE 2. The boundary-layer system (8) has a Lyapunov function W(x, y) : R n x R m -> R+ such that for all x E Bx and y E By ('\7yW(X,y)fg(X,y,O)::;-0:2¢?(y-h(x)) 0:2>0 (10) where ¢>(y - h(x)) is a scalar-valued function (y - h(x)) E R m that vanishes at y = h(x) and is different from zero for all other x E Bx and y E By. This condition guarantees that y = h(x) is an asymptotically stable equilibrium point of the boundary-layer system (8). 3. The following three inequalities hold "Ix E Bx and Vy E By: a.) ('\7 ,..W(x, y)ff(x, y) ::; C1¢>2(y - h(x)) + C21/J(X)¢>(Y - h(x)) (11) b.) ('\7,.. V(x)f[f(x, y) - f(x, h(x))] ::; /311/J(X)¢>(y - h(x)) (12) c.) ('\7yW(x,y)f[g(x,y,()-g(x,y,O)] < (K1¢>2(y - h(x)) + (K21/J(X)¢>(Y - h(x)) (13) The constants C1, C2, /31 ,K1 and K2 are nonnegative. The inequalities above determine the permissible interaction between the slow and fast variables. They are basically smoothness requirements of f and g. After these introductory remarks the stability criterion is now stated: Theorem: Suppose that conditions 1-3 hold; let d be a positive number such that 0< d < 1, and let c*(d) be the positive number given by (14) where Ih = f{2 + G2, 'Y = f{l + Gl , then for all c < c*(d), the origin (x = y = 0) is an asymptotically stable equilibrium point of (3) and (.0 and v(x, y) = (1 - d)V(x) + dW(x, y) (15) is a Lyapunov function of (3) and (4). If we put c = t as a global neural time constant in equation (1) then we have to determine two Lyapunov functions: one for the boundary-layer system and the other for the reduced-order system. In [CG83] is mentioned a global Lyapunov function for a competitive neural network with only an activation dynamics. (16) under the constraints: mij = mji, ai(xi) 2: 0, fj(xj) 2: O. This Lyapunov-function can be t·aken as one for the boundary-layer system (STMequation) , if the LTM contribution Si is considered as a fixed unknown parameter: Quadratic-type Lyapunov Functions for Competitive Neural Networks 341 N r i N (Xi 1 N W(x, S) = L 10 aj((j)!;((j)d(j-L BjSj 10 f;((j)d(j-2 L Dij!i(Xj)!k(Xk) j=l 0 j=l 0 j=l (17) For the reduced-order system (LTM- equation) we can take as a Lyapunov-function: N V(S) = ~STS = L S; i=l (18) The Lyapunov-function for the coupled STM and LTM dynamics is the sum of the two Lyapunov-function: vex, S) = (1 - d)V(S) + dW(x, S) 4 DESIGN OF STABLE COMPETITIVE NEURAL NETWORKS (19) Competitive neural networks with learning rules have moving equilibria during the learning process. The concept of asymptotic stability derived from matrix perturbation theory can capture this phenomenon. We design in this section a competitive neural network that is able to store a desired pattern as a stable equilibrium. The theoretical implications are illustrated in an example of a two neuron network. Example: Let N = 2, ai = A, Bj = B, Dii = a > 0, Dij = -(3 < 0 and the nonlinearity be a linear function f(xj) = Xj in equations (1) and (2). We get for the boundary-layer system: N Xj = -Axj + L Dijf(xd + BSj (20) i=l and for the reduced-order system: . B C S· = S·[-- -1] --J lA-a A-a (21) Then we get for the Lyapunov-functions: (22) and (23) 342 A. MEYER-BASE -0.2 . \ -0.4 \ [JJ OJ .IJ lIS .IJ -0.6 [JJ ~ / U) -0.8 \J -1 -1.2 ~ __ ~ __ ~ ____ ~ __ ~ __ -L __ ~~ __ ~ __ -L __ ~~~ o 1 2 3 4 5 6 7 8 9 10 time in msec Figure 1: Time histories of the neural network with the origin as an equilibrium point: STM states. For the nonnegative constants we get: al = 1 A~a' a2 = (A - a)2, Cl = 'Y = - B, with B < 0 , and C2 = i3l = i32 = 1 and I<l = I<2 = O. We get some interesting implications from the above results as: A-a> B , A-a> 0 and B < o. The above impications can be interpreted as follows: To achieve a stable equilibrium point (0,0) we should have a negative contribution of the external stimulus term and the sum of the excitatory and inhibitory contribution of the neurons should be less than the time constant of a neuron. An evolution of the trajectories of the STM and LTM states for a two neuron system is shown in figure 1 and 2. The STM states exhibit first an oscillation from the expected equilibrium point, while the LTM states reach monotonically the equilibrium point. We can see from the pictures that the equilibrium point (0,0) is reached after 5 msec by the STM- and LTM-states. Choosing B = -5, A = 1 and a = 0.5 we obtain for f*(d) : f*(d) = ll.of . 55+ .d(1-d) From the above formula we can see that f*(d) has a maximum at d = d* = 0.5. 5 CONCLUSIONS We presented in this paper a quadratic-type Lyapunov function for analyzing the stability of equilibrium points of competitive neural networks with fast and slow dynamics. This global stability analysis method is interpreting neural networks as nonlinear singularly perturbed systems. The equilibrium point is constrained to a neighborhood of (0,0). This technique supposes a monotonically increasing non-linearity and a symmetric lateral inhibition matrix. The learning rule is a deterministic Hebbian. This method gives an upper bound on the perturbation Quadratic-type Lyapunov Functions for Competitive Neural Networks 343 0.6 ~--~--~----~--~--~----~--'---~--~r---. 0.5 0.4 III <II .j.J III .j.J 0.3 III ~ ~ 0.2 0.1 o L-__ ~~~~~~~ ____ ~ __ ~ __ ~ __ -L __ ~ o 1 2 3 4 5 6 7 8 9 10 time in msec Figure 2: Time histories of the neural network with the origin as an equilibrium point: LTM states. parameter and such an estimation of a maximal positive neural time-constant. The practical implication ofthe theoretical problem is the design of a competitive neural network that is able to store a desired pattern as a stable equilibrium. References [Ama82] S. Amari. Competitive and cooperative aspects in dynamics of neural excitation and self-organization. Competition and cooperation in neural networks, 20:1-28, 7 1982. [Ama83] S. Amari. Field theory of self-organizing neural nets. IEEE Transactions on systems, machines and communication, SMC-13:741-748, 7 1983. [CG83] A. M. Cohen und S. Grossberg. Absolute Stability of Global Pattern Formation and Parallel Memory Storage by Competitive Neural Networks. IEEE Transactions on Systems, Man and Cybernetics, SMC-13:815-826, 9 1983. [Gro76] S. Grossberg. Adaptive Pattern Classification and Universal Recording. Biological Cybernetics, 23:121-134, 1 1976. [Heb49] D. O. Hebb. The Organization of Behavior. J. Wiley Verlag, 1949. [SK84] Ali Saberi und Hassan Khalil. Quadratic-Type Lyapunov Functions for Singularly Perturbed Systems. IEEE Transactions on A utomatic Control, pp. 542-550, June 1984.
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Learning long-term dependencies is not as difficult with NARX networks Tsungnan Lin* Department of Electrical Engineering Princeton University Princeton, N J 08540 Peter Tiiio Dept. of Computer Science and Engineering Slovak Technical University Ilkovicova 3, 812 19 Bratislava, Slovakia Abstract Bill G. Horne NEC Research Institute 4 Independence Way Princeton, NJ 08540 c. Lee Gilest NEC Research Institute 4 Independence Way Princeton, N J 08540 It has recently been shown that gradient descent learning algorithms for recurrent neural networks can perform poorly on tasks that involve long-term dependencies. In this paper we explore this problem for a class of architectures called NARX networks, which have powerful representational capabilities. Previous work reported that gradient descent learning is more effective in NARX networks than in recurrent networks with "hidden states". We show that although NARX networks do not circumvent the problem of long-term dependencies, they can greatly improve performance on such problems. We present some experimental 'results that show that NARX networks can often retain information for two to three times as long as conventional recurrent networks. 1 Introduction Recurrent Neural Networks (RNNs) are capable of representing arbitrary nonlinear dynamical systems [19, 20]. However, learning simple behavior can be quite "Also with NEC Research Institute. tAlso with UMIACS, University of Maryland, College Park, MD 20742 578 T. LIN, B. G. HORNE, P. TINO, C. L. GILES difficult using gradient descent. For example, even though these systems are 'lUring equivalent, it has been difficult to get them to successfully learn small finite state machines from example strings encoded as temporal sequences. Recently, it has been demonstrated that at least part of this difficulty can be attributed to long-term dependencies, i.e. when the desired output at time T depends on inputs presented at times t « T. In [13] it was reported that RNNs were able to learn short term musical structure using gradient based methods, but had difficulty capturing global behavior. These ideas were recently formalized in [2], which showed that if a system is to robustly latch information, then the fraction of the gradient due to information n time steps in the past approaches zero as n becomes large. Several approaches have been suggested to circumvent this problem. For example, gradient-based methods can be abandoned in favor of alternative optimization methods [2, 15]. However, the algorithms investigated so far either perform just as poorly on problems involving long-term dependencies, or, when they are better, require far more computational resources [2]. Another possibility is to modify conventional gradient descent by more heavily weighing the fraction of the gradient due to information far in the past, but there is no guarantee that such a modified algorithm would converge to a minima of the error surface being searched [2]. Another suggestion has been to alter the input data so that it represents a reduced description that makes global features more explicit and more readily detectable [7, 13, 16, 17]. However, this approach may fail if short term dependencies are equally as important. Finally, it has been suggested that a network architecture that operates on multiple time scales might be useful [5, 6]. In this paper, we also propose an architectural approach to deal with long-term dependencies [11]. We focus on a class of architectures based upon Nonlinear AutoRegressive models with eXogenous inputs (NARX models), and are therefore called NARX networks [3, 14]. This is a powerful class of models which has recently been shown to be computationally equivalent to 'lUring machines [18]. Furthermore, previous work has shown that gradient descent learning is more effective in NARX networks than in recurrent network architectures with "hidden states" when applied to problems including grammatical inference and nonlinear system identification [8]. Typically, these networks converge much faster and generalize better than other networks. The results in this paper give an explanation of this phenomenon. 2 Vanishing gradients and long-term dependencies Bengio et al. [2] have analytically explained why learning problems with long- term dependencies is difficult. They argue that for many practical applications the goal of the network must be to robustly latch information, i.e. the network must be able to store information for a long period of time in the presence of noise. More specifically, they argue that latching of information is accomplished when the states of the network stay within the vicinity of a hyperbolic attractor, and robustness to noise is accomplished if the states of the network are contained in the reduced attracting set of that attractor, i.e. those set of points at which the eigenvalues of the Jacobian are contained within the unit circle. In algorithms such as Backpropagation Through Time (BPTT), the gradient of the cost function function C is written assuming that the weights at different time Learning Long-term Dependencies Is Not as Difficult with NARX Networks 579 u(k) u(k-l) u(k-2) y(k-3) y(k-2) y(k-l) Figure 1: NARX network. indices are independent and computing the partial gradient with respect to these weights. The total gradient is then equal to the sum of these partial gradients. It can be easily shown that the weight updates are proportional to where Yp(T) and d p are the actual and desired (or target) output for the pth pattern!, x(t) is the state vector of the network at time t and Jx(T,T - T) = \l xC-r)x(T) denotes the Jacobian of the network expanded over T T time steps. In [2], it was shown that if the network robustly latches information, then Jx(T, n) is an exponentially decreasing function of n, so that limn-too Jx(T, n) = 0 . This implies that the portion of \l we due to information at times T « T is insignificant compared to the portion at times near T. This vanishing gradient is the essential reason why gradient descent methods are not sufficiently powerful to discover a relationship between target outputs and inputs that occur at a much earlier time. 3 NARX networks An important class of discrete- time nonlinear systems is the Nonlinear AutoRegressive with eXogenous inputs (NARX) model [3, 10, 12, 21]: y(t) = f (u(t - Du ), ... ,u(t - 1), u(t), y(t - D y ),'" ,y(t - 1)) , where u(t) and y(t) represent input and output ofthe network at time t, Du and Dy are the input and output order, and f is a nonlinear function. When the function f can be approximated by a Multilayer Perceptron, the resulting system is called a NARX network [3, 14]. In this paper we shall consider NARX networks with zero input order and a one dimensional output. However there is no reason why our results could not be extended to networks with higher input orders. Since the states of a discrete-time lWe deal only with problems in which the target output is presented at the end of the sequence. 580 0 9 De (a) t]., .... 0.3 ,- '· 0.6 60 0' 009 DOS "5 ~ OO 7 ~ O 06 Ii JOO5 :i ..Q 004 "0 h 03 ~ "' 002 DO' 0 0 T. LIN, B. G. HORNE, P. TINO, C. L. GILES :' 1 ~ I ., , j , i I '...\ 10 20 30 n (b) . 0 Em" 0.3 - ·0. 6 60 Figure 2: Results for the latching problem. (a) Plots of J(t,n) as a function of n. (b) Plots of the ratio E~~(ltJ(tr) as a function of n. dynamical system can always be associated with the unit-delay elements in the realization of the system, we can then describe such a network in a state space form i=l i = 2, ... ,D (1) with y(t) = Xl (t + 1) . If the Jacobian of this system has all of its eigenvalues inside the unit circle at each time step, then the states of the network will be in the reduced attracting set of some hyperbolic attractor, and thus the system will be robustly latched at that time. As with any other RNN, this implies that limn-too Jx(t, n) = o. Thus, NARX networks will also suffer from vanishing gradients and the long- term dependencies problem. However, we find in the simulation results that follow that NARX networks are often much better at discovering long-term dependencies than conventional RNNs. An intuitive reason why output delays can help long-term dependencies can be found by considering how gradients are calculated using the Backpropagation Through Time algorithm. BPTT involves two phases: unfolding the network in time and backpropagating the error through the unfolded network. When a NARX network is unfolded in time, the output delays will appear as jump-ahead connections in the unfolded network. Intuitively, these jump-ahead connections provide a shorter path for propagating gradient information, thus reducing the sensitivity of the network to long- term dependencies. However, this intuitive reasoning is only valid if the total gradient through these jump- ahead pathways is greater than the gradient through the layer-to-layer pathways. It is possible to derive analytical results for some simple toy problems to show that NARX networks are indeed less sensitive to long-term dependencies. Here we give one such example, which is based upon the latching problem described in [2]. Consider the one node autonomous recurrent network described by, x(t) = tanh(wx(t - 1)) where w = 1.25, which has two stable fixed points at ±0.710 and one unstable fixed point at zero. The one node, autonomous NARX network x(t) = tanh (L:~=l wrx(t - r)) has the same fixed points as long as L:?:l Wi = w. Learning Long-tenn Dependencies Is Not as Difficult with NARX Networks 581 Assume the state of the network has reached equilibrium at the positive stable fixed point and there are no external inputs. For simplicity, we only consider the Jacobian J(t, n) = 8~{t~~)' which will be a component of the gradient 'ilwC. Figure 2a shows plots of J(t, n) with respect to n for D = 1, D = 3 and D = 6 with Wi = wiD. These plots show that the effect of output delays is to flatten out the curves and place more emphasis on the gradient due to terms farther in the past. Note that the gradient contribution due to short term dependencies is deemphasized. In Figure 2b we show plots of the ratio L::~\tj(t,r) , which illustrates the percentage of the total gradient that can be attributed to information n time steps in the past. These plots show that this percentage is larger for the network with output delays, and thus one would expect that these networks would be able to more effectively deal with long-term dependencies. 4 Experimental results 4.1 The latching problem We explored a slight modification on the latching problem described in [2], which is a minimal task designed as a test that must necessarily be passed in order for a network to robustly latch information. In this task there are three inputs Ul(t), U2(t), and a noise input e(t), and a single output y(t) . Both Ul(t) and U2(t) are zero for all times t> 1. At time t = 1, ul(l) = 1 and u2(1) = 0 for samples from class 1, and ul(l) = 0 and u2(1) = 1 for samples from class 2. The noise input e(t) is drawn uniformly from [-b, b] when L < t S T, otherwise e(t) = 0 when t S L. This network used to solve this problem is a NARX network consisting of a single neuron, where the parameters h{ are adjustable and the recurrent weights Wr are fixed 2 . We fixed the recurrent feedback weight to Wr = 1.251 D, which gives the autonomous network two stable fixed points at ±0.710, as described in Section 3. It can be shown [4] that the network is robust to perturbations in the range [-0.155,0.155]. Thus, the uniform noise in e(t) was restricted to this range. For each simulation, we generated 30 strings from each class, each with a different e(t). The initial values of h{ for each simulation were also chosen from the same distribution that defines e(t). For strings from class one, a target value of 0.8 was chosen, for class two -0.8 was chosen. The network was run using a simple BPTT algorithm with a learning rate of 0.1 for a maximum of 100 epochs. (We found that the network converged to some solution consistently within a few dozen epochs.) If the simulation exceeded 100 epochs and did not correctly classify all strings then the simulation was ruled a failure. We varied T from 10 to 200 in increments of 2. For each value of T, we ran 50 simulations. Figure 3a shows a plot of the percentage of those runs that were successful for each case. It is clear from these plots that 2 Although this description may appear different from the one in [2], it can be shown that they are actually identical experiments for D = 1. 582 0 9 09 . 0 1 ~ ~06 ~ ~05 1 0 • " 03 02 0' ....... '.J "60J' . Ii' , , \ ! , ~ i ! , , ~ ., ... . 0.3 ·-·-0.6 T. LIN, B. G. HORNE, P. TINO, C. L. GILES , _ .. ..,.. .......... : . .: . 09 " , . ...... ' :~ .. ,.'< ...... . ... ... 06 04 02 \ .. '" ·.~Il~ ~~~~~~~~~~~~'6~ 0 ~'M~~200 ~~~~'0~~'5~~ro--~25--=OO~3~ 5 ~4~0~4~5~50 · 00 20 40 Langlh 01 InPJI nC.IIIe (a) (b) Figure 3: (a) Plots of percentage of successful simulations as a function of T, the length of the input strings. (b) Plots of the final classification rate with respect to different length input strings. the NARX networks become increasingly less sensitive to long- term dependencies as the output order is increased. 4.2 The parity problem In the parity problem, the task is to classify sequences depending on whether or not the number of Is in the input string is odd. We generated 20 strings of different lengths from 3 to 5 and added uniformly distributed noise in the range [-0.2,0.2] at the end of each string. The length of input noise varied from 0 to 50. We arbitrarily chose 0.7 and -0.7 to represent the symbol "1" and "0". The target is only given at the end of each string. Three different networks with different number of output delays were run on this problem in order to evaluate the capability of the network to learn long-term dependencies. In order to make the networks comparable, we chose networks in which the number of weights was roughly equal. For networks with one to three delays, 5, 4 and 3 hidden neurons were chosen respectively, giving 21, 21, and 19 trainable weights. Initial weight values were randomly generated between -0.5 and 0.5 for 10 trials. Fig. 3b shows the average classification rate with respect to different length of input noise. When the length of the noise is less than 5, all three of the networks can learn all the sequences with the classification rate near to 100%. When the length increases to between 10 and 35, the classification rate of networks with one feedback delay drops quickly to about 60% while the rate of those networks with two or three feedback delays still remains about 80%. 5 Conclusion In this paper we considered an architectural approach to dealing with the problem of learning long-term dependencies. We explored the ability of a class of architectures called NARX networks to solve such problems. This has been observed previously, in the sense that gradient descent learning appeared to be more effective in NARX Learning Long-tenn Dependencies Is Not as Difficult with NARX Networks 583 networks than in RNNs [8]. We presented an analytical example that showed that the gradients do not vanish as quickly in NARX networks as they do in networks without multiple delays when the network is operating at a fixed point. We also presented two experimental problems which show that NARX networks can outperform networks with single delays on some simple problems involving long-term dependencies. We speculate that similar results could be obtained for other networks. In particular we hypothesize that any network that uses tapped delay feedback [1, 9] would demonstrate improved performance on problems involving long-term dependencies. Acknowledgements We would like to thank A. Back and Y. Bengio for many useful suggestions. References (1] A.D. Back and A.C. Tsoi. FIR and IIR synapses, a new neural network architecture for time series modeling. Neural Computation, 3(3):375-385, 1991. (2] Y. Bengio, P. Simard, and P. Frasconi. Learning long-term dependencies with gradient is difficult. IEEE Trans. on Neural Networks, 5(2):157- 166, 1994. (3] S. Chen, S.A. Billings, and P.M. Grant. Non-linear system identification using neural networks. International Journal of Control, 51(6):1191-1214, 1990. (4] P. Frasconi, M. Gori, M. Maggini, and G. Soda. Unified integration of explicit knowledge and learning by example in recurrent networks. IEEE Trans. on Know. and Data Eng.,7(2):340-346, 1995. (5] M. Gori, M. Maggini, and G. Soda. Scheduling of modular architectures for inductive inference of regular grammars. In ECAI'94 Work. on Comb. Sym. and Connectionist Proc., pages 78-87. (6J S. EI Hihi and Y. Bengio. Hierarchical recurrent neural networks for long-term dependencies. In NIPS 8, 1996. (In this Proceedings.) (7] S. Hochreiter and J. Schmidhuber. Long short term memory. Technical Report FKI-207-95, Technische Universitat Munchen, 1995. (8] B.G. Horne and C.L. Giles. An experimental comparison of recurrent neural networks. In NIPS 7, pages 697-704, 1995. (9J R.R. Leighton and B.C. Conrath. The autoregressive backpropagation algorithm. In Proceedings of the International Joint Conference on Neural Networks, volume 2, pages 369-377, July 1991. (10] I.J. Leontaritis and S.A. Billings. Input-output parametric models for non-linear systems: Part I: deterministic non- linear systems. International Journal of Control, 41(2):303-328, 1985. (ll] T .N. Lin, B.G. Horne, P.Tino and C.L. Giles. Learning long-term dependencies is not as difficult with NARX recurrent neural networks. Technical Report UMIACS-TR-95-78 and CS-TR-3500, Univ. Of Maryland, 1995. (12] L. Ljung. System identification: Theory for the user. Prentice-Hall, 1987. [13] M. C. Mozer. Induction of multiscale temporal structure. In J.E. Moody, S. J. Hanson, and R.P. Lippmann, editors, NIPS 4, pages 275-282, 1992. (14] K.S. Narendra and K. Parthasarathy. Identification and control of dynamical systems using neural networks. IEEE Trans. on Neural Networks, 1:4-27, March 1990. (15] G.V. Puskorius and L.A. Feldkamp. Recurrent network training with the decoupled extended Kalman filter. In Proc. 1992 SPIE Con/. on the Sci. of ANN, Orlando, Florida, April 1992. (16] J . Schmidhu ber. Learning complex, extended sequences using the principle of history compression. In Neural Computation, 4(2):234-242, 1992. (17] J. Schmidhuber. Learning unambiguous reduced sequence descriptions. In NIPS 4, pages 291298,1992. (18] H.T. Siegelmann, B.G. Horne, and C.L. Giles. Computational capabilities of NARX neural networks. In IEEE Trans. on Systems, Man and Cybernetics, 1996. Accepted. (19] H.T. Siegel mann and E.D. Sontag. On the computational power of neural networks. Journal of Computer and System Science, 50(1):132-150, 1995. [20] E.D. Sontag. Systems combining linearity and saturations and relations to neural networks. Technical Report SYCON-92- 01, Rutgers Center for Systems and Control, 1992. (21] H. Su, T. McAvoy, and P. Werbos. Long-term predictions of chemical processes using recurrent neural networks: A parallel training approach. Ind. Eng. Chem. Res., 31:1338, 1992.
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Learning the structure of similarity Joshua B. Tenenbaum Department of Brain and Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 jbt~psyche.mit.edu Abstract The additive clustering (ADCL US) model (Shepard & Arabie, 1979) treats the similarity of two stimuli as a weighted additive measure of their common features. Inspired by recent work in unsupervised learning with multiple cause models, we propose anew, statistically well-motivated algorithm for discovering the structure of natural stimulus classes using the ADCLUS model, which promises substantial gains in conceptual simplicity, practical efficiency, and solution quality over earlier efforts. We also present preliminary results with artificial data and two classic similarity data sets. 1 INTRODUCTION The capacity to judge one stimulus, object, or concept as similado another is thought to play a pivotal role in many cognitive processes, including generalization, recognition, categorization, and inference. Consequently, modeling subjective similarity judgments in order to discover the underlying structure of stimulus representations in the brain/mind holds a central place in contemporary cognitive science. Mathematical models of similarity can be divided roughly into two families: spatial models, in which stimuli correspond to points in a metric (typically Euclidean) space and similarity is treated as a decreasing function of distance; and set-theoretic models, in which stimuli are represented as members of salient subsets (presumably corresponding to natural classes or features in the world) and similarity is treated as a weighted sum of common and distinctive subsets. Spatial models, fit to similarity judgment data with familiar multidimensional scaling (MDS) techniques, have yielded concise descriptions of homogeneous, perceptual domains (e.g. three-dimensional color space), often revealing the salient dimensions of stimulus variation (Shepard, 1980). Set-theoretic models are more general, in principle able to accomodate discrete conceptual structures typical of higher-level cognitive domains, as well as dimensional stimulus structures more common in per4 1. B. TENENBAUM ception (Tversky, 1977). In practice, however, the utility of set-theoretic models is limited by the hierarchical clustering techniques that underlie conventional methods for discovering the discrete features or classes of stimuli. Specifically, hierarchical clustering requires that any two classes of stimuli correspond to disjoint or properly inclusive subsets, while psychologically natural classes may correspond in general to arbitrarily overlapping subsets of stimuli. For example, the subjective similarity of two countries results from the interaction of multiple geographic and cultural factors, and there is no reason a priori to expect the subsets of communist, African, or French-speaking nations to be either disjoint or properly inclusive. In this paper we consider the additive clustering (ADCL US) model (Shepard & Arabie, 1979), the simplest instantiation of Tversky 's (1977) general contrast model that accommodates the arbitrarily overlapping class structures associated with multiple causes of similarity. Here, the similarity of two stimuli is modeled as a weighted additive measure of their common clusters: K Sij = I: wkfikfJk + C, (1) k=l where Sij is the reconstructed similarity of stimuli i and j, the weight Wk captures the salience of cluster k, and the binary indicator variable fik equals 1 if stimulus i belongs to cluster k and 0 otherwise. The additive constant c is necessary because the similarity data are assumed to be on an interval scale. 1 As with conventional clustering models, ADCLUS recovers a system of discrete subsets of stimuli, weighted by salience, and the similarity of two stimuli increases with the number (and weight) of their common subsets. ADCLUS, however, makes none of the structural assumptions (e.g. that any two clusters are disjoint or properly inclusive) which limit the applicability of conventional set-theoretic models. Unfortunately this flexibility also makes the problem of fitting the ADCL US model to an observed similarity matrix exceedingly difficult. Previous attempts to fit the model have followed a heuristic strategy to minimize a squared-error energy function, E = I:(Sij - Sij)2 = I:(Sij - I: wklikfJk)2, (2) itj itj k by alternately solving for the best cluster configurations fik given the current weights Wk and solving for the best weights given the current clusters (Shepard & Arabie, 1979; Arabie & Carroll, 1980). This strategy is appealing because given the cluster configuration, finding the optimal weights becomes a simple linear least-squares problem.2 However, finding good cluster configurations is a difficult problem in combinatorial optimization, and this step has always been the weak point in previous work. The original ADCLUS (Shepard & Arabie, 1979) and later MAPCLUS (Arabie & Carroll, 1980) algorithms employ ad hoc techniques of combinatorial optimization that sometimes yield unexpected or uninterpretable final results. Certainly, no rigorous theory exists that would explain why these approaches fail to discover the underlying structure of a stimulus set when they do. Essentially, the ADCL US model is so challenging to fit because it generates similarities from the interaction of many independent underlying causes. Viewed this way, modeling the structure of similarity looks very similar to the multiple-cause learning 1 In the remainder of this paper, we absorb c into the sum over k, taking the sum over k = 0, ... , K , defining Wo == c, and fixing !iO = 1, (Vi). 2Strictly speaking, because the weights are typically constrained to be nonnegative, more elaborate techniques than standard linear least-squares procedures may be required. Learning the Structure of Similarity 5 problems that are currently a major focus of study in the neural computation literature (Ghahramani, 1995; Hinton, Dayan, et al., 1995; Saund, 1995; Neal, 1992). Here we propose a novel approach to additive clustering, inspired by the progress and promise of work on multiple-cause learning within the Expectation-Maximization (EM) framework (Ghahramani, 1995; Neal, 1992). Our BM approach still makes use of the basic insight behind earlier approaches, that finding {wd given {lid is easy, but obtains better performance from treating the unknown cluster memberships probabilistically as hidden variables (rather than parameters of the model), and perhaps more importantly, provides a rigorous and well-understood theory. Indeed, it is natural to consider {/ik} as "unobserved" features of the stimuli, complementing the observed data {Sij} in the similarity matrix. Moreover, in some experimental paradigms, one or more of these features may be considered observed data, if subjects report using (or are requested to use) certain criteria in their similarity judgments. 2 ALGORITHM 2.1 Maximum likelihood formulation We begin by formulating the additive clustering problem in terms of maximum likelihood estimation with unobserved data. Treating the cluster weights w = {Wk} as model parameters and the unobserved cluster memberships I = {lik} as hidden causes for the observed similarities S = {Sij}, it is natural to consider a hierarchical generative model for the "complete data" (including observed and unobserved components) of the form p(s, Ilw) = p(sl/, w)p(flw). In the spirit of earlier approaches to ADCLUS that seek to minimize a squared-error energy function, we take p(sl/, w) to be gaussian with common variance u 2 : p(sl/, w) ex: exp{ -~ 'L:(Sij - Sij )2} = exp{ -~ 'L:(Sij - 'L: wklik/ik)2}. (3) 2u itj 2u itj k Note that logp(sl/, w) is equivalent to -E/(2u2 ) (ignoring an additive constant), where E is the energy defined above. In general, priors p(flw) over the cluster configurations may be useful to favor larger or smaller clusters, induce a dependence between cluster size and cluster weight, or bias particular kinds of class structures, but only uniform priors are considered here. In this case -E /(2u2 ) also gives the "complete data" loglikelihood logp(s, Ilw). 2.2 The EM algorithm for additive clustering Given this probabilistic model, we can now appeal to the EM algorithm as the basis for a new additive clustering technique. EM calls for iterating the following twostep procedure, in order to obtain successive estimates of the parameters w that are guaranteed never to decrease in likelihood (Dempster et al., 1977). In the E-step, we calculate Q(wlw(n)) = L,: p(f' Is, wen)) logp(s,f/lw) = 2 \ (-E}3,w(n). (4) l' u Q(wlw(n) is equivalent to the expected value of E as a function of w, averaged over all possible configurations I' of the N K binary cluster memberships, given the observed data s and the current parameter estimates wen). In the M-step, we maximize Q(wlw(n) with respect to w to obtain w(n+l). Each cluster configuration I' contributes to the mean energy in proportion to its probability under the gaussian generative model in (3). Thus the number of configurations making significant contributions depends on the model variance u 2 . For large 6 J. B. TENENBAUM U 2 , the probability is spread over many configurations. In the limiting case u 2 ---+ 0, only the most likely configuration contributes, making EM effectively equivalent to the original approaches presented in Section 1 that use only the single best cluster configuration to solve for the best cluster weights at each iteration. In line with the basic insight embodied less rigorously in these earlier algorithms, the M-step still reduces to a simple (constrained) linear least-squares problem, because the mean energy (E} = L:i#j (srj - 2Sij L:k Wk(fik!ik} + L:kl WkWl(fik!jk!il!il}) , like the energy E, is quadratic in the weights Wk. The E-step, which amounts to computing the expectations mijk = (fik!ik} and mijkl = (fik !ik!il/j I} , is much more involved, because the required sums over all possible cluster configurations f' are intractable for any realistic case. We approximate these calculations using Gibbs sampling, a Monte Carlo method that has been successfully applied to learning similar generative models with hidden variables (Ghahramani, 1995; Neal 1992).3 Finally, the algorithm should produce not only estimates of the cluster weights, but also a final cluster configuration that may be interpreted as the psychologically natural features or classes of the relevant domain. Consider the expected cluster memberships Pik = (fik}$ w(n) , which give the probability that stimulus i belongs to cluster k, given the observed similarity matrix and the current estimates of the weights. Only when all Pik are close to 0 or 1, i.e. when u2 is small enough that all the probability becomes concentrated on the most likely cluster configuration and its neighbors, can we fairly assert which stimuli belong to which classes. 2.3 Simulated annealing Two major computational bottlenecks hamper the efficiency of the algorithm as described so far. First, Gibbs sampling may take a very long time to converge to the equilibrium distribution, particularly when u 2 is small relative to the typical energy difference between neighboring cluster configurations. Second, the likelihood surfaces for realistic data sets are typically riddled with local maxima. We solve both problems by annealing on the variance. That is, we run Gibbs sampling using an effective variance u;" initially much greater than the assumed model variance u2 , and decrease u;" towards u 2 according to the following two-level scheme. We anneal within the nth iteration of EM to speed the convergence of the Gibbs sampling E-step (Neal, 1993), by lowering u;jJ from some high starting value down to a target U~arg(n) for the nth EM iteration. We also anneal between iterations of EM to avoid local maxima (Ros~ et al., 1990), by intializing U~arg(o) at a high value and taking U~arg(n) ---+ u2 as n Increases. 3 RESULTS In all of the examples below, one run of the algorithm consisted of 100-200 iterations of EM, annealed both within and between iterations. Within each E-step, 10-100 cycles of Gibbs sampling were carried out at the target temperature UTarg while the statistics for mik and mijk were recorded. These recorded cycles were preceeded by 20-200 unrecorded cycles, during which the system was annealed from a higher temperature (e.g. 8u~arg) down to U~arg, to ensure that statistics were collected as close to equilibrium as possible. The precise numbers of recorded and unrecorded iterations were chosen as a compromise between the need for longer samples as the 3We generally also approximate miJkl ~ miJkmi;"l, which usually yields satisfactory results with much greater efficiency. Learning the Structure of Similarity 7 Table 1: Classes and weights recovered for the integers 0-9. Rank Weight Stimuli in class Interpretation 1 .444 2 4 8 powers of two 2 .345 012 small numbers 3 .331 3 6 9 multiples of three 4 .291 6 789 large numbers 5 .255 2 345 6 middle numbers 6 .216 1 3 5 7 9 odd numbers 7 .214 1 2 3 4 smallish numbers 8 .172 4 5 6 7 8 largish numbers Variance accounted for = 90.9% with 8 clusters (additive constant = .148). number of hidden variables is increased and the need to keep computation times practical. 3.1 Artificial data We first report results with artificial data, for which the true cluster memberships and weights are known, to verify that the algorithm does in fact find the desired structure. We generated 10 data sets by randomly assigning each of 12 stimuli independently and with probability 1/2 to each of 8 classes, and choosing random weights for the classes uniformly from [0.1,0.6]. These numbers are grossly typical of the real data sets we examine later in this section. We then calculated the observed similarities from (1), added a small amount of random noise (with standard deviation equal to 5% of the mean noise-free similarity), and symmeterized the similarity matrix. The crucial free parameter is K, the assumed number of stimulus classes. When the algorithm was configured with the correct number of clusters (K = 8), the original classes and weights were recovered during the first run of the algorithm on all 10 data sets, after an average of 58 EM iterations (low 30, high 92). When the algorithm was configured with K = 7 clusters, one less than the correct number, the seven classes with highest weight were recovered on 9/10 first runs. On these runs, the recovered weights and true weights had a mean correlation of 0.948 (p < .05 on each run). When configured with K = 5, the first run recovered either four of the top five classes (6/10 trials) or three of the top five (4/10 trials). When configured with too many clusters (K = 12), the algorithm typically recovered only 8 clusters with significantly non-zero weights, corresponding to the 8 correct classes. Comparable results are not available for ADCLUS or MAPCLUS, but at least we can be satisfied that our algorithm achieves a basic level of competence and robustness. 3.2 Judged similarities of the integers 0-9 Shepard et al. (1975) had subjects judge the similarities of the integers 0 through 9, in terms of the "abstract concepts" of the numbers. We analyzed the similarity matrix (Shepard, personal communication) obtained by pooling data across subjects and across three conditions of stimulus presentation (verbal, written-numeral, and written-dots). We chose this data set because it illustrates the power of additive clustering to capture a complex, overlapping system of classes, and also because it serves to compare the performance of our algorithm with the original ADCL US algorithm. Observe first that two kinds of classes emerge in the solution. Classes 1, 3, and 6 represent familiar arithmetic concepts (e.g. "multiples of three", "odd numbers"), while the remaining classes correspond to subsets of consecutive integers 8 1. B. TENENBAUM Table 2: Classes and weights recovered for the 16 consonant phonemes. Rank Weight Stimuli in class Interpretation 1 .800 f 0 front unvoiced fricatives 2 .572 dg back voiced stops 3 .463 p k unvoiced stops (omitting t) 4 .424 b v {t front voiced 5 .357 p t k unvoiced stops 6 .292 mn nasals 7 .169 dgvCTz2 voiced (omitting b) 8 .132 ptkfOs unvoiced (omittings) Variance accounted for = 90.2% with 8 clusters (additive constant = .047). and thus together represent the dimension of numerical magnitude. In general, both arithmetic properties and numerical magnitude contribute to judged similarity, as every number has features of both types (e.g. 9 is a "large" "odd" "multiple of three"), except for 0, whose only property is "small." Clearly an overlapping clustering model is necessary here to accomodate the multiple causes of similarity. The best solution reported for these data using the original ADCLUS algorithm consisted of 10 classes, accounting for 83.1% of the variance of the data (Shepard & Arabie, 1979).4 Several of the clusters in this solution differed by only one or two members (e.g. three of the clusters were {0,1}, {0,1,2}, and {0,1,2,3,4}), which led us to suspect that a better fit might be obtained with fewer than 10 classes. Table 2 shows the best solution found in five runs of our algorithm, accounting for 90.9% of the variance with eight classes. Compared with our solution, the original ADCLUS solution leaves almost twice as much residual variance unaccounted for, and with 10 classes, is also less parsimonious. 3.3 Confusions between 16 consonant phonemes Finally, we examine Miller & Nicely's (1955) classic data on the confusability of 16 consonant phonemes, collected under varying signal/noise conditions with the original intent of identifying the features of English phonology (compiled and reprinted in Carroll & Wish, 1974). Note that the recovered classes have reasonably natural interpretations in terms of the basic features of phonological theory, and a very different overall structure from the classes recovered in the previous example. Quite significantly, the classes respect a hierarchical structure almost perfectly, with class 3 included in class 5, classes 1 and 5 included in class 8, and so on. Only the absence of /b / in class 7 violates the strict hierarchy. These data also provide the only convenient oppportunity to compare our algorithm with the MAPCLUS approach to additive clustering (Arabie & Carroll, 1980). The published MAPCLUS solution accounts for 88.3% of the variance in this data, using eight clusters. Arabie & Carroll (1980) report being "substantively pe ... turbed" (p. 232) that their algorithm does not recover a distinct cluster for the nasals /m n/, which have been considered a very salient subset in both traditional phonology (Miller & Nicely, 1955) and other clustering models (Shepard, 1980). Table 3 presents our eight-cluster solution, accounting for 90.2% of the variance. While this represents only a marginal improvement, our solution does contain a cluster for the nasals, as expected on theoretical grounds. 4Variance accounted for = 1- Ej Ei#j(SiJ - 8)2, where s is the mea.n of the set {Sij}. Learning the Structure of Similarity 9 3.4 Conclusion These examples show that ADCLUS can discover meaningful representations of stimuli with arbitrarily overlapping class structures (arithmetic properties), as well as dimensional structure (numerical magnitude) or hierarchical structure (phoneme families) when appropriate. We have argued that modeling similarity should be a natural application of learning generative models with multiple hidden causes, and in that spirit, presented a new probabilistic formulation of the ADCLUS model and an algorithm based on EM that promises better results than previous approaches. We are currently pursuing several extensions: enriching the generative model, e.g. by incorporating significant prior structure, and improving the fitting process, e.g. by developing efficient and accurate mean field approximations. More generally, we hope this work illustrates how sophisticated techniques of computational learning can be brought to bear on foundational problems of structure discovery in cognitive science. Acknowledgements I thank P. Dayan, W. Richards, S. Gilbert, Y. Weiss, A. Hershowitz, and M. Bernstein for many helpful discussions, and Roger Shepard for generously supplying inspiration and unpublished data. The author is a Howard Hughes Medical Institute Predoctoral Fellow. References Arabie, P. & Carroll, J. D. (1980). MAPCLUS: A mathematical programming approach to fitting the ADCLUS model. Psychometrika 45, 211-235. Carroll, J. D. & Wish, M. (1974) Multidimensional perceptual models and measurement methods. In Handbook of Perception, Vol. 2. New York: Academic Press, 391-447. Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood estimation from incomplete data via the EM Algorithm (with discussion). J. Roy. Stat. Soc. B39, 1-38. Ghahramani, Z. (1995). Factorial learning and the EM algorithm. In G. Tesauro, D. S. Touretzky, & T . K. Leen (eds.), Advances in Neural Information Processing Systems 7. Cambridge, MA: MIT Press, 617-624. Hinton, G. E., Dayan, P., Frey, B. J., & Neal, R. M. (1995) The ((wake-sleep" algorithm for unsupervised neural networks. Science 268, 1158-1161. Miller, G. A. & Nicely, P. E. (1955). An analysis of perceptual confusions among some English consonants. J. Ac. Soc. Am. 27, 338-352. Neal, R. M. (1992). Connectionist learning of belief networks. Arti/. Intell. 56, 71-113. Neal, R. M. (1993). Probabilistic inference using Markov chain Monte Carlo methods. Technical Report CRG-TR-93-1, Dept. of Computer Science, U. of Toronto. Rose, K., Gurewitz, F., & Fox, G. (1990). Statistical mechanics and phase transitions in clustering. Physical Review Letters 65, 945-948. Saund, E. (1995). A multiple cause mixture model for unsupervised learning. Neural Computation 7, 51-71. Shepard, R. N. & Arabie, P. (1979). Additive clustering: Representation of similarities as combinations of discrete overlapping properties. Psychological Review 86, 87-123. Shepard, R. N., Kilpatric, D. W., & Cunningham, J. P., (1975). The internal representation of numbers. Cognitive Psychology 7, 82-138. Shepard, R. N. (1980). Multidimensional scaling, tree-fitting, and clustering. Science 210, 390-398. Tversky, A. (1977). Features of similarity. Psychological Review 84, 327-352.
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Investment Learning with Hierarchical PSOMs Jorg Walter and Helge Ritter Department of Information Science University of Bielefeld, D-33615 Bielefeld, Germany Email: {walter.helge}@techfak.uni-bielefeld.de Abstract We propose a hierarchical scheme for rapid learning of context dependent "skills" that is based on the recently introduced "Parameterized SelfOrganizing Map" ("PSOM"). The underlying idea is to first invest some learning effort to specialize the system into a rapid learner for a more restricted range of contexts. The specialization is carried out by a prior "investment learning stage", during which the system acquires a set of basis mappings or "skills" for a set of prototypical contexts. Adaptation of a "skill" to a new context can then be achieved by interpolating in the space of the basis mappings and thus can be extremely rapid. We demonstrate the potential of this approach for the task of a 3D visuomotor map for a Puma robot and two cameras. This includes the forward and backward robot kinematics in 3D end effector coordinates, the 2D+2D retina coordinates and also the 6D joint angles. After the investment phase the transformation can be learned for a new camera set-up with a single observation. 1 Introduction Most current applications of neural network learning algorithms suffer from a large number of required training examples. This may not be a problem when data are abundant, but in many application domains, for example in robotics, training examples are costly and the benefits of learning can only be exploited when significant progress can be made within a very small number of learning examples. Investment Learning with Hierarchical PSOMs 571 In the present contribution, we propose in section 3 a hierarchically structured learning approach which can be applied to many learning tasks that require system identification from a limited set of observations. The idea builds on the recently introduced "Parameterized Self-Organizing Maps" ("PSOMs"), whose strength is learning maps from a very small number of training examples [8, 10, 11]. In [8], the feasibility of the approach was demonstrated in the domain of robotics, among them, the learning of the inverse kinematics transform of a full 6-degree of freedom (DOF) Puma robot. In [10], two improvements were introduced, both achieve a significant increase in mapping accuracy and computational efficiency. In the next section, we give a short summary of the PSOM algorithm; it is decribed in more detail in [11] which also presents applications in the domain of visual learning. 2 The PSOM Algorithm A Parameterized Self-Organizing Map is a parametrized, m-dimensional hyper-surface M = {w(s) E X ~ rn.dls E S ~ rn.m} that is embedded in some higher-dimensional vector space X. M is used in a very similar way as the standard discrete self-organizing map: given a distance measure dist(x, x') and an input vector x, a best-match location s*(x) is determined by minimizing s*:= argmin dist(x, w(s)) (1) SES The associated "best-match vector" w(s*) provides the best approximation of input x in the manifold M. If we require dist(·) to vary only in a subspace X in of X (i.e., dist( x, x') = dist(Px, Px/), where the diagonal matrix P projects into xin), s* (x) actually will only depend on Px. The projection (l-P)w(s* (x)) E x out ofw(s* (x)) lies in the orthogonal subspace x out can be viewed as a (non-linear) associative completion of a fragmentary input x of which only the part Px is reliable. It is this associative mapping that we will exploit in applications of the PSOM. X out 3 D aeA;;S Figure 1: Best-match s* and associative completion w(s*(x)) of input Xl, X2 (Px) given in the input subspace Xin. Here in this simple case, the m = 1 dimensional manifold M is constructed to pass through four data vectors (square marked). The left side shows the d = 3 dimensional embedding space X = xin X X out and the right side depicts the best match parameter s* (x) parameter manifold S together with the "hyper-lattice" A of parameter values (indicated by white squares) belonging to the data vectors. M is constructed as a manifold that passes through a given set D of data examples (Fig. I depicts the situation schematically). To this end, we assign to each data sample a point a E Sand denote the associated data sample by Wa. The set A of the assigned parameter values a should provide a good discrete "model" of the topology of our data set (Fig. I right). The assignment between data vectors and points a must be made in a topology preserving fashion to ensure good interpolation by the manifold M that is obtained by the following steps. 572 1. WALTER, H. RITTER For each point a E A, we construct a "basis function" H(·, a; A) or simplified I H(·, a) : S ~ 1R that obeys ( i) H (ai, aj) = 1 for i = j and vanishes at all other points of A i =J j (orthonormality condition,) and (ii) EaEA H (a, s) = 1 for '<Is ("partition of unity" condition.) We will mainly be concerned with the case of A being a m-dimensional rectangular hyper-lattice; in this case, the functions H(·, a) can be constructed as products of Lagrange interpolation polynomials, see [11] . Then, W(s) = L H(s, a) Wa· aEA (2) defines a manifold M that passes through all data examples. Minimizing dist(·) in Eq. 1 can be done by some iterative procedure, such as gradient descent or - preferably - the Levenberg-Marquardt algorithm [11]. This makes M into the attractor manifold of a (discrete time) dynamical system. Since M contains the data set D, any at least m-dimensional "fragment" of a data example x = wED will be attracted to the correct completion w. Inputs x ¢ D will be attracted to some approximating manifold point. This approach is in many ways the continuous analog of the standard discrete selforganizing map. Particularly attractive features are (i) that the construction of the map manifold is direct from a small set of training vectors, without any need for time consuming adaptation sequences, (ii) the capability of associative completion, which allows to freely redefine variables as inputs or outputs (by changing dist(·) on demand, e.g. one can reverse the mapping direction), and (iii) the possibility of having attractor manifolds instead of just attractor points. 3 Hierarchical PSOMs: Structuring Learning Rapid learning requires that the structure of the learner is well matched to his task. However, if one does not want to pre-structure the learner by hand, learning again seems to be the only way to achieve the necessary pre-structuring. This leads to the idea of structuring learning itself and motivates to split learning into two stages: (i) The earlier stage is considered as an "investment stage" that may be slow and that may require a larger number of examples. It has the task to pre-structure the system in such a way that in the later stage, (ii) the now specialized system can learn fast and with extremely few examples. To be concrete, we consider specialized mappings or "skills", which are dependent on the state of the system or system environment. Pre-structuring the system is achieved by learning a set of basis mappings, each in a prototypical system context or environment state ("investment phase".) This imposes a strong need for an efficient learning tool- efficient in particular with respect to the number of required training data points. The PSOM networks appears as a very attractive solution: Fig. 2 shows a hierarchical arrangement of two PSOM. The task of mapping from input to output spaces is learned and performed, by the "Transformation-PSOM" ("T-PSOM"). During the first learning stage, the investment learning phase the T-PSOM is used to learn a set of basis mappings Tj : Xl +-t X2 or context dependent "skills" is constructed in the "T-PSOM", each of which gets encoded as a internal parameter or "weight" set Wj . The 1 In contrast to kernel methods, the basis functions may depend on the relative POSition to all other knots. However, we drop in our notation the dependency H (a, s) = H (a, s; A) on the latter. Investment Learning with Hierarchical PSOMs 573 Context .~~~~ .• (Meta-PSOM) ......... ~ CO. : weIghts -.. ---.~ ( T-P!OM ) Figure 2: The transforming ''T-PSOM'' maps between input and output spaces (changing direction on demand). In a particular environmental context, the correct transformation is learned, and encoded in the internal parameter or weight set w. Together with an characteristic environment observation Uref, the weight set w is employed as a training vector for the second level "Meta-PSOM". After learning a structured set of mappings, the Meta-PSOM is able to generalizing the mapping for a new environment. When encountering any change, the environment observation Uref gives input to the Meta-PSOM and determines the new weight set w for the basis T-PSOM. second level PSOM ("Meta-PSOM") is responsible for learning the association between the weight sets Wj of the first level T-PSOM and their situational contexts. The system context is characterized by a suitable environment observation, denoted ure/' see Fig. 2. The context situations are chosen such that the associated basis mappings capture already a significant amount of the underlying model structure, while still being sufficiently general to capture the variations with respect to which system environment identification is desired. For the training of the second level Meta-PSOM each constructed T-PSOM weight set Wj serves together with its associated environment observation ure/,j as a high dimensional training data vector. Rapid learning is the return on invested effort in the longer pre-training phase. As a result, the task of learning the "skill" associated with an unknown system context now takes the form of an immediate Meta-PSOM --+ T-PSOM mapping: the Meta-PSOM maps the new system context observation ure/,new into the parameter set Wnew for the T-PSOM. Equipped with Wnew , the T-PSOM provides the desired mapping Tnew. 4 Rapid Learning of a Stereo Visuo-motor Map In the following, we demonstrate the potential of the investment learning approach, with the task of fast learning of 3D vi suo-motor maps for a robot manipulator seen by a pair of movable cameras. Thus, in this demonstration, each situated context is given by a particular camera arrangement, and the assicuated "skill" is the mapping between camera and robot coordinates. The Puma robot is positioned behind a table and the entire scene is displayed on two windows on a computer monitor. By mouse-pointing, a user can, for example, select on the monitor one point and the position on a line appearing in the other window, to indicate a good position for the robot end effector, see Fig. 3. This requires to compute the transformation T between pixel coordinates U = (uL , uR ) on the monitor images and corresponding world coordinates if in the robot reference frame - or alternatively - the corresponding six robot joint angles (j (6 DOF). Here we demonstrate an integrated solution, offering both solutions with the same network. The T-PSOM learns each individual basis mapping Tj by visiting a rectangular grid set of end effector positions ei (here a 3x3x3 grid in if of size 40 x 40 x 30cm3) jointly with 574 J. WALTER, H. RITTER (OL weights Figure 3: Rapid learning of the 3D visuo-motor coordination for two cameras. The basis T-PSOM (m = 3) is capable of mapping to and from three coordinate systems: Cartesian robot world coordinates, the robot joint angles (6-DOF), and the location of the end-effector in coordinates of the two camera retinas. Since the left and right camera can be relocated independently, the weight set of T-PSOM is split, and parts W L, W R are learned in two separate Meta-PSOMs ("L" and "R"). the joint angle tuple ~ and the location in camera retina coordinates (2D in each camera) ut, uf· Thus the training vectors wai for the construction of the T-PSOM are the tuples ( ~ ~ -+L ..... R Xi, (}i, U i 'Ui ). However, each Tj solves the mapping task only for the current camera arrangement, for which Tj was learned. Thus there is not yet any particular advantage to other, specialized methods for camera calibration [1]. The important point is, that we now will employ the Meta-PSOM to interpolate in the space of the mappings {Tj }. To keep the number of prototype mappings manageable, we reduce some DOFs of the cameras by calling for fixed focal length. camera tripod height. and twist joint. To constrain the elevation and azimuth viewing angle. we require one land mark f.!ix to remain visible in a constant image position. This leaves two free parameters per camera, that can now be determined by one extra observation of a chosen auxiliary world reference point f.re!. We denote the camera image coordinates of f.re! by Ure! = (u~e! ' u::e!). By reuse of the cameras as "environment sensor", Ure! now implicitly encodes the two camera positions. In the investing pre-training phase, nine mappings Tj are learned by the T-PSOM, each camera visiting a 3 x 3 grid. sharing the set of visited robot positions f.i. As Fig. 2 suggests, normal1y the entire weight set w serves as part of the training vector to the Meta-PSOM. Here the problem becomes factorized since the left and right camera change tripod place independently: the weight set of the T-PSOM is split, and the two parts can be learned in separate Meta-PSOMs. Each training vector Waj for the left camera Meta-PSOM consists of the context observation u~e! and the T-PSOM weight set part w L = (uf,···, U~7) (analogous the right camera Meta-PSOM.) This enables in the following phase the rapid learning, for new. unknown camera places. On the basis of one single observation Ure!. the desired transformation T is constructed. As visualized in Fig. 3. Ure! serves as the input to the second level Meta-PSOMs. Their outputs are interpolations between previously learned weight sets and they project directly into the weight set of the basis level T-PSOM. The resulting T-PSOM can map in various directions. This is achieved by specifying a suitable distance function dist(·) via the projection matrix P, e.g.: i(u) DUI-tX ( .... I'T-PSOM u; (3) Investment Learning with Hierarchical PSOMs 8(u) u(x) wL(u~el ) FT~tSOM (u; wL( u~e/)' WR (u~/)) Ff~PSOM(X; wL(u~e/),WR(U~e/)) FM7t~-PSOM,L(U~e/; OL); analog WR(u~/) 575 (4) (5) (6) Table 1 shows experimental results averaged over 100 random locations ~ (from within the range of the training set) seen in 10 different camera set-ups, from within the 3 x 3 square grid of the training positions, located in a normal distance of about 125 cm (center to work space center, 1 m2 , total range of about 55-21Ocm), covering a disparity angle range of 25°-150°. For identification of the positions ~ in image coordinates, a tiny light source was installed at the manipulator tip and a simple procedure automated the finding of u with about ±0.8 pixel accuracy. For the achieved precision it is important to share the same set of robot positions ~i' and that the sets are topologically ordered, here as a 3x3x3 goal position grid (i) and two 3 x 3 camera location (j) grids. Direct trained T-PSOMwith Mapping Direction T-PSOM Meta-PSOM pixel Ul--t Xrobot => Cartesian error !:1x 1.4mm 0.008 4.4mm 0.025 Cartesian x I--t u => pixel error 1.2pix 0.010 3.3 pix 0.025 pixel Ul--t o"obot => Cartesian error !:1x 3.8mm 0.023 5.4mm 0.030 Table 1: Mean Euclidean deviation (mm or pixel) and normalized root mean square error (NRMS) for 1000 points total in comparison of a direct trained T-PSOM and the described hierarchical MetaPSOM network, in the rapid learning mode after one single observation. 5 Discussion and Conclusion A crucial question is how to structure systems, such that learning can be efficient. In the present paper, we demonstrated a hierarchical approach that is motivated by a decomposition of the learning phase into two different stages: A longer, initial learning phase "invests" effort into a gradual and domain-specific specialization of the system. This investment learning does not yet produce the final solution, but instead pre-structures the system such that the subsequently final specialization to a particular solution (within the chosen domain) can be achieved extremely rapidly. To implement this approach, we used a hierarchical architecture of mappings. While in principle various kinds of network types could be used for this mappings, a practically feasible solution must be based on a network type that allows to construct the required basis mappings from rather small number of training examples. In addition, since we use interpolation in weight space, similar mappings should give rise to similar weight sets to make interpolation meaningful. PSOM meat this requirements very well, since they allow a direct non-iterative construction of smooth mappings from rather small data sets. They achieve this be generalizing the discrete self-organizing map [3, 9] into a continuous map manifold such that interpolation for new data points can benefit from topology information that is not available to most other methods. While PSOMs resemble local models [4, 5, 6] in that there is no interference between different training points, their use of a orthogonal set of basis functions to construct the 576 J. WALTER, H. RIITER map manifold put them in a intennediate position between the extremes of local and of fully distributed models. A further very useful property in the present context is the ability of PSOMs to work as an attractor network with a continuous attractor manifold. Thus a PSOM needs no fixed designation of variables as inputs and outputs; Instead the projection matrix P can be used to freely partition the full set of variables into input and output values. Values of the latter are obtained by a process of associative completion. Technically, the investment learning phase is realized by learning a set of prototypical basis mappings represented as weight sets of a T-PSOM that attempt to cover the range of tasks in the given domain. The capability for subsequent rapid specialization within the domain is then provided by an additional mapping that maps a situational context into a suitable combination of the previously learned prototypical basis mappings. The construction of this mapping again is solved with a PSOM ("Meta"-PSOM) that interpolates in the space oJprototypical basis mappings that were constructed during the "investment phase". We demonstrated the potential of this approach with the task of 3D visuo-motor mapping, learn-able with a single observation after repositioning a pair of cameras. The achieved accuracy of 4.4 mm after learning by a single observation, compares very well with the distance range 0.5-2.1 m of traversed positions. As further data becomes available, the T-PSOM can certainly be fine-tuned to improve the perfonnance to the level of the directly trained T-PSOM. The presented arrangement of a basis T-PSOM and two Meta-PSOMs demonstrates further the possibility to split hierarchical learning in independently changing domain sets. When the number of involved free context parameters is growing, this factorization is increasingly crucial to keep the number of pre-trained prototype mappings manageable. References [1] K. Fu, R. Gonzalez and C. Lee. Robotics: Control, Sensing, Vision, and Intelligence. McGrawHill, 1987 [2] F. Girosi and T. Poggio. Networks and the best approximation property. BioI. Cybem., 63(3):169-176,1990. [3] T. Kohonen. Self-Organization and Associative Memory. Springer, Heidelberg, 1984. [4] 1. Moody and C. Darken. Fast learning in networks of locally-tuned processing units. Neural Computation, 1:281-294, 1989. [5] S. Omohundro. Bumptrees for efficient function, constraint, and classification learning. In NIPS*3, pages 693-699. Morgan Kaufman Publishers, 1991. [6] 1. Platt. A resource-allocating network for function interpolation. Neural Computation, 3:213255,1991 [7] M. Powell. Radial basis functions for multivariable interpolation: A review, pages 143-167. Clarendon Press, Oxford, 1987. [8] H. Ritter. Parametrized self-organizing maps. In S. Gielen and B. Kappen; editors, ICANN'93Proceedings, Amsterdam, pages 568-575. Springer Verlag, Berlin, 1993. [9] H. Ritter, T. Martinetz, and K. Schulten. Neural Computation and Self-organizing Maps. Addison Wesley, 1992. [10] 1. Walter and H. Ritter. Local PSOMs and Chebyshev PSOMs - improving the parametrised self-organizing maps. In Proc. ICANN, Paris, volume 1, pages 95-102, October 1995. [11] 1. Walter and H. Ritter. Rapid learning with parametrized self-organizing maps. Neurocomputing, Special Issue, (in press), 1996.
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How Perception Guides Production Birdsong Learning Christopher L. Fry cfry@cogsci.ucsd.edu Department of Cognitive Science University of California at San Diego La Jolla, CA 92093-0515 Abstract A c.:omputational model of song learning in the song sparrow (M elospiza melodia) learns to categorize the different syllables of a song sparrow song and uses this categorization to train itself to reproduce song. The model fills a crucial gap in the computational explanation of birdsong learning by exploring the organization of perception in songbirds. It shows how competitive learning may lead to the organization of a specific nucleus in the bird brain, replicates the song production results of a previous model (Doya and Sejnowski, 1995), and demonstrates how perceptual learning can guide production through reinforcement learning. 1 INTRODUCTION • In The passeriformes or songbirds make up more than half of all bird species and are divided into two groups: the os cines which learn their songs and sub-oscines which do not. Oscines raised in isolation sing degraded species typical songs similar to wild song. Deafened oscines sing completely degraded songs (Konishi, 1965) , while deafened sub-oscines develop normal songs (Kroodsma and Konishi, 1991) indicating that auditory feedback is crucial in oscine song learning. Innate structures in the bird brain regulate song learning. For example, song sparrows show innate preferences for their own species' songs and song structure (Marler, 1991). Innate preferences are thought to be encoded in an auditory template which limits the sounds young birds may copy. According to the auditory template hypothesis birds go through two phases during song learning, a memorization phase and a motor phase. In the memorization phase, which lasts from approximately 20 to 50 days after birth in the song sparrow, the bird selects which sounds to copy based on an innate template and refines the template based How Perception Guides Production in Birdsong Learning POSTERIOR 10 •• chea • SJrIn I ANTERIOR --+ ...... ing Pall..,. ~l'IodUclcn Pall ... , 111 Figure 1: A simplified sketch of a saggital section of the songbird brain. Field L (Field L) receives auditory input and projects to the production pathway: HVc (formerly the caudal nucleus of the hyperstriatum), RA (robust nucleus of archistriatum), nXIIts (hypoglossal nerve), the syrinx (vocal organ) and the learning pathway: X (area X), DLM (medial nucleus of the dorsolateral thalamus), LMAN (lateral magnocellular nucleus of the anterior neostriatum), RA (Konishi, 1989; Vicario, 1994). V is the lateral ventricle. on the sounds it hears. In the motor phase (from approximately 272 to 334 days after birth) the template provides feedback during singing. Learning to sing the memorized, template song is a gradual process of refining the produced song to match memory (Marler, 1991). A song is made up of phrases, phrases of syllables and syllables of notes. Syllables, usually separated by periods of silence, are the main units of analysis. Notes typically last from 10-100 msecs and are used to construct syllables (100-200 msecs) which are reused to produce trills and other phrases. 2 NEUROBIOLOGY OF SONG The two main neural pathways that govern song are the motor and learning pathways seen in figure 1 (Konishi, 1989). Lesions to the motor pathway interrupt singing throughout life while lesions to the learning pathway disrupt early song learning. Although these pathways seem to have segregated functions , recordings of neurons during song playback have shown that cells throughout the song system respond to song (Konishi, 1989). Studies of song perception have shown the best auditory stimulus that will evoke a response in the song system is the bird's own song (Margoliash, 1986). The song specific neurons in HV c of the white-crowned sparrow often require a sequence of two syllables to respond (Margoliash, 1986; Margoliash and Fortune, 1992) and are made up of two main types in HV c. One type is sensitive to temporal combinations of stimuli while the other is sensitive to harmonic characteristics (Margoliash and Fortune, 1992). 3 COMPUTATION Previous computational work on birdsong learning predicted individual neural responses using back-propagation (Margoliash and Bankes, 1993) and modelled motor mappings for song production (Doya and Sejnowski, 1995). The current work de112 C.L.FRY 1 2 00 8 Kohonen Neuron InpulLayer .. Sliding ~ _____ _ Windo"",. ~ 1000 Figure 2: Perceptual network input encoding. The song is converted into frequency bins which are presented to the Kohonen layer over four time steps. velops a model of birdsong syllable perception which extends Doya and Sejnowski's (1995) model of birdsong learning. Birdsong syllable segmentation is accomplished using an unsupervised system and this system is used to train the network to reproduce its input using reinforcement learning. The model implements the two phases of the auditory template hypothesis, memorization and motor. In the first phase the template song is segmented into syllables by an unsupervised Kohonen network (Kohonen, 1984). In the second phase the syllables are reproduced using a reinforcement learning paradigm based on Doya and Sejnowski (1995). The model extends previous work in three ways: 1) a self-organizing network picks out syllables in the song; 2) the self-organizing network provides feedback during song production; and 3) a more biologically plausible model of the syrinx is used to generate song. 3.1 Perception Recognizing a syllable involves identifying a short sequence of notes. Kohonen networks use an unsupervised learning method to categorize an input space based on similar neural responses. Thus a Kohonen network is a natural candidate for identifying the syllables in a song. One song from the repertoire of a song sparrow was chosen as the training song for the network. The song was encoded by passing a sliding window across the training waveform (sampled at 22.255 kHz) of the selected song. At each time step, a non-overlapping 256 point (~ .011 sec) fast fourier transform (FFT) was used to generate a power spectrum (figure 2). The power spectrum was divided into 8 bins. Each bin was mapped to a real number using a gaussian summation procedure with the peak of the gaussian at the center of each frequency bin. Four time-steps were passed to each Kohonen neuron. The network's task was to identify similar syllables in the input song. The input song was broken down into syllables by looking for points where the power at all How Perception Guides Production in Birdsong Learning 10 >. u .: " g. 5 " " ... a kH< s n1 " n2 CD .. n3 ii: S n4 .. n5 :I n6 CD z n7 n8 t -/>, 0.0 0.5 1.0 time 113 20 Figure 3: Categorization of song syllables by a Kohonen network. The power-spectrum of the training song is at the top. The responses of the Kohonen neurons are at the bottom. For each time-step the winning neuron is shown with a vertical bar. The shaded areas indicate the neuron that fired the most during the presentation of the syllable. frequencies dropped below a threshold. A syllable was defined as sound of duration greater than .011 seconds bounded by two low-power points. The network was not trained on the noise between syllables. The song was played for the network ten times (1050 training vectors), long enough for a stable response pattern to emerge. The activation of a neuron was: N etj = 'ExiWij' Where: N etj = output of neuron j , Wij = the weight connecting inputi to neuronj , Xi = inputi. The Kohonen network was trained by initializing the connection weights to 1/Jnumber of neurons + small random component (r S; .01), normalizing the inputs, and updating the weights to the winning neuron by the following rule: W n ew = W old + a(x W old) where: a = training rate = .20. If the same neuron won twice in a row the training rate was decreased by 1/2. Only the winning neuron was reinforced resulting in a non-localized feature map. 3.1.1 Perceptual Results The Kohonen network was able to assign a unique neuron to each type of syllable (figure 3). Of the eight neurons in the network. the one that fired the most frequently during the presentation of a syllable uniquely identified the type of syllable. The first four syllables of the input song sound alike, contain similar frequencies, and are coded by the first neuron (N1). The last three syllables sound alike, contain similar frequencies , and are coded by the fourth neuron (N4). Syllable five was coded by neuron six (N6), syllable six by neuron two (N2) and syllable seven by neuron eight (N8). Figure 4 shows the frequency sensitivity of each neuron (1-8, figure 3) plotted against each time step (1-4). This plot shows the harmonic and temporally sensitive neurons that developed during the learning phase of the Kohonen network. Neuron 2 is sensitive to only one frequency at approximately 6-7 kHz, indicated by the solid white band across the 6-7 kHz frequency range in figure 4. Neuron 4 is sensitive to mid-range frequencies of short duration. Note that in figure 4 N4 responds 114 N5 o 1 2 3 4 N6 01 23 4 N7 Time S t ep C. L. FRY N8 Figure 4: The values of the weights mapping frequency bins and time steps to Kohonen neurons. White is maximum, Black is minimum. maximally to mid-range frequencies only in the first two time steps. It uses this temporal sensitivity to distinguish between the last three syllables and the fifth syllable (figure 3) by keying off the length of time mid-range frequencies are present. Contrast this early response sensitivity with neuron 6, which is sensitive to midrange frequencies of long duration, but responds only after one time step. It uses this temporal sensitivity to respond to the long sustained frequency of syllable four . Considered together, neurons 2,4,6 and 8 illustrate the two types of neurons (temporal and harmonic) found in HVc by Margoliash and Fortune (1993). Competitive learning may underly the formation of these neurons in HV c. 3.2 Production After competitive learning trains the perceptual part of the network to categorize the song into syllables, the perceptual network can be used to train the production side of the network to sing. The first step in modelling song production is to create a model of the avian vocal apparatus, the syrinx. In the syrinx sounds arise when air flows through the syringeal passage and causes the tympanic membrane to vibrate. The frequency is controlled by the tension of the membrane controlled by the syringeal musculature. The amplitude is dependent on the area of the syringeal orifice which is dependent on the tension of the labium. The interactions of this system were modelled by modulated sine waves. Four parameters governed the fundamental frequency(p) , frequency modulation(tm), amplitude (ex) and frequency of amplitude modulation(I). The range of the parameters was set according to calculations in Greenwalt (1968). The parameters were combined in the following equation (based on Greenwalt, 1968), f(ex , l,p, tm, t) = excos(21l"t 1) cos(21l"t p + cos(21l"t tm)) . Using this equation song can be generated over time by making assumptions about the response properties of neurons in RA. Following Doya and Sejnowski (1995) it was assumed that pools of RA neurons have different temporal response profiles. Syllable like temporal responses can be generated by modifying the weights from the Kohonell layer (HV c) to the production layer (RA). How Perception Guides Production in Birdsong Learning kHz 10 Tnnni~ Song , .f . ·~:if>vr. ... i i '0 15 Tim. . .. . I 20 J'etworl< Song trained with Spectmgmm Target Ti'llll! J'etworl< Song trained with J'euraJ. Activation Target S a a as 10 15 20 115 Figure 5: Training song and two songs produced with different representations of the training song. The production side of the network was trained using the reinforcement learning paradigm described in Doya and Sejnowski (1995). Each syllable was presented in the order it occurred in the training song to the Kohonen layer, which turned on a single neuron. A random vector was added to the weights from the Kohonen layer to the output layer and a syllable was produced. The produced syllable was compared to the stored representation of the template song which was used to generate an error signal and an estimate of the gradient. If the evaluation of the produced syllable was better than a threshold the weights were kept, otherwise they were discarded. Two experiments were done using different representations of the template song. In the first experiment the template song was the stored power spectrum of each syllable and the error signal was the cosine of the angle between the power spectrum of the produced syllable and the template syllable. In the second experiment the template song was the stored neural responses to song (recorded during the memorization phase) and the error signal was the Euclidean distance between neural responses to the produced syllable and the neural responses to the template song. 3.2.1 Production Results Figure 5 shows the output of the production network after training with different representations of the training song. The network was able to replicate the major frequency components of the training song to a high degree of accuracy. The song trained with the spectrogram target was learned to a 90% average cosine between the spectrograms of the produced song and the training song on each syllable with the best syllable learned to 100% accuracy and the worst to 85% after 1000 trials. A crucial aspect to achieving performance was smoothing the template spectrogram. The third song shows that the network was able to learn the template song using the neural responses of the perceptual system to generate the reinforcement signal. The average distance between the initial randomly produced syllables and the training 116 C. L.FRY song was reduced by 50%. 4 DISCUSSION This work fills a crucial gap in the computational explanation of song learning left by prior work. Doya and Sejnowski (1995) showed how song could be produced but left unanswered the questions of how song is perceived and how the perceptual system provides feedback during song production. This study shows a time-delay Kohonen network can learn to categorize the syllables of a sample song and this network can train song production with no external teacher. The Kohonen network explains how neurons sensitive to temporal and harmonic structure could arise in the songbird brain through competitive learning. Taken as a whole, the model presents a concrete proposal of the computational principles governing the Auditory Template Hypothesis and how a song is memorized and used to train song production. Future work will flesh out the effects of innate structure on learning by examining how the settings of the initial weights on the network affect song learning and predict experimental effects of deafening and isolation. Acknowledgements Thanks to S. Vehrencamp for providing the song data, J . Batali, J. Elman, J. Bradbury and T. Sejnowski for helpful comments, and K. Doya for advice on replicating his model. References Doya, K. and Sejnowski, T .J. (1995). A novel reinforcement model of bird song vocalization learning. In Tesauro, G., Touretzky, D. S. and Leen, T.K., editors, Advances in Neural Information Processing Systems 7. MIT Press, Cambridge, MA. Greenwalt, C.H. (1968). Bird Song: Acoustics and Physiology. Smithsonian Institution Press. Wash., D.C. Kohonen, T . (1984). Self-organization and Associative Memory, Vol. 8. Springer-Verlag, Berlin. Konishi, M. (1965). The role of auditory feedback in the control of vocalization in the white-crowned sparrow. Zeitschrijt fur Tierpsychogie, 22,770-783. Konishi, M. (1989). Birdsong for Neurobiologists. Neuron, 3, 541-549. Kroodsma, D.E. and Konishi, M. (1991). A suboscine bird (eastern phoebe, Sayonoris phoebe) develops normal song without auditory feedback. Animal Behavior, 42, 477-487. Marler, P. (1991). The instinct to learn. In The Epigenesis of Mind: Essays on Biology and Cognition, eds. S. Carey and R. Gelman. Lawrence Erlbaum Associates. Margoliash, D. (1986). Preference for autogenous song by auditory neurons in a song system nucleus of the white-crowned sparrow. Journal of Neuroscience, 6,1643-1661. Margoliash, D. and Bankes, S.C. (1993). Computations in the Ascending Auditory Pathway in Songbirds Related to Song Learning. American Zoologist, 33, 94-103. Margoliash, D. and Fortune, E. (1992). Temporal and Harmonic Combination-Sensitive Neurons in the Zebra Finch's HVc. Journal of Neuroscience, 12, 4309-4326. Vicario, D. (1994). Motor Mechanisms Relevant to Auditory-Vocal Interactions in Songbirds. Brain, Behavior and Evolution,44, 265-278.
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Stable Fitted Reinforcement Learning Geoffrey J. Gordon Computer Science Department Carnegie Mellon University Pittsburgh PA 15213 ggordon@cs.cmu.edu Abstract We describe the reinforcement learning problem, motivate algorithms which seek an approximation to the Q function, and present new convergence results for two such algorithms. 1 INTRODUCTION AND BACKGROUND Imagine an agent acting in some environment. At time t, the environment is in some state Xt chosen from a finite set of states. The agent perceives Xt, and is allowed to choose an action at from some finite set of actions. The environment then changes state, so that at time (t + 1) it is in a new state Xt+1 chosen from a probability distribution which depends only on Xt and at. Meanwhile, the agent experiences a real-valued cost Ct, chosen from a distribution which also depends only on Xt and at and which has finite mean and variance. Such an environment is called a Markov decision process, or MDP. The reinforcement learning problem is to control an MDP to minimize the expected discounted cost Lt ,tCt for some discount factor, E [0,1]. Define the function Q so that Q(x, a) is the cost for being in state x at time 0, choosing action a, and behaving optimally from then on. If we can discover Q, we have solved the problem: at each step, we may simply choose at to minimize Q(xt, at). For more information about MDPs, see (Watkins, 1989, Bertsekas and Tsitsiklis, 1989). We may distinguish two classes of problems, online and offline. In the offline problem, we have a full model of the MDP: given a state and an action, we can describe the distributions of the cost and the next state. We will be concerned with the online problem, in which our knowledge of the MDP is limited to what we can discover by interacting with it. To solve an online problem, we may approximate the transition and cost functions, then proceed as for an offline problem (the indirect approach); or we may try to learn the Q function without the intermediate step (the direct approach). Either approach may work better for any given problem: the Stable Fitted Reinforcement Learning 1053 direct approach may not extract as much information from each observation, but the indirect approach may introduce additional errors with its extra approximation step. We will be concerned here only with direct algorithms. Watkins' (1989) Q-Iearning algorithm can find the Q function for small MDPs, either online or offline. Convergence with probability 1 in the online case was proven in (Jaakkola et al., 1994, Tsitsiklis, 1994). For large MDPs, exact Q-Iearning is too expensive: representing the Q function requires too much space. To overcome this difficulty, we may look for an inexpensive approximation to the Q function. In the offline case, several algorithms for this purpose have been proven to converge (Gordon, 1995a, Tsitsiklis and Van Roy, 1994, Baird, 1995). For the online case, there are many fewer provably convergent algorithms. As Baird (1995) points out, we cannot even rely on gradient descent for large, stochastic problems, since we must observe two independent transitions from a given state before we can compute an unbiased estimate of the gradient. One of the algorithms in (Tsitsiklis and Van Roy, 1994), which uses state aggregation to approximate the Q function, can be modified to apply to online problems; the resulting algorithm, unlike Q-Iearning, must make repeated small updates to its control policy, interleaved with comparatively lengthy periods of evaluation of the changes. After submitting this paper, we were advised of the paper (Singh et al., 1995), which contains a different algorithm for solving online MDPs. In addition, our newer paper (Gordon, 1995b) proves results for a larger class of approximators. There are several algorithms which can handle restricted versions of the online problem. In the case of a Markov chain (an MDP where only one action is available at any time step), Sutton's TD('\') has been proven to converge for arbitrary linear approximators (Sutton, 1988, Dayan, 1992). For decision processes with linear transition functions and quadratic cost functions (the so-called linear quadratic regulation problem), the algorithm of (Bradtke, 1993) is guaranteed to converge. In practice, researchers have had mixed success with approximate reinforcement learning (Tesauro, 1990, Boyan and Moore, 1995, Singh and Sutton, 1996). The remainder of the paper is divided into four sections. In section 2, we summarize convergence results for offline Q-Iearning, and prove some contraction properties which will be useful later. Section 3 extends the convergence results to online algorithms based on TD(O) and simple function approximators. Section 4 treats nondiscounted problems, and section 5 wraps up. 2 OFFLINE DISCOUNTED PROBLEMS Standard offline Q-Iearning begins with an MDP M and an initial Q fUnction q(O) . Its goal is to learn q(n), a good approximation to the optimal Q function for M. To accomplish this goal, it performs the series of updates q(i+1) = TM(q(i»), where the component of TM(q(i») corresponding to state x and action a is defined to be [T ( (i) )] ~ p . (i) M q xa = Cxa + "t ~ xay mln qyb y Here Cxa is the expected cost of performing action a in state x; Pxay is the probability that action a from state x will lead to state y; and"t is the discount factor. Offline Q-Iearning converges for discounted MDPs because TM is a contraction in max norm. That is, for all vectors q and r, II TM(q) - TM(r) II ~ "til q - r II where II q 1\ == maxx,a I qxa I· Therefore, by the contraction mapping theorem, TM has a unique fixed point q* , and the sequence q(i) converges linearly to q* . 1054 G.J.OORDON It is worth noting that a weighted version of offline Q-Iearning is also guaranteed to converge. Consider the iteration q(i+l) = (I + aD(TM - I))(q(i)) where a is a positive learning rate and D is an arbitrary fixed nonsingular diagonal matrix of weights. In this iteration, we update some Q valnes more rapidly than others, as might occur if for instance we visited some states more frequently than others. (We will come back to this possibility later.) This weighted iteration is a max norm contraction, for sufficiently small a: take two Q functions q and r, with II q - r II = I. Suppose a is small enough that the largest element of aD is B < 1, and let b > 0 be the smallest diagonal element of aD. Consider any state x and action a, and write dxa for the corresponding element of aD. We then have [(1 - aD)q - (1 - aD)r]xa [TMq - TMr]xa [aDTMq - aDTMr]xa [(I - aD + aDTM)q - (1 - aD + aDTM )r]xa < (1 - dxa)1 < ,I < dxa,l < (1 - dxa)1 + dxa,l < (l-b(l-,))1 so (1 - aD + aDTM) is a max norm contraction with factor (1 - b(l - ,)). The fixed point of weighted Q-Iearning is the same as the fixed point of unweighted Q-Iearning: TM(q*) = q* is equivalent to aD(TM - l)q* = O. The difficulty with standard (weighted or unweighted) Q-Iearning is that, for MDPs with many states, it may be completely infeasible to compute TM(q) for even one value of q. One way to avoid this difficulty is fitted Q-Iearning: if we can find some function MA so that MA 0 TM is much cheaper to compute than TM, we can perform the fitted iteration q(Hl) = MA(TM(q(i))) instead of the standard offline Qlearning iteration. The mapping MA implements a function approximation scheme (see (Gordon, 1995a)); we assume that qeD) can be represented as MA(q) for some q. The fitted offline Q-Iearning iteration is guaranteed to converge to a unique fixed point if MA is a nonexpansion in max norm, and to have bounded error if MA(q*) is near q* (Gordon, 1995a). Finally, we can define a fitted weighted Q-Iearning iteration: q(Hl) = (1 + aMAD(TM - I))(q(i)) If MA is a max norm nonexpansion and M1 = MA (these conditions are satisfied, for example, by state aggregation), then fitted weighted Q-Iearning is guaranteed to converge: ((1 - MA) + MA(1 + aD(TM - I)))q MA(1 + aD(TM - 1)))q since MAq = q for q in the range of MA. (Note that q(i+l) is guaranteed to be in the range of MA if q(i) is.) The last line is the composition of a max norm nonexpansion with a max norm contraction, and so is a max norm contraction. The fixed point of fitted weighted Q-Iearning is not necessarily the same as the fixed point of fitted Q-Iearning, unless MA can represent q* exactly. However, if MA is linear, we have that (1 + aMAD(TM - I))(q + c) = c + MA(I + aD(TM - I)))(q + c) for any q in the range of MA and c perpendicular to the range of MA. In particular, if we take c so that q* - c is in the range of MA, and let q = MAq be a fixed point Stable Fitted Reinforcement Learning 1055 of the weighted fitted iteration, then we have II (I + aMAD(TM - I))q* - (I + aMAD(TM - I))q II < II c + MA(I + aD(TM - I)))q* - MA(I + aD(TM - I)))q II II c II + (1 - b(l - ,))11 q* - q II II q* - q II < IIcll b(l -,) That is, if MA is linear in addition to the conditions for convergence, we can bound the error for fitted weighted Q-Iearning. For offline problems, the weighted version of fitted Q-Iearning is not as useful as the unweighted version: it involves about the same amount of work per iteration, the contraction factor may not be as good, the error bound may not be as tight, and it requires M1 = MA in addition to the conditions for convergence of the unweighted iteration. On the other hand, as we shall see in the next section, the weighted algorithm can be applied to online problems. 3 ONLINE DISCOUNTED PROBLEMS Consider the following algorithm, which is a natural generalization of TD(O) (Sutton, 1988) to Markov decision problems. (This algorithm has been called "sarsa" (Singh and Sutton, 1996).) Start with some initial Q function q(O). Repeat the following steps for i from 0 onwards. Let 1l'(i) be a policy chosen according to some predetermined tradeoff between exploration and exploitation for the Q function q(i). Now, put the agent in M's start state and allow it to follow the policy 1l'(i) for a random number of steps L(i) . If at step t of the resulting trajectory the agent moves from the state Xt under action at with cost Ct to a state Yt for which the action bt appears optimal, compute the estimated Bellman error - ( + [(i) 1 ) [( i) 1 et Ct , q Ytbt q Xtat After observing the entire trajectory, define e(i) to be the vector whose xa-th component is the sum of et for all t such that Xt = x and at = a. Then compute the next weight vector according to the TD(O)-like update rule with learning rate a(i) q(i+l) = q(i) + a (i) MAe(i) See (Gordon, 1995b) for a comment on the types of mappings MA which are appropriate for online algorithms. We will assume that L(i) has the same distribution for all i and is independent of all other events related to the i-th and subsequent trajectories, and that E(L(i») is bounded. Define d~il to be the expected number of times the agent visited state x and chose action a during the i-th trajectory, given 1l'(i). We will assume that the policies are such that d~il > € for some positive € and for all i, x, and a. Let D(i) be the diagonal matrix with elements d~il. With this notation, we can write the expected update for the sarsa algorithm in matrix form: E(q(i+l) I q(i») = (I + a(i) MAD(i)(TM - I))q(i) With the exception of the fact that D(i) changes from iteration to iteration, this equation looks very similar to the offline weighted fitted Q-Iearning update. However, the sarsa algorithm is not guaranteed to converge even in the benign case 1056 G. J. GORDON (a) (b) Figure 1: A counterexample to sarsa. (a) An MDP: from the start state, the agent may choose the upper or the lower path, but from then on its decisions are forced. Next to each arc is its expected cost; the actual costs are randomized on each step. Boxed pairs of arcs are aggregated, so that the agent must learn identical Q values for arcs in the same box. We used a discount, = .9 and a learning rate a = .1. To ensure sufficient exploration, the agent chose an apparently suboptimal action 10% of the time. (Any other parameters would have resulted in similar behavior. In particular, annealing a to zero wouldn't have helped.) (b) The learned Q value for the right-hand box during the first 2000 steps. where the Q-function is approximated by state aggregation: when we apply sarsa to the MDP in figure 1, one of the learned Q values oscillates forever. This problem happens because the frequency-of-update matrix D(i) can change discontinuously when the Q function fluctuates slightly: when, by luck, the upper path through the MDP appears better, the cost-l arc into the goal will be followed more often and the learned Q value will decrease, while when the lower path appears better the cost-2 arc will be weighted more heavily and the Q value will increase. Since the two arcs out of the initial state always have the same expected backed-up Q value (because the states they lead to are constrained to have the same value), each path will appear better infinitely often and the oscillation will continue forever. On the other hand, if we can represent the optimal Q function q*, then no matter what D(i) is, the expected sarsa update has its fixed point at q*. Since the smallest diagonal element of D(i) is bounded away from zero and the largest is bounded above, we can choose an a and a " < 1 so that (I + aMAD(i)(TM I)) is a contraction with fixed point q* and factor " for all i. Now if we let the learning rates satisfy Ei a(i) = 00 and Ei(a(i»)2 < 00, convergencew.p.l to q* is guaranteed by a theorem of (Jaakkola et al., 1994). (See also the theorem in (Tsitsiklis, 1994).) More generally, if MA is linear and can represent q* - c for some vector c, we can bound the error between q* and the fixed point of the expected sarsa update on iteration i: if we choose an a and a " < 1 as in the previous paragraph, II E(q(Hl) I q(i») - q* II ~ ,'II q(i) - q* II + 211 ell for all i. A minor modification of the theorem of (Jaakkola et al., 1994) shows that the distance from q(i) to the region { q III q - q* II ~ 211 c 111 ~ " } converges w.p.l to zero. That is, while the sequence q(i) may not converge, the worst it will do is oscillate in a region around q* whose size is determined by how Stable Fitted Reinforcement Learning 1057 accurately we can represent q* and how frequently we visit the least frequent (state, action) pair. Finally, if we follow a fixed exploration policy on every trajectory, the matrix D( i) will be the same for every i; in this case, because of the contraction property proved in the previous section, convergence w.p.1 for appropriate learning rates is guaranteed again by the theorem of (Jaakkola et al., 1994). 4 NONDISCOUNTED PROBLEMS When M is not discounted, the Q-Iearning backup operator TM is no longer a max norm contraction. Instead, as long as every policy guarantees absorption w.p.1 into some set of cost-free terminal states, TM is a contraction in some weighted max norm. The proofs of the previous sections still go through, if we substitute this weighted max norm for the unweighted one in every case. In addition, the random variables L(i) which determine when each trial ends may be set to the first step t so that Xt is terminal, since this and all subsequent steps will have Bellman errors of zero. This choice of L(i) is not independent of the i-th trial, but it does have a finite mean and it does result in a constant D(i). 5 DISCUSSION We have proven new convergence theorems for two online fitted reinforcement learning algorithms based on Watkins' (1989) Q-Iearning algorithm. These algorithms, sarsa and sarsa with a fixed exploration policy, allow the use of function approximators whose mappings MA are max norm nonexpansions and satisfy M~ = MA. The prototypical example of such a function approximator is state aggregation. For similar results on a larger class of approximators, see (Gordon, 1995b). Acknowledgements This material is based on work supported under a National Science Foundation Graduate Research Fellowship and by ARPA grant number F33615-93-1-1330. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the National Science Foundation, ARPA, or the United States government. References L. Baird. Residual algorithms: Reinforcement learning with function approximation. In Machine Learning (proceedings of the twelfth international conference), San Francisco, CA, 1995. Morgan Kaufmann. D. P. Bertsekas and J. N. Tsitsiklis. Parallel and Distributed Computation: Numerical Methods. Prentice Hall, 1989. J. A. Boyan and A. W. Moore. Generalization in reinforcement learning: safely approximating the value function. In G. Tesauro and D. Touretzky, editors, Advances in Neural Information Processing Systems, volume 7. Morgan Kaufmann, 1995. S. J. Bradtke. Reinforcement learning applied to linear quadratic regulation. In S. J. Hanson, J. D. Cowan, and C. L. Giles, editors, Advances in Neural Information Processing Systems, volume 5. Morgan Kaufmann, 1993. P. Dayan. The convergence of TD(A) for general lambda. Machine Learning, 8(34):341-362, 1992. 1058 G. J. GOROON G. J. Gordon. Stable function approximation in dynamic programming. In Machine Learning (proceedings of the twelfth international conference), San Francisco, CA, 1995. Morgan Kaufmann. G. J. Gordon. Online fitted reinforcement learning. In J. A. Boyan, A. W. Moore, and R. S. Sutton, editors, Proceedings of the Workshop on Value Function Approximation, 1995. Proceedings are available as tech report CMU-CS-95-206. T. Jaakkola, M.I. Jordan, and S. P. Singh. On the convergence of stochastic iterative dynamic programming algorithms. Neural Computation, 6(6):1185- 1201, 1994. S. P. Singh, T. Jaakkola, and M. I. Jordan. Reinforcement learning with soft state aggregation. In G. Tesauro and D. Touretzky, editors, Advances in Neural Information Processing Systems, volume 7. Morgan Kaufmann, 1995. S. P. Singh and R. S. Sutton. Reinforcement learning with replacing eligibility traces. Machine Learning, 1996. R. S. Sutton. Learning to predict by the methods of temporal differences. Machine Learning, 3(1):9- 44, 1988. G. Tesauro. Neurogammon: a neural network backgammon program. In IJCNN Proceedings III, pages 33-39, 1990. J. N. Tsitsiklis and B. Van Roy. Feature-based methods for large-scale dynamic programming. Technical Report P-2277, Laboratory for Information and Decision Systems, 1994. J. N. Tsitsiklis. Asynchronous stochastic approximation and Q-Iearning. Machine Learning, 16(3):185-202, 1994. C. J. C. H. Watkins. Learning from Delayed Rewards. PhD thesis, King's College, Cambridge, England, 1989.
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Information through a Spiking Neuron Charles F. Stevens and Anthony Zador Salk Institute MNL/S La J olIa, CA 92037 zador@salk.edu Abstract While it is generally agreed that neurons transmit information about their synaptic inputs through spike trains, the code by which this information is transmitted is not well understood. An upper bound on the information encoded is obtained by hypothesizing that the precise timing of each spike conveys information. Here we develop a general approach to quantifying the information carried by spike trains under this hypothesis, and apply it to the leaky integrate-and-fire (IF) model of neuronal dynamics. We formulate the problem in terms of the probability distribution peT) of interspike intervals (ISIs), assuming that spikes are detected with arbitrary but finite temporal resolution. In the absence of added noise, all the variability in the ISIs could encode information, and the information rate is simply the entropy of the lSI distribution, H (T) = (-p(T) log2 p(T)}, times the spike rate. H (T) thus provides an exact expression for the information rate. The methods developed here can be used to determine experimentally the information carried by spike trains, even when the lower bound of the information rate provided by the stimulus reconstruction method is not tight. In a preliminary series of experiments, we have used these methods to estimate information rates of hippocampal neurons in slice in response to somatic current injection. These pilot experiments suggest information rates as high as 6.3 bits/spike. 1 Information rate of spike trains Cortical neurons use spike trains to communicate with other neurons. The output of each neuron is a stochastic function of its input from the other neurons. It is of interest to know how much each neuron is telling other neurons about its inputs. How much information does the spike train provide about a signal? Consider noise net) added to a signal set) to produce some total input yet) = set) + net). This is then passed through a (possibly stochastic) functional F to produce the output spike train F[y(t)] --+ z(t). We assume that all the information contained in the spike train can be represented by the list of spike times; that is, there is no extra information contained in properties such as spike height or width. Note, however, that many characteristics of the spike train such as the mean or instantaneous rate 76 C. STEVENS, A. ZADOR can be derived from this representation; if such a derivative property turns out to be the relevant one, then this formulation can be specialized appropriately. We will be interested, then, in the mutual information 1(S(t); Z(t» between the input signal ensemble S(t) and the output spike train ensemble Z(t) . This is defined in terms of the entropy H(S) of the signal, the entropy H(Z) of the spike train, and their joint entropy H(S, Z), 1(S; Z) = H(S) + H(Z) - H(S, Z). (1) Note that the mutual information is symmetric, 1(S; Z) = 1(Z; S), since the joint entropy H(S, Z) = H(Z, S). Note also that if the signal S(t) and the spike train Z(t) are completely independent, then the mutual information is 0, since the joint entropy is just the sum of the individual entropies H(S, Z) = H(S) + H(Z). This is completely in lin'e with our intuition, since in this case the spike train can provide no information about the signal. 1.1 Information estimation through stimulus reconstruction Bialek and colleagues (Bialek et al., 1991) have used the reconstruction method to obtain a strict lower bound on the mutual information in an experimental setting. This method is based on an expression mathematically equivalent to eq. (1) involving the conditional entropy H(SIZ) of the signal given the spike train, 1(S; Z) H(S) - H(SIZ) > H(S) - Hest(SIZ), (2) where Hest(SIZ) is an upper bound on the conditional entropy obtained from a reconstruction sest< t) of the signal. The entropy is estimated from the second order statistics of the reconstruction error e(t) ~ s(t)-sest (t); from the maximum entropy property of the Gaussian this is an upper bound. Intuitively, the first equation says that the information gained about the spike train by observing the stimulus is just the initial uncertainty of the signal (in the absence of knowledge of the spike train) minus the uncertainty that remains about the signal once the spike train is known, and the second equation says that this second uncertainty must be greater for any particular estimate than for the optimal estimate. 1.2 Information estimation through spike train reliability We have adopted a different approach based an equivalent expression for the mutual information: 1(S; Z) = H(Z) - H(ZIS). (3) The first term H(Z) is the entropy of the spike train, while the second H(ZIS) is the conditional entropy of the spike train given the signal; intuitively this like the inverse repeatability of the spike train given repeated applications of the same signal. Eq. (3) has the advantage that, if the spike train is a deterministic function of the input, it permits exact calculation of the mutual information. This follows from an important difference between the conditional entropy term here and in eq. 2: whereas H(SIZ) has both a deterministic and a stochastic component, H(ZIS) has only a stochastic component. Thus in the absence of added noise, the discrete entropy H(ZIS) = 0, and eq. (3) reduces to 1(S; Z) = H(Z). If ISIs are independent, then the H(Z) can be simply expressed in terms of the entropy of the (discrete) lSI distribution p(T), 00 H(T) = - LP(1'i) 10g2P(1'i) (4) i=O Infonnation Through a Spiking Neuron 77 as H(Z) = nH(T), where n is the number of spikes in Z. Here p('li) is the probability that the spike occurred in the interval (i)~t to (i + l)~t. The assumption of finite timing precision ~t keeps the potential information finite. The advantage of considering the lSI distribution peT) rather than the full spike train distribution p(Z) is that the former is univariate while the latter is multivariate; estimating the former requires much less data. Under what conditions are ISIs independent? Correlations between ISIs can arise either through the stimulus or the spike generation mechanism itself. Below we shall guarantee that correlations do not arise from the spike-generator by considering the forgetful integrate-and-fire (IF) model, in which all information about the previous spike is eliminated by the next spike. If we further limit ourselves to temporally uncorrelated stimuli (i. e. stimuli drawn from a white noise ensemble), then we can be sure that ISIs are independent, and eq. (4) can be applied. In the presence of noise, H(ZIT) must also be evaluated, to give f(S; T) = H(T) - H(TIS). (5) H(TIS) is the conditional entropy of the lSI given the signal, H(TIS) = - / t p(1j ISi(t)) log2 p(1j ISi(t))) \J=1 3;(t) (6) where p(1j ISi(t)) is the probability of obtaining an lSI of 1j in response to a particular stimulus Si(t) in the presence of noise net). The conditional entropy can be thought of as a quantification of the reliability of the spike generating mechanism: it is the average trial-to-trial variability of the spike train generated in response to repeated applications of the same stimulus. 1.3 Maximum spike train entropy In what follows, it will be useful to compare the information rate for the IF neuron with the limiting case of an exponential lSI distribution, which has the maximum entropy for any point process of the given rate (Papoulis, 1984). This provides an upper bound on the information rate possible for any spike train, given the spike rate and the temporal precision. Let f(T) = re-rr be an exponential distribution with a mean spike rate r. Assuming a temporal precision of ~t, the entropy/spike is H(T) = log2 r~t' and the entropy/time for a rate r is rH(T) = rlog2 -~ . For example, if r = 1 Hz and ~t = 0.001 sec, this gives (11.4 bits/second) (1 spike/second) = 11.4 bits/spike. That is, if we discretize a 1 Hz spike train into 1 msec bins, it is nof possible for it to transmit more than 11.4 bits/second. If we reduce the bin size two-fold, the rate increases by log2 1/2 = 1 bit/spike to 12.4 bits/spike, while if we double it we lose one bit/s to get 10.4 bit/so Note that at a different firing rate, e.g. r = 2 Hz, halving the bin size still increases the entropy/spike by 1 bit/spike, but because the spike rate is twice as high, this becomes a 2 bit/second increase in the information rate. 1.4 The IF model Now we consider the functional :F describing the forgetful leaky IF model of spike generation. Suppose we add some noise net) to a signal set), yet) = net) + set), and threshold the sum to produce a spike train z(t) = :F[s(t) + net)]. Specifically, suppose the voltage vet) of the neuron obeys vet) = -v(t)/r + yet), where r is the membrane time constant, both s(t~ and net) have a white Gaussian distributions and yet) has mean I' and variance (T • If the voltage reaches the threshold ()o at some time t, the neuron emits a spike at that time and resets to the initial condition Vo. 78 c. STEVENS, A. ZAOOR In the language of neurobiology, this model can be thought of (Tuckwell, 1988) as the limiting case of a neuron with a leaky IF spike generating mechanism receiving many excitatory and inhibitory synaptic inputs. Note that since the input yet) is white, there are no correlations in the spike train induced by the signal, and since the neuron resets after each spike there are no correlations induced by the spikegenerating mechanism. Thus ISIs are independent, and eq. (4) r.an be applied. We will estimate the mutual information I(S, Z) between the ensemble of input signals S and the ensemble of outputs Z. Since in this model ISIs are independent by construction, we need only evaluate H(T) and H(TIS); for this we must determine p(T), the distribution of ISIs, and p(Tlsi), the conditional distribution of ISIs for an ensemble of signals Si(t). Note that peT) corresponds to the first passage time distribution of the Ornstein-Uhlenbeck process (Tuckwell, 1988). The neuron model we are considering has two regimes determined by the relation of the asymptotic membrane potential (in the absence of threshold) J.l.T and the threshold (J. In the suprathreshold regime, J.l.T > (J, threshold crossings occur even if the signal variance is zero (0-2 = 0). In the subthreshold regime, J.l.T ~ (J, threshold crossings occur only if 0-2 > O. However, in the limit that E{T} ~ T, i.e. the mean firing rate is low compared with the integration time constant (this can only occur in the subthreshold regime), the lSI distribution is exponential, and its coefficient of variation (CV) is unity (cf. (Softky and Koch, 1993)). In this low-rate regime the firing is deterministically Poisson; by this we mean to distinguish it from the more usual usage of Poisson neuron, the stochastic situation in which the instantaneous firing rate parameter (the probability of firing over some interval) depends on the stimulus (i.e. f ex: set)). In the present case the exponential lSI distribution arises from a deterministic mechanism. At the border between these regimes, when the threshold is just equal to the asymptotic potential, (Jo = J.l.T, we have an explicit and exact solution for the entire lSI distribution (Sugiyama et al., 1970) peT) = (J.l.T)(T/2)-3/2 [e2T1T _ 1]-3/2exp(2T/T _ (J.l.T? ). (7) (211")1/20(0-2T)(e2TIT - 1) This is the special case where, in the absence of fluctuations (0-2 = 0), the membrane potential hovers just subthreshold. Its neurophysiological interpretation is that the excitatory inputs just balance the inhibitory inputs, so that the neuron hovers just on the verge of firing. 1.5 Information rates for noisy and noiseless signals Here we compare the information rate for a IF neuron at the "balance point" J.l.T = (J with the maximum entropy spike train. For simplicity and brevity we consider only the zero-noise case, i.e. net) = O. Fig. 1A shows the information per spike as a function of the firing rate calculated from eq. (7), which was varied by changing the signal variance 0-2 . We assume that spikes can be resolved with a temporal resolution of 1 msec, i. e. that the lSI distribution has bins 1 msec wide. The dashed line shows the theoretical upper bound given by the exponential distribution; this limit can be approached by a neuron operating far below threshold, in the Poisson limit. For both the IF model and the upper bound, the information per spike is a monotonically decreasing function of the spike rate; the model almost achieves the upper bound when the mean lSI is just equal to the membrane time constant. In the model the information saturates at very low firing rates, but for the exponential distribution the information increases without bound. At high firing rates the information goes to zero when the firing rate is too fast for individual ISIs to be resolved at the temporal resolution. Fig. 1B shows that the information rate (information per second) when the neuron is at the balance point goes through a Infonnation Through a Spiking Neuron 79 maximum as the firing rate increases. The maximum occurs at a lower firing rate than for the exponential distribution (dashed line). 1.6 Bounding information rates by stimulus reconstruction By construction, eq. (3) gives an exact expression for the information rate in this model. We can therefore compare the lower bound provided by the stimulus reconstruction method eq. (2) (Bialek et aI., 1991). That is, we can assess how tight a lower bound it provides. Fig. 2 shows the lower bound provided by the reconstruction (solid line) and the reliability (dashed line) methods as a function of the firing rate. The firing rate was increased by increasing the mean p. of the input stimulus yet), and noise was set to O. At low firing rates the two estimates are nearly identical, but at high firing rates the reconstruction method substantially underestimates the information rate. The amount of the underestimate depends on the model parameters, and decreases as noise is added to the stimulus. The tightness of the bound is therefore an empirical question. While Bialek and colleagues (1996) show that under the conditions of their experiments the underestimate is less than a factor of two, it is clear that the potential for underestimate under different conditions or in different systems is greater. 2 Discussion While it is generally agreed that spike trains encode information about a neuron's inputs, it is not clear how that information is encoded. One idea is that it is the mean firing rate alone that encodes the signal, and that variability about this mean is effectively noise. An alternative view is that it is the variability itself that encodes the signal, i. e. that the information is encoded in the precise times at which spikes occur. In this view the information can be expressed in terms of the interspike interval (lSI) distribution of the spike train. This encoding scheme yields much higher information rates than one in which only the mean rate (over some interval longer than the typical lSI) is considered. Here we have quantified the information content of spike trains under the latter hypothesis for a simple neuronal model. We consider a model in which by construction the ISIs are independent, so that the information rate (in bits/sec) can be computed directly from the information per spike (in bits/spike) and the spike rate (in spikes/sec). The information per spike in turn depends on the temporal precision with which spikes can be resolved (if precision were infinite, then the information content would be infinite as well, since any message could for example be encoded in the decimal expansion of the precise arrival time of a single spike), the reliability of the spike transduction mechanism, and the entropy of the lSI distribution itself. For low firing rates, when the neuron is in the subthreshold limit, the lSI distribution is close to the theoretically maximal exponential distribution. Much of the recent interest in information theoretic analyses of the neural code can attributed to the seminal work of Bialek and colleagues (Bialek et al., 1991; Rieke et al., 1996), who measured the information rate for sensory neurons in a number of systems. The present results are in broad agreement with those of DeWeese (1996) , who considered the information rate of a linear-filtered threshold crossing! (LFTC) model. DeWeese developed a functional expansion, in which the first term describes the limit in which spike times (not ISIs) are independent, and the second term is a correction for correlations. The LFTC model differs from the present IF model mainly in that it does not "reset" after each spike. Consequently the "natural" 1 In the LFTC model, Gaussian signal and noise are convolved with a linear filter; the times at which the resulting waveform crosses some threshold are called "spikes". 80 C. STEVENS, A. ZADOR representation of the spike train in the LFTC model is as a sequence to .. . tn of firing times, while in the IF model the "natural" representation is as a sequence Tl .. . Tn of ISIs. The choice is one of convenience, since the two representations are equivalent. The two models are complementary. In the LFTC model, results can be obtained for colored signals and noise, while such conditions are awkward in the IF model. In the IF model by contrast, a class of highly correlated spike trains can be conveniently considered that are awkward in the LFTC model. That is, the indendent-ISI condition required in the IF model is less restrictive than the independent-spike condition of the LFTC model-spikes are independent iff ISIs are indepenndent and the lSI distribution p(T) is exponential. In particular, at high firing rates the lSI distribution can be far from exponential (and therefore the spikes far from independent) even when the ISIs themselves are independent. Because we have assumed that the input s(t) is white, its entropy is infinite, and the mutual information can grow without bound as the temporal precision with which spikes are resolved improves. Nevertheless, the spike train is transmitting only a minute fraction of the total available information. The signal thereby saturates the capacity of the spike train. While it is not at all clear whether this is how real neurons actually behave, it is not implausible: a typical cortical neuron receives as many as 104 synaptic inputs, and if the information rate of each input is the same as the target, then the information rate impinging upon the target is 104-fold greater (neglecting synaptic unreliability, which could decrease this substantially) than its capacity. In a preliminary series of experiments, we have used the reliability method to estimate the information rate of hippocampal neuronal spike trains in slice in response to somatic current injection (Stevens and Zador, unpublished). Under these conditions ISIs appear to be independent, so the method developed here can be applied. In these pilot experiments, an information rates as high as 6.3 bits/spike was observed. References Bialek, W., Rieke, F., de Ruyter van Steveninck, R., and Warland, D. (1991). Reading a neural code. Science, 252:1854- 1857. DeWeese, M. (1996). Optimization principles for the neural code. In Hasselmo, M., editor, Advances in Neural Information Processing Systems, vol. 8. MIT Press, Cambridge, MA. Papoulis, A. (1984). Probability, random variables and stochastic processes, 2nd edition. McGraw-Hill. Rieke, F., Warland, D., de Ruyter van Steveninck, R., and Bialek, W. (1996). Neural Coding. MIT Press. Softky, W. and Koch, C. (1993). The highly irregular firing of cortical cells is inconsistent with temporal integration of random epsps. J. Neuroscience., 13:334-350. Sugiyama, H., Moore, G., and Perkel, D. (1970). Solutions for a stochastic model of neuronal spike production. Mathematical Biosciences, 8:323-34l. Tuckwell, H. (1988). Introduction to theoretical neurobiology (2 vols.). Cambridge. Infonnation Through a Spiking Neuron Information at balance point § 1000 ~ 15 500 / / / / / / , \ \ \ \ \ \ oL-~~====~--~~~~--~~~~~~~~~~ 1~ 1~ 1~ 1~ 1~ firing rate (Hz) 81 Figure 1: Information rate at balance point. (A; top) The information per spike decreases monotonically with the spike rate (solid line). It is bounded above by the entropy of the exponential limit (dashed line), which is the highest entropy lSI distribution for a given mean rate; this limit is approached for the IF neuron in the subthreshold regime. The information rate goes to 0 when the firing rate is of the same order as the temporal resolution tit. The information per spike at the balance point is nearly optimal when E{T} ::::::: T. (T = 50 msec; tit = 1 msec); (B; bottom) Information per second for above conditions. The information rate for both the balance point (solid curve) and the exponential distribution (dashed curve) pass through a maximum, but the maximum is greater and occurs at an higher rate for the latter. For firing rates much smaller than T, the rates are almost indistinguishable. (T = 50 msec; tit = 1 msec) ~r-----~----~----~----~-----'~----~----~----, 250 200 1150 D 100 ",'" ,,,," " ~V----0 0 10 20 30 40 50 eo 70 80 spike rate (Hz) Figure 2: Estimating information by stimulus reconstruction. The information rate estimated by the reconstruction method solid line and the exact information rate dashed line are shown as a function of the firing rate. The reconstruction method significantly underestimates the actual information, particularly at high firing rates. The firing rate was varied through the mean input p. The parameters were: membrane time constant T = 20 msec; spike bin size tit = 1 msec; signal variance 0"; = 0.8; threshold Q = 10.
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Generating Accurate and Diverse Members of a Neural-Network Ensemble David w. Opitz Computer Science Department University of Minnesota Duluth, MN 55812 opitz@d.umn.edu Jude W. Shavlik Computer Sciences Department University of Wisconsin Madison, WI 53706 shavlik@cs.wisc.edu Abstract Neural-network ensembles have been shown to be very accurate classification techniques. Previous work has shown that an effective ensemble should consist of networks that are not only highly correct, but ones that make their errors on different parts of the input space as well. Most existing techniques, however, only indirectly address the problem of creating such a set of networks. In this paper we present a technique called ADDEMUP that uses genetic algorithms to directly search for an accurate and diverse set of trained networks. ADDEMUP works by first creating an initial population, then uses genetic operators to continually create new networks, keeping the set of networks that are as accurate as possible while disagreeing with each other as much as possible. Experiments on three DNA problems show that ADDEMUP is able to generate a set of trained networks that is more accurate than several existing approaches. Experiments also show that ADDEMUP is able to effectively incorporate prior knowledge, if available, to improve the quality of its ensemble. 1 Introduction Many researchers have shown that simply combining the output of many classifiers can generate more accurate predictions than that of any of the individual classifiers (Clemen, 1989; Wolpert, 1992). In particular, combining separately trained neural networks (commonly referred to as a neural-network ensemble) has been demonstrated to be particularly successful (Alpaydin, 1993; Drucker et al., 1994; Hansen and Salamon, 1990; Hashem et al., 1994; Krogh and Vedelsby, 1995; Maclin and Shavlik, 1995; Perrone, 1992). Both theoretical (Hansen and Salamon, 1990; Krogh and Vedelsby, 1995) and empirical (Hashem et al., 1994; 536 D. W. OPITZ, J. W. SHA VLIK Maclin and Shavlik, 1995) work has shown that a good ensemble is one where the individual networks are both accurate and make their errors on different parts of the input space; however, most previous work has either focussed on combining the output of multiple trained networks or only indirectly addressed how we should generate a good set of networks. We present an algorithm, ADDEMUP (Accurate anD Diverse Ensemble-Maker giving United Predictions), that uses genetic algorithms to generate a population of neural networks that are highly accurate, while at the same time having minimal overlap on where they make their error. Thaditional ensemble techniques generate their networks by randomly trying different topologies, initial weight settings, parameters settings, or use only a part of the training set in the hopes of producing networks that disagree on where they make their errors (we henceforth refer to diversity as the measure of this disagreement). We propose instead to actively search for a good set of networks. The key idea behind our approach is to consider many networks and keep a subset of the networks that minimizes our objective function consisting of both an accuracy and a diversity term. In many domains we care more about generalization performance than we do about generating a solution quickly. This, coupled with the fact that computing power is rapidly growing, motivates us to effectively utilize available CPU cycles by continually considering networks to possibly place in our ensemble. ADDEMUP proceeds by first creating an initial set of networks, then continually produces new individuals by using the genetic operators of crossover and mutation. It defines the overall fitness of an individual to be a combination of accuracy and diversity. Thus ADDEMUP keeps as its population a set of highly fit individuals that will be highly accurate, while making their mistakes in a different part of the input space. Also, it actively tries to generate good candidates by emphasizing the current population's erroneous examples during backpropagation training. Experiments reported herein demonstrate that ADDEMUP is able to generate an effective set of networks for an ensemble. 2 The Importance of an Accurate and Diverse Ensemble Figure 1 illustrates the basic framework of a neural-network ensemble. Each network in the ensemble (network 1 through network N in this case) is first trained using the training instances. Then, for each example, the predicted output of each of these networks (Oi in Figure 1) is combined to produce the output of the ensemble (0 in Figure 1). Many researchers (Alpaydin, 1993; Hashem et al., 1994; Krogh and Vedelsby, 1995; Mani, 1991) have demonstrated the effectiveness of combining schemes that are simply the weighted average of the networks (Le., 0 = L:iEN Wi ·Oi and L:iEN Wi = 1), and this is the type of ensemble we focus on in this paper. Hansen and Salamon (1990) proved that for a neural-network ensemble, if the average error rate for a pattern is less than 50% and the networks in the ensemble are independent in the production of their errors, the expected error for that pattern can be reduced to zero as the number of networks combined goes to infinity; however, such assumptions rarely hold in practice. Krogh and Vedelsby (1995) later proved that if diversity! Di of network i is measured by: Di = I)Oi(X) - o(xW, (1) x then the ensemble generalization error CE) consists of two distinct portions: E = E - D, (2) 1 Krogh and Vedelsby referred to this term as ambiguity. Generating Accurate and Diverse Members of a Neural-network Ensemble 537 " o •• ensemble output InW$lllnW$21-lnW$NI n ~ 1j ••• input Figure 1: A neural-network ensemble. where [) = Li Wi· Di and E = Li Wi· Ei (Ei is the error rate of network i and the Wi'S sum to 1). What the equation shows then, is that we want our ensemble to consist of highly correct networks that disagree as much as possible. Creating such a set of networks is the focus of this paper. 3 The ADDEMUP Algorithm Table 1 summarizes our new algorithm, ADDEMUP, that uses genetic algorithms to generate a set of neural networks that are accurate and diverse in their classifications. (Although ADDEMUP currently uses neural networks, it could be easily extended to incorporate other types of learning algorithms as well.) ADDEMUP starts by creating and training its initial population of networks. It then creates new networks by using standard genetic operators, such as crossover and mutation. ADDEMUP trains these new individuals, emphasizing examples that are misclassified by the current population, as explained below. ADDEMUP adds these new networks to the population then scores each population members with the fitness function: Fitnessi = AccuracYi + A DiversitYi = (1 - Ei) + A Di , (3) where A defines the tradeoff between accuracy and diversity. Finally, ADDEMUP prunes the population to the N most-fit members, which it defines to be its current ensemble, then repeats this process. We define our accuracy term, 1 - Ei , to be network i's validation-set accuracy (or training-set accuracy if a validation set is not used), and we use Equation lover this validation set to calculate our diversity term Di . We then separately normalize each term so that the values range from 0 to 1. Normalizing both terms allows A to have the same meaning across domains. Since it is not always clear at what value one should set A, we have therefore developed some rules for automatically setting A. First, we never change A if the ensemble error E is decreasing while we consider new networks; otl.!erwise we change A if one of following two things happen: (1) population error E is not increasing and the population diversity D is decreasing; diversity seems to be under-emphasized and we increase A, or (2) E is increasing and [) is not decreasing; diversity seems to be over-emphasized and we decrease A. (We started A at 0.1 for the results in this paper.) A useful network to add to an ensemble is one that correctly classifies as many examples as possible while making its mistakes primarily on examples that most 538 D. W. OPITZ. 1. W. SHA VLIK Table 1: The ADDEMUP algorithm. GOAL: Genetically create an accurate and diverse ensemble of networks. 1. Create and train the initial population of networks. 2. Until a stopping criterion is reached: (a) Use genetic operators to create new networks. (b) Thain the new networks using Equation 4 and add them to the population. (c) Measure the diversity of each network with respect to the current population (see Equation 1). (d) Normalize the accuracy scores and the diversity scores of the individual networks. (e) Calculate the fitness of each population member (see Equation 3). (f) Prune the population to the N fittest networks. (g) Adjust oX (see the text for an explanation). (h) Report the current population of networks as the ensemble. Combine the output of the networks according to Equation 5. of the current population members correctly classify. We address this during backpropagation training by multiplying the usual cost function by a term that measures the combined population error on that example: ..2.Cost = L It(k) ~O(k)I>-'+l [t(k) -a(kW, kET E (4) where t(k) is the target and a(k) is the network activation for example k in the training set T. Notice that since our network is not yet a member of the ensemble, o(k) and E are not dependent on our network; our new term is thus a constant when calculating the derivatives during backpropagation. We normalize t(k) -o(k) by the ensemble error E so that the average value of our new term is around 1 regardless of the correctness of the ensemble. This is especially important with highly accurate populations, since tk - o(k) will be close to 0 for most examples, and the network would only get trained on a few examples. The exponent A~l represents the ratio of importance of the diversity term in the fitness function. For instance, if oX is close to 0, diversity is not considered important and the network is trained with the usual cost function; however, if oX is large, diversity is considered important and our new term in the cost function takes on more importance. We combine the predictions of the networks by taking a weighted sum of the output of each network, where each weight is based on the validation-set accuracy of the network. Thus we define our weights for combining the networks as follows: (5) While simply averaging the outputs generates a good composite model (Clemen, 1989), we include the predicted accuracy in our weights since one should believe accurate models more than inaccurate ones. Generating Accurate and Diverse Members of a Neural-network Ensemble 539 4 Experimental Study The genetic algorithm we use for generating new network topologies is the REGENT algorithm (Opitz and Shavlik, 1994). REGENT uses genetic algorithms to search through the space of knowledge-based neural network (KNN) topologies. KNNs are networks whose topologies are determined as a result of the direct mapping of a set of background rules that represent what we currently know about our task. KBANN (Towell and Shavlik, 1994), for instance, translates a set of propositional rules into a neural network, then refines the resulting network's weights using backpropagation. Thained KNNs, such as KBANN'S networks, have been shown to frequently generalize better than many other inductive-learning techniques such as standard neural networks (Opitz, 1995; Towell and Shavlik, 1994). Using KNNs allows us to have highly correct networks in our ensemble; however, since each network in our ensemble is initialized with the same set of domain-specific rules, we do not expect there to be much disagreement among the networks. An alternative we consider in our experiments is to randomly generate our initial population of network topologies, since domain-specific rules are sometimes not available. We ran ADDEMUP on NYNEX's MAX problem set and on three problems from the Human Genome Project that aid in locating genes in DNA sequences (recognizing promoters, splice-junctions, and ribosome-binding sites - RBS). Each of these domains is accompanied by a set of approximately correct rules describing what is currently known about the task (see Opitz, 1995 or Opitz and Shavlik, 1994 for more details). Our experiments measure the test-set error of ADDEMUP on these tasks. Each ensemble consists of 20 networks, and the REGENT and ADDEMUP algorithms considered 250 networks during their genetic search. Table 2a presents the results from the case where the learners randomly create the topology of their networks (Le., they do not use the domain-specific knowledge). Table 2a's first row, best-network, results from a single-layer neural network where, for each fold, we trained 20 networks containing between 0 and 100 (uniformly) hidden nodes and used a validation set to choose the best network. The next row, bagging, contains the results of running Breiman's (1994) bagging algorithm on standard, single-hidden-Iayer networks, where the number of hidden nodes is randomly set between 0 and 100 for each network.2 Bagging is a "bootstrap" ensemble method that trains each network in the ensemble with a different partition of the training set. It generates each partition by randomly drawing, with replacement, N examples from the training set, where N is the size of the training set. Breiman (1994) showed that bagging is effective on "unstable" learning algorithms, such as neural networks, where small changes in the training set result in large changes in predictions. The bottom row of Table 2a, AOOEMUP, contains the results of a run of ADDEMUP where its initial population (of size 20) is randomly generated. The results show that on these domains combining the output of mUltiple trained networks generalizes better than trying to pick the single-best network. While the top table shows the power of neural-network ensembles, Table 2b demonstrates ADDEMUP'S ability to utilize prior knowledge. The first row of Table 2b contains the generalization results of the KBANN algorithm, while the next row, KBANN-bagging, contains the results of the ensemble where each individual network in the ensemble is the KBANN network trained on a different partition of the training set. Even though each of these networks start with the same topology and 2We also tried other ensemble approaches, such as randomly creating varying multilayer network topologies and initial weight settings, but bagging did significantly better on all datasets (by 15-25% on all three DNA domains). 540 D. W. OPITZ. J. W. SHA VLlK Table 2: Test-set error from a ten-fold cross validation. Table (a) shows the results from running three learners without the domain-specific knowledge; Table (b) shows the results of running three learners with this knowledge. Pairwise, one-tailed t-tests indicate that AOOEMUP in Table (b) differs from the other algorithms in both tables at the 95% confidence level, except with REGENT in the splice-junction domain. I Standard neural networks (no domain-specific knowledge used) I Promoters Splice Junction RBS MAX best-network 6.6% 7.8% 10.7% 37.0% bagging 4.6% 4.5% 9.5% 35.7% AOOEMUP 4.6% 4.9% 9.0% 34.9% (a) ·Knowledge-based neural networks (domain-specific knowledge used) Promoters Splice Junction RBS MAX KBANN 6.2% 5.3% 9.4% 35.8% KBANN-bagging 4.2% 4.5% 8.5% 35.6% REGENT-Combined 3.9% 3.9% 8.2% 35.6% AOOEMUP 2.9% 3.6% 7.5% 34.7% (b) "large" initial weight settings (Le., the weights resulting from the domain-specific knowledge), small changes in the training set still produce significant changes in predictions. Also notice that on all datasets, KBANN-bagging is as good as or better than running bagging on randomly generated networks (Le., bagging in Table 2a). The next row, REGENT-Combined, contains the results of simply combining, using Equation 5, the networks in REGENT'S final population. AOOEMUP, the final row of Table 2b, mainly differs from REGENT-Combined in two ways: (a) its fitness function (Le., Equation 3) takes into account diversity rather than just network accuracy, and (b) it trains new networks by emphasizing the erroneous examples of the current ensemble. Therefore, comparing AOOEMUP with REGENT-Combined helps directly test ADDEMUP'S diversity-achieving heuristics, though additional results reported in Opitz (1995) show ADDEMUP gets most of its improvement from its fitness function. There are two main reasons why we think the results of ADDEMUP in Table 2b are especially encouraging: (a) by comparing ADDEMUP with REGENT-Combined, we explicitly test the quality of our heuristics and demonstrate their effectiveness, and (b) ADDEMUP is able to effectively utilize background knowledge to decrease the error of the individual networks in its ensemble, while still being able to create enough diversity among them so as to improve the overall quality of the ensemble. 5 Conclusions Previous work with neural-network ensembles have shown them to be an effective technique if the classifiers in the ensemble are both highly correct and disagree with each other as much as possible. Our new algorithm, ADDEMUP, uses genetic algorithms to search for a correct and diverse population of neural networks to be used in the ensemble. It does this by collecting the set of networks that best fits an objective function that measures both the accuracy of the network and the disagreement of that network with respect to the other members of the set. ADDEMUP tries Generating Accurate and Diverse Members of a Neural-network Ensemble 541 to actively generate quality networks during its search by emphasizing the current ensemble's erroneous examples during backpropagation training. Experiments demonstrate that our method is able to find an effective set of networks for our ensemble. Experiments also show that ADDEMUP is able to effectively incorporate prior knowledge, if available, to improve the quality of this ensemble. In fact, when using domain-specific rules, our algorithm showed statistically significant improvements over (a) the single best network seen during the search, (b) a previously proposed ensemble method called bagging (Breiman, 1994), and (c) a similar algorithm whose objective function is simply the validation-set correctness of the network. In summary, ADDEMUP is successful in generating a set of neural networks that work well together in producing an accurate prediction. Acknowledgements This work was supported by Office of Naval Research grant N00014-93-1-0998. References Alpaydin, E. (1993). Multiple networks for function learning. In Proceedings of the 1993 IEEE International Conference on Neural Networks, vol I, pages 27-32, San Fransisco. Breiman, L. (1994). Bagging predictors. Technical Report 421, Department of Statistics, University of California, Berkeley. Clemen, R. (1989). Combining forecasts: A review and annotated bibliography. International Journal of Forecasting, 5:559-583. Drucker, H., Cortes, C., Jackel, L., LeCun, Y., and Vapnik, V. (1994). Boosting and other machine learning algorithms. In Proceedings of the Eleventh International Conference on Machine Learning, pages 53-61, New Brunswick, NJ. Morgan Kaufmann. Hansen, L. and Salamon, P. (1990). Neural network ensembles. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12:993-100l. Hashem, S., Schmeiser, B., and Yih, Y. (1994). Optimal linear combinations of neural networks: An overview. In Proceedings of the 1994 IEEE International Conference on Neural Networks, Orlando, FL. Krogh, A. and Vedelsby, J. (1995). Neural network ensembles, cross validation, and active learning. In Tesauro, G., Touretzky, D., and Leen, T., editors, Advances in Neural Information Processing Systems, vol 7, Cambridge, MA. MIT Press. Maclin, R. and Shavlik, J. (1995). Combining the predictions of multiple classifiers: Using competitive learning to initialize neural networks. In Proceedings of the Fourteenth International Joint Conference on Artificial Intelligence, Montreal, Canada. Mani, G. (1991). Lowering variance of decisions by using artificial neural network portfolios. Neural Computation, 3:484-486. Opitz, D. (1995). An Anytime Approach to Connectionist Theory Refinement: Refining the Topologies of Knowledge-Based Neural Networks. PhD thesis, Computer Sciences Department, University of Wisconsin, Madison, WI. Opitz, D. and Shavlik, J. (1994). Using genetic search to refine knowledge-based neural networks. In Proceedings of the Eleventh International Conference on Machine Learning, pages 208-216, New Brunswick, NJ. Morgan Kaufmann. Perrone, M. (1992). A soft-competitive splitting rule for adaptive tree-structured neural networks. In Proceedings of the International Joint Conference on Neural Networks, pages 689-693, Baltimore, MD. Towell, G. and Shavlik, J. (1994). Knowledge-based artificial neural networks. Artificial Intelligence, 70(1,2):119- 165. Wolpert, D. (1992). Stacked generalization. Neural Networks, 5:241- 259.
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Discriminant Adaptive Nearest Neighbor Classification and Regression Trevor Hastie Department of Statistics Sequoia Hall Stanford University California 94305 trevor@playfair.stanford.edu Abstract Robert Tibshirani Department of Statistics University of Toronto tibs@utstat.toronto.edu Nearest neighbor classification expects the class conditional probabilities to be locally constant, and suffers from bias in high dimensions We propose a locally adaptive form of nearest neighbor classification to try to finesse this curse of dimensionality. We use a local linear discriminant analysis to estimate an effective metric for computing neighborhoods. We determine the local decision boundaries from centroid information, and then shrink neighborhoods in directions orthogonal to these local decision boundaries, and elongate them parallel to the boundaries. Thereafter, any neighborhood-based classifier can be employed, using the modified neighborhoods. We also propose a method for global dimension reduction, that combines local dimension information. We indicate how these techniques can be extended to the regression problem. 1 Introduction We consider a discrimination problem with J classes and N training observations. The training observations consist of predictor measurements x:::: (Xl,X2," .xp) on p predictors and the known class memberships. Our goal is to predict the class membership of an observation with predictor vector Xo Nearest neighbor classification is a simple and appealing approach to this problem. We find the set of J{ nearest neighbors in the training set to Xo and then classify Xo as the most frequent class among the J{ neighbors. Cover & Hart (1967) show that the one nearest neighbour rule has asymptotic error rate at most twice the Bayes rate. However in finite samples the curse of 410 T. HASTIE, R. TIBSHIRANI dimensionality can severely hurt the nearest neighbor rule. The relative radius of the nearest-neighbor sphere grows like r 1/ p where p is the dimension and r the radius for p = 1, resulting in severe bias at the target point x. Figure 1 (left panel) illustrates the situation for a simple example. Nearest neighbor techniques are , , Figure 1: In the left panel, the vertical strip denotes the NN region using only horizontal coordinate to find the nearest neighbor for the target point (solid dot). The sphere shows the NN region using both coordinates, and we see in this case it has extended into the class 1 region (and found the wrong class in this instance). The middle panel shows a spherical neighborhood containing 25 points, for a two class problem with a circular decision boundary. The right panel shows the ellipsoidal neighborhood found by the DANN procedure, also containing 25 points. The latter is elongated in a direction parallel to the true decision boundary (locally constant posterior probabilities), and flattened orthogonal to it. based on the assumption that locally the class posterior probabilities are constant. While that is clearly true in the vertical strip using only the vertical coordinate, using both this is no longer true. Figure 1 (middle and right panels) shows how we locally adapt the metric to overcome this problem, in a situation where the decision boundary is locally linear. 2 Discriminant adaptive nearest neighbors Consider first a standard linear discriminant (LDA) classification procedure with J{ classes. Let Band W denote the between and within sum of squares matrices. In LDA the data are first sphered with respect to W, then the target point is classified to the class of the closest centroid (with a correction for the class prior membership probabilities). Since only relative distances are relevant, any distances in the complement of the subspace spanned by the sphered centroids can be ignored. This complement corresponds to the null space of B. We propose to estimate Band W locally, and use them to form a local metric that approximately behaves like the LDA metric. One such candidate is 1: W-1BW- 1 W-l/2(W-l/2BW-l/2)W-l/2 W-1/2B*W-1/2. (1) where B* is the between sum-of-squares in the sphered space. Consider the action of 1: as a metric for computing distances (x - xo?1:(x - xo) : (2) Discriminant Adaptive Nearest Neighbor Classification and Regression 411 • it first spheres the space using W; • components of distance in the null space of B* are ignored; • other components are weighted according to the eigenvalues of B* when there are more than 2 classes directions in which the centroids are more spread out are weighted more than those in which they are close Thus this metric would result in neighborhoods similar to the narrow strip in figure l(left figure): infinitely long in the null space of B, and then deformed appropriately in the centroid subspace according to how they are placed. It is dangerous to allow neighborhoods to extend infinitely in any direction, so we need to limit this stretching. Our proposal is W-l/2[W-l/2BW-l/2 + d]W- 1/ 2 W-l/2[B* + d]W- 1/ 2 (3) where f. is some small tuning parameter to be determined. The metric shrinks the neighborhood in directions in which the local class centroids differ, with the intention of ending up with a neighborhood in which the class centroids coincide (and hence nearest neighbor classification is appropriate). Given E we use perform K-nearest neighbor classification using the metric (2). There are several details that we briefly describe here and in more detail in Hastie & Tibshirani (1994): • B is defined to be the covariance of the class centroids, and W the pooled estimate of the common class covariance matrix. We estimate these locally using a spherical, compactly supported kernel (Cleveland 1979), where the bandwidth is determined by the distance of the KM nearest neighbor. • KM above has to be supplied, as does the softening parameter f. . We somewhat arbitrarily use KM = max(N /5,50); so we use many more neighbors (50 or more) to determine the metric, and then typically K = 1, ... ,5 nearest neighbors in this metric to classify. We have found that the metric is relatively insensitive to different values of 0 < f. < 5, and typically use f. = 1. • Typically the data do not support the local calculation of W (p(p + 1)/2 entries), and it can be argued that this is not necessary. We mostly resort to using the diagonal of W instead, or else use a global estimate. Sections 4 and 5 illustrate the effectiveness of this approach on some simulated and real examples. 3 Dimension Reduction using Local Discriminant Information The technique described above is entirely "memory based" , in that we locally adapt a neighborhood about a query point at the time of classification. Here we describe a method for performing a global dimension reduction, by pooling the local dimension information over all points in the training set. In a nutshell we consider subspaces corresponding to eigenvectors oj the average local between sum-oj-squares matrices. Consider first how linear discriminant analysis (LDA) works. After sphering the data, it concentrates in the space spanned by the class centroids Xj or a reduced rank space that lies close to these centroids. If x denote the overall centroid, this 412 T. HASTIE, R. TIBSHIRANI subspace is exactly a principal component hyperplane for the data points Xj - X, weighted by the class proportions, and is given by the eigen-decomposition of the between covariance B. Our idea to compute the deviations Xj - x locally in a neighborhood around each of the N training points, and then do an overall principal components analysis for the N x J deviations. This amounts to an eigen-decomposition of the average between sum of squares matrix 2:~1 B (i) / N. LOA and local Slbapaces K _ 25 , , 2 • I ./ 2 , , Local e.tween Directions " ' " 8 • • • • • • • • . .. 10 "'de, Figure 2: [Left Panel] Two dimensional gaussian data with two classes and correlation 0.65. The solid lines are the LDA decision boundary and its equivalent subspace for classification, computed using both the between and (crucially) the within class covariance. The dashed lines were produced by the local procedure described in this section, without knowledge of the overall within covariance matrix. [Middle panel] Each line segment represents the local between information centered at that point. [Right panel] The eigenvalues of the average between matrix for the 4D sphere in 10D problem. Using these first four dimensions followed by our DANN nearest neighbor routine, we get better performance than 5NN in the real 4D subspace. Figure 2 (left two panels) demonstrates by a simple illustrative example that our subspace procedure can recover the correct LDA direction without making use of the within covariance matrix. Figure 2 (right panel) represents a two class problem with a 4-dimensional spherical decision boundary. The data for the two classes lie in concentric spheres in 4D, the one class lying inside the other with some overlap (a 4D version of the same 2D situation in figure 1.) In addition the are an extra 6 noise dimensions, and for future reference we denote such a model as the "4D spheres in lOD" problem. The decision boundary is a 4 dimensional sphere, although locally linear. The eigenvalues show a distinct change after 4 (the correct dimension), and using our DANN classifier in these four dimensions actually beats ordinary 5NN in the known 4D discriminant subspace. 4 Examples Figure 3 su·mmarizes the results of a number of simulated examples designed to test our procedures in both favorable and unfavorable situations. In all the situations DANN outperforms 5-NN. In the cases where 5NN is provided with the known lowerdimensional discriminant subspace, our subspace technique subDANN followed by DANN comes close to the optimal performance. Discriminant Adaptive Nearest Neighbor Classification and Regression !i! o re o '" o o o :g o ci '" o Two Gaussians wnh Noise [5 I ~I~ -1-10M=; 8 ~LU ~L ~I~ 4·0 Sphere in 10·0 T E$J o T gY T T i S Q . o '" o o o '" o '" o Unstructured with Noise T T I B I T IBy I I 1 1 j ~8 : 1 T og 10·0 sphere in 10·0 413 Figure 3: Boxplots of error rates over 20 simulations. The top left panel has two gaussian distributions separated in two dimensions, with 14 noise dimensions. The notation red-LDA and red-5NN refers to these procedures in the known lower dimensional space. iter-DANN refers to an iterated version of DANN (which appears not to help), while sub-DANN refers to our global subspace approach, followed by DANN. The top right panel has 4 classes, each of which is a mixture of 3-gaussians in 2-D; in addition there are 8 noise variables. The lower two panels are versions of our sphere example. 5 Image Classification Example Here we consider an image classification problem. The data consist of 4 LANDSAT images in different spectral bands of a small area of the earths surface, and the goal is to classify into soil and vegetation types. Figure 4 shows the four spectral bands, two in the visible spectrum (red and green) and two in the infra red spectrum. These data are taken from the data archive of the STATLOG (Michie et al. 1994)1. The goal is to classify each pixel into one of 7 land types: red soil, cotton, vegetation stubble, mixture, grey soil, damp grey soil, very damp grey soil. We extract for each pixel its 8-neighbors, giving us (8 + 1) x 4 = 36 features (the pixel intensities) per pixel to be classified. The data come scrambled, with 4435 training pixels and 2000 test pixels, each with their 36 features and the known classification. Included in figure 4 is the true classification, as well as that produced by linear discriminant analysis. The right panel compares DANN to all the procedures used in STATLOG, and we see the results are favorable. 1 The authors thank C. Taylor and D. Spiegelhalter for making these images and data available 414 T. HASTIE, R. TIBSHIRANI ST ATLOG results Spectral band 1 Spectral band 2 Spectral band 3 LDA Spectral band 4 Land use (Actual) Land use (Predicted) o ci LVQ K-NN ANN 2 Nm"I[C.4.5 C}\fH NUf:r<,1 ALLOCif) HBF 4 6 8 10 Method SMA~g!st!C OOf" 12 Figure 4: The first four images are the satellite images in the four spectral bands. The fifth image represents the known classification, and the final image is the classification map produced by linear discriminant analysis. The right panel shows the misclassification results of a variety of classification procedures on the satellite image test data (taken from Michie et al. (1994)). DANN is the overall winner. 6 Local Regression Near neighbor techniques are used in the regression setting as well. Local polynomial regression (Cleveland 1979) is currently very popular, where, for example, locally weighted linear surfaces are fit in modest sized neighborhoods. Analogs of K-NN classification for small J{ are used less frequently. In this case the response variable is quantitative rather than a class label. Duan & Li (1991) invented a technique called sliced inverse regression, a dimension reduction tool for situations where the regression function changes in a lowerdimensional space. They show that under symmetry conditions of the marginal distribution of X, the inverse regression curve E(XIY) is concentrated in the same lower-dimensional subspace. They estimate the curve by slicing Y into intervals, and computing conditional means of X in each interval, followed by a principal component analysis. There are obvious similarities with our DANN procedure, and the following generalizations of DANN are suggested for regression: • locally we use the B matrix of the sliced means to form our DANN metric, and then perform local regression in the deformed neighborhoods . • The local B(i) matrices can be pooled as in subDANN to extract global subspaces for regression. This has an apparent advantage over the Duan & Li (1991) approach: we only require symmetry locally, a condition that is locally encouraged by the convolution of the data with a spherical kernel 2 7 Discussion Short & Fukanaga (1980) proposed a technique close to ours for the two class problem. In our terminology they used our metric with W = I and ( = 0, with B determined locally in a neighborhood of size J{M. In effect this extends the 2We expect to be able to substantiate the claims in this section by the time of the NIPS995 meeting. 14 Discriminant Adaptive Nearest Neighbor Classification and Regression 415 neighborhood infinitely in the null space of the local between class directions, but they restrict this neighborhood to the original KM observations. This amounts to projecting the local data onto the line joining the two local centroids. In our experiments this approach tended to perform on average 10% worse than our metric, and we did not pursue it further. Short & Fukanaga (1981) extended this to J > 2 classes, but here their approach differs even more from ours. They computed a weighted average of the J local centroids from the overall average, and project the data onto it, a one dimensional projection. Myles & Hand (1990) recognized a shortfall of the Short and Fukanaga approach, since the averaging can cause cancellatlOn, and proposed other metrics to avoid this, different from ours. Friedman (1994) proposes a number of techniques for flexible metric nearest neighbor classification (and sparked our interest in the problem.) These techniques use a recursive partitioning style strategy to adaptively shrink and shape rectangular neighborhoods around the test point. Acknowledgement The authors thank Jerry Friedman whose research on this problem was a source of inspiration, and for many discussions. Trevor Hastie was supported by NSF DMS-9504495. Robert Tibshirani was supported by a Guggenheim fellowship, and a grant from the National Research Council of Canada. References Cleveland, W. (1979), 'Robust locally-weighted regression and smoothing scatterplots', Journal of the American Statistical Society 74, 829-836. Cover, T. & Hart, P. (1967), 'Nearest neighbor pattern classification', Proc. IEEE Trans. Inform. Theory pp. 21- 27. Duan, N. & Li, K.-C. (1991), 'Slicing regression: a link-free regression method', Annals of Statistics pp. 505- 530. Friedman, J. (1994), Flexible metric nearest neighbour classification, Technical report, Stanford U ni versity. Hastie, T . & Tibshirani, R. (1994), Discriminant adaptive nearest neighbor classification, Technical report, Statistics Department, Stanford University. Michie, D., Spigelhalter, D. & Taylor, C., eds (1994), Machine Learning, Neural and Statistical Classification, Ellis Horwood series in Artificial Intelligence, Ellis Horwood. Myles, J. & Hand, D. J. (1990), 'The multi-class metric problem in nearest neighbour discrimination rules', Pattern Recognition 23, 1291-1297. Short, R. & Fukanaga, K. (1980), A new nearest neighbor distance measure, in 'Proc. 5th IEEE Int. Conf. on Pattern Recognition', pp. 81- 86. Short, R. & Fukanaga, K. (1981), 'The optimal distance measure for nearest neighbor classification', IEEE transactions of Information Theory IT-27, 622-627.
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A Predictive Switching Model of Cerebellar Movement Control Andrew G. Barto .J ay T. Buckingham Department of Computer Science University of Massachusetts Amherst, MA 01003-4610 barto@cs.umass.edu .J ames C. Houk Department of Physiology Northwestern University Medical School 303 East Chicago Ave Chicago, Illinois 60611-3008 houk@acns.nwu.edu Abstract We present a hypothesis about how the cerebellum could participate in regulating movement in the presence of significant feedback delays without resorting to a forward model of the motor plant. We show how a simplified cerebellar model can learn to control endpoint positioning of a nonlinear spring-mass system with realistic delays in both afferent and efferent pathways. The model's operation involves prediction, but instead of predicting sensory input, it directly regulates movement by reacting in an anticipatory fashion to input patterns that include delayed sensory feedback. 1 INTRODUCTION The existence of significant delays in sensorimotor feedback pathways has led several researchers to suggest that the cerebellum might function as a forward model of the motor plant in order to predict the sensory consequences of motor commands before actual feedback is available; e.g., (Ito, 1984; Keeler, 1990; Miall et ai., 1993). While we agree that there are many potential roles for forward models in motor control systems, as discussed, e.g., in (Wolpert et al., 1995), we present a hypothesis about how the cerebellum could participate in regulating movement in the presence of significant feedback delays without resorting to a forward model. We show how a very simplified version of the adjustable pattern generator (APG) model being developed by Houk and colleagues (Berthier et al., 1993; Houk et al., 1995) can learn to control endpoint positioning of a nonlinear spring-mass system with significant delays in both afferent and efferent pathways. Although much simpler than a multilink dynamic arm, control of this spring-mass system involves some of the challenges critical in the control of a more realistic motor system and serves to illustrate the principles we propose. Preliminary results appear in (Buckingham et al., 1995). A Predictive Switching Model of Cerebellar Movement Control A 0.5 0.4 ~ 0.3 ~ i 0.2 "g ~ 0.1 o -0.1 B /I •• • T 139 • p foooo:t_~'_p_-_---.-_--_-",,-==--~--_-_-_- -r- -_ -_-_--~--------~-----~ - - ~ ~ . -- -,--~-----,-----,----- ~ -0 .2+----~-----.---~~--~~ ~ ~ u u u U tIs.:} -0.1 -0.05 o 0.05 Position (m) 0.1 Figure 1: Pulse-step control of a movement from initial position :zJo = 0 to target endpoint position :zJT = .05. Panel A: Top-The pulse-step command. MiddleVelocity as a function of time. Bottom-Position as a function of time. Panel B: Switching curve. The dashed line plots states of the spring-mass system at which the command should switch from pulse to step so that the mass will stick at the endpoint :zJT = .05 starting from different initial states. The bold line shows the phase-plane trajectory of the movement shown in Panel A. 2 NONLINEAR VISCOSITY An important aspect of the model is that the plant being contolled has a form of nonlinear viscosity, brought about in animals through a combination of muscle and spinal reflex properties. To illustrate this, we use a nonlinear spring-mass model based on studies of human wrist movement (Wu et al., 1990): mz + bzt + k(:zJ :zJeq ) = 0, (1) where :zJ is the position (in meters) of an object of mass m (kg) attached to the spring, :zJeq is the resting, or equilibrium, position, b is a damping coefficient, and k is the spring's stiffness. Setting m = I, b = 4, and k = 60 produces trajectories that are qualitatively similar to those observed in human wrist movement (Wu et al., 1990). This one-fifth power law viscosity gives the system the potential to produce fast movements that terminate with little or no oscillation. However, the principle of setting the equilibrium position to the desired movement endpoint does not work in practice because the system tends to "stick" at non-equilibrium positions, thereafter drifting extremely slowly toward the equilibrium position, :zJeq • We call the position at which the mass sticks (which we define as the position at which its absolute velocity falls and remains below .005mjs) the endpoint of a movement, denoted :zJ e . Thus, endpoint control of this system is not entirely straightforward. The approach taken by our model is to switch the value of the control signal, :zJ eq , at a preciselyplaced point during a movement. This is similar to virtual trajectory control, except that here the commanded equilibrium position need not equal the desired endpoint either before or after the switch. Panel A of Fig. 1 shows an example of this type of control. The objective is to move the mass from an initial position :zJo = 0 to a target endpoint :zJT = .05. The control signal is the pulse-step shown in the top graph, where :zJp = .1 and :zJ. = .04 140 target sparse expansive encoding A. G. BARTO, J. T. BUCKINGHAM, J. C. HOUK ..---------i~ ..... -------. exlrllcerebella, "",""",ve convnand command spring-mass system state Figure 2: The simplified model. PC, Purkinje cell; MFs, mossy fibers; PFs, parallel fibers; CF, climbing fiber. The labels A and B mark places in the feedback loop to which we refer in discussing the model's behavior. respectively denote the pulse and step values, and d denotes the pulse duration. The mass sticks near the target endpoint ZT = .05, which is different from both equilibrium positions. If the switch had occurred sooner (later), the mass would have undershot (overshot) the target endpoint. The bold trajectory in Panel B of Fig. 1 is the phase-plane portrait of this movement. During its initial phase, the state follows the trajectory that would eventually lead to equilibrium position zp' When the pulse ends, the state switches to the trajectory that would eventually lead to equilibrium position z" which allows a rapid approach to the target endpoint ZT = .05, where the mass sticks before reaching z,. The dashed line plots pairs of positions and velocities at which the switch should occur so that movements starting from different initial states will reach the endpoint ZT = .05. This switching curve has to vary as a function of the target endpoint. 3 THE MODEL'S ARCHITECTURE The simplified model (Fig. 2) consists of a unit representing a Purkinje cell (PC) whose input is derived from a sparse expansive encoding of mossy fiber (MF) input representing the target position, ZT, which remains fixed throughout a movement, delayed information about the state of the spring-mass system, and the current motor command, Zeq.l Patterns of MF activity are recoded to form sparse activity patterns over a large number (here 8000) of binary parallel fibers (PFs) which synapse upon the PC unit, along the lines suggested by Man (Marr, 1969) and the CMAC model of Albus (Albus, 1971). While some liberties have been taken with this representation, the delay distributions are within the range observed for the intermediate cerebellum of the monkey (Van Kan et 01., 1993). Also as in Man and Albus, the PC unit is trained by a signal representing the activity of a climbing fiber (CF), whose response properties are described below. Occasional corrective commands, also discussed below, are assumed to be generated 1 In this model, 256 Gaussian radial basis function (RBF) units represent the target position, 400 RBF units represent the position of the mass (i.e., the length of the spring), with centers distributed uniformly across an appropriate range of positions and with delays distributed according to a Gaussian of mean 15msec and standard deviation 6msec. This distribution is truncated so that the minimum delay is 5msec. This delay distribution is represented by 71 in Fig. 2. Another 400 RBF units similarly represent mass velocity. An additional 4 MF inputs are efference copy signals that simply copy the current motor command. A Predictive Switching Model of Cerebellar Movement Control 141 by an extracerebellar system. The PC's output determines the motor command through a simple transformation. The model includes an efferent and CF delays, both equal to 20msec (T2 and T3, respectively, in Fig. 2). These delays are also within the physiological range for these pathways (Gellman et al., 1983). How this model is related to the full APG model and its justification in terms of the anatomy and physiology of the cerebellum and premotor circuits are discussed extensively elsewhere (Berthier et al., 1993; Houk et al., 1995). The PC unit is a linear threshold unit with hysteresis. Let s(t) = I:i Wi(t)4>i(t), where 4>i(t) denotes the activity of PF i at time t and Wi(t) is the weight at time step t of the synapse by which PF i influences the PC unit. The output of the PC unit at time t, denoted y(t), is the PC's activity state, high or low, at time t, which represents a high or a low frequency of simple spike activity. PC activation depends on two thresholds: (Jhigh and (J,01D < (Jhigh. The activity state switches from low to high when s(t) > (Jhigh, and it switches from high to low when s(t) < (J,01lJ. If (Jhigh = (J,01D' the PC unit is the usual linear threshold unit. Although hysteresis is not strictly necessary for the control task we present here, it accelerates learning: A PC can more easily learn when to switch states than it can learn to maintain the correct output on a moment-to-moment basis. The bistability of this PC unit is a simplified representation of multistability that could be produced by dendritic zones of hysteresis arising from ionic mechanisms (Houk et al., 1995). Because PC activity inhibits premotor circuits, PC state low corresponds to the pulse phase ofthe motor command, which sets a "far" equilibrium position, zp; PC state high corresponds to the step phase, which sets a "near" equilibrium position, z,. Thus, the pulse ends when the PC state switches from low to high. Because the precise switching point determines where the mass sticks, this single binary PC can bring the mass to any target endpoint in a considerable range by switching state at the right moment during a movement. 4 LEARNING Learning is based on the idea that corrective movements following inaccurate movements provide training information by triggering CF responses. These responses are presumed to be proprioceptively triggered by the onset of a corrective movement, being suppressed during the movement itself. Corrective movements can be generated when a cerebellar module generates an additional pulse phase of the motor command, or through the action of a system other than the cerebellum. The second, extracerebellar, source of corrective movements only needs to operate when small corrections are needed. The learning mechanism has to adjust the PC weights, Wi, so that the PC switches state at the correct moment during a movement. This is difficult because training information is significantly delayed due to the combined effects of movement duration and delays in the relevant feedback pathways. The relevant PC activity is completed well before a corrective movement triggers a CF response. To learn under these conditions, the learning mechanism needs to modify synaptic actions that occurred prior to the CF's discharge. The APG model adopts Klopf's (Klopf, 1982) idea of a synaptic "eligibility trace" whereby appropriate synaptic activity sets up a synaptically-local memory trace that renders the synapse "eligible" for modification if and when the appropriate training information arrives within a short time period. The learning rule has two components: one implments a form of long-term depression (LTD); the other implements a much weaker form of long-term potentiation 142 A. G. BARTO, J. T. BUCKINGHAM, J. C. HOUK (LTP). It works as follows. Whenever the CF fires (c(t) = 1), the weights of all the eligible synapses decrease. A synapse is eligible if its presynaptic parallel fiber was active in the past when the PC switched from low to high, with the degree of eligibility decreasing with the time since that state switch. This makes the PC less likely to switch to high in future situations represented by patterns of PF activity similar to the pattern present when the eligibility-initiating switch occurred. This has the effect of increasing the duration of the PC pause, which increases the duration of the pulse phase of the motor command. Superimposed on weight decreases are much smaller weight increases that occur for any synapse whose presynaptic PF is active when the PC switches from low to high, irrespective of CF activity. This makes the PC more likely to switch to high under similar circumstances in the future, which decreases the duration of the pulse phase of the movement command. To define this mathematically, let 11(t) detect when the PC's activity state switches from low to high: 11(t) = 0 unless y( t 1) = low and y(t) = high, in which case 11(t) = 1. The eligibility trace for synapse i at time step t, denoted ei (t), is set to 1 whenever 11(t) = 1 and thereafter decays geometrically toward zero until it is reset to 1 when 11 is again set to 1 by another upward switch of PC activity level. Then the learning rule is given for t = 1,2, ... , by: A 201 ---w_2~~---o 0.1 0.2 0.3 0.4 ~ high 1 lowJ----1. ___ __ _ o 0.1 0.2 0.3 0.4 ~D':l :c: -------_. o 0.1 0.2 0.3 0.4 !D.5j to.~ > 0 0.1 0.2 0.3 0.4 t (sec) B 2D~ w 0 __ __ -20 o 0.1 0.2 0.3 0.4 ~ hi9hl lowJ-· ........ -----o 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 r]~ > 0 D.l D.2 D.3 D.4 t(sec) LlWi(t) = -o:c(t)ei(t)+,811(t)¢i(t), where 0: and ,8, with 0: » ,8, are positive parameters respectively determining the rate of LTD and LTP. See (Houk et al., 1995) for a discussion Figure 3: Model behavior. Panel A: early in learning; Panel B: late in learning. Assume that at time step 0, ZT has just been switched from 0 to .05. Shown are the time courses of the PC's weighted sum, s, activation state, y, and the position and velocity of the mass. of this learning rule in light of physiological data and cellular mechanisms. 5 SIMULATIONS We performed a number of simulations of the simplified APG model learning to control the nonlinear spring-mass system. We trained each version of the model to move the mass from initial positions selected randomly from the interval [-.02, .02] to a target position randomly set to .03, .04, or .05. We set the pulse height, zP' and the step height, z" to .1 and .04 respectively. Each simulation consisted of a series of trial movements. The parameters of the learning rule, which were not optimized, were 0: = .0004 and ,8 = .00004. Eligibility traces decayed 1% per time step. Figure 3 shows time courses of relevant variables at different stages in learning to move to target endpoint ZT = .05 from initial position Zo = O. Early in learning (Panel A), the PC has learned to switch to low at the beginning of the trial but A Predictive Switching Model of Cerebellar Movement Control 143 switches back to high too soon, which causes the mass to undershoot the target. Because of this undershoot, the CF fires at the end of the movement due to a final very small corrective movement generated by an extracerebellar system. The mass sticks at Ze = .027. Late in learning (Panel B), the mass sticks at Ze = .049, and the CF does not fire. Note that to accomplish this, the PC state has to switch to high well before (about 150ms) the endpoint is reached. Figure 4 shows three representations of the switching curve learned by a version of the model for target ZT = .05. As an aid to understanding the model's behavior, all the proprioceptive signals in this version of the model had the same delay of 30ms (Tl in Fig. 2) instead of the more realistic distribution of delays described above. Hence the total loop delay (Tl + T2) was 50ms. The curve labeled "spring switch", which closely coincides with the optimal switching curve (also shown), plots states that the spring-mass system passes through when the command input to the spring switches. In other words, this is the switching curve as seen from the point marked A in Fig. 2. That this coincides with the optimal switching curve shows that the model learned to behave correctly. The movement trajectory crosses this curve about 150ms before the movement ends. 0.6 0.5 0.2 0.' ' ... _.-. ' ....... '._._. o 0.02 P_lm) • _ .• Prap_opdvo ..... -PCs..ild\ - Spring SWItch ..... ~ Troj.-y 0.04 0.06 Figure 4: Phase-plane portraits of switching curves implemented by the model after learning. Four switching curves and one movement trajectory are shown. See text for explanation. The curve labeled "PC switch", on the other hand, plots states that the spring-mass system passes through when the PC unit switches state: it is the switching curve as seen from the point marked B in Fig. 2 (assuming the expansive encoding involves no delay). The state of the spring-mass system crosses this curve 20ms before it reaches the "spring switch" curve. One can see, therefore, that the PC unit learned to switch its activity state 20ms before the motor command must switch state at the spring itself, appropriately compensating for the 20ms latency of the efferent pathway. We can also ask what is the state of the spring-mass system that the PC actually "sees", via proprioceptive signals, when it has to switch state. When the PC has to switch states, that is, when the spring-mass state reaches switching curve "PC switch", the PC is actually receiving via its PF input a description of the system state that occurred a significant time earlier (Tl = 30ms in Fig. 2). Switching curve "proprioceptive input" in Fig. 4 is the locus of system states that the PC is sensing when it has to switch. The PC has learned to do this by learning, on the basis of delayed CF training information, to switch when it sees PF patterns that code spring-mass states that lie on curve "proprioceptive input". 6 DISCUSSION The model we have presented is most closely related to adaptive control methods known as direct predictive adaptive controllers (Goodwin & Sin, 1984). Feedback delays pose no particular difficulties despite the fact that no use is made of a forward model of the motor plant. Instead of producing predictions of proprioceptive 144 A. G. BARTO, 1. T. BUCKINGHAM, J. C. HOUK feedback, the model uses its predictive capabilities to directly produce appropriately timed motor commands. Although the nonlinear viscosity of the spring-mass system renders linear control principles inapplicable, it actually makes the control problem easier for an appropriate controller. Fast movements can be performed with little or no oscillation. We believe that similar nonlinearities in actual motor plants have significant implications for motor control. A critical feature of this model's learning mechanism is its use of eligibility traces to bridge the temporal gap between a PC's activity and the consequences of this activity on the movement endpoint. Cellular studies are needed to explore this important issue. Although nothing in the present paper suggests how this might extend to more complex control problems, one of the objectives of the full APG model is to explore how the collective behavior of multiple APG modules might accomplish more complex control. Acknowledgements This work was supported by NIH 1-50 MH 48185-04. References Albus, JS (1971). A theory of cerebellar function. Mathematical Biosciences, 10, 25-61. Berthier, NE, Singh, SP, Barto, AG, & Houk, JC (1993). Distributed representations of limb motor programs in arrays of adjustable pattern generators. Cognitive Neuroscience, 5, 56-78. Buckingham, JT, Barto, AG, & Houk, JC (1995). Adaptive predictive control with a cerebellar model. In: Proceedings of the 1995 World Congress on Neural Networks, 1-373-1-380. Gellman, R, Gibson, AR, & Houk, JC (1983). Somatosensory properties of the inferior olive of the cat. J. Compo Neurology, 215, 228-243. Goodwin, GC & Sin, KS (1984). Adaptive Filtering Prediction and Control. Englewood Cliffs, N.J.: Prentice-Hall. Houk, JC, Buckingham, JT, & Barto, AG (1995). Models of the cerebellum and motor learning. Brain and Behavioral Sciences, in press. Ito, M (1984). The Cerebellum and Neural Control. New York: Raven Press. Keeler, JD (1990). A dynamical system view of cerebellar function. Physica D, 42, 396-410. Klopf, AH (1982). The Hedonistic Neuron: A Theory of Memory, Learning, and Intelligence. Washington, D.C.: Hemishere. Marr, D (1969). A theory of cerebellar cortex. J. Physiol. London, 202, 437-470. Miall, RC, Weir, DJ, Wolpert, DM, & Stein, JF (1993). Is the cerebellum a smith predictor? Journal of Motor Behavior, 25, 203-216. Van Kan, PLE, Gibson, AR, & Houk, JC (1993). Movement-related inputs to intermediate cerebellum ofthe monkey. Journal of Physiology, 69, 74-94. Wolpert, DM, Ghahramani, Z, & Jordan, MI (1995). Foreward dynamic models in human motor control: Psychophysical evidence. In: Advances in Neural Information Processing Systems 7, (G Tesauro, DS Touretzky, & TK Leen, eds) , Cambridge, MA: MIT Press. Wu, CH, Houk, JC, Young, KY, & Miller, LE (1990). Nonlinear damping of limb motion. In: Multiple Muscle Systems: Biomechanics and Movement Organization, (J Winters & S Woo, eds). 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Recurrent Neural Networks for Missing or Asynchronous Data Yoshua Bengio Dept. Informatique et Recherche Operationnelle Universite de Montreal Montreal, Qc H3C-3J7 bengioy~iro.umontreal.ca Abstract Francois Gingras Dept. Informatique et Recherche Operationnelle Universite de Montreal Montreal, Qc H3C-3J7 gingra8~iro.umontreal.ca In this paper we propose recurrent neural networks with feedback into the input units for handling two types of data analysis problems. On the one hand, this scheme can be used for static data when some of the input variables are missing. On the other hand, it can also be used for sequential data, when some of the input variables are missing or are available at different frequencies. Unlike in the case of probabilistic models (e.g. Gaussian) of the missing variables, the network does not attempt to model the distribution of the missmg variables given the observed variables. Instead it is a more "discriminant" approach that fills in the missing variables for the sole purpose of minimizing a learning criterion (e.g., to minimize an output error). 1 Introduction Learning from examples implies discovering certain relations between variables of interest. The most general form of learning requires to essentially capture the joint distribution between these variables. However, for many specific problems, we are only interested in predicting the value of certain variables when the others (or some of the others) are given. A distinction IS therefore made between input variables and output variables. Such a task requires less information (and less p'arameters, in the case of a parameterized model) than that of estimating the full joint distrIbution. For example in the case of classification problems, a traditional statistical approach is based on estimating the conditional distribution of the inputs for each class as well as the class prior probabilities (thus yielding the full joint distribution of inputs and classes). A more discriminant approach concentrates on estimating the class boundaries (and therefore requires less parameters), as for example with a feedforward neural network trained to estimate the output class probabilities given the observed variables. However, for many learning problems, only some ofthe input variables are given for each particular training case, and the missing variables differ from case to case. The simplest way to deal with this mIssing data problem consists in replacing the missing values by their unconditional mean. It can be used with "discriminant" training algorithms such as those used with feedforward neural networks. However, in some problems, one can obtain better results by taking advantage of the dependencies between the input variables. A simple idea therefore consists -also, AT&T Bell Labs, Holmdel, NJ 07733 396 Y. BENGIO, F. GINGRAS Figure 1: Architectures of the recurrent networks in the experiments. On the left a 90-3-4 architecture for static data with missing values, on the right a 6-3-2-1 architecture with multiple time-scales for asynchronous sequential data. Small squares represent a unit delay. The number of units in each layer is inside the rectangles. The time scale at which each layer operates is on the right of each rectangle. in replacing the missing input variables by their conditional expected value, when the observed input variables are given. An even better scheme is to compute the expected output given the observed inputs, e.g. with a mixture of Gaussian. Unfortunately, this amounts to estimating the full joint distribution of all the variables. For example, with ni inputs, capturing the possible effect of each observed variable on each missing variable would require O(nl) parameters (at least one parameter to capture some co-occurrence statistic on each pair of input variables). Many related approaches have been proposed to deal with missing inputs using a Gaussian (or Gaussian mixture) model (Ahmad and Tresp, 1993; Tresp, Ahmad and Neuneier, 1994; Ghahramani and Jordan, 1994). In the experiments presented here, the proposed recurrent network is compared with a Gaussian mixture model trained with EM to handle missing values (Ghahramani and Jordan, 1994). The approach proposed in section 2 is more economical than the traditional Gaussian-based approaches for two reasons. Firstly, we take advantage of hidden units in a recurrent network, which might be less numerous than the inputs. The number of parameters depends on the product of the number of hidden units and the number of inputs. The hidden units only need to capture the dependencies between input variables which have some dependencies, and which are useful to reducing the output error. The second advantage is indeed that training is based on optimizing the desired criterion (e.g., reducing an output error), rather than predIcting as well as possible the values of the missmg inputs. The recurrent network is allowed to relax for a few iterations (typically as few as 4 or 5) in order to fill-in some values for the missing inputs and produce an output. In section 3 we present experimental results with this approach, comparing the results with those obtained with a feedforward network. In section 4 we propose an extension of this scheme to sequential data. In this case, the network is not relaxing: inputs keep changing with time and the network maps an input sequence (with possibly missing values) to an output sequence. The main advantage of this extension is that It allows to deal with sequential data in which the variables occur at different frequencies. This type of problem is frequent for example with economic or financial data. An experiment with asynchronous data is presented in section 5. 2 Relaxing Recurrent Network for Missing Inputs Networks with feedback such as those proposed in (Almeida, 1987; Pineda, 1989) can be applied to learning a static input/output mapping when some of the inputs are missing. In both cases, however, one has to wait for the network to relax either to a fixed point (assuming it does find one) or to a "stable distribution" (in the case of the Boltzmann machine). In the case of fixedpoint recurrent networks, the training algorithm assumes that a fixed point has been reached. The gradient with respect to the weIghts is then computed in order to move the fixed point to a more desirable position. The approach we have preferred here avoids such an assumption. Recurrent Neural Networks for Missing or Asynchronous Data 397 Instead it uses a more explicit optimization of the whole behavior of the network as it unfolds in time, fills-in the missing inputs and produces an output. The network is trained to minimize some function of its output by back-propagation through time. Computation of Outputs Given Observed Inputs Given: input vector U = [UI, U2, ..• , un,] Resul t: output vector Y = [YI, Y2, .. . , Yn.l 1. Initialize for t = 0: For i = 1 ... nu,xo,; f- 0 For i = 1 . . . n;, if U; is missing then xO,1(;) f- E( i), . Else XO,1(i) f- Ui· 2. Loop over tl.me: For t = 1 to T For i = 1 ... nu If i = I(k) is an input unit and Uk is not missing then Xt if-Uk Else ' Xt,i f- (1- "Y)Xt-I,i + "YfCEles, WIXt_d/,p/) where Si is a set of links from unit PI to unit i, each with weight WI and a discrete delay dl (but terms for which t - dl < 0 were not considered). 3. Collect outputs by averaging at the end of the sequence: Y; f- 'L;=I Vt Xt,O(i) Back-Propagation The back-propagation computation requireE! an extra set of variables Xt and W, which will contain respectively g~ and ~~ after this computation. Given: output gradient vector ~; Resul t: input gradient ~~ and parameter gradient ae aw' 1. Initialize unit gradients using outside gradient: Initialize Xt,; = 0 for all t and i. For i = 1 . . . no, initialize Xt,O(;) f- Vt Z~ 2. Backward loop over time: For t = T to 1 For i = nu ... 1 If i = I(k) is an input unit and Uk is not missing then no backward propagation Else For IE S; 1ft - d! > 0 Xt-d/,p/ f- Xt-d/,p/ + (1 - "Y)Xt-d/+1 3. Collect input For i = 1 .. . ni, + "YwIXt,d'('L/es, WIXt_d/,p/) WI f- WI + "Yf'CLAes, WIXt-d/ ,p/)Xt-d/,p/ gradients: If U; is missing, then ae f- 0 au; Else ae ". au, f- l..Jt Xt,1(;) The observed inputs are clamped for the whole duration of the sequence. The missing units corresponding to missing inputs are initialized to their unconditional expectation and their value is then updated using the feedback links for the rest of the sequence (just as if they were hidden units). To help stability of the network and prevent it from finding periodic solutions (in which the outputs have a correct output only periodically), output supervision is given for several time steps. A fixed vector v, with Vt > 0 and I':t Vt = 1 specifies a weighing scheme that distributes 398 Y. BENGIO, F. GINGRAS the responsibility for producing the correct output among different time steps. Its purpose is to encourage the network to develop stable dynamics which gradually converge toward the correct output tthus the weights Vt were chosen to gradually increase with t) . The neuron transfer function was a hyperbolic tangent in our experiments. The inertial term weighted by , (in step 3 of the forward propagation algorithm below) was used to help the network find stable solutions. The parameter, was fixed by hand. In the experiments described below, a value of 0.7 was used, but near values yielded similar results. This module can therefore be combined within a hybrid system composed of several modules by propagating gradient through the combined system (as in (Bottou and Gallinari, 1991)). For example, as in Figure 2, there might be another module taking as input the recurrent network's output. In this case the recurrent network can be seen as a feature extractor that accepts data with missing values in input and computes a set of features that are never missing. In another example of hybrid system the non-missing values in input of the recurrent network are computed by another, upstream module (such as the preprocessing normalization used in our experiments), and the recurrent network would provide gradients to this upstream module (for example to better tune its normalization parameters) . 3 Experiments with Static Data A network with three layers (inputs, hidden, outputs) was trained to classify data with missing values from the audiolD9Y database. This database was made public thanks to Jergen and Quinlan, was used by (Barelss and Porter, 1987), and was obtained from the UCI Repository of machine learning databases (ftp. ies . ueL edu: pub/maehine-learning-databases). The original database has 226 patterns, with 69 attributes, and 24 classes. Unfortunately, most of the classes have only 1 exemplar. Hence we decided to cluster the classes into four groups. To do so, the average pattern for each of the 24 classes was computed, and the K-Means clustering algorithm was then applied on those 24 prototypical class "patterns", to yield the 4 "superclasses" used in our experiments. The multi-valued input symbolic attributes (with more than 2 possible values) where coded with a "one-out-of-n" scheme, using n inputs (all zeros except the one corresponding to the attribute value). Note that a missing value was represented with a special numeric value recognized by the neural network module. The inputs which were constant over the training set were then removed. The remaining 90 inputs were finally standardized (by computing mean and standard deviation) and transformed by a saturating non-linearity (a scaled hyperbolic tangent). The output class ~s coded with a "one-out-of-4" scheme, and the recognized class is the one for which the corresponding output has the largest value. The architecture of the network is depicted in Figure 1 (left) . The length of each relaxing sequence in the experiments was 5. Higher values would not bring any measurable improvements, whereas for shorter sequences performance would degrade. The number of hidden units was varied, with the best generalization performance obtained using 3 hidden units. The recurrent network was compared with feedforward networks as well as with a mixture of Gaussians. For the feedforward networks, the missing input values were replaced by their unconditional expected value. They were trained to minimize the same criterion as the recurrent networksl i.e., the sum of squared differences between network output and desired output. Several feedtorward neural networks with varying numbers of hidden units were trained. The best generalization was obtained with 15 hidden units. Experiments were also performed with no hidden units and two hidden layers (see Table 1). We found that the recurrent network not only generalized better but also learned much faster (although each pattern required 5 times more work because of the relaxation), as depicted in Figure 3. The recurrent network was also compared with an approach based on a Gaussian and Gaussian mixture model of the data. We used the algorithm described in (Ghahramani and Jordan, 1994) for supervised leaning from incomplete data with the EM algorithm. The whole joint input/output distribution is modeled using a mixture model with Gaussians (for the inputs) and multinomial (outputs) components: P(X = x, C = c) = E P(Wj) (21r)S;I~jll/2 exp{ -~(x _lJj)'Ejl(X -lJj)} j where x is the input vector, c the output class, and P(Wj) the prior probability of component j of the mixture. The IJjd are the multinomial parameters; IJj and Ej are the Gaussian mean vector Recurrent Neural Networks for Missing or Asynchronous Data coat -down atllNlm static module UpA'ellm normalization module 399 Figure 2: Example of hybrid modular system, using the recurrent network (middle) to extract features from patterns which may have missing values. It can be combined with upstream modules (e.g., a normalizing preprocessor, right) and downstream modules (e.g., a static classifier, left). Dotted arrows show the backward flow of gradients. training eet 50r-----~----~----~----~----_, 45 40 35 30 ~ 25 .... 20 15 10 5 40 teat .et 50r-----~----~----~----~----_, 45 40 35 30 ~ 25 .... 20 15 f •• dforvvard recurrent 40 Figure 3: Evolution of training and test error for the recurrent network and for the best of the feedforward networks (90-15-4): average classification error w.r.t. training epoch, (with 1 standard deviation error bars, computed over 10 trials). and covariance matrix for component j. Maximum likelihood training is applied as explained in (Ghahramani and Jordan, 1994), taking missing values into account (as additional missing variables of the EM algorithm). For each architecture in Table 1, 10 trainin~ trials were run with a different subset of 200 training and 26 test patterns (and different initial weights for the neural networks). The recurrent network was dearlr superior to the other architectures, probably for the reasons discussed in the conclusion. In addItion, we have shown graphically the rate of convergence during training of the best feedforward network (90-15-4) as well as the best recurrent network (90-3-4), in Figure 3. Clearly, the recurrent network not only performs better at the end of traming but also learns much faster. 4 Recurrent Network for Asynchronous Sequential Data An important problem with many sequential data analysis problems such as those encountered in financial data sets is that different variables are known at different frequencies, at different times (phase), or are sometimes missing. For example, some variables are given daily, weekly, monthly, quarterly, or yearly. Furthermore, some variables may not even be given for some of the periods or the precise timing may change (for example the date at which a company reports financial performance my vary). Therefore, we propose to extend the algorithm presented above for static data with missing values to the general case of sequential data with missing values or asynchronous variables. For time steps at which a low-frequency variable is not given, a missing value is assumed in input. Again, the feedback links from the hidden and output units to the input units allow the network 400 Y. BENGIO, F. GINGRAS Table 1: Comparative performances of recurrent network, feedforward network, and Gaussian mixture density model on audiology data. The average percentage of classification error is shown after training, for both training and test sets, and tlie standard deviation in parenthesis, for 10 trials. Trammg set error Test set error 90-3-4 Recurrent net 0.3(~ . 6 2.~(?(j 90-6-4 Recurrent net 0(0 3.8(4 90-25-4 Feedforward net °r 6 15(7.3 90-15-4 Feedforward net 0.80.4 13.8(7 90-10-6-4 Feedforward net 1 0.9 16f5.3 90-6-4 Feedforward net 64.9 298.9 90-2-4 Feedforward net 18.5? 27(10 90-4 Feedforward net 22 1 33(8 1 Gaussian 35 1.6 38 9.3 4 Gaussians Mixture 36 1.5 38 9.2 8 Gaussians Mixture 36 2.1 38 9.3 to "complete" the missing data. The main differences with the static case are that the inputs and outputs vary with t (we use Ut and Yt at each time step instead of U and y). The training algorithm is otherwise the same. 5 Experiments with Asynchronous Data To evaluate the algorithm, we have used a recurrent network with random weights, and feedback links on the input units to generate artificial data. The generating network has 6 inputs 3 hidden and 1 outputs. The hidden layer is connected to the input layer (1 delay). The hidden layer receives inputs with delays 0 and 1 from the input layer and with delay 1 from itself. The output layer receives inputs from the hidden layer. At the initial time step as well as at 5% of the time steps (chosen randomly), the input units were clamped with random values to introduce some further variability. The mlssing values were then completed by the recurrent network. To generate asynchronous data, half of the inputs were then hidden with missing values 4 out of every 5 time steps. 100 training sequences and 50 test sequences were generated. The learning problem is therefore a sequence regression problem with mlssing and asynchronous input variables. Preliminary comp'arative experiments show a clear advantage to completing the missing values (due to the the dlfferent frequencies of the input variables) wlth the recurrent network, as shown in Figure 4. The recognition recurrent network is shown on the right of Figure 1. It has multiple time scales (implemented with subsampling and oversampling, as in TDNNs (Lang, Waibel and Hinton, 1990) and reverse-TDNNs (Simard and LeCun, 1992)), to facilitate the learning of such asynchronous data. The static network is a time-delay neural network with 6 input, 8 hidden, and 1 output unit, and connections with delays 0,2, and 4 from the input to hidden and hidden to output units. The "missing values" for slow-varying variables were replaced by the last observed value in the sequence. Experiments with 4 and 16 hidden units yielded similar results. 6 Conclusion When there are dependencies between input variables, and the output prediction can be improved by taking them into account, we have seen that a recurrent network with input feedback can perform significantly better than a simpler approach that replaces missing values by their unconditional expectation. According to us, this explains the significant improvement brought by using the recurrent network instead of a feedforward network in the experiments. On the other hand, the large number of input variables (n; = 90, in the experiments) most likely explains the poor performance of the mixture of Gaussian model in comparison to both the static networks and the recurrent network. The Gaussian model requires estimating O(nn parameters and inverting large covariance matrices. The aPl?roach to handling missing values presented here can also be extended to sequential data with mlssing or asynchronous variables. As our experiments suggest, for such problems, using recurrence and multiple time scales yields better performance than static or time-delay networks for which the missing values are filled using a heuristic. Recurrent Neural Networks for Missing or Asynchronous Data 401 "().18 0.18 0.104 f 0 .12 i tlme-delay network j 0 .1 0.08 0.08 ~ ___ -lecurrent network 0 .040 2 ~ 8 8 10 12 1~ 18 18 20 training 4tPOCh Figure 4: Test set mean squared error on the asynchronous data. Top: static network with time delays. Bottom: recurrent network with feedback to input values to complete missing data. References Ahmad, S. and Tresp, V. (1993) . Some solutions to the missing feature problem in vision. In Hanson, S. J., Cowan, J. D., and Giles, C. L., editors, ACivances in Neural Information Processing Systems 5, San Mateo, CA. Morgan Kaufman Publishers. Almeida, L. (1987). A learning rule for asynchronous perceptrons with feedback in a combinatorial environment. In Caudill, M. and Butler, C., editors, IEEE International Conference on Neural Networks, volume 2, pages 609- 618, San Diego 1987. IEEE, New York. Bareiss, E. and Porter, B. (1987). Protos: An exemplar-based learning apprentice. In Proceedings of the 4th International Workshop on Machine Learning, pages 12-23, Irvine, CA. Morgan Kaufmann. Bottou, L. and Gallinari, P. (1991). A framework for the cooperation of learning algorithms. In Lippman, R. P., Moody, R., and Touretzky, D. S., editors, Advances in Neural Information Processing Systems 3, pages 781-788, Denver, CO. Ghahramani, Z. and Jordan, M. I. (1994). Supervised learning from incomplete data via an EM approach. In Cowan, J. , Tesauro, G., and Alspector, J., editors, Advances in Neural Information Processing Systems 6, page ,San Mateo, CA. Morgan Kaufmann. Lang) K. J ., Waibel, A. H., and Hinton, G. E. (1990). A time-delay neural network architecture tor isolated word recognition. Neural Networks, 3:23- 43. Pineda, F. (1989). Recurrent back-propagation and the dynamical approach to adaptive neural computation. Neural Computation, 1:161- 172. Simard, P. and LeCun, Y. (1992) . Reverse TDNN: An architecture for trajectory generation. In Moody, J ., Hanson, S., and Lipmann, R. , editors, Advances in Neural Information Processing Systems 4, pages 579- 588, Denver, CO. Morgan Kaufmann, San Mateo. Tresp, V., Ahmad, S., and Neuneier, R. (1994). Training neural networks with deficient data. In Cowan, J., Tesauro, G., and Alspector, J., editors, Advances in Neural Information Processing Systems 6, pages 128-135. Morgan Kaufman Publishers, San Mateo, CA.
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A Model of Spatial Representations in Parietal Cortex Explains Hemineglect Alexandre Pouget Dept of Neurobiology UCLA Los Angeles, CA 90095-1763 alex@salk.edu Terrence J. Sejnowski Howard Hughes Medical Institute The Salk Institute Abstract La Jolla, CA 92037 terry@salk.edu We have recently developed a theory of spatial representations in which the position of an object is not encoded in a particular frame of reference but, instead, involves neurons computing basis functions of their sensory inputs. This type of representation is able to perform nonlinear sensorimotor transformations and is consistent with the response properties of parietal neurons. We now ask whether the same theory could account for the behavior of human patients with parietal lesions. These lesions induce a deficit known as hemineglect that is characterized by a lack of reaction to stimuli located in the hemispace contralateral to the lesion. A simulated lesion in a basis function representation was found to replicate three of the most important aspects of hemineglect: i) The models failed to cross the leftmost lines in line cancellation experiments, ii) the deficit affected multiple frames of reference and, iii) it could be object centered. These results strongly support the basis function hypothesis for spatial representations and provide a computational theory of hemineglect at the single cell level. 1 Introduction According to current theories of spatial representations, the positions of objects are represented in multiple modules throughout the brain, each module being specialized for a particular sensorimotor transformation and using its own frame of reference. For instance, the lateral intraparietal area (LIP) appears to encode the location of objects in oculocentric coordinates, presumably for the control of saccadic eye movements. The ventral intraparietal cortex (VIP) and the premotor cortex, on the other hand, seem to use head-centered coordinates and might be A Model of Spatial Representations in Parietal Cortex Explains Hemineglect A ..... -----.a... Right Stimulus ~ Left .-----Stimulus Cl C2 B it~ Cl C2 C3 FP Target Distractors ! \~ Cl + ••• " C2 + ••• " C3 11 Figure 1: A. Retinotopic neglect modulated by egocentric position. B. Stimuluscentered neglect involved in the control of hand movements toward the face. This modular theory of spatial representations is not fully consistent with the behavior of patients with parietal or frontal lesions. Such lesions causes a syndrome known as hemineglect which is characterized by a lack of response to sensory stimuli appearing in the hemispace contralateral to the lesion [3]. According to the modular view, the deficit should be behavior dependent, e.g., oculocentric for eye movements, head-centered for reaching. However, experimental and clinical studies show that this is not the case. Instead, neglect affects multiple frames of reference simultaneously, and to a first approximation, independently of the task. This point is particularly clear in an experiment by Karnath et al (1993) (Figure 1A). Subjects were asked to identify a stimulus that can appear on either side of the fixation point. In order to test whether the position of the stimuli with respect to the body affects performance, two conditions were tested: a control condition with head straight ahead (C1), and a second condition with head rotated 20 degrees on the right (or equivalently, with the trunk rotated 20 degrees on the left, see figure) (C2). In C2, both stimuli appeared further to the right ofthe trunk while being at the same location with respect to the head and retina than in Cl. Moreover, the trunk-centered position of the left stimulus in C2 was the same than the trunk-centered position of the right stimulus in C1. As expected, subjects with right parietal lesions performed better on the right stimulus in the control condition, a result consistent with both, retinotopic and trunk-centered neglect. To distinguish between the two frames of reference, one needs to compare performance across conditions. If the deficit is purely retinocentric, the results should be identical in both conditions, since the retinotopic location of the stimuli does not vary. If, on the other hand, the deficit is purely trunk-centered, the performance on the left stimulus should improve when the head is turned right since the stimulus now appears further toward the right of the trunk-centered hemispace. Furthermore, performance on the right stimulus in the control condition should be the same as performance on the left stimulus in the rotated condition, since they share the same trunk-centered position in both cases. 12 A. POUGET, T. J. SEJNOWSKI N either of these hypotheses can fully account for the data. As expected from a retinotopic neglect, subjects always performed better on the right stimulus in both conditions. However, performance on the left stimulus improved when the head was turned right (C2), though not sufficiently to match the level of performance on the right stimulus in the control condition (C1). Therefore, these results suggest a retinotopic neglect modulated by trunk-centered factors. In addition, Karnath et al (1991) tested patients on a similar experiment in which subjects were asked to generate a saccade toward the target. The analysis of reaction time revealed the same type of results than the one found in the identification task, thereby demonstrating that the spatial deficit is, to a first approximation, independent of the task. An experiment by Arguin and Bub (1993) suggests that neglect can be objectcentered as well. As shown in figure 1B, they found that reaction times were faster when the target appeared on the right of a set of distractors (C2), as opposed to the left (C1), even though the target is at the same retinotopic location in both conditions. Interestingly, moving the target further to the right leads to even faster reaction times (C3), showing that hemineglect is not only object-centered but retinotopic as well in this task. These results strongly support the existence of spatial representations using multiple frames of reference simultaneously shared by several behaviors. We have recently developed a theory [6] which has precisely these properties and we ask here whether a simulated lesion would lead to a deficit similar to hemineglect. Our theory posits that parietal neurons computes basis function (BF) of sensory signals, such as visual, or auditory inputs, and posture signals, such as eye or head position. The resulting representation, which we called a basis function map, can be used for performing nonlinear transformations of the sensory inputs, the type of transformations required for sensorimotor coordination. 2 Model Organization The model contains two distinct parts: a network for performing sensorimotor transformations and a selection mechanism. 2.1 Network Architecture We implemented a network using basis function units in the intermediate layer to perform a transformation from a visual retinotopic map to two motor maps in, respectively, head-centered and oculocentric coordinates (Figure 2). The input contains a retinotopic visual map analog to the one found in the early stages of visual processing, and a set of units encoding eye position, similar to the neurons found in the intralaminar nucleus of the thalamus. These input units project to a set of intermediate units shared by both transformations. Each intermediate unit computes a gaussian of the retinal location of object, rx , multiplied by a sigmoid of eye position, ex: Oi=----1 + e- e:Z:-/;rj (1) These units are organized in a map covering all possible combinations of retinal and eye position selectivities. As we have shown elsewhere [6], this type of response function is consistent with the response of single parietal neurons found in area 7a. A Model of Spatial Representations in Parietal Cortex Explains Hemineglect A Saccadic Eye Movements Reaching t t (00000000000000) (0000000000009) Retinotopic map (Superior CollicuJus) Head-centered map (Premotor Cortex) ,. 00000000000000 00000000000000 ~ , 8 00000000000000 ] . :::::::::::::: .~ 0 00000000000000 & ., :::::::::::::: Q) -,0 00000000000000 ~ .1' :::::::::::::0 Retinal position (0) 000000000 BFmap (7a) Retinotopic map (VI) Eye position cells (Thalamus) B t&:-6 .. t .,', J0,\~ Retinal position (0) Head-centered position (0) ° c:: BFmap ,g l (7a) ~ '" " 7~i1i tJ\,L\. ~ ~ _20 _10 0 10 .. Retinal position (0) . Figure 2: A. Network architecture B. Typical pattern of activity 13 The resulting map forms a basis function map which encodes the location of objects in head-centered and retinotopic coordinates simultaneously. The activity of the unit in the output maps is computed by a simple linear combination of the BF unit activities. Appropriate values of the weights were found by using linear regression techniques. This architecture mimics the pattern of projections of the parietal area 7a. 7a is known to project to, both, the superior colliculus and the premotor cortex (via the ventral parietal area, VIP) , in which neurons have, respectively, retinotopic and head-centered visual receptive fields. Figure 2B shows a typical pattern of activity in the network when two stimuli are presented simultaneously while the eye fixated 10 degrees toward the right. 2.2 Hemispheric Biases and Lesion Model Neurophysiological data indicate that both hemispheres contain neurons with all possible combinations of retinal and eye position selectivities, but with a contralateral bias. Hence, most neurons in the right parietal cortex (resp. left) have their retinal receptive field on the left hemiretina (resp. right) . The bias for eye position is much weaker but a trend has been reported in several studies [1] . Therefore, spatial representations in a patient with a right parietal lesions are biased toward the right side of space. We modeled such a lesion by using a similar bias in the intermediate layer of our network. The BF map simply has more neurons tuned to right retinal and eye positions. We found that the exact profile of the neuronal gradient across the basis function maps did not matter as long as it was monotonic and contralateral for both eye position and retinal location. 2.3 Selection model We also developed a selection mechanism to model the behavior of patients when presented with several stimuli simultaneously. The simultaneous presentation of 14 A. POUGET, T. J. SEJNOWSKI stimuli induces multiple hills of activity in the network (see for instance the pattern of activity shown in figure IB for two visual stimuli). Our selection mechanism operates on the peak values of these hills. At each time step, the most active stimulus is selected according to a winner-takeall and its corresponding activity is set to zero (inhibition of return). At the next time step, the second highest stimuli is selected while the previously selected item is allowed to recover slowly. This procedure ensures that the most active item is not selected twice in a row, but because of the recovery process, stimulus with high activity might be selected again if displayed long enough. This mechanism is such that the probability of selecting an item is proportional to two factors: the absolute amount of activity associated with the item, and the relative activity with respect to other competing items. 2.4 Evaluating network performance We used this model to simulate several experiments in which patient performance was evaluated according to reaction time or percent of correct response. Reaction time in the model was taken to be proportional to the number of time steps required by our selection mechanism to select a particular target. Performance on identification task was assumed to be proportional to the strength of the activity generated by the stimuli in the BF map. 3 Results 3.1 Line cancellation We first tested the network on the line cancellation test, a test in which patients are asked to cross out short line segments uniformly spread over a page. To simulate this test, we presented the display shown in figure 3A and we ran the selection mechanism to determine which lines get selected by the network. As illustrated in figure 3A, the network crosses out only the lines located in the right half of the display, just as left neglect patients do in the same task. The rightward gradient introduced by the lesion biases the selection mechanism in favor of the most active lines, i.e., the ones on the right. As a result, the rightmost lines win the competition over and over, preventing the network from selecting the left lines. 3.2 Mixture of frames of reference Next, we sought to determine the frame of reference of neglect in the model. Since Karnath et al (1993) manipulated head position, we simulated their experiment by using a BF map integrating visual inputs with head position, rather than eye position. We show in figure 3B the pattern of activity obtained in the retinotopic output layer of the network in the various experimental conditions (the other maps behaved in a similar way). In both conditions, head straight ahead (dotted lines) or turned on the side (solid lines), the right stimulus is associated with more activity than the left stimulus. This is the consequence of the larger number of cells in the basis function map for rightward position. In addition, the activity for the left stimulus increases when the head is turned to the right. This effect is related to the larger number of cells in the basis function maps tuned to right head positions. Since network performance is proportional to activity strength, the overall pattern of performance was found to be similar to what has been reported in human patients A Model of Spatial Representations in Parietal Cortex Explains Hemineglect A B Left Stimulus Right Stimulus c a1 ___________ _ ., , , ,., !, ,. " I , ! ',I· '_.' '\ I .1 '\ + )( ••• C1 FP Target Distractors a21-----:--;-~-~ ." .... ,. '.' ... ~ + ••• )( a3 ----------------------,', ,.. " " . . 1". I· './ ' , ... C2 + ••• )( C3 15 Figure 3: Network behavior in line cancellation task (A). Activity patterns in the retinotopic output layer when simulating the experiments by Karnath et al (1993) (B) and Arguin et al (1993) (C) (figure lA), namely: the right stimulus was better processed than the left stimulus and performance on the left stimulus increases when the head is rotated toward the right. Therefore, just like in human, neglect in the model is neither retinocentric nor trunk-centered alone, but both at the same time. 3.3 Object-centered effect When simulating Arguin et al (1993) experiments, the network reaction times were found to follow the same trends than for human patients. Figure 3C illustrates the patterns of activity in the retinotopic output layer of the network when simulating the three conditions of Arguin experiments. Notice that the absolute activity associated with the target (solid lines) in conditions 1 and 2 is the same, but the activity of the distractors (dotted lines) differs in the two conditions. In condition 1, they have higher relative activity and thereby strongly delay the detection of the target by the selection mechanism. In condition 2, the distractors are now less active than the target and do not delay target processing as much as they do in condition 1. The reaction time decreases even more in condition 3, due to a higher absolute activity associated with the target. Therefore, the network exhibits retinocentric and object-centered neglect, just like parietal patients [2]. 4 Discussion The model of parietal cortex presented here was originally developed by considering the response properties of parietal neurons and the computational constraints inherent in sensorimotor transformations. It was not designed to model neglect, so its ability to account for a wide range of deficits is additional evidence in favor of the basis function hypothesis. As we have shown, our model captures three essential aspects of the neglect syndrome: 1) It reproduces the pattern of line crossing reported in patients in linecancellation experiments, 2) the deficit coexists in multiple frames of reference simultaneously, and 3) the model accounts for some of the object-based effects. 16 A. POUGET, T. J. SEJNOWSKI We can account for a very large number of studies beyond the ones we have considered here, using very similar computational principles. We can reproduce, in particular, the behavior of patients in line-bisection experiments and we can explain why neglect affects multiple cartesian frames of reference such as retinotopic, head-centered, trunk-centered, environment-centered (i.e. with respect to gravity), and object-centered. It must be emphasized that these results have been obtained without using explicit representations of these various cartesian frames of reference (except for the retinotopy of the BF map). In fact, this is precisely because the lesion affected noncartesian representations that we have been able to reproduce these results. We have assumed that the lesion affects the functional space in which the basis functions are defined. This functional space shares common dimensions with cartesian spaces, but cannot be reduced to the latter. Hence, a basis function map integrating retinal location and head position is retinotopic, but not solely retinotopic. Consequently, any attempts to determine the cartesian space in which hemineglect operates is bound to lead to inconclusive results in which cartesian frames of reference appear to be mixed. This study and previous research [6] suggests that the parietal cortex represents the position of objects by computing basis functions of the sensory and posture inputs. It would now be interesting to see if this hypothesis could also account for sensorimotor adaptation, such as learning to reach properly when wearing visual prisms. We predict that adaptation takes place in several frames of reference simultaneously, a prediction that is testable and would provide further support for the basis function framework. References [1] R.A. Andersen, C. Asanuma, G. Essick, and R.M. Siegel. Corticocortical connections of anatomically and physiologically defined subdivisions within the inferior parietal lobule. Journal of Comparative Neurology, 296(1):65-113,1990. [2] M. Arguin and D.N. Bub. Evidence for an independent stimulus-centered reference frame from a case of visual hemineglect. Cortex, 29:349-357, 1993. [3] K.M. Heilman, R.T. Watson, and E. Valenstein. Neglect and related disorders. In K.M. Heilman and E. Valenstein, editors, Clinical Neuropsychology, pages 243-294. Oxford University Press, New York, 1985. [4] H.O. Karnath, K. Christ, and W. Hartje. Decrease of contralateral neglect by neck muscle vibration and spatial orientation of trunk midline. Brain, 116:383396, 1993. [5] H.O. Karnath, P. Schenkel, and B. Fischer. Trunk orientation as the determining factor of the 'contralateral' deficit in the neglect syndrome and as the physical anchor of the internal representation of body orientation in space. Brain, 114:1997-2014, 1991. [6] A. Pouget and T.J. Sejnowski. Spatial representations in the parietal cortex may use basis functions. In G. Tesauro, D.S. Touretzky, and T.K. Leen, editors, Advances in Neural Information Processing Systems, volume 7. MIT Press, Cambridge, MA, 1995.
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Temporal Difference Learning in Continuous Time and Space Kenji Doya doya~hip.atr.co.jp ATR Human Information Processing Research Laboratories 2-2 Hikaridai, Seika.-cho, Soraku-gun, Kyoto 619-02, Japan Abstract A continuous-time, continuous-state version of the temporal difference (TD) algorithm is derived in order to facilitate the application of reinforcement learning to real-world control tasks and neurobiological modeling. An optimal nonlinear feedback control law was also derived using the derivatives of the value function. The performance of the algorithms was tested in a task of swinging up a pendulum with limited torque. Both the "critic" that specifies the paths to the upright position and the "actor" that works as a nonlinear feedback controller were successfully implemented by radial basis function (RBF) networks. 1 INTRODUCTION The temporal-difference (TD) algorithm (Sutton, 1988) for delayed reinforcement learning has been applied to a variety of tasks, such as robot navigation, board games, and biological modeling (Houk et al., 1994). Elucidation of the relationship between TD learning and dynamic programming (DP) has provided good theoretical insights (Barto et al., 1995). However, conventional TD algorithms were based on discrete-time, discrete-state formulations. In applying these algorithms to control problems, time, space and action had to be appropriately discretized using a priori knowledge or by trial and error. Furthermore, when a TD algorithm is used for neurobiological modeling, discrete-time operation is often very unnatural. There have been several attempts to extend TD-like algorithms to continuous cases. Bradtke et al. (1994) showed convergence results for DP-based algorithms for a discrete-time, continuous-state linear system with a quadratic cost. Bradtke and Duff (1995) derived TD-like algorithms for continuous-time, discrete-state systems (semi-Markov decision problems). Baird (1993) proposed the "advantage updating" algorithm by modifying Q-Iearning so that it works with arbitrary small time steps. 1074 K.DOYA In this paper, we derive a TD learning algorithm for continuous-time, continuousstate, nonlinear control problems. The correspondence of the continuous-time version to the conventional discrete-time version is also shown. The performance of the algorithm was tested in a nonlinear control task of swinging up a pendulum with limited torque. 2 CONTINUOUS-TIME TD LEARNING We consider a continuous-time dynamical system (plant) x(t) = f(x(t), u(t)) (1) where x E X eRn is the state and u E U C Rm is the control input (action). We denote the immediate reinforcement (evaluation) for the state and the action as r(t) = r(x(t), u(t)). (2) Our goal is to find a feedback control law (policy) u(t) = JL(x(t)) (3) that maximizes the expected reinforcement for a certain period in the future. To be specific, for a given control law JL, we define the "value" of the state x(t) as 1 00 1 ,-t V!L(x(t)) = -e-T r(x(s), u(s))ds, t r (4) where x(s) and u(s) (t < s < 00) follow the system dynamics (1) and the control law (3). Our problem now is to find an optimal control law JL* that maximizes V!L(x) for any state x E X. Note that r is the time scale of "imminence-weighting" and the scaling factor ~ is used for normalization, i.e., ftOO ~e- ':;:t ds = 1. 2.1 TD ERROR The basic idea in TD learning is to predict future reinforcement in an on-line manner. We first derive a local consistency condition for the value function V!L(x). By differentiating (4) by t, we have d r dt V!L(x(t)) = V!L(x(t)) - r(t). (5) Let P(t) be the prediction of the value function V!L(x(t)) from x(t) (output of the "critic"). If the prediction is perfect, it should satisfy rP(t) = P(t) - r(t). If this is not satisfied, the prediction should be adjusted to decrease the inconsistency f(t) = r(t) - P(t) + rP(t). (6) This is a continuous version of the temporal difference error. 2.2 EULER DIFFERENTIATION: TD(O) The relationship between the above continuous-time TD error and the discrete-time TD error (Sutton, 1988) f(t) = r(t) + ,,(P(t) - P(t ~t) (7) can be easily seen by a backward Euler approximation of p(t). By substituting p(t) = (P(t) - P(t ~t))/~t into (6), we have f=r(t)+ ~t [(1- ~t)P(t)-P(t-~t)] . Temporal Difference in Learning in Continuous Time and Space 1075 This coincides with (7) if we make the "discount factor" '"Y = 1- ~t ~ e-'¥, except for the scaling factor It' Now let us consider a case when the prediction of the value function is given by (8) where biO are basis functions (e.g., sigmoid, Gaussian, etc) and Vi are the weights. The gradient descent of the squared TD error is given by ~Vi ex: _ o~r2(t) ex: - r et) [(1 _ ~t) oP(t) _ oP(t ~t)] . OVi T OVi OVi In order to "back-up" the information about the future reinforcement to correct the prediction in the past, we should modify pet ~t) rather than pet) in the above formula. This results in the learning rule ~Vi ex: ret) OP(~~ ~t) = r(t)bi(x(t - ~t)) . (9) This is equivalent to the TD(O) algorithm that uses the "eligibility trace" from the previous time step. 2.3 SMOOTH DIFFERENTIATION: TD(-\) The Euler approximation of a time derivative is susceptible to noise (e.g., when we use stochastic control for exploration). Alternatively, we can use a "smooth" differentiation algorithm that uses a weighted average of the past input, such as pet) ~ pet) - Pet) where Tc dd pet) = pet) - pet) ~ t and Tc is the time constant of the differentiation. The corresponding gradient descent algorithm is ~Vi ex: _ O~;2(t) ex: ret) o~(t) = r(t)bi(t) , Vi UVi (10) where bi is the eligibility trace for the weight d Tc dtbi(t) = bi(x(t)) - bi(t). (11) Note that this is equivalent to the TD(-\) algorithm (Sutton, 1988) with -\ = 1- At T c if we discretize the above equation with time step ~t. 3 OPTIMAL CONTROL BY VALUE GRADIENT 3.1 HJB EQUATION The value function V * for an optimal control J..L* is defined as V*(x(t)) = max -e-T r(x(s), u(s))ds . [1 00 1 . -t ] U[t,oo) t T (12) According to the principle of dynamic programming (Bryson and Ho, 1975), we consider optimization in two phases, [t, t + ~t] and [t + ~t , 00), resulting in the expression V*(x(t)) = max _e- · :;:-t r(x(s), u(s))ds + e--'¥V*(x(t + ~t)) . [I t+At 1 1 . U[t,HAt) t T 1076 By Taylor expanding the value at t + f:l.t as av* V*(x(t + f:l.t)) = V*(x(t)) + ax(t) f(x(t), u(t))f:l.t + O(f:l.t) K.DOYA and then taking f:l.t to zero, we have a differential constraint for the optimal value function [ av* ] V*(t) = max r(x(t), u(t)) + Ta f(x(t), u(t)) . (13) U(t)EU x This is a variant of the Hamilton-Jacobi-Bellman equation (Bryson and Ho, 1975) for a discounted case. 3.2 OPTIMAL NONLINEAR FEEDBACK CONTROL When the reinforcement r(x, u) is convex with respect to the control u, and the vector field f(x, u) is linear with respect to u, the optimization problem in (13) has a unique solution. The condition for the optimal control is ar(x, u) av* af(x, u) _ 0 au +T ax au -. (14) Now we consider the case when the cost for control is given by a convex potential function GjO for each control input f(x, u) = rx(x) - 2:= Gj(Uj), j where reinforcement for the state r x (x) is still unknown. We also assume that the input gain of the system b -(x) = af(x, u) J au-J is available. In this case, the optimal condition (14) for Uj is given by av* -Gj(Uj) + T ax bj(x) = O. Noting that the derivative G'O is a monotonic function since GO is convex, we have the optimal feedback control law ( av* ) Uj = (G')-1 T ax b(x) . (15) Particularly, when the amplitude of control is bounded as IUj I < uj&X, we can enforce this constraint using a control cost ~ Gj(Uj) = Cj IoUi g-l(s)ds, (16) where g-10 is an inverse sigmoid function that diverges at ±1 (Hopfield, 1984). In this case, the optimal feedback control law is given by ( umax av* ) Uj = ujaxg ~j T ax bj(x) . (17) In the limit of Cj -70, this results in the "bang-bang" control law max' [av* b ( )] Uj = Uj SIgn ax j x . (18) Temporal Difference in Learning in Continuous Time and Space 1077 Figure 1: A pendulum with limited torque. The dynamics is given by m18 -f-tiJ + mglsinO + T. Parameters were m = I = 1, 9 = 9.8, and f-t = 0.0l. trials th (a) (b) 20 17 .5 ~\~ 15 12.5 iii 0. i~! I " 10 .' :1 7 . 5 I, trials th (c) (d) Figure 2: Left: The learning curves for (a) optimal control and (c) actor-critic. Lup: time during which 101 < 90°. Right: (b) The predicted value function P after 100 trials of optimal control. (d) The output of the controller after 100 trials with actor-critic learning. The thick gray line shows the trajectory of the pendulum. th: o (degrees), om: iJ (degrees/sec). 1078 K.DOYA 4 ACTOR-CRITIC When the information about the control cost, the input gain of the system, or the gradient of the value function is not available, we cannot use the above optimal control law. However, the TD error (6) can be used as "internal reinforcement" for training a stochastic controller, or an "actor" (Barto et al., 1983). In the simulation below, we combined our TD algorithm for the critic with a reinforcement learning algorithm for real-valued output (Gullapalli, 1990). The output of the controller was given by u;(t) = ujUg (~W;,b'(X(t)) + <1n;(t)) , (19) where nj(t) is normalized Gaussian noise and Wji is a weight. The size of this perturbation was changed based on the predicted performance by (Y = (Yo exp( -P(t)). The connection weights were changed by !:l.Wji ex f(t)nj(t)bi(x(t)). (20) 5 SIMULATION The performance of the above continuous-time TD algorithm was tested on a task of swinging up a pendulum with limited torque (Figure 1). Control of this onedegree-of-freedom system is trivial near the upright equilibrium. However, bringing the pendulum near the upright position is not if we set the maximal torque Tmax smaller than mgl. The controller has to swing the pendulum several times to build up enough momentum to bring it upright. Furthermore, the controller has to decelerate the pendulum early enough to avoid falling over. We used a radial basis function (RBF) network to approximate the value function for the state of the pendulum x = (8,8). We prepared a fixed set of 12 x 12 Gaussian basis functions. This is a natural extension of the "boxes" approach previously used to control inverted pendulums (Barto et al., 1983). The immediate reinforcement was given by the height of the tip of the pendulum, i.e., rx = cos 8. 5.1 OPTIMAL CONTROL First, we used the optimal control law (17) with the predicted value function P instead of V·. We added noise to the control command to enhance exploration. The torque was given by ( Tmax aP(x) ) T = Tmaxg --r--b + (Yn(t) , c ax where g(x) = ~ tan-1 ( ~x) (Hopfield, 1984). Note that the input gain b = (0, 1/mI2)T was constant. Parameters were rmax = 5, c = 0.1, (Yo = 0.01, r = 1.0, and rc = 0.1. Each run was started from a random 8 and was continued for 20 seconds. Within ten trials, the value function P became accurate enough to be able to swing up and hold the pendulum (Figure 2a). An example of the predicted value function P after 100 trials is shown in Figure 2b. The paths toward the upright position, which were implicitly determined by the dynamical properties of the system, can be seen as the ridges of the value function. We also had successful results when the reinforcement was given only near the goal: rx = 1 if 181 < 30°, -1 otherwise. Temporal Difference in Learning in Continuous Time and Space 1079 5.2 ACTOR-CRITIC Next, we tested the actor-critic learning scheme as described above. The controller was also implemented by a RBF network with the same 12 x 12 basis functions as the critic network. It took about one hundred trials to achieve reliable performance (Figure 2c). Figure 2d shows an example of the output of the controller after 100 trials. We can see nearly linear feedback in the neighborhood of the upright position and a non-linear torque field away from the equilibrium. 6 CONCLUSION We derived a continuous-time, continuous-state version of the TD algorithm and showed its applicability to a nonlinear control task. One advantage of continuous formulation is that we can derive an explicit form of optimal control law as in (17) using derivative information, whereas a one-ply search for the best action is usually required in discrete formulations. References Baird III, L. C. (1993). Advantage updating. Technical Report WL-TR-93-1146, Wright Laboratory, Wright-Patterson Air Force Base, OH 45433-7301, USA. Barto, A. G., Bradtke, S. J., and Singh, S. P. (1995). Learning to act using real-time dynamic programming. Artificial Intelligence, 72:81-138. Barto, A. G., Sutton, R. S., and Anderson, C. W. (1983). Neuronlike adaptive elements that can solve difficult learning control problems. IEEE Transactions on System, Man, and Cybernetics, SMC-13:834-846. Bradtke, S. J. and Duff, M. O. (1995). Reinforcement learning methods for continuous-time Markov decision problems. In Tesauro, G., Touretzky, D. S., and Leen, T. K., editors, Advances in Neural Information Processing Systems 7, pages 393-400. MIT Press, Cambridge, MA. Bradtke, S. J., Ydstie, B. E., and Barto, A. G. (1994). Adaptive linear quadratic control using policy iteration. CMPSCI Technical Report 94-49, University of Massachusetts, Amherst, MA. Bryson, Jr., A. E .. and Ho, Y.-C. (1975). Applied Optimal Control. Hemisphere Publishing, New York, 2nd edition. GuUapalli, V. (1990). A stochastic reinforcement learning algorithm for learning real-valued functions. Neural Networks, 3:671-192. Hopfield, J. J. (1984). Neurons with graded response have collective computational properties like those of two-state neurons. Proceedings of National Academy of Science, 81:3088-3092. Houk, J. C., Adams, J. L., and Barto, A. G. (1994). A model of how the basal ganglia generate and use neural signlas that predict renforcement. 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. Sutton, R. S. (1988). Learning to predict by the methods of temporal difference. Machine Learning, 3:9--44.
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Tempering Backpropagation Networks: Not All Weights are Created Equal Nicol N. Schraudolph EVOTEC BioSystems GmbH Grandweg 64 22529 Hamburg, Germany nici@evotec.de Terrence J. Sejnowski Computational Neurobiology Lab The Salk Institute for BioI. Studies San Diego, CA 92186-5800, USA terry@salk.edu Abstract Backpropagation learning algorithms typically collapse the network's structure into a single vector of weight parameters to be optimized. We suggest that their performance may be improved by utilizing the structural information instead of discarding it, and introduce a framework for ''tempering'' each weight accordingly. In the tempering model, activation and error signals are treated as approximately independent random variables. The characteristic scale of weight changes is then matched to that ofthe residuals, allowing structural properties such as a node's fan-in and fan-out to affect the local learning rate and backpropagated error. The model also permits calculation of an upper bound on the global learning rate for batch updates, which in turn leads to different update rules for bias vs. non-bias weights. This approach yields hitherto unparalleled performance on the family relations benchmark, a deep multi-layer network: for both batch learning with momentum and the delta-bar-delta algorithm, convergence at the optimal learning rate is sped up by more than an order of magnitude. 1 Introduction Although neural networks are structured graphs, learning algorithms typically view them as a single vector of parameters to be optimized. All information about a network's architecture is thus discarded in favor of the presumption of an isotropic weight space the notion that a priori all weights in the network are created equal. This serves to decouple the learning process from network design and makes a large body of function optimization techniques directly applicable to backpropagation learning. But what if the discarded structural information holds valuable clues for efficient weight optimization? Adaptive step size and second-order gradient techniques (Battiti, 1992) may 564 N. N. SCHRAUDOLPH. T. J. SEJNOWSKI recover some of it, at considerable computational expense. Ad hoc attempts to incorporate structural information such as the fan-in (Plaut et aI., 1986) into local learning rates have become a familiar part of backpropagation lore; here we deri ve a more comprehensi ve framework which we call tempering and demonstrate its effectiveness. Tempering is based on modeling the acti vities and error signals in a backpropagation network as independent random variables. This allows us to calculate activity- and weightinvariant upper bounds on the effect of synchronous weight updates on a node's activity. We then derive appropriate local step size parameters by relating this maximal change in a node's acti vi ty to the characteristic scale of its residual through a global learning rate. Our subsequent derivation of an upper bound on the global learning rate for batch learning suggests that the d.c. component of the error signal be given special treatment. Our experiments show that the resulting method of error shunting allows the global learning rate to approach its predicted maximum, for highly efficient learning performance. 2 Local Learning Rates Consider a neural network with feedforward activation given by x j = /j (Yj) , Yj = L Xi Wij , iEAj (1) where Aj denotes the set of anterior nodes feeding directly into node j, and /j is a nonlinear (typically sigmoid) activation function. We imply that nodes are activated in the appropriate sequence, and that some have their values clamped so as to represent external inputs. With a local learning rate of'1j for node j, gradient descent in an objective function E produces the weight update (2) Linearizing Ij around Yj approximates the resultant change in activation Xj as (3) iEAj iEAj Our goal is to put the scale of ~Xj in relation to that of the error signal tSj . Specifically, when averaged over many training samples, we want the change in output activity of each node in response to each pattern limited to a certain proportion given by the global learning rate '1 of its residual. We achieve this by relating the variation of ~X j over the training set to that of the error signal: (4) where (.) denotes averaging over training samples. Formally, this approach may be interpreted as a diagonal approximation of the inverse Fischer information matrix (Amari, 1995). We implement (4) by deriving an upper bound for the left-hand side which is then equated with the right-hand side. Replacing the acti vity-dependent slope of Ij by its maximum value s(/j) == maxl/j(u)1 u (5) and assuming that there are no correlations! between inputs Xi and error tSj ' we obtain (~x}):::; '1} s(/j)2 (tS})f.j (6) 1 Note that such correlations are minimized by the local weight update. Tempering Backpropagation Networks: Not All Weights Are Created Equal 565 from (3), provided that ej ~ e; == ([,Lxlf) , lEA] (7) We can now satisfy (4) by setting the local learning rate to TJ' = TJ J 8 (fj ).j[j . (8) There are several approaches to computing an upper bound ej on the total squared input power e;. One option would be to calculate the latter empirically during training, though this raises sampling and stability issues. For external inputs we may precompute e; orderive an upper bound based on prior knowledge of the training data. For inputs from other nodes in the network we assume independence and derive ej from the range of their activation functions: ej = L p(fd 2 , where p(fd == ffiuax/i(u)2. iEAj (9) Note that when all nodes use the same activation function I, we obtain the well-known Vfan-in heuristic (Plaut et al., 1986) as a special case of (8). 3 Error Backpropagation In deriving local learning rates above we have tacitly used the error signal as a stand-in for the residual proper, i.e. the distance to the target. For output nodes we can scale the error to never exceed the residual: (10) Note that for the conventional quadratic error this simplifies to <Pj = s(/j). What about the remainder of the network? Unlike (Krogh et aI., 1990), we do not wish to prescribe definite targets (and hence residuals) for hidden nodes. Instead we shall use our bounds and independence arguments to scale backpropagated error signals to roughly appropriate magnitude. For this purpose we introduce an attenuation coefficient aj into the error backpropagation equation: c5j = aj II (Yi) L Wjj c5j , jEP, (11) where Pi denotes the set of posterior nodes fed directly from node i. We posit that the appropriate variation for c5i be no more than the weighted average of the variation of backpropagated errors: (12) whereas, assuming independence between the c5j and replacing the slope of Ii by its maximum value, (11) gives us (c5?) ~ a? 8(f;)2 L wi / (c5/) . (13) jEP, Again we equate the right-hand sides of both inequalities to satisfy (12), yielding 1 ai == (14) 8(fdJiP;T . 566 N. N. SCHRAUDOLPH, T. J. SEJNOWSKI Note that the incorporation ofthe weights into (12) is ad hoc, as we have no a priori reason to scale a node's step size in proportion to the size of its vector of outgoing weights. We have chosen (12) simply because it produces a weight-invariant value for the attenuation coefficient. The scale of the backpropagated error could be controlled more rigorously, at the expense of having to recalculate ai after each weight update. 4 Global Learning Rate We now derive the appropriate global learning rate for the batch weight update LiWij == 1]j L dj (t) Xi (t) (15) tET over a non-redundant training sample T. Assuming independent and zero-mean residuals, we then have (16) by virtue of (4). Under these conditions we can ensure ~ 2 2) /).Xj ~ (dj , (17) i.e. that the variation of the batch weight update does not exceed that of the residual, by using a global learning rate of 1] ~ 1]* == l/JiTf. (18) Even when redundancy in the training set forces us to use a lower rate, knowing the upper bound 1]* effectively allows an educated guess at 1], saving considerable time in practice. 5 Error Shunting It remains to deal with the assumption made above that the residuals be zero-mean, i.e. that (dj) = O. Any d.c. component in the error requires a learning rate inversely proportional to the batch size far below 1]* , the rate permissible for zero-mean residuals. This suggests handling the d.c. component of error signals separately. This is the proper job of the bias weight, so we update it accordingly: (19) In order to allow learning at rates close to 1]* for all other weights, their error signals are then centered by subtracting the mean: (20) tET T/j (L dj (t) Xi (t) - (Xi) L dj (t)) (21) tET tET Note that both sums in (21) must be collected in batch implementations of back propagation anyway the only additional statistic required is the average input activity (Xi)' Indeed for batch update centering errors is equivalent to centering inputs, which is known to assist learning by removing a large eigenvalue of the Hessian (LeCun et al., 1991). We expect online implementations to perform best when both input and error signals are centered so as to improve the stochastic approximation. Tempering Backpropagation Networks: Not All Weights Are Created Equal 567 2 OO~O~OOOOOOO TJeff ~ person 000000000000 '""<Lt tr,: j 1*'j:'i~ 1.5 TJ 000000 A "~t{(d$ ·.· .. ·:· .. ·.·.d+;BI». .25 TJ 000000000000 ~£;;?dt!i """" "' ~El~ .10TJ 000000 000000 person 1 ~if;i · ' :r:.· . ; .. 0;:....", ..<1! . (i+ 1 ~ S·!· .·· .. ;~ .05 TJ OOOOO~OOOOOO OOOOOOOOOO~O 000000000000 relationship Figure 1: Backpropagation network for learning family relations (Hinton, 1986). 6 Experimental Setup We tested these ideas on the family relations task (Hinton, 1986): a backpropagation network is given examples of a family member and relationship as input, and must indicate on its output which family members fit the relational description according to an underlying family tree. Its architecture (Figure 1) consists of a central association layer of hidden units surrounded by three encoding layers that act as informational bottlenecks, forcing the network to make the deep structure of the data explicit. The input is presented to the network in a canonical local encoding: for any given training example, exactly one input in each of the two input layers is active. On account of the always active bias input, the squared input power for tempering at these layers is thus C = 4. Since the output uses the same local code, only one or two targets at a time will be active; we therefore do not attenuate error signals in the immediately preceding layer. We use crossentropy error and the logistic squashing function (1 + e-Y)-l at the output (giving ¢> = 1) but prefer the hyperbolic tangent for hidden units, with p(tanh) = s(tanh) = 1. To illustrate the impact of tempering on this architecture we translate the combined effect of local learning rate and error attenuation into an effective learning rate2 for each layer, shown on the right in Figure 1. We observe that effective learning rates are largest near the output and decrease towards the input due to error attenuation. Contrary to textbook opinion (LeCun, 1993; Haykin, 1994, page 162) we find that such unequal step sizes are in fact the key to efficient learning here. We suspect that the logistic squashing function may owe its popUlarity largely to the error attenuation side-effect inherent in its maximum slope of 114We expect tempering to be applicable to a variety of backpropagation learning algorithms; here we present first results for batch learning with momentum and the delta-bar-delta rule (Jacobs, 1988). Both algorithms were tested under three conditions: conventional, tempered (as described in Sections 2 and 3), and tempered with error shunting. All experiments were performed with a customized simulator based on Xerion 3.1.3 For each condition the global learning rate TJ was empirically optimized (to single-digit precision) for fastest reliable learning performance, as measured by the sum of empirical mean and standard deviation of epochs required to reach a given low value of the cost function. All other parameters were held in variant across experiments; their values (shown in Table 1) were chosen in advance so as not to bias the results. 2This is possible only for strictly layered networks, i.e. those with no shortcut (or "skip-through") connections between topologically non-adjacent layers. 3 At the time of writing, the Xerion neural network simulator and its successor UTS are available by anonymous file transfer from ai.toronto.edu, directory pub/xerion. 568 N. N. SCHRAUDOLPH. T. 1. SEJNOWSKI Parameter Val ue II Parameter I Value I training set size (= epoch) 100 zero-error radius around target 0.2 momentum parameter 0.9 acceptable error & weight cost 1.0 uniform initial weight range ±0.3 delta-bar-delta gain increment 0.1 weight decay rate per epoch 10-4 delta-bar-delta gain decrement 0.9 Table 1: Invariant parameter settings for our experim~nts. 7 Experimental Results Table 2 lists the empirical mean and standard deviation (over ten restarts) of the number of epochs required to learn the family relations task under each condition, and the optimal learning rate that produced this performance. Training times for conventional backpropagation are quite long; this is typical for deep multi-layer networks. For comparison, Hinton reports around 1,500 epochs on this problem when both learning rate and momentum have been optimized (personal communication). Much faster convergence though to a far looser criterion has recently been observed for online algorithms (O'Reilly, 1996). Tempering, on the other hand, is seen here to speed up two batch learning methods by almost an order of magnitude. It reduces not only the average training time but also its coefficient of variation, indicating a more reliable optimization process. Note that tempering makes simple batch learning with momentum run about twice as fast as the delta-bar-delta algorithm. This is remarkable since delta-bar-delta uses online measurements to continually adapt the learning rate for each individual weight, whereas tempering merely prescales it based on the network's architecture. We take this as evidence that tempering establishes appropriate local step sizes upfront that delta-bar-delta must discover empirically. This suggests that by using tempering to set the initial (equilibrium) learning rates for deltabar-delta, it may be possible to reap the benefits of both prescaling and adaptive step size control. Indeed Table 2 confirms that the respective speedups due to tempering and deltabar-delta multiply when the two approaches are combined in this fashion. Finally, the addition of error shunting increases learning speed yet further by allowing the global learning rate to be brought close to the maximum of 7]* = 0.1 that we would predict from (18). 8 Discussion In our experiments we have found tempering to dramatically improve speed and reliability of learning. More network architectures, data sets and learning algorithms will have to be "tempered" to explore the general applicability and limitations of this approach; we also hope to extend it to recurrent networks and online learning. Error shunting has proven useful in facilitating of near-maximal global learning rates for rapid optimization. Algorithm batch & momentum delta-bar-delta Condition 7]= mean st.d. 7]= mean st.d. conventional 3.10- 3 2438 ± 1153 3.10-4 696± 218 with tempering 1.10-2 339 ± 95.0 3.10- 2 89.6 ± 11.8 tempering & shunting 4.10- 2 142±27.1 9.10-2 61.7±8.1 Table 2: Epochs required to learn the family relations task. Tempering Backpropagation Networks: Not All Weights Are Created Equal 569 Although other schemes may speed up backpropagation by comparable amounts, our approach has some unique advantages. It is computationally cheap to implement: local learning and error attenuation rates are invariant with respect to network weights and activities and thus need to be recalculated only when the network architecture is changed. More importantly, even advanced gradient descent methods typically retain the isotropic weight space assumption that we improve upon; one would therefore expect them to benefit from tempering as much as delta-bar-delta did in the experiments reported here. For instance, tempering could be used to set non-isotropic model-trust regions for conjugate and second-order gradient descent algorithms. Finally, by restricting ourselves to fixed learning rates and attenuation factors for now we have arrived at a simplified method that is likely to leave room for further improvement. Possible refinements include taking weight vector size into account when attenuating error signals, or measuring quantities such as (62 ) online instead of relying on invariant upper bounds. How such adaptive tempering schemes will compare to and interact with existing techniques for efficient backpropagation learning remains to be explored. Acknowledgements We would like to thank Peter Dayan, Rich Zemel and Jenny Orr for being instrumental in discussions that helped shape this work. Geoff Hinton not only offered invaluable comments, but is the source of both our simulator and benchmark problem. N. Schraudolph received financial support from the McDonnell-Pew Center for Cognitive Neuroscience in San Diego, and the Robert Bosch Stiftung GmbH. References Amari, S.-1. (1995). Learning and statistical inference. In Arbib, M. A., editor, The Handbook of Brain Theory and Neural Networks, pages 522-526. MIT Press, Cambridge. Battiti, T. (1992). First- and second-order methods for learning: Between steepest descent and Newton's method. Neural Computation,4(2):141-166. Haykin, S. (1994). Neural Networks: A Comprehensive Foundation. Macmillan, New York. Hinton, G. (1986). Learning distributed representations of concepts. In Proceedings of the Eighth Annual Conference of the Cognitive Science Society, pages 1-12, Amherst 1986. Lawrence Erlbaum, Hillsdale. Jacobs, R. (1988). Increased rates of convergence through learning rate adaptation. Neural Networks,1:295-307. Krogh, A., Thorbergsson, G., and Hertz, J. A. (1990). A cost function for internal representations. In Touretzky, D. S., editor,Advances in Neural Information Processing Systems, volume 2, pages 733-740, Denver, CO, 1989. Morgan Kaufmann, San Mateo. LeCun, Y. (1993). Efficient learning & second-order methods. Tutorial given at the NIPS Conference, Denver, CO. LeCun, Y., Kanter, I., and Solla, S. A. (1991). Second order properties of error surfaces: Learning time and generalization. In Lippmann, R. P., Moody, J. E., and Touretzky, D. S., editors, Advances in Neural Information Processing Systems, volume 3, pages 918-924, Denver, CO, 1990. Morgan Kaufmann, San Mateo. O'Reilly, R. C. (1996). Biologically plausible error-driven learning using local activation differences: The generalized recirculation algorithm. Neural Computation, 8. Plaut, D., Nowlan, S., and Hinton, G. (1986). Experiments on learning by back propagation. Technical Report CMU-CS-86-126, Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA.
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Improving Policies without Measuring Merits Peter Dayan! CBCL E25-201, MIT Cambridge, MA 02139 dayan~ai.mit.edu Abstract Satinder P Singh Harlequin, Inc 1 Cambridge Center Cambridge, MA 02142 singh~harlequin.com Performing policy iteration in dynamic programming should only require knowledge of relative rather than absolute measures of the utility of actions (Werbos, 1991) - what Baird (1993) calls the advantages of actions at states. Nevertheless, most existing methods in dynamic programming (including Baird's) compute some form of absolute utility function. For smooth problems, advantages satisfy two differential consistency conditions (including the requirement that they be free of curl), and we show that enforcing these can lead to appropriate policy improvement solely in terms of advantages. 1 Introd uction In deciding how to change a policy at a state, an agent only needs to know the differences (called advantages) between the total return based on taking each action a for one step and then following the policy forever after, and the total return based on always following the policy (the conventional value of the state under the policy). The advantages are like differentials - they do not depend on the local levels of the total return. Indeed, Werbos (1991) defined Dual Heuristic Programming (DHP), using these facts, learning the derivatives of these total returns with respect to the state. For instance, in a conventional undiscounted maze problem with a lWe are grateful to Larry Saul, Tommi Jaakkola and Mike Jordan for comments, and Andy Barto for pointing out the connection to Werbos' DHP. This work was supported by NSERC, MIT, and grants to Professor Michael I Jordan from ATR Human Information Processing Research and Siemens Corporation. 1060 P. DAYAN, S. P. SINGH penalty for each move, the advantages for the actions might typically be -1,0 or 1, whereas the values vary between 0 and the maximum distance to the goal. Advantages should therefore be easier to represent than absolute value functions in a generalising system such as a neural network and, possibly, easier to learn. Although the advantages are differential, existing methods for learning them, notably Baird (1993), require the agent simultaneously to learn the total return from each state. The underlying trouble is that advantages do not appear to satisfy any form of a Bellman equation. Whereas it is clear that the value of a state should be closely related to the value of its neighbours, it is not obvious that the advantage of action a at a state should be equally closely related to its advantages nearby. In this paper, we show that under some circumstances it is possible to use a solely advantage-based scheme for policy iteration using the spatial derivatives of the value function rather than the value function itself. Advantages satisfy a particular consistency condition, and, given a model of the dynamics and reward structure of the environment, an agent can use this condition to directly acquire the spatial derivatives of the value function. It turns out that the condition alone may not impose enough constraints to specify these derivatives (this is a consequence of the problem described above) - however the value function is like a potential function for these derivatives, and this allows extra constraints to be imposed. 2 Continuous DP, Advantages and Curl Consider the problem of controlling a deterministic system to minimise V"'(xo) = minu(t) Jo= r(y(t), u(t»)dt, where y(t) E Rn is the state at time t, u(t) E Rm is the control, y(O) = xo, and y(t) = f((y(t), u(t)). This is a simplified form of a classic variational problem since rand f do not depend on time t explicitly, but only through y(t) and there are no stopping time or terminal conditions on y(t) (see Peterson, 1993; Atkeson, 1994, for recent methods for solving such problems). This means that the optimal u(t) can be written as a function of y(t) and that V(xo) is a function of Xo and not t. We do not treat the cases in which the infinite integrals do not converge comfortably and we will also assume adequate continuity and differentiability. The solution by advantages: This problem can be solved by writing down the Hamilton-Jacobi-Bellman (HJB) equation (see Dreyfus, 1965) which V"'(x) satisfies: 0= mJn [r(x, u) + f(x, u) . V' x V"'(x)] (1) This is the continuous space/time analogue of the conventional Bellman equation (Bellman, 1957) for discrete, non-discounted, deterministic decision problems, which says that for the optimal value function V"', 0 = mina [r(x, a) + V'" (f(x, a)) - V"'(x)] , where starting the process at state x and using action a incurs a cost r(x, a) and leaves the process in state !(x, a). This, and its obvious stochastic extension to Markov decision processes, lie at the heart of temporal difference methods for reinforcement learning (Sutton, 1988; Barto, Sutton & Watkins, 1989; Watkins, 1989). Equation 1 describes what the optimal value function must satisfy. Discrete dynamic programming also comes with a method called value iteration which starts with any function Vo(x), improves it sequentially, and converges to the optimum. The alternative method, policy iteration (Howard, 1960), operates in the space of Improving Policies without Measuring Merits 1061 policies, ie functions w(x). Starting with w(x), the method requires evaluating everywhere the value function VW(x) = 1000 r(y(t), w(y(t))dt, where y(O) = x, and y(t) = f(y(t), w(y(t)). It turns out that VW satisfies a close relative of equation 1: 0= r(x, w(x)) + f(x, w(x)) . V' x VW(x) (2) In policy iteration, w(x) is improved, by choosing the maximising action: Wi (x) = argm~ [r(x, u) + f(x, u) . V' x VW (x)] (3) as the new action. For discrete Markov decision problems, the equivalent of this process of policy improvement is guaranteed to improve upon w. In the discrete case and for an analogue of value iteration, Baird (1993) defined the optimal advantage function A*(x, a) = [Q*(x, a) - maxb Q*(x, b)] jM, where 6t is effectively a characteristic time for the process which was taken to be 1 above, and the optimal Q function (Watkins, 1989) is Q*(x, a) = r(x, a) + V*(f(x, a)), where V* (y) = maxb Q* (y, b). It turns out (Baird, 1993) that in the discrete case, one can cast the whole of policy iteration in terms of advantages. In the continuous case, we define advantages directly as (4) This equation indicates how the spatial derivatives of VW determine the advantages. Note that the consistency condition in equation 2 can be written as AW(x, w(x)) = O. Policy iteration can proceed using w'(x) = argmaxuAW(x, u). (5) Doing without VW: We can now state more precisely the intent of this paper: a) the consistency condition in equation 2 provides constraints on the spatial derivatives V' x VW(x), at least given a model of rand f; b) equation 4 indicates how these spatial derivatives can be used to determine the advantages, again using a model; and c) equation 5 shows that the advantages tout court can be used to improve the policy. Therefore, one apparently should have no need to know Vv.' (x) but just its spatial derivatives in order to do policy iteration. Didactic Example LQR: To make the discussion more concrete, consider the case of a one-dimensional linear quadratic regulator (LQR). The task is to minimise V*(xo) = It o:x(t)2 + (3u(t)2dt by choosing u(t), where 0:,(3 > O,±(t) = -[ax(t) + u(t)] and x(O) = Xo. It is well known (eg Athans & Falb, 1966) that the solution to this problem is that V*(x) = k*x2 j2 where k* = (0: + (3(u*)2)j(a + u*) and u(t) = (-a + Ja2 + o:j (3)x(t). Knowing the form of the problem, we consider policies w that make u(t) = wx(t) and require h(x,k) == V'" VW(x) = kx, where the correct value of k = (0: + (3w2)j(a + w). The consistency condition in equation 2 evaluated at state x implies that 0 = (0: + (3w2)X2 - h(x, k)(a + w)x. Doing online gradient descent in the square inconsistency at samples Xn gives kn+l = kn -fa [(0: + (3W2)x~ - knXn(a + W)Xn]2 jakn, which will reduce the square inconsistency for small enough f unless x = O. As required, the square inconsistency can only be zero for all values of x if k = (0: + (3w2)j((a + w)). The advantage of performing action v (note this is not vx) at state x is, from equation 4, AW (x, v) = o:x2 + (3v2 - (ax + v)(o: + (3w2)xj(a + w), which, minimising over v (equation 5) gives u(x) = w'x where Wi = (0: + (3w2)j(2(3(a+ w)) , which is the Newton-Raphson iteration to solve the quadratic equation that determines the optimal policy. In this case, without ever explicitly forming VW (x), we have been able to learn an optimal 1062 P. DAYAN, S. P. SINGH policy. This was based, at least conceptually, on samples Xn from the interaction of the agent with the world. The curl condition: The astute reader will have noticed a problem. The consistency condition in equation 2 constrains the spatial derivatives \7 x VW in only one direction at every point - along the route f(x, w(x)) taken according to the policy there. However, in evaluating actions by evaluating their advantages, we need to know \7 x VW in all the directions accessible through f(x, u) at state x. The quadratic regulation task was only solved because we employed a function approximator (which was linear in this case h(x, k) = kx). For the case of LQR, the restriction that h be linear allowed information about f(X', w(x')) . \7 x' VW (x') at distant states x' and for the policy actions w(x') there to determine f(x, u) . \7 x VW(x) at state x but for non-policy actions u. If we had tried to represent h(x, k) using a more flexible approximator such as radial basis functions, it might not have worked. In general, if we didn't know the form of \7 x VW (x), we cannot rely on the function approximator to generalize correctly. There is one piece of information that we have yet to use - function h(x, k) == \7 x VW (x) (with parameters k, and in general non-linear) is the gradient of something - it represents a conservative vector field. Therefore its curl should vanish (\7 x x h(x, k) = 0). Two ways to try to satisfy this are to represent h as a suitably weighted combination of functions that satisfy this condition or to use its square as an additional error during the process of setting the parameters k. Even in the case of the LQR, but in more than one dimension, it turns out to be essential to use the curl condition. For the multi-dimensional case we know that VW (x) = x T KWx/2 for some symmetric matrix KW, but enforcing zero curl is the only way to enforce this symmetry. The curl condition says that knowing how some component of \7 x VW(x) changes in some direction (eg 8\7 x VW(xh/8xl) does provide information about how some other component changes in a different direction (eg 8\7 x vw (xh /8X2). This information is only useful up to constants of integration, and smoothness conditions will be necessary to apply it. 3 Simulations We tested the method of approximating hW(x) = \7 x VW(x) as a linearly weighted combination of local conservative vector fields hW(x) = L~=l ci\7 x <p(x, Zi), where ci are the approximation weights that are set by enforcing equation 2, and </J(x, Zi) = e-a:lx-z;l2 are standard radial basis functions (Broomhead & Lowe, 1988; Poggio & Girosi, 1990). We enforced this condition at a discrete set {xd of 100 points scattered in the state space, using as a policy, explicit vectors Uk at those locations, and employed 49 similarly scattered centres Zi. Issues of learning to approximate conservative and non-conservative vector fields using such sums have been discussed by Mussa-Ivaldi (1992). One advantage of using this representation is that 1jJ(x) = L~=l ci <p(x, Zi) can be seen as the system's effective policy evaluation function VW(x), at least modulo an arbitrary constant (we call this an un-normalised value function). We chose two 2-dimensional problems to prove that the system works. They share the same dynamics x(t) = -x(t) + u(t), but have different cost functions: Improving Policies without Measuring Merits 1063 TLQR(X(t), U(t)) = 5lx(tW + lu(tW , TSp(X(t), U(t)) = Ix(tW + \/1 + IU(t)12 TLQR makes for a standard linear quadratic regulation problem, which haJ6:l quadratic optimal value function and a linear optimal controller as before (although now we are using limited range basis functions instead of using the more appropriate linear form). TSp has a mixture of a quadratic term in x(t), which encourages the state to move towards the origin, and a more nearly linear cost term in u(t), which would tend to encourage a constant speed. All the sample points Xk and radial basis function centres Zi were selected within the {-I, IF square. We started from a randomly chosen policy with both components of Uk being samples from the uniform distribution U( -.25, .25). This was chosen so that the overall dynamics of the system, including the -x(t) component should lead the agent towards the origin. Figure Ia shows the initial values of Uk in the regulator case, where the circles are at the leading edges of the local policies which point in the directions shown with relative magnitudes given by the length of the lines, and (for scale) the central object is the square {-O.I,O.IF. The 'policy' lines are centred at the 100 Xk points. Using the basis function representation, equation 2 is an over-determined linear system, and so, the standard Moore-Penrose pseudo-inverse was used to find an approximate solution. The un-normalised approximate value function corresponding to this policy is shown in figure lb. Its bowl-like character is a feature of the optimal value function. For the LQR case, it is straightforward to perform the optimisation in equation 5 analytically, using the values for h W (Xk) determined by the ci. Figure Ic,d show the policy and its associated un-normalised value function after 4 iterations. By this point, the policy and value functions are essentially optimal - the policy shows the agent moves inwards from all Xk and the magnitudes are linearly related to the distances from the centre. Figure Ie,f show the same at the end point for TSp. One major difference is that we performed the optimisation in equation 5 over a discrete set of values for Uk rather than analytically. The tendency for the agent to maintain a constant speed is apparent except right near the origin. The bowl is not centred exactly at (0,0) - which is an approximation error. 4 Discussion This paper has addressed the question of whether it is possible to perform policy iteration using just differential quantities like advantages. We showed that using a conventional consistency condition and a curl constraint on the spatial derivatives of the value function it is possible to learn enough about the value function for a policy to improve upon that policy. Generalisation can be key to the whole scheme. We showed this working on an LQR problem and a more challenging non-LQR case. We only treated 'smooth' problems - addressing discontinuities in the value function, which imply un differentiability, is clearly key. Care must be taken in interpreting this result. The most challenging problem is the error metric for the approximation. The consistency condition may either under-specify or over-specify the parameters. In the former case, just as for standard approximation theory, one needs prior information to regularise the gradient surface. For many problems there may be spatial discontinuities in the policy evaluation, and therefore this is particularly difficult. IT the parameters are over-specified (and, for good generalisation, one would generally be working in this regime), we need to evaluate inconsistencies. Inconsistencies cost exactly to the degree that the optimisation in equation 5 is compromised - but this is impossible to quantify. Note that this problem is not 1064 P. DAYAN, S. P. SINOH a c e b d f Figure 1: a-d) Policies and un-normalised value functions for the rLQR and e-f) for the rsp problem. confined to the current scheme of learning the derivatives of the value function it also impacts algorithms based on learning the value function itself. It is also unreasonable to specify the actions Uk only at the points Xk. In general, one would either need a parameterised function for u(x) whose parameters would be updated in the light of performing the optimisations in equation 5 (or some sort of interpolation scheme), or alternatively one could generate u on the fly using the learned values of h(x). If there is a discount factor, ie V*(xo) = minu(t) fooo e-Atr(y(t), u(t»dt, then 0 = r(x, w(x» - AVw (x) + f(x, w(x»· \7 x VW (x) is the equivalent consistency condition to equation 2 (see also Baird, 1993) and so it is no longer possible to learn \7 x VW (x) without ever considering VW(x) itself. One can still optimise parameterised forms for VW as in section 3, except that the once arbitrary constant is no longer free. The discrete analogue to the differential consistency condition in equation 2 amounts to the tautology that given current policy 7r, 't/x, A7r(x,7r(x» = O. As in the continuous case, this only provides information about V7r(f(x, 7r(x») - V7r(x) and not V7r(f(x, a»-V 7r(x) for other actions a which are needed for policy improvement. There is an equivalent to the curl condition: if there is a cycle in the undirected transition graph, then the weighted sum of the advantages for the actions along the cycle is equal to the equivalently weighted sum of payoffs along the cycle, where the weights are + 1 if the action respects the cycle and -1 otherwise. This gives a consistency condition that A 7r has to satisfy - and, just as in the constants of integration for the differential case, it requires grounding: A 7r (z, a) = 0 for some z in the cycle. It is certainly not true that all discrete problems will have sufficient cycles to specify A 7r completely - in an extreme case, the undirected version of the directed transition graphs might contain no cycles at all. In the continuous case, if the updates are sufficiently smooth, this is not possible. For stochastic problems, the consistency condition equivalent to equation 2 will involve an integral, which, Improving Policies without Measuring Merits 1065 if doable, would permit the application of our method. Werbos's (1991) DHP and Mitchell and Thrun's (1993) explanation-based Qlearning also study differential forms of the Bellman equation based on differentiating the discrete Bellman equation (or its Q-function equivalent) with respect to the state. This is certainly fine as an additional constraint that V* or Q* must satisfy (as used by Mitchell and Thrun and Werbos' Globalized version of DHP), but by itself, it does not enforce the curl condition, and is insufficient for the whole of policy improvement. References Athans, M & Falb, PL (1966). Optimal Control. New York, NY: McGraw-Hill. Atkeson, CG (1994). Using Local Trajectory Optimizers To Speed Up Global Optimization in Dynamic Programming. In NIPS 6. Baird, LC, IIIrd (1993). Advantage Updating. Technical report, Wright Laboratory, Wright-Patterson Air Force Base. Barto, AG, Bradtke, SJ & Singh, SP (1995). Learning to act using real-time dynamic programming. Artificial Intelligence, 72, 81-138. Barto, AG, Sutton, RS & Watkins, CJCH (1990). Learning and sequential decision making. In M Gabriel & J Moore, editors, Learning and Computational Neuroscience: Foundations of Adaptive Networks. Cambridge, MA: MIT Press, Bradford Books. Bellman, RE (1957). Dynamic Programming. Princeton, NJ: Princeton University Press. Broomhead, DS & Lowe, D (1988). Multivariable functional interpolation and adaptive networks. Complex Systems, 2, 321-55. Dreyfus, SE (1965). Dynamic Programming and the Calculus of Variations. New York, NY: Academic Press. Howard, RA (1960). Dynamic Programming and Markov Processes. New York, NY: Technology Press & Wiley. Mitchell, TM & Thrun, SB (1993). Explanation-based neural network learning for robot control. In NIPS 5. Mussa-Ivaldi, FA (1992). From basis functions to basis fields: Vector field approximation from sparse data. Biological Cybernetics, 67, 479-489. Peterson, JK (1993). On-Line estimation of optimal value functions. In NIPS 5. Poggio, T & Girosi, F (1990). A theory of networks for learning. Science, 247, 978-982. Sutton, RS (1988). Learning to predict by the methods of temporal difference. Machine Learning, 3, pp 9-44. Watkins, CJCH (1989). Learning from Delayed Rewards. PhD Thesis. University of Cambridge, England. Werbos, P (1991). A menu of designs for reinforcement learning over time. In WT Miller IIIrd, RS Sutton & P Werbos, editors, Neural Networks for Control. Cambridge, MA: MIT Press, 67-96.
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Onset-based Sound Segmentation Leslie S. Smith CCCN jDepartment of Computer Science University of Stirling Stirling FK9 4LA Scotland Abstract A technique for segmenting sounds using processing based on mammalian early auditory processing is presented. The technique is based on features in sound which neuron spike recording suggests are detected in the cochlear nucleus. The sound signal is bandpassed and each signal processed to enhance onsets and offsets. The onset and offset signals are compressed, then clustered both in time and across frequency channels using a network of integrateand-fire neurons. Onsets and offsets are signalled by spikes, and the timing of these spikes used to segment the sound. 1 Background Traditional speech interpretation techniques based on Fourier transforms, spectrum recoding, and a hidden Markov model or neural network interpretation stage have limitations both in continuous speech and in interpreting speech in the presence of noise, and this has led to interest in front ends modelling biological auditory systems for speech interpretation systems (Ainsworth and Meyer 92; Cosi 93; Cole et al 95). Auditory modelling systems use similar early auditory processing to that used in biological systems. Mammalian auditory processing uses two ears, and the incoming signal is filtered first by the pinna (external ear) and the auditory canal before it causes the tympanic membrane (eardrum) to vibrate. This vibration is then passed on through the bones of the middle ear to the oval window on the cochlea. Inside the cochlea, the pressure wave causes a pattern of vibration to occur on the basilar membrane. This appears to be an active process using both the inner and outer hair cells of the organ of Corti. The movement is detected by the inner hair cells and turned into neural impulses by the neurons of the spiral ganglion. These pass down the auditory nerve, and arrive at various parts of the cochlear nucleus. From there, nerve fibres innervate other areas: the lateral and medial nuclei of the superior olive, 730 L.S.SMITH and the inferior colliculus, for example. (See (Pickles 88)). Virtually all modern sound or speech interpretation systems use some form of bandpass filtering, following the biology as far as the cochlea. Most use Fourier transforms to perform a calculation of the energy in each band over some time period, usually between 25 and 75 ms. This is not what the cochlea does. Auditory modelling front ends differ in the extent and length to which they follow animal early auditory processing, but the term generally implies at least that wideband filters are used, and that high temporal resolution is maintained in the initial stages. This means the use of filtering techniques. rather than Fourier transforms in the bandpass stage. Such filtering systems have been implemented by Patterson and Holdsworth (Patterson and Holdsworth 90; Slaney 93), and placed directly in silicon (Lazzaro and Mead 89; Lazzaro et al 93; Liu et al 93; Fragniere and van Schaik 94). Some auditory models have moved beyond cochlear filtering. The inner hair cell has been modelled by either simple rectification (Smith 94) or has been based on the work of (Meddis 88) for example (Patterson and Holdsworth 90; Cosi 93; Brown 92). Lazzaro has experimented with a silicon version of Licklider's autocorrelation processing (Licklider 51; Lazzaro and Mead 89). Others such as (Wu et al 1989: Blackwood et al1990; Ainsworth and Meyer 92; Brown 92; Berthommier 93; Smith 94) have considered the early brainstem nuclei, and their possible contribution, based on the neurophysiology of the different cell types (Pickles 88; Blackburn and Sachs 1989; Kim et al 90). Auditory model-based systems have yet to find their way into mainstream speech recognition systems (Cosi 93). The work presented here uses auditory modelling up to onset cells in the cochlear nucleus. It adds a temporal neural network to clean up the segmentation produced. This part has been filed as a patent (Smith 95). Though the system has some biological plausibility, the aim is an effective data-driven segmentation technique implement able in silicon. 2 Techniques used Digitized sound was applied to an auditory front end, (Patterson and Holdsworth 90), which bandpassed the sound into channels each with bandwidth 24.7{4.37Fr; + I)Hz, where Fe is the centre frequency (in KHz) of the band (Moore and Glasberg 83). These were rectified, modelling the effect of the inner hair cells. The signals produced bear some resemblance to that in the auditory nerve. The real system has far more channels and each nerve channel carries spike-coded information. The coding here models the signal in a population of neighboring auditory nerve fibres. 2.1 The onset-offset filter The signal present in the auditory nerve is stronger near the onset of a tone than later (Pickles 88). This effect is much more pronounced in certain cell types of the cochlear nucleus. These fire strongly just after the onset of a sound in the band to which they are sensitive, and are then silent. This emphasis on onsets was modelled by convolving the signal in each band with a filter which computes two averages, a more recent one, and a less recent one, and subtracts the less recent one from the more recent one. One biologically possible justification for this is to consider that a neuron is receiving the same driving input twice, one excitatorily, and the other inhibitorily; the excitatory input has a shorter time-constant than the inhibitory input. Both exponentially weighted averages, and averages formed using a Gaussian filter have been tried (Smith 94), but the former place too much emphasis on the most recent part of the signal, making the latter more effective. Onset-based Sound Segmentation 731 The filter output for input signal s(x) is O(t. k, 'f') = lot (f(t - x, k) - f(t - x, k/,r))s(x)dx (1) where f(x, y) = vY exp( -yx2 ). k and 'r determine the rise and fall times of the pulses of sOlmd that the system is sensitive to. We used A: = 1000, 'r = 1.2, so that the SD of the Gaussians are 24.49ms and 22.36ms. The convolving filter has a positive peak at O. crosses 0 at 22.39ms. and is then negative. With these values. the system is sensitive to energy rises and falls which occm in the envelopes of everyday sounds. A positive onset-offset signal implies that the bandpassed signal is increasing in intensity, and a negative onset-offset signal implies that it is decreasing in intensity. The convolution used is a sound analog of the difference of Gaussians operator used to extract black/white and white/black edges in monochrome images (MalT and Hildreth 80). In (Smith 94) we performed sOlmd segmentation directly on this signal. 2.2 Compressing the onset-offset signal The onset-offset signal was divided into two positive-going signals, an onset signal consisting of the positive-going part, and an offset signal consisting of the inverted negative-going part. Both were compressed logarithmically (where log(x) was taken as 0 for 0 S x S 1). This increases the dynamical range of the system, and models compressive biological effects. The compressed onset signal models the output of a population of onset cells. This technique for producing an onset signal is related to that of (Wu et al 1989: Cosi 93). 2.3 The integrate-and-fire neural network To segment the sound using the onset and offset signals, they need to be integrated across frequency bands and across time. This temporal and tonotopic clustering was achieved using a network of integrate-and-fire units. An integrate-and-fire unit accumulates its weighted input over time. The activity of the unit A. is initially O. and alters according to dA = I(t) - "YA dt (2) where I(t) is the input to the nemon and "Y, the dissipation, describes the leakiness of the integration. When A reaches a threshold. the unit fires (i.e. emits a pulse). and A is reset to O. After firing, there is a period of insensitivity to input, called the refractory period. Such nemons are discussed in. e.g. (Mirolla and Strogatz 90). One integrate-and-fire neuron was used per charmel: this neuron received input either from a single charmel, or from a set of adjacent charmels. all with equal positive weighting. The output of each neuron was fed back to a set of adjacent neurons, again with a fixed positive weight, one time step (here 0.5ms) later. Because of the leaky nature of the accumulation of activity, excitatory input to the neuron arriving when its activation is near' threshold has a lar'ger effect on the next firing time than excitatory input arriving when activation is lower. Thus, if similar input is applied to a set of neurons in adjacent charmels. the effect of the inter-neuron connections is that when the first one fires, its neighbors fire almost immediately. This allows a network of such neurons to cluster the onset or offset signals, producing a sharp burst of spikes across a number of charmels providing unambiguous onsets or offsets. The external and internal weights of the network were adjusted so that onset or offset input alone allowed neurons to fire, while internal input alone was not enough 732 L. S. SMITH to cause firing. The refractory period used was set to 50ms for the onset system, and 5ms for the offset system. For the onset system, the effect was to produce sharp onset firing responses across adjacent channels in response to a sudden increase in energy in some channels, thus grouping onsets both tonotopically and temporally. This is appropriate for onsets, as these are generally brief and clearly marked. The output of this stage we call the onset map. Offsets tend to be more gradual. This is due to physical effects: for example, a percussive sound will start suddenly, as the vibrating element starts to move. but die away slowly as the vibration ceases (see (Gaver 93) for a discussion). Even when the vibration does stop suddenly. the sound will die away more slowly due to echoes. Thus we cannot reliably mark the offset of a sound: instead. we reduce the refractory period of the offset neurons, and produce a train of pulses marking the duration of the offset in this channel. We call the output of this stage the offset map. 3 Results As the technique is entirely data-driven. it can be applied to sound from any source. It has been applied to both speech and musical sounds. Figure 1 shows the effect of applying the techniques discussed to a short piece of speech. Fig lc shows that the neural network integrates the onset timings across the channels, allowing these onsets to be used for segmentation. The simplest technique is to divide up the continuous speech at each onset: however,to ensure that the occasional onset in a single channel does not confuse the system. and that onsets which occur near to each other do not result in very short segments we demanded that a segmentation boundary have at least 6 onsets inside a period of lOms. and the minimum segment length was set to 25ms. The utterance Ne'Uml information processing systems has phonetic representation: / njtlrl: anfarme Ian prosc:salJ ststalllS / and is segmented into the following 19 segments: /n/, jtl/, /r/, /la/, /a/, /nf/. /arm/, /e/, /I/, /an/, /pro/, /os/, /c:s/, /aIJ/, /s/, /t/, /st/, /am/, /s/ The same text spoken more slowly (over 4.38s, rather than 2.31s) has phonetic representation: / njural:anftrmeIanprosc:stIJ ststams / Segmenting using this technique gives the following 25 segments: /n/, /ju/, /u/, /r/. /a/, /al/, /1/, / /, /an/, /f/, /um/, /e/, /I/. /an/, /n:/, /pr/, /ro/, /os/, /c:s/, /tIJ/, /s/, /t/, /st/, /am/, /s/ Although some phonemes are broken between segments, the system provides effective segmentation, and is relatively insensitive to speech rate. The system is also effective at finding speech inside certain types of noise (such as motor-bike noise) , as can be seen in fig Ie and f. The system has been used to segment sound from single musical instruments. Where these have clear breaks between notes this is straightforward: in (Smith 94) correct segmentation was achieved directly from the onset-offset signal but was not achieved for slurred sounds, in which the notes change smoothly. As is visible in figure 2c, the onsets here are clear using the network, and the segmentation produced is nearOnset-based Sound Segmentation [E]J :'. . .. ... ' " .. .. . , " .. .. . GJ . . . .... '"\N',,,,,,.J/'I~'/w,. -"·' ''''''''''~ v..flt'''''I/!fII~~ ... A..it.-./'~f\It'''v,i~~~~';''o~\-J..J{iII'' r ~'if'I{I/'}/i'...J"''' ~ 'W ' hlJ.,j.~~..f"'\/' JJ.A~ "'v.""./fI/<II'~rJ~ M '\-'.'~'f.."""v/ "'I'jNflNlI/V '1'#.~N~I{'f!II/W/W ·" /")~,\/,, ' . . .. ; 733 Figure 1: (a-d):Onset and Offset maps from author saying Neural information processing systems rapidly. a: envelope of original sound. b: onset map. from 28 channels. from 100Hz-6KHz. Onset filter parameters as in text; one neuron per channel, with no interconnection. Neuron refractory period is 50ms. c: as b, but network has input applied to 6 adjacent channels, and internal feedback to 10 channels. d: offset map produced similarly, with refractory period 5ms. e: envelope of say, that's a nice bike with motorbike noise in background (lines mark utterance). f, g: onset, offset maps for e. perfect. Best results were obtained here when the input to the network is not spread across channels. 4 Conclusions and further work An effective data driven segmentation technique based on onset feature detection and using integrate-and-fire neurons has been demonstrated. The system is relatively immune to broadband noise. Segmentation is not an end in itself: the effectiveness of any technique will depend on the eventual application. 734 L. S.SMITH .. '~---.' Figure 2: a: slurred flute sound. with vertical lines showing boundary between notes. b: onsets found using a single neuron per channel, and no interconnection. c: as b, but with internal feedback from each channel to 16 adjacent channels d: offsets found with refractory period 5ms. The segmentation is currently not using the information on which bands the onsets occur in. We propose to extend this work by combining the segmentation described here with work streaming bands sharing same-frequency amplitude modulation. The aim of this is to extract sound segments from some subset of the bands, allowing segmentation and streaming to run concurrently. Acknowledgements Many thanks are due to the members of the Centre for Cognitive and Computational Neuroscience at the University of Stirling. References Ainsworth W. Meyer G. Speech analysis by means of a physiologically-based model of the cochlear nerve and cochlear nucleus. in Visual r'epresentations of speech signals. Cooke M. Beet S. eds. 1992. Berthommier F .. Modelling nelll'rul'eSpOllSes of t.he int.ermediate auditory system, in Mathematics applied to biology and medicine, Demongeot .I, Capa..'!so V, Wuertz Publishing, Canada, 1993. Blackburn C.C .. Sachs M.B. Classification of unit types in the anteroventral cochlear nucleus: PST hist.ograms and regularity analysis, . J. Neurophys'iology, 62, 6, 1989. Onset-based Sound Segmentation 735 Blackwood N .. Meyer G., Aimsworth W. A Model of the processing of voiced plosives in the auditory nerve and cochlear nucleus, Proceedings Inst of Acoustics, 12, 10, 1990. Brown G. Computational Auditory Scene Analysis, TR CS-92-22, Department of Computing Science, University of Sheffield, England, 1992. Cole R .. et al, The challenge of spoken language systems: research directions of the 90's. IEEE Trans Speech and Audio Pmcessing, 3. 1, 1995. Cosi P. On the use of auditory models in speech technology, in Intelligent Perceptual Models, LNCS 745, Springer Verlag, 1993. Fragniere E., van Schaik A .. Lineal' predictive coding of the speech signal using an analog cochlear modeL MANTRA Internal Report, 94/2, MANTRA Center for Neuro-mimetic systems, EPFL, Lausanne, Switzerland, 1994. Gaver W.W. What in the world do we hear?: an ecological approach to auditory event perception. Ecological Psychology, 5(1). 1-29, 1993. Kim D.O. ,Sirianni .T.G., Chang S.O .. Responses of DCN-PVCN neurons and auditory nerve fibres in lmanesthetized decerebrate cats to AM and pure tones: analysis with autocorrelation/power-spectrum, Hearing Research. 45, 95-113. 1990. Lazzaro .T., Mead C., Silicon modelling of pitch perception, Proc Natl. Acad Sciences, USA, 86. 9597-9601, 1989. Lazzaro .T., Wawrzynek .T .. Mahowald M., Sivilotti M., Gillespie D .. Silicon auditory processors as computer peripherals. IEEE Trans on Neural Networks, 4, 3, May 1993. Licklider .T.C.R, A Duplex theory of pitch perception, Experentia, 7. 128-133, 1951. Liu W .. Andreou A.G., Goldstein M.H., Analog cochlear model for multiresolution speech analysis, Advances in Neural Information Processing Systems 5, Hanson S . .T., Cowan .T.D., Lee Giles C. (eds), Morgan Kaufmann, 1993. Marl' D., Hildreth E. Theory of edge detection, Proc. Royal Society of London B, 207. 187-217, 1980. Meddis R .. Simulation of auditory-neural transduction: further studies. J. Acollst Soc Am. 83. 3, 1988. Moore B.C . .J.. Glasberg B.R. Suggested formulae for calculating auditory-filter bandwidths and excitation patterns, J Acoust Soc America, 74. 3, 1983. Mirollo RE., Strogatz S.H. Synchronization of pulse-coupled biological oscillators, SIAM J. Appl Math, 50, 6, 1990. Patterson R. Holdsworth .T. (1990). An Introd'IJ,ction to A1J,ditory Sensation Processing. in AAM HAP. Vol 1. No 1. Pickles .T.O. (1988). An Introd'uction to the PhyS'iology of Hearing, 2nd Edition, Academic Press. Slaney M .. An efficient implementation of the Patterson-Holdsworth auditory filter bank, Apple technical report No 35, Apple Computer Inc, 1993. Smith L.S. SO\illd segmentation using onsets and offsets, J of New Music Research, 23, 1, 1994. Smith L.S. Onset/offset coding for interpretation and segmentation of sound, UK patent no 9505956.4. March 1995. Wu Z.L., Schwartz .T.L .. Escudier P. A theoretical study of neural mechanisms specialized in the detection of articulatory-acoustic events, Proc Eurospeech 89. ed Tubach .T.P., Mariani .T . .T., Paris, 1989.
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Selective Attention for Handwritten Digit Recognition Ethem Alpaydm Department of Computer Engineering Bogazi<1i U ni versi ty Istanbul, TR-SOS15 Turkey alpaydin@boun.edu.tr Abstract Completely parallel object recognition is NP-complete. Achieving a recognizer with feasible complexity requires a compromise between parallel and sequential processing where a system selectively focuses on parts of a given image, one after another. Successive fixations are generated to sample the image and these samples are processed and abstracted to generate a temporal context in which results are integrated over time. A computational model based on a partially recurrent feedforward network is proposed and made credible by testing on the real-world problem of recognition of handwritten digits with encouraging results. 1 INTRODUCTION For all-parallel bottom-up recognition, allocating one separate unit for each possible feature combination, i.e., conjunctive encoding, implies combinatorial explosion. It has been shown that completely parallel, bottom-up visual object recognition is NP-complete (Tsotsos, 1990). By exchanging space with time, systems with much less complexity may be designed. For example, to phone someone at the press of a button, one needs 107 buttons on the phone; the sequential alternative is to have 10 buttons on the phone and press one at a time, seven times. We propose recognition based on selective attention where we analyze only a small part of the image in detail at each step, combining results in time. N oton and Stark's (1971) "scanpath" theory advocates that each object is internally represented as a feature-ring which is a temporal sequence of features extracted at each fixation and the positions or the motor commands for the eye movements in between. In this approach, there is an "eye" that looks at an image but which can really see only a small part of it. This part of the image that is examined in detail is the fovea. The 772 Class Probabilities (lOx!) P ~r------7-"7 L-L ______ / ' softmax t Class Units (lOxI) 0/ 7 T1 Hidden Units (s x I) H L..../~_....;...._-_-_-_-~_-_-_-_-_-~_-_-~7-." ASSOCIATIVE LEVEL E. ALPAYDIN I~-----------------;t~---------- ----;;------------------PRE-ATTENTIVE LEVEL ATTENTIVE LEVEL ,-------------------- -----, ------ -------------------, I : Feature Map I Eye Position Map (rxI): (pxp) F Fovea //p 1 subsample and blur 1-------'---1- ~ -I WTA I - -:- - - - - - - ~ M Bitmap Image (n x n) Saliency Map (n x n) Figure 1: The block diagram of the implemented system. fovea's content is examined by the pre-attentive level where basic feature extraction takes place. The features thus extracted are fed to an a660ciative part together with the current eye position. If the accumulated information is not sufficient for recognition, the eye is moved to another part of the image, making a saccade. To minimize recognition time, the number of saccades should be minimized. This is done through defining a criterion of being "interesting" or saliency and by fixating only at the most interesting. Thus sucessive fixations are generated to sample the image and these samples are processed and abstracted to generate a temporal context in which results are integrated over time. There is a large amount of literature on selective attention in neuroscience and psychology; for reviews see respectively (Posner and Peterson, 1990) and (Treisman, 1988). The point stressed in this paper is that the approach is also useful in engineering. 2 AN EXAMPLE SYSTEM FOR OCR The structure of the implemented system for recognition of handwritten digits is given in Fig. 1. Selective Attention for Handwritten Digit Recognition 773 We have an n x n binary image in which the fovea is m x m with m < n. To minimize recognition time, the system should only attend to the parts of the image that carry discriminative information. We define a criterion of being "interesting" or saliency which is applied to all image locations in parallel to generate a 8aliency map, S. The saliency measure should be chosen to draw attention to parts that have the highest information content. Here, the saliency criterion is a low-pass filter which roughly counts the number of on pixels in the corresponding m x m region of the input image M. As the strokes in handwritten digits are mostly one or two pixels wide, a count of the on pixels is a good measure of the discontinuity (and thus information). It is also simple to compute: i+lm/2J HLm/2J Sij = L L MkIN2((i,jl, (Lm/6J)2 *1), i,j = 1. .. n k=i-Lm/2J l=j-Lm/2J where N2(p., E) is the bivariate normal with mean p. and the covariance E. Note that we want the convolution kernel to have effect up to L m/2 J and also that the normal is zero after p.± 30-. In our simulations where n is 16 and m is 5 (typical for digit recognition), 0- ~ 1. The location that is most salient is the position ofthe next fixation and as such defines the new center of the fovea. A location once attended to is no longer interesting; after each fixation, the saliency of all the locations that currently are in the scope of the fovea are set to 0 to inhibit another fixation there. The attentive level thus controls the scope of the pre-attentive level. The maximum of the saliency map through a winner-take-all gives the eye position (i*, j*) at fixation t. (i*(t),j*(t)) = arg~B:XSij ',J By thus following the salient regions, we get an input-dependent emergent sequence in time. Eye-Position Map The eye p08ition map, P, stores the position of the eye in the current fixation. It is p x p. p is chosen to be smaller than n for dimensionality reduction for decreasing complexity and introducing an effect of regularization (giving invariance to small translations). When p is a factor of n, computations are also simpler. We also blur the immediate neighbors for a smoother representation: P( t) = blur( subsample( winner-take-all( S))) Pre-Attentive Level: Feature Extraction The pre-attentive level extracts detailed features from the fovea to generate a feature map. This information and the current eye position is passed to the associative system for recognition. There is a trade-off between the fovea size and the number of saccades required for recognition: As the operation in the pre-attentive level is carried out in parallel, to minimize complexity the features extracted there should not be many and the fovea should not be large: Fovea is where the expensive computation takes place. On the other hand, the fovea should be large enough to extract discriminative features and thus complete recognition in a small amount of time. The features to be extracted can be learned through an supervised method when feedback is available. 774 E. ALPAYDIN The m x m region symmetrically around (i*, j*) is extracted as the fovea I and is fed to the feature extractors. The r features extracted there are passed on to the associative level as the feature map, F. r is typically 4 to 8. Ug denote the weights of feature 9 and Fg is the value of feature 9 that is found by convolving the fovea input with the feature weight vector (1(.) is the sigmoid function): M io(t)-Lm/2J+i,jo(t)-Lm/2J+j, i,j = 1 ... m f ( ~ ~ U"jI,j(t») , g = 1. .. r Associative Level: Classification At each fixation, the associative level is fed the feature map from the pre-attentive level and the eye position map from the attentive level. As a number of fixations may be necessary to recognize an image, the associative system should have a shortterm memory able to accumulate inputs coming through time. Learning similarly should be through time. When used for classification, the class units are organized so as to compete and during recognition the activations of the class units evolve till one class gets sufficiently active and suppresses the others. When a training set is available, a temporal supervised method can be used to train the associative level. Note that there may be more than one scanpath for each object and learning one sequence for each object fails. We see it is a task of accumulating two types of information through time: the "what" (features extracted) and the "where" (eye position). The fovea map, F, and the eye position map, P, are concatenated to make a r + p X P dimensional input that is fed to the associative level. Here we use an artificial neural network with one hidden layer of 8 units. We have experimented with various architectures and noticed that recurrency at the output layer is the best. There are 10 output units. f (L VhgFg(t) + L L WhabPab(t)) , h = 1. .. s gab LTchHh + L RckPk(t - 1), c = 1. .. 10 h k exp[Oc(t)] Lk exp[Ok(t)] where P denotes the "softmax"ed output probabilities (Bridle, 1990) and P(t - 1) are the values in the preceding fixation (initially 0). We use the cross-entropy as the goodness measure: 1 C = L t L Dk 10gPc(t), t ~ 1 t c Dc is the required output for class c. Learning is gradient-ascent on this goodness measure. The fraction lit is to give more weight to initial fixations than later ones. Connections to the output units are updated as follows (11 is the learning factor): Selective Attention for Handwritten Digit Recognition 775 Note that we assume 8PIc(t -1)/8Rclc = o. For the connections to the hidden units we have: c We can back-propagate one step more to train the feature extractors. Thus the update equations for the connections to feature units are: Cg(t) = L Ch(t)Vhg h A series of fixations are made until one of the class units is sufficiently active: 3c, Pc > 8 (typically 0.99), or when the most salient point has a saliency less than a certain threshold (this condition is rarely met after the first few epochs). Then the computed changes are summed up and the updates are made like the exaple below: Backpropagation through time where the recurrent connections are unfolded in time did not work well in this task because as explained before, for the same class, there is more than one scanpath. The above-mentioned approach is like real-time recurrent learning (Williams and Zipser, 1989) where the partial derivatives in the previous time step is 0, thus ignoring this temporal dependence. 3 RESULTS AND DISCUSSION We have experimented with various parameter settings and finally chose the architecture given above: When input is 16 x 16 and there are 10 classes, the fovea is 5 x 5 with 8 features and there are 16 hidden units. There are 1,934 images for training, 946 for cross-validation and 943 for testing. Results are given in Table 1. ( It can be seen that by scanning less than half of the image, we get 80% generalization. Additional to the local high-resolution image provided by the fovea, a low-resolution image of the surrounding parafovea can be given to the associative level for better recognition. For example we low-pass filtered and undersampled the original image to get a 4 x 4 image which we fed to the class units additional to the attention-based hidden units. Success went up quite high and fewer fixations were necessary; compare rows 1 and 2 of the Table. The information provided by the 4 x 4 map is actually not much as can be seen from row 3 of the table where only that is given as input. Thus the idea is that when we have a coarse input, looking only at a quarter of the image in detail is sufficient to get 93% accuracy. Both features (what) and eye positions (where) are necessary for good recognition. When only one is used without the other, success is quite low as can be seen in rows 4 and 5. In the last row, we see the performance of a multi layer percept ron with 10 hidden units that does all-parallel recognition. Beyond a certain network size, increasing the number of features do not help much. Decreasing 8, the certainty threshold, decreases the number of fixations necessary 776 E. ALPAYDIN Table 1: Results of handwritten digit recognition with selective attention. Values given are average and standard deviation of 10 independent runs. See text for comments. NO OF TEST TRAINING NO OF METHOD PARAMS SUCCESS EPOCHS FIXATIONS SA system 878 79.7, 1.8 74.5, 17.1 6.5,0.2 SA+parafovea 1,038 92.5,0.8 54.2, 10.2 3.9,0.3 Only parafovea 170 86.9,0.2 52.3,8.2 1.0, 0.0 Only what info 622 49.0,21.0 66.6, 30.6 7.5,0.1 Only where info 440 54.2, 1.4 92.9,6.5 7.6,0.0 MLP, 10 hiddens 2,680 95.1, 0.6 13.5,4.1 1.0,0.0 which we want, but decreases success too which we don't. Smaller foveas decrease the number of free parameters but decrease success and require a larger number of fixations. Similarly larger foveas decrease the number of fixations but increase complexity. The simple low-pass filter used here as a saliency measure is the simplest measure. Previously it has been used by Fukushima and Imagawa (1993) for finding the next character, i.e., segmentation, and also by Olshausen et al. (1992) for translation invariance. More robust measures at the expense of more computations, are possible; see (Rimey and Brown, 1990; Milanese et al., 1993). Salient regions are those that are conspicious, i.e., different from their surrounding where there is a change in X where X can be brightness or color (edges), orientation (corners), time (motion), etc. It is also possible that top-down, task-dependent saliency measures be integrated to minimize further recognition time implying a remembered explicit sequence analogous to skilled motor behaviour (probably gained after many repetitions). Here a partially recurrent network is used for temporal processing. Hidden Markov Models like used in speech recognition are another possibility (Rimey and Brown, 1990; Haclsalihzade et al., 1992). They are probabilistic finite automata which can be trained to classify sequences and one can have more than one model for an object. It should be noted here that better approaches for the same problem exists (Le Cun et al., 1989). Here we advocate a computational model and make it plausible by testing it on a real-world problem. It is necessary for more complicated problems where an all-parallel approach would not work. For example Le Cun et al. 's model for the same type of inputs has 2,578 free parameters. Here there are (mx m+1) x r+(r+pxp+ 1) x 8+(S+ 1) x 10+10 x 10 , #' #~~ iT v';w T R free parameters which make 878 when m = 5, r = 8, S = 16. This is the main advantage of selective attention which is that the complexity of the system is heavily reduced at the expense of slower recognition, both in overt form of attention through foveation and in its covert form, for binding features For this latter type of attention not discussed here, see (Ahmad, 1992). Also note that low-level feature extraction operations like carried out in the pre-attentive level are local convolutions Selective Attention for Handwritten Digit Recognition 777 and are appropriate for parallel processing, e.g., on a SIMD machine. Higherlevel operations require larger connectivity and are better carried out sequentially. Nature also seems to have taken this direction. Acknowledgements This work is supported by Tiibitak Grant EEEAG-143 and Bogazi<;;i University Research Funds 95HA108. Cenk Kaynak prepared the handwritten digit database based on the programs provided by NIST (Garris et al., 1994). References S. Ahmad. (1992) VISIT: A Neural Model of Covert Visual Attention. In J. Moody, S. Hanson, R. Lippman (Eds.) Advances in Neural Information Processing Systems 4,420-427. San Mateo, CA: Morgan Kaufmann. J.S. Bridle. (1990) Probabilistic Interpretation of Feedforward Classification Network Outputs with Relationships to Statistical Pattern Recognition. In Neurocomputing, F. Fogelman-Soulie, J. Herault, Eds. Springer, Berlin, 227-236. K. Fukushima, T. Imagawa. (1993) Recognition and Segmentation of Connected Characters with Selective Attention, Neural Networks, 6: 33-41. M.D. Garris et al. (1994) NIST Form-Based Handprint Recognition System, NISTIR 5469, NIST Computer Systems Laboratory. S.S. Haclsalihzade, L.W. Stark, J .S. Allen. (1992) Visual Perception and Sequences of Eye Movement Fixations: A Stochastic Modeling Approach, IEEE SMC, 22, 474-481. Y. Le Cun et al. (1991) Handwritten Digit Recognition with a Back-Propagation Network. In D.S. Touretzky (ed.) Advances in Neural Information Processing Systems 2, 396-404. San Mateo, CA: Morgan Kaufmann. R. Milanese et al. (1994) Integration of Bottom-U p and Top-Down Cues for Visual Attention using Non-Linear Relaxation IEEE Int'l Conf on CVPR, Seattle, WA, USA. D. Noton and L. Stark. (1971) Eye Movements and Visual Perception, Scientific American, 224: 34-43. B. Olshausen, C. Anderson, D. Van Essen. (1992) A Neural Model of Visual Attention and Invariant Pattern Recognition, CNS Memo 18, CalTech. M.L Posner, S.E. Petersen. (1990) The Attention System of the Human Brain, Ann. Rev. Neurosci., 13:25-42. R.D. Rimey, C.M. Brown. (1990) Selective Attention as Sequential Behaviour: Modelling Eye Movements with an Augmented Hidden Markov Model, TR-327, Computer Science, Univ of Rochester. A. Treisman. (1988) Features and Objects, Quarterly Journ. of Ezp. Psych., 40: 201-237. J.K. Tsotsos. (1990) Analyzing Vision at the Complexity Level, Behav. and Brain Sci. 13: 423-469. R.J. Williams, D. Zipser. (1989) A Learning Algorithm for Continually Running Fully Recurrent Neural Networks Neural Computation, 1, 270-280.
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Learning Model Bias Jonathan Baxter Department of Computer Science Royal Holloway College, University of London jon~dcs.rhbnc.ac.uk Abstract In this paper the problem of learning appropriate domain-specific bias is addressed. It is shown that this can be achieved by learning many related tasks from the same domain, and a theorem is given bounding the number tasks that must be learnt. A corollary of the theorem is that if the tasks are known to possess a common internal representation or preprocessing then the number of examples required per task for good generalisation when learning n tasks simultaneously scales like O(a + ~), where O(a) is a bound on the minimum number of examples requred to learn a single task, and O( a + b) is a bound on the number of examples required to learn each task independently. An experiment providing strong qualitative support for the theoretical results is reported. 1 Introduction It has been argued (see [6]) that the main problem in machine learning is the biasing of a learner's hypothesis space sufficiently well to ensure good generalisation from a small number of examples. Once suitable biases have been found the actual learning task is relatively trivial. Exisiting methods of bias generally require the input of a human expert in the form of heuristics, hints [1], domain knowledge, etc. Such methods are clearly limited by the accuracy and reliability of the expert's knowledge and also by the extent to which that knowledge can be transferred to the learner. Here I attempt to solve some of these problems by introducing a method for automatically learning the bias. The central idea is that in many learning problems the learner is typically embedded within an environment or domain of related learning tasks and that the bias appropriate for a single task is likely to be appropriate for other tasks within the same environment. A simple example is the problem of handwritten character recognition. A preprocessing stage that identifies and removes any (small) rotations, dilations and translations of an image of a character will be advantageous for 170 J.BAXTER recognising all characters. If the set of all individual character recognition problems is viewed as an environment of learning tasks, this preprocessor represents a bias that is appropriate to all tasks in the environment. It is likely that there are many other currently unknown biases that are also appropriate for this environment. We would like to be able to learn these automatically. Bias that is appropriate for all tasks must be learnt by sampling from many tasks. If only a single task is learnt then the bias extracted is likely to be specific to that task. For example, if a network is constructed as in figure 1 and the output nodes are simultaneously trained on many similar problems, then the hidden layers are more likely to be useful in learning a novel problem of the same type than if only a single problem is learnt. In the rest of this paper I develop a general theory of bias learning based upon the idea of learning multiple related tasks. The theory shows that a learner's generalisation performance can be greatly improved by learning related tasks and that if sufficiently many tasks are learnt the learner's bias can be extracted and used to learn novel tasks. Other authors that have empirically investigated the idea of learning multiple related tasks include [5] and [8]. 2 Learning Bias For the sake of argument I consider learning problems that amount to minimizing the mean squared error of a function h over some training set D. A more general formulation based on statistical decision theory is given in [3]. Thus, it is assumed that the learner receives a training set of (possibly noisy) input-output pairs D = {(XI, YI), ... , (xm' Ym)}, drawn according to a probability distribution P on X X Y (X being the input space and Y being the output space) and searches through its hypothesis space 1l for a function h: X --+ Y minimizing the empirical error, 1 m E(h, D) = 2)h(xd - yd 2. m i=1 The true error or generalisation error of h is the expected error under P: E(h, P) = r (h(x) - y)2 dP(x, y). ixxY (1) (2) The hope of course is that an h with a small empirical error on a large enough training set will also have a small true error, i.e. it will generalise well. I model the environment of the learner as a pair (P, Q) where P = {P} is a set of learning tasks and Q is a probability measure on P. The learner is now supplied not with a single hypothesis space 1l but with a hypothesis space family IHI = {1l}. Each 1l E IHI represents a different bias the learner has about the environment. For example, one 1l may contain functions that are very smooth, whereas another 1l might contain more wiggly functions. Which hypothesis space is best will depend on the kinds of functions in the environment. To determine the best 1l E !HI for (P, Q), we provide the learner not with a single training set D but with n such training sets DI , ... , Dn. Each Di is generated by first sampling from 'P according to Q to give Pi and then sampling m times from X x Y according to Pi to give Di = {(XiI, Yil), ... , (Xim, Yim)}. The learner searches for the hypothesis space 1l E IHI with minimal empirical error on D I , ... , Dn , where this is defined by ~ 1 2: n ~ E*(1l, DI , ... , Dn) = inf E(h, Dd. n hE1/. i=1 (3) Learning Model Bias ... 9 1 ... ... ___ L __ , I I f 171 \ - - -'- - -, I I I I Figure 1: Net for learning multiple tasks. Input Xij from training set Di is propagated forwards through the internal representation f and then only through the output network gi. The error [gi(l(Xij)) - Yij]2 is similarly backpropagated only through the output network gi and then f. Weight updates are performed after all training sets D1 , ... , Dn have been presented. The hypothesis space 1l with smallest empirical error is the one that is best able to learn the n data sets on average. There are two ways of measuring the true error of a bias learner. The first is how well it generalises on the n tasks PI,"" Pn used to generate the training sets. Assuming that in the process of minimising (3) the learner generates n functions hI, ... , hn E 1l with minimal empirical error on their respective training setsI , the learner's true error is measured by: (4) Note that in this case the learner's empirical error is given by En(hI, ... , hn' Db ... , Dn) = ~ 2:~=I E(hi' Dt}. The second way of measuring the generalisation error of a bias learner is to determine how good 1l is for learning novel tasks drawn from the environment (P, Q): E*(1l, Q) = 1 inf E(h, P) dQ(P) 1> hEll (5) A learner that has found an 1l with a small value of (5) can be said to have learnt to learn the tasks in P in general. To state the bounds ensuring these two types of generalisation a few more definitions must be introduced. Definition 1 Let !HI = {1l} be a hypothesis space family. Let ~ = {h E 1l: 1l E lHt}. For any h:X -+ Y, define a map h:X X Y -+ [0,1] by h(x,y) = (h(x) _ y)2. Note the abuse of notation: h stands for two different functions depending on its argument. Given a sequence of n functions h = (hI, . .. , hn) let h: (X x y)n -+ [0,1] be the function (XI, YI, ... , Xn , Yn) H ~ 2:~=I hi (Xi, yt). Let 1ln be the set of all such functions where the hi are all chosen from 1l. Let JHr = {1ln: 1l E H}. For each 1l E !HI define 1l*:P -+ [0,1] by1l*(P) = infhEll E(h, P} and let:HI* = {1l*:1l E !HI}. 1 This assumes the infimum in (3) is attained. 172 J. BAXTER Definition 2 Given a set of function s 1i from any space Z to [0, 1], and any probability measure on Z, define the pseudo-metric dp on 1i by dp(h, hI) = l lh(Z) - hl(z)1 dP(z). Denote the smallestE-cover of (1i,dp ) byJV{E,1i,dp ). Define the E-capacity of1i by C(E, 1i) = sup JV (E, 1i, dp ) p where the supremum is over all discrete probability measures P on Z. Definition 2 will be used to define the E-capacity of spaces such as !HI* and [IHF ]"., where from definition 1 the latter is [IHF ]". = {h E 1in :1i E H}. The following theorem bounds the number of tasks and examples per task required to ensure that the hypothesis space learnt by a bias learner will, with high probability, contain good solutions to novel tasks in the same environment2 . Theorem 1 Let the n training sets DI , ... , Dn be generated by sampling n times from the environment P according to Q to give PI"", Pn, and then sampling m times from each Pi to generate Di . Let !HI = {1i} be a hypothesis space family and suppose a learner chooses 1£ E 1HI minimizing (3) on D I , ... , Dn. For all E > 0 and 0 < 8 < 1, if n and m then The bound on m in theorem 1 is the also the number of examples required per task to ensure generalisation of the first kind mentioned above. That is, it is the number of examples required in each data set Di to ensure good generalisation on average across all n tasks when using the hypothesis space family 1HI. If we let m(lHI, n, E, 8) be the number of examples required per task to ensure that Pr {Db"" Dn: IEn(hI"' " hn, DI"'" Dn) - En(hI, . . . , hn, PI" " , Pn)1 > E} < 8, where all hi E 1i for some fixed 1i E IHI, then G(IHI, n, E, 8) = m(lHI, 1, E, 8) m{lHI, n, E, 8) represents the advantage in learning n tasks as opposed to one task (the ordinary learning scenario). Call G(IHI, n, E, 8) the n-task gain of IHI. Using the fact [3] that C (E, lHl". ) :::; C (E, [IHr]".) :::; C (E, IHl". t , and the formula for m from theorem 1, we have, 1 :::; G{IHI, n, E, 8) :::; n. 2The bounds in theorem 1 can be improved to 0 (~) if all 11. E H are convex and the error is the squared loss [7]. Learning Model Bias 173 Thus, at least in the worst case analysis here, learning n tasks in the same environment can result in anything from no gain at all to an n-fold reduction in the number of examples required per task. In the next section a very intuitive analysis of the conditions leading to the extreme values of G(H, n, c, J) is given for the situation where an internal representation is being learnt for the environment. I will also say more about the bound on the number of tasks (n) in theorem 1. 3 Learning Internal Representations with Neural Networks In figure 1 n tasks are being learnt using a common representation f. In this case [JHF]". is the set of all possible networks formed by choosing the weights in the representation and output networks. IHl". is the same space with a single output node. If the n tasks were learnt independently (i.e. without a common representation) then each task would use its own copy of H"., i.e. we wouldn't be forcing the tasks to all use the same representation. Let W R be the total number of weights in the representation network and W 0 be the number of weights in an individual output network. Suppose also that all the nodes in each network are Lipschitz boundecP. Then it can be shown [3] that InC(c, [IHr]".):::: 0 ((Wo + ~)In~) and InC(c,IHr):::: 0 (WRln~). Substituting these bounds into theorem 1 shows that to generalise well on average on n tasks using a common representation requires m :::: 0 (/2 [( W 0 + ~) In ~ + ~ In } ]) :::: o (a + .~J examples of each task. In addition, if n :::: 0 CI, WR In ~) then with high probability the resulting representation will be good for learning novel tasks from the same environment. Note that this bound is very large. However it results from a worst-case analysis and so is highly likely to be beaten in practice. This is certainly borne out by the experiment in the next section. The learning gain G(H, n, c) satisfies G(H, n, c) ~ Wo±'!a. Thus, if WR » Wo, Wo±.:::.B. G ~ n, while if Wo » W R then G ~ 1. This is perfectly intuitive: when Wo » W R the representation network is hardly doing any work, most of the power of the network is in the ouput networks and hence the tasks are effectively being learnt independently. However, if WR » Wo then the representation network dominates; there is very little extra learning to be done for the individual tasks once the representation is known, and so each example from every task is providing full information to the representation network. Hence the gain of n. Note that once a representation has been learnt the sampling burden for learning a novel task will be reduced to m:::: 0 (e1, [Wo In ~ + In}]) because only the output network has to be learnt. If this theory applies to human learning then the fact that we are able to learn words, faces, characters, etcwith relatively few examples (a single example in the case offaces) indicates that our "output networks" are very small, and, given our large ignorance concerning an appropriate representation, the representation network for learning in these domains would have to be large, so we would expect to see an n-task gain of nearly n for learning within these domains. 3 A node a : lR P -t lR is LipJChitz bounded if there exists a constant e such that la( x) a(x'}1 < ellx - x'il for all x, x' E lR P• Note that this rules out threshold nodes, but sigmoid squashing functions are okay as long as the weights are bounded. 174 J. BAXTER 4 Experiment: Learning Symmetric Boolean Functions In this section the results of an experiment are reported in which a neural network was trained to learn symmetric4 Boolean functions. The network was the same as the one in figure 1 except that the output networks 9i had no hidden layers. The input space X = {O, Ipo was restricted to include only those inputs with between one and four ones. The functions in the environment of the network consisted of all possible symmetric Boolean functions over the input space, except the trivial "constant 0" and "constant 1" functions. Training sets D 1 , ..• ,Dn were generated by first choosing n functions (with replacement) uniformly from the fourteen possible, and then choosing m input vectors by choosing a random number between 1 and 4 and placing that many l's at random in the input vector. The training sets were learnt by minimising the empirical error (3) using the backpropagation algorithm as outlined in figure 1. Separate simulations were performed with n ranging from 1 to 21 in steps of four and m ranging from 1 to 171 in steps of 10. Further details of the experimental procedure may be found in [3], chapter 4. Once the network had sucessfully learnt the n training sets its generalization ability was tested on all n functions used to generate the training set. In this case the generalisation error (equation (4)) could be computed exactly by calculating the network's output (for all n functions) for each of the 385 input vectors. The generalisation error as a function of nand m is plotted in figure 2 for two independent sets of simulations. Both simulations support the theoretical result that the number of examples m required for good generalisation decreases with increasing n (cf theorem 1). For training sets D1 , ... , Dn that led to a generalisation error of less than G&nerailsahon Error Figure 2: Learning surfaces for two independent simulations. 0.01, the representation network f was extracted and tested for its true error, where this is defined as in equation (5) (the hypothesis space 1£ is the set of all networks formed by attaching any output network to the fixed representation network f). Although there is insufficient space to show the representation error here (see [3] for the details), it was found that the representation error monotonically decreased with the number of tasks learnt, verifying the theoretical conclusions. The representation's output for all inputs is shown in figure 3 for sample sizes (n, m) = (1,131), (5, 31) and (13,31). All outputs corresponding to inputs from the same category (i.e. the same number of ones) are labelled with the same symbol. The network in the n = 1 case generalised perfectly but the resulting representation does not capture the symmetry in the environment and also does not distinguish the inputs with 2,3 and 4 "I's" (because the function learnt didn't), showing that 4 A symmetric Boolean function is one that is invariant under interchange of its inputs, or equivalently, one that only depends on the number of "l's" in its input (e.g. parity). Learning Model Bias 175 learning a single function is not sufficient to learn an appropriate representation. By n = 5 the representation's behaviour has improved (the inputs with differing numbers of l's are now well separated, but they are still spread around a lot) and by n = 13 it is perfect. As well as reducing the sampling burden for the n tasks in ( 1, HII ( 5, J I I ( I), ll) node J node.8~\ 16 o d node ~ no e I I 0 Figure 3: Plots of the output of a representation generated from the indicated (n, m) sample. the training set, a representation learnt on sufficiently many tasks should be good for learning novel tasks and should greatly reduce the number of examples required for new tasks. This too was experimentally verified although there is insufficient space to present the results here (see [3]). 5 Conclusion I have introduced a formal model of bias learning and shown that (under mild restrictions) a learner can sample sufficiently many times from sufficiently many tasks to learn bias that is appropriate for the entire environment. In addition, the number of examples required per task to learn n tasks independently was shown to be upper bounded by O(a + bin) for appropriate environments. See [2] for an analysis of bias learning within an Information theoretic framework which leads to an exact a + bin-type bound. References [1] Y. S. Abu-Mostafa. Learning from Hints in Neural Networks. Journal of Complecity, 6:192-198, 1989. [2] J. Baxter. A Bayesian Model of Bias Learning. Submitted to COLT 1996, 1995. [3] J. Baxter. Learning Internal Representations. PhD thesis, Department of Mathematics and Statistics, The Flinders University of South Australia, 1995. Draft copy in Neuroprose Archive under "/pub/neuroprose/Thesis/baxter.thesis.ps.Z" . [4] J. Baxter. Learning Internal Representations. In Proceedings of the Eighth International Conference on Computational Learning Theory, Santa Cruz, California, 1995. ACM Press. [5] R. Caruana. Learning Many Related Tasks at the Same Time with Backpropagation. In Advances in Neural Information Processing 5, 1993. [6] S. Geman, E. Bienenstock, and R. Doursat. Neural networks and the bias/variance dilemma. Neural Comput., 4:1-58, 1992. [7] W. S. Lee, P. L. Bartlett, and R. C. Williamson. Sample Complexity of Agnostic Learning with Squared Loss. In preparation, 1995. [8] T. M. Mitchell and S. Thrun. Learning One More Thing. Technical Report CMU-CS-94-184, CMU, 1994.
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Estimating the Bayes Risk from Sample Data Robert R. Snapp· and Tong Xu Computer Science and Electrical Engineering Department University of Vermont Burlington, VT 05405 Abstract A new nearest-neighbor method is described for estimating the Bayes risk of a multiclass pattern claSSification problem from sample data (e.g., a classified training set). Although it is assumed that the classification problem can be accurately described by sufficiently smooth class-conditional distributions, neither these distributions, nor the corresponding prior probabilities of the classes are required. Thus this method can be applied to practical problems where the underlying probabilities are not known. This method is illustrated using two different pattern recognition problems. 1 INTRODUCTION An important application of artificial neural networks is to obtain accurate solutions to pattern classification problems. In this setting, each pattern, represented as an n-dimensional feature vector, is associated with a discrete pattern class, or state of nature (Duda and Hart, 1973). Using available information, (e.g., a statistically representative set of labeled feature vectors {(Xi, fin, where Xi E Rn denotes a feature vector and fi E l:::: {Wl,W2, ... ,we}, its correct pattern class), one desires a function (e.g., a neural network claSSifier) that assigns new feature vectors to pattern classes with the smallest possible misclassification cost. If the classification problem is stationary, such that the patterns from each class are generated according to known probability distributions, then it is possible to construct an optimal clasSifier that assigns each pattern to a class with minimal expected risk. Although our method can be generalized to problems in which different types of classification errors incur different costs, we shall simplify our discussion by assuming that all errors are equal. In this case, a Bayes claSSifier assigns each feature vector to a class with maximum posterior probability. The expected risk of this classifier, or Bayes risk then reduces to the probability of error RB = r [1 - SUPPCf!X)] JCx)dx, Js fEL (1) • E-mail:snapp<Demba.uvm.edu Estimating the Bayes Risk from Sample Data 233 (Duda and Hart, 1973). Here, P( fix) denotes the posterior probability of class f conditioned on observing the feature vector x, f(x) denotes the unconditional mixture density of the feature vector x, and S C Rn denotes the probability-one support of f. Knowing how to estimate the value of the Bayes risk of a given classification problem with a specific input representation, may facilitate the design of more accurate classifiers. For example, since the value of RB depends upon the set of features chosen to represent each pattern (e.g., the significance of the input units in a neural network classifier), one might compare estimates of the Bayes risk for a number of different feature sets, and then select the representation that yields the smallest value. Unfortunately, it is necessary to know the explicit probability distributions to evaluate (1). Thus with the possible exception of trivial examples, the Bayes risk cannot be determined exactly for practical classification problems. Lacking the means to evaluate the Bayes risk exactly, motivates the development of statistical estimators of RB. In this paper, we use a recent asymptotic analysis of the finite-sample risk of the k-nearest-neighbor classifier to obtain a new procedure for estimating the Bayes risk from sample data. Section 2 describes the k-nearest-neighbor algorithm, and briefly describes how estimates of its finite-sample risk have been used to estimate RB. Section 3 describes how a recent asymptotic analysis of the finite-sample risk can be applied to obtain new statistical estimators of the Bayes risk. In Section 4 the k-nearest-neighbor algorithm is used to estimate the Bayes risk of two example problems. Section 5 contains some concluding remarks. 2 THE k-NEAREST-NEIGHBOR CLASSIFIER Due to its analytic tractability, and its nearly optimal performance in the large sample limit, the k-nearest-neighbor classifier has served as a useful framework for estimating the Bayes risk from classified samples. Recall, that the k-nearest-neighbor algorithm (Fix and Hodges, 1951) clasSifies an n-dimensional feature vector x by consulting a reference sample of m correctly clasSified feature vectors Xm = {(Xi, f i ) : i = 1, ... m}. First, the algorithm identifies the k nearest neighbors of x, Le., the k feature vectors within Xm that lie closest to x with respect to a given metric. Then, the classifier assigns x to the most frequent class label represented by the k nearest neighbors. (A variety of procedures can be used to resolve ties.) In the following, C denotes the number of pattern classes. The finite-sample risk of this algorithm, Rm , equals the probability that the k-nearestneighbor classifier assigns x to an incorrect class, averaged over all input vectors x, and all m-samples, Xm . The following properties have been shown to be true under weak assumptions: Property 1 (Cover and Hart, 1967): For.fixed k, Rm -+ Roo(k), as m -+ 00 with RB ~ Roo(1) ~ RB (2 - C ~ 1 R B). (2) Property 2 (Devroye, 1981): If k ~ 5, and C = 2, then there exist universal constants a = 0.3399· .. , and (3 = 0.9749 ... such that Roo(k) is bounded by a..(f ( (3) RB ~ Roo(k) ~ (1 + ak)RB, where ak = k _ 3.25 1 + ~ . More generally, ifC = 2, then HB '" Roo(k) '" (1 + If) RB(3) 234 R. R. SNAPP, T. XU By the latter property, this algorithm is said to be Bayes consistent in that for any c > 0, it is possible to construct a k-nearest-neighbor classifier such that IRm - RBI < c if m and k are sufficiently large. Bayes consistency is also evident in other nonparametric pattern classifiers. Several methods for estimating RB from sample data have previously been proposed, e.g., (Devijver, 1985), (Fukunaga, 1985), (Fukunaga and Hummels, 1987), (Garnett and Yau, 1977), and (Loizou and Maybank, 1987). Typically, these methods involve constructing sequences of k-nearest neighbor classifiers, with increasing values of k and m. The misclassification rates are estimated using an independent test sample, from which upper and lower bounds to RB are obtained. Because these experiments are necessarily performed with finite reference samples, these bounds are often imprecise. This is especially true for problems in which Rm converges to Roo(k) at a slow rate. In order to remedy this deficiency, it is necessary to understand the manner in which the limit in Property 1 is achieved. In the next section we describe how this information can be used to construct new estimators for the Bayes risk of sufficiently smooth claSSification problems. 3 NEW ESTIMATORS OF THE BAYES RISK For a subset of multiclass classification problems that can be described by probability densities with uniformly bounded partial derivatives up through order N + 1 (with N 2: 2), the finite-sample risk of a k-nearest-neighbor classifier that uses a weighted Lp metric can be represented by the truncated asymptotic expansion N Rm = Roo(k) + 2:::Cjm-j/n + 0 (m- CN+1)/n) , (4) j=2 (Psaltis, Snapp, and Venkatesh, 1994), and (Snapp and Venkatesh, 1995). In the above, n equals the dimensionality of the feature vectors, and Roo(k), C2, ... ,CN, are the expansion coefficients that depend upon the probability distributions that define the pattern classification problem. This asymptotic expansion provides a parametric description of how the finite-sample risk Rm converges to its infinite sample limit Roo(k). Using a large sample of classified data, one can obtain statistical estimates of the finite-sample risk flm for different values of m. Specifically, let {md denote a sequence of M different sample sizes, and select fixed values for k and N. For each value of mi, construct an ensemble of k-nearest-neighbor classifiers, i.e., for each classifier construct a random reference sample Xmi by selecting mi patterns with replacement from the original large sample. Estimate the empirical risk of each classifier in the ensemble with an independently drawn set of "test" vectors. Let flmi denote the average empirical risk of the i-th ensemble. Then, using the resulting set of data points {(mi, RmJ}, find the values of the coefficients Roo(k), and C2 through CN, that minimizes the sum of the squares: M ( N)2 8 flmi - Roo(k) - ~ Cjm;j/n (5) Several inequalities can then be used obtain approximations of RB from the estimated value of Roo(k). For example, if k = 1, then Cover and Hart's inequality in Property 1 implies that Roo(l) < R < R (1). 2 B_ 00 To enable an estimate of RB with preciSion c, choose k > 2/c2, and estimate Roo(k) by the above methOd. Then Devroye's inequality (3) implies Roo(k) - c ~ Roo(k)(1 - c) ~ RB ~ Roo(k). Estimating the Bayes Risk from Sample Data 235 4 EXPERIMENTAL RESULTS The above procedure for estimating RB was applied to two pattern recognition problems. First consider the synthetic, two-class problem with prior probabilities PI = P2 = 1/2, and normally distributed, class-conditional densities f(x)= 1 e-H(Xl+(-1)t)2+I:~=2xn I (27r)n/2 ' for f = 1 and 2. Pseudorandom labeled feature vectors (x, f) were numerically generated in accordance with the above for dimensions n = 1 and n = 5. Twelve sample sizes between 10 and 3000 were examined. For each dimension and sample size the risks Rm of many independent k-nearest-neighborclassifiers with k = 1,7, and 63 were empirically estimated. (Because the asymptotic expansion (4) does not accurately describe the very small sample behavior of the k-nearest-neighbor classifier, sample sizes smaller than 2k were not included in the fit.) Estimates of the coefficients in (5) for six different fits appear in the first equation of each cell in the third and fourth columns of Table 1. For reference, the second column contains the values of RooCk) that were obtained by numerically evaluating an exact integral expression (Cover and Hart, 1967). Estimates of the Bayes risk appear in the second equation of each cell in the third and fourth columns. Cover and Hart's inequality (2) was used for the experiments that assumed k = 1, and Devroye's inequality (3) was used if k ~ 7. For thiS problem, formula (1) evaluates to RB = (l/2)erfc(I/V2) = 0.15865. Table 1: Estimates of the model coefficients and Bayes error for a classification problem with two normal classes. k Roo(k) n=1 (N=2) n=5 (N =6) R m =0.2287 + 0.6536 Rm =0.2287 + 0.1121 0.2001 0.0222 + 1 0.2248 m2 m2/ 5 m4/ 5 m6/ 5 RB =0.172 ± 0.057 RB =0.172 ± 0.057 4.842 Rm =0.1700+ 0.2218 1.005 3.782 Rm =0.1744 + -2---+-7 0.1746 m m 2/ 5 m4/ 5 m6/ 5 RB =0.152 ±0.023 RB =0.148 ± 0.022 20.23 Rm =0.1595 + 0.1002 1.426 10.96 Rm =0.1606 + -2---+-63 0.1606 m m2/ 5 m4/ 5 m6/ 5 RB =0.157 ± 0.004 RB =0.156 ± 0.004 The second pattern recognition problem uses natural data; thus the underlying probability distributions are not known. A pool of 222 classified multispectral pixels were was extracted from a seven band satellite image. Each pixel was represented by five spectral components, x = (Xl, .. . ,X5), each in the range 0 ~ X" ~ 255. (Thus, n = 5.) The class label of each pixel was determined by one of the remaining spectral components, 0 ~ y ~ 255. Two pattern classes were then defined: Wl = {y < B}, and W2 = {y ~ B}, where B was a predetermined threshOld. (This particular problem was chosen to test the feasibility of this method. In future work, we will examine more interesting pixel claSsification problems.) 236 R. R. SNAPP, T. XU Table 2: Coefficients that minimize the squared error fit for different N. Note that C3 = 0 and Cs = 0 in (2) ifn ~ 4 (Psaltis, Snapp, and Venkatesh, 1994). N Roo(l) 2 0.0757133 0.126214 4 6 0.0757846 0.0766477 0.124007 0.0785847 0.0132804 0.689242 -2.68818 With k = 1, a large number of Bernoulli trials (e.g., 2~1000) were performed for each value of mi . Each trial began by constructing a reference sample of mi classified pixels chosen at random from the pool. The risk of each reference sample was then estimated by classifying t pixels with the nearest-neighbor algorithm under a Euclidean metric. Here, the t pixels, with 2000 ~ t ~ 20000, were chosen independently, with replacement, from the pool. The risk 11m. was then estimated as the average risk of each reference sample of size mi . (The number of experiments performed for each value of mi, and the values oft, were chosen to ensure that the variance of Hm. was sufficiently small, less than 10- 4 in this case.) This process was repeated for M = 33 different values of mi in the range 1 00 ~ mi ~ 15000. Results of these experiments are displayed in Table 2 and Figure 1 for three different values of N. Note that the robustness of the fit begins to dissolve, for this data, at N = 6, either the result of overfitting, or insuffiCient smoothness in the underlying probability distributions. However, the estimate for Roo(l) appears to be stable. For this claSSification problem, we thus obtain RB = 0.0568 ± 0.0190. 5 CONCLUSION The described method for estimating the Bayes risk is based on a recent asymptotic analysis of the finite-sample risk of the k-nearest-neighbor classifier (Snapp and Venkatesh, 1995). Representing the finite-sample risk as a truncated asymptotic series enables an efficient estimation of the infinite-sample risk Roo(k) from the classifier's finite-sample behavior. The Bayes risk can then be estimated by the Bayes consistency of the k-nearest-neighbor algorithm. Because such finite-sample analyses are difficult, and consequently rare, this new method has the potential to evolve into a useful algorithm for estimating the Bayes risk. Further improvements in efficiency may be obtained by incorporating principles of optimal experimental deSign, cf., (Elfving, 1952) and (Federov, 1972). It is important to emphasize, however, that the validity of (4) rests on several rather strong smoothness assumptions, including a high-degree of differentiability of the class-conditional probability densities. For problems that do not satisfy these conditions, other finite-sample descriptions need to be constructed before this method can be applied. Nevertheless, there is much evidence that nature favors smoothness. Thus, these restrictive assumptions may still be applicable to many important problems. Acknowledgments The work reported here was supported in part by the National Science Foundation under Grant No. NSF OSR-9350540 and by Rome Laboratory, Air Force Material Command, USAF, under grant number F30602-94-1-OOlO. Estimating the Bayes Risk from Sample Data -1.8 -rl -2.0 I t: 0.:: -0 -2.2 'ol) 0 -2.4 100 1000 m 10000 237 Figure 1: The best fourth-order (N = 4) fit of Eqn. (5) to 33 empirical estimates of Hmo for a pixel classification problem obtained from a multispectral Landsat image. Using RXI = 0.0758, the fourth-order fit, Rm = 0.0758 + 0.124m- 2/ 5 + 0.0133m - 4/5, is plotted on a log-log scale to reveal the significance of the j = 2 term. References T. M. Cover and P. E. Hart, "Nearest neighbor pattern classification," IEEE Trans. Inform. Theory,vol.IT-13,1967,pp.21-27. P. A. Devijver, "A multiclass, k - N N approach to Bayes risk estimation," Pattern Recognition Letters, vol. 3, 1985, pp. 1-6. L. Devroye, "On the asymptotic probability of error in nonparametric discrimination," Annals Of Statistics, vol. 9, 1981, pp. 1320-1327. R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis. New York, New York: John Wiley & Sons, 1973. G. Elfving, "Optimum allocation in linear regression theory," Ann. Math. Statist., vol. 23, 1952,pp.255-262. V. V. Federov, Theory Of Optimal Experiments, New York, New York: Academic Press, 1972. E. Fix and J. L. Hodges, "Discriminatory Analysis: Nonparametric Discrimination: Consistency Properties," from Project 21-49-004, Repon Number 4, UASF School of Aviation Medicine, Randolf Field, Texas, 1951, pp. 261-279. 238 R. R. SNAPP, T. XU K. Fukunaga, "The estimation of the Bayes error by the k-nearest neighbor approach," in L. N. Kanal and A. Rosenfeld (ed.), Progress in Pattern Recognition, vol. 2, Elesvier Science Publishers B.V. (North Holland), 1985, pp. 169-187. K. Fukunaga and D. Hummels, "Bayes error estimation using Parzen and k-NN procedures," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 9, 1987, pp. 634-643. J. M. Garnett, III and S. S. Yau, "Nonparametric estimation of the Bayes error of feature extractors using ordered nearest neighbor sets," IEEE Transactions on Computers, vol. 26, 1977,pp.46-54. G. Loizou and S. J. Maybank, "The nearest neighbor and the Bayes error rate," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 9, 1987, pp. 254-262. D. Psaltis, R. R. Snapp, and S. S. Venkatesh, "On the finite sample performance of the nearest neighbor classifier," IEEE Trans. Inform. Theory, vol. IT-40, 1994, pp. 820--837. R. R. Snapp and S. S. Venkatesh, "k Nearest Neighbors in Search of a Metric," 1995, (submitted).
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A MODEL OF AUDITORY STREAMING Susan L. McCabe & Michael J. Denham Neurodynamics Research Group School of Computing University of Plymouth Plymouth PL4 8AA, u.K. ABSTRACT An essential feature of intelligent sensory processing is the ability to focus on the part of the signal of interest against a background of distracting signals, and to be able to direct this focus at will. In this paper the problem of auditory scene segmentation is considered and a model of the early stages of the process is proposed. The behaviour of the model is shown to be in agreement with a number of well known psychophysical results. The principal contribution of this model lies in demonstrating how streaming might result from interactions between the tonotopic patterns of activity of input signals and traces of previous activity which feedback and influence the way in which subsequent signals are processed. 1 INTRODUCTION The appropriate segmentation and grouping of incoming sensory signals is important in enabling an organism to interact effectively with its environment (Llinas, 1991). The formation of associations between signals, which are considered to arise from the same external source, allows the organism to recognise significant patterns and relationships within the signals from each source without being confused by accidental coincidences between unrelated signals (Bregman, 1990). The intrinsically temporal nature of sound means that in addition to being able to focus on the signal of interest, perhaps of equal significance, is the ability to predict how that signal is expected to progress; such expectations can then be used to facilitate further processing of the signal. It is important to remember that perception is a creative act (Luria, 1980). The organism creates its interpretation of the world in response to the current stimuli, within the context of its current state of alertness, attention, and previous experience. The creative aspects of perception are exemplified in the auditory system where peripheral processing decomposes acoustic stimuli. Since the frequency spectra of complex sounds generally A Model of Auditory Streaming 53 overlap, this poses a complicated problem for the auditory system : which parts of the signal belong together, and which of the subgroups should be associated with each other from one moment to the next, given the extra complication of possible discontinuities and occlusion of sound signals? The process of streaming effectively acts to to associate those sounds emitted from the same source and may be seen as an accomplishment, rather than the breakdown of some integration mechanism (Bregman, 1990). The cognitive model of streaming, proposed by (Bregman, 1990), is based primarily on Gestalt principles such as common fate, proximity, similarity and good continuation. Streaming is seen as a mUltistage process, in which an initial, preattentive process partitions the sensory input, causing successive sounds to be associated depending on the relationship between pitch proximity and presentation rate. Further refinement of these sound streams is thought to involve the use of attention and memory in the processing of single streams over longer time spans. Recently a number of computational models which implement these concepts of streaming have been developed. A model of streaming in which pitch trajectories are used as the basis of sequential grouping is proposed by (Cooke, 1992). In related work, (Brown, 1992) uses data-driven grouping schema to form complex sound groups from frequency components with common periodicity and simultaneous onset. Sequential associations are then developed on the basis of pitch trajectory. An alternative approach suggests that the coherence of activity within networks of coupled oscillators, may be interpreted to indicate both simultaneous and sequential groupings (Wang, 1995), (Brown, 1995), and can, therefore, also model the streaming of complex stimuli. Sounds belonging to the same stream, are distinguished by synchronous activity and the relationship between frequency proximity and stream formation is modelled by the degree of coupling between oscillators. A model, which adheres closely to auditory physiology, has been proposed by (Beauvois, 1991). Processing is restricted to two frequency channels and the streaming of pure tones. The model uses competitive interactions between frequency channels and leaky integrator model neurons in order to replicate a number of aspects of human psychophysical behaviour. The model, described here, used Beauvois' work as a starting point, but has been extended to include multichannel processing of complex signals. It can account for the relationship streaming and frequency difference and time interval (Beauvois, 1991), the temporal development and variability of streaming perceptions (Anstis, 1985), the influence of background organisation on foreground perceptions (Bregman, 1975), as well as a number of other behavioural results which have been omitted due to space limitations. 2 THEMODEL We assume the existence of tonotopic maps, in which frequency is represented as a distributed pattern of activity across the map. Interactions between the excitatory tonotopic patterns of activity reflecting stimulus input, and the inhibitory tonotopic masking patterns, resulting from previous activity, form the basis of the model. In order to simulate behavioural experiments, the relationship between characteristic frequency and position across the arrays is determined by equal spacing within the ERB scale (Glasberg, 1990). The pattern of activation across the tonotopic axis is represented in terms of a Gaussian function with a time course which reflects the onset-type activity found frequently within the auditory system. 54 s. L. MCCABE, M. J. DENHAM Input signals therefore take the form : i(x,t) = CI (t - t Onset)e-c2(t-tOrtut)e 2~2lfc(x}-r.)2 [1] where i(x.t) is the probability of input activity at position x, time t. C} and C; are constants, tan.m is the starting time of the signal, fc (x) is the characteristic frequency at position x,/. is the stimulus frequency, and a determines the spread of the activation. In models where competitive interactions within a single network are used to model the streaming process, such as (Beauvois, 1991), it is difficult to see how the organisation of background sounds can be used to improve foreground perceptions (Bregman, 1975) since the strengthening of one stream generally serves to weaken others. To overcome this problem, the model of preattentive streaming proposed here, consists of two interacting networks, the foreground and background networks, F and B; illustrated in figure 1. The output from F indicates the activity, if any, in the foreground, or attended stream, and the output from B reflects any other activity. The interaction between the two eventually ensures that those signals appearing in the output from F, i.e. in the foreground stream, do not appear in the output from B, the background; and vice versa. In the model, strengthening of the organisation of the background sounds, results in the 'sharpening' of the foreground stream due to the enhanced inhibition produced by a more coherent background. rre rnR Figure 1 : Connectivity of the Streaming Networks. Neurons within each array do not interact with each other but simply perform a summation of their input activity. A simplified neuron model with low-pass filtering of the inputs, and output representing the probability offiring, is used: p(x, t) = cr[~ Vj(x, t)], where cr(y) = I+~_Y [2] J The inputs to the foreground net are : VI (x,t) = (1- ::)VI (x,t-dt) + VI . ~(i(x,t».dt [3] V2(X,t) = (1- ::)V2(X, t- dt) + V2¥mFi(x,t- dt» .dt [4] V3(X, t) = (1- ~)v3(x,t-dt) + V3 . ~(mB(x,t- dt».dt [5] where x is the position across the array, time t, sampling rate dt. "tj are time constants which determine the rate of decay of activity, V; are weights on each of the inputs, and t/J(y) is a function used to simulate the stochastic properties of nerve firing which returns a value of J or 0 with probability y. A Model of Auditory Streaming 55 The output activity pattern in the foreground net and its 'inverse', mF(x,f) and mFi(x,t), are found by : mF(x, t) = cr[v\ (x, t) -l'\(V2(X, t), n) -l'\(V3(X, t), n)] [6] N mFi(x, t) = max { [~ ~ mF(xi' t - dt)] - mF(x, t- dt), O} [7] i=\ where 17(v(x,f),n) is the mean of the activity within neighbourhood n of position x at time t and N is the number of frequency channels. Background inputs are similarly calculated. To summarise, the current activity in response to the acoustic stimulus forms an excitatory input to both the foreground and background streaming arrays, F and B. In addition, F receives inhibitory inputs reflecting the current background activity, and the inverse of the current foreground activity. The interplay between the excitatory and inhibitory activities causes the model to gradually focus the foreground stream and exclude extraneous stimuli. Since the patterns of inhibitory input reflect the distributed patterns of activity in the input, the relationship between frequency difference and streaming, results simply from the graded inhibition produced by these patterns. The relationship between tone presentation rate and streaming is determined by the time constants in the model which can be tuned to alter the rate of decay of activity. To enable comparisons with psychophysical results, we view the judgement of coherence or streaming made by the model as the difference between the strength of the foreground response to one set of tones compared to the other. The strength of the response to a given frequency, Resp(f,t), is a weighted sum of the activity within a window centred on the frequency : W _k... Respif, t) = ~ mF(x(j) + i, t) * e 2(12 [8] i=-W where W determines the size of the window centred on position, x(/), the position in the map corresponding to frequency f, and a determines the spread of the weighting function about position x(/). The degree of coherence between two tones, say hand h' is assumed to depend on the difference in strength of foreground response to the two : C hif I: t) = 1 _/ Resp(fj .t)--Resp{j2,t) / [9] o \,j 2, Resp(fj . t}+Resp(/2,t) where Coh(f;,h,t) ranges between 0, when Resp(f;,t) or Resp(h,t) vanishes and the difference between the responses is a maximum, indicating maximum streaming, and 1, when the responses are equal and maximally coherent. Values between these limits are interpreted as the degree of coherence, analogous to the probability of human subjects making ajudgement of coherence (Anstis, 1985), (Beauvois, 1991). 3 RESULTS Experiments exploring the effect of frequency interval and tone presentation rate and streaming are described in (Beauvois, 1991). Subjects were required to listen to an alternating sequence of tones, ABABAB ... for 15 seconds, and then to judge whether at the end of the sequence they perceived an oscillating, trill-like, temporally coherent sequence, or two separate streams, one of interrupted high tones, the other of interrupted 56 s. L. MCCABE, M. J. DENHAM low tones. Their results showed clearly an increasing tendency towards stream segmentation both with increasing frequency difference between A and B, and increasing tone presentation rate, results the model manages substantially to reproduce; as may be seen in figure 2. 100r---~c_--~--~--------_, 100r---~~--------~--~----' 4.76 tones/sec I eo ~ 60 60 11 ~ ~ E20 0 1000 OL---~----~----~--~--~ 1100 1200 1300 1<400 1500 1000 1100 1200 1300 1«>0 1500 100 100~~~----------~--------' 7.69 tones/sec 5.88 tones/sec ~ 80 80 ~ ! 60 'li «> · ~ E 20 20 0 1000 oL---~----~----~--~----~ 1100 1200 1300 1«Xl 1500 1000 1100 1200 1300 1«>0 1 500 100 100r---~----------~--~----' 20 tones/sec 11.11 tones/sec · 80 l! ~ S 60 1J .co · ~ eo ~ 20 20 o OL---~----~----~--~--~ 1000 1100 1200 1300 1«Xl 1500 1000 1100 1200 1300 1.ao 1500 Figure 2 : Mean Psychophysical '0' and Model ,*, Responses to the Stimulus ABAB ... (A=lOOO Hz, B as indicated along X axis (Hz), tone presentation rates, as shown.) In investigating the temporal development of stream segmentation, (Anstis, 1985) used a similar stimulus to the experiment described above, but in this case subjects were required to indicate continuously whether they were perceiving a coherent or streaming signal. As can be seen in figure 3, the model clearly reproduces the principal features found in their experiments, i.e. the probability of hearing a single, fused, stream declines during each run, the more rapid the tone presentation rate, the quicker stream segmentation occurs, and the judgements made were quite variable during each run. In an experiment to investigate whether the organisation of the background sounds affects the foreground, subjects were required to judge whether tone A was higher or lower than B (Bregman, 1975). This judgement was easy when the two tones were presented in isolation, but performance degraded significantly when the distractor tones, X, were included. However, when a series of 'captor' tones, C, with frequency close to X were added, the judgement became easier, and the degree of improvement was inversely related to the difference in frequency between X and C. In the experiment, subjects received an initial priming AB stimulus, followed by a set of 9 tones : CCCXABXCC. The frequency of the captor tones, was manipulated to investigate how the proximity of 'captor' to 'distractor' tones affected the required AB order judgement. A Model of Auditory Streaming 57 Figure 3 : The Probability of Perceptual Coherence as a Function of Time in Response to Two Alternating Tones. Symbols: '.' 2 tones/s, '0' 4 tones/s, '+' 8 tones/so In order to model this experiment and the effect of priming, an 'attentive' input, focussed on the region of the map corresponding to the A and B tones, was included. We assume, as argued by Bregman, that subjects' performance in this task is related to the degree to which they are able to stream the AB pair separately. His D parameter is a measure of the degree to which ABIBA can be discriminated. The model's performance is then given by the strength of the foreground response to the AB pair as compared to the distractor tones, and Coh([A B],X) is used to measure this difference. The model exhibits a similar sensitivity to the distractor/captor frequency difference to that of human subjects, and it appears that the formation of a coherent background stream allows the model to distinguish the foreground group more clearly. A) B)'~------~------~----~ 2S00 N I!OO E G····· .. ··c·········(···· .. •· .... · .... _ .......... ··· co .... · .. ··, TIME 09 t.4eM toherente XAB.X. 08 0) 06 05 0 4 _____ . . _____ e -- -03 " .. · .. ·· .... &egm ..... Op...."...,. 02 0 1 o 500 ' 000 '500 Cap'Of hoquoncy jHz) Figure 4 : A) Experiment to Demonstrate the Formation ofMuItiple Streams, (Bregman, 1975). B) Model Response; '·'Mean Degree of Doherence to XABX, '0', Bregman's D Parameter, '+' Model's Judgement of Coherence. 4 DISCUSSION The model of streaming which we have presented here is essentially a very simple one, which can, nevertheless, successfully replicate a wide range of psychophysical experiments. Embodied in the model is the idea that the characteristics of the incoming sensory signals result in activity which modifies the way in which subsequent incoming 58 s. L. MCCABE, M. J. DENHAM signals are processed. The inhibitory feedback signals effectively comprise expectations against which later signals are processed. Processing in much of the auditory system seems to be restricted to processing within frequency 'channels'. In this model, it is shown how local interactions, restricted almost entirely to within-channel activity, can form a global computation of stream formation. It is not known where streaming occurs in the auditory system, but feedback projections both within and between nuclei are extensive, perhaps allowing an iterative refinement of streams. Longer range projections, originating from attentive processes or memory, may modify local interactions to facilitate the extraction of recognised or interesting sounds. The relationship between streaming and frequency interval, could be modelled by systematically graded inhibitory weights between frequency channels. However, in the model this relationship arises directly from the distributed incoming activity patterns, which seems a more robust and plausible solution, particularly if one takes the need to cope with developmental changes into account. Although to simplify the simulations peripheral auditory processing was not included in the model, the activity patterns assumed as input can be produced by the competitive processing of the output from a cochlear model. An important aspect of intelligent sensory processing is the ability to focus on signals of interest against a background of distracting signals, thereby enabling the perception of significant temporal patterns. Artificial sensory systems, with similar capabilities, could act as robust pre-processors for other systems, such as speech recognisers, fault detection systems, or any other application which required the dynamic extraction and temporal linking of subsets of the overall signal. Values Used For Model Parameters a=.005, c)=75, c2=100, V=[lOO 5 5 5 5], T=[.05 .6 .6 .6 .6], n=2, N=lOO References Anstis, S., Saida, S., J. (1985) Exptl Psych, 11(3), pp257-271 Beauvois, M.W., Meddis, R (1991) J. Exptl Psych, 43A(3), pp517-541 Bregman, AS., Rudnicky, AI. (1975) J. ExptJ Psych, 1(3), pp263-267 Bregman, A.S. (1990) 'Auditory scene analysis', MIT Press Brown, GJ. (1992) University of Sheffield Research Reports, CS-92-22 Brown, GJ., Cooke, M. (1995) submitted to IJCAI workshop on Computational Auditory Scene Analysis Cooke, M.P. (1992) Computer Speech and Language 6, pp 153-173 Glasberg, B.R., Moore, B.C.J. (1990) Hearing Research, 47, pp103-138 L1inas, RR, Pare, D. (1991) Neuroscience, 44(3), pp521-535 Luria, A (1980) 'Higher cortical functions in man', NY:Basic van Noorden, L.P.AS. (1975) doctoral dissertation, published by Institute for Perception Research, PO Box 513, Eindhoven, NL Wang, D.L. (1995) in 'Handbook of brain theory and neural networks', MIT Press PART II NEUROSCIENCE
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Context-Dependent Classes in a Hybrid Recurrent Network-HMM Speech Recognition System Dan Kershaw Tony Robinson Mike Hochberg • Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ, England. Tel: [+44]1223332800, Fax: [+44]1223332662. Email: djk.ajr@eng.cam.ac.uk Abstract A method for incorporating context-dependent phone classes in a connectionist-HMM hybrid speech recognition system is introduced. A modular approach is adopted, where single-layer networks discriminate between different context classes given the phone class and the acoustic data. The context networks are combined with a context-independent (CI) network to generate context-dependent (CD) phone probability estimates. Experiments show an average reduction in word error rate of 16% and 13% from the CI system on ARPA 5,000 word and SQALE 20,000 word tasks respectively. Due to improved modelling, the decoding speed of the CD system is more than twice as fast as the CI system. INTRODUCTION The ABBOT hybrid connectionist-HMM system performed competitively with many conventional hidden Markov model (HMM) systems in the 1994 ARPA evaluations of speech recognition systems (Hochberg, Cook, Renals, Robinson & Schechtman 1995). This hybrid framework is attractive because it is compact, having far fewer parameters than conventional HMM systems, whilst also providing the discriminative powers of a connectionist architecture. It is well established that particular phones vary acoustically when they occur in different phonetic contexts. For example a vowel may become nasalized when following a nasal sound. The short-term contextual influence of co-articulation is ·Mike Hochberg is now at Nuance Communications, 333 Ravenswood Avenue, Building 110, Menlo Park, CA 94025, USA. Tel: [+1] 415 6148260. Context-dependent Classes in a Speech Recognition System 751 handled in HMMs by creating a model for all sufficiently differing phonetic contexts with enough acoustic evidence. This modelling of phones in their particular phonetic contexts produces sharper probability density functions. This approach vastly improves HMM recognition accuracy over equivalent context-independent systems (Lee 1989). Although the recurrent neural network (RNN) model acoustic context internally (within the state vector) , it does not model phonetic context. This paper presents an approach to improving the ABBOT system through phonetic context-dependent modelling. In Cohen, Franco, Morgan, Rumelhart & Abrash (1992) separate sets of contextdependent output layers are used to model context effects in different states ofHMM phone models. A set of networks discriminate between phones in 8 different broadclass left and right contexts. Training time is reduced by initialising from a CI multilayer perceptron (MLP) and only changing the hidden-to-output weights during context-dependent training. This system performs well on the DARPA Resource Management Task. The work presented in Zhoa, Schwartz, Sroka & Makhoul (1995) followed along similar work to Cohen et al. (1992). A context-dependent mixture of experts (ME) system (Jordan & Jacobs 1994) based on the structure of the context-independent ME was built. For each state, the whole training data was divided into 46 parts according to its left or right context. Then, a separate ME model was built for each context. Another approach to phonetic context-dependent modelling with MLPs was proposed by Bourlard & Morgan (1993) . It was based on factoring the conditional probability of a phone-in-context given the data in terms of the phone given the data, and its context given the data and the phone. The approach taken in this paper is a mixture of the above work. However, this work augments a recurrent network (rather than an MLP) and concentrates on building a more compact system, which is more suited to our requirements. As a result, the context training scheme is fast and is implemented on a workstation (rather than a parallel processing machine as is used for training the RNN) . OVERVIEW OF THE ABBOT HYBRID SYSTEM The basic framework of the ABBOT system is similar to the one described in Bourlard & Morgan (1994) except that a recurrent network is used as the acoustic model for the within the HMM framework. A more detailed description of the recurrent network for phone probability estimation is given in Robinson (1994). At each 16ms time frame, the acoustic vector u(t) is mapped to an output vector y(t), which represents an estimate of the posterior probability of each of the phone classes Yi(t) ~ Pr(qi(t)luiH ) , (1) where qi(t) is phone class i at time t , and ul = {u(l), .. . , u(t)} is the input from time 1 to t. Left (past) acoustic context is modelled internally by a 256 dimensional state vector x(t) , which can be envisaged as "storing" the information that has been presented at the input. Right (future) acoustic context is given by delaying the posterior probability estimation until four frames of input have been seen by the network. The network is trained using a modified version of error back-propagation through time (Robinson 1994). Decoding with the hybrid connectionist-HMM approach is equivalent to conventional HMM decoding, with the difference being that the RNN models the state observations. Like typical HMM systems, the recognition process is expressed as finding the maximum a posteriori state sequence for the utterance. The decoding criterion specified above requires the computation of the likelihood of the acoustic 752 D. KERSHAW, T. ROBINSON, M. HOCHBERG data given a phone (state) sequence, ( (t)1 .(t)) = Pr(qi(t)lu(t))p(u(t)) p u q, Pr(qi)' (2) where p(u(t)) is the same for all phones, and hence drops out of the decoding process. Hence, the network outputs are mapped to scaled likelihoods by, Yi(t) p(U(t)lqi(t)) ::: Pr(qd ' (3) where the priors Pr(qi) are estimated from the training data. Decoding uses the NOWAY decoder (Renals & Hochberg 1995) to compute the utterance model that is most likely to have generated the observed speech signal. CONTEXT-DEPENDENT PROBABILITY ESTIMATION The approach taken by this work is to augment the CI RNN, in a similar vein to Bourlard & Morgan (1993). The context-dependent likelihood, p(UtICt, Qd, can be factored as, (u IC Q) = Pr(Ct!Ut, Qt)p(Ut/Qt) p t t, t Pr(Ct!Qd' (4) where C is a set of context classes and Q is a set of context-independent phones or monophones. Substituting for the context independent probability density function, p(Ut IQt), using (2), this becomes (u IC Q) = Pr(Ct IUt, Qd Pr(Qt!Ut) (U) p t t, t Pr(CtIQt) Pr(Qt) Pt· (5) The term p(U t} is constant for all frames, so this drops out of the decoding process and is ignored for all further purposes. This format is extremely appealing since Pr(Ct IQt) and Pr(Qt) are estimated from the training data and the CI RNN estimates Pr(QtIUt). All that is then needed is an estimate of Pr(CtIUt, Qt). The approach taken in this paper uses a set of context experts or modules for each monophone class to augment the existing CI RNN. TRAINING ON THE STATE VECTOR An estimate of Pr(Ct!Ut, Qt) can be obtained by training a recurrent network to discriminate between contexts Cj(t) for phone class qi(t), such that (6) where Yjli (t) is an estimate of the posterior probability of context class j given phone class i. However, training recurrent neural networks in this format would be expensive and difficult. For a recurrent format, the network must contain no discontinuities in the frame-by-frame acoustic input vectors. This implies all recurrent networks for all the phone classes i must be "shown" all the data. Instead, the assumption is made that since the state vector x = f(u), then x(t + 4) is a good representation for uiH . Hence, a single-layer perceptron is trained on the state vectors corresponding to each monophone, qi, to classify the different phonetic context classes. Finally, Context-dependent Classes in a Speech Recognition System 753 the likelihood estimates for the phonetic context class j for phone class i used in decoding are given by, Pr(qi(t)lui+4) Pr(cj (t)lx(t + 4) , qi(t)) Pr(cj(t)lqi(t)) Pr(qi(t)) Yi (t)Yjli (t) Pr( Cj Iqi) Pr( qd . (7) Embedded training is used to estimate the parameters of the CD networks and the training data is aligned using a Viterbi segmentation. Each context network is trained on a non-overlapping subset of the state vectors generated from all the Viterbi aligned training data. The context networks were trained using the RProp training procedure (Robinson 1994). 01ime 0 O~elayO '---------fO ~ 0 11----' => => => => '~ yj1i(t) ,',.-.._,1 , , ~ _______ oJ' o o :I ~ I ~ i :I C. CD :I "'tJ i CD ~. o ""I "'tJ a C" D) g: Figure 1: The Phonetic Context-Dependent RNN Modular System. The frame-by-frame phonetic context posterior probabilities are required as input to the NOWAY decoder, i.e. all the outputs from the context modules on the right hand side of Figure 1. These posterior probabilities are calculated from the numerator of (7). The CI RNN stage operates in its normal fashion, generating frame-by-frame monophone posterior probabilities. At the same time the CD modules take the state vector generated by the RNN as input, in order to classify into a context class. The 754 D. KERSHAW, T. ROBINSON, M. HOCHBERG RNN posterior probability outputs are multiplied by the module outputs to form context-dependent posterior probability estimates. RELATIONSHIP WITH MIXTURE OF EXPERTS This architecture has similarities with mixture of experts (Jordan & Jacobs 1994). During training, rather than making a "soft" split of the data as in the mixture of experts case, the Viterbi segmentation selects one expert at every exemplar. This means only one expert is responsible for each example in the data. This assumes that the Viterbi segmentation is a good approximation to tjle segmentation/selection process. Hence, each expert is trained on a small subset of the training data, avoiding the computationally expensive requirement for each expert to "see" all the data. During decoding, the RNN is treated as a gating network, smoothing the predictions of the experts, in an analogous manner to a standard mixture of experts gating network. For further description of the system see Kershaw, Hochberg & Robinson (1995) . CLUSTERING CONTEXT CLASSES One of the problems faced by having a context-dependent system is to decide which context classes are to be included in the CD system. A method for overcoming this problem is a decision-tree based approach to cluster the context classes. This guarantees a full coverage of all phones in any context with the context classes being chosen using the acoustic evidence available. The tree clustering framework also allows for the building of a small number of context-dependent phones, keeping the new context-dependent connectionist system architecture compact. The tree building algorithm was based on Young, Odell & Woodland (1994), and further details can be found in Kershaw et al. (1995). Once the trees were built, they were used to relabel the training data and the pronunciation lexicon. EVALUATION OF THE CONTEXT SYSTEM The context-independent networks were trained on the ARPA Wall Street Journal S184 Corpus. The phonetic context-dependent classes were clustered on the acoustic data according to the decision tree algorithm. Running the data through a recurrent network in a feed-forward fashion to obtain three million frames with 256 dimensional state vectors took approximately 8 hours on an HP735 workstation. Training all the context-dependent networks on all the training data takes between 4- 6 hours (in total) on an HP735 workstation. The context-dependent modules were cross-validated on a development set at the word level. Results for two context-dependent systems, compared with the context-independent baseline are shown in Table 1, where the 1993 spoke 5 test is used for cross-validation and development purposes. The context-dependent systems were also applied to larger tasks such as the recent 1995 SQALE (a European multi-language speech recognition evaluation) 20,000 word development and evaluation sets. The American English context-dependent system (CD527) was extended to include a set of modules trained backwards in time (which were log-merged with the forward context), to augment a four way logmerged context-independent system (Hochberg, Cook, Renals & Robinson 1994). Context-dependent Classes in a Speech Recognition System 755 Table 1: Comparison Of The CI System With The CD205 And CD527 Systems, For 5000 Word, Bigram Language Model Tasks. 1993 CI System CD205 System CD527 System Test Sets WER WER I Red!!. WER WER I Red!!. WER Spoke 5 16.0 14.0 12.7 13.6 14.9 Spoke 6 14.6 12.2 16.3 11.7 19.8 Eval. 15.7 14.3 8.4 13.7 12.6 Table 2: Comparison Of The Merged CI Systems With The CD527US And CD465UK Systems, For 20,000 Word Tasks. All Tests Use A Trigram Language Model. The CD527US And CD465UK Evaluation Results Have Been Officially Adjudicated. 1995 Test Sets CI System CD System Red!!. WER WER WER US English dev _test 12.8 11.3 12.2 US English evLtest 14.5 12.9T 9.8 UK English dev _test 15.6 12.7 18.9 UK English evLtest 16.4 13.8T 15.7 Table 3: Comparison Of Average Utterance Decode Speed Of The CI Systems With The CD527US And CD465UK Systems On An HP735, For 20,000 Word Tasks. All Tests Use A Trigram Language Model, And The Same Pruning Levels. CI CD Speedup Tests Utterance Av. Utterance Av. Decode Speed (s) Decode Speed (s) American English 67 31 2.16 British English 131 48 2.73 Table 4: The Number Of Parameters Used For The CI Systems As Compared With The CD527US And CD465UK Systems. System # CI #CD 'fo Increase In Parameters Parameters Parameters American English 341,000 612,000 79.0 British English 331,000 570,000 72.2 A similar system was built for British English (CD465). Table 2 shows the improvement gained by using context models. The daggers indicate the official entries for the 1995 SQALE evaluation. These figures represent the lowest reported word error rate for both the US and UK English tasks. As a result of improved phonetic modelling and class discrimination the search space was reduced. This meant that decoding speed was over twice as fast as the context-dependent system, Table 3, even though there were roughly ten times as many context-dependent phones compared to the monophones. The increase in the number of parameters due to the introduction of the context models for the SQALE evaluation system are shown in Table 4. Although this seems a large increase in the number of system parameters, it is still an order of magnitude less than any equivalent HMM system built for this task. 756 D. KERSHAW, T. ROBINSON, M. HOCHBERG CONCLUSIONS This paper has discussed a successful way of integrating phonetic context-dependent classes into the current ABBOT hybrid system. The architecture followed a modular approach which could be used to augment any current RNN-HMM hybrid system. Fast training of the context-dependent modules was achieved. Training on all of the SI84 corpus took between 4 and 6 hours. Utterance decoding was performed using the standard NOWAY decoder. The word error was significantly reduced, whilst the decoding speed of the context system was over twice as fast as the baseline system (for 20,000 word tasks). References Bourlard, H. & Morgan, N. (1993), 'Continuous Speech Recognition by Connectionist Statistical Methods', IEEE Transactions on Neural Networks 4(6), 893- 909. Bourlard, H. & Morgan, N. (1994), Connectionist Speech Recognition: A Hybrid Approach, Kluwer Acedemic Publishers. Cohen, M., Franco, H., Morgan, N., Rumelhart, D. & Abrash, V. (1992), ContextDependent Multiple Distribution Phonetic Modeling with MLPs, in 'NIPS 5'. Hochberg, M., Cook, G., Renals, S. & Robinson, A. (1994), Connectionist Model Combination for Large Vocabulary Speech Recognition, in 'Neural Networks for Signal Processing', Vol. IV, pp. 269-278. Hochberg, M., Cook, G., Renals, S., Robinson, A. & Schechtman, R. (1995), The 1994 ABBOT Hybrid Connectionist-HMM Large-Vocabulary Recognition System, in 'Spoken Language Systems Technology Workshop', ARPA, pp. 170-6. Jordan, M. & Jacobs, R. (1994), 'Hierarchical Mixtures of Experts and the EM Algorithm', Neural Computation 6, 181-214. Kershaw, D., Hochberg, M. & Robinson, A. (1995), Incorporating ContextDependent Classes in a Hybrid Recurrent Network-HMM Speech Recognition System, F-INFENG TR217, Cambridge University Engineering Department. Lee, K.-F. (1989), Automatic Speech Recognition; The Development of the SPHINX System, Kluwer Acedemic Publishers. Renals, S. & Hochberg, M. (1995), Efficient Search Using Posterior Phone Probability Estimates, in 'ICASSP', Vol. 1, pp. 596-9. Robinson, A. (1994), 'An Application of Recurrent Nets to Phone Probability Estimation.', IEEE Transactions on Neural Networks 5(2),298-305. Young, S., Odell, J. & Woodland, P. (1994), 'Tree-Based State Tying for High Accuracy Acoustic Modelling', Spoken Language Systems Technology Workshop. Zhoa, Y., Schwartz, R., Sroka, J. & Makhoul, J. (1995), Hierarchical Mixtures of Experts Methodology Applied to Continuous Speech Recognition, in 'NIPS 7'.
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Exploiting Tractable Substructures in Intractable Networks Lawrence K. Saul and Michael I. Jordan {lksaul.jordan}~psyche.mit.edu Center for Biological and Computational Learning Massachusetts Institute of Technology 79 Amherst Street, ElO-243 Cambridge, MA 02139 Abstract We develop a refined mean field approximation for inference and learning in probabilistic neural networks. Our mean field theory, unlike most, does not assume that the units behave as independent degrees of freedom; instead, it exploits in a principled way the existence of large substructures that are computationally tractable. To illustrate the advantages of this framework, we show how to incorporate weak higher order interactions into a first-order hidden Markov model, treating the corrections (but not the first order structure) within mean field theory. 1 INTRODUCTION Learning the parameters in a probabilistic neural network may be viewed as a problem in statistical estimation. In networks with sparse connectivity (e.g. trees and chains), there exist efficient algorithms for the exact probabilistic calculations that support inference and learning. In general, however, these calculations are intractable, and approximations are required. Mean field theory provides a framework for approximation in probabilistic neural networks (Peterson & Anderson, 1987). Most applications of mean field theory, however, have made a rather drastic probabilistic assumption-namely, that the units in the network behave as independent degrees of freedom. In this paper we show how to go beyond this assumption. We describe a self-consistent approximation in which tractable substructures are handled by exact computations and only the remaining, intractable parts of the network are handled within mean field theory. For simplicity we focus on networks with binary units; the extension to discrete-valued (Potts) units is straightforward. Exploiting Tractable Substructures in Intractable Networks 487 We apply these ideas to hidden Markov modeling (Rabiner & Juang, 1991). The first order probabilistic structure of hidden Markov models (HMMs) leads to networks with chained architectures for which efficient, exact algorithms are available. More elaborate networks are obtained by introducing couplings between multiple HMMs (Williams & Hinton, 1990) and/or long-range couplings within a single HMM (Stolorz, 1994). Both sorts of extensions have interesting applications; in speech, for example, multiple HMMs can provide a distributed representation of the articulatory state, while long-range couplings can model the effects of coarticulation. In general, however, such extensions lead to networks for which exact probabilistic calculations are not feasible. One would like to develop a mean field approximation for these networks that exploits the tractability of first-order HMMs. This is possible within the more sophisticated mean field theory described here. 2 MEAN FIELD THEORY We briefly review the basic methodology of mean field theory for networks of binary (±1) stochastic units (Parisi, 1988). For each configuration {S} = {Sl, S2, ... , SN}, we define an energy E{S} and a probability P{S} via the Boltzmann distribution: e-.8E {S} P{S} = Z ' (1) where {3 is the inverse temperature and Z is the partition function. When it is intractable to compute averages over P{S}, we are motivated to look for an approximating distribution Q{S}. Mean field theory posits a particular parametrized form for Q{S}, then chooses parameters to minimize the Kullback-Liebler (KL) divergence: [Q{S}] KL(QIIP) = L Q{S} In P{S} . is} (2) Why are mean field approximations valuable for learning? Suppose that P{S} represents the posterior distribution over hidden variables, as in the E-step of an EM algorithm (Dempster, Laird, & Rubin, 1977). Then we obtain a mean field approximation to this E-step by replacing the statistics of P{S} (which may be quite difficult to compute) with those of Q{S} (which may be much simpler). If, in addition, Z represents the likelihood of observed data (as is the case for the example of section 3), then the mean field approximation yields a lower bound on the loglikelihood. This can be seen by noting that for any approximating distribution Q{S}, we can form the lower bound: In Z = In L e-.8E {S} {S} [ e-.8E {S} ] In L Q{S}· Q{S} {S} > L Q{ S}[ - {3E {S} - In Q{ S}], {S} (3) (4) (5) where the last line follows from Jensen's inequality. The difference between the left and right-hand side of eq. (5) is exactly KL( QIIP); thus the better the approximation to P {S}, the tighter the bound on In Z. Once a lower bound is available, a learning procedure can maximize the lower bound. This is useful when the true likelihood itself cannot be efficiently computed. 488 L. K. SAUL, M.I. JORDAN 2.1 Complete Factorizability The simplest mean field theory involves assuming marginal independence for the units Si. Consider, for example, a quadratic energy function (6) i<j and the factorized approximation: Q{ S} = IJ (1 + :i Si ) . I (7) The expectations under this mean field approximation are (Si) = mi and (Si Sj) = mimj for i =1= j. The best approximation of this form is found by minimizing the KL-divergence, KL(QIIP) = ~[(1+2mi)ln(I+2mi)+(1-2mi)ln(I-2mi)] (8) I L Jijmimj - L himi + InZ, i<j i with respect to the mean field parameters mi. Setting the gradients of eq. (8) equal to zero, we obtain the (classical) mean field equations: tanh- 1(mi) = L Jijmj + hi· (9) j 2.2 Partial Factorizability We now consider a more structured model in which the network consists of interacting modules that, taken in isolation, define tractable substructures. One example of this would be a network of weakly coupled HMMs, in which each HMM, taken by itself, defines a chain-like substructure that supports efficient probabilistic calculations. We denote the interactions between these modules by parameters K~v, where the superscripts J.' and 1/ range over modules and the subscripts i and j index units within modules. An appropriate energy function for this network is: - ,6E{S} = L {LJ~srSf + LhfSr} + L K~vSrS'f. (10) /J i<j i /J<V ij The first term in this energy function contains the intra-modular interactions; the last term, the inter-modular ones. We now consider a mean field approximation that maintains the first sum over modules but dispenses with the inter-modular corrections: Q{S} = }Q exp {L [~J~srSf + ~Hisr]} (11) /J I <1 I The parameters of this mean field approximation are Hi; they will be chosen to provide a self-consistent model of the inter-modular interactions. We easily obtain the following expectations under the mean field approximation, where J.' =1= 1/: (Sr Sj) 8/Jw (Sf Sj) + (1 - 8/Jw)(Sr)(Sj), (12) (Sr sy Sk) 8/Jw(Sf Sk)(S'f} + 8vw(Sj Sk)(Sf) + (13) (1- 8vw )(I- 8w/J)(Sf}{S'f} (Sk)· Exploiting Tractable Substructures in Intractable Networks 489 Note that units in the same module are statistically correlated and that these correlations are assumed to be taken into account in calculating the expectations. We assume that an efficient algorithm is available for handling these intra-modular correlations. For example, if the factorized modules are chains (e.g. obtained from a coupled set of HMMs), then computing these expectations requires a forwardbackward pass through each chain. The best approximation of the form, eq. (11), is found by minimizing the KLdivergence, KL(QIIP) = In(ZjZQ) + L (Hf - hf) (Sn - L Kt'/ (Sf Sf), (14) JJ<V ij with respect to the mean field parameters HI:. To compute the appropriate gradients, we use the fact that derivatives of expectations under a Boltzmann distribution (e.g. a(Sn jaHk ) yield cumulants (e.g. (Sf Sk) - (Sf)(Sk)) . The conditions for stationarity are then: JJ<V ij Substituting the expectations from eqs. (12) and (13), we find that K L( QIIP) is minimized when o = ~ {Hi - hi - L ~Kit(Sj)} [(S'i Sk) - (S,:)(SI:)]. (16) , V~W J The resulting mean field equations are: Hi = L LK~V(Sj) + hi· (17) V~W j These equations may be solved by iteration, in which the (assumed) tractable algorithms for averaging over Q{S} are invoked as subroutines to compute the expectations (Sj) on the right hand side. Because these expectations depend on Hi, these equations may be viewed as a self-consistent model of the inter-modular interactions. Note that the mean field parameter Hi plays a role analogous to-tanh- 1(mi) in eq. (9) of the fully factorized case. 2.3 Inducing Partial Factorizability Many interesting networks do not have strictly modular architectures and can only be approximately decomposed into tractable core structures. Techniques are needed in such cases to induce partial factorizability. Suppose for example that we are given an energy function (18) i<j i<j for which the first two terms represent tractable interactions and the last term, intractable ones. Thus the weights Jij by themselves define a tractable skeleton network, but the weights Kij spoil this tractability. Mimicking the steps of the previous section, we obtain the mean field equations: 0= L ((SiSk) - (Si)(Sk)) [Hi - hi] - L Kij [(SiSj Sk) - (SiSj )(Sk)] . (19) i<j 490 L. K. SAUL. M. I. JORDAN In this case, however, the weights Kij couple units in the same core structure. Because these units are not assumed to be independent, the triple correlator (SiSjSk) does not factorize, and we no longer obtain the decoupled update rules of eq. (17). Rather, for these mean field equations, each iteration requires computing triple correlators and solving a large set of coupled linear equations. To avoid this heavy computational load, we instead manipulate the energy function into one that can be partially factorized. This is done by introducing extra hidden variables Wij = ±1 on the intractable links of the network. In particular, consider the energy function -I3E{S, W} = ~JijSiSj + ~hiSi + ~ [KH)Si + Kfj)Sj] Wij. (20) i<j i i<j The hidden variables Wij in eq. (20) serve to decouple the units connected by the intractable weights Kij. However, we can always choose the new interactions, K (l) d jA2) h ij an '"ij' so t at e-.BE{S} = ~ e-.BE{S,W}. (21) {W} Eq. (21) states that the marginal distribution over {S} in the new network is identical to the joint distribution over {S} in the original one. Summing both sides of eq. (21) over {S}, it follows that both networks have the same partition function. The form of the energy function in eq. (20) suggests the mean field approximation: where the mean field parameters Hi have been augmented by a set of additional mean field parameters Hij that account for the extra hidden variables. In this expression, the variables Si and Wij act as decoupled degrees of freedom and the methods of the preceding section can be applied directly. We consider an example of this reduction in the following section. 3 EXAMPLE Consider a continuous-output HMM in which the probability of an output Xt at time t is dependent not only on the state at time t, but also on the state at time t +~. Such a context-sensitive HMM may serve as a flexible model of anticipatory coarticulatory effects in speech, with ~ ~ 50ms representing a mean phoneme lifetime. Incorporating these interactions into the basic HMM probability model, we obtain the following joint probability on states and outputs: T-l T-~ 1 {I 2} P{S, X} = II aSjSt +1 II (211")D/2 exp -2" [Xt US j VSj+~] . t=l t=l Denoting the likelihood of an output sequence by Z, we have Z = P{X} = ~ P{S, X}. {S} (23) (24) We can represent this probability model using energies rather than transition probabilities (Luttrell, 1989; Saul and Jordan, 1995). For the special case of binary Exploiting Tractable Substructures in Intractable Networks 491 Here, a++ is the probability of transitioning from the ON state to the ON state (and similarly for the other a parameters), while 0+ and V+ are the mean outputs associated with the ON state at time steps t and t + .6. (and similarly for 0_ and V_). Given these definitions, we obtain an equivalent expression for the likelihood: Z = Lexp {-gO + 'E JStSt+1 + thtSt + 'E KStSt+t:..} , {S} t=l t=l t=l (27) where go is a placeholder for the terms in InP{S,X} that do not depend on {S}. We can interpret Z as the partition function for the chained network of T binary units that represents the HMM unfolded in time. The nearest neighbor connectivity of this network reflects the first order structure of the HMM; the long-range connectivity reflects the higher order interactions that model sensitivity to context. The exact likelihood can in principle be computed by summing over the hidden states in eq. (27), but the required forward-backward algorithm scales much worse than the case of first-order HMMs. Because the likelihood can be identified as a partition function, however, we can obtain a lower bound on its value from mean field theory. To exploit the tractable first order structure of the HMM, we induce a partially factorizable network by introducing extra link variables on the long-range connections, as described in section 2.3. The resulting mean field approximation uses the chained structure as its backbone and should be accurate if the higher order effects in the data are weak compared to the basic first-order structure. The above scenario was tested in numerical simulations. In actuality, we implemented a generalization of the model in eq. (23): our HMM had non-binary hidden states and a coarticulation model that incorporated both left and right context. This network was trained on several artificial data sets according to the following procedure. First, we fixed the "context" weights to zero and used the Baum-Welch algorithm to estimate the first order structure of the HMM. Then, we lifted the zero constraints and re-estimated the parameters of the HMM by a mean field EM algorithm. In the E-step of this algorithm, the true posterior P{SIX} was approximated by the distribution Q{SIX} obtained by solving the mean field equations; in the M-step, the parameters of the HMM were updated to match the statistics of Q{SIX}. Figure 1 shows the type of structure captured by a typical network. 4 CONCLUSIONS Endowing networks with probabilistic semantics provides a unified framework for incorporating prior knowledge, handling missing data, and performing inferences under uncertainty. Probabilistic calculations, however, can quickly become intractable, so it is important to develop techniques that both approximate probability distributions in a flexible manner and make use of exact techniques wherever possible. In IThere are boundary corrections to ht (not shown) for t = 1 and t> T - A. 492 -5 -,0 .~.; .. : .. ' .. " . "' .. L. K. SAUL, M. I. JORDAN • :., ..... .. '. :;. ',; ... ~ ... } Figure 1: 2D output vectors {Xt } sampled from a first-order HMM and a contextsensitive HMM, each with n = 5 hidden states. The latter's coarticulation model used left and right context, coupling Xt to the hidden states at times t and t ± 5. At left: the five main clusters reveal the basic first-order structure. At right: weak modulations reveal the effects of context. this paper we have developed a mean field approximation that meets both these objectives. As an example, we have applied our methods to context-sensitive HMMs, but the methods are general and can be applied more widely. Acknowledgements The authors acknowledge support from NSF grant CDA-9404932, ONR grant NOOOI4-94-1-0777, ATR Research Laboratories, and Siemens Corporation. References A. Dempster, N. Laird, and D. Rubin. (1977) Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Stat. Soc. B39:1-38. B. H. Juang and L. R. Rabiner. (1991) Hidden Markov models for speech recognition, Technometrics 33: 251-272. S. Luttrell. (1989) The Gibbs machine applied to hidden Markov model problems. Royal Signals and Radar Establishment: SP Research Note 99. G. Parisi. (1988) Statistical field theory. Addison-Wesley: Redwood City, CA. C. Peterson and J. R. Anderson. (1987) A mean field theory learning algorithm for neural networks. Complex Systems 1:995-1019. L. Saul and M. Jordan. (1994) Learning in Boltzmann trees. Neural Compo 6: 1174-1184. L. Saul and M. Jordan. (1995) Boltzmann chains and hidden Markov models. In G. Tesauro, D. Touretzky, and T . Leen, eds. Advances in Neural Information Processing Systems 7. MIT Press: Cambridge, MA. P. Stolorz. (1994) Recursive approaches to the statistical physics of lattice proteins. In L. Hunter, ed. Proc. 27th Hawaii Inti. Conf on System Sciences V: 316-325. C. Williams and G. E. Hinton. (1990) Mean field networks that learn to discriminate temporally distorted strings. Proc. Connectionist Models Summer School: 18-22.
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Statistical Theory of Overtraining - Is Cross-Validation Asymptotically Effective? s. Amari, N. Murata, K.-R. Miiller* Dept. of Math. Engineering and Inf. Physics, University of Tokyo Hongo 7-3-1, Bunkyo-ku, Tokyo 113, Japan M. Finke Inst. f. Logik, University of Karlsruhe 76128 Karlsruhe, Germany H. Yang Lab. f. Inf. Representation, RIKEN, Wakoshi, Saitama, 351-01, Japan Abstract A statistical theory for overtraining is proposed. The analysis treats realizable stochastic neural networks, trained with KullbackLeibler loss in the asymptotic case. It is shown that the asymptotic gain in the generalization error is small if we perform early stopping, even if we have access to the optimal stopping time. Considering cross-validation stopping we answer the question: In what ratio the examples should be divided into training and testing sets in order to obtain the optimum performance. In the non-asymptotic region cross-validated early stopping always decreases the generalization error. Our large scale simulations done on a CM5 are in nice agreement with our analytical findings. 1 Introduction Training multilayer neural feed-forward networks, there is a folklore that the generalization error decreases in an early period of training, reaches the minimum and then increases as training goes on, while the training error monotonically decreases. Therefore, it is considered advantageous to stop training at an adequate time or to use regularizers (Hecht-Nielsen [1989), Hassoun [1995), Wang et al. [1994)' Poggio and Girosi [1990), Moody [1992)' LeCun et al. [1990] and others). To avoid overtraining, the following stopping rule has been proposed based on cross-validation: *Permanent address: GMD FIRST, Rudower Chaussee 5, 12489 Berlin, Germany. E-mail: Klaus@first.gmd.de Statistical Theory of Overtraining-Is Cross-Validation Asymptotically Effective? 177 Divide all the available examples into two disjoint sets. One set is used for training. The other set is used for testing such that the behavior of the trained network is evaluated by using the test examples and training is stopped at the point that minimizes the testing error. The present paper gives a mathematical analysis of the so-called overtraining phenomena to elucidate the folklore. We analyze the asymptotic case where the number t of examples are very large. Our analysis treats 1) a realizable stochastic machine, 2) Kullback-Leibler loss (negative ofthe log likelihood loss), 3) asymptotic behavior where the number t of examples is sufficiently large (compared with the number m of parameters). We firstly show that asymptotically the gain of the generalization error is small even if we could find the optimal stopping time. We then answer the question: In what ratio, the examples should be divided into training and testing sets in order to obtain the optimum performance. We give a definite answer to this problem. When the number m of network parameters is large, the best strategy is to use almost all t examples in the training set and to use only l/v2m examples in the testing set, e.g. when m = 100, this means that only 7% of the training patterns are to be used in the set determining the point for early stopping. Our analytic results were confirmed by large-scale computer simulations of threelayer continuous feedforward networks where the number m of modifiable parameters are m = 100. When t > 30m, the theory fits well with simulations, showing cross-validation is not necessary, because the generalization error becomes worse by using test examples to obtain an adaequate stopping time. For an intermediate range, where t < 30m overtraining occurs surely and the cross-validation stopping improves the generalization ability strongly. 2 Stochastic feedforward networks Let us consider a stochastic network which receives input vector x and emits output vector y. The network includes a modifiable vector parameter w = (WI,"', wm ) and is denoted by N(w). The input-output relation of the network N(w) is specified by the conditional probability p(Ylx; w). We assume (a) that there exists a teacher network N(wo) which generates training examples for the student N(w). And (b) that the Fisher information matrix Gij(w) = E [a~. logp(x, y; w) a~j logp(x, y; w)] exists, is non-degenerate and is smooth in w, where E denotes the expectation with respect to p(x, Y; w) = q(x)p(Ylx; w). The training set Dt = {(Xl, YI), ... , (Xt, Yt)} consists of t independent examples generated by the distribution p(x, Y; wo) of N(wo). The maximum likelihood estimator (m.l.e.) Vi is the one that maximizes the likelihood of producing Dt , or equivalently minimizes the training error or empirical risk function 1 t Rtrain(w) = -i I:logp(xi,Yi;w). (2.1) i=l The generalization error or risk function R(w) of network N(w) is the expectation with respect to the true distribution, R(w) = -Eo[logp(x, Y; w)] = Ho+D(wo II w) = Ho+Eo [log p~x, Y; wojJ, (2.2) p x,y;w where Eo denotes the expectation with respect to p(x, Y; wo), Ho is the entropy of the teacher network and D(wo II w) is the Kullback-Leibler divergence from probability distribution p(x,y;wo) to p(x,y;w) or the divergence of N(w) from N(wo). Hence, minimizing R(w) is equivalent to minimizing D(wo II w), and the 178 S. AMARI, N. MURATA, K. R. MULLER, M. FINKE, H. YANG minimum is attained at w = Wo. The asymptotic theory of statistics proves that the m.l.e. Wt is asymptotically subject to the normal distribution with mean Wo and variance G-1 It, where G-1 is the inverse of the Fisher information matrix G. We can expand for example the risk R(w) = Ho+ t(w -wo)TG(wo)(w -wo) + 0 (/2) to obtain (Rgen(w)) = Ho + ~ + 0 C~ ), (Rtrain(w)) = Ho - ~ + 0 C~), (2.3) as asymptotic result for training and test error (see Murata et al. [1993] and Amari and Murata [1990)). An extension of (2.3) including higher order corrections was recently obtained by M liller et al. [1995]. Let us consider the gradient descent learning rule (Amari [1967], Rumelhart et al. [1986], and many others), where the parameter w(n) at the nth step is modified by w(n + 1) = w(n) _ € f)Rtr~~(wn) , (2.4) and where € is a small positive constant. This is batch learning where all the training examples are used for each iteration of modifying w( n).l The batch process is deterministic and w( n) converges to W, provided the initial w(O) is included in its basin of attraction. For large n we can argue, that w(n) is approaching w isotropically and the learning trajectory follows a linear ray towards w (for details see Amari et al. [1995]). 3 Virtual optimal stopping rule During learning as the parameter w(n) approaches W, the generalization behavior of network N {w(n)} is evalulated by the sequence R(n) = R{w(n)}, n = 1,2, . .. The folklore says that R(n) decreases in an early period oflearning but it increases later. Therefore, there exists an optimal stopping time n at which R(n) is minimized. The stopping time nopt is a random variable depending on wand the initial w(O). We now evaluate the ensemble average of (R(nopd). The true Wo and the m.l.e. ware in general different, and they are apart of order 1/Vt. Let us compose a sphere S of which the center is at (1/2)(wo+w) and which passes through both Wo and W, as shown in Fig.1b. Its diameter is denoted by d, where d2 = Iw - Wo 12 and Eo [d2] Eo[(w - wo? G- 1(w - wo)] = ~tr(G-1G) = m. (3.1) t t Let A be the ray, that is the trajectory w(n) starting at w(O) which is not in the neighborhood of Wo . The optimal stopping point w" that minimizes R(n) = Ho + ~Iw(n) - wol2 (3.2) is given by the first intersection of the ray A and the sphere S. Since w" is the point on A such that Wo - w" is orthogonal to A, it lies on the sphere S (Fig.1b). When ray A' is approaching w from the opposite side ofwo (the right-hand side in the figure), the first intersection point is w itself. In this case, the optimal stopping never occurs until it converges to W. Let () be the angle between the ray A and the diameter Wo - w of the sphere S. We now calculate the distribution of () when the rays are isotropically distributed. lWe can alternatively use on-line learning, studied by Amari [1967], Heskes and Kappen [1991], and recently by Barkai et al. [1994] and SolI a and Saard [1995]. Statistical Theory of Overtraining-Is Cross-Validation Asymptotically Effective? 179 Lemma 1. When ray A is approaching V. from the side in which Wo is included, the probability density of 0, 0 :::; 0 :::; 7r /2, is given by 1 17r/2 reO) = -- sinm- 2 0, where 1m = sinm OdO. 1m-2 0 (3.3) The det,ailed proof of this lemma can be found in Amari et aI. [1995]. Using the density of 0 given by Eq.(3.3) and we arrive at the following theorem. Theorem 1. The average generalization error at the optimal stopping point is given by (3.4) Proof When ray A is at angle 0, 0 :::; 0 < 7r /2, the optimal stopping point w* is on the sphere S. It is easily shown that Iw* - wol = dsinO. This is the case where A is from the same side as Wo (from the left-hand side in Fig.l b), which occurs with probability 0.5, and the average of (d sin 0)2 is Eo[(dsinO?] Eo[d2] r/\in2 Osinm- 2 OdO = m ~ = m (1- ~). 1m - 2 Jo t 1m-2 t m When 0 is 7r/2 :::; 0 :::; 7r, that is A approaches V. from the opposite side, it does not stop until it reaches V., so that Iw* - Wo 12 = IV. - Wo I = d2 • This occurs with probability 0.5. Hence, we proved the theorem. The theorem shows that, if we could know the optimal stopping time nopt for each trajectory, the generalization error decreases by 1/2t, which has an effect of decreasing the effective dimensions by 1/2. This effect is neglegible when m is large. The optimal stopping time is of the order logt. However, it is impossible to know the optimal stopping time. If we stop learning at an estimated optimal time nopt, we have a small gain when the ray A is from the same side as Wo but we have some loss when ray A is from the opposite direction. This shows that the gain is even smaller if we use a common stopping time iiopt independent of V. and w(O) as proposed by Wang et aI. [1994]. However, the point is that there is neither direct means to estimate nopt nor iiopt rather than for example cross-validation. Hence, we analyze cross-validation stopping in the following. 4 Optimal stopping by cross-validation The present section studies asymptotically two fundamental problems: 1) Given t examples, how many examples should be used in the training set and how many in the testing set? 2) How much gain can one expect by the above cross-validated stopping? Let us divide t examples into rt examples of the training set and r't examples of the testing set, where r + r' = 1. Let V. be the m.I.e. from rt training examples, and let w be the m.I.e. from the other r't testing examples. Since the training examples and testing examples are independent, V. and ware subject to independent normal distributions with mean Wo and covariance matrices G-1/(rt) and G-l/(r't), respecti vely. Let us compose the triangle with vertices Wo, V. and w. The trajectory A starting at w(O) enters V. linearly in the neighborhood. The point w" on the trajectory A which minimizes the testing error is the point on A that is closest to W, since the testing error defined by 1 Rtest(w) = r't ~{-logp(xi'Yi; w)}, (4.1) t 180 S. AMARI, N. MURATA, K. R. MULLER, M. FINKE, H. YANG where summation is taken over r't testing examples, can be expanded as Rtest(w) == Ho - ~Iw - wol 2 + ~Iw - w1 2 . (4.2) Let S be the sphere centered at (w + w)/2 and passing through both wand w. It 's diameter is given by d == Iw - wi. Then, the optimal stopping point w* is given by the intersection of the trajectory A and sphere S . When the trajectory comes from the opposite side of W, it does not intersect S until it converges to w, so that the optimal point is w* == w in this case. Omitting the detailed proof, the generalization error of w* is given by Eq.(??) , so that we calculate the expectation E[lw* -woI 2] == m _ ~ (~_~). tr 2t l' 1" Lemma 2. The average generalization error by the optimal cross-validated stopping IS * 2m - 1 1 (R(w ,1')) = Ho + 4rt + 4r't We can then calculate the optimal division rate J2m -1-1 ropt = 1 2(m _ 1) 1 and ropt = 1 - J2m (large m limit). ( 4.3) ( 4.4) of examples, which minimizes the generalization error. So for large m only (1/J2m) x 100% of examples should be used for testing and all others for training. For example, when m = 100, this shows that 93% of examples are to be used for training and only 7% are to be kept for testing. From Eq.( 4.4) we obtain as optimal generalization error for large m (R(w', ropt» = Ho +; (1 + If) . ( 4.5) This shows that the generalization error asymptotically increases slightly by crossvalidation compared with non-stopped learning which is using all the examples for training. 5 Simulations We use standard feed-forward classifier networks with N inputs, H sigmoid hidden units and M softmax outputs (classes). The output activity 0/ of the lth output unit is calculated via the softmax squashing function _ . _ _ exp(h/) p(y·-GI!x,w)-O/-l 2: (h )' /=l ,·· ·,M, + k exp k where h? = Lj wg Sj - '19? is the local field potential. Each output 0/ codes the aposteriori probability of being in class G/, 0 0 denotes a zero class for normalization purposes. The m network parameters consist of biases '19 and weights w . When x is input, the activity of the j-th hidden unit is N Sj = [1 + exp( - L Wf{:Xk - 'I9.f)]-I , j = 1, .. " H . k=1 The input layer is connected to the hidden layer via w H , the hidden layer is connected to the output layer via wo, but no short-cut connections are present. Although the network is completely deterministic, it is constructed to approximate Statistical Theory of Overtraining-Is Cross-Validation Asymptotically Effective? 181 class conditional probabilities (Finke and Miiller [1994]). The examples {(x}, yd, .. " (Xt , Yt)} are produced randomly, by drawing Xi, i = 1, .. . , t, from a uniform distribution independently and producing the labels Yi stochastically from the teacher classifier. Conjugate gradient learning with linesearch on the empirical risk function Eq.(2.1) is applied, starting from some random initial vector. The generalization ability is measured using Eq. (2.2) on a large test set (50000 patterns). Note that we use Eq. (2.1) on the cross-validation set, because only the empirical risk is available on the cross-validation set in a practical situation. We compare the generalisation error for the settings: exhaustive training (no stopping), early stopping (controlled by the cross-validation set) and optimal stopping (controlled by the large testset). The simulations were performed on a parallel computer (CM5). Every curve in the figures takes about 8h of computing time on a 128 respectively 256 partition of the CM5, i.e. we perform 128-256 parallel trials. This setting enabled us to do extensive statistics (cf. Amari et al. [1995]) . Fig. la shows the results of simulations, where N = 8, H = 8, M = 4, so that the number m of modifiable parameters is m = (N + I)H + (H + I)M = 108. We observe clearly, that saturated learning without early stopping is the best in the asymptotic range of t > 30m, a range which is due to the limited size of the data sets often unaccessible in practical applications. Cross-validated early stopping does not improve the generalization error here, so that no overtraining is observed on the average in this range. In the asymptotic area (figure 1) we observe that the smaller the percentage of the training set, which is used to determine the point of early stopping, the better the performance of the generalization ability. When we use cross-validation, the optimal size of the test set is about 7% of all the examples, as the theory predicts. Clearly, early stopping does improve the generalization ability to a large extent in an intermediate range for t < 30m (see Miiller et al. [1995]). Note, that our theory also gives a good estimate of the optimal size of the early stopping set in this intermediate range. 'i Cit 0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 om " opt,420.% -+--,3:l% .. E} ... / 42% ........ n9,gtopping -.. >A ;' ~ . / "". >:i<:;_=-~;:~~::::~::~~~:-----.~~;;?;;;;»/ ~A.~~.::.../ " , 0.005 L....I..._--'-_-'-_-'--_-'------'_--'-_--'-_-'-----l 5e-5 le-4 1.5e-4 2e-4 2.5e-4 3e-4 3.5e-4 4e-4 4.5e-4 5e-4 lIt (a) (b) Figure 1: (a) R(w) plotted as a function of lit for different sizes r' of the early stopping set for an 8-8-4 classifier network. opt. denotes the use of a very large cross-validation set (50000) and no stopping adresses the case where 100% of the training set is used for exhaustive learning. (b) Geometrical picture to determine the optimal stopping point w* . 182 s. AMARI. N. MURATA. K. R. MOLLER. M. FINKE. H. YANG 6 Conclusion We proposed an asymptotic theory for overtraining. The analysis treats realizable stochastic neural networks, trained with Kullback-Leibler loss. It is demonstrated both theoretically and in simulations that asymptotically the gain in the generalization error is small if we perform early stopping, even if we have access to the optimal stopping time. For cross-validation stopping we showed for large m that optimally only r~pt = 1/ J2m examples should be used to determine the point of early stopping in order to obtain the best performance. For example, if m = 100 this corresponds to using 93% of the t training patterns for training and only 7% for testing where to stop. Yet, even if we use rapt for cross-validated stopping the generalization error is always increased comparing to exhaustive training. Nevertheless note, that this range is due to the limited size of the data sets often unaccessible in practical applications. In the non-asymptotic region simulations show that cross-validated early stopping always helps to enhance the performance since it decreases the generalization error. In this intermediate range our theory also gives a good estimate of the optimal size of the early stopping set. In future we will consider higher order correction terms to extend our theory to give also a quantitative description of the non-asymptotic regIOn. Acknowledgements: We would like to thank Y. LeCun, S. Bos and K Schulten for valuable discussions. K -R. M. thanks K Schulten for warm hospitality during his stay at the Beckman Inst. in Urbana, Illinois. We acknowledge computing time on the CM5 in Urbana (NCSA) and in Bonn, supported by the National Institutes of Health (P41RRO 5969) and the EC S & T fellowship (FTJ3-004, K. -R. M.). References Amari, S. [1967], IEEE Trans., EC-16, 299- 307. Amari, S., Murata, N. [1993], Neural Computation 5, 140 Amari, S., Murata, N., Muller, K-R., Finke, M., Yang, H. [1995], Statistical Theory of Overtraining and Overfitting, Univ. of Tokyo Tech. Report 95-06, submitted Barkai, N. and Seung, H. S. and Sompolinski, H. [1994], On-line learning of dichotomies, NIPS'94 Finke, M. and Muller, K-R. [1994] in Proc. of the 1993 Connectionist Models summer school, Mozer, M., Smolensky, P., Touretzky, D.S., Elman, J.L. and Weigend, A.S. (Eds.), Hillsdale, NJ: Erlenbaum Associates, 324 Hassoun, M. H. [1995], Fundamentals of Artificial Neural Networks, MIT Press. Hecht-Nielsen, R. [1989], Neurocomputing, Addison-Wesley. Heskes, T. and Kappen, B. [1991]' Physical Review, A44, 2718- 2762. LeCun, Y., Denker, J .S., Solla, S. [1990], Optimal brain damage, NIPS'89 Moody, J . E. [1992]' The effective number of parameters: An analysis of generalization and regularization in nonlinear learning systems, NIPS 4 Murata, N., Yoshizawa, S., Amari, S. [1994], IEEE Trans., NN5, 865-872. Muller, K-R., Finke, M., Murata, N., Schulten, K and Amari, S. [1995] A numerical study on learning curves in stochastic multilayer feed-forward networks, Univ. of Tokyo Tech. Report METR 95-03 and Neural Computation in Press Poggio, T. and Girosi, F. [1990], Science, 247, 978- 982. Rissanen, J. [1986], Ann, Statist., 14, 1080- 1100. Rumelhart, D., Hinton, G. E., Williams, R. J. [1986], in PDP, Vol.1, MIT Press. Saad, D., Solla, S. A. [1995], PRL, 74,4337 and Phys. Rev. E, 52,4225 Wang, Ch., Venkatesh, S. S., Judd, J. S. [1994], Optimal stopping and effective machine complexity in learning, to appear, (revised and extended version of NIPS'93).
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Primitive Manipulation Learning with Connectionism Yoky Matsuoka The Artificial Intelligence Laboratory NE43-819 Massachusetts Institute of Techonology Cambridge, MA 02139 Abstract Infants' manipulative exploratory behavior within the environment is a vehicle of cognitive stimulation[McCall 1974]. During this time, infants practice and perfect sensorimotor patterns that become behavioral modules which will be seriated and imbedded in more complex actions. This paper explores the development of such primitive learning systems using an embodied light-weight hand which will be used for a humanoid being developed at the MIT Artificial Intelligence Laboratory[Brooks and Stein 1993]. Primitive grasping procedures are learned from sensory inputs using a connectionist reinforcement algorithm while two submodules preprocess sensory data to recognize the hardness of objects and detect shear using competitive learning and back-propagation algorithm strategies, respectively. This system is not only consistent and quick during the initial learning stage, but also adaptable to new situations after training is completed. 1 INTRODUCTION Learning manipulation in an unpredictable, changing environment is a complex task. It requires a nonlinear controller to respond in a nonlinear system that contains a significant amount of sensory inputs and noise [Miller , et al 1990]. Investigating the human manipulation learning system and implementing it in a physical system has not been done due to its complexity and too many unknown parameters. Conventional adaptive control theory assumes too many parameters that are constantly changing in a real environment [Sutton, et al 1991, Williams 1988]. For an embodied hand, even the simplest form of learning process requires a more intelligent control network. Wiener [Wiener 1948] has proposed the idea of "Connectionism" , which suggests that a muscle is controlled by affecting the gain of the "efferent
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Silicon Models for A uditory Scene Analysis John Lazzaro and John Wawrzynek CS Division UC Berkeley Berkeley, CA 94720-1776 lazzaroOcs.berkeley.edu. johnvOcs.berkeley.edu Abstract We are developing special-purpose, low-power analog-to-digital converters for speech and music applications, that feature analog circuit models of biological audition to process the audio signal before conversion. This paper describes our most recent converter design, and a working system that uses several copies ofthe chip to compute multiple representations of sound from an analog input. This multi-representation system demonstrates the plausibility of inexpensively implementing an auditory scene analysis approach to sound processing. 1. INTRODUCTION The visual system computes multiple representations of the retinal image, such as motion, orientation, and stereopsis, as an early step in scene analysis. Likewise, the auditory brainstem computes secondary representations of sound, emphasizing properties such as binaural disparity, periodicity, and temporal onsets. Recent research in auditory scene analysis involves using computational models of these auditory brainstem representations in engineering applications. Computation is a major limitation in auditory scene analysis research: the complete auditory processing system described in (Brown and Cooke, 1994) operates at approximately 4000 times real time, running under UNIX on a Sun SPARCstation 1. Standard approaches to hardware acceleration for signal processing algorithms could be used to ease this computational burden in a research environment; a variety of parallel, fixed-point hardware products would work well on these algorithms. 700 J. LAZZARO, J. WAWRZYNEK However, hardware solutions appropriate for a research environment may not be well suited for accelerating algorithms in cost-sensitive, battery-operated consumer products. Possible product applications of auditory algorithms include robust pitch-tracking systems for musical instrument applications, and small-vocabulary, speaker-independent wordspotting systems for control applications. In these applications, the input takes an analog form: a voltage signal from a microphone or a guitar pickup. Low-power analog circuits that compute auditory representations have been implemented and characterized by several research groups - these working research prototypes include several generation of cochlear models (Lyon and Mead, 1988), periodicity models, and binaural models. These circuits could be used to compute auditory representations directly on the analog signal, in real-time, using these low-power, area-efficient analog circuits. Using analog computation successfully in a system presents many practical difficulties; the density and power advantages of the analog approach are often lost in the process of system integration. One successful Ie architecture that uses analog computation in a system is the special-purpose analog to digital converter, that includes analog, non-linear pre-processing before or during data conversion. For example, converters that include logarithmic waveform compression before digitization are commercially viable components. Using this component type as a model, we have been developing special-purpose, low-power analog-to-digital converters for speech and audio applications; this paper describes our most recent converter design, and a working system that uses several copies of the chip to compute multiple representations of sound. 2. CONVERTER DESIGN Figure 1 shows an architectural block diagram of our current converter design. The 35,000 transistor chip was fabricated in the 2pm, n-well process of Orbit Semiconductor, broke red through MOSIS; the circuit is fully functional. Below is a summary of the general architectural features ofthis chip; unless otherwise referenced, circuit details are similar to the converter design described in (Lazzaro et al., 1994). • An analog audio signal serves as input to the chip; dynamic range is 40dB to 60dB (l-lOmV to IV peak, dependent on measurement criteria). • This signal is processed by analog circuits that model cochlear processing (Lyon and Mead, 1988) and sensory transduction; the audio signal is transformed into 119 wavelet-filtered, half-wave rectified, non-linearly compressed audio signals. The cycle-by-cycle waveform of each signal is preserved; no temporal smoothing is performed. • Two additional analog processing blocks follow this initial cochlear processing, a temporal autocorrelation processor and a temporal adaptation processor. Each block transforms the input array into a new representation of equal size; alternatively, the block can be programmed to pass its input vector to its output without alteration. • The output circuits of the final processing block are pulse generators, which code the signal as a pattern of fixed-width, fixed-height spikes. All the information in the representation is contained in the onset times of the pulses. Silicon Models for Auditory Scene Analysis 701 • The activity on this array is sent off-chip via an asynchronous parallel bus. The converter chip acts as a sender on the bus; a digital host processor is the receiver. The converter initiates a transaction on the bus to communicate the onset of a pulse in the array; the data value on the bus is a number indicating which unit in the array pulsed. The time of transaction initiation carries essential information. This coding method is also known as the address-event representation. • Many converters can be used in the same system, sharing the same asynchronous output bus (Lazzaro and Wawrzynek, 1995). No extra components are needed to implement bus sharing; the converter bus design includes extra signals and logic that implements multi-chip bus arbitration. This feature is a major difference between this design and (Lazzaro et at., 1994). • The converter includes a digitally-controllable parameter storage and generation system; 25 tunable parameters control the behavior of the analog processing blocks. Programmability supports the creation of multi-converter systems that use a single chip design: each chip receives the same analog signal, but processes the signal in different ways, as determined by the parameter values for each chip. • Non-volatile analog storage elements are used to store the parameters; parameters are changeable via Fowler-Nordhiem tunneling, using a 5V control input bus. Many converters can share the same control bus. Parameter values can be sensed by activating a control mode, which sends parameter information on the converter output bus. Apart from two high-voltage power supply pins, and a trimming input pin for tunneling pulse width, all control voltages used in this converter are generated on-chip. 21V-----. 15V Trim ~ DO ~ Dl -; D2 D3 ~ D4 '0 D5 ~ D6 § CS U WR VDD GND VDD GND VDD GND Audio In ... -------1 R A DO Dl D2 D3 D4 D5 D6 AR rJl RR ] AL b() RL en AM ~ RM "i: DL ~ DR til KO ~ KM ~ Figure 1. Block diagram of the converter chip. Most of the 40 pins of the chip are dedicated to the data output and control input buses, and to the control signals for coordinating bus sharing in multi-converter systems. 702 J. LAZZAR9. J. WAWRZYNEK 3. SYSTEM DESIGN Figure 2 shows a block diagram of a system that uses three copies of the converter chip to compute multiple representations of sound; the system acts as a real-time audio input device to a Sun workstation. An analog audio input connects to each converter; this input can be from a pre-amplified microphone, for spontaneous input, or from the analog audio signal of the workstation, for controlled experiments. The asynchronous output buses from the three chips are connected together, to produce a single output address space for the system; no external components are needed for output bus sharing and arbitration. The onset time of a transaction carries essential information on this bus; additional logic on this board adds a 16bit timestamp to each bus transaction, coding the onset time with 20ps resolution. The control input buses for the three chips are also connected together to produce a single input address space, using external logic for address decoding. We use a commercial interface board to link the workstation with these system buses. 4. SYSTEM PERFORMANCE We designed a software environment, Aer, to support real-time, low-latency data visualization of the multi-converter system. Using Aer, we can easily experiment with different converter tunings. Figure 3 shows a screen from Aer, showing data from the three converters as a function of time; the input sound for this screen is a short 800 Hz tone burst, followed by a sinusoid sweep from 300 Hz to 3 Khz. The top ("Spectral Shape") and bottom ("Onset") representations are raw data from converters 1 and 3, as marked on Figure 2, tuned for different responses. The output channel number is plotted vertically; each dot represents a pulse. The top representation codes for periodicity-based spectral shape; for this representation, the temporal autocorrelation block (see Figure 1) is activated, and the temporal adaptation block is inactivated. Spectral frequency is mapped logarithmically on the vertical dimension, from 300 Hz to 4 Khz; the activity in each channel is the periodic waveform present at that frequency. The difference between a periodicity-based spectral method and a resonant spectral method can be seen in the response to the 800 Hz sinusoid onset: the periodicity representation shows activity only around the 800 Hz channels, whereas a spectral representation would show broadband transient activity at tone onset. Multi-Converter System In~~----.. ----~--------~ O~ Bus Out Sound Input Figure 2. Block diagram of the multi-converter system. aa aaa aa aaaaaaaaaD aaa :: ::::::::ga ::0 aa aaaaaaaaaD aao aa a aa Silicon Models for Auditory Scene Analysis 4 Khz (log) 300 Hz Oms (linear) 12.5 ms 4 Khz (log) 300 Hz . ~;.;\: : ;;~~if~ :r;,;~ .. , .•. :, . 1 .... 1 gj~;':~_ I «>'~': : ·;:.:mT.:·~~ i ttJrC; f;:.~ .. ~.~~~~ . . · ~ •• "' 1,~ _ 200 ms 703 Spectral Shape Summary Auto Corr. Onset Figure 3. Data from the multi-converter system, in response to a 800-Hz pure tone, followed by a sinusoidal sweep from 300Hz to 3Khz. 704 .": ', "' , -: ~ : . .... ", r: D • 100 ms J. LAZZARO, J. WAWRZYNEK • De 300 Oms ... ... o o o ..., :l < » ~ e e :l rJ) 12. 4 Khz ..., II> til C o Figure 4. Data from the multi-converter system, in response to the word "five" followed by the word "nine". Silicon Models for Auditory Scene Analysis 705 The bottom representation codes for temporal onsets; for this representation, the temporal adaptation block is activated, and the temporal autocorrelation block is inactivated. The spectral filtering of the representation reflects the silicon cochlea tuning: a low-pass response with a sharp cutoff and a small resonant peak at the best frequency of the filter. The black, wideband lines at the start of the 800 Hz tone and the sinusoid sweep illustrate the temporal adaptation. The middle ("Summary Auto Corr.") representation is a summary autocorrelogram, useful for pitch processing and voiced/unvoiced decisions in speech recognition. This representation is not raw data from a converter; software post-processing is performed on the converter output to produce the final result. The frequency response of converter 2 is set as in the bottom representation; the temporal adaptation response, however, is set to a 100 millisecond time constant. The converter output pulse rates are set so that the cycle-by-cycle waveform information for each output channel is preserved in the output. To complete the representation, a set of running autocorrelation functions x(t)X(t-T) is computed for T = k 105tLs, k = 1 ... 120, for each of the 119 output channels. These autocorrelation functions are summed over all output channels to produce the final representation; T is plotted as a linear function of time on the vertical axis. The correlation multiplication can be efficiently implemented by integer subtraction and comparison of pulse timestamps; the summation over channels is simply the merging of lists of bus transactions. The middle representation in Figure 3 shows the qualitative characteristics of the summary autocorrelogram: a repetitive band structure in response to periodic sounds. Figure 'f shows the output response of the multi-converter system in response to telephone-bandwidth-limited speech; the phonetic boundaries of the two words, "five" and "nine", are marked by arrows. The vowel formant information is shown most clearly by the strong peaks in the spectral shape representation; the wideband information in the "f" offive is easily seen in the onset representation. The summary autocorrelation representation shows a clear texture break between vowels and the voiced "n" and "v" sounds. Acknowledgements Thanks to Richard Lyon and Peter Cariani for summary autocorrelogram discussions. Funded by the Office of Naval Research (URI-N00014-92-J-1672). References Brown, G.J. and Cooke, M. (1994). Computational auditory scene analysis. Computer Speech and Language, 8:4, pp. 297-336. Lazzaro, J. P. and Wawrzynek, J. (1995). A multi-sender asynchronous extension to the address-event protocol. In Dally, W. J., Poulton, J. W., Ishii, A. T. (eds), 16th Conference on Advanced Research in VLSI, pp. 158-169. Lazzaro, J. P., Wawrzynek, J., and Kramer, A (1994). Systems technologies for silicon auditory models. IEEE Micro, 14:3. 7-15. Lyon, R. F., and Mead, C. (1988). An analog electronic cochlea. IEEE Trans. Acoust., Speech, Signal Processing vol. 36, pp. 1119-1134.
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Human Face Detection in Visual Scenes Henry A. Rowley Shumeet Baluja Takeo Kanade har@cs.cmu.edu baluja@cs.cmu.edu tk@cs.cmu.edu School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213, USA Abstract We present a neural network-based face detection system. A retinally connected neural network examines small windows of an image, and decides whether each window contains a face. The system arbitrates between multiple networks to improve performance over a single network. We use a bootstrap algorithm for training, which adds false detections into the training set as training progresses. This eliminates the difficult task of manually selecting non-face training examples, which must be chosen to span the entire space of non-face images. Comparisons with another state-of-the-art face detection system are presented; our system has better performance in terms of detection and false-positive rates. 1 INTRODUCTION In this paper, we present a neural network-based algorithm to detect frontal views of faces in gray-scale images. The algorithms and training methods are general, and can be applied to other views of faces, as well as to similar object and pattern recognition problems. Training a neural network for the face detection task is challenging because of the difficulty in characterizing prototypical "non-face" images. Unlike in face recognition, where the classes to be discriminated are different faces, in face detection, the two classes to be discriminated are "images containing faces" and "images not containing faces". It is easy to get a representative sample of images which contain faces, but much harder to get a representative sample of those which do not. The size of the training set for the second class can grow very quickly. We avoid the problem of using a huge training set of non-faces by selectively adding images to the training set as training progresses [Sung and Poggio, 1994]. This "bootstrapping" method reduces the size of the training set needed. Detailed descriptions of this training method, along with the network architecture are given in Section 2. In Section 3 the performance of the system is examined. We find that the system is able to detect 92.9% of faces with an acceptable number of false positives. Section 4 compares this system with a similar system. Conclusions and directions for future research are presented in Section 5. 2 DESCRIPTION OF THE SYSTEM Our system consists of two major parts: a set of neural network-based filters, and a system to combine the filter outputs. Below, we describe the design and training of the filters, 876 H. A. ROWLEY, S. BALUJA, T. KANADE which scan the input image for faces. This is followed by descriptions of algorithms for arbitrating among multiple networks and for merging multiple overlapping detections. 2.1 STAGE ONE: A NEURAL -NETWORK-BASED FILTER The first component of our system is a filter that receives as input a small square region of the image, and generates an output ranging from 1 to -1, signifying the presence or absence of a face, respectively. To detect faces anywhere in the input, the filter must be applied at every location in the image. To allow detection of faces larger than the window size, the input image is repeatedly reduced in size (by subsampling), and the filter is applied at each size. The set of scaled input images is known as an "image pyramid", and is illustrated in Figure 1. The filter itself must have some invariance to position and scale. The amount of invariance in the filter determines the number of scales and positions at which the filter must be applied. With these points in mind, we can give the filtering algorithm (see Figure I). It consists of two main steps: a preprocessing step, followed by a forward pass through a neural network. The preprocessing consists of lighting correction, which equalizes the intensity values across the window, followed by histogram equalization, which expands the range of intensities in the window [Sung and Poggio, 19941 The preprocessed window is used as the input to the neural network. The network has retinal connections to its input layer; the receptive fields of each hidden unit are shown in the figure. Although the figure shows a single hidden unit for each subregion of the input, these units can be replicated. Similar architectures are commonly used in speech and character recognition tasks [Waibel et al., 1989, Le Cun et al., 19891 Input image pyramid Extracted window Correct lighting / Preprocessing Neural network Figure 1: The basic algorithm used for face detection. Examples of output from a single filter are shown in Figure 2. In the figure, each box represents the position and size of a window to which the neural network gave a positive response. The network has some invariance to position and scale, which results in mUltiple boxes around some faces. Note that there are some false detections; we present methods to eliminate them in Section 2.2. We next describe the training of the network which generated this output. 2.1.1 Training Stage One To train a neural network to serve as an accurate filter, a large number of face and non-face images are needed. Nearly 1050 face examples were gathered from face databases at CMU and Harvard. The images contained faces of various sizes, orientations, positions, and intensities. The eyes and upper lip of each face were located manually, and these points were used to normalize each face to the same scale, orientation, and position. A 20-by-20 pixel region containing the face is extracted and preprocessed (by apply lighting correction and histogram equalization). In the training set, 15 faces were created from each original image, by slightly rotating (up to 10°), scaling (90%-110%), translating (up to half a pixel), Human Face Detection in Visual Scenes Figure 2: Images with all the above threshold detections indicated by boxes. Figure 3: Example face images, randomly mirrored, rotated, translated. and scaled by small amounts. and mirroring each face. A few example images are shown in Figure 3. 877 It is difficult to collect a representative set of non-faces. Instead of collecting the images before training is started, the images are collected during training, as follows [Sung and Poggio, 1994]: I. Create 1000 non-face images using random pixel intensities. 2. Train a neural network to produce an output of 1 for the face examples, and -I for the non-face examples. 3. Run the system on an image of scenery which contains no faces. Collect subimages in which the network incorrectly identifies a face (an output activation> 0). 4. Select up to 250 of these subimages at random, and add them into the training set. Go to step 2. Some examples of non-faces that are collected during training are shown in Figure 4. We used 120 images for collecting negative examples in this bootstrapping manner. A typical training run selects approximately 8000 non-face images from the 146,212,178 subimages that are available at all locations and scales in the scenery images. Figure 4: Some non-face examples which are collected during training. 2.2 STAGE TWO: ARBITRATION AND MERGING OVERLAPPING DETECTIONS The examples in Figure 2 showed that just one network cannot eliminate all false detections. To reduce the number of false positives, we apply two networks, and use arbitration to produce the final decision. Each network is trained in a similar manner, with random initial weights, random initial non-face images, and random permutations of the order of presentation of the scenery images. The detection and false positive rates of the individual networks are quite close. However, because of different training conditions and because of self-selection of negative training examples, the networks will have different biases and will make different errors. For the work presented here, we used very simple arbitration strategies. Each detection by a filter at a particular position and scale is recorded in an image pyramid. One way to 878 H. A. ROWLEY, S. BALUJA, T. KANADE combine two such pyramids is by ANDing. This strategy signals a detection only if both networks detect a face at precisely the same scale and position. This ensures that, if a particular false detection is made by only one network, the combined output will not have that error. The disadvantage is that if an actual face is detected by only one network, it will be lost in the combination. Similar heuristics, such as ORing the outputs, were also tried. Further heuristics (applied either before or after the arbitration step) can be used to improve the performance of the system. Note that in Figure 2, most faces are detected at multiple nearby positions or scales, while false detections often occur at single locations. At each location in an image pyramid representing detections, the number of detections within a specified neighborhood of that location can be counted. If the number is above a threshold, then that location is classified as a face. These detections are then collapsed down to a single point, located at their centroid. when this is done before arbitration, the centroid locations rather than the actual outputs from the networks are ANDed together. If we further assume that a position is correctly identified as a face, then all other detections which overlap it are likely to be errors, and can therefore be eliminated. There are relatively few cases in which this heuristic fails; however, one such case is illustrated in the left two faces in Figure 2B, in which one face partially occludes another. Together, the steps of combining multiple detections and eliminating overlapping detections will be referred to as merging detections. In the next section, we show that by merging detections and arbitrating among multiple networks, we can reduce the false detection rate significantly. 3 EMPIRICAL RESULTS A large number of experiments were performed to evaluate the system. Because of space . restrictions only a few results are reported here; further results are presented in [Rowley et al., 1995]. We first show an analysis of which features the neural network is using to detect faces, and then present the error rates of the system over two large test sets. 3.1 SENSITIVITY ANALYSIS In order to determine which part of the input image the network uses to decide whether the input is a face, we performed a sensitivity analysis using the method of [Baluja and Pomerleau, 1995]. We collected a test set of face images (based on the training database, but with different randomized scales, translations, and rotations than were used for training), and used a set of negative examples collected during the training of an earlier version of the system. Each of the 20-by-20 pixel input images was divided into 100 two-by-two pixel subimages. For each subimage in turn, we went through the test set, replacing that subimage with random noise, and tested the neural network. The sum of squared errors made by the network is an indication of how important that portion of the image is for the detection task. Plots of the error rates for two networks we developed are shown in Figure 5. FigureS: Sum of squared errors (zaxis) on a small test resulting from adding noise to various portions of the input image (horizontal plane), for two networks. Network 1 uses two sets of the hidden units illustrated in Figure 1, while network 2 uses three sets. The networks rely most heavily on the eyes, then on the nose, and then on the mouth (Figure 5). Anecdotally, we have seen this behavior on several real test images: the network's accuracy decreases more when an eye is occluded than when the mouth is occluded. Further, when both eyes of a face are occluded, it is rarely detected. Human Face Detection in Visual Scenes 879 3.2 TESTING The system was tested on two large sets of images. Test Set A was collected at CMU, and consists of 42 scanned photographs, newspaper pictures, images collected from the World Wide Web, and digitized television pictures. Test set B consists of 23 images provided by Sung and Poggio; it was used in [Sung and Poggio, 1994] to measure the accuracy of their system. These test sets require the system to analyze 22,053,124 and 9,678,084 windows, respectively. Table 1 shows the performance for the two networks working alone, the effect of overlap elimination and collapsing multiple detections, and the results of using ANDing and ~Ring for arbitration. Each system has a better false positive rate (but a worse detection rate) on Test Set A than on Test Set B, because of differences in the types of images in the two sets. Note that for systems using arbitration, the ratio of false detections to windows examined is extremely low, ranging from 1 in 146,638 to 1 in 5,513,281, depending on the type of arbitration used. Figure 6 shows some example output images from the system, produced by merging the detections from networks 1 and 2, and ANDing the results. Using another neural network to arbitrate among the two networks gives about the same performance as the simpler schemes presented above [Rowley et ai., 1995]. Table 1: Detection and Error Rates Test Set A Test Set B # miss 1 Detect rate # miss 1 Detect rate Type System False detects 1 Rate False detects 1 Rate 0) Ideal System 0/169 100.0% 01155 100.0% 0 0/22053124 0 0/9678084 Single 1) Network 1 (52 hidden 17 89.9% 11 92.9% network, units, 2905 connections) 507 1143497 353 1127417 no 2) Network 2 (7~ hidden 20 ~~.2% 10 93.5% heuristics units, 4357 connections) 385 1157281 347 1127891 Single 3) Network 1 4- merge 24 85.8% 12 92.3% network, detections 222 1199338 126 1176810 with 4) Network 2 4- merge 27 84.0% 13 91.6% heuristics detections 179 11123202 123 1178684 Arbitrating 5) Networks 1 and 2 4- AND 52 69.2% 34 78.1% among 4- merge detections 4 115513281 3 113226028 two 6) Networks 1 and 2 436 78.7% 20 ~7.1% networks merge detections 4- AND 15 111470208 15 11645206 7) Networks 1 and 2 426 84.6% 11 92.9% merge 4- OR 4- merge 90 11245035 64 11151220 4 COMPARISON TO OTHER SYSTEMS [Sung and Poggio, 1994] reports a face-detection system based on clustering techniques. Their system, like ours, passes a small window over all portions of the image, and determines whether a face exists in each window. Their system uses a supervised clustering method with six "face" and six "non-face" clusters. Two distance metrics measure the distance of an input image to the prototype clusters. The first metric measures the "partial" distance between the test pattern and the cluster's 75 most significant eigenvectors. The second distance metric is the Euclidean distance between the test pattern and its projection in the 75 dimensional subspace. These distance measures have close ties with Principal Components Analysis (PeA), as described in [Sung and Poggio, 1994]. The last step in their system is to use either a perceptron or a neural network with a hidden layer, trained to classify points using the two distances to each of the clusters (a total of 24 inputs). Their system is trained with 4000 positive examples, and nearly 47500 negative examples collected in the "bootstrap" manner. In comparison, our system uses approximately 16000 positive examples and 8000 negative examples. Table 2 shows the accuracy of their system on Test Set B, along with the results of our 880 H. A. ROWLEY, S. BALUJA, T. KANADE Figure 6: Output produced by System 6 in Table 1. For each image, three numbers are shown: the number of faces in the image, the number of faces detected correctly, and the number of false detections. Some notes on specific images: Although the system was not trained on hand-drawn faces, it detects them in K and R. One false detect is present in both D and R. Faces are missed in D (removed because a false detect overlapped it), B (one due to occlusion, and one due to large angle), and in N (babies with fingers in their mouths are not well represented in training data). Images B, D, F, K, L, and M were provided by Sung and Poggio at MIT. Images A, G, 0, and P were scanned from photographs, image R was obtained with a CCD camera, images J and N were scanned from newspapers, images H, I, and Q were scanned from printed photographs, and image C was obtained off of the World Wide Web. Images P and B correspond to Figures 2A and 2B. Human Face Detection in Visual Scenes 881 system using a variety of arbitration heuristics. In [Sung and Poggio, 1994], only 149 faces were labelled in the test set, while we labelled 155 (some are difficult for either system to detect). The number of missed faces is therefore six more than the values listed in their paper. Also note that [Sung and Poggio, 1994] check a slightly smaller number of windows over the entire test set; this is taken into account when computing the false detection rates. The table shows that we can achieve higher detection rates with fewer false detections. Table 2: Comparison of [Sung and Poggio, 1994] and Our System on Test Set B System II Missed faces Detect I False rate detects Rate 5) Networks 1 and 2 ~ AND ~ merge 34 78.l% 3 113226028 6) Networks 1 and 2 ~ merge ~ AND 20 87.l% 15 11645206 7) Networks 1 and 2 ~ merge ~ OR ~ merge 11 92.9% 64 11151220 [Sung and Poggio, 1994] (Multi-layer network) 36 76.8% 5 111929655 [Sung and Poggio, 1994] (Perceptron) 28 81.9% 13 11742175 5 CONCLUSIONS AND FUTURE RESEARCH Our algorithm can detect up to 92.9% of faces in a set of test images with an acceptable number of false positives. This is a higher detection rate than [Sung and Poggio, 1994]. The system can be made more conservative by varying the arbitration heuristics or thresholds. Currently, the system does not use temporal coherence to focus attention on particular portions of the image. In motion sequences, the location of a face in one frame is a strong predictor of the location of a face in next frame. Standard tracking methods can be applied to focus the detector's attention. The system's accuracy might be improved with more positive examples for training, by using separate networks to recognize different head orientations, or by applying more sophisticated image preprocessing and normalization techniques. Acknowledgements The authors thank Kah-Kay Sung and Dr. Tomaso Poggio (at MIT), Dr. Woodward Yang (at Harvard), and Michael Smith (at CMU) for providing training and testing images. We also thank Eugene Fink, Xue-Mei Wang, and Hao-Chi Wong for comments on drafts of this paper. This work was partially supported by a grant from Siemens Corporate Research, Inc., and by the Department of the Army, Army Research Office under grant number DAAH04-94-G-0006. Shu meet Baluja was supported by a National Science Foundation Graduate Fellowship. The views and conclusions in this document are those of the authors, and should not be interpreted as necessarily representing official policies or endorsements, either expressed or implied, of the sponsoring agencies. References [Baluja and Pomerleau, 1995] Shumeet Baluja and Dean Pomerleau. Encouraging distributed input reliance in spatially constrained artificial neural networks: Applications to visual scene analysis and control. Submitted, 1995. [Le Cun et al., 1989] Y. Le Cun, B. Boser, 1. S. Denker, D. Henderson, R. E. Howard, W. Hubbard, and L. D. Jackel. Backpropogation applied to handwritten zip code recognition. Neural Computation, 1:541-551, 1989. [Rowley et al., 1995] Henry A. Rowley, Shumeet Baluja, and Takeo Kanade. Human face detection in visual scenes. CMU-CS-95-158R, Carnegie Mellon University, November 1995. Also available at http://www.cs.cmu.edul11ar/faces.html. [Sung and Poggio, 1994] Kah-Kay Sung and Tomaso Poggio. Example-based learning for viewbased human face detection. A.I. Memo 1521, CBCL Paper 112, MIT, December 1994. [Waibel et al., 1989] Alex Waibel, Toshiyuki Hanazawa, Geoffrey Hinton, Kiyohiro Shikano, and Kevin J. Lang. Phoneme recognition using time-delay neural networks. Readings in Speech Recognition, pages 393-404, 1989.
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Beating a Defender in Robotic Soccer: Memory-Based Learning of a Continuous FUnction Peter Stone Department of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Manuela Veloso Department of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Abstract Learning how to adjust to an opponent's position is critical to the success of having intelligent agents collaborating towards the achievement of specific tasks in unfriendly environments. This paper describes our work on a Memory-based technique for to choose an action based on a continuous-valued state attribute indicating the position of an opponent. We investigate the question of how an agent performs in nondeterministic variations of the training situations. Our experiments indicate that when the random variations fall within some bound of the initial training, the agent performs better with some initial training rather than from a tabula-rasa. 1 Introduction One of the ultimate goals subjacent to the development of intelligent agents is to have multiple agents collaborating in the achievement of tasks in the presence of hostile opponents. Our research works towards this broad goal from a Machine Learning perspective. We are particularly interested in investigating how an intelligent agent can choose an action in an adversarial environment. We assume that the agent has a specific goal to achieve. We conduct this investigation in a framework where teams of agents compete in a game of robotic soccer. The real system of model cars remotely controlled from off-board computers is under development. Our research is currently conducted in a simulator of the physical system. Both the simulator and the real-world system are based closely on systems designed by the Laboratory for ComputationalIntelligence at the University of British Columbia [Sahota et a/., 1995, Sahota, 1993]. The simulator facilitates the control of any number of cars and a ball within a designated playing area. Care has been taken to ensure that the simulator models real-world responses (friction, conservaMemory-based Learning of a Continuous Function 897 tion of momentum, etc.) as closely as possible. Figure l(a) shows the simulator graphics. (j <0 IJ I IJ 0 ~ ~ <:P (a) (b) Figure 1: (a) the graphic view of our simulator. (b) The initial position for all of the experiments in this paper. The teammate (black) remains stationary, the defender (white) moves in a small circle at different speeds, and the ball can move either directly towards the goal or towards the teammate. The position of the ball represents the position of the learning agent. We focus on the question of learning to choose among actions in the presence of an adversary. This paper describes our work on applying memory-based supervised learning to acquire strategy knowledge that enables an agent to decide how to achieve a goal. For other work in the same domain, please see [Stone and Veloso, 1995b]. For an extended discussion of other work on incremental and memorybased learning [Aha and Salzberg, 1994, Kanazawa, 1994, Kuh et al., 1991, Moore, 1991, Salganicoff, 1993, Schlimmer and Granger, 1986, Sutton and Whitehead, 1993, Wettschereck and Dietterich, 1994, Winstead and Christiansen, 1994], particularly as it relates to this paper, please see [Stone and Veloso, 1995a]. The input to our learning task includes a continuous-valued range of the position of the adversary. This raises the question of how to discretize the space of values into a set of learned features. Due to the cost of learning and reusing a large set of specialized instances, we notice a clear advantage to having an appropriate degree of generalization. For more details please see [Stone and Veloso, 1995a]. Here, we address the issue of the effect of differences between past episodes and the current situation. We performed extensive experiments, training the system under particular conditions and then testing it (with learning continuing incrementally) in nondeterministic variations of the training situation. Our results show that when the random variations fall within some bound of the initial training, the agent performs better with some initial training rather than from a tabula-rasa. This intuitive fact is interestingly well- supported by our empirical results. 2 Learning Method The learning method we develop here applies to an agent trying to learn a function with a continuous domain. We situate the method in the game of robotic soccer. We begin each trial by placing a ball and a stationary car acting as the "teammate" in specific places on the field. Then we place another car, the "defender," in front of the goal. The defender moves in a small circle in front of the goal at some speed and begins at some random point along this circle. The learning agent must take one of two possible actions: shoot straight towards the goal, or pass to the teammate so 898 P. STONE, M. VELOSO that the ball will rebound towards the goal. A snapshot of the experimental setup is shown graphically in Figure 1 (b). The task is essentially to learn two functions, each with one continuous input variable, namely the defender's position. Based on this position, which can be represented unambiguously as the angle at which the defender is facing, ¢, the agent tries to learn the probability of scoring when shooting, Ps* (¢), and the probability of scoring when passing, P; (¢ ).1 If these functions were learned completely, which would only be possible if the defender's motion were deterministic, then both functions would be binary partitions: Ps*, P; : [0.0,360.0) f--.+ {-1 , I}. 2 That is, the agent would know without doubt for any given ¢ whether a shot, a pass, both, or neither would achieve its goal. However, since the agent cannot have had experience for every possible ¢, and since the defender may not move at the same speed each time, the learned functions must be approximations: Ps,Pp : [0.0,360.0) f--.+ [-1.0,1.0]. In order to enable the agent to learn approximations to the functions Ps* and P*, we gave it a memory in which it could store its experiences and from which it coufd retrieve its current approximations Ps (¢) and Pp( ¢). We explored and developed appropriate methods of storing to and retrieving from memory and an algorithm for deciding what action to take based on the retrieved values. 2.1 Memory Model Storing every individual experience in memory would be inefficient both in terms of amount of memory required and in terms of generalization time. Therefore, we store Ps and Pp only at discrete, evenly-spaced values of ¢. That is, for a memory of size M (with M dividing evenly into 360 for simplicity), we keep values of Pp(O) and Ps(O) for 0 E {360n/M I 0 ~ n < M}. We store memory as an array "Mem" of size M such that Mem[n] has values for both Pp(360n/M) and Ps(360n/M) . Using a fixed memory size precludes using memory-based techniques such as KNearest-Neighbors (kNN) and kernel regression which require that every experience be stored, choosing the most relevant only at decision time. Most of our experiments were conducted with memories of size 360 (low generalization) or of size 18 (high generalization), i.e. M = 18 or M = 360. The memory size had a large effect on the rate of learning [Stone and Veloso, 1995a]. 2.1.1 Storing to Memory With M discrete memory storage slots, the problem then arises as to how a specific training example should be generalized. Training examples are represented here as E.p,a,r, consisting of an angle ¢, an action a, and a result r where ¢ is the initial position of the defender, a is "s" or "p" for "shoot" or "pass," and r is "I" or "-I" for "goal" or " miss" respectively. For instance, E 72 .345 ,p ,1 represents a pass resulting in a goal for which the defender started at position 72.345 0 on its circle. Each experience with 0 - 360/2M :::; ¢ < 0 + 360/2M affects Mem[O] in proportion to the distance 10 - ¢I. In particular, Mem[O] keeps running sums of the magnitudes of scaled results, Mem[O]. total-a-results, and of scaled positive results, Mem[O].positive-a-results, affecting Pa(O), where "a" stands for "s" or "p" as before. Then at any given time P (0) = -1 + 2 * positive-a-results The "-I" is for , a total-a-results . 1 As per convention, P * represents the target (optimal) function. 2 Although we think of P; and P; as functions from angles to probabilities, we will use -1 rather than 0 as the lower bound of the range. This representation simplifies many of our illustrative calculations. Memory-based Learning of a Continuous Function 899 the lower bound of our probability range, and the "2*" is to scale the result to this range. Call this our adaptive memory storage technique: Adaptive Memory Storage of E4>,a,r in Mem 0 I _ (1 _ 14>-01) • r r * 360/M . • Mem[O].total-a-results += r'o • If r' > 0 Then Mem[O].positive-a-results += r'o • P (0) = -1 + 2 * posittve-a-results. a total-a-resuLts For example, EllO,p,l wOilld set both total-p-results and positive-p-results for Mem[120] (and Mem[100]) to 0.5 and consequently Pp(120) (and Pp(100)) to 1.0. But then E l25,p,-1 would increment total-p-resultsfor Mem[120] by .75, while leaving positive-p-results unchanged. Thus Pp(120) becomes -1 + 2 * 1:~5 = -.2. This method of storing to memory is effective both for time-varying concepts and for concepts involving random noise. It is able to deal with conflicting examples within the range of the same memory slot. Notice that each example influences 2 different memory locations. This memory storage technique is similar to the kNN and kernel regression function approximation techniques which estimate f( ¢) based on f( 0) possibly scaled by the distance from o to ¢ for the k nearest values of O. In our linear continuum of defender position, our memory generalizes training examples to the 2 nearest memory locations.3 2.1.2 Retrieving from Memory Since individual training examples affect multiple memory locations, we use a simple technique for retrieving Pa (¢) from memory when deciding whether to shoot or to pass. We round ¢ to the nearest 0 for which Mem[O] is defined, and then take Pa (0) as the value of Pa(¢). Thus, each Mem[O] represents Pa(¢) for 0 - 360/2M ~ ¢ < o + 360 /2M. Notice that retrieval is much simpler when using this technique than when using kNN or kernel regression: we look directly to the closest fixed memory position, thus eliminating the indexing and weighting problems involved in finding the k closest training examples and (possibly) scaling their results. 2.2 Choosing an Action The action selection method is designed to make use of memory to select the action most probable to succeed, and to fill memory when no useful memories are available. For example, when the defender is at position ¢, the agent begins by retrieving Pp (¢) and Ps( ¢) as described above. Then, it acts according to the following function: If Pp_(<fJ) = P.(<fJ) (no basis for a decision), shoot or pass randomly. else If Pp(<fJ) > 0 and Pp(<fJ) > Ps(<fJ), pass. else If P.(<fJ) > 0 and P.(<fJ) > Pp(<fJ), shoot. else If Pp(<fJ) = 0, (no previous passes) pass. else If P.(<fJ) = 0, (no previous shots) shoot. else (Pp(<fJ),P.(<fJ) < 0) shoot or pass randomly. An action is only selected based on the memory values if these values indicate that one action is likely to succeed and that it is better than the other. If, on the other hand, neither value Pp(¢) nor Ps(¢) indicate a positive likelihood of success, then an action is chosen randomly. The only exception to this last rule is when one of 3For particularly large values of M it is useful to generalize training examples to more memory locations, particularly at the early stages of learning. However for the values of M considered in this paper, we always generalize to the 2 nearest memory locations. 900 P.STONE,M.VELOSO the values is zero,4 suggesting that there has not yet been any training examples for that action at that memory location. In this case, there is a bias towards exploring the untried action in order to fill out memory. 3 Experiments and Results In this section, we present the results of our experiments. We explore our agent's ability to learn time-varying and nondeterministic defender behavior. While examining the results, keep in mind that even if the agent used the functions P; and P; to decide whether to shoot or to pass, the success rate would be significantly less than 100% (it would differ for different defender speeds): there were many defender starting positions for which neither shooting nor passing led to a goal (see Figure 2). For example, from our experiments with the defender moving I I D ..... · · (b) Cd) Figure 2: For different defender starting positions (solid rectangle), the agent can score when (a) shooting, (b) passing, (c) neither, or (d) both. at a constant speed of 50,5 we found that an agent acting optimally scores 73.6% of the time; an agent acting randomly scores only 41.3% of the time. These values set good reference points for evaluating our learning agent's performance. 3.1 Coping with Changing Concepts Figure 3 demonstrates the effectiveness of adaptive memory when the defender's speed changes. In all of the experiments represented in these graphs, the agent Success Rate vs. Defender Speed: Memory Size = 360 80r-~~--~~~--~~--~ 75 65 60 55 50 45 40 First 1000 trials Next 1000 trials .. - .. TheoreHcal optimum ..... 35L-~~--~~~--~~~~ 10 20 30 40 50 60 70 80 90 100 Defender Speed Success Rate vs. Defender Speed: Memory Size = 18 BOr-~~--~~~--~~~--' 75 70 ~ ... - -".""':::::'" .:.::.-~ ......... ~._. ~ 65 60 55 50 45 40 First 1000 trials Next 1000 trials .-.. Theoretical optimum .. .. 35L-~~--~~~--~~~~ 10 20 30 40 50 60 70 80 90 100 Defender Speed Figure 3: For all trials shown in these graphs, the agent began with a memory trained for a defender moving at constant speed 50. started with a memory trained by attempting a single pass and a single shot with the defender starting at each position 0 for which Mem[O] is defined and moving in 4Recall that a memory value of 0 is equivalent to a probability of .5, representing no reason to believe that the action will succeed or fail. SIn the simulator, "50" represents 50 cm/s. Subsequently, we omit the units. Memory-based Learning of a Continuous Function 901 its circle at speed 50. We tested the agent's performance with the defender moving at various (constant) speeds. With adaptive memory, the agent is able to unlearn the training that no longer applies and approach optimal behavior: it re-Iearns the new setup. During the first 1000 trials the agent suffers from having practiced in a different situation (especially for the less generalized memory, M = 360) , but then it is able to approach optimal behavior over the next 1000 trials. Remember that optimal behavior, represented in the graph, leads to roughly a 70% success rate, since at many starting positions, neither passing nor shooting is successful. From these results we conclude that our adaptive memory can effectively deal with time-varying concepts. It can also perform well when the defender's motion is nondeterministic, as we show next. 3.2 Coping with Noise To model nondeterministic motion by the defender, we set the defender's speed randomly within a range. For each attempt this speed is constant, but it varies from attempt to attempt. Since the agent observes only the defender's initial position, from the point of view of the agent, the defender's motion is nondeterministic. This set of experiments was designed to test the effectiveness of adaptive memory when the defender's speed was both nondeterministic and different from the speed used to train the existing memory. The memory was initialized in the same way as in Section 3.1 (for defender speed 50). We ran experiments in which the defender's speed varied between 10 and 50. We compared an agent with trained memory against an agent with initially empty memories as shown in Figure 4. Success Rate VS . Trial #: M=18, Defender speed 10-50 70 r-~---.--~--~--r--.--~---.--. 55 50 45 No initial memory Full initial memory - 40 L-~ __ -L __ ~ __ ~ __ L-~ __ ~ __ ~~ 50 100 1 50 200 250 300 350 400 450 500 Trial Number Figure 4: A comparison of the effectiveness of starting with an empty memory versus starting with a memory trained for a constant defender speed (50) different from that used during testing. Success rate is measured as goal percentage thus far. The agent with full initial memory outperformed the agent with initially empty memory in the short run. The agent learning from scratch did better over time since it did not have any training examples from when the defender was moving at a fixed speed of 50; but at first, the training examples for speed 50 were better than no training examples. Thus, when you would like to be successful immediately upon entering a novel setting, adaptive memory allows training in related situations to be effective without permanently reducing learning capacity. 902 P. STONE, M. VELOSO 4 Conclusion Our experiments demonstrated that online, incremental, supervised learning can be effective at learning functions with continuous domains. We found that adaptive memory made it possible to learn both time-varying and nondeterministic concepts. We empirically demonstrated that short-term performance was better when acting with a memory trained on a concept related to but different from the testing concept, than when starting from scratch. This paper reports experimental results on our work towards multiple learning agents, both cooperative and adversarial, III a continuous environment. Future work on our research agenda includes simultaneous learning of the defender and the controlling agent in an adversarial context. We will also explore learning methods with several agents where teams are guided by planning strategies. In this way we will simultaneously study cooperative and adversarial situations using reactive and deliberative reasoning. Acknow ledgements We thank Justin Boyan and the anonymous reviewers for their helpful suggestions. This research is sponsored by the Wright Laboratory, Aeronautical Systems Center, Air Force Materiel Command, USAF, and the Advanced Research Projects Agency (ARPA) under grant number F33615-93-1-1330. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of Wright Laboratory or the U. S. Government. References [Aha and Salzberg, 1994] David W . Aha and Steven L. Salzberg. Learning to catch: Applying nearest neighbor algorithms to dynamic control tasks. In P. Cheeseman and R. W. Oldford, editors, Selecttng Models from Data: Artificial Intelhgence and StattStics IV. SpringIer-Verlag, New York, NY, 1994. [Kanazawa, 1994] Keiji Kanazawa. Sensible decisions: Toward a theory of decision-theoretic information invariants. In Proceedings of the Twelfth National Conference on Art~ficial Intelligence, pages 973978, 1994 . [Kuh et al., 1991] A. Kuh, T. Petsche, and R.L. Rivest. Learning time-varying concepts. In Advances in Neural Information Processing Systems 3, pages 183-189. Morgan Kaufman, December 1991. [Moore, 1991] A.W. Moore. Fast, robust adaptive control by learning only forward models. In Advances in Neural Information Processing Systems 3. Morgan Kaufman, December 1991. [Sahota et al., 1995] Michael K. Sahota, Alan K. Mackworth, Rod A. Barman, and Stewart J. Kingdon. Real-time control of soccer-playing robots using off-board vision : the dynamite testbed. In IEEE Internahonal Conference on Systems, Man, and Cybernetics, pages 3690-3663, 1995. [Sahota, 1993] Michael K . Sahota. Real-time intelligent behaviour in dynamic environments: Soccerplaying robots. Master's thesis, University of British Columbia, August 1993. [Salganicoff, 1993] Marcos Salganicoff. Density-adaptive learning and forgetting. In Proceedmgs of the Tenth International Conference on Machine Learning, pages 276-283, 1993. [Schlimmer and Granger, 1986] J.C. Schlimmer and R.H. Granger. Beyond incremental processing: Tracking concept drift. In Proceedings of the Fiffth National Conference on Artifictal Intelligence, pages 502-507. Morgan Kaufman, Philadelphia, PA, 1986. [Stone and Veloso, 1995a] Peter Stone and Manuela Veloso. Beating a defender in robotic soccer: Memory-based learning of a continuous function. Technical Report CMU-CS-95-222, Computer Science Department, Carnegie Mellon University, 1995. [Stone and Veloso, 1995b] Peter Stone and Manuela Veloso. Broad learning from narrow training: A case study in robotic soccer. Technical Report CMU-CS-95-207, Computer Science Department, Carnegie Mellon University, 1995. [Sutton and Whitehead, 1993] Richard S. Sutton and Steven D. Whitehead. Online learning with random representations. In ProceedIngs of the Tenth International Conference on Machine Learnmg, pages 314-321, 1993. [Wettschereck and Dietterich, 1994] Dietrich Wettschereck and Thomas Dietterich. Locally adaptive nearest neighbor algorithms. In J. D . Cowan, G. Tesauro, and J. Alspector, editors, Advances in Neural Informatton Processing Systems 6, pages 184-191, San Mateo, CA, 1994. Morgan Kaufmann. [Winstead and Christiansen, 1994] Nathaniel S. Winstead and Alan D. Christiansen. Pinball: Planning and learning in a dynamic real-time environment. In AAAI-9-4 Fall Symposium on Control of the Physical World by Intelligent Agents, pages 153-157, New Orleans, LA, November 1994.
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SPERT-II: A Vector Microprocessor System and its Application to Large Problems in Backpropagation Training John Wawrzynek, Krste Asanovic, & Brian Kingsbury University of California at Berkeley Department of Electrical Engineering and Computer Sciences Berkeley, CA 94720-1776 {johnw ,krste,bedk }@cs.berkeley.edu James Beck, David Johnson, & Nelson Morgan International Computer Science Institute 1947 Center Street, Suite 600 Berkeley, CA 94704-1105 {beck,davidj,morgan}@icsi.berkeley.edu Abstract We report on our development of a high-performance system for neural network and other signal processing applications. We have designed and implemented a vector microprocessor and packaged it as an attached processor for a conventional workstation. We present performance comparisons with commercial workstations on neural network backpropagation training. The SPERT-II system demonstrates significant speedups over extensively handoptimization code running on the workstations. 1 Introduction We are working on pattern recognition problems using neural networks with a large number of parameters. Because of the large computational requirements of our area of research, we set out to design an integrated circuit that would serve as a good building block for our systems. Initially we considered designing extremely specialized chips, as this would maximize performance for a particular algorithm. However, the algorithms we use undergo considerable change as our research progresses. Still, we needed to provide some specialization if our design was to offer significant improvement over commercial workstation systems. Competing with workstations is 620 J. WAWRZYNEK, K. ASANOVIC, B. KINGSBURY, J. BECK, D. JOHNSON, N. MORGAN a challenge to anyone designing custom programmable processors, but as will be shown in this paper, one can still provide a performance advantage by focusing on one general class of computation. Our solution was to design a vector microprocessor, TO, optimized for fixed-point computations, and to package this as an inexpensive workstation accelerator board. In this manner, we gain a considerable performance/cost advantage for neural network and other signal processing algorithms, while leveraging the commercial workstation environment for software development and I/O services. In this paper, we focus on the neural network applications ofthe SPERT-II system. We are also investigating other applications in the areas of hum an-machine interface and multimedia processing, as we believe vector microprocessors show promise in providing the flexible, cost-effective, high-performance computing required. Section 2 discusses the design of the hardware, followed in Section 3 by a discussion of the software environment we are developing and a discussion of related systems in Section 4. In Section 5 we discuss how we map a backpropagation training task to the system and in Section 6 we compare the resulting performance with two commercial workstation systems. 2 SPERT -II System SPERT-II is a double slot SEus card for use in Sun compatible workstations and is shown in Figure 1. The board contains a TO vector microprocessor and its memory, a Xilinx FPGA device for interfacing with the host, and various system support devices. Host Wor1<station SPERT·11 Board TO Chip Xilinx FPGA Figure 1: SPERT-II System Organization 2.1 The TO vector microprocessor Data 8MBSRAM Development of the TO vector microprocessor follows our earlier work on the original SPERT VLIW /SIMD neuro-microprocessor (Wawrzynek, 1993). The most significant change we have made to the architecture is to move to a vector instruction set architecture (IS A) , based on the industry standard MIPS RISe scalar ISA (Kane, 1992) extended with vector coprocessor instructions. The resulting ISA, which we call Torrent, offers important advantages over our previous design. We gain access to existing software tools for the MIPS architecture, including optimizing e compilers, assemblers, linkers, and debuggers. VLIW machines expose details of the hardware implementation at the instruction set level, and so must change instruction sets SPERT-II: A Vector Microprocessor System 621 ~hen scaling to higher degrees of on-chip parallelism. In contrast, vector ISAs provide a simple abstraction of regular data parallelism that enables different hardware implementations to make different trade-offs between cost and performance while remaining software compatible. Compared with the VLIW /SIMD design, the vector ISA reduces requirements on instruction cache space and fetch bandwidth. It also makes it easier to write optimized library routines in assembly language, and these library routines will still run well on future devices with greater on-chip parallelism. In the design of the TO vector microprocessor, the main technique we employ to improve cost-performance over a commercial general purpose processor is to integrate multiple fixed-point datapaths with a high-bandwidth memory system. Fast digital arithmetic units, multipliers in particular, require chip area proportional to the square of the number of operand bits. In modern microprocessors and digital signal processors a single floating-point unit takes up a significant portion ofthe chip area. High-precision arithmetic units also requires high memory bandwidth to move large operands. However, for a wide class of problems, full-precision floating-point, or even high-precision fixed-point arithmetic, is not needed. Studies by ourselves and others have shown that for error back-propagation training of neural networks, 16-bit weights and 8-bit activation values provide similar training performance to IEEE single-precision floating-point (Asanovic, 1991). However, fast fixed-point multiply-adds alone are not sufficient to increase performance on a wide range of problems. Other components of a complete application may dominate total compute time if only multiply-add operations are accelerated. Our processor integrates a fast general-purpose RISC core, and includes general purpose operations in its vector instruction set to obtain a balanced design. The TO processor is a complete single chip implementation of the Torrent architecture. It was fabricated in Hewlett-Packard's CMOS26B process using 1.0 pm scalable CMOS design rules and two layers of metal. The die measures 16.75mm x 16.75mm, and contains 730,701 transistors. TO runs at an internal clock rate of 40MHz. The main components of TO are the MIPS-II compatible RISC CPU with an onchip instruction cache, a vector unit coprocessor, a 128-bit wide external memory interface, and an 8-bit wide serial host interface (TSIP) and control unit. The external memory interface supports up to 4 GB of memory over a 128-bit wide data bus. The current SPERT-II board uses 16, 4 Mb SRAM parts to provide 8 MB of mam memory. At the core of the TO processor is a MIPS-II compatible 32-bit integer RISC processor with a 1 KB instruction cache. The system coprocessor provides a 32-bit counter/timer and registers for host synchronization and exception handling. The vector unit contains a vector register file with 16 vector registers, each holding 32 elements of 32 bits each, and three vector functional units, VPO, VP1, and VMP. VPO and VPl are vector arithmetic functional units. With the exception of multiplies, that must execute in VPO, either pipeline can execute any arithmetic operation. The multipliers perform 16-bit x 16-bit multiplies producing 32-bit results. All other arithmetic, logical and shift functions operate on 32 bits. VMP is the vector memory unit, and it handles all vector load/store operations, scalar load/store operations, and the vector insert/extract operations. All three vector functional units are composed of 8 parallel pipelines, and so can each produce up to 8 results per cycle. The TO memory interface has a single memory address port, therefore non-unit stride and indexed memory operations are limited to a rate of one element per cycle. 622 J. WAWRZYNEK, K. ASANOVIC, B. KINGSBURY, J. BECK, D. JOHNSON, N. MORGAN The elements of a vector register are striped across all 8 pipelines. With the maximum vector length of 32, a vector functional unit can accept a new instruction every 4 cycles. TO can saturate all three vector functional units by issuing one instruction per cycle to each, leaving a single issue slot every 4 cycles for the scalar unit. In this manner, TO can sustain up to 24 operations per cycle. Several important library routines, such as matrix-vector and matrix-matrix multiplies, have been written which achieve this level of performance. All vector pipeline hazards are fully interlocked in hardware, and so instruction scheduling is only required to improve performance, not to ensure correctness. 3 SPERT-II Software Environment The primary design goal for the SPERT-II software environment was that it should appear as similar as possible to a conventional workstation environment. This should ease the task of porting existing workstation applications, as well as provide a comfortable environment for developing new code. The Torrent instruction set architecture is based on the MIPS-II instruction set, with extra coprocessor instructions added to access the vector unit functionality. This compatibility allows us to base our software environment on the GNU tools which already include support for MIPS based machines. We have ported the gee C/C++ compiler, modified the gdb symbolic debugger to debug TO programs remotely from the host, enhanced the gas assembler to understand the new vector instructions and to schedule code to avoid interlocks, and we also employ the GNU linker and other library management utilities. Currently, the only access to the vector unit we provide is either through library routines or directly via the scheduling assembler. We have developed an extensive set of optimized vector library routines including fixed-point matrix and vector operations, function approximation through linear interpolation, and IEEE single precision floating-point emulation. The majority of the routines are written in Torrent assembler, although a parallel set of functions have been written in ANSI C to allow program development and execution on workstations. Finally, there is a standard C library containing the usual utility, I/O and scalar math routines. After compilation and linking, a TO executable is run on the SPERT-II board by invoking a "server" program on the host. The server loads a small operating system "kernel" into TO memory followed by the TO executable. While the TO application runs, the server services I/O requests on behalf of the TO process. 4 Related Systems Several programmable digital neurocomputers have been constructed, most notably systems based on the CNAPS chip from Adaptive Solutions (Hammerstrom, 1990) and the SYNAPSE-I, based on the MA-16 chip from Siemens (Ramacher, 1991). The Adaptive Solutions CNAPS-I064 chip contains a SIMD array with 64 16-bit processing elements (PEs) per chip. Systems require an external microcode sequencer. The PEs have 16-bit datapaths with a single 32-bit accumulator, and are less flexible than the TO datapaths. This chip provides on-chip memory for 128K 16-bit weights, distributed among the individual PEs. Off-chip memory bandwidth is limited by an 8-bit port. In contrast, TO integrates an on-chip CPU that acts as controller, and provides fast access to a external memory equally accessible by all datapaths thereby increasing the range of applications that can be run efficiently. SPERT-n: A Vector Microprocessor System 623 Like SPERT-II, the SYNAPSE-l leverages commercial memory parts. It features an array of MA-16 chips connected to interleaved DRAM memory banks. The MA16 chips require extensive external circuitry, including 68040 CPUs with attached arithmetic pipelines, to execute computations not supported by the MA-16 itself. The SYNAPSE-l system is a complex and expensive multi-board design, containing several different control streams that must be carefully orchestrated to run an application. However, for some applications the MA-16 could potentially provide greater throughput than TO as the former's more specialized architecture permits more multiply-add units on each chip. 5 Mapping Backpropagation to TO One artificial neural network (ANN) training task that we have done is taken from a speaker-independent continuous speech recognition system. The ANN is a simple feed-forward multi-layer percept ron (MLP) with three layers. Typical MLPs have between 100-400 input units. The input layer is fully connected to a hidden layer of 100-4000 hidden units. The hidden layer is fully connected to an output layer that contains one output per phoneme, typically 56-61. The hidden units incorporate a standard sigmoid activation function. The output units compute a "soft-max" activation function. All training is "on-line", with the weight matrices updated after each pattern presentation. All of the compute-intensive sections can be readily vectorized on TO. Three operations are performed on the weight matrices: forward propagation, error back-propagation, and weight update. These operations are available as three standard linear algebra routines in the TO library: vector-matrix multiply, matrix-vector multiply, and scaled outer-product accumulation, respectively. TO can sustain one multiply-add per cycle in each of the 8 datapath slices, and can support this with one 16-bit memory access per cycle to each datapath slice provided that vector accesses have unit stride. The loops for the matrix operations are rearranged to perform only unit-stride memory accesses, and memory bandwidth requirements are further reduced by tiling matrix accesses and reusing operands from the vector registers whenever possible. There are a number of other operations required while handling input and output vectors and activation values. While these require only O(n) computation versus the O(n2) requirements of the matrix operations, they would present a significant overhead on smaller networks if not vectorized. The sigmoid activation function is implemented using a library piecewise-linear function approximation routine. The function approximation routine makes use of the vector indexed load operations to perform the table lookups. Although TO can only execute vector indexed operations at the rate of one element transfer per cycle, the table lookup routine can simultaneously perform all the arithmetic operations for index calculation and linear interpolation in the vector arithmetic units, achieving a rate of one 16-bit sigmoid result every 2 cycles. Similarly, a table based vector logadd routine is used to implement the soft-max function, also producing one result every 2 cycles. To simplify software porting, the MLP code uses standard IEEE single-precision floating-point for input and output values. Vector library routines convert formats to the internal fixed-point representation. These conversion routines operate at the rate of up to 1 conversion every 2 cycles. 624 J. WAWRZYNEK, K. ASANOVIC, B. KINGSBURY, J. BECK, D. JOHNSON, N. MORGAN 6 Performance Evaluation We chose two commercial RISC workstations against which to compare the performance of the SPERT-II system. The first is a SPARCstation-20/61 containing a single 60 MHz SuperSPARC+ processor with a peak performance of60 MFLOPS, 1 MB of second level cache, and 128 MB of DRAM main memory. The SPARCstation20/61 is representative of a current mid-range workstation. The second is an IBM RS/6000-590, containing the RIOS-2 chipset running at 72 MHz with a peak performance of 266 MFLOPS, 256 KB of primary cache, and 768 MB of DRAM main memory. The RS/6000 is representative of a current high-end workstation. The workstation version of the code performs all input and output and all computation using IEEE single precision floating-point arithmetic. The matrix and vector operations within the back prop algorithm have been extensively hand optimized, using manual loop unrolling together with register and cache blocking. The SPERT-II numbers are obtained for a single TO processor running at 40 MHz with 8 MB of SRAM main memory. The SPERT-II version of the application maintains the same interface, with input and output in IEEE single precision floatingpoint format, but performs all MLP computation using saturating fixed-point arithmetic with 16-bit weights, 16-bit activation values, and 32-bit intermediate results. The SPERT-II timings below include the time for conversion between floating-point and fixed-point for input and output. Figure 2 shows the performance of the three systems for a set of three-layer networks on both backpropagation training and forward propagation. For ease of presentation we use networks with the same number of units per layer. Table 1 presents performance results for two speech network architectures. The general trend we observe in these evaluations is that for small networks the three hardware systems exhibit similar performance, while for larger network sizes the SPERT-II system demonstrates a significant performance advantage. For large networks the SPERT-II system demonstrates roughly 20-30 times the performance of a SPARC20 workstation and 4-6 times the performance of the IBM RS/6000-590 workstation. Acknowledgements Thanks to Jerry Feldman for his contribution to the design of the SPERT-II system, Bertrand Irrisou for his work on the TO chip, John Hauser for Torrent libraries, and John Lazzaro for his advice on chip and system building. Primary support for this work was from the ONR, URI Grant N00014-92-J-1617 and ARPA contract number N0001493-C0249. Additional support was provided by the NSF and ICS!. SPERT-II: A Vector Microprocessor System Forward Pass 300.-------------------------~ <:I""'i-\\ S~<250 .. ......... . .. ... .. . ............. •.. •. .. ... ... •. . . .. ... .. .. .. fii200 Il. o ~150 "C CD CD c%loo 'BM RS/6000 Training 80 ..... .. ........ ......... .............. . . . .. . .. .... . St>Ef\i-\\ ~80 C/) Il. B :::!; ~40 al CD a. C/) 625 20 ·········································· ·'BM·RSisooo ..... . 50 . ••.......•...........•. •.... .... ....... " -------------------SPARC20/61 oL=~==~~~==~==~ L::======~====:L===S=PA~R~C~2~~6~1~ o o 200 400 600 800 1,000 o 200 400 600 800 1,000 Layer Size Layer Size Figure 2: Performance Evaluation Results (all layers the same size). Table 1: Performance Evaluation for Selected Net Sizes. net size IBM net type (in x hidden x out) SPERT-II SPARC20 RS/6000-590 Forward Pass (MCPS) small speech net 153 x 200 x 56 181 17.6 43.0 large speech net 342 x 4000 x 61 276 11.3 45.1 Training (MCUPS) small speech net 153 x 200 x 56 55.8 7.00 16.7 large speech net 342 x 4000 x 61 78.7 4.18 17.2 References Krste Asanovic and Nelson Morgan. Experimental Determination of Precision Requirements for Back-Propagation Training of Artificial Neural Networks. In Proc. 2nd Inti. Conf. on Microelectronics for Neural Networks, Munich, Oct. 1991. D. Hammerstrom. A VLSI architecture for High-Performance, Low-Cost, On-Chip Learning. In Proc. Intl. Joint Cant on Neural Networks, pages 11-537-543, 1990. G. Kane, and Heinrich, J . MIPS RISC Architecture. Prentice Hall, 1992. U. Ramacher, J. Beichter, W. Raab, J. Anlauf, N. Bruls, M. Hachmann, and M. Wesseling. Design of a 1st Generation Neurocomputer. In VLSI Design of Neural Networks. Kluwer Academic, 1991. J. Wawrzynek, K. Asanovic, and N. Morgan. The Design ofa Neuro-Microprocessor. IEEE Journal on Neural Networks, 4(3), 1993.
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Stochastic Hillclimbing as a Baseline Method for Evaluating Genetic Algorithms Ari Juels Department of Computer Science University of California at Berkeley· Martin Wattenberg Department of Mathematics University of California at Berkeleyt Abstract We investigate the effectiveness of stochastic hillclimbing as a baseline for evaluating the performance of genetic algorithms (GAs) as combinatorial function optimizers. In particular, we address two problems to which GAs have been applied in the literature: Koza's ll-multiplexer problem and the jobshop problem. We demonstrate that simple stochastic hillclimbing methods are able to achieve results comparable or superior to those obtained by the GAs designed to address these two problems. We further illustrate, in the case of the jobshop problem, how insights obtained in the formulation of a stochastic hillclimbing algorithm can lead to improvements in the encoding used by a GA. 1 Introduction Genetic algorithms (GAs) are a class of randomized optimization heuristics based loosely on the biological paradigm of natural selection. Among other proposed applications, they have been widely advocated in recent years as a general method for obtaining approximate solutions to hard combinatorial optimization problems using a minimum of information about the mathematical structure of these problems. By means of a general "evolutionary" strategy, GAs aim to maximize an objective or fitness function 1 : 5 --t R over a combinatorial space 5, i.e., to find some state s E 5 for which 1(s) is as large as possible. (The case in which 1 is to be minimized is clearly symmetrical.) For a detailed description of the algorithm see, for example, [7], which constitutes a standard text on the subject. In this paper, we investigate the effectiveness of the GA in comparison with that of stochastic hillclimbing (SH), a probabilistic variant of hillclimbing. As the term ·Supported in part by NSF Grant CCR-9505448. E-mail: juels@cs.berkeley.edu fE-mail: wattenbe@math.berkeley.edu Stochastic Hillclimbing as a Baseline Method for Evaluating Genetic Algorithms 431 "hillclimbing" suggests, if we view an optimization problem as a "landscape" in which each point corresponds to a solution s and the "height" of the point corresponds to the fitness of the solution, f(s), then hillclimbing aims to ascend to a peak by repeatedly moving to an adjacent state with a higher fitness. A number of researchers in the G A community have already addressed the issue of how various versions of hillclimbing on the space of bitstrings, {O, l}n, compare with GAs [1] [4] [9] [18] [15]. Our investigations in this paper differ in two important respects from these previous ones. First, we address more sophisticated problems than the majority of these studies, which make use of test functions developed for the purpose of exploring certain landscape characteristics. Second, we consider hillclimbing algorithms based on operators in some way "natural" to the combinatorial structures of the problems to which we are seeking solutions, very much as GA designers attempt to do. In one of the two problems in this paper, our SH algorithm employs an encoding exactly identical to that in the proposed GA. Consequently, the hillclimbing algorithms we consider operate on structures other than bitstrings. Constraints in space have required the omission of a great deal of material found in the full version of this paper. This material includes the treatment of two additional problems: the NP-complete Maximum Cut Problem [11] and an NP-complete problem known as the multiprocessor document allocation problem (MDAP). Also in the full version of this paper is a substantially more thorough exposition of the material presented here. The reader is encouraged to refer to [10], available on the World Wide Web at http://www.cs.berkeley.edu/,,-,juelsj. 2 Stochastic Hillclimbing The SH algorithm employed in this paper searches a discrete space S with the aim of finding a state whose fitness is as high (or as low) as possible. The algorithm does this by making successive improvements to some current state 0" E S. As is the case with genetic algorithms, the form of the states in S depends upon how the designer of the SH algorithm chooses to encode the solutions to the problems to be solved: as bitstrings, permutations, or in some other form. The local improvements effected by the SH algorithm are determined by the neighborhood structure and the fitness function f imposed on S in the design of the algorithm. We can consider the neighborhood structure as an undirected graph G on vertex set S. The algorithm attempts to improve its current state 0" by making a transition to one of the neighbors of 0" in G. In particular, the algorithm chooses a state T according to some suitable probability distribution on the neighbors of 0". If the fitness of T is as least as good as that of 0" then T becomes the new current state, otherwise 0" is retained. This process is then repeated 3 GP and J obshop 3.1 The Experiments In this section, we compare the performance of SH algorithms with that of GAs proposed for two problems: the jobshop problem and Koza's 11-multiplexer problem. We gauge the performance of the GA and SH algorithms according to the fitness of the best solution achieved after a fixed number of function evaluations, rather than the running time of the algorithms. This is because evaluation of the fitness function generally constitutes the most substantial portion of the execution time of the optimization algorithm, and accords with standard practice in the GA community. 432 A. JUELS, M. WATIENBERG 3.2 Genetic Programming "Genetic programming" (GP) is a method of enabling a genetic algorithm to search a potentially infinite space of computer programs, rather than a space of fixedlength solutions to a combinatorial optimization problem. These programs take the form of Lisp symbolic expressions, called S-expressions. The S-expressions in GP correspond to programs which a user seeks to adapt to perform some pre-specified task. Details on GP, an increasingly common GA application, and on the 11multiplexer problem which we address in this section, may be found, for example, in [13J [12J [14J. The boolean 11-multiplexer problem entails the generation of a program to perform the following task. A set of 11 distinct inputs is provided, with labels ao, aI, a2, do, dl , ... , d7, where a stands for "address" and d for "data". Each input takes the value 0 or 1. The task is to output the value dm , where m = ao + 2al + 4a2. In other words, for any 11-bit string, the input to the "address" variables is to be interpreted as an index to a specific "data" variable, which the program then yields as output. For example, on input al = 1, ao = a2 = 0, and d2 = l,do = dl = d3 = ... = d7 = 0, a correct program will output a '1', since the input to the 'a' variables specifies address 2, and variable d2 is given input 1. The GA Koza's GP involves the use of a GA to generate an S-expression corresponding to a correct ll-multiplexer program. An S-expression comprises a tree of LISP operators and operands, operands being the set of data to be processed the leaves of the tree and operators being the functions applied to these data and internally in the tree. The nature of the operators and operands will depend on the problem at hand, since different problems will involve different sets of inputs and will require different functions to be applied to these inputs. For the ll-multiplexer problem in particular, where the goal is to create a specific boolean function, the operands are the input bits ao, al, a2, do, dl , ... , d7, and the operators are AND, OR, NOT, and IF. These operators behave as expected: the subtree (AND al a2), for instance, yields the value al A a2. The subtree (IF al d4 d3) yields the value d4 if al = 0 and d3 if al = 1 (and thus can be regarded as a "3-multiplexer"). NOT and OR work similarly. An S-expression constitutes a tree of such operators, with operands at the leaves. Given an assignment to the operands, this tree is evaluated from bottom to top in the obvious way, yielding a 0 or 1 output at the root. Koza makes use of a "mating" operation in his GA which swaps subexpressions between two such S-expressions. The sub expressions to be swapped are chosen uniformly at random from the set of all subexpressions in the tree. For details on selection in this GA, see [13J. The fitness of an S-expression is computed by evaluating it on all 2048 possible inputs, and counting the number of correct outputs. Koza does not employ a mutation operator in his GA. The SH Algorithm For this problem, the initial state in the SH algorithm is an S-expression consisting of a single operand chosen uniformly at random from { ao, al, a2, do, ... , d7}. A transition in the search space involves the random replacement of an arbitrary node in the S-expression. In particular, to select a neighboring state, we chose a node uniformly at random from the current tree and replace it with a node selected randomly from the set of all possible operands and operators. With probability ~ the replacement node is drawn uniformly at random from the set of operands {ao, al, a2, do, ... , d7}, otherwise it is drawn uniformly at random from the set of operators, {AND, OR, NOT, IF}. In modifying the nodes of the S-expression in this way, we may change the number of inputs they require. By changing an AND node to a NOT node, for instance, we reduce the number of inputs taken by the node from 2 to 1. In order to accommodate such changes, we do Stochastic Hillclimbing as a Baseline Method for Evaluating Genetic Algorithms 433 the following. Where a replacement reduces the number of inputs taken by a node, we remove the required number of children from that node uniformly at random. Where, on the other hand, a replacement increases the number of inputs taken by a node, we add the required number of children chosen uniformly at random from the set of operands {ao, at, a2, do, ... , d7}. A similar, though somewhat more involved approach of this kind, with additional experimentation using simulated annealing, may be found in [17]. Experimental Results In the implementation described in [14], Koza performs experiments with a GA on a pool of 4000 expressions. He records the results of 54 runs. These results are listed in the table below. The average number of function evaluations required to obtain a correct program is not given in [14]. In [12], however, where Koza performs a series of 21 runs with a slightly different selection scheme, he finds that the average number of function evaluations required to find a correct S-expression is 46,667. In 100 runs of the SH algorithm, we found that the average time required to obtain a correct S-expression was 19,234.90 function evaluations, with a standard deviation of 5179.45. The minimum time to find a correct expression in these runs was 3733, and the maximum, 73,651. The average number of nodes in the correct Sexpression found by the SH algorithm was 88.14; the low was 42, the high, 242, and the standard deviation, 29.16. The following table compares the results presented in [14], indicated by the heading "GP", with those obtained using stochastic hillclimbing, indicated by "SH". We give the fraction of runs in which a correct program was found after a given number of function evaluations. (As this fraction was not provided for the 20000 iteration mark in [14], we omit the corresponding entry.) I Functionevaluations II GP SH 20000 61 % 40000 28 % 98 % 60000 78 % 99 % 80000 90 % 100% We observe that the performance of the SH is substantially better than that of the GA. It is interesting to note - perhaps partly in explanation of the SH algorithm's success on this problem - that the SH algorithm formulated here defines a neighborhood structure in which there are no strict local minima. Remarkably, this is true for any boolean formula. For details, as well as an elementary proof, see the full version of this paper [10]. 3.3 Jobshop Jobshop is a notoriously difficult NP-complete problem [6] that is hard to solve even for small instances. In this problem, a collection of J jobs are to be scheduled on M machines (or processors), each of which can process only one task at a time. Each job is a list of M tasks which must be performed in order. Each task must be performed on a specific machine, and no two tasks in a given job are assigned to the same machine. Every task has a fixed (integer) processing time. The problem is to schedule the jobs on the machines so that all jobs are completed in the shortest overall time. This time is referred to as the makespan. Three instances formulated in [16] constitute a standard benchmark for this problem: a 6 job, 6 machine instance, a 10 job, 10 machine instance, and a 20 job, 5 434 A. JUELS. M. WA TIENBERG machine instance. The 6x6 instance is now known to have an optimal makespan of 55. This is very easy to achieve. While the optimum value for the 10x10 problem is known to be 930, this is a difficult problem which remained unsolved for over 20 years [2]. A great deal of research has also been invested in the similarly challenging 20x5 problem, for which an optimal value of 1165 has been achieved, and a lower bound of 1164 [3]. A number of papers have considered the application of GAs to scheduling problems. We compare our results with those obtained in Fang et al. [5], one of the more recent of these articles. The GA Fang et al. encode a jobshop schedule in the form of a string of integers, to which their GA applies a conventional crossover operator. This string contains JM integers at, a2,' .. , aJM in the range 1..J. A circular list C of jobs, initialized to (1,2, ... , J) is maintained. For i = 1,2, ... , JM, the first uncompleted task in the (ai mod ICl)th job in C is scheduled in the earliest plausible timeslot. A plausible timeslot is one which comes after the last scheduled task in the current job, and which is at least as long as the processing time of the task to be scheduled. When a job is complete, it is removed from C. Fang et al. also develop a highly specialized GA for this problem in which they use a scheme of increasing mutation rates and a technique known as GVOT (Gene-Variance based Operator Targeting). For the details see [5]. The SH Algorithm In our SH algorithm for this problem, a schedule is encoded in the form of an ordering U1,U2, ... ,UJM of JM markers. These markers have colors associated with them: there are exactly M markers of each color of 1, ... , J. To construct a schedule, U is read from left to right. Whenever a marker with color k is encountered, the next uncompleted task in job k is scheduled in the earliest plausible timeslot. Since there are exactly M markers of each color, and since every job contains exactly M tasks, this decoding of U yields a complete schedule. Observe that since markers of the same color are interchangeable, many different ordering U will correspond to the same scheduling of tasks. To generate a neighbor of U in this algorithm, a marker Ui is selected uniformly at random and moved to a new position j chosen uniformly at random. To achieve this, it is necessary to shift the subsequence of markers between Ui and (J'j (including Uj) one position in the appropriate direction. Ifi < j, then Ui+1,Ui+2, ... ,(J'j are shifted one position to the left in u. If i > j, then (J'j, (J'j+l, ... ,Ui-1 are shifted one position to the right. (If i = j, then the generated neighbor is of course identical to u.) For an example, see the full version of this paper [10]. Fang et al. consider the makespan achieved after 300 iterations of their GVOTbased GA on a population of size 500. We compare this with an SH for which each experiment involves 150,000 iterations. In both cases therefore, a single execution of the algorithm involves a total of 150,000 function evaluations. Fang et al. present their average results over 10 trials, but do not indicate how they obtain their "best". We present the statistics resulting from 100 executions of the SH algorithm. IIIOXIO Jobshop GA I SH II 20x5 Jobshop II Mean 977 966.96 1215 1202.40 SD 13.15 12.92 High 997 1288 Low 949 938 1189 1173 Best Known 930 1165 Stochastic Hillclimbing as a Baseline Method for Evaluating Genetic Algorithms 435 As can be seen from the above table, the performance of the SH algorithm appears to be as good as or superior to that of the GA. 3.4 A New Jobshop GA In this section, we reconsider the jobshop problem in an attempt to formulate a new GA encoding. We use the same encoding as in the SH algorithm described above: (7 is an ordering (71, (72, •.. , (7 J M of the J M markers, which can be used to construct a schedule as before. We treated markers of the same color as effectively equivalent in the SH algorithm. Now, however, the label of a marker (a unique integer in {I, . . . ,J M}) will playa role. The basic step in the crossover operator for this GA as applied to a pair (7, T) of orderings is as follows. A label i is chosen uniformly at random from the set {I, 2, ... , J M}. In (7, the marker with label i is moved to the position occupied by i in T; conversely, the marker with label i in T is moved to the position occupied by that marker in (7. In both cases, the necessary shifting is performed as before. Hence the idea is to move a single marker in (7 (and in T) to a new position as in the SH algorithm; instead of moving the marker to a random position, though, we move it to the position occupied by that marker in T (and (7, respectively). The full crossover operator picks two labels j ~ k uniformly at random from {I, 2, . .. , J M}, and performs this basic operation first for label j, then j + 1, and so forth, through k. The mutation operator in our GA performs exactly the same operation as that used to generate a neighbor in the SH algorithm. A marker (7 i is chosen uniformly at random and moved to a new position j, chosen uniformly at random. The usual shifting operation is then performed. Observe how closely the crossover and mutation operators in this GA for the jobshop problem are based on those in the corresponding SH algorithm. Our GA includes, in order, the following phases: evaluation, elitist replacement, selection, crossover, and mutation. In the evaluation phase, the fitnesses of all members of the population are computed. Elitist replacement substitutes the fittest permutation from the evaluation phase of the previous iteration for the least fit permutation in the current population (except, of course, in the first iteration, in which there is no replacement). Because of its simplicity and its effectiveness in practice, we chose to use binary stochastic tournament selection (see [8] for details). The crossover step in our GA selects f pairs uniformly at random without replacement from the population and applies the mating operator to each of these pairs independently with probability 0.6. The number of mutations performed on a given permutation in a single iteration is binomial with parameter p = *. The population in our GA is initialized by selecting every individual uniformly at random from Sn. We execute this GA for 300 iterations on a population of size 500. Results of 100 experiments performed with this GA are indicated in the following table by "new GA". For comparison, we again give the results obtained by the GA of Fang et al. and the SH algorithm described in this paper. II IOxlO Jobshop 20x5 Jobshop II new GA GA SH new GA GA SH II Mean 956.22 977 965.64 1193.21 1215 1204.89 SD 8.69 10.56 7.38 12.92 High 976 996 1211 1241 Low 937 949 949 1174 1189 1183 Best Known 930 1165 436 A. JUELS, M. WAllENBERG 4 Conclusion As black-box algorithms, GAs are principally of interest in solving problems whose combinatorial structure is not understood well enough for more direct, problemspecific techniques to be applied. As we have seen in regard to the two problems presented in this paper, stochastic hill climbing can offer a useful gauge of the performance of the GA. In some cases it shows that a GA-based approach may not be competitive with simpler methods; at others it offers insight into possible design decisions for the G A such as the choice of encoding and the formulation of mating and mutation operators. In light of the results presented in this paper, we hope that designers of black-box algorithms will be encouraged to experiment with stochastic hillclimbing in the initial stages of the development of their algorithms. References [1] D. Ackley. A Connectionist Machine for Genetic Hillclimbing. Kluwer Academic Publishers, 1987. [2] D. Applegate and W. Cook. A computational study of the job-shop problem. ORSA Journal of Computing, 3(2), 1991. [3] J. Carlier and E. Pinson. An algorithm for solving the jobshop problem. Mngmnt. Sci., 35:(2):164-176, 1989. [4] L. Davis. Bit-climbing, representational bias, and test suite design. In Belew and Booker, editors, ICGA-4, pages 18-23, 1991. [5] H. Fang, P. Ross, and D. Corne. A promising GA approach to job-shop scheduling, rescheduling, and open-shop scheduling problems. In Forrest, editor, ICGA-5, 1993. [6] M. Garey and D. Johnson. Computers and Intractability. W .H. Freeman and Co., 1979. [7] D. Goldberg. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison Wesley, 1989. [8] D. Goldberg and K. Deb. A comparative analysis of selection schemes used in GAs. In FOGA-2, pages 69-93, 1991. [9] K. De Jong. An Analysis of the Behavior of a Class of Genetic Adaptive Systems. PhD thesis, University of Michigan, 1975. [10] A. Juels and M. Wattenberg. Stochastic hillclimbing as a baseline method for evaluating genetic algorithms. Technical Report CSD-94-834, UC Berkeley, CS Division, 1994. [11] S. Khuri, T. Back, and J. Heitk6tter. An evolutionary approach to combinatorial optimization problems. In Procs. of CSC 1994, 1994. [12] J. Koza. FOGA, chapter A Hierarchical Approach to Learning the Boolean Multiplexer Function, pages 171-192. 1991. [13] J. Koza. Genetic Programming. MIT Press, Cambridge, MA, 1991. [14] J. Koza. The GP paradigm: Breeding computer programs. In Branko Soucek and the IRIS Group, editors, Dynamic, Genetic, and Chaotic Prog., pages 203-221. John Wiley and Sons, Inc., 1992. [15] M. Mitchell, J. Holland, and S. Forrest. When will a GA outperform hill-climbing? In J.D. Cowen, G. Tesauro, and J. Alspector, editors, Advances in Neural Inf. Processing Systems 6, 1994. [16] J. Muth and G. Thompson. Industrial Scheduling. Prentice Hall, 1963. [17] U. O'Reilly and F. Oppacher. Program search with a hierarchical variable length representation: Genetic programing, simulated annealing and hill climbing. In PPSN3, 1994. [18] S. Wilson. GA-easy does not imply steepest-ascent optimizable. In Belew and Booker, editors, ICGA-4, pages 85-89, 1991.
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Using the Future to "Sort Out" the Present: Rankprop and Multitask Learning for Medical Risk Evaluation Rich Caruana, Shumeet Baluja, and Tom Mitchell School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213 (caruana, baluja, mitchell)@cs.cmu.edu Abstract A patient visits the doctor; the doctor reviews the patient's history, asks questions, makes basic measurements (blood pressure, ... ), and prescribes tests or treatment. The prescribed course of action is based on an assessment of patient risk-patients at higher risk are given more and faster attention. It is also sequential- it is too expensive to immediately order all tests which might later be of value. This paper presents two methods that together improve the accuracy of backprop nets on a pneumonia risk assessment problem by 10-50%. Rankprop improves on backpropagation with sum of squares error in ranking patients by risk. Multitask learning takes advantage of future lab tests available in the training set, but not available in practice when predictions must be made. Both methods are broadly applicable. 1 Background There are 3,000,000 cases of pneumonia each year in the U.S., 900,000 of which are admitted to the hospital for treatment and testing. Most pneumonia patients recover given appropriate treatment, and many can be treated effectively without hospitalization. Nonetheless, pneumonia is serious: 100,000 of those hospitalized for pneumonia die from it, and many more are at elevated risk if not hospitalized. 1.1 The Problem A primary goal of medical decision making is to accurately, swiftly, and economically identify patients at high risk from diseases like pneumonia so they may be hospitalized to receive aggressive testing and treatment; patients at low risk may be more comfortably, safely, and economically treated at home. Note that the diagno960 R. CARUANA, S. BALUJA, T. MITCHELL sis of pneumonia has already been made; the goal is not to determine the illness, but how much risk the illness poses to the patient. Some of the most useful tests for doing this require hospitalization and will be available only if preliminary assessment indicates it is warranted. Low risk patients can safely be treated as outpatients and can often be identified using measurements made prior to admission. The problem considered in this paper is to learn to rank pneumonia patients according to their probability of mortality. We present two learning methods that combined outperform standard backpropagation by 10-50% in identifying groups of patients with least mortality risk. These methods are applicable to domains where the goal is to rank instances according to a probability function and where useful attributes do not become available until after the prediction must be made. In addition to medical decision making, this class includes problems as diverse as investment analysis in financial markets and autonomous vehicle navigation. 1.2 The Pneumonia Database The Medis Pneumonia Database [6] contains 14,199 pneumonia cases collected from 78 hospitals in 1989. Each patient in the database was diagnosed with pneumonia and hospitalized. 65 measurements are available for most patients. These include 30 basic measurements typically acquired prior to hospitalization such as age, sex, and pulse, and 35 lab results such as blood counts or gases not available until after hospitalization. The database indicates how long each patient was hospitalized and whether the patient lived or died. 1,542 (10.9%) of the patients died. 1.3 The Performance Criterion The Medis database indicates which patients lived or died. The most useful decision aid for this problem would predict which patients will live or die. But this is too difficult. In practice, the best that can be achieved is to estimate a probability of death (POD) from the observed symptoms. In fact, it is sufficient to learn to rank patients by POD so lower risk patients can be discriminated from higher risk patients. The patients at least risk may then be considered for outpatient care. The performance criterion used by others working with the Medis database [4] is the accuracy with which one can select a prespecified fraction of the patient population that do not die. For example, given a population of 10,000 patients, find the 20% of this population at least risk. To do this we learn a risk model and a threshold for this model that allows 20% of the population (2000 patients) to fall below it. If 30 of the 2000 patients below this threshold died, the error rate is 30/2000 = 0.015. We say that the error rate for FOP 0.20 is 0.015 for this model ("FOP" stands for fraction of population). In this paper we consider FOPs 0.1, 0.2, 0.3, 0.4, and 0.5. Our goal is to learn models and model thresholds, such that the error rate at each FOP is minimized. Models with acceptably low error rates might then be employed to help determine which patients do not require hospitalization. 2 Methodology The Medis database is unusually large, with over 14K training patterns. Because we are interested in developing methods that will be effective in other domains where databases of this size are not available, we perform our experiments using small training sets randomly drawn from the 14K patterns and use the remaining patterns as test sets. For each method we run ten trials. For each trial we randomly sample 2K patterns from the 14K pool for training. The 2K training sample is further split into a 1K backprop set used to train the net and a 1K halting set used to determine Rankprop and Multitask Learning for Medical Risk Evaluation 961 when to halt training.! Once the network is trained, we run the 1K halt set through the model again to find the threshold that passes 10%,20%,30%,40%, and 50% of the halt set. The performance ofthe model is evaluated on the 12K unused patterns by determining how many of the cases that fall below threshold in this test set die. This is the error rate for that model at that FOP. 3 The Traditional Approach: SSE on 0/1 Targets Sections 3-5 present three neural net approaches to pneumonia risk prediction. This section presents the standard approach: using backpropagation on sum of squares errors (SSE) with 0=lives/1=dies to predict mortality. This works well if early stopping is used to prevent overfitting. Section 4 presents rank prop (SSE on ranks instead of 0/1 targets). Rankprop, which learns to rank patients by risk instead of directly predicting mortality, works better. Section 5 uses multitask learning (MTL) to benefit from tests in the database that in practice will not be available until after deciding to admit the patient. Rankprop with MTL works even better. The straightforward approach to this problem is to use backprop to train a net to learn to predict which patients live or die, and then use the real-valued predictions of this net to sort patients by risk. This net has 30 inputs, 1 for each of the observed patient measurements, a hidden layer with 8 units2 , and a single output trained with O=lived, 1=died.3 Given an infinite training set, a net trained this way should learn to predict the probability of death for each patient, not which patients live or die. In the real world, however, where we rarely have an infinite number of training cases, a net will overtrain and begin to learn a very nonlinear function that outputs values near 0/1 for cases in the training set, but which does not generalize well. In this domain it is critical to use early stopping to halt training before this happens. Table 1 shows the error rates of nets trained with SSE on 0/1 targets for the five FOPs. Each entry is the mean of ten trials. The first entry in the table indicates that on average, in the 10% of the test population predicted by the nets to be at least risk, 1.4% died. We do not know the best achievable error rates for this data. Table 1: Error Rates of SSE on 0/1 Targets FOP Error Rate 4 Using Rankprop to Rank Cases by Risk Because the goal is to find the fraction of the population least likely to die, it is sufficient just to learn to rank patients by risk. Rankprop learns to rank patients without learning to predict mortality. "Rankprop" is short for "backpropagation using sum of squares errors on estimated ranks". The basic idea is to sort the training set using the target values, scale the ranks from this sort (we scale uniformly to [0.25,0.75] with sigmoid output units), and use the scaled ranks as target values for standard backprop with SSE instead of the 0/1 values in the database. lperformance at different FOPs sometimes peaks at different epochs. We halt training separately for each FOP in all the experiments to insure this does not confound results. 2To make comparisons between methods fair, we first found hidden layer sizes and learning parameters that performed well for each method. 3Different representations such as 0.15/0.85 and different error metrics such as cross entropy did not perform better than SSE on 0/1 targets. 962 R. CARUANA, S. BALUJA, T. MITCHELL Ideally, we'd rank the training set by the true probabilities of death. Unfortunately, all we know is which patients lived or died. In the Medis database, 89% of the target values are O's and 11% are l's. There are many possible sorts consistent with these values. Which sort should backprop try to fit? It is the large number of possible sorts of the training set that makes backpropagating ranks challenging. Rankprop solves this problem by using the net model as it is being learned to order the training set when target values are tied. In this database, where there are many ties because there are only two target values, finding a proper ranking of the training set is a serious problem. Rankprop learns to adjust the target ranks of the training set at the same time it is learning to predict ranks from that training set. How does rankprop do this? Rankprop alternates between rank passes and backprop passes. On the rank pass it records the output of the net for each training pattern. It then sorts the training patterns using the target values (0 or 1 in the Medis database), but using the network's predictions for each pattern as a secondary sort key to break ties. The basic idea is to find the legal rank of the target values (0 or 1) maximally consistent with the ranks the current model predicts. This closest match ranking of the target values is then used to define the target ranks used on the next backprop pass through the training set. Rankprop's pseudo code is: foreach epoch do { foreach pattern do { network_output[pattern] = forward_pass(pattern)} target_rank = sort_and_scale_patterns(target_value, network_output) foreach pattern do { backprop(target_rank[pattern] - network_output[pattern])}} where "sorkand..scale_patterns" sorts and ranks the training patterns using the sort keys specified in its arguments, the second being used to break ties in the first. Table 2 shows the mean rankprop performance using nets with 8 hidden units. The bottom row shows improvements over SSE on 0/1 targets. All differences are statistically significant. See Section 7.1 for discussion of why rank prop works better. Table 2: Error Rates of Rankprop and Improvement Over Standard Backprop FOP Error Rate % Change 5 Learning From the Future with Multitask Learning The Medis database contains results from 36 lab tests that will be available only after patients are hospitalized. Unfortunately, these results will not be available when the model is used because the patients will not yet have been admitted. Multitask learning (MTL) improves generalization by having a learner simultaneously learn sets of related tasks with a shared representation; what is learned for each task might benefit other tasks. In this application, we use MTL to benefit from the future lab results. The extra lab values are used as extra backprop outputs as shown in Figure 1. The extra outputs bias the shared hidden layer towards representations that better capture important features of the domain. See [2][3][9] for details about MTL and [1] for other ways of using extra outputs to bias learning. The MTL net has 64 hidden units. Table 3 shows the mean performance of ten runs of MTL with rankprop. The bottom row shows the improvement over rankprop Rankprop and Multitask Learning for Medical Risk Evaluation RANKPROP OUTPUT ~--~ Mortality Hematocnt While Blood Pn t.a.<i1ilUm -- FUT\JRE LABS Rank Cell ("ounl 1 1 1 1 ~ ~~o~ OUTPUT LAYER SHAREDHIDOEN LAYER INPUT LAYER INPUTS Figure 1: Using Future Lab Results as Extra Outputs To Bias Learning 963 alone. Although MTL lowers error at each FOP, only the differences at FOP = 0.3, 0.4, and 0.5 are statistically significant with ten trials. Feature nets [7], a competing approach that trains nets to predict the missing future labs and uses the predictions as extra net inputs does T}ot yield benefits comparable to MTL on this problem. Table 3: Error Rates of Rankprop+MTL and Improvement Over Rankprop Alone FOP Error Rate % Change 6 Comparison of Results Table 4 compares the performance of backprop using SSE on 0/1 targets with the combination of rankprop and multitask learning. On average, Rankprop+MTL reduces error more than 25%. This improvement is not easy to achieve-experiments with other learning methods such as Bayes Nets, Hierarchical Mixtures of Experts, and K-Nearest Neighbor (run not by us, but by experts in their use) indicate SSE on 0/1 targets is an excellent performer on this domain[4]. Table 4: Comparison Between SSE on 0/1 Targets and Rankprop+MTL FOP 0.1 0.2 0.3 0.4 0.5 SSE on 0/1 .0140 .0190 .0252 .0340 .0421 Rankprop+ MTL .0074 .0127 .0197 .0269 .0364 % Change -47.1% I -33.2% I -21.8% I -20.9% I -13.5% 7 Discussion 7.1 Why Does Rankprop Work? We are given data from a target function f (x). Suppose the goal is not to learn a model of f(x), but to learn to sort patterns by f(x). Must we learn a model of f(x) and use its predictions for sorting? No. It suffices to learn a function g( x) such that for all Xl ,X2, [g(xd::; g(X2)]- [J(xd::; f(X2)]. There can be many such functions g(x) for a given f(x), and some of these may be easier to learn than f(x). . 964 R. CARUANA, S. BALUJA, T. MITCHELL Consider the probability function in Figure 2.1 that assigns to each x the probability p = f(x) that the outcome is 1; with probability 1 - p the outcome is O. Figure 2.2 shows a training set sampled from this distribution. Where the probability is low, there are many O's. Where the probability is high, there are many l 's. Where the probability is near 0.5, there are O's and 1 'so This region causes problems for backprop using SSE on 0/1 targets: similar inputs are mapped to dissimilar targets. .8 i 08 I" 111111 111111 I 000 0 011001010110001101111' 2 f •• i :: llllllllllll t 0 2 0... 0.6 08 02 0 4 06 08 0 2 04 06 0 8 Figure 2: SSE on 0/1 Targets and on Ranks for a Simple Probability Function Backprop learns a very nonlinear function if trained on Figure 2.2. This is unfortunate: Figure 2.1 is smooth and maps similar inputs to similar outputs. If the goal is to learn to rank the data, we can learn a simpler, less nonlinear function instead. There exists a ranking of the training data such that if the ranks are used as backprop target values, the resulting function is less nonlinear than the original target function. Figure 2.3 shows these target rank values. Similar input patterns have more similar rank target values than the original target values. Rankprop tries to learn simple functions that directly support ranking. One difficulty with this is that rankprop must learn a ranking of the training data while also training the model to predict ranks. We do not yet know under what conditions this parallel search will converge. We conjecture that when rankprop does converge, it will often be to simpler models than it would have learned from the original target values (0/1 in Medis), and that these simpler models will often generalize better. 7.2 Other Applications of Rankprop and Learning From the Future Rankprop is applicable wherever a relative assessment is more useful or more learnable than an absolute one. One application is domains where quantitative measurements are not available, but relative ones are[8]. For example, a game player might not be able to evaluate moves quantitatively , but might excel at relative move evaluation[10]. Another application is where the goal is to learn to order data drawn from a probability distribution, as in medical risk prediction. But it can also be applied wherever the goal is to order data. For example, in information filtering it is usually important to present more useful information to the user first, not to predict how important each is[5]. MTL is a general method for using related tasks. Here the extra MTL tasks are future measurements. Future measurements are available in many offline learning problems where there is opportunity to collect the measurements for the training set. For example, a robot or autonomous vehicle can more accurately measure the size, location, and identity of objects when it passes near them-road stripes can be detected reliably as a vehicle passes alongside them, but detecting them far ahead of a vehicle is hard. Since driving brings future road into the car's present, stripes can be measured accurately when passed and used as extra features in the training set. They can't be used as inputs for learning to drive because they will not be available until too late when driving. As MTL outputs, though, they provide information Rankprop and Multitask Learning for Medical Risk Evaluation 965 that improves learning without requiring they be available at run time[2] . 8 Summary This paper presents two methods that can improve generalization on a broad class of problems. This class includes identifying low risk pneumonia patients. The first method, rankprop, tries to learn simple models that support ranking future cases while simultaneously learning to rank the training set. The second, multitask learning, uses lab tests available only during training, as additional target values to bias learning towards a more predictive hidden layer. Experiments using a database of pneumonia patients indicate that together these methods outperform standard backpropagation by 10-50%. Rankprop and MTL are applicable to a large class of problems in which the goal is to learn a relative ranking over the instance space, and where the training data includes features that will not be available at run time. Such problems include identifying higher-risk medical patients as early as possible, identifying lower-risk financial investments, and visual analysis of scenes that become easier to analyze as they are approached in the future. Acknowledgements We thank Greg Cooper, Michael Fine, and other members of the Pitt/CMU Cost-Effective Health Care group for help with the Medis Database. This work was supported by ARPA grant F33615-93-1-1330, NSF grant BES-9315428, Agency for Health Care Policy and Research grant HS06468, and an NSF Graduate Student Fellowship (Baluja). References [1] Y.S. Abu-Mostafa, "Learning From Hints in Neural Networks," Journal of Complexity 6:2, pp. 192-198, 1989. [2] R. Caruana, "Learning Many Related Tasks at the Same Time With Backpropagation," Advances in Neural Information Processing Systems 7, pp. 656-664, 1995. [3] R. Caruana, "Multitask Learning: A Knowledge-Based Source of Inductive Bias," Proceedings of the 10th International Conference on Machine Learning, pp. 41-48, 1993. [4] G. Cooper, et al., "An Evaluation of Machine Learning Methods for Predicting Pneumonia Mortality," submitted to AI in Medicine, 1995. [5] K. Lang, "NewsWeeder: Learning to Filter News," Proceedings of the 12th International Conference on Machine Learning, pp. 331-339, 1995. [6] M. Fine, D. Singer, B. Hanusa, J. Lave, and W. Kapoor, "Validation of a Pneumonia Prognostic Index Using the MedisGroups Comparative Hospital Database," American Journal of Medicine, 94 1993. [7] I. Davis and A. Stentz, "Sensor Fusion For Autonomous Outdoor Navigation Using Neural Networks," Proceedings of IEEE 's Intelligent Robots and Systems Conference, 1995. [8] G.T. Hsu, and R. Simmons, "Learning Footfall Evaluation for a Walking Robot," Proceedings of the 8th International Conference on Machine Learning, pp. 303-307, 1991. [9] S.C. Suddarth and A.D.C. Holden, "Symbolic-neural Systems and the Use of Hints for Developing Complex Systems," International Journal of Man-Machine Studies 35:3, pp. 291-311, 1991. [10] P. Utgoff and S. Saxena, "Learning a Preference Predicate," Proceedings of the 4th International Conference on Machine Learning, pp. 115-121, 1987.
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Exponentially many local minima for single neurons Peter Auer Mark Herbster Manfred K. Warmuth Department of Computer Science Santa Cruz, California {pauer,mark,manfred} @cs.ucsc.edu Abstract We show that for a single neuron with the logistic function as the transfer function the number of local minima of the error function based on the square loss can grow exponentially in the dimension. 1 INTRODUCTION Consider a single artificial neuron with d inputs. The neuron has d weights w E Rd. The output of the neuron for an input pattern x E Rd is y = ¢(x· w), where ¢ : R -+ R is a transfer function. For a given sequence of training examples ((Xt, Yt))I<t<m, each consisting of a pattern Xt E R d and a desired output Yt E R, the goal of the training phase for neural networks consists of minimizing the error function with respect to the weight vector w E Rd. This function is the sum of the losses between outputs of the neuron and the desired outputs summed over all training examples. In notation, the error function is m E(w) = L L(Yt, ¢(Xt . w)) , t=1 where L : R x R -+ [0,00) is the loss function. A common example of a transfer function is the logistic function logistic( z) = I+!-' which has the bounded range (0, 1). In contrast, the identity function id(z) = z has unbounded range. One of the most common loss functions is the square loss L(y, y) = (y - Y)2. Other examples are the absolute loss Iy - yl and the entropic1oss yin? + (1 - y) In ::::l We show that for the square loss and the logistic function the error function of a single neuron for n training examples may have L n / dJ d local minima. More generally, this holds for any loss and transfer function for which the composition of the loss function with the transfer function (in notation L(y, ¢(x . w)) is continuous and has bounded range. This Exponentially Many Local Minima for Single Neurons 317 Figure 1: Error Function with 25 Local Minima (16 Visible), Generated by 10 TwoDimensional Examples. proves that for any transfer function with bounded range exponentially many local minima can occur when the loss function is the square loss. The sequences of examples that we use in our proofs have the property that they are nonrealizable in the sense that there is no weight vector W E R d for which the error function is zero, i.e. the neuron cannot produce the desired output for all examples. We show with some minimal assumptions on the loss and transfer functions that for a single neuron there can be no local minima besides the global minimum if the examples are realizable. If the transfer function is the logistic function then it has often been suggested in the literature to use the entropic loss in artificial neural networks in place of the square loss [BW88, WD88, SLF88, Wat92]. In that case the error function of a single neuron is convex and thus has only one minimum even in the non-realizable case. We generalize this observation by defining a matching loss for any differentiable increasing transfer functions ¢: ,p-l(y) L</>(y, f)) = 1 (¢(z) - y) dz . </>-l(y) The loss is the area depicted in Figure 2a. If ¢ is the identity function then L</> is the square loss likewise if ¢ is the logistic function then L</> is the entropic loss. For the matching loss the gradient descent update for minimizing the error function for a sequence of examples is simply Wnew := Wold -1] (f)¢(Xt . Wold) - Yt)Xt) , t=1 where 1] is a positive learning rate. Also the second derivatives are easy to calculate for this general setting: L4>(Y~:v~<:Wt;W)) = ¢'(Xt . W)Xt,iXt,j. Thus, if Ht(w) is the Hessian of L</>(Yt, ¢(Xt . w)) with respect to W then vT Ht(w)v = ¢'(Xt . w)(v . Xt)2. Thus 318 0.8 wO.I 0.4 0.2 .-1 (9) = w · x (a) ... P. AUER. M. HERBSTER, M. K. WARMUTH - 2 o ... (b) Figure 2: (a) The Matching Loss Function L</>. (b) The Square Loss becomes Saturated, the Entropic Loss does not. H t is positive semi-definite for any increasing differentiable transfer function. Clearly L:~I Ht(w) is the Hessian of the error function E(w) for a sequence of m examples and it is also positive semi-definite. It follows that for any differentiable increasing transfer function the error function with respect to the matching loss is always convex. We show that in the case of one neuron the logistic function paired with the square loss can lead to exponentially many minima. It is open whether the number of local minima grows exponentially for some natural data. However there is another problem with the pairing of the logistic and the square loss that makes it hard to optimize the error function with gradient based methods. This is the problem of flat regions. Consider one example (x, y) consisting of a pattern x (such that x is not equal to the all zero vector) and the desired output y. Then the square loss (Iogistic(x . w) - y)2, for y E [0, I] and w E R d , turns flat as a function of w when f) = logistic( x . w) approaches zero or one (for example see Figure 2b where d = I and y = 0). It is easy to see that for all bounded transfer functions with a finite number of minima and corresponding bounded loss functions, the same phenomenon occurs. In other words, the composition L(y, ¢(x . w» of the square loss with any bounded transfer function ¢ which has a finite number of extrema turns flat as Ix . w I becomes large. Similarly, for multiple examples the error function E( w) as defined above becomes flat. In flat regions the gradients with respect to the weight vector w are small, and thus gradient-based updates of the weight vector may have a hard time moving the weight vector out of these flat regions. This phenomenon can easily be observed in practice and is sometimes called "saturation" [Hay94]. In contrast, if the logistic function is paired with the entropic loss (see Figure 2b), then the error function turns flat only at the global minimum. The same holds for any increasing differentiable transfer function and its matching loss function. A number of previous papers discussed conditions necessary and sufficient for mUltiple local minima of the error function of single neurons or otherwise small networks [WD88, SS89, BRS89, Blu89, SS91, GT92]. This previous work only discusses the occurrence of multiple local minima whereas in this paper we show that the number of such minima can grow exponentially with the dimension. Also the previous work has mainly been limited to the demonstration of local minima in networks or neurons that have used the hyperbolic tangent or logistic function with the square loss. Here we show that exponentially many minima occur whenever the composition of the loss function with the transfer function is continuous and bounded. The paper is outlined as follows. After some preliminaries in the next section, we gi ve formal Exponentially Many Local Minima for Single Neurons 04$ O. 036 03 026 02 0.t5 0.1 0.06 0 -2 11 0.9 0.1 07 WOI as -- --- ------------ __ ,O. , , , ' \ / ' .. " , , ' 03 02 01 L......L-~~-~~~~-'--~-~ -1 -8-e-~-2 0 log .. (a) (b) Figure 3: (a) Error Function for the Logistic Transfer Function and the Square Loss with Examples ((10, .55), (.7, .25») (b) Sets of Minima can be Combined. 319 statements and proofs of the results mentioned above in Section 3. At first (Section 3.1) we show that n one-dimensional examples might result in n local minima of the error function (see e.g. Figure 3a for the error function of two one-dimensional examples). From the local minima in one dimension it follows easily that n d-dimensional examples might result in L n/ dJ d local minima of the error function (see Figure 1 and discussion in Section 3.2). We then consider neurons with a bias (Section 4), i.e. we add an additional input that is clamped to one. The error function for a sequence of examples S = ((Xt, Yt»)I<t<m is now m Es(B, w) = I: L(Yt, r/>(B + WXt», t=1 where B denotes the bias, i.e. the weight of the input that is clamped to one. We can prove that the error function might have L n/2dJ d local minima if loss and transfer function are symmetric. This holds for example for the square loss and the logistic transfer function. The proofs are omitted due to space constraints. They are given in the full paper [AHW96], together with additional results for general loss and transfer functions. Finally we show in Section 5 that with minimal assumptions on transfer and loss functions that there is only one minimum of the error function if the sequence of examples is realizable by the neuron. The essence of the proofs is quite simple. At first observe that ifloss and transfer function are bounded and the domain is unbounded, then there exist areas of saturation where the error function is essentially flat. Furthermore the error function is "additive" i.e. the error function produced by examples in SUS' is simply the error function produced by the examples in S added to the error function produced by the examples in S', Esusl = Es + ESI. Hence the local minima of Es remain local minima of Esus 1 if they fall into an area of saturation of Es. Similarly, the local minima of ESI remain local minima of Esusl as well (see Figure 3b). In this way sets of local minima can be combined. 2 PRELIMINARIES ' We introduce the notion of minimum-containing set which will prove useful for counting the minima of the error function. 320 P. AUER, M. HERBSTER, M. K. WARMUTH Definition 2.1 Let f : Rd_R be a continuous function. Then an open and bounded set U E Rd is called a minimum-containing set for f if for each w on the boundary of U there is a w'" E U such that f(w"') < f(w). Obviously any minimum-containing set contains a local minimum of the respecti ve function. Furthermore each of n disjoint minimum-containing sets contains a distinct local minimum. Thus it is sufficient to find n disjoint minimum-containing sets in order to show that a function has at least n local minima. 3 MINIMA FOR NEURONS WITHOUT BIAS We will consider transfer functions ¢ and loss functions L which have the following property: (PI): The transfer function ¢ : R-R is non-constant. The loss function L : ¢(R) x ¢(R)-[O, 00) has the property that L(y, y) = 0 and L(y, f)) > 0 for all y f. f) E ¢(R). FinallythefunctionL(·,¢(·)): ¢(R) x R-[O,oo) is continuous and bounded. 3.1 ONE MINIMUM PER EXAMPLE IN ONE DIMENSION Theorem 3.1 Let ¢ and L satisfy ( PI). Then for all n ~ I there is a sequence of n examples S = (XI, y), ... , (xn, y)), Xt E R, y E ¢(R), such that Es(w) has n distinct local minima. Since L(y, ¢( w)) is continuous and non-constant there are w- , w"', w+ E R such that the values ¢( w-), ¢( w"'), ¢( w+) are all distinct. Furthermore we can assume without loss of generality that 0 < w- < w'" < w+. Now set y = ¢(w"'). If the error function L(y, ¢(w)) has infinitely many local minima then Theorem 3.1 follows immediately, e.g. by setting XI = ... = Xn = 1. If L(y, ¢(w)) has only finitely many minima then limw ..... oo L(y, ¢(w)) = L(y, ¢(oo)) exists since L(y, ¢(w)) is bounded and continuous. We use this fact in the following lemma. It states that we get a new minimum-containing set by adding an example in the area of saturation of the error function. Lemma 3.2 Assume that limw ..... oo L(y, ¢( w)) exists. Let S = (XI, YI), ... , (xn, Yn)) be a sequence of examples and 0 < WI < wi < wt < ... < w;; < w~ < w~ such that Es(wt ) > Es(wn and Es(wn < Es(wt) for t = 1, ... , n. Let S' = (xo, y} (XI, Yd, ... , (xn, Yn)) where Xo is sufficiently large. Furthermoreletwo = w'" /xo and Wo = w± /xo (where w-, w"', w+, Y = ¢(w"') are as above). Then 0 < we; < Wo < wt < WI < wi < wt < ... < w;; < w~ < w~ and Proof. We have to show that for all Xo sufficiently large condition (l) is satisfied, i.e. that We get lim ESI(WO) = L(y, ¢(w"')) + lim Es(w'" /xo) = L(y, ¢(w"')) + Es(O), ~ ..... oo ~-oo recalling that Wo = w'" /xo and S' = S u (xo, y). Analogously lim ESI(w~) = L(y,¢(w±)) + Es(O). x 0"'" 00 (2) Exponentially Many Local Minima for Single Neurons 321 Thus equation (2) holds for t = 0. For t = 1, ... , n we get lim ESI(w;) = lim L(y, ¢(w;xo)) + Es(wn = L(y, ¢(oo)) + Es(wn :1:0-+00 :1:0-+00 and Since Es (w;) < Es (w;) for t = 1, ... , n, the lemma follows. o Proof of Theorem 3.1. The theorem follows by induction from Lemma 3.2 since each interval ( wi, wi) is a minimum-containing set for the error function. 0 Remark. Though the proof requires the magnitude of the examples to be arbitrarily large I in practice local minima show up for even moderately sized w (see Figure 3a). 3.2 CURSE OF DIMENSIONALITY: THE NUMBER OF MINIMA MIGHT GROW EXPONENTIALLY WITH THE DIMENSION We show how the I-dimensional minima of Theorem 3.1 can be combined to obtain ddimensional minima. Lemma 3.3 Let I : R -+ R be a continuous function with n disjoint minimum-containing sets UI , .•. ,Un. Then the sets UtI x ... X Utd , tj E {I, ... , n}, are n d disjoint minimumcontaining sets for the function 9 : Rd -+ R, g(XI, . .. , Xd) = l(xI) + ... + I(xd). Proof. Omitted. o Theorem 3.4 Let ¢ and L satisfy ( PI). Then for all n ~ 1 there is a sequence of examples S = (XI,Y),""(xn,y)), Xt E Rd, y E ¢(R), such that Es(w) has l~Jd distinct local minima. Proof. By Lemma 3.2 there exists a sequence of one-dimensional examples S' = (xI,y)"" ,(xLcrJ'Y)) such that ESI(w) has L~J disjoint minimum-containing sets. Thus by Lemma 3.3 the error function Es (w) has l ~ J d disjoint minimum-containing sets where S = ((XI, 0, .. . ,0), y), ... , «xLcrJ' 0, ... ,0), y), .. . , «0, ... , xI), y), .. . , «0, .. . , xLcrJ), y)). 0 4 MINIMA FOR NEURONS WITH A BIAS Theorem 4.1 Let the transfer function ¢ and the loss function L satisfy ¢( Bo + z) - ¢o = ¢o - ¢(Bo - z) and L(¢o + y, ¢o + y) = L(¢o - y, ¢o - y)for some Bo, ¢o E R and all z E R, y, Y E ¢(R). Furthermore let ¢ have a continuous second derivative and assume that the first derivative of ¢ at Bo is non-zero. At last let ~L(y, y) be continuous in y and y, L(y, y) = 0for all y E ¢(R), and (~L(Y, y)) (¢o, ¢o) > 0. Then for all n ~ 1 there is a sequence of examples S = (XI, YI), . .. , (xn, Yn)), Xt E Rd, Yt E ¢(R), such that Es (B, w) has l ~ J d distinct local minima. Note that the square loss along with either the hyperbolic or logistic transfer function satisfies the conditions of the theorem. IThere is a parallel proof where the magnitudes of the examples may be arbitrarily small. 322 P. AUER, M. HERBSTER, M. K. WARMUTH 5 ONE MINIMUM IN THE REALIZABLE CASE We show that when transfer and loss function are monotone and the examples are realizable then there is only a single minimal surface. A sequence of examples S is realizable if Es(w) = 0 for some wE Rd. Theorem 5.1 Let 4> and L satisfy (P 1). Furthermore let 4> be mOriotone and L such that L(y, y + rl) ~ L(y, y + r2) for 0 ~ rl ~ r2 or 0 ~ rl ~ r2. Assume that for some sequence of examples S there is a weight vectorwo E Rd such that Es(wo) = O. Thenfor each WI E Rd the function h( a) = Es (( 1 - a )wo + aWl) is increasing for a ~ O. Thus each minimum WI can be connected with Wo by the line segment WOWI such that Es(w) = 0 for all W on WOWI. Proof of Theorem 5.1. Let S = ((XI, yd, ... , (xn, Yn)). Then h(a) E~=I L(yt, 4>(WOXt + a(wl - wo)xt}). Since Yt = 4>(WOXt) it suffices to show that L(4)(z), 4>(z+ar)) is monotonically increasing in a ~ o for all Z, r E R. Let 0 ~ al ~ a2. Since 4> is monotone we get 4>(z + aIr) = 4>(z) + rl, 4>(z + a2r) = 4>(z) + r2 where o ~ rl ~ r2 or 0 ~ rl ~ r2· Thus L(4)(z), 4>(z + aIr)) ~ L(4)(z), 4>(z + a2r)). 0 Acknowledgments We thank Mike Dooley, Andrew Klinger and Eduardo Sontag for valuable discussions. Peter Auer gratefully acknowledges support from the FWF, Austria, under grant J01028-MAT. Mark Herbster and Manfred Warmuth were supported by NSF grant IRI-9123692. References [AHW96] P. Auer, M. Herbster, and M. K. Warmuth. Exponentially many local minima for single neurons. Technical Report UCSC-CRL-96-1, Univ. of Calif. Computer Research Lab, Santa Cruz, CA, 1996. In preperation. [Blu89] E.K. Blum. Approximation of boolean functions by sigmoidal networks: Part i: Xor and other two-variable functions. Neural Computation, 1 :532-540, February 1989. [BRS89] M.L. Brady, R. Raghavan, and J. Slawny. Back propagation fails to separate where perceptrons succeed. IEEE Transactions On Circuits and Systems, 36(5):665-674, May 1989. [BW88] E. Baum and F. Wilczek. Supervised learning of probability distributions by neural networks. In D.Z. Anderson, editor, Neural Information Processing Systems, pages 5261, New York, 1988. American Insitute of Physics. [GT92] Marco Gori and Alberto Tesi. On the problem of local minima in backpropagation. IEEE Transaction on Pattern Analysis and Machine Intelligence, 14(1):76-86, 1992. [Hay94] S. Haykin. Neural Networks: a Comprehensive Foundation. Macmillan, New York, NY, 1994. [SLF88] S. A. Solla, E. Levin, and M. Fleisher. Accelerated learning in layered neural networks. Complex Systems, 2:625-639,1988. [SS89] E.D. Sontag and H.l. Sussmann. Backpropagation can give rise to spurious local minima even for networks without hidden layers. Complex Systems, 3(1):91-106, February 1989. [SS91] E.D. Sontag and H.l. Sussmann. Back propagation separates where perceptrons do. Neural Networks,4(3),1991. [Wat92] R. L. Watrous. A comparison between squared error and relative entropy metrics using several optimization algorithms. Complex Systems, 6:495-505, 1992. [WD88] B.S. Wittner and J .S. Denker. Strategies for teaching layered networks classification tasks. In D.Z. Anderson, editor, Neural Information Processing Systems, pages 850--859, New York, 1988. American Insitute of Physics.
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An Information-theoretic Learning Algorithm for Neural Network Classification David J. Miller Department of Electrical Engineering The Pennsylvania State University State College, Pa: 16802 Ajit Rao, Kenneth Rose, and Allen Gersho Department of Electrical and Computer Engineering University of California Santa Barbara, Ca. 93106 Abstract A new learning algorithm is developed for the design of statistical classifiers minimizing the rate of misclassification. The method, which is based on ideas from information theory and analogies to statistical physics, assigns data to classes in probability. The distributions are chosen to minimize the expected classification error while simultaneously enforcing the classifier's structure and a level of "randomness" measured by Shannon's entropy. Achievement of the classifier structure is quantified by an associated cost. The constrained optimization problem is equivalent to the minimization of a Helmholtz free energy, and the resulting optimization method is a basic extension of the deterministic annealing algorithm that explicitly enforces structural constraints on assignments while reducing the entropy and expected cost with temperature. In the limit of low temperature, the error rate is minimized directly and a hard classifier with the requisite structure is obtained. This learning algorithm can be used to design a variety of classifier structures. The approach is compared with standard methods for radial basis function design and is demonstrated to substantially outperform other design methods on several benchmark examples, while often retaining design complexity comparable to, or only moderately greater than that of strict descent-based methods. 592 D. NnLLER.A.RAO.K. ROSE.A. GERSHO 1 Introduction The problem of designing a statistical classifier to minimize the probability of misclassification or a more general risk measure has been a topic of continuing interest since the 1950s. Recently, with the increase in power of serial and parallel computing resources, a number of complex neural network classifier structures have been proposed, along with associated learning algorithms to design them. While these structures offer great potential for classification, this potenl ial cannot be fully realized without effective learning procedures well-matched to the minimllm classificationerror oh.iective. Methods such as back propagation which approximate class targets in a sqllared error sense do not directly minimize the probability of error. Rather, it has been shown that these approaches design networks to approximate the class a posteriori probabilities. The probability estimates can then be used to form a decision rule. While large networks can in principle accurately approximate the Bayes discriminant, in practice the network size must be constrained to avoid overfitting the (finite) training set. Thus, discriminative learning techniques, e.g. (Juang and Katagiri, 1992), which seek to directly minimize classification error may achieve better results. However, these methods may still be susceptible to finding shallow local minima far from the global minimum. As an alternative to strict descent-based procedures, we propose a new deterministic learning algorithm for statistical classifier design with a demonstrated potential for avoiding local optima of the cost. Several deterministic, annealing-based techniques have been proposed for avoiding nonglobal optima in computer vision and image processing (Yuille, 1990), (Geiger and Girosi,1991), in combinatorial optimization, and elsewhere. Our approach is derived based on ideas from information theory and statistical physics, and builds on the probabilistic framework of the deterministic annealing (DA) approach to clustering and related problems (Rose et al., 1990,1992,1993). In the DA approach for data clustering, the probability distributions are chosen to minimize the expected clustering cost, given a constraint on the level of randomness, as measured by Shannon's entropy 1. In this work, the DA approach is extended in a novel way, most significantly to incorporate structural constraints on data assignments, but also to minimize the probability of error as the cost. While the general approach we suggest is likely applicable to problems of structured vector quantization and regression as well, we focus on the classification problem here. Most design methods have been developed for specific classifier structures. In this work, we will develop a general approach but only demonstrate results for RBF classifiers. The design of nearest prototype and MLP classifiers is considered in (Miller et al., 1995a,b). Our method provides substantial performance gains over conventional designs for all of these structures, while retaining design complexity in many cases comparable to the strict descent methods. Our approach often designs small networks to achieve training set performance that can only be obtained by a much larger network designed in a conventional way. The design of smaller networks may translate to superior performance outside the training set. INote that in (Rose et al., 1990,1992,1993), the DA method was formally derived using the maximum entropy principle. Here we emphasize the alternative, but mathematically equivalent description that the chosen distributions minimize the expected cost given constrained entropy. This formulation may have more intuitive appeal for the optimization problem at hand. An Information-theoretic Learning Algorithm for Neural Network Classification 593 2 Classifier Design Formulation 2.1 Problem Statement Let T = {(x,c)} be a training set of N labelled vectors, where x E 'Rn is a feature vector and c E I is its class label from an index set I. A classifier is a mapping C : 'Rn _ I, which assigns a class label in I to each vector in 'Rn. Typically, the classifier is represented by a set of model parameters A. The classifier specifies a partitioning of the feature space into regions Rj = {x E 'Rn : C(x) = j}, where U Rj = 'Rn and n Rj = 0. It also induces a partitioning of the training set into j j sets 7j C T, where 7j = {{x,c} : x E Rj,(x,c) E T}. A training pair (x,c) E T is misc1assified if C(x) "# c. The performance measure of primary interest is the empirical error fraction Pe of the classifier, i.e. the fraction of the training set (for generalization purposes, the fraction of the test set) which is misclassified: Pe = 2. L 6(c,C(x» = ~L L 6(c,j), (1) N (X,c)ET jEI (X,C)ETj where 6( c, j) = 1 if c "# j and 0 otherwise. In this work, we will assume that the classifier produces an output Fj(x) associated with each class, and uses a "winnertake-all" classification fll Ie: Rj == {x E'Rn : Fj (x) ~ Fk(X) "Ik E I}. (2) This rule is consistent with MLP and RBF-based classification. 2.2 Randomized Classifier Partition As in the original DA approach for clustering (Rose et aI., 1990,1992), we cast the optimization problem in a framework in which data are assigned to classes in probability. Accordingly, we define the probabilities of association between a feature x and the class regions, i.e. {P[x E R j ]}. As our design method, which optimizes over these probabilities, must ultimately form a classifier that makes "hard" decisions based on a specified network model, the distributions must be chosen to be consistent with the decision rule of the model. In other words, we need to introduce randomness into the classifier's partition. Clearly, there are many ways one could define probability distributions which are consistent with the hard partition at some limit. We use an information-theoretic approach. We measure the randomness or uncertainty by Shannon's entropy, and determine the distribution for a given level of entropy. At the limit of zero entropy we should recover a hard partition. For now, suppose that the values of the model parameters A have been fixed. We can then write an objective function whose maximization determines the hard partition for a given A: 1 Fh = N ~ L Fj(x). (3) JEI (X,c)ETj Note specifically that maximizing (3) over all possible partitions captures the decision rule of (2). The probabilistic generalization of (3) is 1 F = N L LP[x E Rj]Fj(x), (4) (X,c)ET j where the (randomized) partition is now represented by association probabilities, and the corresponding entropy is 1 H = - N L LP[x E Rj)logP[x E Rj). (5) (X,c)ET j 594 D. MILLER, A. RAO, K. ROSE, A. GERSHO We determine the distribution at a given level of randomness as the one which maximizes F while maintaining H at a prescribed level iI: max F subject to H = iI. {P[XERj]} (6) The result is the best probabilistic partition, in the sense of F, at the specified level of randomness. For iI = 0 we get back the hard partition maximizing (3). At any iI, the solution of(6) is the Gibbs distribution e'YFj(X) P[x E Rj] == Pjl~(A) = E e'YF" (X) , k (7) where 'Y is the Lagrange multiplier. For 'Y --t 0, the associations become increasingly uniform, while for 'Y --t 00, they revert to hard classifications, equivalent to application of the rule in (2). Note that the probabilities depend on A through the network outputs. Here we have emphasized this dependence through our choice of concise notation. 2.3 Information-Theoretic Classifier Design Until now we have formulated a controlled way of introducing randomness into the classifier's partition while enforcing its structural constraint. However, the derivation assumed that the model parameters were given, and thus produced only the form of the distribution Pjl~(A), without actually prescribing how to choose the valw's of its parameter set. Moreover the derivation did not consider the ultimate goal of minimizing the probability of error. Here we remedy both shortcomings. The method we suggest gradually enforces formation of a hard classifier minimizing the probability of error. We start with a highly random classifier and a high expected misclassification cost. We then gradually reduce both the randomness and the cost in a deterministic learning process which enforces formation of a hard classifier with the requisite structure. As before, we need to introduce randomness into the partition while enforcing the classifier's structure, only now we are also interested in minimizing the expected misclassification cost. While satisfying these multiple objectives may appear to be a formidable task, the problem is greatly simplified by restricting the choice of random classifiers to the set of distributions {Pjl~(A)} as given in (7) - these random classifiers naturally enforce the structural constraint through 'Y. Thus, from the parametrized set {Pjl~(A)}, we seek that distribution which minimizes the average misclassification cost while constraining the entropy: (8) subject to The solution yields the best random classifier in the sense of minimum < Pe > for a given iI. At the limit of zero entropy, we should get the best hard classifier in the sense of Pe with the desired structure, i.e. satisfying (2). The constrained minimization (8) is equivalent to the unconstrained minimization of the Lagrangian: min L == minfj < Pe > -H, A,'Y A,'Y (9) An Infonnation-theoretic Learning Algorithm for Neural Network Classification 595 where {3 is the Lagrange multiplier associated with (8). For {3 = 0, the sole objective is entropy maximization, which is achie\"ed by the uniform distribution. This solution, which is the global minimum for L at {3 = 0, can be obtained by choosing , = O. At the other end of the spectrum, for {3 00, the sole objective is to minimize < Pe >, and is achieved by choosing a non-random (hard) classifier (hence minimizing Pe ). The hard solution satisfies the classification rule (2) and is obtained for , 00. Motivation for minimizing the Lagrangian can be obtained from a physical perspective by noting that L is the Helmholtz free energy of a simulated system, with < Pe > the "energy", H the system entropy, and ~ the "temperature". Thus, from this physical view we can suggest a deterministic annealing (DA) process which involves minimizing L starting at the global minimum for {3 = 0 (high temperature) and tracking the solution while increasing {3 towards infinity (zero temperature). In this way, we obtain a sequence of solutions of decreasing entropy and average misclassification cost. Each such solution is the best random classifier in the sense of < Pe > for a given level of randomness. The annealing process is useful for avoiding local optima of the cost < Pe >, and minimizes < Pe > directly at low temperature. While this annealing process ostensibly involves the quantities Hand < Pe >, the restriction to {PjIAA)} from (7) ensures that the process also enforces the structural constraint on the classifier in a controlled way. Note in particular that, has not lost its interpretation as a Lagrange multiplier determining F. Thus, , = 0 means that F is unconstrained - we are free to choose the uniform distribution. Similarly, sending, 00 requires maximizing F - hence the hard solution. Since, is chosen to minimize L, this parameter effectively determines the level of F - the level of structural constraint - consistent with Hand < Pe > for a given {3. As {3 is increased, the entropy constraint is relaxed, allowing greater satisfaction of both the minimum < Pe > and maximum F objectives. Thus, annealing in {3 gradually enforces both the structural constraint (via ,) and the minimum < Pe > objective 2. Our formulation clearly identifies what distinguishes the annealing approach from direct descent procedures. Note that a descent method could be obtained by simply neglecting the constraint on the entropy, instead choosing to directly minimize < Pe > over the parameter set. This minimization will directly lead to a hard classifier, and is akin to the method described in (Juang and Katagiri, 1992) as well as other related approaches which attempt to directly minimize a smoothed probability of error cost. However, as we will experimentally verify through simulations, our annealing approach outperforms design based on directly minimizing < Pe >. For conciseness, we will not derive necessary optimality conditions for minimizing the Lagrangian at a give temperature, nor will we specialize the formulation for individual classification structures here. The reader is referred to (Miller et al., 1995a) for these details. 3 Experimental Comparisons We demonstrate the performance of our design approach in comparison with other methods for the normalized RBF structure (Moody and Darken, 1989). For the DA method, steepest descent was used to minimize L at a sequence of exponentially increasing {3, given by (3(n + 1) = a:{3(n) , for a: between 1.05 and 1.1. We have found that much of the optimization occurs at or near a critical temperature in the 2While not shown here, the method does converge directly for f3 00, and at this limit enforces the classifier's structure. 596 D.~LLER,A.RAO,K.ROSE,A.GERSHO Method DA TR-RHF MU-ltBJ<' \jPe M 4 30 4 10 30 50 10 50 10 Pe (tram) 0.11 0.028 0.33 0.162 0.145 0.129 0.3 0.19 0.18 Pe (test) 0.13 0.167 0.35 0.165 0.168 0.179 0.37 0.18 0.20 Table 1: A comparison of DA with known design techniques for RBF classification on the 40-dimensional noisy waveform data from (Breiman et al., 1980). solution process. Beyond this critical temperature, the annealing process can often be "quenched" to zero temperature by sending I ---+ 00 without incurring significant performance loss. Quenching the process often makes the design complexity of our method comparable to that of descent-based methods such as back propagation or gradient descent on < Pe >. We have compared our RBF design approach with the mf,thod in (Moody and Darken, 1989) (MD-RBF), with a method described ill (Tarassenko and Roberts,1994) (TR-RBF), with the approach in (Musavi et al., 1992), and with steepest descent on < Pe > (G-RBF). MD-RBF combines unsupervised learning of receptive field parameters with supervis,'d learning of the weights from the receptive fields so as to minimize the squared distance to target class outputs. The primary advantage of this approach is its modest design complexity. However, the recept i\"c fields are not optimized in a supervised fashion, which can cause performance degradation. TR-RBF optimizes all of the RBF parameters to approximate target class outputs. This design is more complex than MD-RBF and achieves better performance for a given model size. However, as aforementioned, the TR-RBF design objective is not equivalent to minimizing Pe , but rather to approximating the Bayes-optimal discriminant. While direct descent on < Pe > may minimize the "right" objective, problems of local optima may be quite severe. In fact, we have found that the performance of all of these methods can be quite poor without a judicious initialization. For all of these methods, we have employed the unsupervised learning phase described in (Moody and Darken, 1989) (based on Isodata clustering and variance estimation) as model initialization. Then, steepest descent was performed on the respective cost surface. We have found that the complexity of our design is typically 1-5 times that of TR-RB F or G-RBF (though occasionally our design is actually faster than G-RBF). Accordingly, we have chosen the best results based on five random initializations for these techniques, and compared with the single DA design run. One example reported here is the 40D "noisy" waveform data used in (Breiman et al., 1980) (obtained from the DC-Irvine machine learning database repository.). We split the 5000 vectors into equal size training and test sets. Our results in Table I demonstrate quite substantial performance gains over all the other methods, and performance quite close to the estimated Bayes rate of 14%. Note in particular that the other methods perform quite poorly for a small number of receptive fields (M), and need to increase M to achieve training set performance comparable to our approach. However, performance on the test set does not necessarily improve, and may degrade for increasing M. To further justify this claim, we compared our design with results reported in (Musavi et al., 1992), for the two and eight dimensional mixture examples. For the 2D example, our method achieved Petro-in = 6.0% for a 400 point training set and Pe, •• , = 6.1% on a 20,000 point test set, using M = 3 units (These results are near-optimal, based on the Bayes rate.). By contrast, the method of Musavi et An Information-theoretic Learning Algorithm for Neural Network Classification 597 al. used 86 receptive fields and achieved P et •• t = 9.26%. For the 8D example and M = 5, our method achieved Petr,.;n = 8% and Pet •• t = 9.4% (again near-optimal), while the method in (Musavui et al., 1992) achieved Pet•5t = 12.0% using M = 128. In summary, we have proposed a new, information-theoretic learning algorithm for classifier design, demonstrated to outperform other design methods, and with general applicability to a variety of structures. Future work may investigate important applications, such as recognition problems for speech and images. Moreover, our extension of DA to incorporate structure is likely applicable to structured vector quantizer design and to regression modelling. These problems will be considered in future work. Acknowledgements This work was supported in part by the National Science Foundation under grant no. NCR-9314335, the University of California M(( 'BO program, DSP Group, Inc. Echo Speech Corporation, Moseley Associates, 1'\ ill ional Semiconductor Corp., Qualcomm, Inc., Rockwell International Corporation, Speech Technology Labs, and Texas Instruments, Inc. References L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone. Classification and Regression Trees. The Wadsworth Statistics/Probability Series, Belmont,CA., 1980. D. Geiger and F. Girosi. Parallel and deterministic algorithms from MRFs: Surface reconstruction. IEEE Trans. on Patt. Anal. and Mach. Intell., 13:401- 412, 1991. B.-H. Juang and S. Katagiri. Discriminative learning for minimum error classification. IEEE Trans. on Sig. Proc., 40:3043-3054, 1992. D. Miller, A. Rao, K. Rose, and A. Gersho. A global optimization technique for statistical classifier design. (Submitted for publication.), 1995. D. Miller, A. Rao, K. Rose, and A. Gersho. A maximum entropy framework for optimal statistical classification. In IEEE Workshop on Neural Networks for Signal Processing.),1995. J. Moody and C. J. Darken. Fast learning in locally-tuned processing units. Neural Comp., 1:281-294, 1989. M. T. Musavi, W. Ahmed, K. H. Chan, K. B. Faris, and D. M. Hummels. On the training of radial basis function classifiers. Neural Networks, 5:595--604, 1992. K. Rose, E. Gurewitz, and G. C. Fox. Statistical mechanics and phase transitions in clustering. Phys. Rev. Lett., 65:945--948, 1990. K. Rose, E. Gurewitz, and G. C. Fox. Vector quantization by deterministic annealing. IEEE Trans. on Inform. Theory, 38:1249-1258, 1992. K. Rose, E. Gurewitz, and G. C. Fox. Constrained clustering as an optimization method. IEEE Trans. on Patt. Anal. and Mach. Intell., 15:785-794, 1993. L. Tarassenko and S. Roberts. Supervised and unsupervised learning in radial basis function classifiers. lEE Proc.- Vis. Image Sig. Proc., 141:210-216, 1994. A. L. Yuille. Generalized deformable models, statistical physics, and matching problems. 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Improving Elevator Performance Using Reinforcement Learning Robert H. Crites Computer Science Department University of Massachusetts Amherst, MA 01003-4610 critesGcs.umass.edu Andrew G. Barto Computer Science Department University of Massachusetts Amherst, MA 01003-4610 bartoGcs.umass.edu Abstract This paper describes the application of reinforcement learning (RL) to the difficult real world problem of elevator dispatching. The elevator domain poses a combination of challenges not seen in most RL research to date. Elevator systems operate in continuous state spaces and in continuous time as discrete event dynamic systems. Their states are not fully observable and they are nonstationary due to changing passenger arrival rates. In addition, we use a team of RL agents, each of which is responsible for controlling one elevator car. The team receives a global reinforcement signal which appears noisy to each agent due to the effects of the actions of the other agents, the random nature of the arrivals and the incomplete observation of the state. In spite of these complications, we show results that in simulation surpass the best of the heuristic elevator control algorithms of which we are aware. These results demonstrate the power of RL on a very large scale stochastic dynamic optimization problem of practical utility. 1 INTRODUCTION Recent algorithmic and theoretical advances in reinforcement learning (RL) have attracted widespread interest. RL algorithms have appeared that approximate dynamic programming (DP) on an incremental basis. Unlike traditional DP algorithms, these algorithms can perform with or without models of the system, and they can be used online as well as offline, focusing computation on areas of state space that are likely to be visited during actual control. On very large problems, they can provide computationally tractable ways of approximating DP. An example of this is Tesauro's TD-Gammon system (Tesauro, 1992j 1994; 1995), which used RL techniques to learn to play strong masters level backgammon. Even the 1018 R . H . CR~.A.G . BARTO best human experts make poor teachers for this class of problems since they do not always know the best actions. Even if they did, the state space is so large that it would be difficult for experts to provide sufficient training data. RL algorithms are naturally suited to this class of problems, since they learn on the basis of their own experience. This paper describes the application of RL to elevator dispatching, another problem where classical DP is completely intractable. The elevator domain poses a number of difficulties that were not present in backgammon. In spite of these complications, we show results that surpass the best of the heuristic elevator control algorithms of which we are aware. The following sections describe the elevator dispatching domain, the RL algorithm and neural network architectures that were used, the results, and some conclusions. 2 THE ELEVATOR SYSTEM The particular elevator system we examine is a simulated 10-story building with 4 elevator cars (Lewis, 1991; Bao et al, 1994). Passenger arrivals at each floor are assumed to be Poisson, with arrival rates that vary during the course of the day. Our simulations use a traffic profile (Bao et al, 1994) which dictates arrival rates for every 5-minute interval during a typical afternoon down-peak rush hour. Table 1 shows the mean number of passengers arriving at each floor (2-10) during each 5-minute interval who are headed for the lobby. In addition, there is inter-floor traffic which varies from 0% to 10% of the traffic to the lobby. Table 1: The Down-Peak Traffic Profile The system dynamics are approximated by the following parameters: • Floor time (the time to move one floor at the maximum speed): 1.45 secs. • Stop time (the time needed to decelerate, open and close the doors, and accelerate again): 7.19 secs. • Turn time (the time needed for a stopped car to change direction): 1 sec. • Load time (the time for one passenger to enter or exit a car): random variable from a 20th order truncated Erlang distribution with a range from 0.6 to 6.0 secs and a mean of 1 sec. • Car capacity: 20 passengers. The state space is continuous because it includes the elapsed times since any hall calls were registered. Even if these real values are approximated as binary values, the size of the state space is still immense. Its components include 218 possible combinations of the 18 hall call buttons (up and down buttons at each landing except the top and bottom), 240 possible combinations of the 40 car buttons, and 184 possible combinations of the positions and directions of the cars (rounding off to the nearest floor). Other parts of the state are not fully observable, for example, the desired destinations of the passengers waiting at each floor. Ignoring everything except the configuration of the hall and car call buttons and the approximate position and direction of the cars, we obtain an extremely conservative estimate of the size of a discrete approximation to the continuous state space: Improving Elevator Performance Using Reinforcement Learning 1019 Each car has a small set of primitive actions. Ifit is stopped at a floor, it must either "move up" or "move down". If it is in motion between floors, it must either "stop at the next floor" or "continue past the next floor". Due to passenger expectations, there are two constraints on these actions: a car cannot pass a floor if a passenger wants to get off there and cannot turn until it has serviced all the car buttons in its present direction. We have added three additional action constraints in an attempt to build in some primitive prior knowledge: a car cannot stop at a floor unless someone wants to get on or off there, it cannot stop to pick up passengers at a floor if another car is already stopped there, and given a choice between moving up and down, it should prefer to move up (since the down-peak traffic tends to push the cars toward the bottom of the building). Because of this last constraint, the only real choices left to each car are the stop and continue actions. The actions of the elevator cars are executed asynchronously since they may take different amounts of time to complete. The performance objectives of an elevator system can be defined in many ways. One possible objective is to minimize the average wait time, which is the time between the arrival of a passenger and his entry into a car. Another possible objective is to minimize the average 6y6tem time, which is the sum of the wait time and the travel time. A third possible objective is to minimize the percentage of passengers that wait longer than some dissatisfaction threshold (usually 60 seconds). Another common objective is to minimize the sum of 6quared wait times. We chose this latter performance objective since it tends to keep the wait times low while also encouraging fair service. 3 THE ALGORITHM AND NETWORK ARCHITECTURE Elevator systems can be modeled as ducrete event systems, where significant events (such as passenger arrivals) occur at discrete times, but the amount oftime between events is a real-valued variable. In such systems, the constant discount factor 'Y used in most discrete-time reinforcement learning algorithms is inadequate. This problem can be approached using a variable discount factor that depends on the amount of time between events (Bradtke & Duff, 1995). In this case, returns are defined as integrals rather than as infinite sums, as follows: becomes where rt is the immediate cost at discrete time t, r.,. is the instantaneous cost at continuous time T (e.g., the sum of the squared wait times of all waiting passengers), and {3 controls the rate of exponential decay. Calculating reinforcements here poses a problem in that it seems to require knowledge of the waiting times of all waiting passengers. There are two ways of dealing with this problem. The simulator knows how long each passenger has been waiting. It could use this information to determine what could be called omnucient reinforcements. The other possibility is to use only information that would be available to a real system online. Such online reinforcements assume only that the waiting time of the first passenger in each queue is known (which is the elapsed button time). If the Poisson arrival rate A for each queue is estimated as the reciprocal of the last inter-button time for that queue, the Gamma distribution can be used to estimate the arrival times of subsequent passengers. The time until the nth. subsequent arrival follows the Gamma distribution r(n, f). For each queue, subsequent 1020 R. H. CRITES, A. G. BARTO arrivals will generate the following expected penalties during the first b seconds after the hall button has been pressed: 00 rb L Jo (prob nth arrival occurs at time r) . (penalty given arrival at time r) dr n=l 0 This integral can be solved by parts to yield expected penalties. We found that using online reinforcements actually produced somewhat better results than using omniscient reinforcements, presumably because the algorithm was trying to learn average values anyway. Because elevator system events occur randomly in continuous time, the branching factor is effectively infinite, which complicates the use of algorithms that require explicit lookahead. Therefore, we employed a team of discrete-event Q-Iearning agents, where each agent is responsible for controlling one elevator car. Q(:z:, a) is defined as the expected infinite discounted return obtained by taking action a in state :z: and then following an optimal policy (Watkins, 1989). Because of the vast number of states, the Q-values are stored in feedforward neural networks. The networks receive some state information as input, and produce Q-value estimates as output. We have tested two architectures. In the parallel architecture, the agents share a single network, allowing them to learn from each other's experiences and forcing them to learn identical policies. In the fully decentralized architecture, the agents have their own networks, allowing them to specialize their control policies. In either case, none of the agents have explicit access to the actions of the other agents. Cooperation has to be learned indirectly via the global reinforcement signal. Each agent faces added stochasticity and nonstationarity because its environment contains other learning agents. Other work on team Q-Iearning is described in (Markey, 1994). The algorithm calls for each car to select its actions probabilistic ally using the Boltzmann distribution over its Q-value estimates, where the temperature is lowered gradually during training. After every decision, error backpropagation is used to train the car's estimate of Q(:z:, a) toward the following target output: where action a is taken by the car from state :z: at time tx , the next decision by that car is required from state y at time ty, and TT and (3 are defined as above. e-tJ(tv-t.) acts as a variable discount factor that depends on the amount of time between events. The learning rate parameter was set to 0.01 or 0.001 and {3 was set to 0.01 in the experiments described in this paper. After considerable experimentation, our best results were obtained using networks for pure down traffic with 47 input units, 20 hidden sigmoid units, and two linear output units (one for each action value). The input units are as follows: • 18 units: Two units encode information about each of the nine down hall buttons. A real-valued unit encodes the elapsed time if the button has been pushed and a binary unit is on if the button has not been pushed. Improving Elevator Performance Using Reinforcement Learning 1021 • 16 units: Each of these units represents a possible location and direction for the car whose decision is required. Exactly one of these units will be on at any given time. • 10 units: These units each represent one of the 10 floors where the other cars may be located. Each car has a "footprint" that depends on its direction and speed. For example, a stopped car causes activation only on the unit corresponding to its current floor, but a moving car causes activation on several units corresponding to the floors it is approachmg, with the highest activations on the closest floors. • 1 unit: This unit is on if the car whose decision is required is at the highest floor with a waiting passenger. • 1 unit: This unit is on if the car whose decision is required is at the floor with the passenger that has been waiting for the longest amount of time. • 1 unit: The bias unit is always on. 4 RESULTS Since an optimal policy for the elevator dispatching problem is unknown, we measured the performance of our algorithm against other heuristic algorithms, including the best of which we were aware. The algorithms were: SECTOR, a sector-based algorithm similar to what is used in many actual elevator systems; DLB, Dynamic Load Balancing, attempts to equalize the load of all cars; HUFF, Highest Unanswered Floor First, gives priority to the highest floor with people waiting; LQF, Longest Queue First, gives priority to the queue with the person who has been waiting for the longest amount of time; FIM, Finite Intervisit Minimization, a receding horizon controller that searches the space of admissible car assignments to minimize a load function; ESA, Empty the System Algorithm, a receding horizon controller that searches for the fastest way to "empty the system" assuming no new passenger arrivals. ESA uses queue length information that would not be available in a real elevator system. ESA/nq is a version of ESA that uses arrival rate information to estimate the queue lengths. For more details, see (Bao et al, 1994). These receding horizon controllers are very sophisticated, but also very computationally intensive, such that they would be difficult to implement in real time. RLp and RLd denote the RL controllers, parallel and decentralized. The RL controllers were each trained on 60,000 hours of simulated elevator time, which took four days on a 100 MIPS workstation. The results are averaged over 30 hours of simulated elevator time. Table 2 shows the results for the traffic profile with down traffic only. Algorithm I AvgWait I SquaredWait I SystemTime I Percent>60 secs I SECTOR 21.4 674 47.7 1.12 DLB 19.4 658 53.2 2.74 BASIC HUFF 19.9 580 47.2 0.76 LQF 19.1 534 46.6 0.89 HUFF 16.8 396 48.6 0.16 FIM 16.0 359 47.9 0.11 ESA/nq 15.8 358 47.7 0.12 ESA 15.1 338 47.1 0.25 RLp 14.8 320 41.8 0.09 RLd 14.7 313 41.7 0.07 Table 2: Results for Down-Peak Profile with Down Traffic Only 1022 R.H.C~.A.G . BARTO Table 3 shows the results for the down-peak traffic profile with up and down traffic, including an average of 2 up passengers per minute at the lobby. The algorithm was trained on down-only traffic, yet it generalizes well when up traffic is added and upward moving cars are forced to stop for any upward hall calls. Algorithm I AvgWait I Squared wait I SystemTime I Percent>60 secs I SECTOR 27.3 1252 54.8 9.24 DLB 21.7 826 54.4 4.74 BASIC HUFF 22.0 756 51.1 3.46 LQF 21.9 732 50.7 2.87 HU ... ·F 19.6 608 50.5 1.99 ESA 18.0 524 50.0 1.56 FIM 17.9 476 48.9 0.50 RLp 16.9 476 42.7 1.53 RLd 16.9 468 42.7 1.40 Table 3: Results for Down-Peak Profile with Up and Down Traffic Table 4 shows the results for the down-peak traffic profile with up and down traffic, including an average of 4 up passengers per minute at the lobby. This time there is twice as much up traffic, and the RL agents generalize extremely well to this new situation. Algorithm I AvgWait I SquaredWait I SystemTime I Percent>60 secs I SECTOR 30.3 1643 59.5 13.50 HUFF 22.8 884 55.3 5.10 DLB 22.6 880 55.8 5.18 LQF 23.5 877 53.5 4.92 BASIC HUFF 23.2 875 54.7 4.94 FIM 20.8 685 53.4 3.10 ESA 20.1 667 52.3 3.12 RLd 18.8 593 45.4 2.40 RLp 18.6 585 45.7 2.49 Table 4: Results for Down-Peak Profile with Twice as Much Up Traffic One can see that both the RL systems achieved very good performance, most notably as measured by system time (the sum of the wait and travel time), a measure that was not directly being minimized. Surprisingly, the decentralized RL system was able to achieve as good a level of performance as the parallel RL system. Better performance with nonstationary traffic profiles may be obtainable by providing the agents with information about the current traffic context as part of their input representation. We expect that an additional advantage of RL over heuristic controllers may be in buildings with less homogeneous arrival rates at each floor, where RL can adapt to idiosyncracies in their individual traffic patterns. 5 CONCLUSIONS These results demonstrate the utility of RL on a very large scale dynamic optimization problem. By focusing computation onto the states visited during simulated trajectories, RL avoids the need of conventional DP algorithms to exhaustively Improving Elevator Performance Using Reinforcement Learning 1023 sweep the state set. By storing information in artificial neural networks, it avoids the need to maintain large lookup tables. To achieve the above results, each RL system experienced 60,000 hours of simulated elevator time, which took four days of computer time on a 100 MIPS processor. Although this is a considerable amount of computation, it is negligible compared to what any conventional DP algorithm would require. The results also suggest that approaches to decentralized control using RL have considerable promise. Future research on the elevator dispatching problem will investigate other traffic profiles and further explore the parallel and decentralized RL architectures. Acknowledgements We thank John McNulty, Christ os Cassandras, Asif Gandhi, Dave Pepyne, Kevin Markey, Victor Lesser, Rod Grupen, Rich Sutton, Steve Bradtke, and the ANW group for assistance with the simulator and for helpful discussions. This research was supported by the Air Force Office of Scientific Research under grant F4962093-1-0269. References G. Bao, C. G. Cassandras, T. E. Djaferis, A. D. Gandhi, and D. P. Looze. (1994) Elevator Di,patcher, for Down Peale Traffic. Technical Report, ECE Department, University of Massachusetts, Amherst, MA. S. J. Bradtke and M. O. Duff. (1995) Reinforcement Learning Methods for Continuous-Time Markov Decision Problems. In: G. Tesauro, D. S. Touretzky and T. K. Leen, eds., Advance, in Neural Information Procelling Sy,tem, 7, MIT Press, Cambridge, MA. J. Lewis. (1991) A Dynamic Load Balancing Approach to the Control of Multuerver Polling Sy,tem, with Applicationl to Elevator Syltem Dupatching. PhD thesis, University of Massachusetts, Amherst, MA. K. L. Markey. (1994) Efficient Learning of Multiple Degree-of-Freedom Control Problems with Quasi-independent Q-agents. In: M. C. Mozer, P. Smolensky, D. S. Touretzky, J. L. Elman and A. S. Weigend, eds., Proceeding' of the 1993 Connectionilt Modell Summer SchooL Erlbaum Associates, Hillsdale, NJ. G. Tesauro. (1992) Practical Issues in Temporal Difference Learning. Machine Learning 8:257-277. G. Tesauro. (1994) TO-Gammon, a Self-Teaching Backgammon Program, Achieves Master-Level Play. Neural Computation 6:215-219. G. Tesauro. (1995) Temporal Difference Learning and TD-Gammon. Communication, of the ACM 38:58-68. C. J. C. H. Watkins. (1989) Learning from Delayed Reward,. 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Optimal Asset Allocation • uSIng Adaptive Dynamic Programming Ralph Neuneier* Siemens AG, Corporate Research and Development Otto-Hahn-Ring 6, D-81730 Munchen, Germany Abstract In recent years, the interest of investors has shifted to computerized asset allocation (portfolio management) to exploit the growing dynamics of the capital markets. In this paper, asset allocation is formalized as a Markovian Decision Problem which can be optimized by applying dynamic programming or reinforcement learning based algorithms. Using an artificial exchange rate, the asset allocation strategy optimized with reinforcement learning (Q-Learning) is shown to be equivalent to a policy computed by dynamic programming. The approach is then tested on the task to invest liquid capital in the German stock market. Here, neural networks are used as value function approximators. The resulting asset allocation strategy is superior to a heuristic benchmark policy. This is a further example which demonstrates the applicability of neural network based reinforcement learning to a problem setting with a high dimensional state space. 1 Introduction Billions of dollars are daily pushed through the international capital markets while brokers shift their investments to more promising assets. Therefore, there is a great interest in achieving a deeper understanding of the capital markets and in developing efficient tools for exploiting the dynamics of the markets. * Ralph.Neuneier@zfe.siemens.de, http://www.siemens.de/zfe.Jlll/homepage.html Optimal Asset Allocation Using Adaptive Dynamic Programming 953 Asset allocation (portfolio management) is the investment of liquid capital to various trading opportunities like stocks, futures, foreign exchanges and others. A portfolio is constructed with the aim of achieving a maximal expected return for a given risk level and time horizon. To compose an optimal portfolio, the investor has to solve a difficult optimization problem consisting of two phases (Brealy, 1991). First, the expected yields are estimated simultaneously with a certainty measure. Second, based on these estimates, a portfolio is constructed obeying the risk level the investor is willing to accept (mean-variance techniques). The problem is further complicated if transaction costs must be considered and if the investor wants to revise the decision at every time step. In recent years, neural networks (NN) have been successfully used for the first task. Typically, a NN delivers the expected future values of a time series based on data of the past. Furthermore, a confidence measure which expresses the certainty of the prediction is provided. In the following, the modeling phase and the search for an optimal portfolio are combined and embedded in the framework of Markovian Decision Problems, MDP. That theory formalizes control problems within stochastic environments (Bertsekas, 1987, Elton, 1971). If the discrete state space is small and if an accurate model of the system is available, MDP can be solved by conventional Dynamic Programming, DP. On the other extreme, reinforcement learning methods, e.g. Q-Learning, QL, can be applied to problems with large state spaces and with no appropriate model available (Singh, 1994). 2 Portfolio Managelnent is a Markovian Decision Problem The following simplifications do not restrict the generalization of the proposed methods with respect to real applications but will help to clarify the relationship between MDP and portfolio optimization. • There is only one possible asset for a Deutsch-Mark based investor, say a foreign currency called Dollar, US-$. • The investor is small and does not influence the market by her/his trading. • The investor has no risk aversion and always invests the total amount. • The investor may trade at each time step for an infinite time horizon. MDP provide a model for multi-stage decision making problems in stochastic environments. MDP can be described by a finite state set S = 1, ... , n, a finite set U (i) of admissible control actions for every state i E S, a set of transition probabilities P0' which describe the dynamics of the system, and a return function 1 r(i,j,u(i)),with i,j E S,u(i) E U(i). Furthermore, there is a stationary policy rr(i), which delivers for every state an admissible action u(i). One can compute the value-function l;j11" of a given state and policy, 00 Vi: = E[I:'"-/R(it,rr(it ))), (1) t=o 1 In the MDP-literature, the return often depends only on the current state i, but the theory extends to the case of r = r(i,j,u(i)) (see Singh, 1994). 954 R. NEUNEIER where E indicates the expected value, 'Y is the discount factor with 0 ~ 'Y < 1, and where R are the expected returns, R = Ej(r(i, j, u(i)). The aim is now to find a policy 71"* with the optimal value-function Vi* = max?!" Vi?!" for all states. In the context discussed here, a state vector consists of elements which describe the financial time series, and of elements which quantify the current value of the investment. For the simple example above, the state vector is the triple of the exchange rate, Xt, the wealth of the portfolio, Ct, expressed in the basis currency (here DM), and a binary variable b, representing the fact that currently the investment is in DM or US-$. Note, that out of the variables which form the state vector, the exchange rate is actually independent of the portfolio decisions, but the wealth and the returns are not. Therefore, asset allocation is a control problem and may not be reduced to pure prediction. 2 This problem has the attractive feature that, because the investments do not influence the exchange rate, we do not need to invest real money during the training phase of QL until we are convinced that our strategy works. 3 Dynamic Programming: Off-line and Adaptive The optimal value function V* is the unique solution of the well-known Bellman equation (Bertsekas, 1987). According to that equation one has to maximize the expected return for the next step and follow an optimal policy thereafter in order to achieve global optimal behavior (Bertsekas, 1987). An optimal policy can be easily derived from V* by choosing a 71"( i) which satisfies the Bellman equation. For nonlinear systems and non-quadric cost functions, V* is typically found by using an iterative algorithm, value iteration, which converges asymptotically to V*. Value iteration applies repeatedly the operator T for all states i, (2) Value iteration assumes that the expected return function R(i, u(i)) and the transition probabilities pij (i. e. the model) are known. Q-Learning, (QL), is a reinforcement-learning method that does not require a model of the system but optimizes the policy by sampling state-action pairs and returns while interacting with the system (Barto, 1989). Let's assume that the investor executes action u(i) at state i, and that the system moves to a new state j. Let r(i, j, u(i)) denote the actual return. QL then uses the update equation Q(i, u(i)) Q(k, v) (1 - TJ)Q(i, u(i)) + TJ(r(i, j, u(i)) + 'Yma:xQ(j, u(j))) u(J ) Q(k, v), for all k oF i and voF u(i) (3) where TJ is the learning rate and Q(i, u(i)) are the tabulated Q-values. One can prove, that this relaxation algorithm converges (under some conditions) to the optimal Q-values (Singh, 1994). 2To be more precise, the problem only becomes a mUlti-stage decision problem if the transaction costs are included in the problem. Optimal Asset Allocation Using Adaptive Dynamic Programming 955 The selection of the action u( i) should be guided by the trade-off between exploration and exploitation. In the beginning, the actions are typically chosen randomly (exploration) and in the course of training, actions with larger Q-values are chosen with increasingly higher probability (exploitation). The implementation in the following experiments is based on the Boltzmann-distribution using the actual Qvalues and a slowly decreasing temperature parameter (see Barto, 1989). 4 Experiment I: Artificial Exchange Rate In this section we use an exchange-rate model to demonstrate how DP and QLearning can be used to optimize asset allocation. The artificial exchange rate Xt is in the range between 1 and 2 representing the value of 1 US-$ in DM. The transition probabilities Pij of the exchange rate are chosen to simulate a situation where the Xt follows an increasing trend, but with higher values of Xt, a drop to very low values becomes more and more probable. A realization of the time series is plotted in the upper part of fig. 2. The random state variable Ct depends on the investor's decisions Ut, and is further influenced by Xt, Xt+b and Ct-l. A complete state vector consists of the current exchange rate Xt and the capital Ct, which is always calculated in the basis currency (DM). Its sign represents the actual currency, i. e., Ct = -1.2 stands for an investment in US-$ worth of 1.2 DM, and Ct = 1.2 for a capital of 1.2 DM. Ct and Xt are discretized in 10 bins each. The transaction costs ~ = 0.1 + Ic/IOOI are a combination of fixed (0.1) and variable costs (Ic/IOOI). Transactions only apply, if the currency is changed from DM to US-$. The immediate return rt(Xt,ct, Xt+1, ut) is computed as in table 1. If the decision has been made to change the portfolio into DM or to keep the actual portfolio in DM, Ut = DM, then the return is always zero. If the decision has been made to change the portfolio into US-$ or to keep the actual portfolio in US-$, Ut = US-$, then the return is equal to the relative change of the exchange rate weighted with Ct. That return is reduced by the transaction costs e, if the investor has to change into US-$. Table 1: The immediate return function. Ut =DM Ut = US-$ Ct E DM o Ct E US-$ o The success of the strategies was tested on a realization (2000 data points) of the exchange rate. The initial investment is 1 DM, at each time step the algorithm has to decide to either change the currency or remain in the present currency. As a reinforcement learning method, QL has to interact with the environment to learn optimal behavior. Thus, a second set of 2000 data was used to learn the Qvalues. The training phase is divided into epochs. Each epoch consists of as many trials as data exist in the training set. At every trial the algorithm looks at Xt, chooses randomly a portfolio value Ct and selects a decision. Then the immediate return and the new state is evaluated to apply eq. 3. The Q-values were initialized with zero, the learning rate T} was 0.1. Convergence was achieved after 4 epochs. 956 a': $ DM 2 2 04 ~03 .s ~02 1t' o 1 o 2 R. NEUNEIER 2 -2 1 Figure 1: The optimal decisions (left) and value function (right). . ... . _. . . _. . . . . . . _. . . . ' . 1 0 10 20" 30' 40 50 60 70 80 90 100 ~::[: : : : ~ o 10 20 30 40 50 60 70 80 90 100 ~] :ONJD:V, U: :IT] o 10 20 30 40 50 60 70 80 90 100 Time Figure 2: The exchange rate (top), the capital and the decisions (bottom). To evaluate the solution QL has found, the DP-algorithm from eq. 2 was implemented using the given transition probabilities. The convergence of DP was very fast. Only 5 iterations were needed until the average difference between successive value functions was lower than 0.01. That means 500 updates in comparison to 8000 updates with QL. The solutions were identical with respect to the resulting policy which is plotted in fig. 1, left. It can clearly be seen, that there is a difference between the policy of a DM-based and a US-$-based portfolio. If one has already changed the capital to US-$, then it is advisable to keep the portfolio in US-$ until the risk gets too high, i. e. Xt E {1.8, 1.9}. On the other hand, if Ct is still in DM, the risk barrier moves to lower values depending on the volume of the portfolio. The reason is that the potential gain by an increasing exchange rate has to cover the fixed and variable transaction costs. For very low values of Ct, it is forbidden to change even at low Xt because the fixed transaction costs will be higher than any gain. Figure 2 plots the Optimal Asset Allocation Using Adaptive Dynamic Programming 957 exchange rate Xt, the accumulated capital Ct for 100 days, and the decisions Ut. Let us look at a few interesting decisions. At the beginning, t = 0, the portfolio was changed immediately to US-$ and kept there for 13 steps until a drop to low rates Xt became very probable. During the time steps 35-45, the 'O'xchange rate oscillated at higher exchange rates. The policy insisted on the DM portfolio, because the risk was too high. In contrary, looking at the time steps 24 to 28, the policy first switched back to DM, then there was a small decrease of Xt which was sufficient to let the investor change again. The following increase justified that decision. The success of the resulting strategy can be easily recognized by the continuous increase of the portfolio. Note, that the ups and downs of the portfolio curve get higher in magnitude at the end because the investor has no risk aversion and always the whole capital is traded. 5 Experiment II: German Stock Index DAX In this section the approach is tested on a real world task: assume that an investor wishes to invest her Ihis capital into a block of stocks which behaves like the German stock index DAX. We based the benchmark strategy (short: MLP) on a NN model which was build to predict the daily changes of the DAX (for details, see Dichtl, 1995). If the prediction of the next day DAX difference is positive then the capital is invested into DAX otherwise in DM. The input vector of the NN model was carefully optimized for optimal prediction. We used these inputs (the DAX itself and 11 other influencing market variables) as the market description part of the state vector for QL. In order to store the value functions two NNs, one for each action, with 8 nonlinear hidden neurons and one linear output are used. The data is split into a training (from 2. Jan. 1986 to 31. Dec. 1992) and a test set (from 2. Jan. 1993 to 18. Oct. 1995). The return function is defined in the same way as in section 4 using 0.4% as proportional costs and 0.001 units as fixed costs, which are realistic for financial institutions. The training proceeds as outlined in the previous section with TJ = 0.001 for 1000 epochs. In fig. 3 the development of a reinvested capital is plotted for the optimized (upper line) and the MLP strategy (middle line). The DAX itself is also plotted but with a scaling factor to fit it into the figure (lower line). The resulting policy by QL clearly beats the benchmark strategy because the extra return amounts to 80% at the end of the training period and to 25% at the end of the test phase. A closer look at some statistics can explain the success. The QL policy proposes almost as often as the MLP policy to invest in DAX, but the number of changes from DM to DAX and v. v. is much lower (see table 2). Furthermore, it seems that the QL strategy keeps the capital out of the market if there is no significant trend to follow and the market shows too much volatility (see fig. 3 with straight horizontal lines of the capital development curve indicating no investments). An extensive analysis of the resulting strategy will be the topic of future research. In a further experiment the NNs which store the Q-values are initialized to imitate the MLP strategy. In some runs the number of necessary epochs were reduced by a factor of 10. But often the QL algorithni took longer to converge because the initialization ignores the input elements which describe the investor's capital and therefore led to a bad starting point in the weight space. 958 R. NEUNEIER 4S,r--------------------------, , 7.-----------------------------, , Jan 1993 18 Od H:195 Figure 3: The development of a reinvested capital on the training (left) and test set (right). The lines from top to bottom: QL-strategy, MLP-strategy, scaled DAX. Table 2: Some statistics of the policies. DAX investments position changes Data MLP Policy QL-Policy MLP Policy QL-Policy Training set 1825 1020 1005 904 284 Test set 729 434 395 344 115 6 Conclusions and Future Work In this paper, the task of asset allocation/portfolio management was approached by reinforcement learning algorithms. QL was successfully utilized in combination with NNs as value function approximators in a high dimensional state space. Future work has to address the possibility of several alternative investment opportunities and to clarify the connection to the classical mean-variance approach of professional brokers. The benchmark strategy in the real world experiment is in fact a neuro-fuzzy model which allows the extraction of useful rules after learning. It will be interesting to use that network architecture to approximate the value function in order to achieve a deeper insight in the resulting optimized strategy. References Barto A. G., Sutton R. S. and Watkins C. J. C. H. (1989), Learning and Sequential Decision Making, COINS TR 89-95. Bertsekas D. P. (1987), Dynamic Programming, NY: Wiley. Singh, P. S. (1993), Learning to Solve Markovian Decision Processes, CMPSCI TR 93-77. Neuneier R. (1995), Optimal Strategies with Density-Estimating Neural Networks, ICANN 95, Paris. Brealy, R. A., Myers, S. C. (1991), Principles of Corporate Finance, McGraw-Hill. Watkins C. J., Dayan, P. (1992), Technical Note: Q-Learning, Machine Learning 8, 3/4. Elton, E. J. , Gruber, M. J. (1971), Dynamic Programming Applications in Finance, The Journal of Finance, 26/2. Dichtl, H. (1995), Die Prognose des DAX mit Neuro-Fuzzy, masterthesis, engl. abstract in preparation.
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Recursive Estimation of Dynamic Modular RBF Networks Visakan Kadirkamanathan Automatic Control & Systems Eng. Dept. University of Sheffield, Sheffield Sl 4DU, UK visakan@acse.sheffield.ac. uk Abstract Maha Kadirkamanathan Dragon Systems UK Cheltenham GL52 4RW, UK maha@dragon.co.uk In this paper, recursive estimation algorithms for dynamic modular networks are developed. The models are based on Gaussian RBF networks and the gating network is considered in two stages: At first, it is simply a time-varying scalar and in the second, it is based on the state, as in the mixture of local experts scheme. The resulting algorithm uses Kalman filter estimation for the model estimation and the gating probability estimation. Both, 'hard' and 'soft' competition based estimation schemes are developed where in the former, the most probable network is adapted and in the latter all networks are adapted by appropriate weighting of the data. 1 INTRODUCTION The problem of learning multiple modes in a complex nonlinear system is increasingly being studied by various researchers [2, 3, 4, 5, 6], The use of a mixture of local experts [5, 6], and a conditional mixture density network [3] have been developed to model various modes of a system. The development has mainly been on model estimation from a given set of block data, with the model likelihood dependent on the input to the networks. A recursive algorithm for this static case is the approximate iterative procedure based on the block estimation schemes [6]. In this paper, we consider dynamic systems - developing a recursive algorithm is difficult since mode transitions have to be detected on-line whereas in the block scheme, search procedures allow optimal detection. Block estimation schemes for general architectures have been described in [2, 4]. However, unlike in those schemes, the algorithm developed here uses relationships based on Bayes law and Kalman filters and attempts to describe the dynamic system explicitly, The modelling is carried out by radial basis function (RBF) networks for their property that by preselecting the centres and widths, the problem can be reduced to a linear estimation. 240 V. KADIRKAMANATHAN. M. KADIRKAMANATHAN 2 DYNAMIC MODULAR RBF NETWORK The dynamic modular RBF network consists of a number of models (or experts) to represent each nonlinear mode in a dynamical system. The models are based on the RBF networks with Gaussian function, where the RBF centre and width parameters are chosen a priori and the unknown parameters are only the linear coefficients w. The functional form of the RBF network can be expressed as, K f(XiP) = L wkgk(X) = w T g k=l where w = [ ... , Wk, .. Y E lRK is the linear weight vector and g [ ... , gk(X), .. . ]T E lR~ are the radial basis functions, where, (1) gk(X) = exp {-O.5r-21Ix - mk112} (2) mk E lRM are the RBF centres or means and r the width. The RBF networks are used for their property that having chosen appropriate RBF centre and width parameters mk, r, only the linear weights w need to be estimated for which fast, efficient and optimal algorithms exist. Each model has an associated probability score of being the current underlying model for the given observation. In the first stage of the development, this probability is not determined from parametrised gating network as in the mixture of local experts [5] and the mixture density network [3], but is determined on-line as it varies with time. In dynamic systems, time information must be taken into account whereas the mixture of local experts use only the state information which is not sufficient in general, unless the states contain the necessary information. In the second stage, the probability is extended to represent both the time and state information explicitly using the expressions from the mixture of local experts. Recently, time and state information have been combined in developing models for dynamic systems such as the mixture of controllers [4] and the Input - Output HMM [2]. However, the scheme developed here is more explicit and is not as general as the above schemes and is recursive as opposed to block estimation. 3 RECURSIVE ESTIMATION The problem of recursive estimation with RBF networks have been studied previously [7, 8] and the algorithms developed here is a continuation of that process. Let the set of input - output observations from which the model is to be estimated be, 2 N = {zn 1 n = 1, ... , N} (3) where, 2N includes all observations upto the Nth data and Zn is the nth data, Zn = {( Xn , Yn) 1 Xn E lRM , Yn E lR} ( 4 ) The underlying system generating the observations are assumed to be multi-modal (with known H modes), with each observation satisfying the nonlinear relation, Y = fh(X) + 1] (5) where 1] is the noise with unknown distribution and fh (.) : lRM 1-+ lR is the unknown underlying nonlinear function for the hth mode which generated the observation. Under assumptions of zero mean Gaussian noise and that the model can approximate the underlying function arbitrarily closely, the probability distribution, ( I h n ) ( _1 -t {1 -11 ( h)12} P Zn W ,M = Mh, 2 n - 1 = 271") 2 Ro exp -"2Ro Yn -!h Xn; W (6) Recursive Estimation of Dynamic Modular RBF Networks 241 is Gaussian. This is the likelihood of the observation Zn for the model Mh, which in our case is the GRBF network, given model parameters wand that the nth observation was generated by Mh. Ro is the variance of the noise TJ. In general however, the model generating the nth observation is unknown and the likelihood of the nth observation is expanded to include I~ the indicator variable, as in [6], H k p(zn"nIW,M,Zn-l) = IT [p(znlw\Mn = Mh,Zn_dp(Mn = Mhlxn,zn-l)r" h=l Bayes law can be applied to the on-line or recursive parameter estimation, p(WIZn,M) = p(znIW,M, Zn-dp(WIZn-l,M) P(ZnIZn-l,M) (7) (8) and the above equation is applied recursively for n = 1, ... , N . The term p(zn IZn-l, M) is the evidence. If the underlying system is unimodal, this will result in the optimal Kalman estimator and if we assign the prior probability distribution for the model parameters p(wh IMhk to be Gaussian with mean Wo and covariance matrix (positive definite) Po E 1R xK, which combines the likelihood and the prior to give the posterior probability distribution which at time n is given by p(whlZn, Mh) which is also Gaussian, p(whIZn,Mh) = (27r)-4Ip~l-t exp { _~(wh - W~fp~-l (wh - w~)} (9) In the multimodal case also, the estimation for the individual model parameters decouple naturally with the only modification being that the likelihood used for the parameter estimation is now based on weighted data and given by, h ' 1 h- 1 1 {1 1 h I h 12} p(znlw ,Mh,Zn-l)=(27r)-~(Roln )-~exp -'2Ro In Yn-ih(Xn;W) (10) The Bayes law relation (8) applies to each model. Hence, the only modification in the Kalman filter algorithm is that the noise variance for each model is set to Roh~ and the resulting equations can be found in [7]. It increases the apparent uncertainty in the measurement output according to how likely the model is to be the true underlying mode, by increasing the noise variance term of the Kalman filter algorithm. Note that the term p(Mn = Mhlxn, zn-l) is a time-varying scalar and does not influence the parameter estimation process. The evidence term can also be determined directly from the Kalman filter, where the e~ is the prediction error and R~ is the innovation variance with, eh n Rh n hT Yn - wn-1gn h- 1 T h ROln + gnP n-lgn (11) (12) (13) This is also the likelihood of the nth observation given the model M and the past observations Zn-l. The above equation shows that the evidence term used in Bayesian model selection [9] is computed recursively, but for the specific priors Ro, Po. On-line Bayesian model selection can be carried out by choosing many different priors, effectively sampling the prior space, to determine the best model to fit the given data, as discussed in [7]. 242 V. KADIRKAMANATHAN. M. KADIRKAMANATHAN 4 RECURSIVE MODEL SELECTION Bayes law can be invoked to perform recursive or on-line model selection and this has been used in the derivation of the multiple model algorithm [1]. The multiple model algorithm has been used for the recursive identification of dynamical nonlinear systems [7]. Applying Bayes law gives the following relation: (14) which can be computed recursively for n = 1, ... , N. p(ZnIMh, Zn-1) is the likelihood given in (11) and p(MhIZn) is the posterior probability of model Mh being the underlying model for the nth data given the observations Zn· The term p(Zn IZn-1) is the normalising term given by, H P(ZnI Zn-1) = Lp(znIMh,Zn-1)p(MhIZn-1) (15) h=l The initial prior probabilities for models are assigned to be equal to 1/ H. The equations (11), (14) combined with the Kalman filter estimation equations is known as the multiple model algorithm [1] . Amongst all the networks that are attempting to identify the underlying system, the identified model is the one with the highest posterior probability p(MhIZn) at each time n, ie., (16) and hence can vary from time to time. This is preferred over the averaging of all the H models as the likelihood is multimodal and hence modal estimates are sought. Predictions are based on this most probable model. Since the system is dynamical, if the underlying model for the dynamics is known, it can be used to predict the estimates at the next time instant based on the current estimates, prior to observing the next data. Here, a first order Markov assumption is made for the mode transitions. Given that at the time instant n - 1 the given mode is j, it is predicted that the probability of the mode at time instant n being h is the transition probability Phj . With H modes, 2: Phj = 1. The predicted probability of the mode being h at time n therefore is given by, H Pnln-l(MhIZn-1) = L Phjp(Mj IZn-1) j=l (17) This can be viewed as the prediction stage of the model selection algorithm. The predicted output of the system is obtained from the output of the model that has the highest predicted probability. Given the observation Zn, the correction is achieved through the multiple model algorithm of (14) with the following modification: p(MhIZn) = p(znIMh, Zn-1)Pnln-1(MhI Zn-1) p(znIZn-d (18) where modification to the prior has been made. Note that this probability is a time-varying scalar value and does not depend on the states. Recursive Estimation of Dynamic Modular RBF Networks 243 5 HARD AND SOFT COMPETITION The development of the estimation and model selection algorithms have thus far assumed that the indicator variable 'Y~ is known. The 'Y~ is unknown and an expected value must be used in the algorithm, which is given by, (3h _ p(znlMn = Mh, Zn-I)Pnln_1(Mn = MhIZn-I) n P(ZnIZn-1) (19) Two possible methodologies can be used for choosing the values for 'Y~. In the first scheme, 'Y~ = 1 if,B~ > ,B~ for all j 1= h, and 0 otherwise (20) This results in 'hard' competition where, only the model with the highest predicted probability undergoes adaptation using the Kalman filter algorithm while all other models are prevented from adapting. Alternatively, the expected value can be used in the algorithm, (21) which results in 'soft' competition and all models are allowed to undergo adaptation with appropriate data weighting as outlined in section 3. This scheme is slightly different from that presented in [7]. Since the posterior probabilities of each mode effectively indicate which mode is dominant at each time n, changes can then be used as means of detecting mode transitions. 6 EXPERIMENTAL RESULTS The problem chosen for the experiment is learning the inverse robot kinematics used in [3]. This is a two link rigid arm manipulator for which, given joint arm angles (01 , O2 ), the end effector position in cartesian co-ordinates is given by, :l:1 L1 COS(Ol) - L2 COS(Ol + O2) :l:2 = L1 sin(Ol) - L2 sin(Ol + O2) (22) L1 = 0.8, L2 = 0.2 being the arm lengths. The inverse kinematics learning problem requires the identification of the underlying mapping from (:l:1' :l:2) (01 , O2), which is bi-modal. Since the algorithm is developed for the identification of dynamical systems, the data are generated with the joint angles being excited sinusoidally with differing frequencies within the intervals [0.3,1.2] x ['71"/2,371"/2]. The first 1000 observations are used for training and the next 1000 observations are used for testing with the adaptation turned off. The models use 28 RBFs chosen with fixed parameters, the centres being uniformly placed on a 7 x 4 grid. , .-'~',"--'-"-':"", (, :',-1 • ;:"';'d:',~r":rbv:~b · ::":~~:-:-:-~W-,~:~,g ::-~~. " :' " " '\'1 " ': '\ . c> g II ~, ... : I. .' 'I ~~ ., ! ' .. ~\ :." ., :" I I ' " ., " " ,I :' 'I :l ,' ." ,.' ~ . ' I I 'I " :. I, I~ : r .::: ~: I ': ::, .. : ::: " I : : ,",: : I : ::~:: :, I :,1 '~, I. i ~ I I ': :: i I :: :: :::: I, 0 __ :: ".. l:: ~: : .... :: I .::: I . :: :: I '1: ::': I ,. I I :: . :,,: I I : 0 _"7 I!: '" t : :: : ~ :: ~ ! ~: ::!! ~;~ I I • " , ~ ~ t ,'I " '0 " I 0 _ e . " ~ i ~ y. ~ , 1 0 . & , -\ 0__ : .~ : :=~I I I : ~ , ,1.111.;, l,!111. ::, tU \",: ,i :,:, , :, ::' L ~ t.U\JJW~ :,' :'llJr: , .l/v :. \ _ 'J I •• t ::::' ::; °0 .., co :zoo 300 _00 .. 00 "TI......,_ ! " ~: : " ' ,,,, I. I' . " : : VI f;' :: .:. I' . ' I. I. ., I' I... ,,' Figure 1: Learning inverse kinematics (,hard' competition): Model probabilities. Figure 1 shows the model probabilities during training and shows the switching taking place between the two modes. 244 V. KADIRKAMANATHAN. M. KADIRKAMANATHAN Modtl1 Teat Ca. errora in the EncI.n.ctcr Poei6oo S~ Modtl 2 Teat Data enoN In ttw End .n.ckH' PoeItion Space 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 \l 0.5 \l OS 0.4 0.4 0.3 0.3 02 02 0.' 0.' ~~~0. '--~02~0~ .3~0~. ' ~0.~5~ 0.6~~OJ~0~ .8~0~ .9~ " ~~~ 0.~'~ 02~0~.3~0~.4~ 0.~5~0.6~~ 0.7~0~ .8~0~. 9 ~ " Figure 2: End effector position errors (test data) ('hard' competition): (a) Model 1 prediction (b) Model 2 prediction. Figure 2 show the end effector position errors on the test data by both models 1 and 2 separately under the 'hard' competition scheme. The figure indicates the errors achieved by the best model used in the prediction - both models predicting in the centre of the input space where the function is multi-modal. This demonstrates the successful operation of the algorithm in the two RBF networks capturing some elements of the two underlying modes of the relationship. The best results on this learning task are: The RMSE on test data for this problem by the Mixture Density Table 1: Learning Inverse Kinematics: Results Hard Competition Soft Competition RMSE (Train) 0.0213 0.0442 RMSE (Test) 0.0084 0.0212 Network is 0.0053 and by a single network is 0.0578 [3]. Note however that the algorithm here did not use state information and used only the time dependency. 7 PARAMETRISED GATING NETWORKS The model parameters were determined explicitly based on the time information in the dynamical system. If the gating model probabilities are expressed as a function of the states, similar to [6], H p(Mhlxn, Zn-l) = exp{ahT g} / L exp{ahT g} = a~ (23) h=l where a h are the gating network parameters. Note that the gating network shares the same basis functions as the expert models. This extension to the gating networks does not affect the model parameter estimation procedure. The likelihood in (7) decomposes into a part for model parameter estimation involving output prediction error and a part for gating parameter estimation involving the indicator variable Tn . The second part can be approximated to a Gaussian of the form, h 1 h-l. {1 h- 1 h h 2} P(Tnlxn,a ,Zn-d ~ (21r)-~RgO ~ exp -"2Rgo I'Yn - ani (24) Recursive Estimation of Dynamic Modular RBF Networks 245 This approximation allows the extended Kalman filter algorithm to be used for gating network parameter estimation. The model selection equations of section 4 can be applied without any modification with the new gating probabilities. The choice of the indicator variable 'Y~ can be based as before, resulting in either hard or soft competition. The necessary expressions in (21) are obtained through the Kalman filter estimates and the evidence values, for both the model and gating parameters. Note that this is different from the estimates used in [6] in the sense that, marginalisation over the model and gating parameters have been done here. 8 CONCLUSIONS Recursive estimation algorithms for dynamic modular RBF networks have been developed. The models are based on Gaussian RBF networks and the gating is simply a time-varying scalar. The resulting algorithm uses Kalman filter estimation for the model parameters and the multiple model algorithm for the gating probability. Both, (hard' and (soft' competition based estimation schemes are developed where in the former, the most probable network is adapted and in the latter all networks are adapted by appropriate weighting of the data. Experimental results are given that demonstrate the capture of the switching in the dynamical system by the modular RBF networks. Extending the method to include the gating probability to be a function of the state are then outlined briefly. Work is currently in progress to experimentally demonstrate the operation of this extension. References [1] Bar-Shalom, Y. and Fortmann, T. E. Tracking and data association, Academic Press, New York, 1988. [2] Bengio, Y. and Frasconi, P. "An input output HMM architecture", In G. Tesauro, D. S. Touretzky and T . K. Leen (eds.) Advances in Neural Information Processing Systems 7, Morgan Kaufmann, CA: San Mateo, 1995. [3] Bishop, C. M. "Mixture density networks", Report NCRG/4288, Computer Science Dept., Aston University, UK, 1994. [4] Cacciatore, C. W. and Nowlan, S. J. "Mixtures of controllers for jump linear and nonlinear plants", In J. Cowan, G. Tesauro, and J. Alspector (eds.) Advances in Neural Information Processing Systems 6, Morgan Kaufmann, CA: San Mateo, 1994. [5] Jacobs, R. A., Jordan, M. I., Nowlan, S. J . and Hinton, G. E. "Adaptive mixtures of local experts", Neural Computation, 9: 79-87, 1991. [6] Jordan, M. I. and Jacobs, R. A. "Hierarchical mixtures of experts and the EM algorithm" , Neural Computation, 6: 181-214, 1994. [7] Kadirkamanathan, V. "Recursive nonlinear identification using multiple model algorithm", In Proceedings of the IEEE Workshop on Neural Networks for Signal Processing V, 171-180, 1995. [8] Kadirkamanathan, V. ((A statistical inference based growth criterion for the RBF network", In Proceedings of the IEEE Workshop on Neural Networks for Signal Processing IV, 12-21, 1994. [9] MacKay, D. J. C. "Bayesian interpolation", Neural Computation, 4: 415-447, 1992.
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A Neural Network Classifier for the 11000 OCR Chip John C. Platt and Timothy P. Allen Synaptics, Inc. 2698 Orchard Parkway San Jose, CA 95134 platt@synaptics.com, tpa@synaptics.com Abstract This paper describes a neural network classifier for the 11000 chip, which optically reads the E13B font characters at the bottom of checks. The first layer of the neural network is a hardware linear classifier which recognizes the characters in this font. A second software neural layer is implemented on an inexpensive microprocessor to clean up the results of the first layer. The hardware linear classifier is mathematically specified using constraints and an optimization principle. The weights of the classifier are found using the active set method, similar to Vapnik's separating hyperplane algorithm. In 7.5 minutes ofSPARC 2 time, the method solves for 1523 Lagrange mUltipliers, which is equivalent to training on a data set of approximately 128,000 examples. The resulting network performs quite well: when tested on a test set of 1500 real checks, it has a 99.995% character accuracy rate. 1 A BRIEF OVERVIEW OF THE 11000 CHIP At Synaptics, we have created the 11000, an analog VLSI chip that, when combined with associated software, optically reads the E13B font from the bottom of checks. This E13B font is shown in figure 1. The overall architecture of the 11000 chip is shown in figure 2. The 11000 recognizes checks hand-swiped through a slot. A lens focuses the image of the bottom of the check onto the retina. The retina has circuitry which locates the vertical position of the characters on the check. The retina then sends an image vertically centered around a possible character to the classifier. The classifier in the nooo has a tough job. It must be very accurate and immune to noise and ink scribbles in the input. Therefore, we decided to use an integrated segmentation and recognition approach (Martin & Pittman, 1992)(Platt, et al., 1992). When the classifier produces a strong response, we know that a character is horizontally centered in the retina. A Neural Network Classifier for the 11000 OCR Chip 939 Figure 1: The E13B font, as seen by the 11000 chip 11000 chip ~----------l I I I linear winner I retina take I mIcroprocessor G __ ~ ~l:i:r __ ~l_ J i Slot for check 18 by 24 image best character vertically positioned hypothesis 42 confidences Figure 2: The overall architecture of the 11000 chip We decided to use analog VLSI to minimize the silicon area of the classifier. Because of the analog implementation, we decided to use a linear template classifier, with fixed weights in silicon to minimize area. The weights are encoded as lengths of transistors acting as current sources. We trained the classifier using only the specification of the font , because we did not have the real E13B data at the time of classifier design. The design of the classifier is described in the next section. As shown in figure 2, the input to the classifier is an 18 by 24 pixel image taken from the retina at a rate of 20 thousand frames per second. The templates in the classifier are 18 by 22 pixels. Each template is evaluated in three different vertical positions, to allow the retina to send a slightly vertically mis-aligned character. The output of the classifier is a set of 42 confidences, one for each of the 14 characters in the font in three different vertical positions. These confidences are fed to a winnertake-all circuit (Lazzaro, et al. , 1989), which finds the confidence and the identity of the best character hypothesis. 2 SPECIFYING THE BEHAVIOR OF THE CLASSIFIER Let us consider the training of one template corresponding to one of the characters in the font. The template takes a vector of pixels as input. For ease of analog implementation, the template is a linear neuron with no bias input: (1) where 0 is the output of the template, Wi are the weights of the template, and Ii are the input pixels of the template. We will now mathematically express the training of the templates as three types of constraints on the weights of the template. The input vectors used by these constraints are the ideal characters taken from the specification of the font. The first type of constraint on the template is that the output of the template should be above 1 when the character that corresponds to the template is centered 940 1. C. PLAIT, T. P. ALLEN I ! I I ! Figure 3: Examples of images from the bad set for the templates trained to detect the zero character. These images are E13B characters that have been horizontally and vertically offset from the center of the image. The black border around each of the characters shows the boundary of the input field. Notice the variety of horizontal and vertical shifts of the different characters. in the horizontal field. Call the vector of pixels of this centered character Gi . This constraint is stated as: (2) The second type of constraint on the template is to have an output much lower than 1 when incorrect or offset characters are applied to the template. We collect these incorrect and offset characters into a set of pixel vectors jjj, which we call the "bad set." The constraint that the output of the template be lower than a constant c for all of the vectors in the bad set is expressed as: L wiBf :s c Vj (3) Together, constraints (2) and (3) permit use of a simple threshold to distinguish between a positive classifier response and a negative one. The bad set contains examples of the correct character for the template that are horizontally offset by at least two pixels and vertically offset by up to one pixel. In addition, examples of all other characters are added to the bad set at every horizontal offset and with vertical offsets of up to one pixel (see figure 3). Vertically offset examples are added to make the classifier resistant to characters whose baselines are slightly mismatched. The third type of constraint on the template requires that the output be invariant to the addition of a constant to all of the input pixels. This constraint makes the classifier immune to any changes in the background lighting level, k. This constraint is equivalent to requiring the sum of the weights to be zero: (4) Finally, an optimization principle is necessary to choose between all possible weight vectors that fulfill constraints (2), (3), and (4). We minimize the perturbation of the output of the template given uncorrelated random noise on the input. This optimization principle is similar to training on a large data set, instead of simply the ideal characters described by the specification. This optimization principle is equivalent to minimizing the sum of the square of the weights: minLWl (5) Expressing the training of the classifier as a combination of constraints and an optimization principle allows us to compactly define its behavior. For example, the combination of constraints (3) and (4) allows the classifier to be immune to situations when two partial characters appear in the image at the same time. The confluence of two characters in the image can be described as: I?verlap = k + B! + B': , 'I (6) A Neural Network Classifier for the 11000 OCR Chip 941 where k is a background value and B! and B[ are partial characters from the bad set that appears on the left side and right side of the image, respectively. The output of the template is then: ooverlap = 2: Wi(k + BI + BD = 2: Wjk + 2: wiBI + 2: WiB[ < 2c (7) Constraints (3) and (4) thus limit the output of the neuron to less than 2c when two partial characters appear in the input. Therefore, we want c to be less than 0.5. In order to get a 2:1 margin, we choose c = 0.25. The classifier is trained only on individual partial characters instead of all possible combinations of partial characters. Therefore, we can specify the classifier using only 1523 constraints, instead of creating a training set of approximately 128,000 possible combinations of partial characters. Applying these constraints is therefore much faster than back-propagation on the entire data set. Equations (2), (3) and (5) describe the optimization problem solved by Vapnik (Vapnik, 1982) for constructing a hyperplane that separates two classes. Vapnik solves this optimization problem by converting it into a dual space, where the inequality constraints become much simpler. However, we add the equality constraint (4), which does not allow us to directly use Vapnik's dual space method. To overcome this limitation, we use the active set method, which can fulfill any extra linear equality or inequality constraints. The active set method is described in the next section. 3 THE ACTIVE SET METHOD Notice that constraints (2), (3), and (4) are all linear in Wi. Therefore, minimizing (5) with these constraints is simply quadratic programming with a mixture of equality and inequality constraints. This problem can be solved using the active set method from optimization theory (Gill, et al., 1981). When the quadratic programming problem is solved, some of the inequality constraints and all of the equality constraints will be "active." In other words, the active constraints affect the solution as equality constraints. The system has "bumped into" these constraints. All other constraints will be inactive; they will not affect the solution. Once we know which constraints are active, we can easily solve the quadratic minimization problem with equality constraints via Lagrange multipliers. The solution is a saddle point of the function: ~ 2: wl + 2: Ak(2: Akj Wj - Ck) (8) i k i where Ak is the Lagrange multiplier of the kth active constraint, and Akj and Ck are the linear and constant coefficients of the kth active constraint. For example, if constraint (2) is the kth active constraint, then Ak = G and Ck = 1. The saddle point can be found via the set of linear equations: Wi - 2: AkAki (9) k -2)2: AjiAki)-lCj (10) j The active set method determines which inequality constraints belong in the active set by iteratively solving equation (10) above. At every step, one inequality constraint is either made active, or inactive. A constraint can be moved to the active 942 J. C. PLAIT, T. P. ALLEN Action: move X to here, make constraint 13 inactive ro • ~ L A13=0 on this line 1 .. solution from \. equation (10) constramt 19 violated on this line X space Figure 4: The position along the step where the constraints become violated or the Lagrange multipliers become zero can be computed analytically. The algorithm then takes the largest possible step without violating constraints or having the Lagrange multipliers become zero. set if the inequality constraint is violated. A constraint can be moved off the active set if its Lagrange multiplier has changed sign 1 . Each step of the active set method attempts to adjust the vector of Lagrange multipliers to the values provided by equation (10). Let us parameterize the step from the old to the new Lagrange multipliers via a parameter a: X = XO + a8X (11) where Xo is the vector of Lagrange multipliers before the step, 8X is the step, and when a = 1, the step is completed. Now, the amount of constraint violation and the Lagrange multipliers are linear functions of this a. Therefore, we can analytically derive the a at which a constraint is violated or a Lagrange multiplier changes sign (see figure 4). For currently inactive constraints, the a for constraint violation is: Ck + Lj AJ Li AjiAki ak = (12) Lj 8Aj Li AjiAki For a currently active constraint, the a for a Lagrange multiplier sign change is simply: (13) We choose the constraint that has a smallest positive ak. If the smallest ak is greater than 1, then the system has found the solution, and the final weights are computed from the Lagrange multipliers at the end of the step. Otherwise, if the kth constraint is active, we make it inactive, and vice versa. We then set the Lagrange multipliers to be the interpolated values from equation (11) with a = ak. We finally re-evaluate equation (10) with the updated active set2 . When this optimization algorithm is applied to the E13B font, the templates that result are shown in figure 5. When applied to characters that obey the specification, the classifier is guaranteed to give a 2:1 margin between the correct peak and any false peak caused by the confluence of two partial characters. Each template has 1523 constraints and takes 7.5 minutes on a SPARe 2 to train. Back-propagation on the 128,000 training examples that are equivalent to the constraints would obviously require much more computation time. IThe sign of the Lagrange multiplier indicates on which side of the inequality constraint the constrained minimum lies. 2 For more details on active set methods, such as how to recognize infeasible constraints, consult (Gill, et al., 1981). A Neural Network Classifier for the 11000 OCR Chip 943 Figure 5: The weights for the fourteen E13B templates. The light pixels correspond to positive weights, while the dark pixels correspond to negative weights. 14 output neurons history of 11000 outputs pinger neuron ~ spatial window of 15 pixels 2 hidden neurons 14 outputs of the 11000 (Every vertical column contains 13 zeros) Figure 6: The software second layer 4 THE SOFTWARE SECOND LAYER As a test of the linear classifier, we fabricated the 11000 and tested it with E 13B characters on real checks. The system worked when the printing on the check obeyed the contrast specification of the font. However, some check printing companies use very light or very dark printing. Therefore, there was no single threshold that could consistently read the lightly printed checks without hallucinating characters on the dark checks. The retina shown in figure 2 does not have automatie gain control (AGC). One solution would have been to refabricate the chip using an AGC retina. However, we opted for a simpler solution. The output of the 11000 chip is a 2-bit confidence level and a character code that is sent to an inexpensive microprocessor every 50 microseconds. Because this output bandwidth is low, it is feasible to put a small software second layer into this microprocessor to post-process and clean up the output of the 11000. The architecture of this software second layer is shown in figure 6. The input to the second layer is a linearly time-warped history of the output of the 11000 chip. The time warping makes the second layer immune to changes in the velocity of the check in the slot. There is one output neuron that is a "pinger." That is, it is trained to turn on when the input to the 11000 chip is centered over any character (Platt, et al. , 1992) (Martin & Pittman, 1992). There are fourteen other neurons that each correspond to a character in the font. These neurons are trained to turn on when the appropriate character is centered in the field, and otherwise turn off. The classification output is the output of the fourteen neurons only when the pinger neuron is on. Thus, the pinger neuron aids in segmentation. Considering the entire network spanning both the hardware first layer and software 944 J. C. PLATT. T. P. ALLEN second layer, we have constructed a non-standard TDNN (Waibel, et. al., 1989) which recognizes characters. We trained the second layer using standard back-propagation, with a training set gathered from real checks. Because the nooo output bandwidth is quite low, collecting the data and training the network was not onerous. The second layer was trained on a data set of approximately 1000 real checks. 5 OVERALL PERFORMANCE When the hardware first layer in the 11000 is combined with the software second layer, the performance of the system on real checks is quite impressive. We gathered a test set of 1500 real checks from across the country. This test set contained a variety of light and dark checks with unusual backgrounds. We swiped this test set through one system. Out of the 1500 test checks, the system only failed to read 2, due to staple holes in important locations of certain characters. As such, this test yielded a 99.995% character accuracy on real data. 6 CONCLUSIONS For the 11000 analog VLSI OCR chip, we have created an effective hardware linear classifier that recognizes the E13B font. The behavior of this classifier was specified using constrained optimization. The classifier was designed to have a predictable margin of classification, be immune to lighting variations, and be resistant to random input noise. The classifier was trained using the active set method, which is an enhancement of Vapnik's separating hyperplane algorithm. We used the active set method to find the weights of a template in 7.5 minutes of SPARC 2 time, instead of training on a data set with 128,000 examples. To make the overall system resistant to contrast variation, we separately trained a software second layer on top of this first hardware layer, thereby constructing a non-standard TDNN. The application discussed in this paper shows the utility of using the active set method to very rapidly create either a stand-alone linear classifier or a first layer of a multi-layer network. References P. Gill, W. Murray, M. Wright (1981), Practical Optimization, Section 5.2, Academic Press. J. Lazzaro, S. Ryckebusch, M. Mahowald, C. Mead (1989), "Winner-Take-All Networks of O(N) Complexity," Advances in Neural Information Processing Systems, 1, D. Touretzky, ed., Morgan-Kaufmann, San Mateo, CA. G. Martin, M. Rashid (1992), "Recognizing Overlapping Hand-Printed Characters by Centered-Object Integrated Segmentation and Recognition," Advances in Neural Information Processing Systems, 4, Moody, J., Hanson, S., Lippmann, R., eds., Morgan-Kaufmann, San Mateo, CA. J. Platt, J. Decker, and J. LeMoncheck (1992), Convolutional Neural Networks for the Combined Segmentation and Recognition of Machine Printed Characters, USPS 5th Advanced Technology Conference, 2, 701-713. V. Vapnik (1982), Estimation of Dependencies Based on Empirical Data, Addendum I, Section 2, Springer-Verlag. A. Waibel, T. Hanazawa, G. Hinton, K. Shikano, K. Lang (1989), "Phoneme Recognition Using Time-Delay Neural Networks," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, pp. 328-339.
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A Neural Network Autoassociator for Induction Motor Failure Prediction Thomas Petsche, Angelo Marcantonio, Christian Darken, Stephen J. Hanson, Gary M. Kuhn and Iwan Santoso [PETSCHE, ANGELO, DARKEN, JOSE, GMK, NIS]@SCR.SIEMENS.COM Siemens Corporate Research, Inc. 755 College Road East Princeton, NJ 08853 Abstract We present results on the use of neural network based autoassociators which act as novelty or anomaly detectors to detect imminent motor failures. The autoassociator is trained to reconstruct spectra obtained from the healthy motor. In laboratory tests, we have demonstrated that the trained autoassociator has a small reconstruction error on measurements recorded from healthy motors but a larger error on those recorded from a motor with a fault. We have designed and built a motor monitoring system using an autoassociator for anomaly detection and are in the process of testing the system at three industrial and commercial sites. 1 Introduction An unexpected breakdown of an electric induction motor can cause financial loss significantly in excess of the cost of the motor. For example, the breakdown of a motor in a production line during a production run can cause the loss of work in progress as well as loss of production time. When a motor does fail, it is not uncommon to replace it with an oversized motor based on the assumption that if a motor is not running at its design limit then it will survive longer. While this is frequently effective, this leads to significantly lower operating efficiencies and higher initial and operating costs. The primary motivation behind this project is the observation that if a motor breakdown and be predicted before the actual breakdown occurs, then the motor can be replaced in a more orderly way, with minimal interruption of the process in which it is involved. The goal is to produce a system that is conceptually similar to a fuel gauge on an automobile. When the system detects conditions that indicate that the motor is approaching its end-of-life, the operators are notified that a replacement is necessary in the near future. A Neural Network Autoassociator for Induction Motor Failure Prediction 925 2 Background At present, motors in critical operations that are subject to mechanical failures - for example, fire pump motors on US Navy vessels - are typically monitored by a human expert who periodically listens to the vibrations of the motor and, based on experience, determines whether the motor sounds healthy or sounds like a problem is developing. Since mechanical probiems in motors typically lead to increased or changed vibrations, this technique can werk well. Unfortunately, it depends on a competent and expensive expert. In an attempt to automate motor monitoring, several vendors have "automated motor monitoring" equipment available. For mechanical failure monitoring, such systems typically rely on several accelerometers to measure the vibration of the motor at various points and along various axes. The systems then display information, primarily about the vibration spectrum, to an operator who determines whether the motor is functioning properly. These systems are expensive since they rely on several accelerometers, each of which is itself expensive, as well as data collection hardware and a computer. Further, the systems require an expert operator and frequently require that the motor be tested only when it is driving a known load. Neither the human motor expert nor the existing motor monitoring systems provide an affordable solution for continuous on-line mechanical failure monitoring. However, the success of the human expert and existing vibration monitors does demonstrate that in fact, there is sufficient information in the vibration of an electric induction motor to detect imminent mechanical failures. Siemens Energy and Automation has proposed a new product, the Siemens Advanced Motor Master System II (SAMMS II), that will continuously monitor and protect an electric induction motor while it is operating on-line. Like the presently available SAMMS, the SAMMS II is designed to provide protection against thermal and electrical overload an, in addition, it will provide detection of insulation deterioration and mechanical fault monitoring. In contrast to existing systems and techniques, the SAMMS II is designed to (1) require no human expert to determine if a motor is developing problems; (2) be inexpensive; and (3) provide continuous, on-line monitoring of the motor in normal operation. The requirements for the SAMMS II, in partiCUlar the cost constraint, require that several issues be resolved. First, in order to produce a low cost system, it is necessary to eliminate the need for expensive accelerometers. Second, wiring should be limited to the motor control center, i.e., it should not be necessary to run new signal wires from the motor control center to the motor. Third, the SAMMS II is to provide continuous on-line monitoring, so the system must adapt to or factor out the effect of changing loads on the motor. Finally since the SAMMS II would not necessarily be bundled with a motor and so might be used to control and monitor an arbitrary motor from an arbitrary manufacturer, the design can not assume that a full description of the motor construction is available. 3 Approach The first task was to determine how to eliminate the accelerometers. Based on work done elsewhere (Schoen, Habetler & Bartheld, 1994), SE&A determined that it might be possible to use measurements of the current on a single phase of the power supply to estimate the vibration of the motor. This depends on the assumption that any vibration of the motor will cause the rotor to move radially relative to the stator which will cause changes in the airgap which, in tum, will induce changes in the current. Experiments were done at the Georgia Institute of Technology to determine the feasibility of this idea using the same sort of data collection system described later. Early experiments indicated that, for a single motor driving a variety of loads, it is possible to distinguish 926 T. PETSCHE, A. MARCANTONIO, C. DARKEN, S. J. HANSON, G. M. KUHN, I. SANTOSO Table 1: Loads for motors #1 and #2. Load type constant sinusoidal oscillation at rotating frequency sinusoidal oscillation at twice the rotating frequency switching load (50% duty cycle) at rotating frequency sinusoidal oscillation 28 Hz sinusoidal oscillation at 30 Hz switching load (50% duty cycle) at 30 Hz Load Magnitude half and full rated half and full rated full rated full rated half and full rated full rated full rated Table 2: Neural network classifier experiment. Features (N) Performance on motor #1 Performance on motor #2 48 100% 63 100% 30% 64 92% 25% 110 100% 55% 320 100% 37% between a current spectrum obtained from the motor while it is healthy and another obtained when the motor contains a fault. Moreover, it is also possible to automatically generate a classifiers that correctly determine the presence or absence of a fault in the motor. The first, obvious approach to this monitoring task would seem to be to build a classifier that would be used to distinguish between a healthy motor and one that has developed a fault that is likely to lead to a breakdown. Unfortunately, this approach does not work. As described above, we have successfully built classifiers of various sorts using manual and automatic techniques to distinguish between current spectra obtained from a motor when it is healthy and those obtained when it contains a fault. However, since the SAMMS II will be connected to a motor before it fails and will be asked to identify a failure without ever seeing a labeled example of a failure from that motor, a classifier can only be used if it can be trained on data collected from one or more motors and then used to monitor the motor of interest. Unfortunately, experiments indicate that this will not work. One of these experiments is illustrated in table 2. Several feedforward neural network classifiers were trained using examples from a single motor under four conditions: (1) healthy, (2) unbalanced, (3) containing a broken rotor bar and (4) containing a hole in the outer bearing race. The ten different loads listed in table 1 were applied to the motor for each of these conditions. The networks contained N inputs (where N is given in table 2); 9 hidden units and 4 outputs. There were 40 training examples where each example is the average of 50 distinct magnitude scaled FFrs obtained from motor #1 from a single load/fault combination. The test data for which the results are reported in the table consisted of 40 averaged FFfs from motor #1 and 20 averaged FFfs (balanced and unbalanced only) from motor #2. The test set for motor #1 is completely distinct from the training set. In the case where n = 110, the FFf components were selected to include the frequencies identified by the theory of motor physics as interesting for the three fault conditions and exclude all other components. This led to an improvement over the other cases where a single contiguous set of components was chosen, but the performance still degrades to about random chance instead of 100%. This experiment clearly illustrates that is is possible to distinguish between healthy and faulty spectra obtained from the same motor. However, it also clearly illustrates that a A Neural Network Autoassociator for Induction Motor Failure Prediction Measurements Adaptation AlgOrithm Novelty detection Novelty Decision Diagnosis Figure 1: The basic form of an anomaly detection system. 927 classifier trained on one motor does not perform well on another motor since the error rates increase immensely. Based on results such as these, we have concluded that it is not feasible to build a single classifier that would be trained once and then placed in the field to monitor a motor. Instead we are pursuing an alternative based on anomaly detection which adapts a monitor to the particular motor for which it is responsible. 4 Anomaly detection The basic notion of anomaly detection for monitoring is illustrated in figure 1. Statistical anomaly detection centers around a model of the data that was seen while the motor was operating normally. This model is produced by collecting spectra from the motor while it is operating normally. Once trained, the system compares each new spectrum to the model to determine how similar to or different from the training set it is. This similarity is described by an "anomaly metric" which, in the simplest case, can be thresholded to determine whether the motor is still normal or has developed a fault. Once the "anomaly metric" has been generated, various statistical techniques can be used to determine if there has been a change in the distribution of values. 5 A Neural Network-based Anomaly Detector The core of the most successful monitoring system we have built to date is a neural network designed to function as an autoassociator (Rumelhart, Hinton & Williams, 1986, called it an "encoder"). We use a simple three layer feedforward network with N inputs, N outputs and K < N hidden units. The input layer is fully connected to the hidden layer which is fully connected to the output layer. Each unit in the hidden and output layers computes Xi = (J ( 2::;0 Wi,jXj) , where Xi is the output of neuron i which receives inputs from Mi other neurons and Wi,j is the weight on the connection from neuron} to neuron i. The network is trained using the backpropagation algorithm to reconstruct the input vector on the output units. Specifically, if Xi is one of n input vectors and Xi is the corresponding output vector, the network is trained to minimize the sum of squared errors E = 2::~1 Ilxi - xdl2. Once training is complete, the anomaly metric is mi = IIXi - xi11 2. 6 Anomaly Detection Test We have tested the effectiveness of the neural network autoassociator as an anomaly detector on several motors. For all these tests, the autoasociator had 20 hidden units. The hidden layer size was chosen after some experimentation and data analysis on motor #1 , but no attempt was made to tune the' hidden layer size for motor #2 or motor #3. Motor #1 was tested using the ten different loads listed in table 1 and four different 928 T. PETSCHE, A. MARCANTONIO, C. DARKEN, S. 1. HANSON, O. M. KUHN, I: SANTOSO <Xl o "! o 0.0 o.oooos 0.0001 Threshold 0.00015 0.0002 q ,---------------------------~ <Xl o C\I o .. ..-. .. ... .... : ......... ,.;-.. , ,_.balanced unbalanced 0.0 0.00002 0.00004 0.00006 0.00008 0.00010 0.00012 Threshold Figure 2: Probability of error as a function of threshold using individual FFfs on (a) motor #1 with 319 inputs and (b) motor #2 with 320 inputs. health/fault conditions: healthy (balanced); unbalanced; broken rotor bar; and a hole in the outer bearing race. Motor #2 was tested while driving the same ten loads, but for one healthy and one faulty condition: healthy (balanced) and unbalanced. For both motors #1 and #2, recordings of a single current phase were made as follows. For each fault condition, a load was selected and applied and the motor was run and the current signal recorded for five minutes. Then a new load was introduced and the motor was run again. The load was constant during any five minute recording session. Motor #3 was tested using thirteen different loads, but only two fault conditions: healthy (balanced) and unbalanced. In this case, however, load changes occurred at random times. We preprocessed this data to to identify where the load changes occurred to generate the training set and the healthy motor test sets. 6.1 Preprocessing Recordings were made on a digital audio tape (OAT). The current on a single phase was measured with a current transformer, amplified, notch filtered to reduce the magnitude of the 60Hz component, amplified again and then applied as input to the OAT. The notch filter was a switched capacitor filter which reduced the magnitude at 60Hz by about 30dB. The time series obtained from the OAT was processed to reduce the sampling rate and then dividing the data into non-overlapping blocks and computing the FFT of each block. A subset of the FFf magnitude coefficients was selected and for each FFT, independent of any other FFf, the components were linearly scaled and translated to the interval [e, 1 e] (typically e = 0.02). That is, for each FFT consisting of coefficients to, ... .tn-t, we selected a subset, F, (the same for all FFTs) of the components and computed a = (l - 2e)(maxiEFh - miniEFh)-t and b = miniEFh. Then the input vector, x, to the network is Xj = a(fij - b) + e where, for allj < k: ij, ik E F and ij < ik. 6.2 Experimental Results In figure 2a, we illustrate the results of a typical anomaly detection experiment on motor #1 using an autoassociator with 319 inputs and 20 hidden units. This graph illustrates the performance (false alarm and miss rates) of a very simple anomaly detection system which thresholds the anomaly metric to determine if the motor is good or bad. The decreasing curve that starts at threshold = 0, P(error) = 1 is the false alarm rate as a function of the threshold. Each increasing curve is the miss rate for a particular fault type. In figure 2b we illustrate the performance of an autoassociator on motor #2 using an A Neural Network Autoassociator for Induction Motor Failure Prediction 929 q I <Xl ci %~ ii iL-.:t: 0 '" ci 0 ci 0.0 , ,,/ .... -. ...... /~',. .... / / .I ..... - ' .. / ------------1 0.0001 0.0002 0.0003 0.0004 0.0005 Threshold Figure 3: Probability of error for motor #3 using individual FFTs and 319 inputs. q ,----------------------------, q ,---------------------------~ <Xl ci 0.0 0.00005 , ......... 0.0001 0.00015 Threshold 0.0002 <Xl o '" ci o .,.-' ,/ balanced unbalanced ci ~---.---,r_--.---~--_r--_.~ 0.0 0.00002 0.00004 0.00006 0.00008 0.00010 0.00012 Threshold Figure 4: Probability of error using averaged FFTs for (a) motor #1 and 319 inputs (b) motor #2 and 320 inputs. autoassociator with 320 inputs and 20 hidden units. Figure 3 shows our results on motor #3 using an autoassociator with 319 inputs. We have found significant performance improvements by averaging several consecutive FFTs. In figure 4 we show the results for motors #1 and #2 when we averaged 11 FFTs to produce the input features. Compare these curves to those in figure 2. In particular, notice that the probability of error is much lower for the averaged FFTs when the good motor curve crosses anyone of the faulty motor curves. 7 Candor System Design Based on our experiments with autoassociators, we designed a prototype mechanical motor condition monitoring system. The functional system architecture is shown in figure 5. In order to control costs, the system is implemented on a PC. The system is designed so that each PC can monitor up to 128 motors using one 16-bit analog to digital converter. The signals are collected, filtered and multiplexed on custom external signal processing cards. Each card supports up to eight motors (with up to 16 cards per PC). The system records current measurements from one motor at a time. For each motor, measurements are collected, four FFTs are computed on non-overlapping time series, and the four FFTs are averaged to produce a vector that is input to the neural network. The system reports that a motor is bad only if more than five of the last ten averaged FFTs produced an anomaly metric more than five standard deviations greater than the mean metric computed on the training set. Otherwise the motor is reported to be normal. In addition to monitoring the motors, the prototype systems are designed to record all measurements on tape to support 930 T. PETSCHE, A. MARCANTONIO, C. DARKEN, S. 1. HANSON, G. M. KUHN, I. SANTOSO Figure 5: Functional architecture of Candor. GOOD BAD future experiments with alternative algorithms and tuning to improve performance. To date, three monitoring systems have been installed: in an oil refinery, in a testing laboratory and on an office building ventilation system. The system has correctly detected the only failure it has seen so far: when a filter on the inlet to a water circulation pump became clogged the spectrum changed so much that the average daily novelty metric jumped from less than one standard deviation above the training set average to more than twenty standard deviations. We hope to have further test results in a year or so. 8 Related work Gluck and Myers (1993) proposed a model oflearning in the hippocampus based in part on an autoassociator which is used to detect novel stimuli and to compress the representation of the stimuli. This model has accurately predicted many of the classical conditioning behaviors that have been observed in normal and hippocampal-damaged animals. Based on this work, Japkowicz, Myers and Gluck (1995) independently derived an autoassociatorbased novelty detector for machine learning tasks similar to that used in our system. Together with Gluck, we have tested an autoassociator based anomaly detector on helicopter gearbox failures for the US Navy. In this case, the autoassociator is given 512 inputs consisting of 64 vibration based features from each of 8 accelerometers mounted at different locations on the gearbox. In a blind test, the autoassociator was able to correctly distinguish between feature vectors taken from a damaged gearbox and other feature vectors taken from normal gearboxes, all recorded in flight. Our anomaly detector will be included in test flights of a gearbox monitoring system later this year. References Gluck, M. A. & Myers, C. E. (1993). Hippocampal mediation of stimulus representation: A compuational theory. Hippocampus, 3(4), 491-561. Japkowicz, N., Myers, c., & Gluck, M. A. (1995). A novelty detection approach to classification. In Proceedings of the Fourteenth International Joint Conference on Artificial Intelligence. Rumelhart, D., Hinton, G., & Williams, R. (1986). Learning internal representations by error propagation. In D. Rumelhart & J. McClelland (Eds.), Parallel Distributed Processing (pp. 318-362). MIT Press. Schoen, R., Habetler, T., & Bartheld, R. (1994). Motor bearing damage detection using stator current monitoring. In Proceedings of the IEEE lAS Annual Meeting.
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Modeling Saccadic Targeting in Visual Search Rajesh P. N. Rao Computer Science Department University of Rochester Rochester, NY 14627 rao@cs.rochester.edu Mary M. Hayhoe Center for Visual Science University of Rochester Rochester, NY 14627 mary@cvs.rochester.edu Gregory J. Zelinsky Center for Visual Science University of Rochester Rochester, NY 14627 greg@cvs.rochester.edu Dana H. Ballard Computer Science Department University of Rochester Rochester, NY 14627 dana@cs.rochester.edu Abstract Visual cognition depends criticalIy on the ability to make rapid eye movements known as saccades that orient the fovea over targets of interest in a visual scene. Saccades are known to be ballistic: the pattern of muscle activation for foveating a prespecified target location is computed prior to the movement and visual feedback is precluded. Despite these distinctive properties, there has been no general model of the saccadic targeting strategy employed by the human visual system during visual search in natural scenes. This paper proposes a model for saccadic targeting that uses iconic scene representations derived from oriented spatial filters at multiple scales. Visual search proceeds in a coarse-to-fine fashion with the largest scale filter responses being compared first. The model was empirically tested by comparing its perfonnance with actual eye movement data from human subjects in a natural visual search task; preliminary results indicate substantial agreement between eye movements predicted by the model and those recorded from human subjects. 1 INTRODUCTION Human vision relies extensively on the ability to make saccadic eye movements. These rapid eye movements, which are made at the rate of about three per second, orient the high-acuity foveal region of the eye over targets of interest in a visual scene. The high velocity of saccades, reaching up to 700° per second for large movements, serves to minimize the time in flight; most of the time is spent fixating the chosen targets. The objective of saccades is currently best understood for reading text [13] where the eyes fixate almost every word, sometimes skipping over smalI function words. In general scenes, however, the purpose of saccades is much more difficult to analyze. It was originally suggested that Modeling Saccadic Targeting in Visual Search 831 (a) (b) Figure 1: Eye Movements in Visual Search. (a) shows the typical pattern of multiple saccades (shown here for two different subjects) elicited during the course of searching for the object composed of the fork and knife. The initial fixation point is denoted by • +'. (b) depicts a summary of such movements over many experiments as a function of the six possible locations of a target object on the table. the movements and their resultant fixations formed a visual-motor memory (or "scan-paths") of objects [11] but subsequent work has suggested that the role of saccades is more tightly coupled to the momentary problem solving strategy being employed by the subject. In chess, it has been shown that saccades are used to assess the current situation on the board in the course of making a decision to move, but the exact information that is being represented is not yet known [5]. In a task involving the copying of a model block pattern located on a board, fixations have been shown to be used in accessing crucial information for different stages of the copying task [2]. In natural language processing, there has been recent evidence that fixations reflect the instantaneous parsing of a spoken sentence [18]. However, none of the above work addresses the important question of what possible computational mechanisms underlie saccadic targeting. The complexity of the targeting problem can be illustrated by the saccades employed by subjects to solve a natural visual search task. In this task, subjects are given a 1 second preview of a single object on a table and then instructed to determine, in the shortest possible amount of time, whether the previewed object is among a group of one to five objects on the same table in a subsequent view. The typical eye movements elicited are shown in Figure 1 (a). Rather than a single movement to the remembered target, several saccades are typical, with each successive saccade moving closer to the goal object (Figure 1 (b» . The purpose of this paper is to describe a mechanism for programming saccades that can appro x imately model the saccadic targeting method used by human SUbjects. Previous models of human visual search have focused on simple search tasks involving elementary features such as horizontaVvertical bars of possibly different color [1,4,8] or have relied exclusively on bottom-up input-driven saliency criteria for generating scan-paths [10, 19]. The proposed model achieves targeting in arbitrary visual scenes by using bottom-up scene representations in conjunction with previously memorized top-down object representations; both of these representations are iconic, based on oriented spatial filters at multiple scales. One of the difficult aspects of modeling saccadic targeting is that saccades are ballistic, i.e., their final location is computed prior to making the movement and the movement trajectory is uninterrupted by incoming visual signals. Furthermore, owing to the structure of the retina, the central 1.50 of the visual field is represented with a resolution that is almost 100 times greater than that of the periphery. We resolve these issues by positing that the targeting computation proceeds sequentially with coarse resolution information being used in the computation of target coordinates prior to fine resolution information. The method is compared to actual eye movements made by human subjects in the visual search task described above; the eye movements predicted by the model are shown to be in close agreement with observed human eye movements. 832 R. P. N. RAO, G. J. ZELINSKY, M. M. HAYHOE, D. H. BALLARD Figure 2: Multiscale Natural Basis Functions. The 10 oriented spatial filters used in our model to generate iconic scene representations, shown here at three octave-separated scales. These filters resemble the receptive field profiles of cells in the primate visual cortex [20] and have been shown to approximate the dominant eigenvectors of natural image distributions as obtained from principal component analysis [7,17]. 2 ICONIC REPRESENTATIONS The current implementation of our model uses a set of non-orthogonal basis functions as given by a zeroth order Gaussian Go and nine of its oriented derivatives as follows [6]: G~n,n = 1,2,3,8n = 0, ... ,m7r/(n + l),m = 1, ... ,n (1) where n denotes the order of the filter and 8n refers to the preferred orientation of the filter (Figure 2). The response of an image patch I centered at (xo, Yo) to a particular basis filter G~; can be obtained by convolving the image patch with the filter: Ti,i(xO,YO) = II G~;(xo-x,yo-y)I(x,y)dxdy (2) The iconic representation for the local image patch centered at (xo, Yo) is formed by combining into a high-dimensional vector the responses from the ten basis filters at different scales: rs(xo,Yo) = h,i,s(XO, YO)] (3) where i = 0, 1, 2, 3 denotes the order of the filter, j = 1, ... , i + 1 denotes the different filters per order, and S = Smin, ... ,Sma., denotes the different scales as given by the levels of a Gaussian image pyramid. The use of multiple scales is crucial to the visual search model (see Section 3). In particular, the larger the number of scales, the greater the perspicuity of the representation as depicted in Figure 3. A multiscale representation also alJows interpolation strategies for scale invariance. The bighdimensionality of the vectors makes them remarkably robust to noise due to the orthogonality inherent in high-dimensional spaces: given any vector, most of the other vectors in the space tend to be relatively uncorrelated with the given vector. The iconic representations can also be made invariant to rotations in the image plane (for a fixed scale) without additional convolutions by exploiting the property of steerability [6]. Rotations about an image plane axis are handled by storing feature vectors from different views. We refer the interested reader to [14] for more details regarding the above properties. 3 THE VISUAL SEARCH MODEL Our model for visual search is derived from a model for vision that we previously proposed in [14]. This model decomposes visual behaviors into sequences of two visual routines, one for identifying the visual image near the fovea (the "what" routine), and another for locating a stored prototype on the retina (the "where" routine). Modeling Saccadic Targeting in Visual Search I "-Q~II G.O'..., I .. i . Con ... .:. (a) j .. i (b) ~ ~ 833 Figure 3: The Effect of Scale. The distribution of distances (in tenns of correlations) between the response vector for a selected model point in the dining table scene and all other points in the scene is shown for single scale response vectors (a) and multiple scale vectors (b). Using responses from multiple scales (five in this case) results in greater perspicuity and a sharper peak near 0.0; only one point (the model point) had a correlation greater than 0.94 in the multiple scale case (b) whereas 936 candidate points fell in this category in the single scale case (a). The visual search model assumes the existence of three independent processes running concurrently: (a) a targeting process (similar to the "where" routine of [14]) that computes the next location to be fixated; (b) an oculomotor process that accepts target locations and executes a saccade to foveate that location (see [16] for more details); and (c) a decision process that models the cortico-cortical dynamics of the VI f+ V2 f+ V 4 f+ IT pathway related to the identification of objects in the fovea (see [15] for more details). Here, we focus on the saccadic targeting process. Objects of interest to the current search task are assumed to be represented by a set of previously memorized iconic feature vectors r-:p where s denotes the scale of the filters. The targeting algorithm computes the next location to be foveated as follows: 1. Initialize the routine by setting the current scale of analysis k to the largest scale i.e. k = max; set Sm(x, y) = 0 for all (x, y). 2. Compute the current saliency image Sm as max Sm(x, y) = L Ilr~(x, y) - r-:Pll2 (4) s=k 3. Find the location to be foveated by using the following weighted population averaging (or soft max) scheme: (x, '0) = L F(S7n(x, y))(x, y) (x,y) where F is an interpolation function. For the experiments, we chose: e-S",(x,y)/)'(k) F(S7n(x, y)) = E e-S",(x,y)/)'(k) (x,y) (5) (6) This choice is attractive since it allows an interpretation of our algorithm as computing maximum likelihood estimates (cf. [12]) of target locations. In the above, >'(k) is decreased with k. 4. Iterate step (2) and (3) above with k = max-I, max-2, . . . until either the target object has been foveated or the number of scales has been exhausted. Figure 4 illustrates the above targeting procedure. The case where multiple model vectors are used per object proceeds in an analogous manner with the target location being averaged over all the model vectors. 834 R. P. N. RAO, G. J. ZELINSKY, M. M. HAYHOE, D. H. BALLARD (c) (d) Figure 4: mustration of Saccadic Targeting. The saliency image after the inclusion of the largest (a), intermediate (b), and smallest scale (c) as given by image distances to the prototype (the fork and knife); the lightest points are the closest matches. (d) shows the predicted eye movements as determined by the weighted population averaging scheme (for comparison, saccades from a human subject are given by the dotted arrows). 4 EXPERIMENTAL RESULTS AND DISCUSSION Eye movements from four human subjects were recorded for the search task described in Section 1 for three different scenes (dining table, work bench, and a crib) using an SRI Dual Purkinje Eyetracker. The model was implemented on a pipeline image processor, the Datacube MV200, which can compute convolutions at frame rate (30/ sec). Figure 5 compares the model's performance to the human data. As the results show, there is remarkably good correspondence between the eye movements observed in human subjects and those generated by the model on the same data sets. The model has only one important parameter: the scaling function used to rate the peaks in the saliency map. In the development of the algorithm, this was adjusted to achieve an approximate fit to the human data. Our model relies crucially on the existence of a coarse-to-fine matching mechanism. The main benefit of a coarse-to-fine strategy is that it allows continuous execution of the decision/oculomotor processes, thereby increasing the probability of an early match. Coarse-to-fine strategies have enjoyed recent popularity in computer vision with the advent of image pyramids in tasks such as motion detection [3]. Although these methods show that considerable speedup can be achieved by decreasing the size of window of analysis as resolution increases, our preliminary experiments suggest that this might be an inappropriate strategy for visual search: limiting search to a small window centered on the coarse location estimate obtained from a larger scale often resulted in significant errors since the targets frequently lay outside the search window. A possible solution is to adaptively select the size of the search window based on the current scene but this would require additional computational machinery. A key question that remains is the source of sequential application of the filters in the human visual system. A possible source is the variation in resolution of the retina. Since only very high resolution information is at the fovea, and since this resolution falls off with distance. fine spatial scales may be ineffective purely because the fixation point is distant from the target. However. our preliminary experiments with modeling the variation in retinal resolution suggest that this is probably not the sole cause. The variations at middle distances from the fovea are too small to explain the dramatic improvement in target location experienced with the second saccade. Thus, Modeling Saccadic Targeting in Visual Search ~ 2D ; IS : 10 First SICCldOl: Hullln 8 11 16 10 1~ 18 51 S.cond SICCOdOl: Hullon I 1'""' I 56 40 ~::I~ .: :, ,., .... ,.~,.,.,., ~,.,.,.,.., 1"",., 4 8 11 16 10 14 18 51 56 40 Third SoccodOl: HUIIIO ~ 50 :0 .:~ l! 40 ; 30 : 20 .: ': ,I,.,.,.~",..""", I I I I '-I 31 36 40 8 '1 16 10 14 18 En~olnt Error (,1 •• 11 • 10) First Socc.d .. : nod.1 ~ 15 '" ~ 20 .. ::1 o 15 ! '~ ,.",., ••• ,III,I,I,I,.,.,.,.,.,.~,., ~ 8 11 16 10 1~ 18 31 56 40 S.clnd SoccodOl: nod.1 ~ 15 '" ~ 20 .. ::l ; IS : 10 .: :,.1,.,11.,1.,.,.,.,.,., ... ,.,., ,..,., ~5D '" l! 40 ; 30 : 20 .. .. '0 4 8 11 16 10 1~ 18 51 36 40 Third Soccldos: nod.1 .. ::1 0'-"1 ,-I""'I I 1""1 I '-1-1-' .-. I I I 1-' ~ 8 '1 16 10 1~ 18 51 36 40 Endpoint Error (,I •• ls • 10) 835 Figure 5: Experimental Results. The graphs compare the distribution of endpoint errors (in terms of frequency histograms) for three consecutive saccades as predicted by the model for 180 trials (on the right) and as observed with four human subjects for 676 trials (left). Each of the trials contained search scenes with one to five objects, one of the objects being the previeWed model. there are two remaining possibilities: (a) the resolution fall-off in the cortex is different from the retinal variation in a way that supports the data, or (b) the cortical machinery is set up to match the larger scales first. In the latter case, the observed data would result from the fact that the oculomotor system is ready to move before all the scales can be matched, and thus the eyes move to the current best target position. This interpretation of the data is appealing in two aspects. First, it reflects a long history of observations on the priority of large scale channels [9], and second, it reflects current thinking about eye movement programming suggesting that fixation times are approximately constant and that the eyes are moved as soon as they can be during the course of visual problem solving. The above questions can however be definitively answered only through additional testing of human subjects followed by subsequent modeling. We expect our saccadic targeting model to playa crucial role in this process. Acknowledgments This research was supported by NIHIPHS research grants 1-P41-RR09283 and 1-R24-RR0685302, and by NSF research grants IRI-9406481 and IRI-8903582. 836 R. P. N. RAO. G. 1. ZELINSKY. M. M. HAYHOE. D. H. BALLARD References [1] Subutai Ahmad and Stephen Omohundro. Efficient visual search: A connectionist solution. In Proceeding of the 13th Annual Conference of the Cognitive Science Society. Chicago, 1991. [2] Dana H. Ballard, Mary M. Hayhoe, and Polly K. Pook. Deictic codes for the embodiment of cognition. Technical Report 95.1, National Resource Laboratory for the study of Brain and Behavior, University of Rochester, January 1995. [3] P.J. Burt. Attention mechanisms for vision in a dynamic world. In ICPR, pages 977-987, 1988. [4] David Chapman. Vision. Instruction. and Action. PhD thesis, MIT Artificial Intelligence Laboratory, 1990. (Technical Report 1204). [5] W.G. Chase and H.A. Simon. Perception in chess. Cognitive Psychology, 4:55-81,1973. [6] William T. Freeman and Edward H. Adelson. The design and use of steerable filters. IEEE PAMI, 13(9):891-906, September 1991. [7] Peter lB. Hancock, Roland J. Baddeley, and Leslie S. Smith. The principal components of natural images. Network, 3:61-70, 1992. [8] Michael C. Mozer. The perception of multiple objects: A connectionist approach. Cambridge, MA: MIT Press, 1991. [9] D. Navon. Forest before trees: The precedence of global features in visual perception. Cognitive Psychology, 9:353-383,1977. [10] Ernst Niebur and ChristofKoch. Control of selective visual attention: Modeling the "where" pathway. This volume, 1996. [11] D. N oton and L. Stark. Scanpaths in saccadic eye movements while viewing and recognizing patterns. Vision Reseach, 11:929-942,1971. [12] Steven 1. Nowlan. Maximum likelihood competitive learning. In Advances in Neural Infonnation Processing Systems 2, pages 574-582. Morgan Kaufmann, 1990. [13] J.K. O'Regan. Eye movements and reading. In E. Kowler, editor, Eye Movements and Their Role in Visual and Cognitive Processes, pages 455-477. New York: Elsevier, 1990. [14] Rajesh P.N. Rao and Dana H. Ballard. An active vision architecture based on iconic representations. Artificial Intelligence (Special Issue on Vision), 78:461-505, 1995. [15] Rajesh P.N. Rao and Dana H. Ballard. Dynamic model of visual memory predicts neural response properties in the visual cortex. Technical Report 95.4, National Resource Laboratory for the study of Brain and Behavior, Computer Sci. Dept., University of Rochester, November 1995. [16] Rajesh P.N. Rao and Dana H. Ballard. Learning saccadic eye movements using multi scale spatial filters. In G. Tesauro, D.S. Touretzky, and T.K. Leen, editors, Advances in Neural Infonnation Processing Systems 7, pages 893-900. Cambridge, MA: MIT Press, 1995. [17] Rajesh P.N. Rao and Dana H. Ballard. Natural basis functions and topographic memory for face recognition. In Proc. ofllCAl, pages 10-17, 1995. [18] M. Tanenhaus, M. Spivey-Knowlton, K. Eberhard, and I Sedivy. Integration of visual and linguistic information in spoken language comprehension. To appear in Science, 1995. [19] Keiji Yamada and Garrison W. Cottrell. A model of scan paths applied to face recognition. In Proc. 17th Annual Conf. of the Cognitive Science Society, 1995. [20] R.A. Young. The Gaussian derivative theory of spatial vision: Analysis of cortical cell receptive field line-weighting profiles. General Motors Research Publication GMR-4920, 1985.
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Plasticity of Center-Surround Opponent Receptive Fields in Real and Artificial Neural Systems of Vision S. Yasui Kyushu Institute of Technology lizuka 820, Japan M. Yamada Electrotechnical Laboratory Tsukuba 305, Japan T. Furukawa Kyushu Institute of Technology lizuka 820, Japan T. Saito Tsukuba University Tsukuba 305, Japan Abstract Despite the phylogenic and structural differences, the visual systems of different species, whether vertebrate or invertebrate, share certain functional properties. The center-surround opponent receptive field (CSRF) mechanism represents one such example. Here, analogous CSRFs are shown to be formed in an artificial neural network which learns to localize contours (edges) of the luminance difference. Furthermore, when the input pattern is corrupted by a background noise, the CSRFs of the hidden units becomes shallower and broader with decrease of the signal-to-noise ratio (SNR). The same kind of SNR-dependent plasticity is present in the CSRF of real visual neurons; in bipolar cells of the carp retina as is shown here experimentally, as well as in large monopolar cells of the fly compound eye as was described by others. Also, analogous SNRdependent plasticity is shown to be present in the biphasic flash responses (BPFR) of these artificial and biological visual systems. Thus, the spatial (CSRF) and temporal (BPFR) filtering properties with which a wide variety of creatures see the world appear to be optimized for detectability of changes in space and time. 1 INTRODUCTION A number of learning algorithms have been developed to make synthetic neural machines be trainable to function in certain optimal ways. If the brain and nervous systems that we see in nature are best answers of the evolutionary process, then one might be able to find some common 'softwares' in real and artificial neural systems. This possibility is examined in this paper, with respect to a basic visual 160 S. Y ASUI, T. FURUKAWA, M. YAMADA, T. SAITO mechanism relevant to detection of brightness contours (edges). In most visual systems of vertebrate and invertebrate, one finds interneurons which possess centersurround opponent receptive fields (CSRFs). CSRFs underlie the mechanism of lateral inhibition which produces edge enhancement effects such as Mach band. It has also been shown in the fly compound eye that the CSRF of large monopolar cells (LMCs) changes its shape in accordance with SNR; the CSRF becomes wider with increase of the noise level in the sensory environment. Furthermore, whereas CSRFs describe a filtering function in space, an analogous observation has been made in LMCs as regards the filtering property in the time domain; the biphasic flash response (BPFR) lasts longer as the noise level increases (Dubs, 1982; Laughlin, 1982). A question that arises is whether similar SNR-dependent spatia-temporal filtering properties might be present in vertebrate visual cells. To investigate this, we made an intracellular recording experiment to measure the CSRF and BPFR profiles of bipolar cells in the carp retina under appropriate conditions, and the results are described in the first part of this paper. In the second part, we ask the same question in a 3-layer feedforward artificial neural network (ANN) trained to detect and localize spatial and temporal changes in simulated visual inputs corrupted by noise. In this case, the ANN wiring structure evolves from an initial random state so as to minimize the detection error, and we look into the internal ANN organization that emerges as a result of training. The findings made in the real and artificial neural systems are compared and discussed in the final section. In this study, the backpropagation learning algorithm was applied to update the synaptic parameters of the ANN. This algorithm was used as a means for the computational optimization. Accordingly, the present choice is not necessarily relevant to the question of whether the error backpropagation pathway actually might exist in real neural systems( d. Stork & Hall, 1989). 2 THE CASE OF A REAL NEURAL SYSTEM: RETINAL BIPOLAR CELL Bipolar cells occur as a second order neuron in the vertebrate retina, and they have a good example of CSRF Here we are interested in the possibility that the CSRF and BPFR of bipolar cells might change their size and shape as a function of the visual environment, particularly as regards the dark- versus light-adapted retinal states which correspond to low versus high SNR conditions as explained later. Thus, the following intracellular recording experiment was carried out. 2.1 MATERIAL AND METHOD The retina was isolated from the carp which had been kept in complete darkness for 2 hrs before being pithed for sacrifice. The specimen was then mounted on a chamber with the receptor side up, and it was continuously superfused with a Ringer solution composed of (in mM) 102 NaCI, 28 NaHC03 , 2.6 KCI, 1 CaCh, 1 MgCh and 5 glucose, maintained at pH=7.6 and aerated with a gas mixture of 95% O2 and 5% CO2 • Glass micropipettes filled with 3M KCI and having tip resistances of about 150 Mn were used to record the membrane potential. Identification of bipolar cell units was made on the basis of presence or absence of CSRF. For this preliminary test, the center and peripheral responses were examined by using flashes of a small centered spot and a narrow annular ring. To map their receptive field profile, the stimulus was given as flashes of a narrow slit presented at discrete positions 60 pm apart on the retina. The slit of white light was 4 mm long and 0.17 mm wide, and its flash had intensity of 7.24 pW /cm2 and duration of 250 msec. The CSRF measurement was made under dark- and light- adapted conditions. A Plasticity of Center-Surround Opponent Receptive Fields (b) 1.0 (a) Lighl t CCnler o . "'/I~~I~~1~!.yvr""'~ -1.0 i -1.0 _0 _ _ ._._._._._ 0_0_0_0_._ "_ "-". (c) n.,k I ~ I ~~H·\J)rl .. i !rt~~\ .. +,~ SIIlV _ ._ . _ ___ 0_0_ "_"_ 0_ ._0_"_._ " GO/1m ~ I Osee i 0 lsec aUght • Dark 161 ".". i 1.0 I 5mV Figure 1: (a) Intracellular recordings from an ON-center bipolar cell of the carp retina with moving slit stimuli under light and dark adapted condition. (b) The receptive field profiles plotted from the recordings. (c) The response recorded when the slit was positioned at the receptive field center. steady background light of 0.29 JJW /cm2 was provided for light adaptation. 2.2 RESULTS Fig.la shows a typical set of records obtained from a bipolar cell. The response to each flash of slit was biphasic (i.e., BPFR), consisting of a depolarization (ON) followed by a hyperpolarization(OFF). The ON response was the major component when the slit was positioned centrally on the receptive field, whereas the OFF response was dominant at peripheral locations and somewhat sluggish. The CSRF pattern was portrayed by plotting the response membrane potential measured at the time just prior to the cessation of each test flash. The result compiled from the data of Figola is presented in Fig.lb, showing that the CSRF of the darkadapted state was shallow and broad as opposed to the sharp profile produced during light adaptation. The records with the slit positioned at the receptive field center are enlarged in Fig.lc, indicating that the OFF part of the BPFR waveform was shallower and broader when the retina was dark adapted than when light adaptedo 3 THE CASE OF ARTIFICIAL NEURAL NETWORKS Visual pattern recognition and imagery data processing have been a traditional application area of ANNs. There are also ANNs that deal with time series signals. These both types of ANNs are considered here, and they are trained to detect and localize spatial or temporal changes of the input signal corrupted by noise. 162 S. Y ASUI, T. FURUKAWA, M. YAMADA, T. SAITO 3.1 PARADIGMS AND METHODS The ANN models we used are illustrated in Figs.2. The model of Fig.2a deals with one-dimensional spatial signals. It consists of three layers (input, hidden, output), each having the same number of 12 or 20 neuronal units. The pattern given to the input layer represents the brightness distribution of light. The network was trained by means of the standard backpropagation algorithm, to detect and localize step-wise changes (edges) which were distributed on each training pattern in a random fashion with respect to the number, position and height. The mean level of the whole pattern was varied randomly as well. In addition, there was a background noise (not illustrated in Figs.2); independent noise signals of the same statistics were given to the all input units, and the maximum noise amplitude (NL: noise level) remained constant throughout each training session. The teacher signal was the "true" edge positions which were subject to obscuration due to the background noise; the learning was supervised such that each output unit would respond with 1 when a step-wise change not due to the background noise occurred at the corresponding position, and respond with -1 otherwise. The value of each synaptic weight parameter was given randomly at the outset and updated by using the backpropagation algorithm after presentation of each training pattern. The training session was terminated when the mean square error stopped decreasing. To process time series inputs, the ANN model of Fig.2b was constructed with the backpropagation learning algorithm. This temporal model also has three layers, but the meaning of this is quite different from the spatial network model of Fig.2a. That is, whereas each unit of each layer in the spatial model is an anatomical entity, this is not the case with respect to the temporal model. Thus, each layer represents a single neuron so that there are actually only three neuronal elements, i.e., a receptor, an interneuron, and an output cell. And, the units in the same layer represent activity states of one neuron at different time slices; the rightmost unit for the present time, the next one for one time unit ago, and so on. As is apparent from Fig.2b, therefore, there is no convergence from the future (right) to the past (left). Each cell has memory of T-units time. Accordingly, the network requires 2T - 1 units in the input layer, T units in the hidden layer and 1 units in the output layer to calculate the output at present time. The input was a discrete time series in which step-wise changes took place randomly in a manner analogous to the spatial input of Fig.2a. As in the spatial case, there was a background noise (b) Input Correetiom Correctiom Figure 2: The neural network architectures. Spatial (a) and temporal model (b). Plasticity of Center-Surround Opponent Receptive Fields 163 (8) 11I1JJ11JJ111 iFJiiii •• Oulput ut,i'" 2000 4000 10000 30000 0.2 0.1 0.0 01.O-=~~~~4~.0~!!iI!!:!II"'''-'l8.0xl04 Iterations Figure 3: Development of receptive fields. Synaptic weights (a) and mean square error (b), both as a function of the number of iterations. added to the input. The network was trained to respond with + 1/ -1 when the original input signal increased/decreased, and to respond with 0 otherwise. 3.2 RESULTS Spatial case: Emergence of CSRFs with SNR-dependent plasticity As regards the edge detection learning by the ANN model of Fig.2a, the results without the background noise are described first (Furukawa & Yasui, 1990; Joshi & Lee, 1993). Fig.3a illustrates how the synaptic connections developed from the initial random state. If the final distribution of synaptic weight parameters is examined from input units to any hidden unit and also from hidden units to any output unit, then it can be seen in either case that the central and peripheral connections are opposite in the polarity of their HIddm Layer Output Layer \'--I I weight parameters; the central group had eiFigure 4: A Sample of activity ther positive (ON-center) or negative (OFFpattern of each layer center) values, but the reversed profiles are shown in the drawing of Fig.3a for the OFF -center case. In any event, CSRFs were formed inside the network as a result of the edge detection learning. Fig.3b shows the performance improvement during a learning session. FigA shows the activation pattern of each layer in response to a sample input, and edge enhancement like the Mach band effect can be observed in the hidden layer. Fig.5a presents sample input patterns corrupted by the background noise of various NL values, and Fig.5b shows how a hidden unit was connected to the input layer at the end of training. CSRFs were still formed when the environment suffered from the noise. However, the structure of the center-surround antagonism changed as a function of NL; the CSRFs became shallow and broad as NL increased, i.e., as the SNR decreased. Temporal case: Emergence of BPFRs with SNR-dependent plasticity With reference to the learning paradigm of Fig.2b, Fig.5c reveals how a representative hidden unit made synaptic connections with the input units as a function of NL; the weight parameters are plotted against the elapsed time. Each trace would correspond to the response of the hidden unit to a flash of light, and it consists of 164 S. Y ASUI, T. FURUKAWA, M. YAMADA, T. SAITO two phases of ON and OFF, i.e., BPFRs (biphasic flash responses) emerged in this ANN as a result of learning, and the biphasic time course changed depending on NL; the negative-going phase became shallower and longer with decrease of SNR. 4 DISCUSSION: Common Receptive Field Properties in Vertebrate, Invertebrate and Artificial Systems A CSRF profile emerges after differentiating twice in space a small patch of light, and CSRF is a kind of point spreading function. Accordingly, the response to any input distribution can be obtained by convolving the input pattern with CSRF. The double differentiation of this spatial filtering acts to locate edge positions. On the other hand, the waveform of BPFR appears by differentiating once in time a short flash of light. Thus, the BPFR is an impulse response function with which to convolve the given input time series to obtain the response waveform. This is a derivative filtering, which subserves detection of temporal changes in the input visual signal. While both CSRF and BPFR occur in visual neurons of a wide variety of vertebrates and invertebrates, the first part of the present study shows that these spatial and temporal filtering functions can develop autonomously in our ANNs. The neural system of visual signal processing encounters various kinds of noise. There are non-biological ones such as a background noise in the visual input itself and the photon noise which cannot be ignored when the light intensity is low. Endogenous sources of noise include spontaneous photoisomerization in photoreceptor cells, quantal transmitter release at synaptic sites, open/close activities of ion channels and so on. Generally speaking, therefore, since the surroundings are dim when the retina is dark adapted, SNR in the neuronal environment tends to be low during dark adaptation. According to the present experiment on the carp retina, the CSRF of bipolar cells widens in space and the BPFR is prolonged in time when the retina is dark adapted, that is, when SNR is presumably low. Interestingly, the same SNR-dependent properties have also been described in connection with the CSRF and BPFR of large monopolar cells in the fly compound eye. These spatial and temporal observations are both in accord with a notion that a method to remove noise is smoothing which requires averaging for a sufficiently long interval. In other words, when SNR is low, the signal averaging takes place over a large portion of the spatio-temporal domain comprised of CSRF and BPFR. Smoothing and differentiation are entirely opposite in the signal processing role. The SNR dependency of the CSRF and BPFR profiles can be viewed as a compromise between these two operations, for the need to detect signal changes in the presence of noise. These (a) (b) (c) ~ 0 10 20 -io 0 10 10 20 Figure 5: (a) A sample set of training patterns with different background noise levels (NLs). The NLs are 0.0, 0.4, 1.0 from bottom to top. The receptive field profiles (b) and flash responses (c) after training with each NL. The ordinate scale is linear but in arbitrary unit, with the zero level indicated by dotted lines. Plasticity of Center-Surround Opponent Receptive Fields 165 points parallel the results of information-theoretic analysis by Atick and Redlich (1992) and by Laughlin (1982). 5 CONCLUDING REMARKS We have learnt from this study that the same software is at work for the SNRdependent control of the spati~temporal visual receptive field in entirely different hardwares; namely, vertebrate, invertebrate and artificial neural systems. In other words, the plasticity scheme represents nature's optimum answer to the visual functional demand, not a result of compromise with other factors such as metabolism or morphology. Some mention needs to be made of the standard regularization theory. If the theory is applied to the edge detection problem, then one obtains the Laplacian-Gaussian filter which is a well-known CSRF example(Torre & Poggio, 1980). And, the shape of this spatial filter can be made wide or narrow by manipulating the value of a constant usually referred to as the regularization parameter. This parameter choice corresponds to the compromise that our ANN finds autonomously between smoothing and differentiation. The present type of research aided by trainable artificial neural networks seems to be a useful top-down approach to gain insight into the brain and neural mechanisms. Earlier, Lehky and Sejnowski (1988) were able to create neuron-like units similar to the complex cells of the visual cortex by using the backpropagation algorithm, however, the CSRF mechanism was given a priori to an early stage in their ANN processor. It should also be noted that Linsker (1986) succeeded in self-organization of CSRFs in an ANN model that operates under the learning law of Hebb. Perhaps, it remains to be examined whether the CSRFs formed in such an unsupervised learning paradigm might also possess an SNR-dependent plasticity similar to that described in this paper. References Atick, J .J. & Redlich, A.N. (1992) What does the retina know about natural scenes? Neural Computation, 4, 196-210. Dubs, A. (1982) The spatial integration of signals in the retina and lamina of the fly compound eye under different conditions of luminance. 1. Compo Physiol A, 146, 321-334. Furukawa, T. & Yasui, S. (1990) Development of center-surround opponent receptive fields in a neural network through backpropagation training. Proc. Int. Con/. Fuzzy Logic & Neural Networks (Iizuka, Japan) 473-490. Joshi, A. & Lee, C.H. (1993) Backpropagation learns Marr's operator Bioi. Cybern., 10, 65-73. Laughlin, S. B. (1982) Matching coding to scenes to enhance efficiency. In Braddick OJ, Sleigh AC(eds) The physical and biological processing of images (pp.42-52). Springer, Berlin, Heidelberg New York. Lehky, S. R. & Sejnowski, T. J. (1988) Network model of shape-from shading: neural function arises from both receptive and projective fields. Nature, 333, 452-454. Linsker, R. (1986) From basic network principles to neural architecture: Emergence of spatial-opponent cells. Proc. Natl. Acad. Sci. USA, 83, 7508-7512. Stork, D. G. & Hall, J. (1989) Is backpropagation biologically plausible? International Join Con/. Neural Networks, II (Washington DC), 241-246. Torre, V. & Poggio, T. A. (1986) On edge detection. IEEE Trans. Pattern Anal. Machine Intel. , PAMI-8, 147-163. PART III THEORY
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Active Gesture Recognition using Learned Visual Attention Trevor Darrell and Alex Pentland Perceptual Computing Group MIT Media Lab 20 Ames Street, Cambridge MA, 02138 trevor,sandy~media.mit.edu Abstract We have developed a foveated gesture recognition system that runs in an unconstrained office environment with an active camera. Using vision routines previously implemented for an interactive environment, we determine the spatial location of salient body parts of a user and guide an active camera to obtain images of gestures or expressions. A hidden-state reinforcement learning paradigm is used to implement visual attention. The attention module selects targets to foveate based on the goal of successful recognition, and uses a new multiple-model Q-Iearning formulation. Given a set of target and distractor gestures, our system can learn where to foveate to maximally discriminate a particular gesture. 1 INTRODUCTION Vision has numerous uses in the natural world. It is used by many organisms in navigation and object recognition tasks, for finding resources or avoiding predators. Often overlooked in computational models of vision, however, and particularly relevant for humans, is the use of vision for communication and interaction. In these domains visual perception is an important communication modality, either in addition to language or when language cannot be used. In general, people place considerable weight on visual signals from another individual, such as facial expression, hand gestures, and body language. We have been developing neurally-inspired methods which combine low-level vision and learning to model these visual abilities. Previously, we presented a method for view-based recognition of spatia-temporal hand gestures [2] and a similar mechanism for the analysis/real-time tracking of facial expressions [4]. These methods offered real-time performance and a relatively high level of accuracy, but required foveated images of the object performing the Active Gesture Recognition Using Learned Visual Attention 859 gesture. There are many domains/tasks for which these are not unreasonable assumptions, such as interaction with a single user workstation or an automobile with a single driver. However the method had limited usefulness in unconstrained domains, such as "intelligent rooms" or interactive virtual environments, when the identity and location of the user are unknown. In this paper, we expand our gesture recognition method to include an active component, utilizing a foveated image sensor that can selectively track a person's hand or face as they walk through a room. The camera tracking and model selection routines are guided by an action-selection system that implements visual attention based on reinforcement learning. Using on a simple reward schedule, this attention system learns the appropriate object (hand, head) to foveate in order to maximize recognition performance. 2 FOVEATED GESTURE ANALYSIS Our system for foveated gesture recognition combines person tracking routines, an active, high-resolution camera, and view-based normalized correlation analysis. First we will briefly describe the person tracking module and view-based analysis, then discuss their use with an active camera. We have implemented vision routines to track a user in in an office setting as part of our ALIVE system, an Artificial Life Interactive Video Environment[3]. This system can track people and identify head/hand locations as they walk about a room, and provides the contextual environment within which view-based gesture analysis methods can be successfully applied. The ALIVE system assumed little prior knowledge of the user, and operated on coarse-scale images. 1 ALIVE allows a user to interact with virtual artificial life creatures, through the use of a "magicmirror" metaphor in which user sees him/herself presented in a video display along with virtual creatures. A wide field-of-view video camera acquires an image of the user, which is then combined with computer graphics imagery and projected on a large screen in front of the user. Vision routines in ALIVE compute figure/ground segmentation and analyze the user's silhouette to determine the location of head, hands, and other salient body features. We use only a single, calibrated, wide fieldof-view camera to determine the 3-D position of these features. 2 For details of our person tracking method see [14]. In our approach to real-time expression matching/tracking, a set of view-based correlation models is used to represent spatio-temporal gesture patterns. We take a sequence of images representing the gesture to be trained, and build a set of view models that are sufficient to track the object as it performs the gesture. Our view models are normalized correlation templates, and can either be intensity-based or based on band-pass or wavelet-based signal representations.3 We applied our model to the problem of hand gesture recognition [2] as well as for tracking facial expressions [4]. For facial tracking, we implemented an interpolation paradigm to map view-based correlation scores to facial motor controls. We used the Radial Basis Function (RBF) method[7]; interpolation was performed using a set of exemplars consisting of pairs of real faces and model faces in different expressions, which were 1 A simple mechanism for recognition of hand gestures was implemented in the original ALIVE system but made no use of high-resolution view models, and could only recognize pointing and waving motions defined by the motion of the centroid of the hand. 2By assuming the the user is sitting or standing on the ground plane, we use the imaging and ground plane geometry to compute the location of the user in 3-D. 3The latter have the advantage of being less dependent on illumination direction. 860 animation / rendering VIEW-BASED GESTURE ANALYSIS T.DARRELL,A.PENTLAND ~VideoWall Figure 1: Overview of system for person tracking and active gesture recognition. Static, wide-field-of-view, camera tracks user's head and hands, which drives gaze control of active narrow-field-of-view camera. Foveated images are used for viewbased gesture analysis and recognition. Graphical objects are rendered on video wall and can react to user's position, pose, and gestures. obtained by generating a 3-D model face and asking the user to match it. With this simple formalism, we were able to track expressions of a real user and interpolate equivalent 3-D model faces in real-time. This view-based analysis requires detailed imagery, which cannot be obtained from a single, fixed camera as the user walks about a room. To provide high resolution images for gesture recognition, we augment the wide field-of-view camera in our interactive environment with an active, narrow-field-of-view camera, as shown in Figure 1. Information about head/hand location from the existing ALIVE routines is used to drive the motor control parameters of the narrow field camera. Currently the camera can be directed to autonomously track head or hands. Using a highly simplified, two expression model offacial expression (neutral and surprised), we have been able to track facial expressions as users move about the room and the narrow angle camera followed the face. For details on this foveated gesture recognition see [5] 3 VISUAL ATTENTION FOR RECOGNITION The visual routines in the ALIVE system can be used to track the head and hands of a user, and the active camera can provide foveated images for gesture recognition. If we know a priori which body part will produce the gesture of interest, or if we have a sufficient number of active cameras to track all body parts, then we have solved the problem. Of course, in practice there are more possible loci of gesture performance than there are active cameras, and we have to address the problem of action selection for visual routines, i.e., attention. In our active gesture recognition system, we have adopted an action selection model based on reinforcement learning. Active Gesture Recognition Using Learned Visual Attention 861 3.1 THE ACTIVE GESTURE RECOGNITION PROBLEM We define an Active Gesture Recognition (AGR) task as follows. First, we assume primitive routines exist to provide the continuous valued control and tracking of the different body parts that perform gestures. Second, we assume that body pose and hand/face state is represented as a feature set, based on the representation produced by our body tracker and view-based recognition system, and we define a gesture to be a configuration of the user's body pose and hand/face expression. Third, we assume that, in addition to there being actions for foveating all the relevant body parts, there is also a special action labeled accept, and that the execution of this action by the AG R system signifies detection of the gesture. Finally, the goal of the AGR task is to execute the accept action whenever the user is in the target gesture state, and not to perform that action when the user is in any other (e.g. distract or) state. The AGR system should use the foveation actions to optimally discriminate the target pattern frqm distractor patterns, even when no single view of the user is sufficient to decide what gesture the user is performing. An important problem in applying reinforcement learning to this task is that our perceptual observations may not provide a complete description of the user's state. Indeed, because we have a foveated image sensor we know that the user's true gestural state will be hidden whenever the user is performing a gesture and the camera is not foveated on the appropriate body part. By definition, a system for perceptual action selection must not assume a full observation of state is available, otherwise there would be no meaningful perception taking place. The AG R task can be considered as a Partially Observable Markov Decision Process (POMDP), which is essentially a Markov Decision Process without direct access to state[ll, 9]. Rather than attempt to solve them explicitly, we look to techniques for hidden state reinforcement learning to find a solution [10, 8, 6, 1]. A POMDP consists of a set of states in the world S, a set of observations 0, a set of actions A, a reward function R. After executing an action a, the likelihood of transitioning between two states s, s' is given by T(s, a, a'), an observation 0 is generated with probability O(s, a, 0). In practice, T and 0 are not easily obtainable, and we use reinforcement learning methods which do not require them a priori. Our state is defined by the users pose, facial expression, and hand configurations, expressed in nine variables. Three are boolean and are provided directly by the person tracker: person-present, left-arm-extended, and right-arm-extended. Three more are provided by the foveated gesture recognition system, (face, left-hand, right-hand), and take on an integer number of values according to the number of view-based expressions/hand-poses: in our first experiments face can be one of neutral, smile, or surprise, and the hands can each be one of neutral, point, or grab. In addition, three boolean features represent the internal state of the vision system: head-foveated, left-hand-foveated, right-hand-foveated. At each time step, the world is defined by a state s E S, which is defined by these features. An observation, 0 E 0, consists of the same feature variables, except that those provided by the foveated gesture system (e.g., head and hands) are only observable when foveated. Thus the face variable is hidden unless the head-foveated variable is set, the left-hand variable hidden unless the left-hand-foveated variable set, and similarly with the right hand. Hidden variables are set to a undefined value. The set of actions, A, available to the AGR system are 4 foveation commands: look-body, look-head, look-left-hand, and look-right-hand plus the special accept action. Each foveation command causes the active camera to follow the respective body part, and sets the internal foveation feature bits accordingly. 862 T. DARRELL, A. PENTLAND The reward function provides a unit positive reward whenever the accept action is performed and the user is in the target state (as defined by an oracle, external to the AGR system), and a fixed negative reward of magnitude a when performed and the user is in a distractor (non-target) state. Zero reward is given whenever a foveation action is performed. 3.2 HIDDEN-STATE REINFORCEMENT LEARNING We have implemented a instance-based method for hidden state reinforcement learning, based on earlier work by McCallum [10]. The instance-based approach to reinforcement learning replaces the absolute state with a distributed memory-based state representation. Given a history of action, reward, and observation tuples, (a[t], r[t], o[t]) , 0 :::; t :::; T, a Q-value is also stored with each time step, q[t], and Q-Iearning[12, 13] is performed by evaluating the similarity of recently observed tuples with sequences farther back in the history chain. Q-values are computed, and the Q-Iearning update rule applied, maintaining this distributed, memory-based representation of Q-values. As in traditional Q-Iearning, at each time step the utility of each action in the current state is evaluated. If full access to the state was available and a table used to represent Q values, this would simply be a table look-up operation, but in a POMDP we do not have full access to state. Using a variation on the instance based approach employed by McAllum's Nearest Sequence Memory (NSM) algorithm, we instead find the I< nearest neighbors in the history list relative to the current time point, and compute their average Q value. For each element on the history list, we compute the sequence match criteria with the current time point, M(i, T), where M(i,j) = S(i,j) + M(i -l,j -1) if S(i,j) > 0 and i> 0 and j > 0 o otherwise. We define Sci, j) to be 1 if o[i] = o[j] or a[i] = a(j], 2 if both are equal, and o otherwise. Using a superscript in parentheses to denote the action index of a Q-value, we then compute T Q(a)[T] = (1/ I<) L v(a)[i]q[t] , (1) i=O where v(a*)[i] indicates whether the history tuple at time step i votes when computing the Q-value of a new action a"': v(a*)[i] is set to 1 when a[i] = a'" and M( i-I, T) is among the I< largest match values for all k which have a[k] = a"', otherwise it is set to O. Given Q values for each action the optimal policy is simply lI"[T] = arg maxQ(a)[T] . aEA (2) The new action a[T + 1] is chosen either according to this policy or based on an exploration strategy. In either case, the action is executed yielding an observation and reward, and a new tuple added to the history. The new Q-value is set to be the Q value of the chosen action, q[T + 1] = Q(a[T+1]) [T]. The update step of Q learning is then computed, evaluating U[T + 1] = maxQ(a)[T + 1] , aEA q[i] +- (1 - fJ)q[i] + fJ(r[i] + ')'U[T + 1]) , for each i such that v(a[T+l])[i] = l. (3) (4) (a) Active Gesture Recognition Using Learned Visual Attention %error 60 50 40 30 20 10 863 0.84% 0.44"10 0.48% 0L---------~---.8----~ 2 4 16 K (\3={).5, "(=0.5, a.= 10, 2500 trialS) Figure 2: (a) Multiple model Q-learning: one Q-learning agent for each target gesture to be recognized, with coupled observation and action but separate reward and Q-value. (b) Results on recognition task with 8 gesture targets; graph shows error rate after convergence plotted as a function of number of nearest neighbors used in learning algorithm. 4 MULTIPLE MODEL Q-LEARNING In general, we have found the simple, instance-based hidden state reinforcement learning described above to be an effective way to perform action selection for foveation when the task is recognition of a single object from a set of distractors. However, we did not find that this type of system performed well when the AG R task was extended to include more than one target gesture. When multiple accept actions were added to enumerate the different targets, we were not able to find exploration strategies that would converge in reasonable time. This is not unexpected, since the addition of multiple causes of positive reward makes the Q-value space considerably more complex. To remedy this problem, we propose a multiple model Q-learning system. In a multiple model approach to the AG R problem, separate learning agents model the task from each targets perspective. Conceptually, a separate Q-learning agent exists for each target, maintains it's own Q-value and history structure, and is coupled to the other agents via shared observations. Since we can interpret the Q-value of an individual AGR agent as a confidence value that its target is present, we can mediate among the actions predicted by the different agents by selecting the action from the agent with highest Q-value (Figure 2). Formally, in our multiple model Q-learning system all agents share the same observation and selected action, but have different reward and Q-values. Thus they can be considered a single Q-learning system, but with vector reward and Q-values. Our multiple model learning system is thus obtained by rewriting Eqs. (1)-(4) with vector q[t] and r[t]. Using a subscript j to indicate the target index, we have T Q;a)[T] = (1/ K) L v(a)[i]qj [t] , i=O 1T[11 = arg max (maxQ;a)[T]) . aEA J (5) Rewards are computed with: if a[T] = accept then rj [T] = R(j, T) else rj [T] = 0; R(j, T) = 1 if gesture j was present at time T, else R(j, T) = -(Y. Further, Uj [T + 1] = maxQ(a)[T + 1] , (6) aEA ] 864 T.DARRELL,A.PENTLAND qj[i] f- (1- ,8)qj[i] + ,8(rj[i] + /'Uj[T+ 1]) Vi s.t. v(a[T+1])[i] = 1 . (7) Note that our sequence match criteria, unlike that in [10], does not depend on r[t]; this allows considerable computational savings in the multiple model system since v(a) need not depend on j. We ran the multiple model learning system on the AGR task using 8 targets, with ,8 = 0.5, /' = 0.5, Q; = 10. Results summed over 2500 trials are shown in Figure 2(b), with classification error plotted against the number of nearest neighbors used in the NSM algorithm. The error rate shown is after convergence; we ran the algorithm with a period of deterministic exploration before following the optimal policy. (The system deterministically explored each action/accept pair.) As can be seen from the graph, for any non-degenerate value of K reasonable performance was obtained; for K > 2, the system performed almost perfectly. References [1] A. Cassandra, L. P. Kaelbling, and M. Littman. Acting optimally in partially observable stochastic domains. In Proc. AAAI-94, pages 1023-1028. Morgan Kaufmann, 1994. [2] T. Darrell and A. P. Pentland. Classification of Hand Gestures using a ViewBased Distributed Representation In Advances in Neural Information Processing Systems 6, Morgan Kauffman, 1994. [3] T. Darrell, P. Maes, B. Blumberg, and A. P. Pentland, A Novel Environment for Situated Vision and Behavior, Proc. IEEE Workshop for Visual Behaviors, IEEE Compo Soc. Press, Los Alamitos, CA, 1994 [4] T. Darrell, I. Essa, and A. P. Pentland, Correlation and Interpolation Networks for Real-time Expression Analysis/Synthesis, In Advances in Neural Information Processing Systems 7, MIT Press, 1995. [5] T. Darrell and A. Pentland, A., Attention-driven Expression and Gesture Analysis in an Interactive Environment, in Proc. Inti. Workshop on A utomatic Face and Gesture Recognition (IWAFGR '95), Zurich, Switzerland, 1995. [6] T. Jaakkola, S. Singh, and M. Jordan. Reinforcement Learning Algorithm for Partially Observable Markov Decision Problems. In Advances In Neural Information Processing Systems 7, MIT Press, 1995. [7] T. Poggio and F. Girosi, A Theory of Networks for Approximation and Learning. MIT AI Lab TR-1140, 1989. [8] 1. Lin and T. Michell. Reinforcement learning with hidden states. In Proc. AAAI-92. Morgan Kaufmann, 1992. [9] W. Lovejoy. A survey of algorithmic methods of partially observed markov decision processes. Annals of Operation Reserach, 28:47-66, 1991. [10] R. A. McCallum. Instance-based State Identification for Reinforcement Learning. In Advances In Neural Information Processing Systems 7, MIT Press, 1995. [11] Edward J. Sondik. The optimal control of partially observable markov processes over the infinite horizon: Discounted costs. Operations Reserach, 26(2):282304, 1978. [12] R. S. Sutton. Learning to predict by the method of temporal differences. Machine Learning, 3:9-44, 1988. [13] C. Watkins and P. Dayan. Q-learning. Machine Learning, 8:279-292, 1992. [14] C. Wren, A. Azarbayejani, T. Darrell, and A. Pentland, Pfinder: Real-Time Tracking of the Human Body, Media Lab Per. Compo TR-353, 1994
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On Neural Networks with Minimal Weights Vasken Bohossian J ehoshua Bruck California Institute of Technology Mail Code 136-93 Pasadena, CA 91125 E-mail: {vincent, bruck }«Iparadise. cal tech. edu Abstract Linear threshold elements are the basic building blocks of artificial neural networks. A linear threshold element computes a function that is a sign of a weighted sum of the input variables. The weights are arbitrary integers; actually, they can be very big integers-exponential in the number of the input variables. However, in practice, it is difficult to implement big weights. In the present literature a distinction is made between the two extreme cases: linear threshold functions with polynomial-size weights as opposed to those with exponential-size weights. The main contribution of this paper is to fill up the gap by further refining that separation. Namely, we prove that the class of linear threshold functions with polynomial-size weights can be divided into subclasses according to the degree of the polynomial. In fact, we prove a more general result- that there exists a minimal weight linear threshold function for any arbitrary number of inputs and any weight size. To prove those results we have developed a novel technique for constructing linear threshold functions with minimal weights. 1 Introduction Human brains are by far superior to computers for solving hard problems like combinatorial optimization and image and speech recognition, although their basic building blocks are several orders of magnitude slower. This observation has boosted interest in the field of artificial neural networks [Hopfield 82]' [Rumelhart 82]. The latter are built by interconnecting multiple artificial neurons (or linear threshold gates), whose behavior is inspired by that of biological neurons. Artificial neural networks have found promising applications in pattern recognition, learning and On Neural Networks with Minimal Weights 247 other data processing tasks. However most of the research has been oriented towards the practical aspect of neural networks, simulating or building networks for particular tasks and then comparing their performance with that of more traditional methods for those particular tasks. To compare neural networks to other computational models one needs to develop the theoretical settings in which to estimate their capabilities and limitations. 1.1 Linear Threshold Gate The present paper focuses on the study of a single linear threshold gate (artificial neuron) with binary inputs and output as well as integer weights (synaptic coefficients). Such a gate is mathematically described by a linear threshold function. Definition 1 (Linear Threshold FUnction) A linear threshold function of n variables is a Boolean function f { -1, I} n ~ { -1, 1} that can be written as n f( .... ) (F( .... » - { 1 ,for F(x) ~ 0 x - sgn x 1 th . ,0 erW1se , where F(x) = tV· x = L WiXi i=1 for any x E {-1, 1}n and a fixed tV E zn. Although we could allow the weights Wi to be real numbers, it is known [Muroga 71), [Raghavan 88) that for a, binary input neuron, one needs O( n log n) bits per weight, where n is the number of inputs. So in the rest ofthe paper, we will assume without loss of generality that all weights are integers. 1.2 Motivation Many experimental results in the area of neural networks have indicated that the magnitudes of the coefficients in the linear threshold elements grow very fast with the size of the inputs and therefore limit the practical use of the network. One natural question to ask is the following. How limited is the computational power of the network if one limits oneself to threshold elements with only "small" growth in the size of the coefficients? To answer that question we have to define a measure of the magnitudes of the weights. Note that, given a function I, the weight vector tV is not unique (see Example 1 below). Definition 2 (Weight Space) Given a lineal' threshold function f we define W as the set of all weights that satisfy Definition 1, that is W = {UI E zn : Vx E {-1, 1}n,sgn(tV· x) = f(x)}. Here follows a measure of the size of the weights. Definition 3 (Minimal Weight Size) We define the size of a weight vector as the sum of the absolute values of the weights. The minimal weight size of a linear threshold function is defined as : n S[j) = ~ia/L IWi I) ,=1 The particular vector that achieves the minimum is called a minimal weight vector. Naturally, S[f) is a function of n. 248 V. BOHOSSIAN, J. BRUCK It has been shown [Hastad 94], [Myhill 61], [Shawe-Taylor 92], (Siu 91] that there exists a linear threshold function that can be implemented by a single threshold element with exponentially growing weights, S[j] '" 2'1, but cannot be implemented by a threshold element with smaller: polynomialy growing weights, S[j] '" nd , d constant. In light of that result the above question was dealt with by defining a class within the set of linear threshold functions: the class of functions with "small" (Le. polynomialy growing) weights [Siu 91]. Most of the recent research focuses on the power of circuits with small weights, relative to circuits with arbitrary weights [Goldmann 92], [Goldman 93]. Rather than dealing with circuits we are interested in studying a single threshold gate. The main contribution of the present paper is to further refine the division of small versus arbitrary weights. We separate the set of functions with small weights into classes indexed by d, the degree of polynomial growth and show that all of them are non-empty. In particular, we develop a technique for proving that a weight vector is minimal. We use that technique to construct a function of size S[j] = s for an arbitrary s. 1.3 Approach The main difficulty in analyzing the size of the weights of a threshold element is due to the fact that a single linear threshold function can be implemented by different sets of weights as shown in the following example. Example 1 (A Threshold FUnction with Minimal Weights) Consider the following two sets of weights (weight vectors). tih = (124), FI(X) = Xl + 2X2 + 4X3 W2 = (248), F2(X) = 2XI + 4X2 + 8X3 They both implement the same threshold function f(X) = sgn(F2(x» = sgn(2FI (x» = sgn(FI (x» A closer look reveals that f(x) = sgn(x3), implying that none of the above weight vectors has minimal size. Indeed, the minimal one is W3 = (00 1) and S(J] = 1. It is in general difficult to determine if a given set of weights is minimal [Amaldi 93], [Willis 63]. Our technique consists of limiting the study to only a particular subset of linear threshold functions, a subset for which it is possible to prove that a given weight vector is minimal. That subset is loosely defined by the requirement that there exist input vectors for which f(x) = f( -x). The existence of such a vector, called a root of f, puts a constraint on the weight vector used to implement f. The larger the set of roots - the larger the constraint on the set of weight vectors, which in turn helps determine the minimal one. A detailed description of the technique is given in Section 2. 1.4 Organization Here follows a brief outline of the rest of the paper. Section 2 mathematically defines the setting of the problem as well as derives some basic results on the properties of functions that admit roots. Those results are used as bUilding blocks for the proof of the main results in Section 3. It also introduces a construction method for functions with minimal weights. Section 3 presents the main result: for any weight size, s, and any nunlber of inputs, n, there exists an n-input linear threshold fllllction that requires weights of size S[f] = s. Section 4 presents some applications of the result of Section 3 and indicates future research directions. On Neural Networks with Minimal Weights 249 2 Construction of Minimal Threshold Functions The present section defines the mathematical tools used to construct functions with minimal weights. 2.1 Mathematical setting We are interested in constructing functions for which the minimal weight is easily determined. Finding the minimal weight involves a search, we are therefore interested in finding functions with a constrained weight spaces. The following tools allows us to put constraints on W. Definition 4 (Root Space of a Boolean Function) A vector v E {-I, 1} n such that 1 (V) = 1 (-V) is called a root of I. We define the root space, R, as the set of all roots of I. Definition 5 (Root Generator Matrix) For a given weight vector w E W and a root v E R, the root generator matrix, G = (gij), is a (n x k)-matrix, with entries in {-I, 0,1}, whose rows 9 are orthogonal to w and equal to vat all non-zero coordinates, namely, 1. Gw = 0 2. 9ij = ° or 9ij = Vj for all i and j. Example 2 (Root Generator Matrix) Suppose that we are given a linear threshold function specified by a weight vector w = (1,1,2,4,1,1,2,4). By inspection we determine one root v = (1,1,1,1, -1, -1, -1, -1). Notice that WI + W2 W7 = ° which can be written as g. w = 0, where 9 = (1,1,0,0,0,0, -1,0) is a row of G. Set r= v - 2g. Since 9 is equal to vat all non-zero coordinates, r E {-I, I} n. Also r· w = v· w + g. w = 0. We have generated a new root : r = (-1, -1, 1, 1, -1, -1, 1, -1). Lemma 6 (Orthogonality of G and W) For a given weight vector w E Wand a root v E R ilGT = 0 holds for any weight vector il E W. Proof. For an arbitrary il E Wand an arbitrary row, gi, of G, let if = v - 2gi. By definition of gi, if E {-I,1}n and if· w = 0. That implies I(if) = I(-if) : if is a root of I. For any weight vector il E W, sgn(il· if) = sgn( -il· if). Therefore il· (v - 2gi) = ° and finally, since v· il = ° we get il· gi = 0. 0 Lemma 7 (Minimality) For a given weight vector w E W and a root v E R if rank( G) = n - 1 (Le. G has n - 1 independent rows) and IWil = 1 for some i, then w is the minimal weight vector. Proof. From Lemma 6 any weight vector il satisfies ilGT = O. rank( G) = n - 1 implies that dim(W) = 1, i.e. all possible weight vectors are integer multiples of each other. Since IWi I = 1, all vectors are of the form il = kw, for k ~ 1. Therefore w has the smallest size. 0 We complete Example 2 with an application of Lemma 7. 250 V. BOHOSSIAN, J. BRUCK Example 3 (Minimality) Given ill = (1,1,2,4,1,1,2,4) and v = (1,1,1,1, -1, -1, -1, -1) we can construct: 1 0 0 0 -1 0 0 0 0 1 0 0 0 -1 0 0 0 0 1 0 0 0 -1 0 G= 0 0 0 1 0 0 0 -1 1 0 0 0 0 -1 0 0 1 1 0 0 0 0 -1 0 1 1 1 0 0 0 0 -1 It is easy to verify that rank( G) = n - 1 = 7 and therefore, by Lemma 7, ill is minimal and 8[/] = 16. 2.2 Construction of minimal weight vectors In Example 3 we saw how, given a weight vector, one can show that it is minimal. In this section we present an example of a linear threshold function with minimal weight size, with an arbitrary number of input variables. We would like to construct a weight vector and show that it is minimal. Let the number of inputs, n, be even. Let ill consist of two identical blocks : (Wl,W2, ... ,Wn /2,Wl,W2, ... ,Wn/2)' Clearly, if = (1,1,; .. ,1,-1,-1, ... ,-1) is a root and G is the corresponding generator matrix. 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 G= 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 3 The Main Result The following theorem states that given an integer s and a number of variables n there exists a function of n variables and minimal weight size s. Theorem 8 (Main Result) For any pair (s,n) that satisfies 2. seven , for n even , for n odd there exists a linear threshold function of n variables, I, with minimal weight size 8[J] = s. Proof. Given a pair (s, n), that satisfies the above conditions we first construct a weight vector w that satisfies E~l IWil = s, then show that it is the minimal weight vector ofthe function I(x) = sgn(w·X). The proof is shown only for n even. CONSTRUCTION. 1. Define (at, a2, ... , an/2) = (1,1, ... , 1). On Neural Networks with Minimal Weights 251 n/2 . 2 2. If L,:::l a, < s/2 then increase by one the smallest a, such that a, < 2'- . (In the case of a tie take the Wi with smallest index i). 3. Repeat the previous step until L~; ai = s /2 or (aI, a2, ... , aN) = (1,1,2,4, ... , 2~ - 2). 4. Set w= (al,a2, ... ,an/2,al,a2, ... ,an/2)' Because we increase the size by one unit at a time the algorithm will converge to the desired result for any integer s that satisfies n ~ s ~ 2~. We have a construction for any valid (s, n) pair. Let us show that w is minimal. MINIMALITY. Given that w = (aI, a2, ... , an/2, aI, a2, ... , aaj2) we find a root v = (1, 1, ... , 1, -1, -1, ... , -1) and n/2 rows of the generator matrix G corresponding to the equations w, = wH ~. To form additional rows note that the first k ais are powers of two (where k depends on sand n). Those can be written as a, = L~:~ aj and generate k - 1 rows. And finally note that all other ai, i > k, are smaller than 2k+l. Hence, they can be written as a binary expansion a, = L~:::l aijaj where aij E {O, I}. There are -r - k such weights. G has a total of n -1 independent rows. rank(G) = n -1 and WI = 1, therefore by Lemma 7, tV is minimal and S[J] = s. 0 Example 4 (A Function of 10 variables and size S[fJ = 26) We start with a = (1,1,1,1,1). We iterate: (1,1,2,1,1), (1,1,2,2,1), (1,1,2,2,2), (1,1,2,3,2), (1,1,2,3,3), (1,1,2,4,3), (1,1,2,4,4), and finally (1,1,2,4,5). The construction algorithm converges to a = (1,1,2,4,5). We claim that tV = (a, a) = (1,1,2,4,5,1,1,2,4,5) is minimal. Indeed, v = (1,1,1,1,1, -1, -1, -1, -1, -1) and 1 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 -1 0 G= 0 0 0 0 1 0 0 0 0 -1 1 0 0 0 0 0 -1 0 0 0 1 1 0 0 0 0 0 -1 0 0 1 1 1 0 0 0 0 0 -1 0 1 0 0 1 0 0 0 0 0 -1 is a matrix of rank 9. Example 5 (Functions with Polynomial Size) This example shows an application of Theorem 8. We define fred) as the set of linear threshold functions for which S[I} ~ nd • The Theorem states that for any even n there exists a function 1 of n variables and minimum weight S[I] = nd • The -- (d- I) -- (d) implication is that for all d, LT is a proper subset of LT 4 Conclusions We have shown that for any reasonable pair of integers (n, s), where s is even, there exists a linear threshold function of n variables with minimal weight size S[J} = s. We have developed a novel technique for constructing linear threshold functions with minimal weights that is based on the existence of root vectors. An interesting application of our method is the computation of a lower bound on the number of linear threshold functions [Smith 66}. In addition, our technique can help in studying the trade-otIs between a number of important parameters associated with 252 V. BOHOSSIAN, 1. BRUCK linear threshold (neural) circuits, including, the munber of elements, the number of layers, the fan-in, fan-out and the size of the weights. Acknow ledgements This work was supported in part by the NSF Young Investigator Award CCR9457811, by the Sloan Research Fellowship, by a grant from the IBM Almaden Research Center, San Jose, California, by a grant from the AT&T Foundation and by the center for Neuromorphic Systems Engineering as a part of the National Science Foundation Engineering Research Center Program; and by the California Trade and Commerce Agency, Office of Strategic Technology. References [Amaldi 93] E. Amaldi and V. Kann. The complexity andapproximabilityoffinding maximum feasible subsystems of linear relations. Ecole Poly technique Federale De Lausanne Technical Report, ORWP 93/11, August 1993. [Goldmann 92] M. Goldmann, J. Hastad, and A. Razborov. Majority gates vs. general weighted threshold gates. Computational Complexity, (2):277-300, 1992. [Goldman 93] M. Goldmann and M. Karpinski. Simulating threshold circuits by majority circuits. In Proc. 25th ACM STOC, pages pp. 551- 560, 1993. [Hastad 94] .1. Hastad. On the size of weights for threshold gates. SIAM. J. Disc. Math., 7:484-492, 1994. [Hopfield 82) .1. Hopfield. Neural networks and physical systems with emergent collective computational abilities. Proc. of the USA National Academy of Sciences, 79:2554- 2558, 1982. [Muroga 71) M. Muroga. Threshold Logic and its Applications. Wiley-Interscience, 1971. [Myhill 61) J. Myhill and W. H. Kautz. On the size of weights required for linearinput switching functions. IRE Trans. Electronic Computers, (EClO):pp. 288290, 1961. [Raghavan 88] P. Raghavan. Learning in threshold networks: a computational model and applications. Technical Report RC 13859, IBM Research, July 1988. [Rumelhart 82] D. Rumelhart and J. McClelland. Parallel distributed processing: Explorations in the microstructure of cognition. MIT Press, 1982. [Shawe-Taylor 92] J. S. Shawe-Taylor, M. H. G. Anthony, and W. Kern. Classes of feedforward neural networks and their circuit complexity. Neural Networks, Vol. 5:pp. 971- 977, 1992. [Siu 91] K. Siu and J. Bruck. On the power of threshold circuits with small weights. SIAM J. Disc. Math., Vol. 4(No. 3):pp. 423-435, August 1991. [Smith 66] D. R. Smith. Bounds on the number of threshold functions. IEEE Transactions on Electronic Computers, June 1966. [Willis 63] D. G. Willis. Minimum weights for threshold switches. In Switching Theory in Space Techniques. Stanford University Press, Stanford, Calif., 1963.
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Neural Networks with Quadratic VC Dimension Pascal Koiran* Lab. de l'Informatique du Paraltelisme Ecole Normale Superieure de Lyon - CNRS 69364 Lyon Cedex 07, France Abstract Eduardo D. Sontagt Department of Mathematics Rutgers University New Brunswick, NJ 08903, USA This paper shows that neural networks which use continuous activation functions have VC dimension at least as large as the square of the number of weights w. This result settles a long-standing open question, namely whether the well-known O( w log w) bound, known for hard-threshold nets, also held for more general sigmoidal nets. Implications for the number of samples needed for valid generalization are discussed. 1 Introduction One of the main applications of artificial neural networks is to pattern classification tasks. A set of labeled training samples is provided, and a network must be obtained which is then expected to correctly classify previously unseen inputs. In this context, a central problem is to estimate the amount of training data needed to guarantee satisfactory learning performance. To study this question, it is necessary to first formalize the notion of learning from examples. One such formalization is based on the paradigm of probably approximately correct (PAC) learning, due to Valiant (1984). In this framework, one starts by fitting some function /, chosen from a predetermined class F, to the given training data. The class F is often called the "hypothesis class" , and for purposes of this discussion it will be assumed that the functions in F take binary values {O, I} and are defined on a common domain X. (In neural networks applications, typically F corresponds to the set of all neural networks with a given architecture and choice of activation functions. The elements of X are the inputs, possibly multidimensional.) The training data consists of labeled samples (Xi,ci), with each Xi E X and each Ci E {O, I}, and *koiranGlip. ens-lyon. fr. tsontagGhilbert.rutgers.edu. 198 P. KOIRAN, E. D. SONTAG "fitting" by an f means that f(xj) = Cj for each i. Given a new example x, one uses f( x) as a guess of the "correct" classification of x. Assuming that both training inputs and future inputs are picked according to the same probability distribution on X, one needs that the space of possible inputs be well-sampled by the training data, so that f is an accurate fit. We omit the details of the formalization of PAC learning, since there are excellent references available, both in textbook (e.g. Anthony and Biggs (1992), Natarajan (1991)) and survey paper (e.g. Maass (1994)) form, and the concept is by now very well-known. After the work of Vapnik (1982) in statistics and of Blumer et. al. (1989) in computationallearning theory, one knows that a certain combinatorial quantity, called the Vapnik-Chervonenkis (VC) dimension VC(F) of the class F of interest completely characterizes the sample sizes needed for learnability in the PAC sense. (The appropriate definitions are reviewed below. In Valiant's formulation one is also interested in quantifying the computational effort required to actually fit a function to the given training data, but we are ignoring that aspect in the current paper.) Very roughly speaking, the number of samples needed in order to learn reliably is proportional to VC(F). Estimating VC(F) then becomes a central concern. Thus from now on, we speak exclusively of VC dimension, instead of the original PAC learning problem. The work of Cover (1988) and Baum and Haussler (1989) dealt with the computation of VC(F) when the class F consists of networks built up from hard-threshold activations and having w weights; they showed that VC(F)= O(wlogw). (Conversely, Maass (1993) showed that there is also a lower bound of this form.) It would appear that this definitely settled the VC dimension (and hence also the sample size) question. However, the above estimate assumes an architecture based on hard-threshold ("Heaviside") neurons. In contrast, the usually employed gradient descent learning algorithms ("backpropagation" method) rely upon continuous activations, that is, neurons with graded responses. As pointed out in Sontag (1989), the use of analog activations, which allow the passing of rich (not just binary) information among levels, may result in higher memory capacity as compared with threshold nets. This has serious potential implications in learning, essentially because more memory capacity means that a given function f may be able to "memorize" in a "rote" fashion too much data, and less generalization is therefore possible. Indeed, Sontag (1992) showed that there are conceivable (though not very practical) neural architectures with extremely high VC dimensions. Thus the problem of studying VC(F) for analog networks is an interesting and relevant issue. Two important contributions in this direction were the papers by Maass (1993) and by Goldberg and Jerrum (1995), which showed upper bounds on the VC dimension of networks that use piecewise polynomial activations. The last reference, in particular, established for that case an upper bound of O(w2), where, as before, w is the number of weights. However it was an open problem (specifically, "open problem number 7" in the recent survey by Maass (1993) if there is a matching w 2 lower bound for such networks, and more generally for arbitrary continuous-activation nets. It could have been the case that the upper bound O( w 2 ) is merely an artifact of the method of proof in Goldberg and Jerrum (1995), and that reliable learning with continuous-activation networks is still possible with far smaller sample sizes, proportional to O( w log w). But this is not the case, and in this paper we answer Maass' open question in the affirmative. Assume given an activation (T which has different limits at ±oo, and is such that there is at least one point where it has a derivative and the derivative is nonzero (this last condition rules out the Heaviside activation). Then there are architectures with arbitrary large numbers of weights wand VC dimension proportional Neural Networks with Quadratic VC Dimension 199 to w 2 • The proof relies on first showing that networks consisting of two types of activations, Heavisides and linear, already have this power. This is a somewhat surprising result, since purely linear networks result in VC dimension proportional to w, and purely threshold nets have, as per the results quoted above, VC dimension bounded by w log w. Our construction was originally motivated by a related one, given in Goldberg and Jerrum (1995), which showed that real-number programs (in the Blum-Shub-Smale (1989) model of computation) with running time T have VC dimension O(T2). The desired result on continuous activations is then obtained, approximating Heaviside gates by IT-nets with large weights and approximating linear gates by IT-nets with small weights. This result applies in particular to the standard sigmoid 1/(1 + e- X ). (However, in contrast with the piecewise-polynomial case, there is still in that case a large gap between our O( w 2 ) lower bound and the O( w4 ) upper bound which was recently established in Karpinski and Macintyre (1995).) A number of variations, dealing with Boolean inputs, or weakening the assumptions on IT, are discussed. The full version of this paper also includes some remarks on thresholds networks with a constant number of linear gates, and threshold-only nets with "shared" weights. Basic Terminology and Definitions Formally, a (first-order, feedforward) architecture or network A is a connected directed acyclic graph together with an assignment of a function to a subset of its nodes. The nodes are of two types: those of fan-in zero are called input nodes and the remaining ones are called computation nodes or gates. An output node is a node of fan-out zero. To each gate g there is associated a function IT g : IR. -!- IR., called the activation or gate function associated to g. The number of weights or parameters associated to a gate 9 is the integer ng equal to the fan-in of 9 plus one. (This definition is motivated by the fact that each input to the gate will be multiplied by a weight, and the results are added together with a "bias" constant term, seen as one more weight; see below.) The (total) number of weights (or parameters) of A is by definition the sum of the numbers n g , over all the gates 9 of A. The number of inputs m of A is the total number of input nodes (one also says that "A has inputs in IR.m,,); it is assumed that m > O. The number of outputs p of A is the number of output nodes (unless otherwise mentioned, we assume by default that all nets considered have one-dimensional outputs, that is, p = 1). Two examples of gate functions that are of particular interest are the identity or linear gate: Id( x) = x for all x, and the threshold or H eaviside function: H (x) = 1 if x ~ 0, H(x) = 0 if x < O. Let A be an architecture. Assume that nodes of A have been linearly ordered as 11"1, ... , 11" m, gl, ... , gl, where the 1I"j 's are the input nodes and the gj 's the gates. For simplicity, write nj := n g., for each i = 1, ... , I. Note that the total number of parameters is n = L:~=1 nj and the fan-in of each gj is nj 1. To each architecture A (strictly speaking, an architecture together with such an ordering of nodes) we associate a function F : ]Rm x ]Rn -!-]RP , where p is the number of outputs of A, defined by first assigning an "output" to each node, recursively on the distance from the the input nodes. Assume given an input x E ]Rm and a vector of weights w E ]Rn. We partition w into blocks (WI , ... , WI) of sizes nl, ... , nl respectively. First the coordinates of x are assigned as the outputs of the input nodes 11"1, ... , 1I"m respectively. For each of the other gates gj, we proceed as follows. Assume that outputs Yl, ... , Yn. -1 have already 200 P. KOIRAN, E. D. SONTAG been assigned to the predecessor nodes of gi (these are input and/or computation nodes, listed consistently with the order fixed in advance). Then the output of gi is by definition (1'g. (Wi,O + Wi ,lYI + Wi,2Y2 + ... + wi,n.-lYn.-d , where we are writing Wi = (Wi,O, Wi,l, Wi ,2, ... , wi,n.-d. The value of F(x, w) is then by definition the vector (scalar if p = 1) obtained by listing the outputs of the output nodes (in the agreed-upon fixed ordering of nodes). We call F the function computed by the architecture A. For each choice of weights W E IRn, there is a function Fw : IRm _ IRP defined by Fw(x) := F(x, w); by abuse of terminology we sometimes call this also the function computed by A (if the weight vector has been fixed). Assume that A is an architecture with inputs in IRm and scalar outputs, and that the (unique) output gate has range {O, 1}. A subset A ~ IR m is said to be shattered by A if for each Boolean function 13 : A {O, 1} there is some weight W E IRn so that Fw(x) = f3(x) for all x EA . The Vapnik-Chervonenkis (VC) dimension of A is the maximal size of a subset A ~ IRm that is shattered by A. If the output gate can take non-binary values, we implicitly assume that the result of the computation is the sign of the output. That is, when we say that a subset A ~ IRm is shattered by A, we really mean that A is shattered by the architecture H(A) in which the output of A is fed to a sign gate. 2 Networks Made up of Linear and Threshold Gates Proposition 1 For every n ;::: 1, there is a network architecture A with inputs in IR 2 and O( VN) weights that can shatter a set of size N = n2. This architecture is made only of linear and threshold gates. Proof. Our architecture has n parameters WI , ... , Wn; each of them is an element ofT = {O.WI . .. Wn ;Wi E {O, 1}}. The shattered set will be S = [n]2 = {1, .. . ,nF. For a given choice of W = (WI' ... ' Wn), A will compute the boolean function fw : S {O, 1} defined as follows: fw(x, y) is equal to the x-th bit of Wy . Clearly, for any boolean function f on S, there exists a (unique) W such that f = fw. We first consider the obvious architecture which computes the function: n flv(Y) = WI + I)Wz - Wz-dH(y - z + 1/2) (1) z=2 sending each point Y E [n] to Wy. This architecture has n - 1 threshold gates, 3(n - 1) + 1 weights, and just one linear gate. Next we define a second multi-output net which maps wET to its binary representation j2(w) = (WI' . .. ' wn ). Assume by induction that we have a net N? that maps W to (WI, ... ,Wi,O.Wi+l ... Wn) . Since Wi+l = H(O.Wi+l . .. Wn -1/2) and o. Wi+2 ... Wn = 2 x o. Wi+1 . .. Wn - Wi+!, .N;;'l can be obtained by adding one threshold gate and one linear gate to .N;2 (as well as 4 weights). It follows that N~ has n threshold gates, n linear gates and 4n weights. Finally, we define a net N3 which takes as input x E [n] and W = (WI , ... , wn) E {O, l}n, and outputs W X • We would like this network to be as follows: n n f3(X , w) = WI + L wzH(x - z + 1/2) - L wz_IH(x - z + 1/2). z=2 z=2 Neural Networks with Quadratic VC Dimension 201 This is not quite possible, because the products between the Wi'S (which are inputs in this context) and the Heavisides are not allowed. However, since we are dealing with binary variables one can write uv = H(u + v - l.5). Thus N3 has one linear gate, 4(n - 1) threshold gates and 12(n - 1) + n weights. Note that fw(x, y) = p (x, P Ulv (y)). This can be realized by means of a net that has n + 2 linear gates, (n-l)+n+4(n-l) = 6n-5 threshold gates, and (3n-2)+4n+(12n-ll) = 19n-13 weights. 0 The following is the main result of this section: Theorem 1 For every n ;::: 1, there is a network architecture A with inputs in IR. and O( VN) weights that can shatter a set of size N = n2. This architecture is made only of linear and threshold gates. Proof. The shattered set will be S = {O, 1, .. . ,n2 -I}. For every xES, there are unique integers x, y E {O, 1, ... , n - I} such that u = nx + y. The idea of the construction is to compute x and y, and then feed (x + 1, y + 1) to the network constructed in Proposition 1. Note that x is the unique integer such that u - nx E {O, 1, .. . , n - I}. It can therefore by computed by brute force search as follows: n-1 X = L kH[H(u - nk) + H(n - 1 - (u - nk)) - l.5]. k=O This network has 3n threshold gates, one linear gate and 8n weights. Then of course y = u - nx. 0 A Boolean version is as follows. Theorem 2 For every d ;::: 1, there is a network architecture A with O( VN) weights that can shatter the N = 22d points of {O, 1 Fd . This architecture is made only of linear and threshold gates. Proof. Given u E {O, IFd, one can compute x = 1 + 2::=1 2i-1ui and y = 1 + 2:1=12i-1Ui+d with two linear gates. Then (x, y) can be fed to the network of Proposition 1 (with n = 2d ). 0 In other words, there is a network architecture with 2d weights that can compute all boolean functions on 2d variables. 3 Arbitrary Sigmoids We now extend the preceding VC dimension bounds to networks that use just one activation function tr (instead of both linear and threshold gates). All that is required is that the gate function have a sigmoidal shape and satisfy a very weak smoothness property: l. tr is differentiable at some point Xo (i.e., tr(xo+h) = tr(xo)+tr'(xo)h+o(h)) where tr'(xo)# 0. 2. limx __ oo tr(x) = ° and limx_+oo tr(x) = 1 (the limits ° and 1 can be replaced by any distinct numbers). A function satisfying these two conditions will be called sigmoidal. Given any such tr, we will show that networks using only tr gates provide quadratic VC dimension. 202 P. KOIRAN, E. D. SONTAG Theorem 3 Let tT be an arbitrary sigmoidal function. There exist architectures Al and A2 with O( VN) weights made only of tT gates such that: • Al can shatter a subset ofIR of cardinality N = n2 ,• A2 can shatter the N = 22d points of {O, 1}2d. This follows directly from Theorems 1 and 2, together with the following simulation result: Theorem 4 Let tT be a an arbitrary sigmoidal function. Let N be a network of T threshold and L linear gates, with a threshold gate at the output. Then N can be simulated on any given finite set of inputs by a network N' of T + L gates that all use the activation function tT (except the output gate which is still a threshold). Moreover, if N has n weights then N' has O( n) weights. Proof. Let S be a finite set of inputs. We can assume, by changing the thresholds of threshold gates if necessary, that the net input Ig (x) to any threshold gate 9 of N is different from ° for all inputs xES. Given € > 0, let N( be the net obtained by replacing the output functions of all gates by the new output function x 1--+ tT( X / €) if this output function is the sign function, and by x 1--+ tT(x) = [tT(xo+€x)-tT(xo))/[€tT'(xo)] ifit is the identity function. Note that for any a > 0, lim(_o+ tT(x/€) = H(x) uniformly for x E) - 00, -a] U [a, +00] and limHo tT(x) = x uniformly for x E [-l/a, l/a]. This implies by induction on the depth of 9 that for any gate 9 of N and any input XES, the net input Ig,(x) to 9 in the transformed net N( satisfies li~_o IgAx) = Ig(x) (here, we use the fact that the output function of every 9 is continuous at Ig(x)). In particular, by taking 9 to be the output gate of N, we see that Nand N( compute the same function on S if € is small enough. Such a net N( can be transformed into an equivalent net N' that uses only tT as gate function by a simple transformation of its weights and thresholds. The number of weights remains the same, except at most for a constant term that must be added to each net input to a gate; thus if N has n weights, N' has at most 2n weights. 0 4 More General Gate Functions The objective of this section is to establish results similar to Theorem 3, but for even more arbitrary gate functions, in particular weakening the assumption that limits exist at infinity. The main result is, roughly, that any tT which is piecewise twice (continuously) differentiable gives at least quadratic VC dimension, save for certain exceptional cases involving functions that are almost everywhere linear. A function tT : IR --+ IR is said to be piecewise C 2 if there is a finite sequence al < a2 < ... < ap such that on each interval I of the form] - 00, al [, )ai, ai+1 [ or ]ap , +00[, tTll is C2. (Note: our results hold even if it is only assumed that the second derivative exists in each of the above intervals; we do not use the continuity of these second derivatives.) Theorem 5 Let tT be a piecewise C2 function. For every n ~ 1, there exists an architecture made of tT-gates, and with O( n) weights, that can shatter a subset of IR 2 of cardinality n2 , except perhaps in the following cases: 1. tT is piecewise-constant, and in this case the VC dimension of any architecture of n weights is O( n log n),Neural Networks with Quadratic VC Dimension 203 2. u is affine, and in this case the VC dimension of any architecture of n weights is at most n. 3. there are constants af; 0 and b such that u( x) = ax + b except at a finite nonempty set of points. In this case, the VC dimension of any architecture of n weights is O(n 2 ), and there are architectures of VC dimension O(nlogn). Due to the lack of space, the proof cannot be included in this paper. Note that the upper bound of the first special case is tight for threshold nets, and that of the second special case is tight for linear functions in ]Rn. Acknowledgements Pascal Koiran was supported by an INRIA fellowship , DIMACS, and the International Computer Science Institute. Eduardo Sontag was supported in part by US Air Force Grant AFOSR-94-0293. References M. ANTHONY AND N.L. BIGGS (1992) Computational Learning Theory: An Introduction, Cambridge U. Press. E .B. BAUM AND D . HAUSSLER (1989) What size net gives valid generalization?, Neural Computation 1, pp. 151-160. L. BLUM, M. SHUB AND S. SMALE (1989) On the theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines, Bulletin of the AMS 21, pp. 1- 46. A. BLUMER, A. EHRENFEUCHT, D . HAUSSLER, AND M. WARMUTH (1989) Learnability and the Vapnik- Chervonenkis dimension, J. of the ACM 36, pp. 929-965. T.M. COVER (1988) Capacity problems for linear machines, in: Pattern Recognition, L. Kanal ed., Thompson Book Co., pp. 283-289. P. GOLDBERG AND M. JERRUM (1995) Bounding the Vapnik-Chervonenkis dimension of concept classes parametrized by real numbers, Machine Learning 18, pp. 131-148. M . KARPINSKI AND A. MACINTYRE (1995) Polynomial bounds for VC dimension of sigmoidal neural networks, in Proc. 27th ACM Symposium on Theory of Computing, pp. 200208. W. MAASS (1993) Bounds for the computational power and learning complexity of analog neural nets, in Proc. of the 25th ACM Symp. Theory of Computing, pp. 335-344. W . MAASS (1994) Perspectives of current research about the complexity of learning in neural nets, in Theoretical Advances in Neural Computation and Learning, V.P. Roychowdhury, K.Y. Siu, and A. Orlitsky, editors, Kluwer, Boston, pp. 295-336. B.K. NATARAJAN (1991) Machine Learning : A Theoretical Approach, M. Kaufmann Publishers, San Mateo, CA. E.D. SONTAG (1989) Sigmoids distinguish better than Heavisides, Neural Computation 1, pp. 470-472. E.D. SONTAG (1992) Feedforward nets for interpolation and classification, J. Compo Syst. Sci 45, pp. 20-48. L.G. VALIANT (1984) A theory of the learnable, Comm. of the ACM 27, pp. 1134-1142 V.N. VAPNIK (1982) Estimation of Dependencies Based on Empirical Data, Springer, Berlin.
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Factorial Hidden Markov Models Zoubin Ghahramani zoubin@psyche.mit.edu Department of Computer Science University of Toronto Toronto, ON M5S 1A4 Canada Michael I. Jordan jordan@psyche.mit.edu Department of Brain & Cognitive Sciences Massachusetts Institute of Technology Cambridge, MA 02139 USA Abstract We present a framework for learning in hidden Markov models with distributed state representations. Within this framework, we derive a learning algorithm based on the Expectation-Maximization (EM) procedure for maximum likelihood estimation. Analogous to the standard Baum-Welch update rules, the M-step of our algorithm is exact and can be solved analytically. However, due to the combinatorial nature of the hidden state representation, the exact E-step is intractable. A simple and tractable mean field approximation is derived. Empirical results on a set of problems suggest that both the mean field approximation and Gibbs sampling are viable alternatives to the computationally expensive exact algorithm. 1 Introduction A problem of fundamental interest to machine learning is time series modeling. Due to the simplicity and efficiency of its parameter estimation algorithm, the hidden Markov model (HMM) has emerged as one of the basic statistical tools for modeling discrete time series, finding widespread application in the areas of speech recognition (Rabiner and Juang, 1986) and computational molecular biology (Baldi et al. , 1994). An HMM is essentially a mixture model, encoding information about the history of a time series in the value of a single multinomial variable (the hidden state). This multinomial assumption allows an efficient parameter estimation algorithm to be derived (the Baum-Welch algorithm). However, it also severely limits the representational capacity of HMMs. For example, to represent 30 bits of information about the history of a time sequence, an HMM would need 230 distinct states. On the other hand an HMM with a distributed state representation could achieve the same task with 30 binary units (Williams and Hinton, 1991). This paper addresses the problem of deriving efficient learning algorithms for hidden Markov models with distributed state representations. Factorial Hidden Markov Models 473 The need for distributed state representations in HMMs can be motivated in two ways. First, such representations allow the state space to be decomposed into features that naturally decouple the dynamics of a single process generating the time series. Second, distributed state representations simplify the task of modeling time series generated by the interaction of multiple independent processes. For example, a speech signal generated by the superposition of multiple simultaneous speakers can be potentially modeled with such an architecture. Williams and Hinton (1991) first formulated the problem of learning in HMMs with distributed state representation and proposed a solution based on deterministic Boltzmann learning. The approach presented in this paper is similar to Williams and Hinton's in that it is also based on a statistical mechanical formulation of hidden Markov models. However, our learning algorithm is quite different in that it makes use of the special structure of HMMs with distributed state representation, resulting in a more efficient learning procedure. Anticipating the results in section 2, this learning algorithm both obviates the need for the two-phase procedure of Boltzmann machines, and has an exact M-step. A different approach comes from Saul and Jordan (1995), who derived a set of rules for computing the gradients required for learning in HMMs with distributed state spaces. However, their methods can only be applied to a limited class of architectures. 2 Factorial hidden Markov models Hidden Markov models are a generalization of mixture models. At any time step, the probability density over the observables defined by an HMM is a mixture of the densities defined by each state in the underlying Markov model. Temporal dependencies are introduced by specifying that the prior probability of the state at time t depends on the state at time t -1 through a transition matrix, P (Figure 1a). Another generalization of mixture models, the cooperative vector quantizer (CVQ; Hinton and Zemel, 1994 ), provides a natural formalism for distributed state representations in HMMs. Whereas in simple mixture models each data point must be accounted for by a single mixture component, in CVQs each data point is accounted for by the combination of contributions from many mixture components, one from each separate vector quantizer. The total probability density modeled by a CVQ is also a mixture model; however this mixture density is assumed to factorize into a product of densities, each density associated with one of the vector quantizers. Thus, the CVQ is a mixture model with distributed representations for the mixture components. Factorial hidden Markov models! combine the state transition structure of HMMs with the distributed representations of CVQs (Figure 1 b). Each of the d underlying Markov models has a discrete state s~ at time t and transition probability matrix Pi. As in the CVQ, the states are mutually exclusive within each vector quantizer and we assume real-valued outputs. The sequence of observable output vectors is generated from a normal distribution with mean given by the weighted combination of the states of the underlying Markov models: where C is a common covariance matrix. The k-valued states Si are represented as 1 We refer to HMMs with distributed state as factorial HMMs as the features of the distributed state factorize the total state representation. 474 Z. GHAHRAMANI. M. I. JORDAN discrete column vectors with a 1 in one position and 0 everywhere else; the mean of the observable is therefore a combination of columns from each of the Wi matrices. a) ~-------.... y p Figure 1. a) Hidden Markov model. b) Factorial hidden Markov model. We capture the above probability model by defining the energy of a sequence of T states and observations, {(st, yt)};=l' which we abbreviate to {s, y}, as: 1l( {s,y}) = ~ t. k -t. w;s:]' C- 1 [yt -t. w;s:]-t. t. sf A;S:-l, (1) where [Ai]jl = logP(s~jls~I-I) such that 2::=1 e[Ai]j/ = 1, and I denotes matrix transpose. Priors for the initial state, sl, are introduced by setting the second term in (1) to - 2:t=1 sf log7ri. The probability model is defined from this energy by the Boltzmann distribution 1 P({s,y}) = Z exp{-ll({s,y})}. (2) Note that like in the CVQ (Ghahramani, 1995), the undamped partition function Z = J d{y} Lexp{-ll({s,y})}, {s} evaluates to a constant, independent of the parameters. This can be shown by first integrating the Gaussian variables, removing all dependency on {y}, and then summing over the states using the constraint on e[A,]j/ . The EM algorithm for Factorial HMMs As in HMMs, the parameters of a factorial HMM can be estimated via the EM (Baum-Welch) algorithm. This procedure iterates between assuming the current parameters to compute probabilities over the hidden states (E-step), and using these probabilities to maximize the expected log likelihood of the parameters (Mstep). Using the likelihood (2), the expected log likelihood of the parameters is Q(4)new l4>) = (-ll({s,y}) -logZ)c, (3) Factorial Hidden Markov Models 475 where </J = {Wi, Pi, C}f=l denotes the current parameters, and Oc denotes expectation given the damped observation sequence and </J. Given the observation sequence, the only random variables are the hidden states. Expanding equation (3) and limiting the expectation to these random variables we find that the statistics that need to be computed for the E-step are (sDc, (s~sj')c, and (S~S~-l\. Note that in standard HMM notation (Rabiner and Juang, 1986), (sDc corresponds to t t I' , It and (SiSi )c corresponds to et, whereas (s~st·)c has no analogue when there is only a single underlying Markov model. The ~-step uses these expectations to maximize Q with respect to the parameters. The constant partition function allowed us to drop the second term in (3). Therefore, unlike the Boltzmann machine, the expected log likelihood does not depend on statistics collected in an undamped phase of learning, resulting in much faster learning than the traditional Boltzmann machine (Neal, 1992). M-step Setting the derivatives of Q with respect to the output weights to zero, we obtain a linear system of equations for W: Wnew = [2:(SS')c] t [2:(S)CY'] , N,t. N,t where sand Ware the vector and matrix of concatenated Si and. Wi, respectively,L:N denotes summation over a data set of N sequences, and t is the Moore-Penrose pseudo-inverse. To estimate the log transition probabilities we solve 8Q/8[Ai ]jl = 0 subject to the constraint L:j e[A,]i l = 1, obtaining [A .]~ew _ I i...JN,t ij il c ( '" (st st-l) ) • JI og t t-l . L:N,t,j(SijSil )c (4) The covariance matrix can be similarly estimated: cnew = 2: YY' - 2: y(s)~(ss')!(s)cy'. N,t N,t The M-step equations can therefore be solved analytically; furthermore, for a single underlying Markov chain, they reduce to the traditional Baum-Welch re-estimation equations. E-step Unfortunately, as in the simpler CVQ, the exact E-step for factorial HMMs is computationally intractable. For example, the expectation of the lh unit in vector i at time step t, given {y}, is: (s!j)c p(sL = II{y}, </J) k 2: P(s~it=I,.",s~j = 1, ... ,s~,jd=ll{y},</J) it,···,jhyt',oo.,jd Although the Markov property can be used to obtain a forward-back ward-like factorization of this expectation across time steps, the sum over all possible configurations of the other hidden units within each time step is unavoidable. For a data set 476 Z. GHAHRAMANI, M. I. JORDAN of N sequences of length T, the full E-ste~ calculated through the forward-backward procedure has time complexity O(NTk2 ). Although more careful bookkeeping can reduce the complexity to O(NTdkd+1), the exponential time cannot be avoided. This intractability of the exact E-step is due inherently to the cooperative nature of the model-the setting of one vector only determines the mean of the observable if all the other vectors are fixed. Rather than summing over all possible hidden state patterns to compute the exact expectations, a natural approach is to approximate them through a Monte Carlo method such as Gibbs sampling. The procedure starts with a clamped observable sequence {y} and a random setting of the hidden states {sj}. At each time step, each state vector is updated stochastically according to its probability distribution conditioned on the setting of all the other state vectors: s~ '" P (s~ I {y }, {sj : j "# i or T "# t}, ¢). These conditional distributions are straightforward to compute and a full pass of Gibbs sampling requires O(NTkd) operations. The first and secondorder statistics needed to estimate (sDc, (s~sj\ and (S~s~-l\ are collected using the S~j'S visited and the probabilities estimated during this sampling process. Mean field approximation A different approach to computing the expectations in an intractable system is given by mean field theory. A mean field approximation for factorial HMMs can be obtained by defining the energy function 1l({s,y}) = ~ L [yt -Itt]' C- 1 [yt -Itt] - Lsf logm}. t t ,i which results in a completely factorized approximation to probability density (2): .P({s,y}) ex II exp{-~ [yt -Itt], C- 1 [yt -Itt]} II (m~j)3:j (5) t t ,i ,j In this approximation, the observables are independently Gaussian distributed with mean Itt and each hidden state vector is multinomially distributed with mean m~. This approximation is made as tight as possible by chosing the mean field parameters Itt and m~ that minimize the ¥ullback-Liebler divergence K.q.PIIP) == (logP)p - (log?)p where Op denotes expectation over the mean field distribution (5). With the observables clamped, Itt can be set equal to the observable yt. Minimizing K£(.PIIP) with respect to the mean field parameters for the states results in a fixed-point equation which can be iterated until convergence: m~ new u{W!C-1 [yt - yt] + W!C-1Wim~ - ~diag{W!C-IWd - 1 (6) +At'm~-l + A~m~+1} t t t where yt == Ei Wim~ and u{-} is the softmax exponential, normalized over each hidden state vector. The first term is the projection of the error in the observable onto the weights of state vector i-the more a hidden unit can reduce this error, the larger its mean field parameter. The next three terms arise from the fact that (s;j) p is equal to mij and not m;j. The last two terms introduce dependencies forward and backward in time. Each state vector is asynchronously updated using (6), at a time cost of O(NTkd) per iteration. Convergence is diagnosed by monitoring the K£ divergence in the mean field distribution between successive time steps; in practice convergence is very rapid (about 2 to 10 iterations of (6)). Factorial Hidden Markov Models 477 Table 1: Comparison of factorial HMM on four problems of varying size d k Alg # Train Test Cycles Time7Cycle 3 2 HMM 5 649 ±8 358 ± 81 33 ± 19 l.ls Exact 877 ±O 768 ±O 22 ±6 3.0 s Gibbs 710 ± 152 627 ± 129 28 ±ll 6.0 s MF 755 ± 168 670 ± 137 32 ± 22 l.2 s 3 3 HMM 5 670 ± 26 -782 ± 128 23 ± 10 3.6 s Exact 568 ± 164 276 ± 62 35 ± 12 5.2 s Gibbs 564 ± 160 305 ± 51 45 ± 16 9.2 s MF 495 ± 83 326 ± 62 38 ± 22 l.6 s 5 2 HMM 5 588 ± 37 -2634 ± 566 18 ± 1 5.2 s Exact 223 ± 76 159 ± 80 31 ± 17 6.9 s Gibbs 123 ± 103 73 ± 95 40 ±5 12.7 s MF 292 ± 101 237 ± 103 54 ± 29 2.2 s 5 3 HMM 3 1671,1678,1690 -00,-00 ,-00 14,14,12 90.0 s Exact -55,-354,-295 -123,-378,-402 90,100,100 5l.Os Gibbs -123,-160,-194 -202,-237,-307 100,73,100 14.2 s MF -287,-286,-296 -364,-370,-365 100,100,100 4.7 s Table 1. Data was generated from a factorial HMM with d underlying Markov models of k states each. The training set was 10 sequences of length 20 where the observable was a 4-dimensional vector; the test set was 20 such sequences. HMM indicates a hidden Markov model with k d states; the other algorithms are factorial HMMs with d underlying k-state models. Gibbs sampling used 10 samples of each state. The algorithms were run until convergence, as monitored by relative change in the likelihood, or a maximum of 100 cycles. The # column indicates number of runs. The Train and Test columns show the log likelihood ± one standard deviation on the two data sets. The last column indicates approximate time per cycle on a Silicon Graphics R4400 processor running Matlab. 3 Empirical Results We compared three EM algorithms for learning in factorial HMMs-using Gibbs sampling, mean field approximation, and the exact (exponential) E step- on the basis of performance and speed on randomly generated problems. Problems were generated from a factorial HMM structure, the parameters of which were sampled from a uniform [0,1] distribution, and appropriately normalized to satisfy the sum-to-one constraints of the transition matrices and priors. Also included in the comparison was a traditional HMM with as many states (k d ) as the factorial HMM. Table 1 summarizes the results. Even for moderately large state spaces (d ~ 3 and k ~ 3) the standard HMM with kd states suffers from severe overfitting. Furthermore, both the standard HMM and the exact E-step factorial HMM are extremely slow on the larger problems. The Gibbs sampling and mean field approximations offer roughly comparable performance at a great increase in speed. 4 Discussion The basic contribution of this paper is a learning algorithm for hidden Markov models with distributed state representations. The standard Baum-Welch procedure is intractable for such architectures as the size of the state space generated from the cross product of d k-valued features is O(kd), and the time complexity of Baum-Welch is quadratic in this size. More importantly, unless special constraints are applied to this cross-product HMM architecture, the number of parameters also 478 z. GHAHRAMANI, M. 1. JORDAN grows as O(k2d), which can result in severe overfitting. The architecture for factorial HMMs presented in this paper did not include any coupling between the underlying Markov chains. It is possible to extend the algorithm presented to architectures which incorporate such couplings. However, these couplings must be introduced with caution as they may result either in an exponential growth in parameters or in a loss of the constant partition function property. The learning algorithm derived in this paper assumed real-valued observables. The algorithm can also be derived for HMMs with discrete observables, an architecture closely related to sigmoid belief networks (Neal, 1992). However, the nonlinearities induced by discrete observables make both the E-step and M-step of the algorithm more difficult. In conclusion, we have presented Gibbs sampling and mean field learning algorithms for factorial hidden Markov models. Such models incorporate the time series modeling capabilities of hidden Markov models and the advantages of distributed representations for the state space. Future work will concentrate on a more efficient mean field approximation in which the forward-backward algorithm is used to compute the E-step exactly within each Markov chain, and mean field theory is used to handle interactions between chains (Saul and Jordan, 1996). Acknowledgements This project was supported in part by a grant from the McDonnell-Pew Foundation, by a grant from ATR Human Information Processing Research Laboratories, by a grant from Siemens Corporation, and by grant N00014-94-1-0777 from the Office of Naval Research. References Baldi, P., Chauvin, Y., Hunkapiller, T ., and McClure, M. (1994). Hidden Markov models of biological primary sequence information. Proc. Nat. Acad. Sci. (USA),91(3):10591063. Ghahramani, Z. (1995). Factorial learning and the EM algorithm. In Tesauro, G., Touretzky, D., and Leen, T., editors, Advances in Neural Information Processing Systems 7. MIT Press, Cambridge, MA. Hinton, G. and Zemel, R. (1994). Autoencoders, minimum description length, and Helmholtz free energy. In Cowan, J., Tesauro, G., and Alspector, J., editors, Advances in Neural Information Processing Systems 6. Morgan Kaufmanm Publishers, San Francisco, CA. Neal, R. (1992). Connectionist learning of belief networks. Artificial Intelligence, 56:71113. Rabiner, 1. and Juang, B. (1986). An Introduction to hidden Markov models. IEEE Acoustics, Speech €1 Signal Processing Magazine, 3:4-16. Saul, 1. and Jordan, M. (1995). Boltzmann chains and hidden Markov models. In Tesauro, G., Touretzky, D., and Leen, T., editors, Advances in Neural Information Processing Systems 7. MIT Press, Cambridge, MA. Saul, 1. and Jordan, M. (1996). Exploiting tractable substructures in Intractable networks. In Touretzky, D., Mozer, M., and Hasselmo, M., editors, Advances in Neural Information Processing Systems 8. MIT Press. Williams, C. and Hinton, G. (1991). Mean field networks that learn to discriminate temporally distorted strings. In Touretzky, D., Elman, J., Sejnowski, T., and Hinton, G., editors, Connectionist Models: Proceedings of the 1990 Summer School, pages 18-22. Morgan Kaufmann Publishers, Man Mateo, CA.
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Extracting Thee-Structured Representations of Thained Networks Mark W. Craven and Jude W. Shavlik Computer Sciences Department University of Wisconsin-Madison 1210 West Dayton St. Madison, WI 53706 craven@cs.wisc.edu, shavlik@cs.wisc.edu Abstract A significant limitation of neural networks is that the representations they learn are usually incomprehensible to humans. We present a novel algorithm, TREPAN, for extracting comprehensible, symbolic representations from trained neural networks. Our algorithm uses queries to induce a decision tree that approximates the concept represented by a given network. Our experiments demonstrate that TREPAN is able to produce decision trees that maintain a high level of fidelity to their respective networks while being comprehensible and accurate. Unlike previous work in this area, our algorithm is general in its applicability and scales well to large networks and problems with high-dimensional input spaces. 1 Introduction For many learning tasks, it is important to produce classifiers that are not only highly accurate, but also easily understood by humans. Neural networks are limited in this respect, since they are usually difficult to interpret after training. In contrast to neural networks, the solutions formed by "symbolic" learning systems (e.g., Quinlan, 1993) are usually much more amenable to human comprehension. We present a novel algorithm, TREPAN, for extracting comprehensible, symbolic representations from trained neural networks. TREPAN queries a given network to induce a decision tree that describes the concept represented by the network. We evaluate our algorithm using several real-world problem domains, and present results that demonstrate that TREPAN is able to produce decision trees that are accurate and comprehensible, and maintain a high level of fidelity to the networks from which they were extracted. Unlike previous work in this area, our algorithm Extracting Tree-structured Representations of Trained Networks 25 is very general in its applicability, and scales well to large networks and problems with high-dimensional input spaces. The task that we address is defined as follows: given a trained network and the data on which it was trained, produce a concept description that is comprehensible, yet classifies instances in the same way as the network. The concept description produced by our algorithm is a decision tree, like those generated using popular decision-tree induction algorithms (Breiman et al., 1984; Quinlan, 1993). There are several reasons why the comprehensibility of induced concept descriptions is often an important consideration. If the designers and end-users of a learning system are to be confident in the performance of the system, they must understand how it arrives at its decisions. Learning systems may also play an important role in the process of scientific discovery. A system may discover salient features and relationships in the input data whose importance was not previously recognized. If the representations formed by the learner are comprehensible, then these discoveries can be made accessible to human review. However, for many problems in which comprehensibility is important, neural networks provide better generalization than common symbolic learning algorithms. It is in these domains that it is important to be able to extract comprehensible concept descriptions from trained networks. 2 Extracting Decision Trees Our approach views the task of extracting a comprehensible concept description from a trained network as an inductive learning problem. In this learning task, the target concept is the function represented by the network, and the concept description produced by our learning algorithm is a decision tree that approximates the network. However, unlike most inductive learning problems, we have available an oracle that is able to answer queries during the learning process. Since the target function is simply the concept represented by the network, the oracle uses the network to answer queries. The advantage of learning with queries, as opposed to ordinary training examples, is that they can be used to garner information precisely where it is needed during the learning process. Our algorithm, as shown in Table 1, is similar to conventional decision-tree algorithms, such as CART (Breiman et al. , 1984), and C4.5 (Quinlan, 1993), which learn directly from a training set. However, TREPAN is substantially different from these conventional algorithms in number of respects, which we detail below. The Oracle. The role of the oracle is to determine the class (as predicted by the network) of each instance that is presented as a query. Queries to the oracle, however, do not have to be complete instances, but instead can specify constraints on the values that the features can take. In the latter case, the oracle generates a complete instance by randomly selecting values for each feature, while ensuring that the constraints are satisfied. In order to generate these random values, TREPAN uses the training data to model each feature's marginal distribution. TREPAN uses frequency counts to model the distributions of discrete-valued features, and a kernel density estimation method (Silverman, 1986) to model continuous features. As shown in Table 1, the oracle is used for three different purposes: (i) to determine the class labels for the network's training examples; (ii) to select splits for each of the tree's internal nodes; (iii) and to determine if a node covers instances of only one class. These aspects of the algorithm are discussed in more detail below. Tree Expansion. Unlike most decision-tree algorithms, which grow trees in a depth-first manner, TREPAN grows trees using a best-first expansion. The notion 26 M. W. CRAVEN, J. W. SHAVLIK Table 1: The TREPAN algorithm. TREPAN(training_examples, features) Queue:= 0 /* sorted queue of nodes to expand * / for each example E E training_examples class label for E := ORACLE(E) initialize the root of the tree, T, as a leaf node put (T, training_examples, {} ) into Queue /* use net to label examples * / while Queue is not empty and size(T) < tree...size_limit /* expand a node * / remove node N from head of Queue examplesN := example set stored with N constraintsN := constraint set stored with N use features to build set of candidate splits use examplesN and calls to ORAcLE(constraintsN) to evaluate splits S := best binary split search for best m-of-n split, S', using 5 as a seed make N an internal node with split S' for each outcome, s, of 5' /* make children nodes * / return T make C, a new child node of N constraintsc := constraintsN U {5' = s} use calls to ORACLE( constraintsc) to determine if C should remain a leaf otherwise examplesc := members of examplesN with outcome s on split S' put (C, examplesc, constraintsc) into Queue of the best node, in this case, is the one at which there is the greatest potential to increase the fidelity of the extracted tree to the network. The function used to evaluate node n is f(n) = reach(n) x (1 - fidelity(n)) , where reach(n) is the estimated fraction of instances that reach n when passed through the tree, and fidelity(n) is the estimated fidelity of the tree to the network for those instances. Split Types. The role of internal nodes in a decision tree is to partition the input space in order to increase the separation of instances of different classes. In C4. 5, each of these splits is based on a single feature. Our algorithm, like Murphy and Pazzani's (1991) ID2-of-3 algorithm, forms trees that use m-of-n expressions for its splits. An m-of-n expression is a Boolean expression that is specified by an integer threshold, m, and a set of n Boolean conditions. An m-of-n expression is satisfied when at least m of its n conditions are satisfied. For example, suppose we have three Boolean features, a, b, and c; the m-of-n expression 2-of-{ a, ....,b, c} is logically equivalent to (a /\ ....,b) V (a /\ c) V (....,b /\ c). Split Selection. Split selection involves deciding how to partition the input space at a given internal node in the tree. A limitation of conventional tree-induction algorithms is that the amount of training data used to select splits decreases with the depth of the tree. Thus splits near the bottom of a tree are often poorly chosen because these decisions are based on few training examples. In contrast, because TREPAN has an oracle available, it is able to use as many instances as desired to select each split. TREPAN chooses a split after considering at least Smin instances, where Smin is a parameter of the algorithm. When selecting a split at a given node, the oracle is given the list of all of the previously selected splits that lie on the path from the root of the tree to that node. These splits serve as constraints on the feature values that any instance generated by the oracle can take, since any example must satisfy these constraints in order to Extracting Tree-structured Representations of Trained Networks 27 reach the given node. Like the ID2-of-3 algorithm, TREPAN uses a hill-climbing search process to construct its m-of-n splits. The search process begins by first selecting the best binary split at the current node; as in C4. 5, TREPAN uses the gain ratio criterion (Quinlan, 1993) to evaluate candidate splits. For two-valued features, a binary split separates examples according to their values for the feature. For discrete features with more than two values, we consider binary splits based on each allowable value of the feature (e.g., color=red?, color=blue?, ... ). For continuous features, we consider binary splits on thresholds, in the same manner as C4.5. The selected binary split serves as a seed for the m-of-n search process. This greedy search uses the gain ratio measure as its heuristic evaluation function, and uses the following two operators (Murphy & Pazzani, 1991): • m-of-n+l : Add a new value to the set, and hold the threshold constant. For example, 2-of-{ a, b} => 2-of-{ a, b, c} . • m+l- of-n+l: Add a new value to the set, and increment the threshold. For example, 2-of-{ a, b, c} => 3-of-{ a, b, c, d}. Unlike ID2-of-3, TREPAN constrains m-of-n splits so that the same feature is not used in two or more disjunctive splits which lie on the same path between the root and a leaf of the tree. Without this restriction, the oracle might have to solve difficult satisfiability problems in order create instances for nodes on such a path. Stopping Criteria. TREPAN uses two separate criteria to decide when to stop growing an extracted decision tree. First, a given node becomes a leaf in the tree if, with high probability, the node covers only instances of a single class. To make this decision, TREPAN determines the proportion of examples, Pc, that fall into the most common class at a given node, and then calculates a confidence interval around this proportion (Hogg & Tanis, 1983). The oracle is queried for additional examples until prob(pc < 1 - f) < 6, where f and 6 are parameters of the algorithm. TREPAN also accepts a parameter that specifies a limit on the number of internal nodes in an extracted tree. This parameter can be used to control the comprehensibility of extracted trees, since in some domains, it may require very large trees to describe networks to a high level of fidelity. 3 Empirical Evaluation In our experiments, we are interested in evaluating the trees extracted by our algorithm according to three criteria: (i) their predictive accuracy; (ii) their comprehensibility; (i) and their fidelity to the networks from which they were extracted. We evaluate TREPAN using four real-world domains: the Congressional voting data set (15 features, 435 examples) and the Cleveland heart-disease data set (13 features, 303 examples) from the UC-Irvine database; a promoter data set (57 features, 468 examples) which is a more complex superset of the UC-Irvine one; and a data set in which the task is to recognize protein-coding regions in DNA (64 features, 20,000 examples) (Craven & Shavlik, 1993b). We remove the physician-fee-freeze feature from the voting data set to make the problem more difficult. We conduct our experiments using a 10-fold cross validation methodology, except for in the proteincoding domain. Because of certain domain-specific characteristics of this data set, we use 4-fold cross-validation for our experiments with it. We measure accuracy and fidelity on the examples in the test sets. Whereas accuracy is defined as the percentage of test-set examples that are correctly classified, fidelity is defined as the percentage of test-set examples on which the classification 28 M. W. CRAVEN, J. W. SHAVLIK Table 2: Test-set accuracy and fidelity. domain accuracy fidelity networks C4.5 ID2-of-3 TREPAN TREPAN heart 84.5% 71.0% 74.6% 81.8% 94.1% promoters 90.6 84.4 83.5 87.6 85.7 protein coding 94.1 90.3 90.9 91.4 92.4 voting 92.2 89.2 87.8 90.8 95.9 made by a tree agrees with its neural-network counterpart. Since the comprehensibility of a decision tree is problematic to measure, we measure the syntactic complexity of trees and take this as being representative of their comprehensibility. Specifically, we measure the complexity of each tree in two ways: (i) the number of internal (i.e., non-leaf) nodes in the tree, and (ii) the number of symbols used in the splits of the tree. We count an ordinary, single-feature split as one symbol. We count an m-of-n split as n symbols, since such a split lists n feature ,-alues. The neural networks we use in our experiments have a single layer of hidden units. The number of hidden units used for each network (0, 5, 10, 20 or 40) is chosen using cross validation on the network's training set, and we use a validation set to decide when to stop training networks. TREPAN is applied to each saved network. The parameters of TREPAN are set as follows for all runs: at least 1000 instances (training examples plus queries) are considered before selecting each split; we set the E and 6 parameters, which are used for the stopping-criterion procedure, to 0.05; and the maximum tree size is set to 15 internal nodes, which is the size of a complete binary tree of depth four. As baselines for comparison, we also run Quinlan'S (1993) C4.5 algorithm, and Murphy and Pazzani's (1991) ID2-of-3 algorithm on the same testbeds. Recall that ID2-of-3 is similar to C4.5, except that it learns trees that use m-of-n splits. We use C4.5's pruning method for both algorithms and use cross validation to select pruning levels for each training set. The cross-validation runs evaluate unpruned trees and trees pruned with confidence levels ranging from 10% to 90%. Table 2 shows the test-set accuracy results for our experiments. It can be seen that, for every data set, neural networks generalize better than the decision trees learned by C4.5 and ID2-of-3. The decision trees extracted from the networks by TREPAN are also more accurate than the C4.5 and ID2-of-3 trees in all domains. The differences in accuracy between the neural networks and the two conventional decision-tree algorithms (C4.5 and ID2-of-3) are statistically significant for all four domains at the 0.05 level using a paired, two-tailed t-test. We also test the significance of the accuracy differences between TREPAN and the other decision-tree algorithms. Except for the promoter domain, these differences are also statistically significant. The results in this table indicate that, for a range of interesting tasks, our algorithm is able to extract decision trees which are more accurate than decision trees induced strictly from the training data. Table 2 also shows the test-set fidelity measurements for the TREPAN trees. These results indicate that the trees extracted by TREPAN provide close approximations to their respective neural networks. Table 3 shows tree-complexity measurements for C4.5, ID2-of-3, and TREPAN. For all four data sets, the trees learned by TREPAN have fewer internal nodes than the trees produced by C4.5 and ID2-of-3. In most cases, the trees produced by TREPAN and ID2-of-3 use more symbols than C4.5, since their splits are more Extracting Tree-structured Representations of Trained Networks 29 Table 3: Tree complexity. domain # internal nodes # symbols C4.5 ID2-of-3 TREPAN II C4.5 ID2-of-3 TREPAN heart 17.5 15.7 11.8 17.5 48.8 20.8 promoters 11.2 12.6 9.2 11.2 47.5 23.8 protein coding 155.0 66.0 10.0 155.0 455.3 36.0 voting 20.1 19.2 11.2 20.1 77.3 20.8 complex. However, for most of the data sets, the TREPAN trees and the C4.5 trees are comparable in terms of their symbol complexity. For all data sets, the ID2-of-3 trees are more complex than the TREPAN trees. Based on these results, we argue that the trees extracted by TREPAN are as comprehensible as the trees learned by conventional decision-tree algorithms. 4 Discussion and Conclusions In the previous section, we evaluated our algorithm along the dimensions of fidelity, syntactic complexity, and accuracy. Another advantage of our approach is its generality. Unlike numerous other extraction methods (Hayashi, 1991; McMillan et al., 1992; Craven & Shavlik, 1993a; Sethi et al., 1993; Tan, 1994; Tchoumatchenko & Ganascia, 1994; Alexander & Mozer, 1995; Setiono & Liu, 1995), the TREPAN algorithm does not place any requirements on either the architecture of the network or its training method. TREPAN simply uses the network as a black box to answer queries during the extraction process. In fact, TREPAN could be used to extract decision-trees from other types of opaque learning systems, such as nearest-neighbor classifiers. There are several existing algorithms which do not require special network architectures or training procedures (Saito & Nakano, 1988; Fu, 1991; Gallant, 1993). These algorithms, however, assume that each hidden unit in a network can be accurately approximated by a threshold unit. Additionally, these algorithms do not extract m-of-n rules, but instead extract only conjunctive rules. In previous work (Craven & Shavlik, 1994; Towell & Shavlik, 1993), we have shown that this type of algorithm produces rule-sets which typically are far too complex to be comprehensible. Thrun (1995) has developed a general method for rule extraction, and has described how his algorithm can be used to verify that an m-of-n rule is consistent with a network, but he has not developed a rule-searching method that is able to find concise rule sets. A strength of our algorithm, in contrast, is its scalability. We have demonstrated that our algorithm is able to produce succinct decision-tree descriptions of large networks in domains with large input spaces. In summary, a significant limitation of neural networks is that their concept representations are usually not amenable to human understanding. We have presented an algorithm that is able to produce comprehensible descriptions of trained networks by extracting decision trees that accurately describe the networks' concept representations. We believe that our algorithm, which takes advantage of the fact that a trained network can be queried, represents a promising advance towards the goal of general methods for understanding the solutions encoded by trained networks. Acknow ledgements This research was partially supported by ONR grant N00014-93-1-0998. 30 M. W. eRA VEN, J. W. SHA VLIK References Alexander, J. A. & Mozer, M. C. (1995). Template-based algorithms for connectionist rule extraction. In Tesauro, G., Touretzky, D., & Leen, T., editors, Advances in Neural Information Processing Systems (volume 7). MIT Press. Breiman, L., Friedman, J., Olshen, R., & Stone, C. (1984). Classification and Regression Trees. Wadsworth and Brooks, Monterey, CA. Craven, M. & Shavlik, J. (1993a). Learning symbolic rules using artificial neural networks. In Proc. of the 10th International Conference on Machine Learning, (pp. 73-80), Amherst, MA. Morgan Kaufmann. Craven, M. W. & Shavlik, J. W. (1993b). Learning to predict reading frames in E. coli DNA sequences. In Proc. of the 26th Hawaii International Conference on System Sciences, (pp. 773-782), Wailea, HI. IEEE Press. Craven, M. W. & Shavlik, J. W. (1994). Using sampling and queries to extract rules from trained neural networks. In Proc. of the 11th International Conference on Machine Learning, (pp. 37- 45), New Brunswick, NJ. Morgan Kaufmann. Fu, L. (1991). Rule learning by searching on adapted nets. In Proc. of the 9th National Conference on Artificial Intelligence, (pp. 590- 595) , Anaheim, CA. AAAI/MIT Press. Gallant, S. I. (1993). Neural Network Learning and Expert Systems. MIT Press. Hayashi, Y. (1991). A neural expert system with automated extraction of fuzzy ifthen rules. In Lippmann, R., Moody, J., & Touretzky, D., editors, Advances in Neural Information Processing Systems (volume 3). Morgan Kaufmann, San Mateo, CA. Hogg, R. V. & Tanis, E. A. (1983). Probability and Statistical Inference. MacMillan. McMillan, C., Mozer, M. C., & Smolensky, P. (1992). Rule induction through integrated symbolic and sub symbolic processing. In Moody, J., Hanson, S., & Lippmann, R., editors, Advances in Neural Information Processing Systems (volume 4). Morgan Kaufmann. Murphy, P. M. & Pazzani, M. J. (1991). ID2-of-3: Constructive induction of M-of-N concepts for discriminators in decision trees. In Proc. of the 8th International Machine Learning Workshop, (pp. 183- 187), Evanston, IL. Morgan Kaufmann. Quinlan, J. (1993). C4.5: Programs for Machine Learning. Morgan Kaufmann. Saito, K. & Nakano, R. (1988). Medical diagnostic expert system based on PDP model. In Proc. of the IEEE International Conference on Neural Networks, (pp. 255- 262), San Diego, CA. IEEE Press. Sethi, I. K., Yoo, J. H., & Brickman, C. M. (1993). Extraction of diagnostic rules using neural networks. In Proc. of the 6th IEEE Symposium on Computer-Based Medical Systems, (pp. 217-222), Ann Arbor, MI. IEEE Press. Setiono, R. & Liu, H. (1995). Understanding neural networks via rule extraction. In Proc. of the 14th International Joint Conference on Artificial Intelligence, (pp. 480- 485), Montreal, Canada. Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman and Hall. Tan, A.-H. (1994). Rule learning and extraction with self-organizing neural networks. In Proc. of the 1993 Connectionist Models Summer School. Erlbaum. Tchoumatchenko, 1. & Ganascia, J.-G. (1994). A Bayesian framework to integrate symbolic and neural learning. In Proc. of the 11th International Conference on Machine Learning, (pp. 302- 308), New Brunswick, NJ. Morgan Kaufmann. Thrun, S. (1995). Extracting rules from artificial neural networks with distributed representations. In Tesauro, G., Touretzky, D., & Leen, T., editors, Advances in Neural Information Processing Systems (volume 7). MIT Press. Towell, G. & Shavlik, J. (1993). Extracting refined rules from knowledge-based neural networks. Machine Learning, 13(1):71-101.
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Improving Committee Diagnosis with Resampling Techniques Bambang Parmanto Department of Information Science University of Pittsburgh Pittsburgh, PA 15260 parmanto@li6.pitt. edu Paul W. Munro Department of Information Science University of Pittsburgh Pittsburgh, PA 15260 munro@li6.pitt. edu Howard R. Doyle Pittsburgh Transplantation Institute 3601 Fifth Ave, Pittsburgh, PA 15213 doyle@vesaliw. tu. med. pitt. edu Abstract Central to the performance improvement of a committee relative to individual networks is the error correlation between networks in the committee. We investigated methods of achieving error independence between the networks by training the networks with different resampling sets from the original training set. The methods were tested on the sinwave artificial task and the real-world problems of hepatoma (liver cancer) and breast cancer diagnoses. 1 INTRODUCTION The idea of a neural net committee is to combine several neural net predictors to perform collective decision making, instead of using a single network (Perrone, 1993). The potential of a committee in improving classification performance has been well documented. Central to this improvement is the extent to which the errors tend to coincide. Committee errors occur where the misclassification sets of individual networks overlap. On the one hand, if all errors of committee members coincide, using a committee does not improve performance. On the other hand, if errors do not coincide, performance of the committee dramatically increases and asymptotically approaches perfect performance. Therefore, it is beneficial to make the errors among the networks in the committee less correlated in order to improve the committee performance. Improving Committee Diagnosis with Resampling Techniques 883 One way of making the networks less correlated is to train them with different sets of data. Decreasing the error correlation by training members of the committee using different sets of data is intuitively appealing. Networks trained with different data sets have a higher probability of generalizing differently and tend to make errors in different places in the problem space. The idea is to split the data used in the training into several sets. The sets are not necessarily mutually exclusive, they may share part of the set (overlap). This idea resembles resampling methods such as cross-validation and bootstrap known in statistics for estimating the error of a predictor from limited sets of available data. In the committee framework, these techniques are recast to construct different training sets from the original training set. David Wolpert (1992) has put forward a general framework of training the committee using different partitions of the data known as stacked generalization. This approach has been adopted to the regression environment and is called stacked regression (Breiman, 1992). Stacked regression uses cross-validation to construct different sets of regression functions. A similar idea of using a bootstrap method to construct different training sets has been proposed by Breiman (1994) for classification and regression trees predictors. 2 THE ALGORITHMS 2.1 BOOTSTRAP COMMITTEE (BOOTC) Consider a total of N items are available for training. The approach is to generate K replicates from the original set, each containing the same number of item as the original set. The replicates are obtained from the original set by drawing at random with replacement. See Efron & Tibshirani (1993) for background on bootstrapping. Use each replicate to train each network in the committee. Using this bootstrap procedure, each replicate is expected to include roughly 36 % duplicates (due to replacement during sampling). Only the distinct fraction is used for training and the leftover fraction for early stopping, if necessary (notice slight difference from the standard bootstrapping and from Breiman's bagging). Early stopping usually requires a fraction of the data to be taken from the original training set, which might degrade the performance of the neural network. The advantage of a BOOTC is that the leftover sample is already available. Algorithm: 1. Generate bootstrap replicates Ll, ... , LK from the original set. 2. For each bootstrap replicate, collect unsampled items into leftover sample t .. l*l l*K se s, gIVIng: , ... , . 3. For each Lk, train a network. Use the leftover set l*k as validation stopping criteria if necessary. Giving K neural net predictors: f(~i Lk) 4. Build a committee from the bootstrap networks using a simple averaging procedure: fcom(~) = ic ~~=l f(~i Lk) There is no rule as to how many bootstrap replicates should be used to achieve a good performance. In error estimation, the number ranges from 20 to 200. It is beneficial to keep the number of replicates, hence the number of networks, small to reduce training time. Unless the networks are trained on a parallel machine, training time increases proportionally to the number of networks in the committee. In this experiment, 20 bootstrap training replicates were constructed for 20 networks in 884 B. PARMANTO, P. W. MUNRO, H. R. DOYLE the committee. Twenty replicates were chosen since beyond this number there is no significant improvement on the performance. 2.2 CROSS-VALIDATION COMMITTEE (CVC) The algorithm is quite similar to the procedure used in prediction error estimation. First, generate replicates from the original training set by removing a fraction of the data. Let D denote the original data, and D- V denote the data with subset v removed. The procedure revolves so that each item is in the removed fraction at least once. Generate replicates D11Jl , ••• Di/Ie and train each network in the committee with one replicate. An important issue in the eve is the degree of data overlap between the replicates. The degree of overlap depends on the number of replicates and the size of a removed fraction from the original sample. For example, if the committee consists of 5 networks and 0.5 of the data are removed for each replicate, the minimum fraction of overlap is 0 (calculation: (v x 2) - 1.0) and the maximum is ~ (calculation: 1.0 - k)' Algorithm: 1. Divide data into v-fractions db . . . , dv 2. Leave one fraction die and train network fie with the rest of the data (D-d le ). 3. Use die as a validation stopping criteria, if necessary. 4. Build a committee from the networks using a simple averaging procedure. The fraction of data overlap determines the trade-off between the individual network performance and error correlation between the networks. Lower correlation can be expected if the networks train with less overlapped data, which means a larger removed fraction and smaller fraction for training. The smaller the training set size, the lower the individual network performance that can be expected. We investigated the effect of data overlap on the error correlations between the networks and the committee performance. We also studied the effect of training size on the individual performance. The goal was to find an optimal combination of data overlap and individual training size. 3 THE BASELINE & PERFORMANCE EVALUATION To evaluate the improvement of the proposed methods on the committee performance, they should be compared with existing methods as the baseline. The common method for constructing a committee is to train an ensemble of networks independently. The networks in the committee are initialized with different sets of weights. This type of committee has been reported as achieving significant improvement over individual network performances in regression (Hashem, 1993) and classification tasks (Perrone, 1993; Parmanto et al., 1994). The baseline, BOOTe, and eve were compared using exactly the same architecture and using the same pair of training-test sets. Performance evaluation was conducted using 4-fold exhaustive cross-validation where 0.25 fraction of the original data is used for the test set and the remainder of the data is used for the training set. The procedure was repeated 4 times so that all items were once on the test set. The performance was calculated by averaging the results of 4 test sets. The simulations Improving Committee Diagnosis with Resampling Techniques 885 were conducted several times using different initial weights to exclude the possibility that the improvement was caused by chance. 4 EXPERIMENTS 4.1 SYNTHETIC DATA: SINWAVE CLASSIFICATION The sinwave task is a classification problem with two classes, a negative class represented as 0 and a positive class represented as 1. The data consist of two input variables, x = (Xli X2). The entire space is divided equally into two classes with the separation line determined by the curve X2 = sin( 2: Xl). The upper half of the rectangle is the positive class, while the lower half is the negative one (see Fig. 1). Gaussian noise along the perfect boundary with variance of 0.1 is introduced to the clean data and is presented in Fig. 1 (middle). Let z be a vector drawn from the Gaussian distribution with variance TI, then the classification rule is given by equation: (1) A similar artificial problem is used to analyze the bias-variance trade-offs by Geman et al. (1992). Figure 1: Complete and clean data/without noise (top), complete data with noise (middle), and a small fraction used for training (bottom). The population contains 3030 data items, since a grid of 0.1 is used for both Xl and X2 . In the real world, we usually have no access to the entire population. To mimic this situation, the training set contained only a small fraction of the population. Fig. 1 (bottom) visualizes a training set that contains 200 items with 100 items for each class. The training set is constructed by randomly sampling the population. The performance of the predictor is measured with respect to the test set. The population (3030 items) is used as the test set. 4.2 HEPATOMA DETECTION Hepatoma is a very important clinical problem in patients who are being considered for liver transplantation for its high probability of recurrence. Early hepatoma detection may improve the ultimate outlook of the patients since special treatment can be carried out. Unfortunately, early detection using non-invasive procedures 886 B. PARMANTO, P. W. MUNRO, H. R. DOYLE can be difficult, especially in the presence of cirrhosis. We have been developing neural network classifiers as a detection system with minimum imaging or invasive studies (Parmanto et al., 1994). The task is to detect the presence or absence (binary output) of a hepatoma given variables taken from an individual patient. Each data item consists of 16 variables, 7 of which are continuous variables and the rest are binary variables, primarily blood measurements. For this experiment, 1172 data items with their associated diagnoses are available. Out of 1172 itmes, 693 items are free from missing values, 309 items contain missing values only on the categorical variables, and 170 items contain missing values on both types of variables. For this experiment, only the fraction without missing values and the fraction with missing values on the categorical variables were used, giving the total item of 1002. Out of the 1002 items, 874 have negative diagnoses and the remaining 128 have positive diagnoses. 4.3 BREAST CANCER The task is to diagnose if a breast cytology is benign or malignant based on cytological characteristics. Nine input variables have been established to differentiate between the benign and malignant samples which include clump thickness, marginal adhesion, the uniformity of cell size and shape, etc. The data set was originally obtained from the University of Wisconsin Hospitals and currently stored at the UCI repository for machine learning (Murphy & Aha, 1994). The current size of the data set is 699 examples. 5 THE RESULTS Committee Performance Indiv. Performance ¥ ~ ~.::.:.:-:~~~«: .:::: . .::.::---.-.-......... ---..... . § :! ---.... _--,I; N ~ ..... 0 bas.an. o • ::-:::. &10. .. ,,,, 4 6 10 12 14 16 4 8 10 12 14 16 /I hidden units /I hidden units Correlation Percent Improvement 0r------------------. .... -... ...... -. -~-------~-------~ .... -......... -" , --. .... _-.. --_._ ... _---. Q -~ .... o & :::: li&>mr", o~ ________________ ~ 4 6 10 12 14 16 8 10 12 14 16 II hidden units /I hidden ...,its Figure 2: Results on the sinwave classif. task. Performances of individual nets and the committee (top); error correlation and committee improvement (bottom). Figure 2. (top) and Table 1. show that the performance of the committee is always better than the average performance of individual networks in all three committees. Improving Committee Diagnosis with Resampling Techniques 887 Task Methods Indiv. Nets Error Committee Improv. Improv. % error Corr % error to Indiv. to baseline Smwave Baseline 13.31 .87 11.8 11 '70 (2 vars ) BOOTC 12.85 .57 8.36 35 % 29 % CVC 15.72 .33 9.79 38 % 17 % Cancer Baseline 2.7 .96 2.5 5% (9 vars) BOOTC 3.14 .83 2.0 34 % 20 % CVC 3.2 .80 1.63 49 % 35 % Hepatoma BaSeline 25.95 .89 23.25 10.5 % (16 vars) BOOTC 26.00 .70 19.72 24 % 15.2 % CVC 26.90 .55 19.05 29 % 18 % Table 1: Error rate, correlation, and performance improvement calculated based on the best architecture for each method. Reduction of misclassification rates compare to the baseline committee Correlation vs. Fraction of Data Overlap 0r-----------------------____ -. m ° N o .,., T ! i ~ , , Fraction 01 data overlap Figure 3: Error correlation and fraction of overlap in training data (results from the sinwave classification task). The CVC and BOOTC are always better than the baseline even when the individual network performance is worse. Figure 2 (bottom) and the table show that the improvement of a committee over individual networks is proportional to the error correlation between the networks in the committee. The CVC consistently produces significant improvement over its individual network performance due to the low error correlation, while the baseline committee only produces modest improvement. This result confirms the basic assumption of this research: committee performance can be improved by decorrelating the errors made by the networks. The performance of a committee depends on two factors: individual performance of the networks and error correlation between the networks. The gain of using BOOTC or CVC depends on how the algorithms can reduce the error correlations while still maintaining the individual performance as good as the individual performance of the baseline. The BOOTC produced impressive improvement (29 %) over the baseline on the sinwave task due to the lower correlation and good individual performance. The performances of the BOOTC on the other two tasks were not as impressive due to the modest reduction of error correlation and slight decrease in individual performance. The performances were still significantly better than the baseline committee. The CVC, on the other hand, consistently reduced the correlation and 888 B. PARMANTO, P. W. MUNRO, H. R. DOYLE improved the committee performance. The improvement on the sinwave task was not as good as the BOOTC due to the low individual performance. The individual performance of the CVC and BOOTC in general are worse than the baseline. The individual performance of CVC is 18 % and 19 % lower than the baseline on the sinwave and cancer tasks respectively, while the BOOTC suffered significant reduction of individual performance only on the cancer task (16 %). The degradation of individual performance is due to the smaller training set for each network on the CVC and the BOOTC. The detrimental effect of a small training set, however, is compensated by low correlation between the networks. The effect of a smaller training set depends on the size of the original training set. If the data size is large, using a smaller set may not be harmful. On the contrary, if the data set is small, using an even smaller data set can significantly degrade the performance. Another interesting finding of this experiment is the relationship between the error correlation and the overlap fraction in the training set. Figure 3 shows that small data overlap causes the networks to have low correlation to each other. 6 SUMMARY Training committees of networks using different set of data resampled from the original training set can improve committee performance by reducing the error correlation among the networks in the committee. Even when the individual network performances of the BOOTC and CVC degrade from the baseline networks, the committee performance is still better due to the lower correlation. Acknowledgement This study is supported in part by Project Grant DK 29961 from the National Institutes of Health, Bethesda, MD. We would like to thank the Pittsburgh Transplantation Institute for providing the data for this study. References Breiman, L, (1992) Stacked Regressions, TR 367, Dept. of Statistics., UC. Berkeley. Breiman, L, (1994) Bagging Predictors, TR 421, Dept. of Statistics, UC. Berkeley. Efron, B., & Tibshirani, R.J. (1993) An Introd. to the Bootstrap. Chapman & Hall. Hashem, S. (1994). Optimal Linear Combinations of Neural Networks. PhD Thesis, Purdue University. Geman, S., Bienenstock, E., and Doursat, R. (1992) Neural networks and the bias/variance dilemma. Neural Computation, 4(1), 1-58. Murphy, P. M., &. Aha, D. W. (1994). UCI Repository of machine learning databases [ftp: ics.uci.edu/pub/machine-Iearning-databases/] Parmanto, B., Munro, P.W., Doyle, H.R., Doria, C., Aldrighetti, 1., Marino, I.R., Mitchel, S., and Fung, J.J. (1994) Neural network classifier for hepatoma detectipn. Proceedings of the World Congress of Neural Networks 1994 San Diego, June 4-9. Perrone, M.P. (1993) Improving Regression Estimation: Averaging Methods for Variance Reduction with Eztension to General Convez Measure Optimization. PhD Thesis, Department of Physics, Brown University. Wolpert, D. (1992). Stacked generalization, Neural Networks, 5, 241-259.
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The Capacity of a Bump Gary William Flake· Institute for Advance Computer Studies University of Maryland College Park, MD 20742 Abstract Recently, several researchers have reported encouraging experimental results when using Gaussian or bump-like activation functions in multilayer perceptrons. Networks of this type usually require fewer hidden layers and units and often learn much faster than typical sigmoidal networks. To explain these results we consider a hyper-ridge network, which is a simple perceptron with no hidden units and a rid¥e activation function. If we are interested in partitioningp points in d dimensions into two classes then in the limit as d approaches infinity the capacity of a hyper-ridge and a perceptron is identical. However, we show that for p ~ d, which is the usual case in practice, the ratio of hyper-ridge to perceptron dichotomies approaches pl2(d + 1). 1 Introduction A hyper-ridge network is a simple perceptron with no hidden units and a ridge activation function. With one output this is conveniently described as y = g(h) = g(w . x - b) where g(h) = sgn(1 - h2). Instead of dividing an input-space into two classes with a single hyperplane, a hyper-ridge network uses two parallel hyperplanes. All points in the interior of the hyperplanes form one class, while all exterior points form another. For more information on hyper-ridges, learning algorithms, and convergence issues the curious reader should consult [3]. We wouldn't go so far as to suggest that anyone actually use a hyper-ridge for a real-world problem, but it is interesting to note that a hyper-ridge can represent linear inseparable mappings such as XOR, NEGATE, SYMMETRY, and COUNT(m) [2, 3]. Moreover, hyper-ridges are very similar to multilayer perceptrons with bump-like activation functions, such as a Gaussian, in the way the input space is partitioned. Several researchers [6, 2,3, 5] have independently found that Gaussian units offer many advantages over sigmoidal units. ·Current address: Adaptive Information and Signal Processing Department, Siemens Corporate Research, 755 College Road East, Princeton, NJ 08540. Email: ftake@scr.siemens.com The Capacity of a Bump 557 In this paper we derive the capacity of a hyper-ridge network. Our first result is that hyper-ridges and simple perceptrons are equivalent in the limit as the input dimension size approaches infinity. However, when the number of patterns is far greater than the input dimension (as is the usual case) the ratio of hyper-ridge to perceptron dichotomies approaches p/2(d + 1), giving some evidence that bump-like activation functions offer an advantage over the more traditional sigmoid. The rest of this paper is divided into three more sections. In Section 2 we derive the number of dichotomies for a hyper-ridge network. The capacities for hyper-ridges and simple perceptrons are compared in Section 3. Finally, in Section 4 we give our conclusions. 2 The Representation Power of a Hyper-Ridge Suppose we have p patterns in the pattern-space, ~d, where d is the number of inputs of our neural network. A dichotomy is a classification of all of the points into two distinct sets. Clearly, there are at most 2P dichotomies that exist. We are concerned with the number of dichotomies that a single hyper-ridge node can represent. Let the number of dichotomies of p patterns in d dimensions be denoted as D(p, d). For the case of D(1, d), when p = 1 there are always two and only two dichotomies since one can trivially include the single point or no points. Thus, D(1, d) = 2. For the case of D(p, 1), all of the points are constrained to fallon a line. From this set pick two points, say Xa and Xb. It is always possible to place a ridge function such that all points between Xa and Xb (inclusive of the end points) are included in one set, and all other points are excluded. Thus, there are p dichotomies consisting of a single point, p - 1 dichotomies consisting of two points, p - 2 dichotomies consisting of three points, and so on. No other dichotomies besides the empty set are possible. The number of possible hyper-ridge dichotomies in one dimension can now be expressed as P 1 D(p, 1)= 2: i + 1 = 2P(P + 1)+ 1, i=1 (1) with the extra dichotomy coming from the empty set. To derive the general form of the recurrence relationship, we would have to resort to techniques similar to those used by Cover [1], Nilsson [7], and Gardner [4]. Because of space considerations, we do not give the full derivation of the general form of the recurrence relationship in this paper, but instead cite the complete derivation given in [3]. The short version of the story is that the general form of the recurrence relationship for hyper-ridge dichotomies is identical to the equivalent expression for simple perceptrons: D(p, d) = D(P 1, d) + D(P 1, d 1). (2) All differences between the capacity of hyper-ridges and simple perceptrons are, therefore, a consequence of the different base cases for the recurrence expression. To get Equation 2 into closed form, we first expand D(p, d) a total of p times, yielding p-l . p-l ( ) D(P,d)=~ i D(I,d-z). (3) For Equation 3 it is possible for the second term of D( 1, d - 1) to become zero or negative. Taking the two identities D(P,O) = p + 1 and D(p, -1) = 1 are the only choices that are consistent with the recurrent relationship expressed in Equation 2. With this in mind, there are three separate cases that we need to be concerned with: p < d + 2, p = d + 2, and 558 G. W.FLAKE p>d+2. Whenp<d+2 p-l () p-l ( ) D(p, d) = ~ P ~ 1 D(1, d - i) = 2 ~ P ~ 1 = 2P, (4) since all of the second tenns in D(I, d - i) are always greater or equal to zero. When p = d + 2, the last tenn in D(I, d - i), in the summation, will be equal to -I. Thus we can expand Equation 3 in this case to DCp,d) = ~ (p ~ 1) D(I,d _ i)= ~ (p ~ 1) D(1,p-2- i) = I: (p ~ 1) D(1, p - 2 - i) + 1 = 2 I: (p ~ I) + 1 ~ l~ = 2(2P- 1 -l)+1=2P -1. (5) Finally, when p > d + 2, some ofthe last terms in D(I, d - i) are always negative. We can disregard all d - i < -1, taking D(1, d - i) equal to zero in these cases (which is consistent with the recurrence relationship), DCp,d) = ~ (p~ 1) D(I, d _ i)= ~ (p ~ 1) D(I,d _ i) ~ (p - 1) . (p - 1) ~ (p - 1) (p - 1) = ~ i D(1, d - z) + d + 1 = 2 ~ i + d + 1 . (6) Combining Equations 4, 5, and 6 gives d 2 L (p ~ 1) + (~~:) for p > d + 2 I~ D~~= m 2P 1 for p = d + 2 2P forp <d+2 3 Comparing Representation Power Cover [1], Nilsson [7], and Gardner [4] have all shown that D(p, ~ for simple perceptrons obeys the rule 2~ (p~ 1) forp >d+2 D(P, d) = (8) 2P - 2 for p=d+2 2P forp <d+2 The interesting case is when p > d + 2, since that is where Equations 7 and 8 differ the most. Moreover, problems are more difficult when the number of training patterns greatly exceeds the number of trainable weights in a neural network. Let Dh(p, d) and Dp(p, d) denote the number of dichotomies possible for hyper-ridge networks and simple perceptrons, respectively. Additionally, Let Ch, and Cp denote the The Capacity of a Bump 559 respective capacities. We should expect both Dh(p, d)/2P and Dp(p, d)/2P to be at or around 1 for small values of p/(d + 1). At some point, for large p/(d + 1), the 2P term should dominate, making the ratio go to zero. The capacity of a network can loosely be defined as the value p/(d + 1) such that D(p, d)/2P = ~. This is more rigorously defined as C= { . l' D(c(d+ 1).d) =~} c . d~~ 2c(d+1) 2' which is the point in which the transition occurs in the limit as the input dimension goes to infinity. Figures 1, 2, and 3 illustrate and compare Cp and Ch at different stages. In Figure 1 the capacities are illustrated for perceptrons and hyper-ridges, respectively, by plotting D(p, d)/LP versus p/(d + 1) for various values of d. On par with our intuition, the ratio D(p, d)/LP equals 1 for small values of p/(d + 1) but decreases to zero as p(d + 1) increases. Figure 2 and the left diagram of Figure 3 plot D(p, d)/2P versus p/(d + 1) for perceptron and hyper-ridges, side by side, with values of d = 5,20, and 100. As d increases, the two curves become more similar. This fact is further illustrated in the right diagram of Figure 3 where the plot is of Dh(p, d)/Dp(P, d) versus p for various values of d. The ratio clearly approaches 1 as d increases, but there is significant difference for smaller values of d. The differences between Dp and Dh can be more explicitly quantified by noting that ( p -1) Dh(p, d) = Dp(p, d) + d + 1 for p > d + 2. This difference clearly shows up in in the plots comparing the two capacities. We will now show that the capacities are identical in the limit as d approaches infinity. To do this, we will prove that the capacity curves for both hyper-ridges and perceptrons crosses ~ at p/(d + 1) = 2. This fact is already widely known for perceptrons. Because of space limitations we will handwave our way through lemma and corollary proofs. The curious reader should consult [3) for the complete proofs. Lemma 3.1 lim (2nn) = O. n-oo 22n Short Proof Since n approaches infinity, we can use Stirling's formula as an approximation of the factorials. Corollary 3.2 For all positive integer constants, a, b, and c, lim _1_ (2n + b) = O. n-oo 22n+a n + c o Short Proof When adding the constants band c to the combination, the whole combination can always be represented as comb(2n, n)· y, where y is some multiplicative constant. Such a constant can always be factored out of the limit. Additionally, large values of a only increase the growth rate of the denominator. o Lemma 3.3 For p/(d + 1) = 2, liffid ..... oo Dp(p, d)/2P = ~. Short Proof Consult any of Cover [1], Nilsson [7], or Gardner [4] for full proof. o 560 " " '\ . \ : \ ' .~ 0.' I d = S d =20 ---d= 100 .. . ~ - ~ --r-- -_ . ..• os - G. W.FLAKE d = 5d= 20 -d=l00 . Figure 1: On the left, Dp(P, tf)12P versus pl(d + 1), and on the right, Dh(p, d)/2P versus pl(d + 1) for various values of d. Notice that for perceptrons the curve always passes through! at pl(d + 1) = 2. For hyper-ridges, the point where the curve passes through! decreases as d increases. 2 p/(d+ I) I perceptmnihyper-ridge --\ o.s ----'r-'--- - \ . . °0L-----~----~ 2 --~~------J pl(d+ J) Figure 2: On the left, capacity comparison for d = 5. There is considerable difference for small values of d, especially when one considers that the capacities are normalized by 2P. On the right, comparison for d = 20. The difference between the two capacities is much more subtle now that d is fairly large. os t perecpuon :hyper.ridge _. ! o L-----~----~~--~----~ o 20 10 d: 1 d", 2 ---d= 5 d", IO d= 100 - _. 10 20 30 40 so 60 70 80 90 100 P Figure 3: On the left, capacity comparison for d = 100. For this value of d, the capacities are visibly indistinguishable. On the right, Dh(P, d)1 Dp(P, tf) versus p for various values of d. For small values of d the capacity of a hyper-ridge is much greater than a perceptron. As d grows, the ratio asymptotically approaches 1. The Capacity of a Bump 561 Theorem 3.4 For pl(d + 1) = 2, lim Dh(p, d) = !. d-oo 2P 2 Proof Taking advantage of the relationship between perceptron dichotomies and hyperridge dichotomies allows us to expand Dh(p, d), 1· Dh(P, d) l' Dp(P, d) l' (p - 1) 1m = 1m + 1m . d-oo 2P d-oo 2P d-oo d + 1 By Lemma 3.3, and substituting 2(d + 1) for p, we get: 1 l' (2d + 1) - + 1m . 2 d-oo d + 1 Finally, by Corollary 3.2 the right limit vanishes leaving us with !. o Superficially, Theorem 3.4 would seem to indicate that there is no difference between the representation power of a perceptron and a hyper-ridge network. However, since this result is only valid in the limit as the number of inputs goes to infinity, it would be interesting to know the exact relationship between Dp(d, p) and Dh(d, p) for finite values of d. In the right diagram of Figure 3 values of Dp(d,p)IDh(d,p) are plotted against various values of p. The figure is slightly misleading since the ratio appears to be linear in p, when, in fact, the ratio is only approximately linear in p. If we normalize the ratio by } and recompute the ratio in the limit as p approaches infinity the ratio becomes linear in d. Theorem 3.5 establishes this rigorously. Theorem 3.5 Proof First, note that we can simplify the left hand side of the expression to . 1 Dh(d,p) . 1 Dp(d,p) + (~~ :) . 1 (~~:) hm = hm = hm (9) p-oopDp(d,p) p_oop Dp(d,p) p_oop Dp(d,p) In the next step, we will invert Equation 9, making it easier to work with. We need to show that the new expression is equal to 2(d + 1). L~ (p~ 1) lim p Dp(d,p) = lim 2p l = p-oo (~ ~ :) p_oo (~ ~ :) . 2: d (P - I)! (d + 1)!(P - d - 2)! . 2: d (d + I)! (P - d - 2)! hm 2p = hm 2p = p_oo . i!(P - i-I)! (P - I)! P_oo. i! (P - i-I)! 1=0 1=0 d d lim p 2(d+l)"d!(p-d-l)!= lim2(d+l)"d!(p-d-l)! (10) p_oo (P - 1 - d) 6 i! (P - i - 1)! p_oo ~ i! (P - i-I)! 1=0 i=O In Equation 10, the summation can be reduced to 1 since 1. d! (P - d - 1 )! _ {O when 0 :5 i < d 1m 1 h . d p-oo i! (P - i-I)! w en l = 562 Thus, Equation 10 is equal to 2(d + 1), which proves the theorem. o G. W.FLAKE Theorem 3.5 is valid only in the case when p ~ d, which is typically true in interesting classification problems. The result of the theorem gives us a good estimate of how many more dichotomies are computable with a hyper-ridge network when compared to a simple perceptron. When p ~ d the equation Dh(d,p) P Dp(d,p) 2(d+ 1) (11) is an accurate estimate of the difference between the capacities of the two architectures. For example, taking d = 4 and p = 60 and applying the values to Equation 11 yields the ratio of 6, which should be interpreted as meaning that one could store six times the number of mappings in a hyper-ridge network than one could in a simple perceptron. Moreover, Equation 11 is in agreement with the right diagram of Figure 3 for all values of p ~ d. 4 Conclusion An interesting footnote to this work is that the VC dimension [8] of a hyper-ridge network is identical to a simple perceptron, namely d. However, the real difference between perceptrons and hyper-ridges is more noticeable in practice, especially when one considers that linear inseparable problems are representable by hyper-ridges. We also know that there is no such thing as a free lunch and that generalization is sure to suffer in just the cases when representation power is increased. Yet given all of the comparisons between Ml.Ps and radial basis functions (RBFs) we find it encouraging that there may be a class of approximators that is a compromise between the local nature of RBFs and the global structure of MLPs. References [l] T.M. Cover. Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition. IEEE Transactions on Electronic Computers, 14:326-334,1965. [2] M.R.W. Dawson and D.P. Schopflocher. Modifying the generalized delta rule to train networks of non-monotonic processors for pattern classification. Connection Science, 4(1), 1992. [3] G. W. Flake. Nonmonotonic Activation Functions in Multilayer Perceptrons. PhD thesis, University of Maryland, College Park, MD, December 1993. [4] E. Gardner. Maximum storage capacity in neural networks. Europhysics Letters, 4:481-485,1987. [5] F. Girosi, M. Jones, and T. Poggio. Priors, stabilizers and basis functions: from regularization to radial, tensor and additive splines. Technical Report A.I. Memo No. 1430, C.B.C.L. Paper No. 75, MIT AI Laboratory, 1993. [6] E. Hartman and J. D. Keeler. Predicting the future: Advanages of semilocal units. Neural Computation, 3:566-578,1991. [7] N.J. Nilsson. Learning Machines: Foundations of Trainable Pattern Classifying Systems. McGraw-Hill, New York, 1965. [8] Y.N. Vapnik and A. Y. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and Its Applications, 16:264-280, 1971.
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From Isolation to Cooperation: An Alternative View of a System of Experts Stefan Schaal:!:* Christopher C. Atkeson:!: sschaal@cc.gatech.edu cga@cc.gatech.edu http://www.cc.gatech.eduifac/Stefan.Schaal http://www.cc.gatech.eduifac/Chris.Atkeson +College of Computing, Georgia Tech, 801 Atlantic Drive, Atlanta, GA 30332-0280 * A TR Human Infonnation Processing, 2-2 Hikaridai, Seiko-cho, Soraku-gun, 619-02 Kyoto Abstract We introduce a constructive, incremental learning system for regression problems that models data by means of locally linear experts. In contrast to other approaches, the experts are trained independently and do not compete for data during learning. Only when a prediction for a query is required do the experts cooperate by blending their individual predictions. Each expert is trained by minimizing a penalized local cross validation error using second order methods. In this way, an expert is able to find a local distance metric by adjusting the size and shape of the receptive field in which its predictions are valid, and also to detect relevant input features by adjusting its bias on the importance of individual input dimensions. We derive asymptotic results for our method. In a variety of simulations the properties of the algorithm are demonstrated with respect to interference, learning speed, prediction accuracy, feature detection, and task oriented incremental learning. 1. INTRODUCTION Distributing a learning task among a set of experts has become a popular method in computationallearning. One approach is to employ several experts, each with a global domain of expertise (e.g., Wolpert, 1990). When an output for a given input is to be predicted, every expert gives a prediction together with a confidence measure. The individual predictions are combined into a single result, for instance, based on a confidence weighted average. Another approach-the approach pursued in this paper-of employing experts is to create experts with local domains of expertise. In contrast to the global experts, the local experts have little overlap or no overlap at all. To assign a local domain of expertise to each expert, it is necessary to learn an expert selection system in addition to the experts themselves. This classifier determines which expert models are used in which part of the input space. For incremental learning, competitive learning methods are usually applied. Here the experts compete for data such that they change their domains of expertise until a stable configuration is achieved (e.g., Jacobs, Jordan, Nowlan, & Hinton, 1991). The advantage of local experts is that they can have simple parameterizations, such as locally constant or locally linear models. This offers benefits in terms of analyzability, learning speed, and robustness (e.g., Jordan & Jacobs, 1994). For simple experts, however, a large number of experts is necessary to model a function. As a result, the expert selection system has to be more complicated and, thus, has a higher risk of getting stuck in local minima and/or of learning rather slowly. In incremental learning, another potential danger arises when the input distribution of the data changes. The expert selection system usually makes either implicit or explicit prior assumptions about the input data distribution. For example, in the classical mixture model (McLachlan & Basford, 1988) which was employed in several local expert approaches, the prior probabilities of each mixture model can be interpreted as 606 S. SCHAAL. C. C. ATKESON the fraction of data points each expert expects to experience. Therefore, a change in input distribution will cause all experts to change their domains of expertise in order to fulfill these prior assumptions. This can lead to catastrophic interference. In order to avoid these problems and to cope with the interference problems during incremental learning due to changes in input distribution, we suggest eliminating the competition among experts and instead isolating them during learning. Whenever some new data is experienced which is not accounted for by one of the current experts, a new expert is created. Since the experts do not compete for data with their peers, there is no reason for them to change the location of their domains of expertise. However, when it comes to making a prediction at a query point, all the experts cooperate by giving a prediction of the output together with a confidence measure. A blending of all the predictions of all experts results in the final prediction. It should be noted that these local experts combine properties of both the global and local experts mentioned previously. They act like global experts by learning independently of each other and by blending their predictions, but they act like local experts by confining themselves to a local domain of expertise, i.e., their confidence measures are large only in a local region. The topic of data fitting with structurally simple local models (or experts) has received a great deal of attention in nonparametric statistics (e.g., Nadaraya, 1964; Cleveland, 1979; Scott, 1992, Hastie & Tibshirani, 1990). In this paper, we will demonstrate how a nonparametric approach can be applied to obtain the isolated expert network (Section 2.1), how its asymptotic properties can be analyzed (Section 2.2), and what characteristics such a learning system possesses in terms of the avoidance of interference, feature detection, dimensionality reduction, and incremental learning of motor control tasks (Section 3). 2. RECEPTIVE FIELD WEIGHTED REGRESSION This paper focuses on regression problems, i.e., the learning of a map from 9tn ~ 9tm • Each expert in our learning method, Receptive Field Weighted Regression (RFWR), consists of two elements, a locally linear model to represent the local functional relationship, and a receptive field which determines the region in input space in which the expert's knowledge is valid. As a result, a given data set will be modeled by piecewise linear elements, blended together. For 1000 noisy data points drawn from the unit interval of the function z == max[exp(-10x 2),exp(-50l),1.25exp(-5(x2 + l)], Figure 1 illustrates an example of function fitting with RFWR. This function consists of a narrow and a wide ridge which are perpendicular to each other, and a Gaussian bump at the origin. Figure 1 b shows the receptive fields which the system created during the learning process. Each experts' location is at the center of its receptive field, marked by a $ in Figure 1 b. The recep0 . 5 0 -0.5 -1 1.5 ,1 10. 5% 0 I 1- 0 .5 1 0 - 0 .5 -1 x (a) 1.5 0.5 ,., 0 -0.5 -1 -1.5 -1.5 (b) -1 -0.5 o x 0.5 1.5 Figure 1: (a) result of function approximation with RFWR. (b) contour lines of 0.1 iso-activation of each expert in input space (the experts' centers are marked by small circles). From Isolation to Cooperation: An Alternative View of a System of Experts 607 tive fields are modeled by Gaussian functions, and their 0.1 iso-activation lines are shown in Figure 1 b as well. As can be seen, each expert focuses on a certain region of the input space, and the shape and orientation of this region reflects the function's complexity, or more precisely, the function's curvature, in this region. It should be noticed that there is a certain amount of overlap among the experts, and that the placement of experts occurred on a greedy basis during learning and is not globally optimal. The approximation result (Figure 1 a) is a faithful reconstruction of the real function (MSE = 0.0025 on a test set, 30 epochs training, about 1 minute of computation on a SPARC1O). As a baseline comparison, a similar result with a sigmoidal 3-layer neural network required about 100 hidden units and 10000 epochs of annealed standard backpropagation (about 4 hours on a SPARC1O). 2.1 THE ALGORITHM li'Iear ..•... '. ~"" " Galng Unrt WeighBd' / ~:~~ ConnectIOn Average centered at e Output y, Figure 2: The RFWR network RFWR can be sketched in network form as shown in Figure 2. All inputs connect to all expert networks, and new experts can be added as needed. Each expert is an independent entity. It consists of a two layer linear subnet and a receptive field subnet. The receptive field subnet has a single unit with a bell-shaped activation profile, centered at the fixed location c in input space. The maximal output of this unit is "I" at the center, and it decays to zero as a function of the distance from the center. For analytical convenience, we choose this unit to be Gaussian: (1) x is the input vector, and D the distance metric, a positive definite matrix that is generated from the upper triangular matrix M. The output of the linear subnet is: A Tb b -Tf3 y=x + o=x (2) The connection strengths b of the linear subnet and its bias bO will be denoted by the d-dimensional vector f3 from now on, and the tilde sign will indicate that a vector has been augmented by a constant "I", e.g., i = (x T , Il. In generating the total output, the receptive field units act as a gating component on the output, such that the total prediction is: (3) The parameters f3 and M are the primary quantities which have to be adjusted in the learning process: f3 forms the locally linear model, while M determines the shape and orientation of the receptive fields. Learning is achieved by incrementally minimizing the cost function: (4) The first term of this function is the weighted mean squared cross validation error over all experienced data points, a local cross validation measure (Schaal & Atkeson, 1994). The second term is a regularization or penalty term. Local cross validation by itself is consistent, i.e., with an increasing amount of data, the size of the receptive field of an expert would shrink to zero. This would require the creation of an ever increasing number of experts during the course of learning. The penalty term introduces some non-vanishing bias in each expert such that its receptive field size does not shrink to zero. By penalizing the squared coefficients of D, we are essentially penalizing the second derivatives of the function at the site of the expert. This is similar to the approaches taken in spline fitting 608 S. SCHAAL, C. C. A TI(ESON (deBoor, 1978) and acts as a low-pass filter: the higher the second derivatives, the more smoothing (and thus bias) will be introduced. This will be analyzed further in Section 2.2. The update equations for the linear subnet are the standard weighted recursive least squares equation with forgetting factor A (Ljung & SOderstrom, 1986): 1 ( pn- -Tpn ) f3n+1 =f3n+wpn+lxe wherepn+1 =_ pn_ xx ande =(y-xT f3n) cv' A Ajw + xTpnx cv (5) This is a Newton method, and it requires maintaining the matrix P, which is size 0.5d x (d + 1) . The update of the receptive field subnet is a gradient descent in J: Mn+l=Mn- a dJ!aM (6) Due to space limitations, the derivation of the derivative in (6) will not be explained here. The major ingredient is to take this derivative as in a batch update, and then to reformulate the result as an iterative scheme. The derivatives in batch mode can be calculated exactly due to the Sherman-Morrison-Woodbury theorem (Belsley, Kuh, & Welsch, 1980; Atkeson, 1992). The derivative for the incremental update is a very good approximation to the batch update and realizes incremental local cross validation. A new expert is initialized with a default M de! and all other variables set to zero, except the matrix P. P is initialized as a diagonal matrix with elements 11 r/, where the ri are usually small quantities, e.g., 0.01. The ri are ridge regression parameters. From a probabilistic view, they are Bayesian priors that the f3 vector is the zero vector. From an algorithmic view, they are fake data points of the form [x = (0, ... , '12 ,o, ... l,y = 0] (Atkeson, Moore, & Schaal, submitted). Using the update rule (5), the influence of the ridge regression parameters would fade away due to the forgetting factor A. However, it is useful to make the ridge regression parameters adjustable. As in (6), rj can be updated by gradient descent: 1'n+1 = 1'n - a aJ/ar I I I (7) There are d ridge regression parameters, one for each diagonal element of the P matrix. In order to add in the update of the ridge parameters as well as to compensate for the forgetting factor, an iterative procedure based on (5) can be devised which we omit here. The computational complexity of this update is much reduced in comparison to (5) since many computations involve multiplications by zero. Initialize the RFWR network. with no expert; For every new training sample (x,y): a) For k= I to #experts: b) c) d) e) end; - calculate the activation from (I) - update the expert's parameters according to (5), (6), and (7) end; Ir no expert was activated by more than W gen: - create a new expert with c=x end; Ir two experts are acti vated more than W pn .. ~ - erase the expert with the smaller receptive field end; calculate the mean, err ""an' and standard de viation errslIl of the incrementally accumulated error er,! of all experts; For k.= I to #experts: Ir (Itrr! - err_I> 9 er'Sld) reinitialize expert k with M = 2 • Mdef end; In sum, a RFWR expert consists of three sets of parameters, one for the locally linear model, one for the size and shape of the receptive fields, and one for the bias. The linear model parameters are updated by a Newton method, while the other parameters are updated by gradient descent. In our implementations, we actually use second order gradient descent based on Sutton (1992), since, with minor extra effort, we can obtain estimates of the second derivatives of the cost function with respect to all parameters. Finally, the logic of RFWR becomes as shown in the pseudo-code above. Point c) and e) of the algorithm introduce a pruning facility. Pruning takes place either when two experts overlap too much, or when an expert has an exceptionally large mean squared error. The latter method corresponds to a simple form of outlier detection. Local optimization of a distance metric always has a minimum for a very large receptive field size. In our case, this would mean that an expert favors global instead of locally linear regression. Such an expert will accumulate a very large error which can easily be detected From Isolation to Cooperation: An Alternative View of a System of Experts 609 in the given way. The mean squared error term, err, on which this outlier detection is based, is a bias-corrected mean squared error, as will be explained below. 2.2 ASYMPTOTIC BIAS AND PENALTY SELECTION The penalty term in the cost function (4) introduces bias. In order to assess the asymptotic value of this bias, the real function f(x) , which is to be learned, is assumed to be represented as a Taylor series expansion at the center of an expert's receptive field. Without loss of generality, the center is assumed to be at the origin in input space. We furthermore assume that the size and shape of the receptive field are such that terms higher than 0(2) are negligible. Thus, the cost (4) can be written as: J ~ (1w(f. +fTX+~XTFX-bo -bTx Y dx )/(1 wdx )+r~Dnm (8) where fo' f, and F denote the constant, linear, and quadratic terms of the Taylor series expansion, respectively. Inserting Equation (1), the integrals can be solved analytically after the input space is rotated by an orthonormal matrix transforming F to the diagonal matrix F'. Subsequently, bo' b, and D can be determined such that J is minimized: 0.25 ( ) ~ b~ = fa + bias = fa + ~075 ~ sgn(F:')~IF;,:I, b' = f, D:: = (2r)2 (9) This states that the linear model will asymptotically acquire the correct locally linear model, while the constant term will have bias proportional to the square root of the sum of the eigenvalues of F, i.e., the F:n • The distance metric D, whose diagonalized counterpart is D', will be a scaled image of the Hessian F with an additional square root distortion. Thus, the penalty term accomplishes the intended task: it introduces more smoothing the higher the curvature at an expert's location is, and it prevents the receptive field of an expert shrinking to zero size (which would obviously happen for r ~ 0). Additionally, Equation (9) shows how to determine rfor a given learning problem from an estimate of the eigenvalues and a permissible bias. Finally, it is possible to derive estimates of the bias and the mean squared error of each expert from the current distance metric D: biasesl = ~0 .5r IJeigenvalues(D)l.; en,,~, = r L D;m (10) n.m The latter term was incorporated in the mean squared error, err, in Section 2.1. Empirical evaluations (not shown here) verified the validity of these asymptotic results. 3. SIMULA TION RESULTS This section will demonstrate some of the properties of RFWR. In all simulations, the threshold parameters of the algorithm were set to e = 3.5, w prune = 0.9, and w min = 0.1. These quantities determine the overlap of the experts as well as the outlier removal threshold; the results below are not affected by moderate changes in these parameters. 3.1 AVOIDING INTERFERENCE In order to test RFWR's sensitivity with respect to changes in input data distribution, the data of the example of Figure 1 was partitioned into three separate training sets 1; = {(x, y, z) 1-1.0 < x < -O.2} , 1; = {(x, y, z) 1-0.4 < x < OA}, 1; = {(x, y, z) I 0.2 < x < 1.0}. These data sets correspond to three overlapping stripes of data, each having about 400 uniformly distributed samples. From scratch, a RFWR network was trained first on I; for 20 epochs, then on T2 for 20 epochs, and finally on 1; for 20 epochs. The penalty was chosen as in the example of Figure 1 to be r = I.e - 7 , which corresponds to an asymptotic bias of 610 S. SCHAAL, C. C. ATKESON 0.1 at the sharp ridge of the function. The default distance metric D was 50*1, where I is the identity matrix. Figure 3 shows the results of this experiment. Very little interference can be found. The MSE on the test set increased from 0.0025 (of the original experiment of Figure 1) to 0.003, which is still an excellent reconstruction of the real function. y 0 .5 -0 . 5 - 0 . 5 (a) (b) (c) -1 Figure 3: Reconstructed function after training on (a) 7;, (b) then ~,(c) and finally 1;. 3.2 LOCAL FEATURE DETECTION The examples of RFWR given so far did not require ridge regression parameters. Their importance, however, becomes obvious when dealing with locally rank deficient data or with irrelevant input dimensions. A learning system should be able to recognize irrelevant input dimensions. It is important to note that this cannot be accomplished by a distance metric. The distance metric is only able to decide to what spatial extent averaging over data in a certain dimension should be performed. However, the distance metric has no means to exclude an input dimension. In contrast, bias learning with ridge regression parameters is able to exclude input dimensions. To demonstrate this, we added 8 purely noisy inputs (N(0,0.3)) to the data drawn from the function of Figure 1. After 30 epochs of training on a 10000 data point training set, we analyzed histograms of the order of magnitude of the ridge regression parameters in all 100bias input dimensions over all the 79 experts that had been generated by the learning algorithm. All experts recognized that the input dimensions 3 to 8 did not contain relevant information, and correctly increased the corresponding ridge parameters to large values. The effect of a large ridge regression parameter is that the associated regression coefficient becomes zero. In contrast, the ridge parameters of the inputs 1, 2, and the bias input remained very small. The MSE on the test set was 0.0026, basically identical to the experiment with the original training set. 3.3 LEARNING AN INVERSE DYNAMICS MODEL OF A ROBOT ARM Robot learning is one of the domains where incremental learning plays an important role. A real movement system experiences data at a high rate, and it should incorporate this data immediately to improve its performance. As learning is task oriented, input distributions will also be task oriented and interference problems can easily arise. Additionally, a real movement system does not sample data from a training set but rather has to move in order to receive new data. Thus, training data is always temporally correlated, and learning must be able to cope with this. An example of such a learning task is given in Figure 4 where a simulated 2 DOF robot arm has to learn to draw the figure "8" in two different regions of the work space at a moderate speed (1.5 sec duration). In this example, we assume that the correct movement plan exists, but that the inverse dynamics model which is to be used to control this movement has not been acquired. The robot is first trained for 10 minutes (real movement time) in the region of the lower target trajectory where it performs a variety of rhythmic movements under simple PID control. The initial performance of this controller is shown in the bottom part of Figure 4a. This training enables the robot to learn the locally appropriate inverse dynamics model, a ~6 ~ ~2 continuous mapping. Subsequent perFrom Isolation to Cooperation: An Alternative View of a System of Experts 611 0.5 0.' t GralMy 0.' 0.2 ~ 8 0.1 ~t Z 8 8 ..,. ~. ·0.4 ~.5 (a) (b) (0) 0 0.1 0.2 0.3 0.4 0.!5 Figure 4: Learning to draw the figure "8" with a 2-joint arm: (a) Performance of a PID controller before learning (the dimmed lines denote the desired trajectories, the solid lines the actual performance); (b) Performance after learning using a PD controller with feedforward commands from the learned inverse model; (c) Performance of the learned controller after training on the upper "8" of (b) (see text for more explanations). formance using this inverse model for control is depicted in the bottom part of Figure 4b. Afterwards, the same training takes place in the region of the upper target trajectory in order to acquire the inverse model in this part of the world. The figure "8" can then equally well be drawn there (upper part of Figure 4a,b). Switching back to the bottom part of the work space (Figure 4c), the first task can still be performed as before. No interference is recognizable. Thus, the robot could learn fast and reliably to fulfill the two tasks. It is important to note that the data generated by the training movements did not always have locally full rank. All the parameters of RFWR were necessary to acquire the local inverse model appropriately. A total of 39 locally linear experts were generated. 4. DISCUSSION We have introduced an incremental learning algorithm, RFWR, which constructs a network of isolated experts for supervised learning of regression tasks. Each expert determines a locally linear model, a local distance metric, and local bias parameters by incrementally minimizing a penalized local cross validation error. Our algorithm differs from other local learning techniques by entirely avoiding competition among the experts, and by being based on nonparametric instead of parametric statistics. The resulting properties of RFWR are a) avoidance of interference in the case of changing input distributions, b) fast incremental learning by means of Newton and second order gradient descent methods, c) analyzable asymptotic properties which facilitate the selection of the fit parameters, and d) local feature detection and dimensionality reduction. The isolated experts are also ideally suited for parallel implementations. Future work will investigate computationally less costly delta-rule implementations of RFWR, and how well RFWR scales in higher dimensions. 5. REFERENCES Atkeson, C. G., Moore, A. W., & Schaal, S. (submitted). "Locally weighted learning." Artificial Intelligence Review. Atkeson, C. G. (1992). "Memory-based approaches to approximating continuous functions." In: Casdagli, M., & Eubank, S. (Eds.), Nonlinear Modeling and Forecasting, pp.503-521. Addison Wesley. Belsley, D. A., Kuh, E., & Welsch, R. E. (1980). Regression diagnostics: Identifying influential data and sources ofcollinearity. New York: Wiley. Cleveland, W. S. (1979). "Robust locally weighted regression and smoothing scatterplots." J. American Stat. Association, 74, pp.829-836. de Boor, C. (1978). A practical guide to splines. New York: Springer. Hastie, T. J., & Tibshirani, R. J. (1990). Generalized additive models. London: Chapman and Hall. 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. (1994). "Hierarchical mixtures of experts and the EM algorithm." Neural Computation, 6, pp.79-87. Ljung, L., & S_derstr_m, T. (1986). Theory and practice of recursive identification. Cambridge, MIT Press. McLachlan, G. J., & Basford, K. E. (1988). Mixture models. New York: Marcel Dekker. Nadaraya, E. A. (1964). "On estimating regression." Theor. Prob. Appl., 9, pp.141-142. Schaal, S., & Atkeson, C. G. (l994b). "Assessing the quality of learned local models." In: Cowan, J. ,Tesauro, G., & Alspector, J. (Eds.), Advances in Neural Information Processing Systems 6. Morgan Kaufmann. Scott, D. W. (1992). Multivariate Density Estimation. New York: Wiley. Sutton, R. S. (1992). "Gain adaptation beats least squares." In: Proc. of 7th Yale Workshop on Adaptive and Learning Systems, New Haven, CT. Wolpert, D. H. (1990). "Stacked genealization." Los Alamos Technical Report LA-UR-90-3460. Boosting Decision Trees Harris Drucker AT&T Bell Laboratories Holmdel, New Jersey 07733 Corinna Cortes AT&T Bell Laboratories Murray Hill, New Jersey 07974 Abstract A new boosting algorithm of Freund and Schapire is used to improve the performance of decision trees which are constructed using the information ratio criterion of Quinlan's C4.5 algorithm. This boosting algorithm iteratively constructs a series of decision trees, each decision tree being trained and pruned on examples that have been filLered by previously trained trees. Examples that have been incorrectly classified by the previous trees in the ensemble are resampled with higher probability to give a new probability distribution for the next tree in the ensemble to train on. Results from optical character recognition (OCR), and knowledge discovery and data mining problems show that in comparison to single trees, or to trees trained independently, or to trees trained on subsets of the feature space, the boosting ensemble is much better. 1 INTRODUCTION A new boosting algorithm termed AdaBoost by their inventors (Freund and Schapire, 1995) has advantages over the original boosting algorithm (Schapire, 1990) and a second version (Freund, 1990). The implications of a boosting algorithm is that one can take a series of learning machines (termed weak learners) each having a poor error rate (but no worse than .5-y, where y is some small positive number) and combine them to give an ensemble that has very good performance (termed a strong learner). The first practical implementation of boosting was in OCR (Drucker, 1993, 1994) using neural networks as the weak learners. In a series of comparisons (Bottou, 1994) boosting was shown to be superior to other techniques on a large OCR problem. The general configuration of AdaBoost is shown in Figure 1. Each box is a decision tree built using Quinlans C4.5 algorithm (Quinlan, 1993) The key idea is that each weak learner is trained sequentially. The first weak learner is trained on a set of patterns picked randomly (with replacement) from a training set. After training and pruning, the training patterns are passed through this first decision tree. In the two class case the hypothesis hi is either class 0 or class 1. Some of the patterns will be in error. The training set for the 480 .5 w ~ a: a: o a: a: w H. DRUCKER. C. CORTES INPUT FEATURES #1 #2 #3 T h 1 h h hT ~2 ~3 ~l ~T 2 3 ~)t log (11 ~t ) FIGURE 1. BOOSTING ENSEMBLE ENSEMBLE TRAINING ERROR RATE WEAK LEARNER WEIGHTED TRAINING ERROR RATE ENSEMBLE TEST ERROR RATE NUMBER OF WEAK LEARNERS FIGURE 2. INDIVIDUAL WEAK LEARNER ERROR RATE AND ENSEMBLE TRAINING AND TEST ERROR RATES Boosting Decision Trees 481 second weak learner will consist of patterns picked from the training set with higher probability assigned to those patterns the first weak learner classifies incorrectly. Since patterns are picked with replacement, difficult patterns are more likely to occur multiple times in the training set. Thus as we proceed to build each member of the ensemble, patterns which are more difficult to classify correctly appear more and more likely. The training error rate of an individual weak learner tends to grow as we increase the number of weak learners because each weak learner is asked to classify progressively more difficult patterns. However the boosting algorithm shows us that the ensemble training and test error rate decrease as we increase the number of weak learners. The ensemble output is determined by weighting the hypotheses with the log of (l!~i) where ~ is proportional to the weak learner error rate. If the weak learner has good error rate performance, it will contribute significantly to the output, because then 1 / ~ will be large. Figure 2 shows the general shape of the curves we would expect. Say we have constructed N weak learners where N is a large number (right hand side of the graph). The N'th weak learner (top curve) will have a training error rate that approaches .5 because it is trained on difficult patterns and can do only sightly better than guessing. The bottom two curves show the test and training error rates of the ensemble using all N weak learners. which decrease as weak learners are added to the ensemble. 2 BOOSTING Boosting arises from the PAC (probably approximately correct) learning model which has as one of its primary interests the efficiency of learning. Schapire was the first one to show that a series of weak learners could be converted to a strong learner. The detailed algorithm is show in Figure 3. Let us call the set of N 1 distinct examples the original training set. We distinguish the original training set from what we will call the filtered training set which consists of N 1 examples picked with replacement from the original training set. Basically each of N 1 original examples is assigned a weight which is proportional to the probability that the example will appear in the filtered training set (these weights have nothing to do with the weights usually associated with neural networks). Initially all examples are assigned a weight of unity so that all the examples are equally likely to show up in the initial set of training examples. However, the weights are altered at each state of boosting (Step 5 of Figure 3) and if the weights are high we may have multiple copies of some of the original examples appearing in the filtered training set. In step three of this algorithm, we calculate what is called the weighted training error and this is the error rate over all the original N 1 training examples weighted by their current respective probabilities. The algorithms terminates if this error rate is .5 (no better than guessing) or zero (then the weights of step 5 do not change). Although not called for in the original C4.5 algorithm, we also have an original set of pruning examples which also are assigned weights to form a filtered pruning set and used to prune the classification trees constructed using the filtered training set. It is known (Mingers, 1989a) that reducing the size of the tree (pruning) improves generalization. 3 DECISION TREES For our implementation of decision trees, we have a set of features (attributes) that specifies an example along with their classification (we discuss the two-class problem primarily). We pick a feature that based on some criterion, best splits the examples into two subsets. Each of these two subsets will usually not contain examples of just one class, so we recursively divide the subsets until the final subsets each contain examples of just one class. Thus, each internal node specifies a feature and a value for that feature that determines whether one should take the left or right branch emanating from that node. At terminal nodes, we make the final decision, class 0 or 1. Thus, in decision trees one starts at a root node and progressively traverses the tree from the root node to one of the 482 H. DRUCKER,C. CORTES Inputs: N I training paUans. N 2 pruning paUems. N 3 test paUans laitialize the weight veco of the N I training pattems: wI = 1 for i = 1 •...• N I laitialize the weight veco of the N 2 pruning paUmls: sl = 1 for i = 1 •...• N 2 laitialize the number of trees in the ensemble to t = 1 Do Vatil weighted training enol' rate is 0 or .5 or ensemble test enoI'rate asymptotes 1. For the training set and pruning sets w' p'=-N. 1:wl i-I a' r' = -w.1:sl Pick N I samples from original training set with probability P(i) to form filtered training set Pick N 2 samples from original pruning set with probability r(i) to form filtered pruning set 2. Train tree t using filtered training set and prune using filtered pruning set 3. Pass the N I mginal training examples through the IRJICd tree whose output h, (i) is either 0 or 1 and classification c(i) is either 0 or 1. Calculate the weighted training error N. rate: E, = 1: pll h, (i) - c(i) I i-I E, 4. Set Ii, = 1 E, 5. Set the new training weight vectm' to be wI+1 = wf{Ii,**(1-lh,(i) - c(i)I») i = 1 •...• N I Pass the N 2 original pruning paUems through the pruned tree and calculate new pruning weight vector: 6. F<r each tree t in the ensemble (total trees 1) • pass the j'th test pattern through and obtain h, (j) for each t The final hypothesis hr(j) for this pattern: hr(j)={I. O. Do for each test paUml and calculate the ensemble test enu rate: 7.t=t+l End Vatil Figure 3: Boosting Algorithm Boosting Decision Trees 483 terminal nodes where a final decision is made. CART (Brei man, 1984) and C4.5 (Quinlan 1993) are perhaps the two most popular tree building algorithms. Here, C4.5 is used. The attraction of trees is that the simplest decision tree can be respecified as a series of rules and for certain potential users this is more appealing than a nonlinear "black box" such as a neural network. That is not to say that one can not design trees where the decision at each node depends on some nonlinear combination of features, but this will not be our implementation. Other attractions of decision trees are speed of learning and evaluation. Whether trees are more accurate than other techniques depends on the application domain and the effectiveness of the particular implementation. In OCR, our neural networks are more accurate than trees but the penalty is in training and evaluation times. In other applications which we will discuss later a boosting network of trees is more accurate. As an initial example of the power of boosting, we will use trees for OCR of hand written digits. The main rationale for using OCR applications to evaluate AdaBoost is that we have experience in the use of a competing technology (neural networks) and we have from the National Institute of Standards and Technology (NISn a large database of 120,000 digits, large enough so we can run multiple experiments. However, we will not claim that trees for OCR have the best error performance. Once the tree is constructed, it is pruned to give hopefully better generalization performance than if the original tree was used. C4.5 uses the original training set for what is called "pessimistic pruning" justified by the fact that there may not be enough extra examples to form a set of pruning examples. However, we prefer to use an independent set of examples to prune this tree. In our case, we have (for each tree in the ensemble) an independent filtered pruning set of examples whose statistical distribution is similar to that of the filtered training set. Since the filtering imposed by the previous members of the ensemble can severely distort the original training distribution, we trust this technique more than pessimistic pruning. In pruning (Mingers, 1989), we pass the pruning set though the tree recording at each node (including non-terminal nodes) how many errors there would be if the tree was terminated there. Then, for each node (except for terminal nodes), we examine the subtree of that node. We then calculate the number of errors that would be obtained if that node would be made a terminal node and compare it to the number of errors at the terminal nodes of that subtree. If the number of errors at the root node of this subtree is less than or equal to that of the subtree, we replace the subtree with that node and make it a terminal node. Pruning tends to substantially reduce the size of the tree, even if the error rates are not substantially decreased. 4 EXPERIMENTS In order to run enough experiments to claim statistical validity we needed a large supply of data and few enough features that the information ratio could be determined in a reasonable amount of time. Thus we used the 120,000 examples in a NIST database of digits subsampled to give us a IOxlO pixel array (100 features) where the features are continuous values. We do not claim that OCR is best done by using classification trees and certainly not in l00-dimensional space. We used 10,000 training examples, 2000 pruning examples and 2000 test examples for a total of 14,000 examples. We also wanted to test our techniques on a wide range of problems, from easy to hard. Therefore, to make the problem reasonably difficult, we assigned class 0 to all digits from o to 4 (inclusive) and assigned class 1 to the remainder of the digits. To vary the difficulty of the problem, we prefiltered the data to form data sets of difficulty f Think of f as the fraction of hard examples generated by passing the 120,000 examples through a poorly trained neural network and accepting the misclassified examples with probability f and the correctly classified examples with probability 1- f. Thus f = .9 means that the training set consists of 10,000 examples that if passed through this neural network would 484 H.DRUCKER,C. CORTES have an error rate of .9. Table I compares the boosting performance with single tree performance. Also indicated is the average number of trees required to reach that performance. Overtraining never seems to be a problem for these weak learners, that is, as one increases the number of trees, the ensemble test error rate asymptotes and never increases. Table 1. For fraction f of difficult examples, the error rate for a single tree and a boosting ensemble and the number of trees required to reach the error rate for that ensemble. f single boosting number of tree trees trees .1 12% 3.5% 25 .3 13 4.5 28 .5 16 7.1 31 .7 21 7.7 60 .9 23 8.1 72 We wanted to compare the boosting ensemble to other techniques for constructing ensembles using 14,000 examples, holding out 2000 for testing. The problem with decision trees is that invariably, even if the training data is different (but drawn from the same distribution), the features chosen for the first few nodes are usually the same (at least for the OCR data). Thus, different decision surfaces are not created. In order to create different decision regions for each tree, we can force each decision tree to consider another attribute as the root node, perhaps choosing that attribute from the first few attributes with largest information ratio. This is similar to what Kwok and Carter (1990) have suggested but we have many more trees and their interactive approach did not look feasible here. Another technique suggested by T.K. Ho (1992) is to construct independent trees on the same 10,000 examples but randomly striking out the use of fifty of the 100 possible features. Thus, for each tree, we randomly pick 50 features to construct the tree. When we use up to ten trees, the results using Ho's technique gives similar results to that of boosting but the asymptotic performance is far better for boosting. After we had performed these experiments, we learned of a technique termed "bagging" (Breiman, 1994) and we have yet to resolve the issue of whether bagging or boosting is better. 5 CONCLUSIONS Based on preliminary evidence, it appears that for these applications a new boosting algorithm using trees as weak learners gives far superior performance to single trees and any other technique for constructing ensemble of trees. For boosting to work on any problem, one must find a weak learner that gives an error rate of less than 0.5 on the filtered training set. An important aspect of the building process is to prune based on a separate pruning set rather than pruning based on a training set. We have also tried this technique on knowledge discovery and data mining problems and the results are better than single neural networks. References L. Bottou, C. Cortes, J.S. Denker, H. Drucker, I. Guyon, L.D. Jackel, Y. LeCun, U.A. Muller, E. Sackinger, P. Simard, and V. Vapnik (1994), "Comparison of Classifier Methods: A Case Study in Handwritten Digit Recognition", 1994 International Conference on Pattern Recognition, Jerusalem. L. Breiman, J. Friedman, R.A. Olshen, and C.J. Stone (1984), Classification and Regression Trees, Chapman and Hall. Boosting Decision Trees 485 L. Breiman, "Bagging Predictors", Technical Report No. 421, Department of Statistics University of California, Berkeley, California 94720, September 1994. H. Drucker (1994), C. Cortes, LD Jackel, Y. LeCun "Boosting and Other Ensemble Methods", Neural Computation, vol 6, no. 6, pp. 1287-1299. H. Drucker, R.E. Schapire, and P. Simard (1993) "Boosting Performance in Neural Networks", International Journal of Pattern Recognition and Artificial Intelligence, Vol 7. N04, pp. 705-719. Y. Freund (1990), "Boosting a Weak Learning Algorithm by Majority", Proceedings of the Third Workshop on Computational Learning Theory, Morgan-Kaufmann, 202-216. Y. Freund and R.E. Schapire (1995), "A decision-theoretic generalization of on-line leaming and an application to boosting", Proceeding of the Second European Conference on Computational Learning. T.K. Ho (1992), A theory of MUltiple Classifier Systems and Its Applications to Visual Word Recognition, Doctoral Dissertation, Department of Computer Science, SUNY at Buffalo. S.W. Kwok and C. Carter (1990), "Multiple Decision Trees", Uncertainty in ArtifiCial Intelligence 4, R.D. Shachter, T.S. Levitt, L.N. Kanal, J.F Lemmer (eds) Elsevier Science Publishers. J.R. Quinlan (1993), C4.5: Programs For Machine Learning, Morgan Kauffman. J. Mingers (1989), "An Empirical Comparison of Pruning Methods for Decision Tree Induction", Machine Learning, 4:227-243. R.E. Schapire (1990), The strength of weak learnability, Machine Learning, 5(2):197227.
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The Gamma MLP for Speech Phoneme Recognition Steve Lawrence~ Ah Chung Tsoi, Andrew D. Back {lawrence,act,back}Oelec.uq.edu.au Department of Electrical and Computer Engineering University of Queensland St. Lucia Qld 4072 Australia Abstract We define a Gamma multi-layer perceptron (MLP) as an MLP with the usual synaptic weights replaced by gamma filters (as proposed by de Vries and Principe (de Vries and Principe, 1992)) and associated gain terms throughout all layers. We derive gradient descent update equations and apply the model to the recognition of speech phonemes. We find that both the inclusion of gamma filters in all layers, and the inclusion of synaptic gains, improves the performance of the Gamma MLP. We compare the Gamma MLP with TDNN, Back-Tsoi FIR MLP, and Back-Tsoi I1R MLP architectures, and a local approximation scheme. We find that the Gamma MLP results in an substantial reduction in error rates. 1 INTRODUCTION 1.1 THE GAMMA FILTER Infinite Impulse Response (I1R) filters have a significant advantage over Finite Impulse Response (FIR) filters in signal processing: the length of the impulse response is uncoupled from the number of filter parameters. The length of the impulse response is related to the memory depth of a system, and hence I1R filters allow a greater memory depth than FIR filters of the same order. However, I1R filters are *http://www.neci.nj.nec.com/homepages/lawrence 786 S. LAWRENCE, A. C. TSOI, A. D. BACK not widely used in practical adaptive signal processing. This may be attributed to the fact that a) there could be instability during training and b) the gradient descent training procedures are not guaranteed to locate the global optimum in the possibly non-convex error surface (Shynk, 1989). De Vries and Principe proposed using gamma filters (de Vries and Principe, 1992), a special case of IIR filters, at the input to an otherwise standard MLP. The gamma filter is designed to retain the uncoupling of memory depth to the number of parameters provided by IIR filters, but to have simple stability conditions. The output of a neuron I f ("",Nr-l I 1-1) Yk = L--i=O WkiYi . term memory with delays: in a multi-layer perceptron is computed using1 De Vries and Principe consider adding short I f ("",Nr-l "",K I (t .) 1-1 (t .)) h Yk L--i=O L--j=O 9kij - J Yi - J were 9~ij = (r!i)! tj-le-/-'~it j = 1, ... , K . The depth of the memory is controlled by J.t, and K is the order of the filter. For the discrete time case, we obtain the recurrence relation: zo(t) = x(t) and Zj(t) = (1 - J.t)Zj(t - 1) + J.tZj-l (t - 1) for j = 1, ... , K. In this form, the gamma filter can be interpreted as a cascaded series of filter modules, where each module is a first order IIR filter with the transfer function q-(I-/-,) , where qZj(t) ~ Zj(t + 1). We have a filter with K poles, all located at 1 - J.t. Thus, the gamma filter may be considered as a low pass filter for J.t < 1. The value of J.t can be fixed, or it can be adapted during training. 2 NETWORK MODELS Figure 1: A gamma filter synapse with an associated gain term 'c'. We have defined a gamma MLP as a multi-layer perceptron where every synapse contains a gamma filter and a gain term, as shown in figure 1. The motivation behind the inclusion of the gain term is discussed later. A separate J.t parameter is used for each filter. Update equations are derived in a manner analogous to the standard MLP and can be found in Appendix A. The model is defined as follows. lwhere yi is the output of neuron k in layer I, Nl is the number of neurons in layer I, Wii is the weight connecting neuron k in layer I to neuron i in layer I - 1, yb = 1 (bias), and / is commonly a sigmoid function. The Gamma MLP for Speech Phoneme Recognition 787 Definition 1 A Gamma MLP with L layers excluding the input layer (0,1, ... , L), gamma filters of order K, and No, N 1 , ... , NL neurons per layer, is defined as Ziij (t) Ziij (t) f (x~ (t)) N'-l K L C~i(t) L wLj (t)Zkij (t) i=O j=O (1- ILL(t))zkij(t -1) + ILL(t)zki(j_I)(t -1) y!-l (t) where y(t) = neuron output, c'ki = synaptic gain, f(a) = 1,2, ... ,N,(neuronindex), I = 0,1, ... ,L(layer), and Ziijli=O = 0, C~ij li=O = 1(bias). 1 ~j ~ K j=O eO / 2 _e- o / 2 eO/2+e 0/2, k 1, W~ij li=O,#O (1) o For comparison purposes, we have used the TDNN (Time Delay Neural Network) architecture2 , the Back-Tsoi FIR3 and I1R MLP architectures (Back and Tsoi, 1991a) where every synapse contains an FIR or I1R filter and a gain term, and the local approximation algorithm used by Casdagli (k-NN LA) (Casdagli, 1991)4. The Gamma MLP is a special case of the I1R MLP. 3 TASK 3.1 MOTIVATION Accurate speech recognition requires models which can account for a high degree of variability in the data. Large amounts of data may be available but it may be impractical to use all of the information in standard neural network models. Hypothesis: As the complexity of a problem increases (higher dimensionality, greater variety of training data), the error surface of a neural network becomes more complex. It may contain a number of local minima5 many of which may be much worse than the global minimum. The training (parameter estimation) algorithms become "stuck" in local minima which may be increasingly poor compared to the global optimum. The problem suffers from the so called "curse of dimenSionality" and the 2We use TDNN to refer to an MLP with a time window of inputs, not the replicated architecture introduced by Lang (Lang et al., 1990). 3We distinguish the Back-Tsoi FIR network from the Wan FIR network in that the Wan architecture has no synaptic gains, and the update algorithms are different. The Back-Tsoi update algorithm has provided better convergence in previous experiments. 4Casdagli created an affine model of the following form for each test pattern: yi = aD + L~=l ai~, where k is the number of neighbors, j = 1, ... , k, and n is the input dimension. The resulting model is used to find y for the test pattern. 5We note that it can be difficult to distinguish a true local minimum from a long plateau in the standard backpropagation algorithm. 788 S. LAWRENCE, A. C. TSOI, A. D. BACK difficulty in optimizing a function with limited control over the nature of the error surface. We can identify two main reasons why the application of the Gamma MLP may be superior to the standard TDNN for speech recognition: a) the gamma filtering operation allows consideration of the input data using different time resolutions and can account for more past history of the signal which can only be accounted for in an FIR or TDNN system by increasing the dimensionality of the model, and b) the low pass filtering nature of the gamma filter may create a smoother function approximation task, and therefore a smoother error surface for gradient descent6 . 3.2 TASK DETAILS Model Input Window Networl( Output 1 [~ Classification 0 Target Function Networl( Output 2 II ; j : ~} ...::.. ...:::'.!'} ~.} ,.;:!.. ""::'I'} . ; i ~ ! l I ~ Frames of RASTA data ~ Sequence End ~ Figure 2: PLP input data format and the corresponding network target functions for the phoneme "aa" . Our data consists of phonemes extracted from the TIMIT database and organized as a number of sequences as shown in figure 2 (example for the phoneme "aa"). One model is trained for each phoneme. Note that the phonemes are classified in context, with a number of different contexts, and that the surrounding phonemes are labelled only as not belonging to the target phoneme class. Raw speech data was pre-processed into a sequence of frames using the RASTA-PLP v2.0 software7. We used the default options for PLP analysis. The analysis window (frame) was 20 ms. Each succeeding frame overlaps with the preceding frame by 10 ms. 9 PLP coefficients together with the signal power are extracted and used as features describing each frame of data. Phonemes used in the current tests were the vowel "aa" and the fricative "s" . The phonemes were extracted from speakers coming from the same demographic region in the TIMIT database. Multiple speakers were used and the speakers used in the test set were not contained in the training set. The training set contained 4000 frames, where each phoneme is roughly 10 frames. The test set contained 2000 frames, and an additional validation set containing 2000 frames was used to control generalization. 6If we consider a very simple network and derive the relationship of the smoothness of the required function approximation to the smoothness of the error surface this statement appears to be valid. However, it is difficult to show a direct relationship for general networks. 7 Obtained from ftp:/ /ftp.icsi.berkeley.edu/pub/speech/rasta2.0.tar.Z. The Gamma MLP for Speech Phoneme Recognition 789 4 RESULTS Two outputs were used in the neural networks as shown by the target functions in figure 2, corresponding to the phoneme being present or not. A confidence criterion was used: Ymax x (Ymax - Ymin) (for soft max outputs). The initial learning rate was 0.1, 10 hidden nodes were used, FIR and Gamma orders were 5 (6 taps), the TDNN and k-NN models had an input window of 6 steps in time, the tanh activation function was used, target outputs were scaled between -0.8 and 0.8, stochastic update was used, and initial weights were chosen from a set of candidates based on training set performance. The learning rate was varied over time according to the schedule: 'TI = 'TIo/ (N/2 + ( ",~!(o .Cj(n C2 N»)) where'TI = learning rate, 'TIo = initial max 1,(cI(I C2)N learning rate, N = total epochs, n = current epoch, Cl = 50, C2 = 0.65. This is similar to the schedule proposed in (Darken and Moody, 1991) with an additional term to decrease the learning rate towards zero over the final epochs8 . I Train Error % I 2-NN I 5-NN I 1st layer I All layers I Gains 1st layer I Gains all layers , , FIR MLP 17.6 0.43 14.5 1.5 27.2 0 .59 40.9 19.8 Gamma MLP 7.78 0.39 5.73 0.88 6 .07 0.12 5.63 1.68 TDNN 14.4 0.86 k-NN LA 0 0 , , I Test Error % I 2-NN I 5-NN list layer I All layers I Gams 1st layer I Gams all layers I FIR MLP 22.2 0.97 20.4 0.61 29 0.14 41 21 Gamma MLP 14.7 0.16 13.5 0.33 12.8 1.0 12.7 0.50 TDNN 24.5 0 .68 k-NN LA 31 28.4 I Test False +ve I 2-NN I 5-NN list layer I All layers I Gams 1st layer I Gams all layers I , , FIR MLP 13.5 0 .67 11.4 2.0 4.5 0 .77 31.3 49.0 Gamma MLP 7.94 0.45 7 .01 0.47 6.83 0 .34 8.05 1.8 TDNN 13 0 .27 k-NN LA 22.6 17.4 , , I Test False -ve I 2-NN I 5-NN list layer I All layers I Gams 1st layer I Gams all layers I FIR MLP 44.9 2.6 44.1 5.6 92.9 2.4 66.4 53 Gamma MLP 32 .2 1.2 30.4 2.2 28.4 2.8 24.7 4.4 TDNN 54.6 1.8 k-NN LA 53 56.8 Table 1: Results comparing the architectures and the use of filters in all layers and synaptic gains for the FIR and Gamma MLP models. The NMSE is followed by the standard deviation. The TDNN results are listed under an arbitrary column heading (gains and 1st layer/alilayers does not apply). The results of the simulations are shown in table 19 . Each result represents an average over four simulations with different random seeds - the standard deviation of the four individual results is also shown. The FIR and Gamma MLP networks have been tested both with and without synaptic gains, and with and without filters in the output layer synapses. These results are for the models trained on the "s" phoneme, results for the "aa" phoneme exhibit the same trend. "Test false negative" is probably the most important result here, and is shown graphically in figure 3. This is the percentage of times a true classification (ie. the current 8Without this term we have encountered considerable parameter fluctuation over the last epoch. 9NMSE = 2:~=1 (d(k) - y(k))2 I (2:~=1 (d(k) (2:~=1 d(k)) INr) IN. 790 60 55 Q) 50 ~ ~ 45 Z Q) 40 .!!2 '" 35 LL i 30 I25 20 --" f------f S. LAWRENCE, A. C. TSOI, A. D. BACK ~ Ga:~~ ~t~ = -=-~-' TDNN -_ .. _.k-NN LA _._._ .. I - - -__ I -r-·---- ----+ --- -1 2-NN 5-NN NG 1 L NG AL G lL GAL Figure 3: Percentage of false negative classifications on the test set. NG=No gains, G=Gains, lL=filters in the first layer only, AL=filters in all layers. The error bars show plus and minus one standard deviation. The synaptic gains case for the FIR MLP is not shown as the poor performance compresses the remainder of the graph. Top to bottom, the lines correspond to: k-NN LA (left), TDNN, FIR MLP, and Gamma MLP. phoneme is present) is incorrectly reported as false. From the table we can see that the Gamma MLP performs Significantly better than the FIR MLP or standard TDNN models for this problem. Synaptic gains and gamma filters in all layers improve the performance of the Gamma MLP, while the inclusion of synaptic gains presented difficulty for the FIR MLP. Results for the IIR MLP are not shown - we have been unable to obtain significant convergencelO. We investigated values of k not listed in the table for the k-NN LA model, but it performed poorly in all cases. 5 CONCLUSIONS We have defined a Gamma MLP as an MLP with gamma filters and gain terms in every synapse. We have shown that the model performs significantly better on our speech phoneme recognition problem when compared to TDNN, Back-Tsoi FIR and IIR MLP architectures, and Casdagli's local approximation model. The percentage of times a phoneme is present but not recognized for the Gamma MLP was 44% lower than the closest competitor, the Back-Tsoi FIR MLP model. The inclusion of gamma filters in all layers and the inclusion of synaptic gains improved the performance of the Gamma MLP. The improvement due to the inclusion of synaptic gains may be considered non-intuitive to many - we are adding degrees of freedom, but no additional representational power. The error surface will be different in each case, and the results indicate that the surface for the synaptic gains case is more amenable to gradient descent. One view of the situation is seen by Back & Tsoi with their FIR and IIR MLP networks (Back and Tsoi, 1991b): From a signal processing perspective the response of each synapse is determined by polezero positions. With no synaptic gains, the weights determine both the static gain and the pole-zero positions of the synapses. In an experimental analysis performed by Back & Tsoi it was observed that some synapses devoted themselves to modellOTheoretically, the IIR MLP model is the most powerful model used here. Though it is prone to stability problems, the stability of the model can and was controlled in the simulations performed here (basically, by reflecting poles that move outside the unit circle back inside). The most obvious hypothesis for the difficulty in training the model is related to the error surface and the nature of gradient descent. We expect the error surface to be considerably more complex for the IIR MLP model, and for gradient descent update to experience increased difficulty optimizing the function. The Gamma MLP for Speech Phoneme Recognition 791 ing the dynamics of the system in question, while others "sacrificed" themselves to provide the necessary static gainsll to construct the required nonlinearity. APPENDIX A: GAMMA MLP UPDATE EQUATIONS ~W~i;(t) ~C~i(t) ~J'~i (t) o = = 8J(t) I I I -'1 8 I () = '16" (t)c", (t)Z"i; (t) w",; t = (1 J'~i(t))a~,;(t -1) + J'~i(t)a~iC;_I)(t - 1) +z~,(;_I)(t -1) Z~i;(t - 1) 1 (1 - J';,,(t)).B;,,;(t -1) + J';,,(t).B~"(;_l) (t - 1) Acknowledgments j=O 1 $j $ K I=L 1 $j $ K j=O 1 $j $K (2) (3) (4) (5) (6) (7) This work has been partially supported by the Australian Research Council (ACT and ADB) and the Australian Telecommunications and Electronics Research Board (SL). References Back, A. and Tsoi, A. (1991a). FIR and IIR synapses, a new neural network architecture for time series modelling. Neural Computation, 3(3):337-350. Back, A. D. and Tsoi, A. C. (1991b). Analysis of hidden layer weights in a dynamic locally recurrent network. In Simula, 0., editor, Proceedings International Conference on Artificial Neural Networks, ICANN-91, volume 1, pages 967-976, Espoo, Finland. Casdagli, M. (1991). Chaos and deterministic versus stochastic non-linear modelling. J.R. Statistical Society B, 54(2):302-328. Darken, C. and Moody, J. (1991). Note on learning rate schedules for stochastic optimization. In Neural Information Processing Systems 3, pages 832-838. Morgan Kaufmann. de Vries, B. and Principe, J. (1992). The gamma model- a new neural network for temporal processing. Neural Networks, 5(4):565-576. Lang, K. J., Waibel, A. H., and Hinton, G. E. (1990). A time-delay neural network architecture for isolated word recognition. Neural Networks, 3:23-43. Shynk, J . (1989). Adaptive IIR filtering. IEEE ASSP Magazine, pages 4-21. llThe neurons were observed to have gone into saturation, providing a constant output. PART VII VISION
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Some results on convergent unlearning algorithm Serguei A. Semenov &: Irina B. Shuvalova Institute of Physics and Technology Prechistenka St. 13/7 Moscow 119034, Russia Abstract In this paper we consider probabilities of different asymptotics of convergent unlearning algorithm for the Hopfield-type neural network (Plakhov & Semenov, 1994) treating the case of unbiased random patterns. We show also that failed unlearning results in total memory breakdown. 1 INTRODUCTION In the past years the unsupervised learning schemes arose strong interest among researchers but for the time being a little is known about underlying learning mechanisms, as well as still less rigorous results like convergence theorems were obtained in this field. One of promising concepts along this line is so called "unlearning" for the Hopfield-type neural networks (Hopfield et ai, 1983, van Hemmen & Klemmer, 1992, Wimbauer et ai, 1994). Elaborating that elegant ideas the convergent unlearning algorithm has recently been proposed (Plakhov & Semenov, 1994), executing without patterns presentation. It is aimed at to correct initial Hebbian connectivity in order to provide extensive storage of arbitrary correlated data. This algorithm is stated as follows. Pick up at iteration step m, m = 0,1,2, ... a random network state s(m) = (S~m), .. . , S~m), with the values sfm) = ±1 having equal probability 1/2, calculate local fields generated by s(m) N h~m) = ~ J~~)S~m) . 1 N , L..J 'J J ' t = , ... , , i=l and then update the synaptic weights by J ~~+1) = J~~) cN-lh~m)h~m) .. 1 N 'J 'J 'J ' t, J = , ... , . (1) Some Results on Convergent Unlearning Algorithm 359 Here C > 0 stands for the unlearning strength parameter. We stress that selfinteractions, Jii , are necessarily involved in the iteration process. The initial condition for (1) is given by the Hebb matrix, J~O) = J!f: (2) with arbitrary (±1)-patterns eJJ , J.l = 1, ... ,p. For C < Ce, the (rescaled) synaptic matrix has been proven to converge with probability one to the projection one on the linear subspace spanned by maximal subset of linearly independent patterns (Plakhov & Semenov, 1994). As the sufficient condition for that convergence to occur, the value of unlearning strength C should be less than Ce = '\;~~x where Amax denotes the largest eigenvalue of the Hebb matrix. Very often in real-world situations there are no means to know Ce in advance, and therefore it is of interest to explore asymptotic behaviour of iterated synaptic matrix for arbitrary values of c. As it is seen, there are only three possible limiting behaviours of the normalized synaptic matrix (Plakhov 1995, Plakhov & Semenov, 1995). The corresponding convergence theorems relate corresponding spectrum dynamics to limiting behaviour of normalized synaptic matrix j = J IIIJII ( IPII = (L:~=1 Ji~)1/2 ) which can be described in terms of A~n;2 the smallest eigenvalues of J(m): I. if A~2 = 0 for every m = 0,1,2, ... , with multiplicity of zero eigenvalue being fixed, then (A) lim j~r:n) = S-1/2 PiJ· m-oo IJ where P marks the projection matrix on the linear subspace CeRN spanned by the nominated patterns set eJJ , J.l = 1, . .. , p, s = dim C ~ p; II. if A~n;2 = 0, m = 0,1,2, ... , besides at some (at least one) steps mUltiplicity of zero eigenvalue increases, then (B) I· J-(m) '-1/2p' 1m .. s .. m-oo IJ IJ where P' is the projector on some subspace C' C C, s' = dimC' < s; III. if A~n;2 < 0 starting from some value of m, then (C) with some (not a ±1) unity random vector e = (6, · .. ,eN). (3) These three cases exhaust all possible asymptotic behaviours of ji~m), that is their total probability is unity: PA + PB + Pc = 1. The patterns set is supposed to be fixed. The convergence theorems say nothing about relative probabilities to have specific asymptotics depending on model parameters. In this paper we present some general results elucidating this question and verify them by numerical simulation. We show further that the limiting synaptic matrix for the case (C) which is the projector on -e E C cannot maintain any associative memory. Brief discussion on the retrieval properties of the intermediate case (B) is also given. 360 S. A. SEMENOV, 1. B. SHUVALOVA 2 PROBABILITIES OF POSSIBLE LIMITING BEHAVIOURS OF j(m) The unlearning procedure under consideration is stochastic in nature. Which result of iteration process, (A), (B) or (C), will realize depends upon the value of €, size and statistical properties of the patterns set {~~, J.l = 1, ... , p}, and realization of unlearning sequence {sCm), m=0,1,2, . .. }. Under fixed patterns set probabilities of appearance of each limiting behaviour of synaptic matrix is determined by the value of unlearning strength E only. In this section we consider these probabilities as a function of E. Generally speaking, considered probabilities exhibit strong dependence on patterns set, making impossible to calculate them explicitly. It is possible however to obtain some general knowledge concerning that probabilities, namely: P A (E) 1 as E 0+, and hence, PB,c(E) 0, otherwise Pc(E) 1 as E 00, and PA,B(£) 0, because of P A + PB + Pc = 1. This means that the risk to have failed unlearning rises when E increases. Specifically, we are able to prove the following: Proposition. There exist positive €l and €2 such that P A (£) = 1, 0 < € < €l, and Pc(€) = 1, £2 < €. Before passing to the proof we bring forward an alternative formulation of the above stated classification. After multiplying both sides of(l) by sIm)sjm) and summing up over all i and j, we obtain in the matrix notation s(m)T J(m+l)s(m) = D.ms(m)T J(m)s(m) (4) where the contraction factor D.m = 1 - EN-ls(m)T J(m)s(m) controls the asymptotics of j(m), as it is suggested by detailed analysis (Plakhov & Semenov, 1995). (Here and below superscript T designates the transpose.) The hypothesis of convergence theorems can be thus restated in terms of D.m , instead of .A~'7~, respectively: I. D.m > 0 'tim; II. D.m = 0 for I steps ml, ... , ml; III. D.m < 0 at some step m. Proof It is obvious that D.m 2 1 - €.Af:a1 where .A~"!1 marks the largest eigenvalue of J(m) . From (4), it follows that the sequence p~"!t, m = 0,1,2, ... } is non increasing, and consequently D.m 2 1 £.A~~x with .A~~x = s~ x T JHx = s~ N- l t (L~rxi)2 Ixl-l Ixl-l ~=1 i P N N =:; sup N- 1 L L)~n2 L xl = p. Ixl=l ~=l i=l i=l From this, it is straightforward to see that, if £ < p-l , then D.m > 0 for any m. By convergence theorem (Plakhov & Semenov, 1995) iteration process (1) thus leads to the limiting relation (A). Let by definition "I = mins N-lsr JHS where minimum is taken over such (±1)vectors S for which JH S =1= 0 (-y > 0, in view of positive semidefiniteness of JH), and put € > "1- 1 . Let us further denote by n the iteration step such that JH sCm) = 0, m = 0,1, ... , n - 1 and JH sen) =1= O. Needless to say that this condition may be satisfied even for the initial step n = 0: JH S(O) =1= O. At step n one has D.n = 1 - EN- 1 s(n)T JH sen) =:; 1 - q < O. Some Results on Convergent Unlearning Algorithm 361 The latter implies loss of positive semidefiniteness of J(m), what results in asymptotics (C) (Plakhov, 1995, Plakhov & Semenov, 1995). By choosing Cl = p-l and C2 = 1'-1 we come to the statement of Proposition. Comparison of numerical estimates of considered probabilities with analytical approximations can be done on simple patterns statistics. In what follows the patterns are assumed to be random and unbiased. The dependence P(c) has been found in computer simulation with unbiased random patterns. It is worth noting, by passing, that calculation Llm using current simulation data supplies a good control of unlearning process owing to an alternative formulation of convergence theorems. In simulation we calculate pf (c) averaged over the sets of unbiased random patterns, as well as over the realizations of unlearning sequence. As N increases, with 0: = piN remaining fixed, the curves slope steeply down approaching step function PA'(c) = O(c - 0:- 1) (Plakhov & Semenov, 1995). Without presenting of derivation or proof we will advance the reasoning suggestive of it. First it can be checked that Llm is a selfaveraging quantity with mean 1 - cN- 1TrJ(m) and variance vanishing as N goes to infinity. Initially one has N- 1TrJ H = 0:, and obviously the sequence {TrJ(m), m = 0,1,2, ... } is nonincreasing. Therefore Llo = 1 - cO:, and all others Llm are not less than Llo. If one chooses c < 0:- 1 , then all Llm will be positive, and the case (A) will realize. On the other hand, when c > 0:- 1, we have Llo < 0, and the case (C) will take place. What is probability for asymptotics (B) to appear? We will adduce an argument (detailed analysis (Plakhov & Semenov, 1995) is rather cumbersome and omitted here) indicating that this probability is quite small. First note that given patterns set it is nonzero for isolated values of c only. Under the assumption that the patterns are random and unbiased, we have calculated probability of I-fold appearance Llm = o summed up over that isolated values of c. Using Gaussian approximation at large N, we have found that probability scales with N as N'/2+2-21+m+l. The total probability can then be obtained through summing up over integer values I: 0 < I < s and all the iteration steps m = 0,1,2, .... As a result, the main contribution to the total probability comes from m = 0 term which is of the order N- 3 / 2 . 3 LIMITING RETRIEVAL PROPERTIES How does reduction of dimension of "memory space" in the case (B), 5 ~ 5' = 5-1, affect retrieval properties of the system? They may vary considerably depending on I. In the most probable case I = 1 it is expected that there will be a slight decrease in storage capacity but the size of attraction basins will change negligibly. This is corroborated by calculating the stability parameter for each pattern J.I. I-' _ cl-' '"' pi cl-' "'i <'i ~ ij<'j· jti (5) Let SemI) be the state vector with normalized projection on C given by V = ps(mI) IIPs(mI)1 such that N IPs(ml)1 = Jo:N, ~ '" N-l/2, L ~~r '" 1. i=1 Then the stability parameter (5) is estimated by ",r = ~r L (Pij - ~Vj)~j = (1- Pii)- (~~r t Vj~j - Vi2) ~ 1-Pii+O(N- 1/ 2 ). j~i j=1 362 S. A. SEMENOV, I. B. SHUV ALOVA Since Pij has mean a and variance vanishing as N --t 00, we thus conclude that the stability parameter only slightly differs from that calculated for the projector rule (s = s') (Kanter & Sompolinsky, 1987). On the other hand, in the situation 0 < s' /s ~ 1 (the possible case i = 0 is trivial) the system will be capable retrieving only a few nominated patterns which ones we cannot specify beforehand. As mentioned above, this case realizes with very small but finite probability. The main effect of self-interactions Jji lies in substantial decrease in storage capacity (Kanter & Sompolinsky, 1987). This is relevant when considering the cases (A) and (B). In the case (C) the system possesses an interesting dynamics exhibiting permanent walk over the state space. There are no fixed points at all. To show this, we write down the fixed point condition for arbitrary state S: Si I:f:l JjjSj > 0, i = 1, ... , N. By using the explicit expression for limiting matrix ~j (3) and summing up over i's, we get as a result (I:j Sj€j)2 < 0, what is impossible. If self-interactions are excluded from local fields at the stage of network dynamics, it is then driven by the energy function of the form H = -(2N)-1 I:itj JjjSjSj. (Zero-temperature sequential dynamics either random or regular one is assumed.) In the rest of this section we examine dynamics of the network equiped with limiting synaptic matrix (C) (3). We will show that in this limit the system lacks any associative memory. There are a single global maximum of H given by Sj = sgn(€d and exponentially many shallow minima concentrated close to the hyperplane orthogonal to €. Moreover it is turned out that all the metastable states are unstable against single spin flip only, whatever the realization of limiting vector €. Therefore after a spin flips the system can relax into a new nearby energy minimum. Through a sequence of steps each consisting of a single spin flip followed by relaxation one can, in principle, pass from one metastable state to the other one. We will prove in what follows that any given metastable state S' one can pass to any other one S through a sequence of steps each consisting of a single spin flip and subsequent relaxation to a some new metastable state. Note that this general statement gives no indications concerning the order of spin flips when moving along a particular trajectory in the state space. Now on we turn to the proof. Let us enumerate the spins in increasing order in absolute values of vector components 0 ~ 161 ~ ... ~ I€NI. The proof is carried out by induction on j = 1, ... , N where j is the maximal index for which SJ 1= Sj. For j = 1 the statement is evident. Assuming that it holds for 1, ... , j - 1 (2 ~ j ~ N), let us prove it for j. One has j = max { i: Sf 1= Sd. With flipping spin j in the state Sl, we next allow relaxation by flipping spins 1, .. . ,j - 1 only. The system finally reaches the state S2 realizing conditional energy minimum under fixed Sj, ... , S N • Show that S2 is true energy minimum. There are two possibilities: (i) For some i, 1 ~ i ~ j - 1, one has sgn (€j Sn = sgn (€T S2) . The fixed point condition for S2 can be then written as I €T S2 I~ min {I€d: 1 ~ i ~ j - 1, sgn(€jS;) = sgn(€T S2)} . l.From this, in view of increasing order of I€i I 's, one gets immediately I €TS2 I~ min {I€d: 1 ~ i ~ N, sgn(€jS;) = sgn(€T S2)} , what implies S2 is true energy minimum. Some Results on Convergent Unlearning Algorithm 363 If ~T S2 = 0, the fixed point condition for S2 is automatically satisfied. Otherwise, for 1 $ i $ j - 1 one has and j-l N ~TS2 = -sgn(~T S2) 2: 1~s:1 + 2:~iSs:, (6) i;;;1 i=j For the sake of definiteness, we set ~T S > O. (The opposite case is treated analogously.) In this case ~T S2 > 0, since otherwise, according to (6), it should be j-I N o ~ ~T S2 = I: l~s:I + 2: ~iSi ~ ~T S, what contradicts our setting. One thus obtains i=1 i=j j-l N ~TS2 = - I: I~d + L~iSi $ ~TS, i=1 i=j and using the fixed point condition for S one gets ~T S $ min {1~s:I: ~iSi > O} $ min {1~s:I: j $ i $ N, ~iSi > O} (7) = min{l~s:I: ~iSf > O}. (8) In the latter inequality of(8) one uses that ~iSf < 0, 1 $ i ~ j-l and Sf = Ss:, j ~ i ~ N. Taking into account (7) and (8), as a result we come to the condition for S2 to be true energy minimum o < ~T S2 ~ min {I~il : ~iSf > O} . According to inductive hypothesis, since S; = Si, j ~ i ~ N, from the state S2 one can pass to S, and therefore from S' through S2 to S. This proves the statement. In general, metastable states may be grouped in clusters surrounded by high energy barriers. The meaning of proven statement resides in excluding the possibility of even such type a memory. Conversely, allowing a sequence of single spin flips (for instance, this can be done at finite temperatures) it is possible to walk through the whole set of metastable states. 4 CONCLUSION In this paper we have begun studying on probabilities of different asymptotics of convergent unlearning algorithm considering the case of unbiased random patterns. We have shown also that failed unlearning results in total memory breakdown. References Hopfield, J.J., Feinstein, D.I. & Palmer, R.G. (1983) "Unlearning" has a stabilizing effect in collective memories. Nature 304:158-159. van Hemmen, J.L. & Klemmer, N. (1992) Unlearning and its relevance to REM sleep: Decorrelating correlated data. In J. G. Taylor et al (eds.) , Neural Network Dynamics, pp. 30-43. London: Springer. 364 S. A. SEMENOV, I. B. SHUV ALOV A Wimbauer, U., Klemmer, N. & van Hemmen, J .L. (1994) Universality of unlearning. Neural Networks 7:261-270. Plakhov, A.Yu. & Semenov, S.A. (1994) Neural networks: iterative unlearning algorithm converging to the projector rule matrix. J. Phys.I France 4:253-260. Plakhov, A.Yu. (1995) private communication Plakhov, A.Yu. & Semenov, S.A. (1995) preprint IPT. Kanter, I. & Sompolinsky, H. (1987) Associative recall of memory without errors. Phys. Rev. A 35:380-392.
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Forward-backward retraining of recurrent neural networks Andrew Senior • Tony Robinson Cambridge University Engineering Department Trumpington Street, Cambridge, England Abstract This paper describes the training of a recurrent neural network as the letter posterior probability estimator for a hidden Markov model, off-line handwriting recognition system. The network estimates posterior distributions for each of a series of frames representing sections of a handwritten word. The supervised training algorithm, backpropagation through time, requires target outputs to be provided for each frame. Three methods for deriving these targets are presented. A novel method based upon the forwardbackward algorithm is found to result in the recognizer with the lowest error rate. 1 Introduction In the field of off-line handwriting recognition, the goal is to read a handwritten document and produce a machine transcription. Such a system could be used for a variety of purposes, from cheque processing and postal sorting to personal correspondence reading for the blind or historical document reading. In a previous publication (Senior 1994) we have described a system based on a recurrent neural network (Robinson 1994) which can transcribe a handwritten document. The recurrent neural network is used to estimate posterior probabilities for character classes, given frames of data which represent the handwritten word. These probabilities are combined in a hidden Markov model framework, using the Viterbi algorithm to find the most probable state sequence. To train the network, a series of targets must be given. This paper describes three methods that have been used to derive these probabilities. The first is a naive bootstrap method, allocating equal lengths to all characters, used to start the training procedure. The second is a simple Viterbi-style segmentation method that assigns a single class label to each of the frames of data. Such a scheme has been used before in speech recognition using recurrent networks (Robinson 1994). This representation, is found to inadequately represent some frames which can represent two letters, or the ligatures between letters. Thus, by analogy with the forward-backward algorithm (Rabiner and Juang 1986) for HMM speech recognizers, we have developed a ·Now at IDM T .J.Watson Research Center, Yorktown Heights NYI0598, USA. 744 A. SENIOR, T. ROBINSON forward-backward method for retraining the recurrent neural network. This assigns a probability distribution across the output classes for each frame of training data, and training on these 'soft labels' results in improved performance of the recognition system. This paper is organized in four sections. The following section outlines the system in which the neural network is used, then section 3 describes the recurrent network in more detail. Section 4 explains the different methods of target estimation and presents the results of experiments before conclusions are presented in the final section. 2 System background The recurrent network is the central part of the handwriting recognition system. The other parts are summarized here and described in more detail in another publication (Senior 1994). The first stage of processing converts the raw data into an invariant representation used as an input to the neural network. The network outputs are used to calculate word probabilities in a hidden Markov model. First, the scanned page image is automatically segmented into words and then normalized. Normalization removes variations in the word appearance that do not affect its identity, such as rotation, scale, slant, slope and stroke thickness. The height of the letters forming the words is estimated, and magnifications, shear and thinning transforms are applied, resulting in a more robust representation of the word. The normalized word is represented in a compact canonical form encoding both the shape and salient features. All those features falling within a narrow vertical strip across the word are termed a frame. The representation derived consists of around 80 values for each of the frames, denoted Xt. The T frames (Xl,' .. , xr ) for a whole word are written xl' Five frames would typically be enough to represent a single character. The recurrent network takes these frames sequentially and estimates the posterior character probability distribution given the data: P(Ai IxD, for each of the letters, a, .. ,z, denoted Ao, ... , A25 • These posterior probabilities are scaled by the prior class probabilities, and are treated as the emission probabilities in a hidden Markov model. A separate model is created for each word in the vocabulary, with one state per letter. Transitions are allowed only from a state to itself or to the next letter in the word. The set of states in the models is denoted Q = {ql, ... , qN} and the letter represented by qi is given by L(qi), L : Q 1-+ Ao, ... , A25 • Word error rates are presented for experiments on a single-writer task tested with a 1330 word vocabulary!. Statistical significance of the results is evaluated using Student's t-test, comparing word recognition rates taken from a number of networks trained under the same conditions but with different random initializations. The results of the t-test are written: T( degrees of freedom) and the tabulated values: tsignificance (degrees of freedom). 3 Recurrent networks This section describes the recurrent error propagation network which has been used as the probability distribution estimator for the handwriting recognition system. Recurrent networks have been successfully applied to speech recognition (Robinson 1994) but have not previously been used for handwriting recognition, on-line or off-line. Here a left-to-right scanning process is adopted to map the frames of a word into a sequence, so adjacent frames are considered in consecutive instants. lThe experimental data are available in ftp:/ /svr-ftp.eng.cam.ac.uk/pub/data Forward-backward Retraining of Recurrent Neural Networks 745 A recurrent network is well suited to the recognition of patterns occurring in a time-series because series of arbitrary length can be processed, with the same processing being performed on each section of the input stream. Thus a letter 'a' can be recognized by the same process, wherever it occurs in a word. In addition, internal 'state' units are available to encode multi-frame context information so letters spread over several frames can be recognized. The recurrent network Input Frames Network Output J :._-------.. (Characlcrprobabllllles) _ IT TT, ; -- JD: , , , ! ( .. vv le e WWW II ... ) i : f inputiOulpul Units ---l-_. ;---------r -ie~~;k-Um, s Untt Time Iklay Figure 1: A schematic of the recurrent error propagation network. For clarity only a few of the units and links are shown. architecture used here is a single layer of standard perceptrons with nonlinear activation functions. The output 0 i of a unit i is a function of the inputs aj and the network parameters, which are the weights of the links Wij with a bias bi : 0i !i({O"j}), (1) O"i bi + Lakwik. (2) The network is fully connected that is, each input is connected to every output. However, some of the input units receive no external input and are connected one-to-one to corresponding output units through a unit time-delay (figure 1). The remaining input units accept a single frame of parametrized input and the remaining 26 output units estimate letter probabilities for the 26 character classes. The feedback units have a standard sigmoid activation function (3), but the character outputs have a 'softmax' activation function (4). eO' • (3) (4) L:j eO" • During recognition ('forward propagation'), the first frame is presented at the input and the feedback units are initialized to activations of 0.5. The outputs are calculated (1 and 2) and read off for use in the Markov model. In the next iteration, the outputs of the feedback units are copied to the feedback inputs, and the next frame presented to the inputs. Outputs are again calculated, and the cycle is repeated for each frame of input, with a probability distribution being generated for each frame. To allow the network to assimilate context information, several frames of data are passed through the network before the probabilities for the first frame are read off, previous output probabilities being discarded. This input/output latency is maintained throughout the input sequence, with extra, empty frames of inputs being presented at the end to give probability distributions for the last frames of true inputs. A latency of two frames has been found to be most satisfactory in experiments to date. 3.1 Training To be able to train the network the target values (j (t) desired for the outputs OJ (Xt) j = 0, ... ,25 for frame Xt must be specified. The target specification is dealt 746 A. SENIOR. T. ROBINSON with in the next section. It is the discrepancy between the actual outputs and these targets which make up the objective function to be maximized by adjusting the internal weights of the network. The usual objective function is the mean squared error, but here the relative entropy, G, of the target and output distributions is used: "" (j(t) - L- L- (j (t) log -.-( -)' t j oJ Xt G (5) At the end of a word, the errors between the network's outputs and the targets are propagated back using the generalized delta rule (Rumelhart et al. 1986) and changes to the network weights are calculated. The network at successive time steps is treated as adjacent layers of a multi-layer network. This process is generally known as 'back-propagation through time' (Werbos 1990). After processing T frames of data with an input/output latency, the network is equivalent to a (T + latency) layer perceptron sharing weights between layers. For a detailed description of the training procedure, the reader is referred elsewhere (Rumelhart et al. 1986; Robinson 1994). 4 Target re-estimation The data used for training are only labelled by word. That is, each image represents a single word, whose identity is known, but the frames representing that word are not labelled to indicate which part of the word they represent. To train the network, a label for each frame's identity must be provided. Labels are indicated by the state St E Q and the corresponding letter L(St) of which a frame Xt is part. 4.1 A simple solution To bootstrap the network, a naive method was used, which simply divided the word up into sections of equal length, one for each letter in the word. Thus, for an Nletter word of T frames, xI, the first letter was assumed to be represented by frames .. 2r xr, the next by xk+1 and so on. The segmentation is mapped into a set of targets as follows: I'J.(t) { 1 if L(St) = Aj (6) .. 0 otherwise. Figure 2a shows such a segmentation for a single word. Each line, representing (j(t) for some j, has a broad peak for the frames representing letter Aj. Such a segmentation is inaccurate, but can be improved by adding prior knowledge. It is clear that some letters are generally longer than others, and some shorter. By weighting letters according to their a priori lengths it is possible to give a better, but still very simple, segmentation. The letters Ii, I' are given a length of ! and 'm, w' a length ~ relative to other letters. Thus in the word 'wig', the first half of the frames would be assigned the label 'w', the next sixth Ii' and the last third the label 'g'. While this segmentation is constructed with no regard for the data being segmented, it is found to provide a good initial approximation from which it is possible to train the network to recognize words, albeit with high error rates. 4.2 Viterbi re-estimation Having trained the network to some accuracy, it can be used to calculate a good estimate of the probability of each frame belonging to any letter. The probability of any state sequence can then be calculated in the hidden Markov model, and the most likely state sequence through the correct word S* found using dynamic programming. This best state sequence S* represents a new segmentation giving a label for each frame. For a network which models the probability distributions well, this segmentation will be better than the automatic segmentation of section 4.1 Forward-backward Retraining of Recurrent Neural Networks " Figure 2: Segmentations of the word 'butler'. Each line represents P(St = AilS) for one letter ~ and is high for framet when S; = Ai. (a) is the equal-length segmentation discussed in section 4.1 (b) is a segmentation of an untrained network. (c) is the segmentation re-estimated with a trained network. 747 since it takes the data into account. Finding the most probable state sequence S· is termed a forced alignment. Since only the correct word model need be considered, such an alignment is faster than the search through the whole lexicon that is required for recognition. Training on this automatic segmentation gives a better recognition rate, but still avoids the necessity of manually segmenting any of the database. Figure 2 shows two Viterbi segmentations of the word 'butler'. First, figure 2b shows the segmentation arrived at by taking the most likely state sequence before training the network. Since the emission probability distributions are random, there is nothing to distinguish between the state sequences, except slight variations due to initial asymmetry in the network, so a poor segmentation results. After training the network (2c), the durations deviate from the prior assumed durations to match the observed data. This re-estimated segmentation represents the data more accurately, so gives better targets towards which to train. A further improvement in recognition accuracy can be obtained by using the targets determined by the reestimated segmentation. This cycle can be repeated until the segmentations do not change and performance ceases to improve. For speed, the network is not trained to convergence at each iteration. It can be shown (Santini and Del Bimbo 1995) that, assuming that the network has enough parameters, the network outputs after convergence will approximate the posterior probabilities P(~lxD. Further, the approximation P(AilxD ~ P(Adxt) is made. The posteriors are scaled by the class priors P(Ai) (Bourlard and Morgan 1993), and these scaled posteriors are used in the hidden Markov model in place of data likelihoods since, by Bayes' rule, P(XtIAi) P(~lxt) ()( P(Ai)· (7) Table 1 shows word recognition error rates for three 80-unit networks trained towards fixed targets estimated by another network, and then retrained, re-estimating the targets at each iteration. The retraining improves the recognition performance (T(2) = 3.91, t.9s(2) = 2.92). 4.3 Forward-backward re-estimation The system described above performs well and is the method used in previous recurrent network systems, but examining the speech recognition literature, a potential method of improvement can be seen. Viterbi frame alignment has so far been used to determine targets for training. This assigns one class to each frame, based on the most likely state sequence. A better approach might be to allow a distribution across all the classes indicating which are likely and which are not, avoiding a 748 A. SENIOR, T. ROBINSON Table 1: Error rates for 3 networks with 80 units trained with fixed alignments, and retrained with re-estimated alignments. Training Error (%) method J.I. (7 Fixed targets 21.2 1.73 Retraining 17.0 0.68 'hard' classification at points where a frame may indeed represent more than one class (such as where slanting characters overlap), or none (as in a ligature). A 'soft' classification would give a more accurate portrayal of the frame identities. <» Such a distribution, 'Yp(t) = P(St = qplxI, W), can be calculated with the forwardbackward algorithm (Rabiner and Juang 1986). To obtain 'Yp(t), the forward probabilities Ctp(t) = P(St = qp, xD must be combined with the backward probabilities f3p(t) = P(St = qp, x;+l)' The forward and backward probabilities are calculated recursively in the same manner. Ctr(t + 1) L Ctp(t)P(xtIL(qp))ap,r, (8) /3p(t - 1) (9) r Suitable initial distributions Ctr(O) = 7l'r and f3r(r + 1) = Pr are chosen, e.g. 7l' and P are one for respectively the first and last character in the word, and zero for the others. The likelihood of observing the data Xl and being in state qp at time t is then given by: e (t) = Ctp(t)/3p(t). (10) Then the probabilities 'Yp(t) of being in state qp at time t are obtained by normalization and used as the targets (j (t) for the recurrent network character probability outputs: ep(t) (11) (j (t) L 'Yp(t). (12) l:r er(t)' p:L(qp)=Aj Figure 3a shows the initial estimate of the class probabilities for a sample of the word' butler'. The probabilities shown are those estimated by the forward-backward algorithm when using an untrained network, for which the P(XtISt = qp) will be independent of class. Despite the lack of information, the probability distributions can be seen to take reasonable shapes. The first frame must belong to the first letter, and the last frame must belong to the last letter, of course, but it can also be seen that half way through the word, the most likely letters are those in the middle of the word. Several class probabilities are non-zero at a time, reflecting the uncertainty caused since the network is untrained. Nevertheless, this limited information is enough to train a recurrent network, because as the network begins to approximate these probabilities, the segmentations become more definite. In contrast, using Viterbi segmentations from an untrained network, the most likely alignment can be very different from the true alignment (figure 2b). The segmentation is very definite though, and the network is trained towards the incorrect targets, reinforcing its error. Finally, a trained network gives a much more rigid segmentation (figure 3b), with most of the probabilities being zero or one, but with a boundary of uncertainty at the transitions between letters. This uncertainty, where a frame might truly represent parts of two letters, or a ligature between two, represents the data better. Just as with Viterbi training, the segmentations can be re-estimated after training and retraining results in improved performance. The final probabilistic segmentation can be stored with the data and used when subsequent networks are trained on the same data. Training is then significantly quicker than when training towards the approximate bootstrap segmentations and re-estimating the targets. Forward-backward Retraining of Recurrent Neural Networks Figure 3: Forward-backward segmentations of the word 'butler'. (a) is the segmentation of an untrained network with a uniform class prior. (b) shows the segmentation after training. 749 The better models obtained with the forward-backward algorithm give improved recognition results over a network trained with Viterbi alignments. The improvement is shown in table 2. It can be seen that the error rates for the networks trained with forward-backward targets are lower than those trained on Viterbi targets (T(2) = 5.24, t.97S(2) = 4.30). Table 2: Error rates for networks with 80 units trained with Viterbi or Forward-Backward alignments. Training Error 1%) method J.I. (7 Viterbi 17.0 0.68 Forward-Backward 15.4 0.74 5 Conclusions This paper has reviewed the training methods used for a recurrent network, applied to the problem of off-line handwriting recognition. Three methods of deriving target probabilities for the network have been described, and experiments conduded using all three. The third method is that of the forward-backward procedure, which has not previously been applied to recurrent neural network training. This method is found to improve the performance of the network, leading to reduced word error rates. Other improvements not detailed here (including duration models and stochastic language modelling) allow the error rate for this task to be brought below 10%. Acknowledgments The authors would like to thank Mike Hochberg for assistance in preparing this paper. References BOURLARD, H. and MORGAN, N. (1993) Connectionist Speech Recognition: A Hybrid Approach. Kluwer. RABINER, L. R. and JUANG, B. H. (1986) An introduction to hidden Markov models. IEEE ASSP magazine 3 (1): 4-16. ROBINSON, A. (1994) The application ofrecuIIent nets to phone probability estimation. IEEE 'lransactions on Neural Networks. RUMELHART, D. E ., HINTON, G. E. and WILLIAMS, R. J. (1986) Learning internal representations by eIIor propagation. In Parallel Distributed Processing: Explorations in the Microstructure of Cognition, ed. by D. E. Rumelhart and J . L. McClelland, volume 1, chapter 8, PE. 318-362. Bradford Books. SANTINI, S. and DEL BIMBO, A. (1995) RecuIIent neural networks can be trained to be maximum a posteriori probability classifiers. Neural Networks 8 (1): 25-29. SENIOR, A . W ., (1994) Off-line Cursive Handwriting Recognition using Recurrent Neural Networks. Cambridge University Engineering Department Ph.D. thesis. URL: ~_~: / / svr-ft.p . enK. cam. ac . uk/pub/reports/senioLthesis . ps . gz. WERBOS, P. J. (1990) Backpropagation through time: What it does and how to do it. Proceedings of the IEEE 78: 1550-60.
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KODAK lMAGELINK™ OCR Alphanumeric Handprint Module Alexander Shustorovich and Christopher W. Thrasher Business Imaging Systems, Eastman Kodak Company, Rochester, NY 14653-5424 ABSTRACT This paper describes the Kodak Imageliok TM OCR alphanumeric handprint module. There are two neural network algorithms at its cme: the first network is trained to find individual characters in an alphamuneric field, while the second one perfmns the classification. Both networks were trained on Gabor projections of the ociginal pixel images, which resulted in higher recognition rates and greater noise immunity. Compared to its purely numeric counterpart (Shusurovich and Thrasher, 1995), this version of the system has a significant applicatim specific postprocessing module. The system has been implemented in specialized parallel hardware, which allows it to run at 80 char/sec/board. It has been installed at the Driver and Vehicle Licensing Agency (DVLA) in the United Kingdom. and its overall success rate exceeds 96% (character level without rejects). which translates into 85% field rate. If approximately 20% of the fields are rejected. the system achieves 99.8% character and 99.5% field success rate. 1 INTRODUCTION The system we describe below was designed to process alphanumeric fields extracted from forms. The major assumptialS were that (1) the form layoot and definition allows the system to capture the field image with a single line of characters, (2) the characters are handprinted capital letters and numerals, with possible addition of several special characters, and (3) the characters may occasimally touch, but generally they do not overlap. We also assume that some additional informatim about the cootents of the field is available to assist in the process of disambiguation. Otherwise, it is virtually impossible to distinguish not only between" 0 " and zero, but also" I " and one, " Z " and two, " S " and five, etc. A good example of such an applicatim is the processing of vehicle registration forms at the Driver and Vehicle Licensing Agency (DVLA) in the United Kingdom. The alphamuneric field in question contains a license plate. There are 29 allowed patterns of character combinations, fran two to seven characters long. For example, " A999AAA " is a valid license, whereas" A9A9A9 " is not (here .. A " stands for any alpha character, " 9 .. - for any numeric character). In addition, every field has a KODAK IMAGELINKTM OCR Alphanumeric Handprint Module 779 control character box on the right. This control cltaracter is computed as a remainder of the integer division by 37 of a linear e<mbination of numeric values of the characters in the main field. Ambiguous cltaracters. namely " 0 ". " I ". and " S " are not allowed in the role of the control character. so they are replaced here by " - ". " + ". and " I " (not a very good choice. and the 37th character used is the " % fl. To make things m<m complicated. sometimes the control character is not available at the moment of filing the form (at a local post dfice). and this lack. of knowledge is indicated by putting an asterisk instead. Later we will discuss possible ways to use this additiooal information in an application specific postprocessing module. 2 SEGMENTATION AND ALTERNATIVE APPROACHES The most challenging problem for handprint OCR. is finding individual characters in a field. A number of approaches to this problem can be found in the literature. the two most common being (1) segmentation (Gupta et al .• 1993. as an example of a recent publication). and (2) combined segmentation and recognition (Keeler and Rumelhart. 1992). The segmentation approach has difficulty separating touching characters. and recendy the consensus of practitioners in the field started shifting towards e<mbined segmentation and recognition. In this scheme. the algmthm moves a window of a certain width along the field. and confidence values of competing classification hypotheses are used (sometimes with a separate centered/noncentered node) to decide if the window is positioned on top of a cltaracter. In the Saccade system (Martin et al .• 1993). for example. the neural network was trained not only to recognize characters in the center of the moving window (and whether there is a character centered in the window). but also to make corrective jumps (saccades) to the nearest character and. after classification. to the next character. Still another variation on the theme is an arrangement when the classification window is duplicated with one- or several-pixel shifts along the field (Benjio et al .• 1994). Then the outputs of the classifiers serve as input for a postprocessing module (in this paper. a IDdden Markov Model) used to decide which of the multitude of processing windows actually have centered cltaracters in them. All these approaches have deficiencies. As we mentioned earlier. touching cltaracters are difficult for autonomous segmenters. The moving (and jumping) window with a sing1e cemered/noncentered node tends to miss narrow characters and sometimes to duplicate wide ones. The replication of a classifier together with postprocessing tends to be quite expensive computationally. 3 POSmONING NETWORK To do the positioning. we decided to introduce an may of output units corresponding to successive pixels in the middle portion of the window. These nodes signal if a center ("heart") of a character lies at the c<rresponding positions. Because the precision with which a human operator can mark the character heart is low (usually within one or two pixels at best). the target activatims of three cmsecutive nodes are set to one if there is a cltaracter heart at a pixel positioo corresponding to the middle node. The rest of the target activations are set to zero. The network is then trained to produce bumps of activation indicating the cltaracter hearts. Two buffer regions on the left and on the right of the window (pixels without COITesponding output nodes) are necessary to allow all or most of the cltaracter centered at each of the output node positions to fit inside the window. The replacement of a single centered/noncentered node by an array allows us to average output activations. generated by different window shifts. while corresponding to the same position. lbis additional procedure allows us to slide the window several pixels 780 A. SHUSTOROVICH, C. W. THRASHER at a time: the appropriate step is a trade-off between the processing speed and the required level of robustness. The final procedure involves thresholding of the activation-wave and the estimation of the predicted character position as the center of mass of the activation-bubble. The resulting algmthm is very effective: touching characters do not present significant problems. and only abnormally wide characters sometimes fool the system into false alarms. The system works with preprocessed images. Each field is divided into subfields of disconnected groups of characters. These subfields are size-normalized to a height of 20 pixels. After that they are reassembled into a single field again. with 6 pixel gaps between them. Two blank rows are added both along the top and the bottom of the recombined field as preferred by the Gabor projection technique (Shustorovich. 1994). In our current system. the input nodes of a sliding window are organized in a 24 x 36 array. The first. intermediary. layer of the network implements the Galxr projections. It has 12 x 12 local receptive fields (LRFs) with fixed precanputed weights. The step between LRFs is 6 pixels in both directions. We work with 16 Gabor basis functions with circular Gaussian envelopes centered within each LRF; they are both sine and cosine wavelets in four mentati(llS and two sizes. All 16 projections fr<m each LRF constitute the input to a column of 20 hidden units. thus the second (first trainable) hidden layer is organized in a three-dimensional array 3 x 5 x 20. The third hidden layer of the network also has local receptive fields. they are three-dimensiooal 2 x 2 x 20 with the step 1 x 1 x O. The units in the third hidden layer are also duplicated 20 times. thus this layer is organized in a three-dimensional array 2 x 4 x 20. The fourth hidden layer has 60 units fully connected to the third layer. Fmally. the output layer has 12 units. also fully connected to the fourth layer. The network was trained using a variant of the Back-Propagation algorithm. Both training and testing sets were drawn from. the field data collected at DVLA. The training set contained approximately 60.000 charactel"s from 8.000 fields. and about 5,000 charactel"s from 650 fields were used for testing. On this test set. more than 92% of all character hearts were found within I-pixel precision, and only 0.4% were missed by more than 4 pixels. 4 CLASSIFICATION NETWORK The structure of the classification network resembles that of the positioning network. The Gabor projection layer w<X'ks in exactly the same way. but the window size is smaller. only 24 x 24 pixels. We chose this size because after height normalization to 20 pixels. only occasionally the charactel"s are wider than 24 pixels. Widening the window complicates training: it increases the dimensionality of the input while providing information. mostly about irrelevant pieces of adjacent characters. As a result. the second layer is organized as a 3 x 3 x 20 array of units with LRFs and shared weights. the third is a 2 x 2 x 20 array of units with LRFs. and there are 37 output units fully connected to the 80 units in the third layer. The number of ouq,ut units in this variant of our system has been determined by the intended application. It was necessary to recognize uppercase letters. numerals. and also five special charactel"s. namely plus (+). minus (-). slash (f). percent (%). and asterisk (*). Since additional information was available for the purposes of disambiguation. we combined .. 0 .. and zero. .. I .. and one. .. Z to and two. .. S .. and five. and so the number of output classes became 26 (alpha) + 6 (numerals 3,4,6.7.8.9) + 5 (special characters) = 37. Because we did not expect any positioning module to provide precision higher than 1 or 2 pixels. the classifier network was trained and tested. on five copies of all centered characters in the database, with shifts of O. 1, and 2 pixels, both left and right On the same test set mentioned in the previous section. the corresponding character recognition rates averaged 93.0%. 955%. and 96.0% for characters normalized to the KODAK IMAGELINKTM OCR Alphanumeric Handprint Module 781 height of 18 to 20 pixels and placed in the middle of the window with shifts of 0 and 1 pixel up and down. S POSTPROCESSING MODULE The postprocessing module is a rule-based algorithm. Fust. it monitors the width of each subfield and rejects it if the number of predicted charactex hearts is inconsistent with the width. For example. if the positioning system cannot find a single character in a subfield. the output of the system bec<mes a question made. Second. the postprocessing module <rganizes competition between predicted character hearts if they are too close to each other. For example. it will kill a predicted center with a lower activation value if its distance from a competitor is Jess than ten pixels. but it may allow both to survive if one of the two labels is "one". It is especially sensitive to closely positioned centers with identical labels. and will remove the weaker one for wide characters such as II W " or " Mil. The rest of the postprocessing had to rely on the applicatioo knowledge. Since the alphanumeric fields on DVLA forms contain license plates. we could use the fact that there ~ exactly 29 allowed patterns of symbol combinations. and that carect strings should match control characters from the box on the right. Because in this applicatioo rejection of individual characters is meaningless. we decided to keep and analyze all possible candidates for each detected positioo. that is. characters with output activations above a certain threshold (currently. 0.1). Of course. special charactexs are not allowed in the main field. The field as a whole is rejected if for any one position there is not even a single candidate cllaracter. All possible COOlbinations of candidate characters are analyzed A candidate string is rejected if it does not conform to any of allowed patterns. or if it does not match any of the candidate control cllaracters. All remaining candidate strings are assigned confidences. Since a chain is no stronger than its weakest link. in the case of an asterisk (no control charactex information). the string confidence equals that of its least confident cllaracter. If there is a valid control character. then we can tolerate one low-confidence cllaracter. and so the string confidence equals that of its charactex with the second lowest individual confidence. If there are two or mme candidate strings. the difference in confidence between the best and the second best is compared to another threshold (currently. 0.7) in order to pass the final round of rejects. 6 CONCLUSIONS Kodak Imagelink™OCR alphanumeric handprint module desaibed in this paper uses one neural network to find individual cllaracters in a field. and then the second network performs the classification. The outputs of both networks are interpreted by a postprocessing module that generates the final label string (Figure 1. Figure 2). The algmthms were designed within the constraints of the planned hardware implementation. At the same time. they provide a high level of positioning accuracy as well as classification ability. One new feature of our approach is the use of an array of centered/noncentered nodes to significantly improve speed and robustness of the positioning scheme. The overall robustness of the system is further improved by noise resistance provided by a layer of Gabor projection units. The positioning module and the classification module are unified by the postprocessing module. System-level testing was performed on a test set mentioned above. The image quality was generally very good. but the data included some fields with touching characters. The character level success rate (without rejects) achieved on this test exceeded 96%. which corresponded to above 85% field rate. With approximately 20% d the fields rejected. the system achieved 99.8% character and 995% field success rate. 782 A. SHUSTOROVICH, C. W. THRASHER In the testing mode, the preprocessing module would separate characters if it can reliably do so, normalize them individually, and place them with gaps of ten blank pixels, in order to simplify the job of both the positioning and the classification modules. When it is impossible to segment individual characters, our system is still able to perform on the level of approximately 94% (since it has beea trained on such data). The robustness of our system is an impOOant factor in its success. Most other systems have substantial difficulties trying to recover from. errors in segmentation. References Benjio, Y., Le Gm, Y., and Henderson, D. (1994) Globally Trained Handwritten Word Recognizer Using Spatial Representation, Space Displacement Neural Networks and Hidden Markov Models. In Cowan, J.D., Tesauro, G., and Alspector, J. (eds.), Advances in Neural Information Processing Systems 6, pp. 937-944. San Mateo, CA: Morgan Kaufmann Publishers. Gupta, A., Nagendraprasad, M.V., Lin. A., Wang, P.S.P., and Ayyadurai, S. (1993) An Integrated Architecture for Recognition of Totally Unconstrained Handwritten Numerals. International Journal of Pattern Recognition and Artificial Intelligence 7 (4), pp. 757-773. Keeler, J. and Rume1hart. DE (1992) A Self-Organizing Integrated Segmentation and Recognition Neural Net. In Moody, J.E., Hanson. S.1., and Lippmann, R.P. (eds.), Advances in Neural Information Processing Systems 4, pp. 496-503. San Mateo, CA: Morgan Kaufmann Publisbers. Martin. G., Mosfeq, R, Otapman. D., and Pittman, J. (1993) Learning to See Where and What: Training a Net to Make Saccades and Recognize Handwritten Otaracters. In Hanson. S.J., Cowan. JD., and Giles, c.L. (eds.), Advances in Neural Information Processing Systems 5, pp. 441-447. San Mateo, CA: Morgan Kallfm8l1D Publishers. Shustorovich, A. (1994) A Subspace Projection Approach to Feature Extraction: the Tw~Dimensianal Gab« Transform for Character Recognition. Neural Networks 7 (8), 1295-1301. Shustorovich, A. and Thrasher, C.W. (1995) KODAK IMAGELINK™OCR Numeric Handprint Module: Neural Network Positioning and Oassification. ~ings of Session 11 (Document Processing) of the industrial conference of ICANN-95 Paris, October 9-13, 1995. KODAK IMAGELINKTM OCR Alphanumeric Handprint Module 783 Original Image with Detected Subimages Scaled Subimages Character Heart Index Waveform Detected Character Hearts Best Guess Characters MY 9 Z B E we Final Character string (After Post-Processing) M7 9 2 B E we Figure 1: An Example of a Field Processed by the System Outline characters indicate low confidence. 784 A. SHUSTOROVICH. C. W. THRASHER Original Image with Detected Subimages Scaled Subimages Olaracter Heart Index Waveform Detected Character Hearts Best Guess Cll81'acters G3S8AAF3 Final 0Iaracter String (MterPost-Processing) G358AAF3 Figure 2: Another Example cI a FJeld Processed by the System.
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Predictive Q-Routing: A Memory-based Reinforcement Learning Approach to Adaptive Traffic Control Samuel P.M. Choi, Dit-Yan Yeung Department of Computer Science Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong {pmchoi,dyyeung}~cs.ust.hk Abstract In this paper, we propose a memory-based Q-Iearning algorithm called predictive Q-routing (PQ-routing) for adaptive traffic control. We attempt to address two problems encountered in Q-routing (Boyan & Littman, 1994), namely, the inability to fine-tune routing policies under low network load and the inability to learn new optimal policies under decreasing load conditions. Unlike other memory-based reinforcement learning algorithms in which memory is used to keep past experiences to increase learning speed, PQ-routing keeps the best experiences learned and reuses them by predicting the traffic trend. The effectiveness of PQ-routing has been verified under various network topologies and traffic conditions. Simulation results show that PQ-routing is superior to Q-routing in terms of both learning speed and adaptability. 1 INTRODUCTION The adaptive traffic control problem is to devise routing policies for controllers (i.e. routers) operating in a non-stationary environment to minimize the average packet delivery time. The controllers usually have no or only very little prior knowledge of the environment. While only local communication between controllers is allowed, the controllers must cooperate among themselves to achieve the common, global objective. Finding the optimal routing policy in such a distributed manner is very difficult. Moreover, since the environment is non-stationary, the optimal policy varies with time as a result of changes in network traffic and topology. In (Boyan & Littman, 1994), a distributed adaptive traffic control scheme based 946 S. P. M. CHOI, D. YEUNG on reinforcement learning (RL), called Q-routing, is proposed for the routing of packets in networks with dynamically changing traffic and topology. Q-routing is a variant of Q-Iearning (Watkins, 1989), which is an incremental (or asynchronous) version of dynamic programming for solving multistage decision problems. Unlike the original Q-Iearning algorithm, Q-routing is distributed in the sense that each communication node has a separate local controller, which does not rely on global information of the network for decision making and refinement of its routing policy. 2 EXPLORATION VERSUS EXPLOITATION As in other RL algorithms, one important issue Q-routing must deal with is the tradeoff between exploration and exploitation. While exploration of the state space is essential to learning good routing policies, continual exploration without putting the learned knowledge into practice is of no use. Moreover, exploration is not done at no cost. This dilemma is well known in the RL community and has been studied by some researchers, e.g. (Thrun, 1992). One possibility is to divide learning into an exploration phase and an exploitation phase. The simplest exploration strategy is random exploration, in which actions are selected randomly without taking the reinforcement feedback into consideration. After the exploration phase, the optimal routing policy is simply to choose the next network node with minimum Q-value (i.e. minimum estimated delivery time). In so doing, Q-routing is expected to learn to avoid congestion along popular paths. Although Q-routing is able to alleviate congestion along popular paths by routing some traffic over other (possibly longer) paths, two problems are reported in (Boyan & Littman, 1994). First, Q-routing is not always able to find the shortest paths under low network load. For example, if there exists a longer path which has a Q-value less than the (erroneous) estimate of the shortest path, a routing policy that acts as a minimum selector will not explore the shortest path and hence will not update its erroneous Q-value. Second, Q-routing suffers from the so-called hysteresis problem, in that it fails to adapt to the optimal (shortest) path again when the network load is lowered. Once a longer path is selected due to increase in network load, a minimum selector is no longer able to notice the subsequent decrease in traffic along the shortest path. Q-routing continues to choose the same (longer) path unless it also becomes congested and has a Q-value greater than some other path. Unless Q-routing continues to explore, the shortest path cannot be chosen again even though the network load has returned to a very low level. However, as mentioned in (Boyan & Littman, 1994), random exploration may have very negative effects on congestion, since packets sent along a suboptimal path tend to increase queue delays, slowing down all the packets passing through this path. Instead of having two separate phases for exploration and exploitation, one alternative is to mix them together, with the emphasis shifting gradually from the former to the latter as learning proceeds. This can be achieved by a probabilistic scheme for choosing next nodes. For example, the Q-values may be related to probabilities by the Boltzmann-Gibbs distribution, involving a randomness (or pseudo-temperature) parameter T. To guarantee sufficient initial exploration and subsequent convergence, T usually has a large initial value (giving a uniform probability distribution) and decreases towards 0 (degenerating to a deterministic minimum selector) during the learning process. However, for a continuously operating network with dynamically changing traffic and topology, learning must be continual and hence cannot be controlled by a prespecified decay profile for T. An algorithm which automatically adapts between exploration and exploitation is therefore necessary. It is this very reason which led us to develop the algorithm presented in this paper. Predictive Q-Routing 947 3 PREDICTIVE Q-ROUTING A memory-based Q-learning algorithm called predictive Q-routing (PQ-routing) is proposed here for adaptive traffic control. Unlike Dyna (Peng & Williams, 1993) and prioritized sweeping (Moore & Atkeson, 1993) in which memory is used to keep past experiences to increase learning speed, PQ-routing keeps the best experiences (best Q-values) learned and reuses them by predicting the traffic trend. The idea is as follows. Under low network load, the optimal policy is simply the shortest path routing policy. However, when the load level increases, packets tend to queue up along the shortest paths and the simple shortest path routing policy no longer performs well. If the congested paths are not used for a period of time, they will recover and become good candidates again. One should therefore try to utilize these paths by occasionally sending packets along them. We refer to such controlled exploration activities as probing. The probing frequency is crucial, as frequent probes will increase the load level along the already congested paths while infrequent probes will make the performance little different from Q-routing. Intuitively, the probing frequency should depend on the congestion level and the processing speed (recovery rate) of a path. The congestion level can be reflected by the current Q-value, but the recovery rate has to be estimated as part of the learning process. At first glance, it seems that the recovery rate can be computed simply by dividing the difference in Q-values from two probes by the elapse time. However, the recovery rate changes over time and depends on the current network traffic and the possibility of link/node failure. In addition, the elapse time does not truly reflect the actual processing time a path needs. Thus this noisy recovery rate should be adjusted for every packet sent. It is important to note that the recovery rate in the algorithm should not be positive, otherwise it may increase the predicted Q-value without bound and hence the path can never be used again. Predictive Q-Routing Algorithm TABLES: Qx(d, y) - estimated delivery time from node x to node d via neighboring node y Bx(d, y) - best estimated delivery time from node x to node d via neighboring node y Rx(d,y) - recovery rate for path from node x to node d via neighboring node y U x (d, y) - last update time for path from node x to node d via neighboring node y TABLE UPDATES: (after a packet arrives at node y from node x) 6.Q = (transmission delay + queueing time at y + minz{Qy(d,z)}) - Qx(d,y) Qx(d,y) ~ Qx(d,y) +O"6.Q Bx(d, y) ~ min(Bx(d, y), Qx(d, y)) if (6.Q < 0) then 6.R ~ 6.Q / (current time - Ux(d, y)) Rx(d,y) ~ Rx(d,y)+f36.R else if (6.Q > 0) then Rx(d,y) ~ -yRx(d,y) end if Ux(d, y) ~ current time ROUTING POLICY: (packet is sent from node x to node y) 6.t = current time - Ux(d,y) Q~(d, y) = max(Qx(d, y) + 6.tRx(d, y), Bx(d, y)) y ~ argminy{Q~(d,y)} There are three learning parameters in the PQ-routing algorithm. a is the Qfunction learning parameter as in the original Q-learning algorithm. In PQ-routing, this parameter should be set to 1 or else the accuracy of the recovery rate may be 948 s. P. M. CHOI. D. YEUNG affected. f3 is used for learning the recovery rate. In our experiments, the value of 0.7 is used. 'Y is used for controlling the decay of the recovery rate, which affects the probing frequency in a congested path. Its value is usually chosen to be larger than f3. In our experiments, the value of 0.9 is used. PQ-Iearning is identical to Q-Iearning in the way the Q-function is updated. The major difference is in the routing policy. Instead of selecting actions based solely on the current Q-values, the recovery rates are used to yield better estimates of the Q-values before the minimum selector is applied. This is desirable because the Q-values on which routing decisions are based may become outdated due to the ever-changing traffic. 4 EMPIRICAL RESULTS 4.1 A 15-NODE NETWORK To demonstrate the effectiveness of PQ-routing, let us first consider a simple 15node network (Figure 1(a)) with three sources (nodes 12 to 14) and one destination (node 15). Each node can process one packet per time step, except nodes 7 to 11 which are two times faster than the other nodes. Each link is bidirectional and has a transmission delay of one time unit. It is not difficult to see that the shortest paths are 12 ---+ 1 ---+ 4 ---+ 15 for node 12, 13 ---+ 2 ---+ 4 ---+ 15 for node 13, and 14 ---+ 3 ---+ 4 ---+ 15 for node 14. However, since each node along these paths can process only one packet per time step, congestion will soon occur in node 4 if all source nodes send packets along the shortest paths. One solution to this problem is that the source nodes send packets along different paths which share no common nodes. For instance, node 12 can send packets along path 12 ---+ 1 ---+ 5 ---+ 6 ---+ 15 while node 13 along 13 ---+ 2 ---+ 7 ---+ 8 ---+ 9 ---+ 10 ---+ 11 ---+ 15 and node 14 along 14 ---+ 3 ---+ 4 ---+ 15. The optimal routing policy depends on the traffic from each source node. If the network load is not too high, the optimal routing policy is to alternate between the upper and middle paths in sending packets. 4.1.1 PERIODIC TRAFFIC PATTERNS UNDER LOW LOAD For the convenience of empirical analysis, we first consider periodic traffic in which each source node generates the same traffic pattern over a period of time. Figure 1(b) shows the average delivery time for Q-routing and PQ-routing. PQ-routing performs better than Q-routing after the initial exploration phase (25 time steps), despite of some slight oscillations. Such oscillations are due to the occasional probing activities of the algorithm. When we examine Q-routing ·more closely, we can find that after the initial learning, all the source nodes try to send packets along the upper (shortest) path, leading to congestion in node 4. When this occurs, both nodes 12 and 13 switch to the middle path, which subsequently leads to congestion in node 5. Later, nodes 12 and 13 detect this congestion and then switch to the lower path. Since the nodes along this path have higher (two times) processing speed, the Q-values become stable and Q-routing will stay there as long as the load level does not increase. Thus, Q-routing fails to fine-tune the routing policy to improve it. PQ-routing, on the other hand, is able to learn the recovery rates and alternate between the upper and middle paths. Predictive Q-Routing Source Source Source ! .. l .! (a) Network 'q' 'pq' ---" " °O~~~~~,~~~--~20~O--~~~-_~~~=O--~~ S,""*lIonr,,,. (c) Aperiodic traffic patterns under high load '" 949 'q' 'pq'---~ 30 I ! .. l .! 20 " \ ,'r,., .... ---;J ':-~.,"" ...... ;-~ .. -=~,""" ,_~----.;-~_-.J;:".-. -. __ .-... ;:-"" ..... :-,.-.--.-... -.--.-:., . .,,= .. -.-_.-_.-.--,-... -... 1 ·O~~2~O~"'~~~~~~~~I~--~12-0~1"'~~I~~~,00~~200 SlITIUM.lIonnM (b) Periodic traffic patterns under low load 30 ,''pq' ---" 20 " 10 O.~~~-2=OO--~~=-~~~~~-.=~--~7~OO--~ SunulatlonT'irl'\tl (d) Varying traffic patterns and network load Figure 1: A 15-Node Network and Simulation Results 4.1.2 APERIODIC TRAFFIC PATTERNS UNDER HIGH LOAD It is not realistic to assume that network traffic is strictly periodic. In reality, the time interval between two packets sent by a node varies. To simulate varying intervals between packets, a probability of 0.8 is imposed on each source node for generating packets. In this case, the average delivery time for both algorithms oscillates, Figure 1( c) shows the performance of Q-routing and PQ-routing under high network load. The difference in delivery time between Q-routing and PQrouting becomes less significant, as there is less available bandwidth in the shortest path for interleaving. Nevertheless, it can be seen that the overall performance of PQ-routing is still better than Q-routing. 4.1.3 VARYING TRAFFIC PATTERNS AND NETWORK LOAD In the more complicated situation of varying traffic patterns and network load, PQ-routing also performs better than Q-routing. Figure 1( d) shows the hysteresis problem in Q-routing under gradually changing traffic patterns and network load. After an initial exploration phase of 25 time steps, the load level is set to medium 950 S. P. M. CHOI, D. YEUNG from time step 26 to 200. From step 201 to 500, node 14 ceases to send packets and nodes 12 and 13 slightly increase their load level. In this case, although the shortest path becomes available again, Q-routing is not able to notice the change in traffic and still uses the same routing policy, but PQ-routing is able to utilize the optimal paths. After step 500, node 13 also ceases to send packets. PQ-routing is successful in adapting to the optimal path 12 -4 1 -4 4 -4 15. 4.2 A 6x6 GRID NETWORK Experiments have been performed on some larger networks, including a 32-node hypercube and some random networks, with results similar to those above. Figures 2(b) and 2( c) depict results for Boyan and Littman's 6x6 grid network (Figure 2( a)) under varying traffic patterns and network load. (a) Network t<" 'q''pq' ---120 100 eo .. 20 °O~--2=OO~~_~--=eoo~~eoo~~I=_~~,,,,~ ........... (b) Varying traffic patterns and network load ~ " 1 .! 700 .~'p« •• eoo 500 ... 300 ... 100 800 800 1000 1200 1..ao 1800 1&00 2000 SlmulaHonTm. (c) Varying traffic patterns and network load Figure 2: A 6x6 Grid Network and Simulation Results In Figure 2(b), after an initial exploration for 50 time steps, the load level is set to low. From step 51 to 300, the load level increases to medium but with the same periodic traffic patterns. PQ-routing performs slightly better. From step 301 to 1000, the traffic patterns change dramatically under high network load. Q-routing cannot learn a stable policy in this (short) period of time, but PQ-routing becomes more stable after about 200 steps. From step 1000 onwards, the traffic patterns change again and the load level returns to low. PQ-routing still performs better. Predictive Q-Routing 951 In Figure 2{ c), the first 100 time steps are for initial exploration. After this period, packets are sent from the bottom right part of the grid to the bottom left part with low network load. PQ-routing is found to be as good as the shortest path routing policy, while Q-routing is slightly poorer than PQ-routing. From step 400 to 1000, packets are sent from both the left and right parts of the grid to the opposite sides at high load level. Both the two bottleneck paths become congested and hence the average delivery time increases for both algorithms. From time step 1000 onwards, the network load decreases to a more manageable level. We can see that PQ-routing is faster than Q-routing in adapting to this change. 5 DISCUSSIONS PQ-Iearning is generally better than Q-Iearning under both low and varying network load conditions. Under high load conditions, they give comparable performance. In general, Q-routing prefers stable routing policies and tends to send packets along paths with higher processing power, regardless of the actual packet delivery time. This strategy is good under extremely high load conditions, but may not be optimal under other situations. PQ-routing, on the contrary, is more aggressive. It tries to minimize the average delivery time by occasionally probing the shortest paths. If the load level remains extremely high with the patterns unchanged, PQ-routing will gradually degenerate to Q-routing, until the traffic changes again. Another advantage PQ-routing has over Q-routing is that shorter adaptation time is generally needed when the traffic patterns change, since the routing policy of PQ-routing depends not only on the current Q-values but also on the recovery rates. In terms of memory requirement, PQ-routing needs more memory for recovery rate estimation. It should be noted, however, that extra memory is needed only for the visited states. In the worst case, it is still in the same order as that of the original Q-routing algorithm. In terms of computational cost, recovery rate estimation is computationally quite simple. Thus the overhead for implementing PQ-routing should be minimal. References J.A. Boyan & M.L. Littman (1994). Packet routing in dynamically changing networks: a reinforcement learning approach. Advances in Neural Information Processing Systems 6, 671-678. Morgan Kaufmann, San Mateo, California. M. Littman & J. Boyan (1993). A distributed reinforcement learning scheme for network routing. Proceedings of the First International Workshop on Applications of Neural Networks to Telecommunications,45- 51. Lawrence Erlbaum, Hillsdale, New Jersey. A.W. Moore & C.G. Atkeson (1993). Memory-based reinforcement learning: efficient computation with prioritized sweeping. Advances in Neural Information Processing Systems 5, 263-270. Morgan Kaufmann, San Mateo, California. A.W. Moore & C.G. Atkeson (1993). Prioritized sweeping: reinforcement learning with less data and less time. Machine Learning, 13:103-130. J. Peng & R.J. Williams (1993). Efficient learning and planning within the Dyna framework. Adaptive Behavior, 1:437- 454. S. Thrun (1992). The role of exploration in learning control. In Handbook of Intelligent Control: Neural, Fuzzy, and Adaptive Approaches, D.A. White & D.A. Sofge (eds). Van Nostrand Reinhold, New York. C.J.C.H. Watkins (1989). Learning from delayed rewards. PhD Thesis, University of Cambridge, England.
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|
1995
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54
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